Isotope Effects In Chemistry and Biology

Isotope Effects in Chemistry and Biology

Isotope Effects in Chemistry and Biology Edited by

Amnon Kohen Hans-Heinrich Limbach

Boca Raton London New York

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

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Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2449-6 (Hardcover) International Standard Book Number-13: 978-0-8247-2449-8 (Hardcover) Library of Congress Card Number 2005041897 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Isotope effects in chemistry and biology / edited by Amnon Kohen, Hans-Heinrich Limbach. p. cm. Includes bibliographical references and index. ISBN 0-8247-2449-6 1. Chemical reaction, Conditions and laws of. 2. Isotopes. 3. Chemical kinetics. 4. Isobaric spin. 5. Chemical equilibrium. I. Kohen, Amnon. II. Limbach, Hans-Heinrich. III. Title. QD501.I8127 2005 541'.394--dc22

2005041897

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Preface The unifying theme of this book is the application of isotopic methods to make significant advances in chemistry and biology. Isotopes are atoms with identical nucleic electrical charges and identical electronic properties. Isotopes contain the same number of protons but a different number of neutrons, hence they exhibit different masses and different nuclear spins. Isotope effects can be classified into three categories, i.e., kinetic isotope effects (KIEs), equilibrium isotope effects (EIEs), and anharmonic isotope effects (AIEs). KIEs are the ratio of reaction rates involving reactants that only differ by their isotopic composition. These are one of the only measures that directly probe the nature of the reaction’s transition state, and thus are very useful tools in studies of reactions’ mechanisms. EIEs are the ratio of two isotopes that are distributed between stable populations in thermodynamic equilibrium. These are a unique measure of the difference in the chemical potential of these two populations. AIEs lead to geometric changes of molecules and molecular systems via the anharmonicity of zero-point vibrations. Isotope effects are of substantial importance and utility in many fields of science and technology. The use of isotope effects is prevalent in a wide variety of disciplines. Scheme 1 below summarizes many of the areas that utilize isotope effects. The book’s nine parts and 42 chapters provide a comprehensive review of developments in isotope effects studies to date. The chapters were written by internationally recognized leading researchers in their fields. Authors from 13 countries contributed to the book: Canada, Denmark, France, Germany, Israel, Japan, Poland, Russia, Spain, Sweden, Switzerland, UK, and USA (by alphabetical order). Geoscience

Marine science

Molecular biology

Organic chemistry

Elucidation of reaction mechanisms

Isotope effects structural dynamic kinetic equilibrium

Physical chemistry Solid state physics

Solid state physics

Cosmology Environmental and ecological science Agricultural science

Engineering applications

Elucidation of molecular structures, interactions dynamics

Inorganic chemistry

Chemical physics

Anthropology

Elucidation of naturally occurring processes

Structural biology Biochemistry Biophysics

Atmospheric science

Medicinal sciences Pharmacological sciences Polymer sciences Material sciences

Solid state physics

Stable isotope separation

Scheme 1. The many applications and implications of isotope effects and their relationship to many fields of science and technology. This Scheme was drawn by Takanobu Ishida and modified by Hans Limbach.

iv

Preface

Subjects range from the physical and theoretical origin of isotope effects to modern uses of these effects in chemical, biological, geological, and other applications. The following Table of Contents clearly emphasizes the multidisciplinary nature of this book. The book starts with the problem of isotope effects on molecular geometries arising from anharmonic vibrations and the consequences for isotope-dependent non-covalent interactions. Chemical bond breaking and formation dynamics are then addressed using the examples of simple molecules in the gas phase, also including the motif of hydrogen transfer. Novel mass independent isotope effects are discussed. The problem of hydrogen transfer, tunneling, and exchange is picked up for condensed matter, ranging from polyatomic molecules to enzymes. When the barrier for hydrogen or proton transfer becomes small, the area of low-barrier hydrogen bonds is reached and explored experimentally and theoretically. A unique application is provided in a chapter devoted to water isotope effects under pressure. Isotope effect studies in organic and organometallic reactions are needed for the understanding of the sessions that follow on isotope effects in more complex enzyme reactions. The book brings together a wide scope of different points of view and practical applications based on our current knowledge at the beginning of the new millennium. Some chapters summarize the perspective of a well-established subject while others review recent findings and ongoing research. It may appear that some of these later items are not consistent with each other. This reflects contemporary conclusions and controversies in the field. We chose to present such studies only in cases where clear scientific arguments and discussion are presented by all relevant authors. This approach demonstrates the way research progresses, and we hope it will enhance the reader’s curiosity and interest.

Editors Amnon Kohen was born in a kibbutz in northern Israel. He received his B.Sc. degree in chemistry in 1989 from the Hebrew University in Jerusalem and his D.Sc. in 1994 from the Technion-Israel Institute of Technology. After that he was a postdoctoral scholar with Judith Klinman at the University of California at Berkeley. In 1999, he moved to the University of Iowa and is currently (2005) an associate professor in the Department of Chemistry. His main interest is bioorganic chemistry, and he enjoys studying the mechanisms by which enzymes activate C– H bonds and N2 triple bonds. His research focuses on the relationship between enzyme structure, dynamics, and catalytic activity. Isotope effects were one of the main tools used by his group in recent years. Hans-Heinrich Limbach was born in Bruehl near Cologne, Germany. He studied chemistry at the Universities of Bonn and Freiburg. He did his doctoral research (Dr. rer. nat.) under the direction of Herbert W. Zimmermann at the University of Freiburg. After his Habilitation he was a visiting scientist with C.S. Yannoni at the IBM Research Laboratory, San Jose and with C.B. Moore at U.C. Berkeley. He is currently a professor of physical chemistry at the Freie Universita¨t Berlin. His research interests include the chemistry of hydrogen and its isotopes in liquids, organic solids, and mesoporous systems up to enzymes, which he is studying by liquid and solid state nuclear magnetic resonance.

Contributors Vernon E. Anderson Department of Biochemistry Case Western Reserve University Cleveland, Ohio

Gleb S. Denisov V.A. Fock Institute of Physics St. Petersburg State University St. Petersburg, Russian Federation

Katsutoshi Aoki Synchrotron Radiation Research Center Kansai Research Establishment Japan Atomic Energy Research Institute Kansai, Japan

Ileana Elder Department of Pharmacology University of Florida Gainesville, Florida

Jaswir Basran Department of Biochemistry University of Leicester Leicester, United Kingdom

Antonio Ferna´ndez-Ramos Department of Physical Chemistry Faculty of Chemistry University of Santiago de Compostela Santiago de Compostela, Spain

Jacob Bigeleisen Department of Chemistry State University of New York Stony Brook, New York

Paul F. Fitzpatrick Department of Biochemistry & Biophysics Texas A&M University College Station, Texas

Adam G. Cassano Center for RNA Molecular Biology Case Western Reserve University Cleveland, Ohio

Perry A. Frey Department of Biochemistry University of Wisconsin-Madison Madison, Wisconsin

W. Wallace Cleland Institute for Enzyme Research and Department of Biochemistry University of Wisconsin Madison, Wisconsin

Yasuhiko Fujii Tokyo Institute of Technology Research Institute for Nuclear Reactors O-okayama, Meguro-ku, Tokyo, Japan

Paul F. Cook Department of Chemistry and Biochemistry University of Oklahoma Norman, Oklahoma Janet E. Del Bene Department of Chemistry Youngstown State University Youngstown, Ohio

Nikolai S. Golubev V.A. Fock Institute of Physics St. Petersburg State University St. Petersburg, Russian Federation Sharon Hammes-Schiffer Department of Chemistry Davey Laboratory Pennsylvania State University University Park, Pennsylvania

viii

Contributors

Poul Erik Hansen Department of Life Sciences and Chemistry Roskilde University Roskilde, Denmark

Amnon Kohen Department of Chemistry University of Iowa Iowa City, Iowa

Michael E. Harris Center for RNA Molecular Biology Case Western Reserve University Cleveland, Ohio

Alexander M. Kuznetsov Department of Chemistry Technical University of Denmark Lyngby, Denmark

Alvan C. Hengge Department of Chemistry and Biochemistry Utah State University Logan, Utah Michael Hippler Department of Chemistry University of Sheffield Sheffield, United Kingdom James T. Hynes Department of Chemistry and Biochemistry University of Colorado Boulder, Colorado Takanobu Ishida Department of Chemistry State University of New York Stony Brook, New York William E. Karsten Department of Chemistry and Biochemistry University of Oklahoma Norman, Oklahoma Philip M. Kiefer Department of Chemistry and Biochemistry University of Colorado Boulder, Colorado

Jonathan S. Lau Department of Chemistry University of California San Diego, California Rene´ Le´tolle l’Universite Pierre et Marie Curie Paris, France Brett E. Lewis The Albert Einstein College of Medicine Bronx, New York Hans-Heinrich Limbach Institut fu¨r Chemie Freie Universita¨t Berlin, Germany John D. Lipscomb Department of Biochemistry, Molecular Biology and Biophysics University of Minnesota Minneapolis, Minnesota Laura Masgrau Department of Biochemistry University of Leicester Leicester, United Kingdom

Judith P. Klinman Department of Chemistry and Department of Molecular and Cell Biology University of California Berkeley, California

Olle Matsson Department of Chemistry Uppsala University Uppsala, Sweden

Heinz F. Koch Department of Chemistry Ithaca College Ithaca, New York

Zofia Mielke Faculty of Chemistry University of Wrocław Wrocław, Poland

Contributors

Dexter B. Northrop Division of Pharmaceutical Sciences School of Pharmacy University of Wisconsin-Madison Madison, Wisconsin Mats H. M. Olsson Department of Chemistry University of Southern California Los Angeles, California Piotr Paneth Institute of Applied Radiation Chemistry Technical University of Lodz Lodz, Poland Charles L. Perrin Department of Chemistry University of California San Diego, California Ehud Pines Department of Chemistry Ben-Gurion University of the Negev Be’er Sheva, Israel Bryce V. Plapp Department of Biochemistry The University of Iowa Iowa City, Iowa Martin Quack Physical Chemistry ETH Zu¨rich Zu¨rich, Switzerland Daniel M. Quinn The University of Iowa Department of Chemistry Iowa City, Iowa Franc¸ois Robert Laboratoire de Mine´ralogie Centre National di Recherche Scientifique Paris, France Emil Roduner Institut fu¨r Physikalische Chemie Universita¨t Stuttgart Stuttgart, Germany

ix

Etienne Roth National des Arts et Me´tiers Paris, France Justine P. Roth Department of Chemistry Johns Hopkins University Baltimore, Maryland Richard L. Schowen Simons Laboratories Higuchi Biosciences Center University of Kansas Lawrence, Kansas Vern L. Schramm The Albert Einstein College of Medicine Bronx, New York Steven D. Schwartz Departments of Biophysics and Biochemistry The Albert Einstein College of Medicine Bronx, New York Nigel S. Scrutton Department of Biochemistry University of Leicester Leicester, United Kingdom Willem Siebrand Steacie Institute for Molecular Sciences National Research Council of Canada Ottawa, Canada David N. Silverman Department of Pharmacology University of Florida Gainesville, Florida Zorka Smedarchina Steacie Institute for Molecular Sciences National Research Council of Canada Ottawa, Canada Lucjan Sobczyk Faculty of Chemistry University of Wrocław Wrocław, Poland C. M. Stevens Naperville, Illinois

x

Contributors

Michael J. Sutcliffe Department of Biochemistry University of Leicester Leicester, United Kingdom

Jordi Villa`-Freixa Grup de Recerca en Informatica Biomedica, IMIM/UPF Barcelona, Spain

Donald G. Truhlar Department of Chemistry and Supercomputing Institute Minneapolis, Minnesota

Arieh Warshel Department of Chemistry University of Southern California Los Angeles, California

Jens Ulstrup Department of Chemistry Technical University of Denmark Lyngby, Denmark

Ralph E. Weston Jr. Chemistry Department Brookhaven National Laboratory Upton, New York

W. Alexander Van Hook Chemistry Department University of Tennessee Knoxville, Tennesse

Max Wolfsberg Chemistry Department University of California Irvine, California

Table of Contents Chapter 1 Theoretical Basis of Isotope Effects from an Autobiographical Perspective ................................................................................................................ 1 Jacob Bigeleisen

Chapter 2 Enrichment of Isotopes .......................................................................................... 41 Takanobu Ishida and Yasuhiko Fujii

Chapter 3 Comments on Selected Topics in Isotope Theoretical Chemistry ........................ 89 Max Wolfsberg

Chapter 4 Condensed Matter Isotope Effects ....................................................................... 119 W. Alexander Van Hook

Chapter 5 Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes ............................................................................. 153 Janet E. Del Bene

Chapter 6 Isotope Effects on Hydrogen-Bond Symmetrization in Ice and Strong Acids at High Pressure ............................................................................. 175 Katsutoshi Aoki

Chapter 7 Hydrogen Bond Isotope Effects Studied by NMR .............................................. 193 Hans-Heinrich Limbach, Gleb S. Denisov, and Nikolai S. Golubev

Chapter 8 Isotope Effects and Symmetry of Hydrogen Bonds in Solution: Single- and Double-Well Potential ...................................................................... 231 Jonathan S. Lau and Charles L. Perrin

Chapter 9 NMR Studies of Isotope Effects of Compounds with Intramolecular Hydrogen Bonds ................................................................................................... 253 Poul Erik Hansen

Chapter 10 Vibrational Isotope Effects in Hydrogen Bonds ................................................ 281 Zofia Mielke and Lucjan Sobczyk

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Table of Contents

Chapter 11 Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics .......... 305 Michael Hippler and Martin Quack

Chapter 12 Nonmass-Dependent Isotope Effects ................................................................. 361 Ralph E. Weston, Jr.

Chapter 13 Isotope Effects in the Atmosphere ..................................................................... 387 Etienne Roth, Rene´ Le´tolle, C. M. Stevens, and Franc¸ois Robert

Chapter 14 Isotope Effects for Exotic Nuclei ....................................................................... 417 Olle Matsson

Chapter 15 Muonium — An Ultra-Light Isotope of Hydrogen ........................................... 433 Emil Roduner

Chapter 16 The Kinetic Isotope Effect in the Photo-Dissociation Reaction of Excited-State Acids in Aqueous Solutions ................................................... 451 Ehud Pines

Chapter 17 The Role of an Internal-Return Mechanism on Measured Isotope Effects .................................................................................................... 465 Heinz F. Koch

Chapter 18 Vibrationally Enhanced Tunneling and Kinetic Isotope Effects in Enzymatic Reactions ......................................................................... 475 Steven D. Schwartz

Chapter 19 Kinetic Isotope Effects for Proton-Coupled Electron Transfer Reactions ............................................................................................................ 499 Sharon Hammes-Schiffer

Chapter 20 Kinetic Isotope Effects in Multiple Proton Transfer ......................................... 521 Zorka Smedarchina, Willem Siebrand, and Antonio Ferna´ndez-Ramos

Chapter 21 Interpretation of Primary Kinetic Isotope Effects for Adiabatic and Nonadiabatic Proton-Transfer Reactions in a Polar Environment ............. 549 Philip M. Kiefer and James T. Hynes

Chapter 22 Variational Transition-State Theory and Multidimensional Tunneling for Simple and Complex Reactions in the Gas Phase, Solids, Liquids, and Enzymes ...................................................................................................... 579 Donald G. Truhlar

Table of Contents

xiii

Chapter 23 Computer Simulations of Isotope Effects in Enzyme Catalysis ....................... 621 Arieh Warshel, Mats H. M. Olsson, and Jordi Villa`-Freixa

Chapter 24 Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen ......................................................................................... 645 Justine P. Roth and Judith P. Klinman

Chapter 25 Solution and Computational Studies of Kinetic Isotope Effects in Flavoprotein and Quinoprotein Catalyzed Substrate Oxidations as Probes of Enzymic Hydrogen Tunneling and Mechanism ................................ 671 Jaswir Basran, Laura Masgrau, Michael J. Sutcliffe, and Nigel S. Scrutton

Chapter 26 Proton Transfer and Proton Conductivity in Condensed Matter Environment ....................................................................................................... 691 Alexander M. Kuznetsov and Jens Ulstrup

Chapter 27 Mechanisms of CH-Bond Cleavage Catalyzed by Enzymes ............................ 725 Willem Siebrand and Zorka Smedarchina

Chapter 28 Kinetic Isotope Effects as Probes for Hydrogen Tunneling in Enzyme Catalysis ............................................................................................... 743 Amnon Kohen

Chapter 29 Hydrogen Bonds, Transition-State Stabilization, and Enzyme Catalysis ........................................................................................ 765 Richard L. Schowen

Chapter 30 Substrate and pH Dependence of Isotope Effects in Enzyme Catalyzed Reactions ........................................................................ 793 William E. Karsten and Paul F. Cook

Chapter 31 Catalysis by Alcohol Dehydrogenases ............................................................... 811 Bryce V. Plapp

Chapter 32 Effects of High Hydrostatic Pressure on Isotope Effects .................................. 837 Dexter B. Northrop

Chapter 33 Solvent Hydrogen Isotope Effects in Catalysis by Carbonic Anhydrase: Proton Transfer through Intervening Water Molecules ................. 847 David N. Silverman and Ileana Elder

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Table of Contents

Chapter 34 Isotope Effects from Partitioning of Intermediates in Enzyme-Catalyzed Hydroxylation Reactions ................................................ 861 Paul F. Fitzpatrick

Chapter 35 Chlorine Kinetic Isotope Effects on Biological Systems .................................. 875 Piotr Paneth

Chapter 36 Nucleophile Isotope Effects ............................................................................... 893 Vernon E. Anderson, Adam G. Cassano, and Michael E. Harris

Chapter 37 Enzyme Mechanisms from Isotope Effects ....................................................... 915 W. Wallace Cleland

Chapter 38 Catalysis and Regulation in the Soluble Methane Monooxygenase System: Applications of Isotopes and Isotope Effects ...................................... 931 John D. Lipscomb

Chapter 39 Secondary Isotope Effects .................................................................................. 955 Alvan C. Hengge

Chapter 40 Isotope Effects in the Characterization of Low Barrier Hydrogen Bonds ........ 975 Perry A. Frey

Chapter 41 Theory and Practice of Solvent Isotope Effects ................................................ 995 Daniel M. Quinn

Chapter 42 Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase ............................................................................ 1019 Brett E. Lewis and Vern L. Schramm

Index .................................................................................................................................. 1055

1

Theoretical Basis of Isotope Effects from an Autobiographical Perspective Jacob Bigeleisen

CONTENTS I. II. III.

From Soddy – Fajans through Urey –Greiff ......................................................................... 1 Equilibrium Systems — General ......................................................................................... 3 Equilibrium in Ideal Gases .................................................................................................. 4 A. Classical and Quantum Mechanical Systems .............................................................. 4 B. The Reduced Partition Function Ratio of an Ideal Gas .............................................. 5 1. Numerical Calculation of the Reduced Partition Function Ratio ........................ 7 C. Corrections to the Bigeleisen –Mayer Equation.......................................................... 8 IV. Isotope Chemistry and Molecular Structure...................................................................... 12 A. The First Order Rules of Isotope Chemistry ............................................................. 12 B. Statistical Mechanical Perturbation Theory .............................................................. 13 C. Polynomial Expansions of the Reduced Partition Function Ratio............................ 14 V. Kinetic Isotope Effects....................................................................................................... 18 VI. Condensed Matter Isotope Effects ..................................................................................... 25 Acknowledgments .......................................................................................................................... 32 References....................................................................................................................................... 33

I. FROM SODDY– FAJANS THROUGH UREY– GREIFF Isotopes were discovered in radiochemical investigations of the decay of the heavy elements. Products were found with different nuclear properties, which could not be separated chemically, but stood in the same place in the Periodic Table; e.g. Radium, 226Ra, an a emitter with a half life of 1600 years, Mesothorium 1, 228Ra, a b emitter with a half life of 5.7 years and Actinium X, 223Ra, an a emitter with a half life 11.7 days. They were named isotopes by Soddy1 from the Greek words isos topos, the same place. Isotopes had the same nuclear charge, but different atomic masses. This was firmly established by the determination of the atomic weights of the lead isotopes which were the end products of the three radioactive series. Lead from the thorium series was found to have an atomic weight of 207.77; lead from the uranium –radium series had an atomic weight of 206.08. Fajans,2 a major figure in radiochemistry, concluded that isotopes had similar, but not identical, chemical properties. Since isotopes have different atomic masses, molecules substituted with sister isotopes (isotopomers) would have different vibrational frequencies. Consequently they would have different heat capacities, entropies, and free energies. After WWI, Lindemann3,4 subsequently known as Lord Cherwell, derived the equations for the differences in vapor pressures 1

2

Isotope Effects in Chemistry and Biology

of isotopes. For a monatomic solid with a Debye frequency distribution, Lindemann found 0

0

‘nðP =PÞ ¼ 9=8ðQ 2 QÞD =T 0

0

2

‘nðP =PÞ ¼ 3=40ðQ 2 2 Q ÞD =T

½QD ¼ hnD =k . 2p 2

½QD ¼ hnD =k , 2p

ð1:1Þ ð1:2Þ

where Q 0 and Q are the Debye temperatures for the light and heavy isotopomers, respectively. Equation 1.1 and Equation 1.2 were derived for the case that there was a zero point energy associated with a vibration. He calculated the difference in vapor pressures of 206Pb and 208Pb and predicted the ratio of the vapor pressures of 206Pb/208Pb to be 1.0002 at 600 K. A much larger effect of the opposite sign was predicted for the case of no zero point energy. Since no such difference was found, Lindemann correctly concluded that there was a zero point energy. In actual fact, Lindemann’s calculation for the zero point energy case is a factor of ten too large; the correct calculation from Equation 1.2 leads to a result of 0.002%; the light isotope, 206Pb, has the higher vapor pressure. Equation 1.2, which had been derived independently by Otto Stern, provided the incentive for Keesom and Van Dijk5 to achieve a partial separation of the neon isotopes by low temperature fractional distillation. In planning a search for an isotope of mass 2, Urey6 decided to carry out an enrichment of hydrogen of natural abundance by a Raleigh distillation. He used Equation 1.1 to estimate the difference in vapor pressures of H2 and HD. Urey then turned his attention to the question of isotope effects in chemical reactions. He had his student, David Rittenberg, calculate, using quantum statistical mechanics, the equilibrium constant for the exchange reaction H2 þ 2DI ¼ D2 þ 2HI

ð1:3Þ

as a function of temperature.7 Their calculations were confirmed by experiment.8 The method of Urey and Rittenberg was extended to the case of polyatomic molecules by Urey and Greiff.9 For the isotopic exchange reaction AX þ A0 Y ¼ A0 X þ AY

ð1:4Þ

they expressed the equilibrium constant in terms of partition function ratios. K ¼ ðAY=A0 YÞ=ðAX=A0 XÞ ¼ ðQ=Q 0 ÞAY =ðQ=Q 0 ÞAX X Q¼ expð21i =kTÞ

ð1:5Þ ð1:6Þ

i

For the partition function ratio of molecules of like symmetry, ðQ=Q 0 Þ; and for which the translation and rotation obeyed classical statistical mechanics, they obtained ðQ1 =Q2 Þ ¼ ðM1 =M2 Þ3=2 ðABCÞ1 =ðABCÞ2

1=2

Y

ðeðu2i 2u1i Þ=2 Þð1 2 e2u2i Þ=ð1 2 e2u1i Þ

ð1:7Þ

i

In Equation 1.7 subscripts 1 and 2 refer to the heavy and light isotopes, respectively; M is the molecular weight, A; B; and C are the principal moments of inertia and ui ¼ hui ¼ hni =kT: The terms eðu2i 2u1i Þ=2 and ð1 2 e2u2i Þ=ð1 2 e2u1i Þ are, respectively, the contributions from the zero point energy differences between the light molecule, u2 ; and the heavy molecule, u1 ; and the Boltzmann excitation factors. Urey and Greiff 9 tabulated values of ðQ1 =Q2 Þ for compounds of the light elements as a function of temperature between 273 and 600 K. The values of ðQ1 =Q2 Þ varied from an order of magnitude for the isotopes of hydrogen to a few percent for isotopes of the elements in the first two rows of the periodic table. The ratios decreased with temperature. The equilibrium constant for an isotopic exchange reaction, which is the quotient of two partition function ratios, is of the order of a few percent excepting those reactions which involve the isotopes of hydrogen. Urey was able to utilize small differences in the chemical properties of the light

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

3

elements to separate the isotopes of nitrogen10 through the exchange reaction 15

14 15 þ NH3 ðgÞ þ 14 NHþ 4 ðsolÞ ¼ NH3 ðgÞ þ NH4 ðsolÞ

ð1:8Þ

He utilized similar acid base exchange reactions to concentrate the isotopes of carbon, oxygen, and sulfur. Lewis and MacDonald11 used the redox reaction, 7

Li ðamalgamÞ þ 6 Li ðEtOHÞþ ¼ 6 Li ðamalgamÞ þ 7 Li ðEtOHÞþ

ð1:9Þ

in a tour de force, to effect a partial separation of the lithium isotopes.

II. EQUILIBRIUM SYSTEMS — GENERAL With the discovery of nuclear fission in 1938, Urey joined with a small group of American scientists to develop the nuclear bomb. He was the logical choice to head up the effort to separate the isotopes required for this purpose. He organized the SAM Laboratory at Columbia University. For this purpose he recruited a staff of scientists, engineers, technicians, and support personnel. The staff grew to approximately 1000 by 1944. Research and development at SAM Laboratory led to the construction of two different types of heavy water plants, a boron isotope separation plant and the large 235U separation plant, K-25, constructed in Oak Ridge Tennessee. A number of other research projects were carried out, including the countercurrent gas centrifuge in collaboration with J. W. Beams, and the feasibility of separation of the uranium isotopes by a photochemical process. Earlier, mercury and chlorine isotopes had been separated by photochemical processes. I was recruited in June 1943 to work on the feasibility of a photochemical process for uranium isotope separation. I brought to this assignment experience in spectroscopy and photochemistry of systems at low temperature. It was in this capacity that my collaboration with Maria Goeppert-Mayer began. The most promising uranium compounds which might exhibit a difference in the absorption spectra were the uranyl salts, which had sharp line spectra. Maria Mayer worked on the theory of the spectra. Along with S. S. Hanna, M. L. Schultz, and D. T. Vier, I determined the spectra of most of the compounds reported in the compilation by Dieke and Duncan.12 In November 1943 I was asked by Urey’s assistant, Isidor Kirschenbaum, to calculate the differences in chemical properties of the uranium isotopes for all compounds of uranium through Equation 1.7. The only compound for which there was sufficient data for this calculation was the uranyl ion, UO2þ 2 . The structure of UF6 was unknown at the time. I started my assignment with the objective to calculate each of the terms Q in Equation 1.7 directly as a difference between two small quantities. I simplified the terms i ðeðu2i 2u1i Þ=2 Þð1 2 e2u2i Þ=ð1 2 e2u1i Þ to give 1 þ ðeui 2 1Þ21 dui 2

ð1:10Þ

where dui ¼ u2i 2 u1i : When I discussed my work with Maria Mayer, she volunteered to join me in this approach. I was pleased to accept her help. Her first suggestion was to remove the classical contributions to Equation 1.7 by adding the term ð21=ui Þ to Equation 1.10. The final result, the reduced partition function ratio to be defined in Section III, ðs=s0 Þf ¼ 1 þ

X i

Gðui Þdðui Þ ¼ 1 þ

X 1 þ ðeui 2 1Þ21 2 1=ui dðui Þ 2 i

ð1:11Þ

was directly applicable for the calculation of the isotope fractionation factor, a; to be discussed below. Based on Equation 1.11, Maria Mayer and I were able to predict that chemical isotope separation factors as large as 1.001 were possible for the isotopes of uranium. We showed that

4

Isotope Effects in Chemistry and Biology

simple acid –base reactions, e.g. reaction (Equation 1.8) and the isotopic exchanges between 22 SiF4 and SiF22 6 and SnCl4 and SnCl6 led to small isotope separation factors. For the purpose of these calculations I prepared a table of GðuÞ values as a function of u: A summary of this development was prepared by Maria Mayer for review by Edward Teller13 and is further discussed by Clyde Hutchinson.14

III. EQUILIBRIUM IN IDEAL GASES In the fall of 1946 a complete exposition of the work by Maria Mayer and myself was prepared for publication in the open literature at the suggestion of Willard F. Libby. What follows is a summary of that landmark paper.15

A. CLASSICAL AND Q UANTUM M ECHANICAL S YSTEMS In classical statistical mechanics, the partition function, Q; is ð h f Q ¼ ð1=sÞ e2Hðp;q=kTÞ dp…dq

ð1:12Þ

where h is Planck’s constant, s is the symmetry number, H is the Hamiltonian, k is Boltmann’s constant and T is the temperature. The integral runs over all the coordinates, q; and the momenta, p: We choose Cartesian coordinates. The integration of the momenta leads to QCl ¼ ð1=sÞ

Y i

ð2pmi kT=h2 Þ3=2

ð

e2Vðq=kTÞ dq1 …dq3n

ð1:13Þ

where V is the potential energy of the system of 3n particles and mi are the atomic masses. Within the Born – Oppenheimer approximation the potential energy of a system is isotope invariant. Within this approximation the classical partition function ratio of two isotopomers, Q=Q 0 ; is Y ð1:14Þ ðQ=Q 0 ÞCl ¼ ðs 0 =sÞ ðmi =m0i Þ3=2 i

This result holds for all phases. Since the m’s in Equation 1.14 are atomic masses, they will cancel identically in any chemical reaction or in any phase change. The equilibrium constant for any equilibrium process calculated by classical statistical mechanics is just the ratios of the symmetry numbers. But this is not the isotope separation factor. We will illustrate this by consideration of the hypothetical exchange reaction between dioxygen, 16O2, with an ideal gas of oxygen atoms, 18O. 16

O2 þ 18 O ¼ 16 O18 O þ 16 O

ð1:15Þ

We now define ðs=s 0 Þf ; the reduced partition function ratio of a pair of isotopomers, RFPR, ðs=s 0 Þf ¼ ðQ=Q 0 Þqm =ðQ=Q 0 ÞCl

ð1:16Þ

ðQ=Q 0 ÞCl ¼ ðs 0 =sÞðm=m 0 Þ3=2

ð1:17Þ

For both the O2 and O species

KCl ¼ sð16 O2 Þ=sð16 O18 OÞ=sð16 OÞ=sð18 OÞ ¼ 2 This equilibrium constant corresponds to random distribution of 16O and 18O atoms between the dioxygen molecules and the oxygen atoms. If the atom fraction of 18O in both the oxygen atoms

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

5

and molecules is x; then the relative numbers of the dioxygen isotopomers are: 16

O2 ¼ ð1 2 xÞ2 ;

16

O18 O ¼ 2ð1 2 xÞx;

18

O2 ¼ x2

The ratio of 18O to 16O atoms in both the atomic and molecular oxygen is x=ð1 2 xÞ ¼ g: The classical equilibrium constant assuming random distribution is 2, just the inverse ratio of symmetry numbers. Now consider the quantum mechanical equilibrium constant. For atomic oxygen ðQ=Q 0 Þqm is equal to ðQ=Q 0 ÞCl . The atom fraction of 18O in the atom is x and the atom ratio is x=ð1 xÞ. Let y be the atom fraction of 18O in the O2 isotopomers. The relative numbers of the dioxygen molecules for the quantum mechanical case is O2 ¼ ð1 2 yÞ2 ,

16

O18 O ¼ 2ð1 2 yÞy,

18

O 2 ¼ y2

The atom ratio 18O/16O in the dioxygen molecules, b; is y=ð1 2 yÞ: The quantum mechanical equilibrium constant is now Kqm ¼ ½2y=ð1 2 yÞ =½x=ð1 2 xÞ ¼ 2b=g ¼ a KCl

ð1:18Þ

where a ¼ ½2y=ð1 yÞ =½x=ð1 xÞ is just the isotope fractionation factor, b=g; between the two chemical species, dioxygen and oxygen atoms. The equilibrium constant is a product of a symmetry number ratio and a fractionation factor. The fractionation factor is independent of the symmetry number. RFPR, ðs=s 0 Þf ; is the isotope fractionation factor between a pair of isotopomers and the free gaseous atoms of the isotopes. Since ðs=s 0 Þf for the gaseous atomic species is identically unity, the free gaseous atom is the logical standard state for isotope fractionation studies. Later we will show that ðs=s 0 Þf is always a positive quantity when the ratio ðQ=Q 0 Þ is defined as the heavy isotope divided by the light isotope, always with the superscript 0 or subscript 2. The derivation of the result in Equation 1.18, Kqm ¼ aKCl ; can be generalized for isotopomers of any symmetry. In general, 0

0

0

0

‘n Kqm ¼ ‘nðs=s Þfproducts 2 ‘nðs=s Þfreactants þ ‘nðs =sÞproducts 2 ‘nðs =sÞreactants

B. THE R EDUCED PARTITION F UNCTION R ATIO OF AN I DEAL G AS To derive a relationship for ðs=s 0 Þf of an ideal gas of a pair of isotopomers, we note that in quantum mechanics the partition function is a product of translation, rotation, and vibration. The nuclear spin can be neglected since it follows classical statistics, except for molecules at very low temperature. Under the Born – Oppenheimer approximation, the electronic energy states are isotope invariant. The translation and rotation of the molecules can be treated classically, except for molecules with very light atoms. In that case a correction needs to be introduced for nonclassical rotation. If the zero of the energy scale is chosen as the minimum in the vibrational potential energy, Qvib;qm ¼

Y i

ðexpð2ui =2ÞÞ=ð1 2 expð2ui ÞÞ

Qvib;Cl ¼

Y i

ð1=ui Þ

ð1:19Þ ð1:20Þ

where ui ¼ hni =kT: The product runs over all 3n 2 6 vibrational frequencies. A degeneracy of order gi is counted gi times. From Equation 1.19 and Equation 1.20 one gets for ðs=s0 Þf and its logarithm Y ðu1i =u2i Þeðu2i 2u1i Þ=2 ð1 2 e2u2i Þ=ð1 2 e2u1i Þ ðs=s 0 Þf ¼ ð1:21Þ i

6

Isotope Effects in Chemistry and Biology 0

‘nðs=s Þf ¼

X

0

‘nðui =ui Þ þ dui =2 þ i

X i

0

‘n 1 2 expð2ui Þ = 1 2 expð2ui Þ

0

0

ð1:23Þ

Gðui Þdðui Þ

ð1:24Þ

‘n cðui Þ ¼ ‘nðui =ui Þ þ dui =2 þ ‘n 1 2 expð2ui Þ = 1 2 expð2ui Þ

0

‘nðs=s Þf ¼

Gðui Þ ¼

X i

‘n cðui Þ ¼

X i

ð1:22Þ

1 þ ðeui 2 1Þ21 2 1=ui 2

ð1:25Þ

Equation 1.21 is frequently referred to as the Bigeleisen –Mayer equation. An equation similar to 1.24 was derived independently by Waldmann, who utilized it to give a qualitative analysis of the fractionation of the nitrogen isotopes in the NH3 – NHþ 4 exchanges and the carbon isotopes in the CN2 – HCN exchanges.16 The nomenclature for dui is dui ¼ u 0i 2 ui : Since prime refers to the light isotope, di is always positive. A plot of Gðui Þ as a function of ui is given in Figure 1.1. It is positive for all values of u and runs from Gðui Þ ¼ 0 at u ¼ 0 to Gðui Þ ¼ 12 at u ¼ 1: Thus ‘nðs=s 0 Þf is a positive number. Equation 1.24 is derived from Equation 1.22 for small dui : This approximation has been found to hold remarkably well for isotopomers of hydrogen, which generally have large values of dui at and above room temperature.17 Stretching vibrations have higher frequencies and larger frequency shifts on isotopic substitution than bending frequencies. Thus it follows from Equation 1.24 that stretching motions will be the major contributors to ‘nðs=s 0 Þf : They need not be the most important factor in determining the fractionation factor between two species. Bigeleisen and Mayer then expanded eui in Equation 1.25 in powers of ui up through u3i to give ðs=s 0 Þf ¼ 1 þ ð1=24Þ

X i

du2i

ð1:26Þ

du2i ¼ u21i 2 u2i : They carried out the summation over the ð3n 2 6Þ vibrational frequencies by the use of Cartesian coordinates to construct the determinant of the H ¼ FG matrix for the molecular

0.5

u /12

0.4

G (u)

0.3 0.2 0.1 0.0

0

10

u

20

FIGURE 1.1 Plot of GðuÞ; Equation 1.25, vs. u: The limiting slope is u=12:

30

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

vibrations. X

7

jH 2 lIj ¼ 0;

½du2 X i ½dli

3n26 3n26

X ¼ ð"=kTÞ2 dli ; X ¼ fij dgij ; ij

X X 0 ½dli 3n26 ¼ ðmi 2 mi Þfii ; i X X 0 dli ¼ ðmi 2 mi Þaii ; i X X dli ¼ ð1=m0i 2 1=mi Þaii i X ðs=s0 Þf ¼ 1 þ ð1=24Þð"=kTÞ2 ð1=m0i 2 1=mi Þaii

ð1:27Þ

i

aii is the three dimensional Cartesian force constant for the displacement of the isotopic atom along the Cartesian axes. They applied Equation 1.27 to calculate the reduced partition function ratios for the isotopomers X0 Yn and XYn simply from the symmetrical stretching vibrational frequency. This was a useful approximation at the time when there was a paucity of detailed knowledge about molecular vibration frequencies; the symmetrical stretching frequency in molecules of the type X0 Yn was frequently known from Raman spectroscopy. It follows from Equation 1.22, Equation 1.24, and Equation 1.27 that plots of ‘n cðui Þ and ‘nðs=s 0 Þf vs. 1=T are linear with positive slow at low temperature or large values of u: They approach u ¼ 0 or T ¼ 1 with zero slope. In the region of high temperature both ‘n cðui Þ and ‘nðs=s 0 Þf are linear in 1=T 2 : While ‘nðs=s 0 Þf is a monotonically increasing function of 1=T; ‘n Kqm and ‘na; the equilibrium constant and fractionation factor between two chemical species, respectively, need not be. This results from the different temperature dependencies of the ‘nðs=s0 Þf values of the two different chemical species. Thus cases are known where the heavy isotopomer concentrates in species A at high temperature but in species B at low temperature. The conditions for such a cross over are spelled out clearly by Urey.18 Additional cases have been found in which the tritium/protium fractionation factor is larger than the deuterium/protium fractionation factor at high temperature but is smaller at low temperature.19 Systems which show the cross over will have either maxima or minima in the fractionation factor as a function of temperature. Finally, in the case of exchange reactions involving the oxides of nitrogen, cases have been found in which there are points of inflection in the temperature dependence.20 The formulation of the concept of the reduced partition function ratio in 1943 and its publication in 194715 led the way toward major developments in isotope effects in the second half of the twentieth century. Among these are the research which led to new processes for separation of isotopes,14,21 – 28 the theory of the kinetic isotope effect,29,30 the use of the kinetic isotope effect in the study of biochemical and organic reaction mechanisms,31,32 the theory of condensed phase isotope effects,33 numerous discoveries relating to quantum effects in condensed matter,34,35 and stable isotope geochemistry.36,37 All of these topics are discussed further in this and subsequent chapters of this book. Through the reduced partition function ratio it has been possible to relate isotope effects directly to molecular structure. A brief description of this development is given in Section IV of this chapter. The Bigeleisen – Mayer paper is the most frequently cited reference in the literature of isotope chemistry. 1. Numerical Calculation of the Reduced Partition Function Ratio Numerical calculations of RFPR are made by calculation of the vibrational frequencies of the isotopomers and their substitution into Equation 1.21 and Equation 1.24. Frequently referenced are the tabulation by Urey18 of values of f ; rather than ðs=s 0 Þf ; for isotopomers of hydrogen, lithium, boron, carbon, nitrogen, oxygen, chlorine, and bromine, all between 273 –600 K. This table was

8

Isotope Effects in Chemistry and Biology

prepared in the summer of 1946 in preparation for the Liversidge Lecture before the Chemical Society of London on December 18, 1946, which preceded the proper treatment of the anharmonic correction to the zero point energy of a polyatomic molecule.38 Thus RFPR tabulations for the isotopomers of hydrogen in Urey’s paper in addition to all publications prior to 1969 fail to account properly for anharmonicity. Urey’s tabulation for the self-exchange reaction between H2O and D2O as a function of temperature conflicts with the theorem in Section IV relating the sign of the deviation from the rule of the geometric mean. He tabulates values for K larger than 4.00; whereas it is less than 4.00 at all temperatures. The Urey tabulation opened up the entire field of isotope geochemistry. Urey’s assistant, Lawrence S. Myers, used Equation 1.24 to calculate the RPFR values of all the isotopomers of the light elements other than hydrogen. To assist him in this effort I supplied them with my unpublished GðuÞ tables. With these tables Myers was able to calculate f values with a precision of ^ 0.0001. and Urey noticed that the fractionation factor for the exchange of 18O and 16O between CO22 3 H2O decreased by 0.00025/8C. Now, if one could measure isotope ratios in paleocarbonate samples with a precision of ^ 0.0002, and if the samples retained the record, one could measure paleotemperatures to ^ 0.88C. Isotope ratio measurements of this precision were an order of magnitude beyond the state of the art at the time. Challenges of this type did not discourage Urey. He had met equal challenges before. He knew how to put together the necessary resources, both material and personnel; he had the drive and persistence to reach the objective. Publications 39 and 40 report the successful attainment of the carbonate– water isotopic temperature scale and the measurement of the temperature variation in the life cycle of a belemnite. Urey was deservedly recognized for this achievement with the Arthur L. Day Medal of the Geological Society of America and the V. M. Goldschmidt Medal of the Geochemical Society. During the 1940s and 1950s RFPR values were generally calculated for symmetrical molecules with the aid of vibrational formulae. Mechanical calculating machines were used. The advent of high-speed digital computers changed all of this. Starting in the early 1960s Max Wolfsberg wrote computer programs to calculate ðs=s 0 Þf from isotopic frequencies derived by solution of the Wilson FG matrices for the isotopomers. The programs were also appropriately modified to calculate kinetic and condensed phase isotope effects. He made these programs widely available and instructed scientists in their use. As a result there are now tabulations of RFPRs which are as good as the F matrices utilized in the calculations. F matrices were constructed to fit experimental spectroscopic data. Some examples are calculations by Bron, Chang, and Wolfsberg41 for the molecular species, H2, H2O, H2S, H2Se, and NH3. These calculations are fully corrected for anharmonicity. In the harmonic approximation there are, among other tabulations, those for carbon and hydrogen isotope RFPRs for a series of simple organic molecules by Hartshorn and Shiner42 and nitrogen and oxygen RFPRs in a series of nitro compounds by Monse, Spindel, and Stern.20,43 Subsequently Wolfsberg and coworkers showed that the necessary vibrational frequencies calculated ab initio from solution of the Schroedinger equation could be used to calculate RFPRs.44,45

C. CORRECTIONS TO THE B IGELEISEN – M AYER E QUATION Two corrections are necessary to the Bigeleisen – Mayer equation. The first has already been alluded to. This is the effect of anharmonicity. The corrections for anharmonicity for diatomic46 and polyatomic molecules38 have been derived by Wolfsberg. The principal correction comes from the fact that the zero point energy of a molecule is mass dependent. This dependence plays no role in spectroscopy, since spectroscopic studies measure vibrational energy differences. This correction is of importance only for the isotopomers of hydrogen. It can lead to corrections by at most 1% in the fractionation factors for isotopic exchange between different molecular species containing hydrogen. The second correction comes from the use of the Born –Oppenheimer approximation. There are two types of corrections. The first correction is due to the fact that the nuclear and electronic wave functions are not completely independent. For nuclei of finite mass there is a

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

9

correction to the electronic energy states which arises from the coupling of the electronic and nuclear motions. These corrections have been worked out by Kleinman and Wolfsberg.47 – 49 The corrections are largest for reactions involving dihydrogen and can amount to a few percent for reactions at 300 K. Bardo and Wolfsberg50 have applied both of these corrections to the exchange reaction HDðgÞ þ H2 OðgÞ ¼ H2 ðgÞ þ HDOðgÞ as a function of temperature. The Bardo – Wolfsberg calculations include the complete corrections for nonclassical rotation. A comparison of their calculations with a variety of experimental data is given in Figure 1.2. The experimental data51 – 53 from three independent laboratories fall on the theoretical curve within the limits of experimental error. A second correction to the Born – Oppenheimer approximation was discovered in the analysis of the isotope fractionation factors involving the isotopes 234U, 235U, and 236U from 238U in the exchange reactions between U(III) and U(IV) and between U(IV) and U(VI).54 My colleagues, Takanobu Ishida and Max Wolfsberg, organized a symposium on 6 May, 1989 at the State University of New York, Stony Brook in celebration of my 70th birthday. Alfred Klemm of the Otto Hahn Institut fu¨r Chemie, who came to the symposium with his wife Hannelore, visited me in my office on the preceding Friday afternoon. He was most interested in my opinion of a manuscript which he had received in his capacity as editor of the Zeitschrift fu¨r Naturforschung. The paper reported an anomalous effect in the fractionation of the isotope 235U from 238U in the chemical exchange reaction between U(IV) in aqueous solution and U(VI) adsorbed on an ion exchange resin. The 235U – 238U fractionation factor was reported to be significantly larger than 1.5 times the 236U – 238U fractionation factor. The nonlinearity of the fractionation factor as a function of the mass number of the isotopomer is shown in Figure 1.3.55 The manuscript had received a favorable review from one referee; a second referee recommended against publication. I read the manuscript and recognized that this was an important experimental finding. I made a back-of-the-envelope calculation and rejected the authors’ explanation that the enhancement of the 235U – 238U fractionation factor was due to the nuclear spin of 235U. I recommended publication on the basis of the established record of the authors in measuring uranium isotope fractionation factors. I suggested that the authors’ explanation of the effect be condensed. In due course someone would figure out the origin of the anomaly.

H2O(g) + HD(g) = HDO(g) + H2(g)

In K

1.25

1.00 Suess (1949) CMRRS&V (1954) RHB (1975) BW (1975)

0.75

0.50 1.5

2.0

2.5 103 /T

3.0

3.5

FIGURE 1.2 Plot of ‘n K for the exchange reaction HDðgÞ þ H2 OðgÞ ¼ HDOðgÞ þ H2 ðgÞ as a function of temperature. Suess,51 CMRRS&V,52 RHB,53 BW.50

Isotope Effects in Chemistry and Biology

5.6

Separation Coefficient e×104

1.8

15

(×10−4)

10

13

10

7.4

5

0

234

235

236

237

238

Mass Number

FIGURE 1.3 Plot of the logarithms of the 234U, 235U, and 236U isotope separation factors vs. 238U. 1 ¼ ‘n 238 U=i U IV; aq: = 238 U=i U VI; resin as a function of the isotope mass, mi ; as reduced to 308 K. (Reproduced from Bigeleisen, J., J. Am. Chem. Soc., 118, 3676– 3680, 1996. With permission.)

The following year I made an accurate calculation of the nuclear spin effect in the hypothetical exchange reaction between U(III) and U(VI). On the basis of the information I received from Clyde Hutchison on the hyperfine splitting in Uþ3, I showed that the nuclear spin effect was almost two orders of magnitude smaller than the observed anomaly for that reaction. It was then that I realized that the effect reported by Fujii et al.55 was due to the nuclear field shift. Due to a number of health problems I did not complete my work on this problem until the summer of 1995.54 The nuclear field shift is a shift in the electron energy states in an atom or molecule due to the perturbation resulting from the interaction of those electrons with a high density at the nucleus with the nuclear charge. Such shifts, which are observed in the differences in the electronic spectra of isotopomers, have been known from the atomic spectra of the heavy elements since the 1920s. For ˚ line in the spectra of 235U and 238U is 0.4098 cm21. A shift instance, the shift between the 5028 A of this magnitude corresponds to a correction of 2 £ 1023 to ‘na; the logarithm of the separation factor at 300 K. This is larger than the observed enrichment factor in the 235 –238 U(IV) – U(VI) exchange,1.3 £ 1023 at 300 K. It is important to call attention to the fact that the heavy isotopomer, 238 U, enriches in the U(IV). The vibrational spectrum and reduced partition function ratio of the 12 – 14 UOþ2 From the known separation factor and the reduced partition function 2 are well known. ratio one calculates that the U(IV) species would have to have the improbable stretching frequency of 2000 cm21 compared with 860 cm21 for the uranyl ion, UOþ2 2 . It is difficult to rationalize such strong bonding in the solvated U(IV) ion when compared with UOþ2 2 , which has two covalent double bonds in addition to the solvation. Since an electron is bound more tightly to a small nucleus, the light isotopomer, the ground state energy of the heavy isotopomer, will be higher than that of the light isotopomer. This is opposite of the ordering of the zero point energies associated with the molecular vibrations. In the exchange reaction between 235U and 238U 235

UðIVÞaq þ 238 UðVIÞresin ¼ 238 UðIVÞaq þ 235 UðVIÞresin

ð1:28Þ

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

11

the logarithm of the fractionation factor, a ¼ ð238 U=235 UÞðIVÞ=ð238 U=235 UÞðVIÞ; is, after neglect of the anharmonic correction to the molecular vibrations and KBOELE ; the usual mass correction to the Born – Oppenheimer approximation,47 – 49 ‘n a ¼ ‘n a0 þ ‘n Kfs

ð1:29Þ

where 0

0

‘n a0 ¼ ‘nðs=s Þf ðUIVÞ 2 ‘nðs=s Þf ðUVIÞ ‘n Kfs ; the correction for the nuclear field shift is ‘n Kfs ¼ ðkTÞ

21

ðE 0

238

U 2 E0

235

UÞðVIÞ 2 ðE 0

238

U 2 E0

235

UÞðIVÞ

ð1:30Þ

From the definition of a ¼ ð238 U=235 UÞðIVÞ=ð238 U=235 UÞðVIÞ it follows that the nuclear field shift will lead to a preference of the heavy isotope for the chemical species with the smallest number of s electrons or s bonding orbitals. The nuclear field shift in any electronic energy state of an atom or molecule can be written as the product of the electron density at the nucleus and the size and shape of the nucleus. The latter is independent of the chemical species in which the isotope is combined. It is a specific nuclear property for each isotope. Thus the shifts in different compounds of a given element can be related to the shifts in the atomic spectra, which are frequently known, by n o ð1:31Þ dEi0 UðVIÞ 2 dEi0 UðIVÞ ¼ dTðUðIÞÞi lPðOÞl2 UðVIÞ 2 lPðOÞl2 UðIVÞ =lPðOÞl2 UðIÞ where dEi0 is the nuclear field shift energy for the ith pair of isotopomers, dTðUðIÞÞi is the shift in the atomic spectrum for that pair of isotopes, and lP(O)l2 is the electron density at the nucleus. The logarithm of the separation factor for ith isotope from 238U is 2

‘n ai ¼ aðhc=kTÞfsi þ b ð1=24Þð"=kTÞ dmi =238mi

ð1:32Þ

˚ line in the atomic spectrum of UI for the ith isotopomer with where fsi is the field shift of the 5028 A 238 respect to U, a is the field shift scaling factor, and b is aii (U(IV)) 2 aii (U(VI)) of Equation 1.27. The parameters a and b were evaluated from the absolute value of ‘n a (235 –238) at 300 K and the ratio of ‘nað234 – 238Þ=‘nað235 – 238Þ at 433 K.55,56 The result of the analysis of the contributions of the field shift and the vibrational reduced partition function ratio to the overall separation factors for the isotopes 233U, 234U, 235U and 236U from 238U is given in Table 1.1. Comparisons are made with the experimental values.55,56 The absolute values of the experimental values are derived from

TABLE 1.1 Contributions of Vibrational Effects (‘n a0 ) and Field Shifts (‘n Kfs ) to the U(IV) – U(VI) Isotope Separation Factors at 433 K Isotope Pair 236–238 235–238 234–238 233–238 232–238 1 ¼ 104 ‘n a:

104‘n a0

104‘n Kfs

104‘n a; Calculated54

104‘n a; Experimental55,56

1i =1234

238 2 mi/4

22.54 23.82 25.12 26.48 27.65

8.76 14.62 17.46 22.83

6.22 10.80 12.34 16.40

6.2 10.8 (12.3) 16.4 18.3

0.50 0.88 1 1.33 1.49

0.50 0.75 1 1.25 1.5

12

Isotope Effects in Chemistry and Biology

the measured experimental values and the absolute value of the 235 –238 separation factor at 433 K calculated from the measured value at 300 K. The calculated separation factors are in quantitative agreement with the experimental ones. The anomalous behavior of the 233U and 235U isotopomers is due to the fact that these nuclei have quadrupole moments. The nuclear field shift is the largest contribution to the separation factor for all the isotopes. The nuclear field shift resolves the 50-year-old puzzle as to why the heavy isotope enriches in the U(IV) species. In accordance with molecular structure predictions the vibrational effect does lead to a preference of the heavy isotope, 238U, for the U(VI) species. Since the nuclear field shift, which leads to a preference of this isotope for the U(IV) species, is much larger than the vibrational effect, the net result is a preference of the heavy isotope for the U(IV) species. It is interesting that the role of the nuclear field shift in the fractionation of the uranium isotopes was overlooked in 1943 when Maria Mayer and I worked at the SAM Laboratory under the direction of Harold C. Urey. We were principally engaged in the search for isotopic shifts in the spectra of 235U and 238U. In addition to the fact that there was no information about the field shift in any compound of uranium at the time, we never made the connection between that work and our effort to estimate uranium isotope separation factors by chemical exchange reactions. If we had, we probably would not have proceeded with the development of the statistical mechanical theory of isotope chemistry. I often reflect on what direction my career would have taken. I take satisfaction from my recognition of the importance of this Born –Oppenheimer correction to the isotope chemistry of the heavy elements. In 1995 I was able to formulate a correct answer to the problem Harold Urey presented me with in 1943, even if for only one chemical reaction.

IV. ISOTOPE CHEMISTRY AND MOLECULAR STRUCTURE Section III provides the information for the calculation of isotope fractionation factors from the vibrational frequencies of the isotopomers involved in an equilibrium chemical reaction. It is instructive to see what forces within the molecules are responsible for the fractionation. From Equation 1.27 we will derive some general rules, the first order rules of isotope chemistry. Developments in the theory and analysis of the mathematical structure of the reduced partition function ratio have led to powerful methods to dissect the reduced partition function ratio and thus the fractionation factor into the respective contributions from each of the forces in each of the molecules participating in an exchange reaction.17,57 – 61 Bigeleisen and Ishida61 showed that, when isotopic substitution is at an end atom in the molecule, only the uncoupled stretching and bending coordinates contribute significantly to the reduced partition function ratio. For such molecules the bend – bend interactions have small F matrix elements and therefore make small contributions to RFPR. Somewhat larger corrections come from the interaction of the stretching and bending coordinate due to their larger F matrix elements. With an approximate correction to Equation 1.27, the first order quantum correction, they were able to obtain reasonable agreement with exact values of ‘nðs=s 0 Þf for hydrogen isotope substitution in a series of hydrocarbons. After the derivation of the first order rules of isotope chemistry we present two independent general methods for the dissection of RFPR.

A. THE F IRST O RDER R ULES OF I SOTOPE C HEMISTRY We rewrite Equation 1.27 in its logarithmic form 0

2

‘nðs=s Þf ¼ ð1=24Þð"=kTÞ

X i

ð1=m0i 2 1=mi Þaii

ð1:33Þ

Although Equation 1.27 was derived for small u and small du; in Section IV.C we will derive Equation 1.33 for large values of u and du: The first order rules follow from Equation 1.33.57,60

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

13

1. Isotope effects, ‘nðs=s 0 Þf ; depend only on the masses of the isotopic atoms and the force constants bonding the atom at the site of isotopic substitution with other atoms in the molecule, aii : 2. Isotope effects between different compounds occur only when there are force constant changes at the site of isotopic substitution. 3. Isotope effects are additive. a. Isotopic additivity 16

18

0

16

0

16

0

18

16

‘nðs=s Þf ðD2 O=H2 OÞ ¼ ‘nðs=s Þf ðD2 O=H2 OÞ þ ‘nðs=s Þf ðH2 O=H2 OÞ

b. Substituent additivity 0

‘nðs=s Þf ðCHDFCl=CH2 FClÞ

¼ ‘nðs=s0 Þf ðCH2 DCl=CH3 ClÞþ ‘nðs=s0 Þf ðCH2 DF=CH3 FÞ2 ‘nðs=s0 Þf ðCH3 D=CH4 Þ c. Isotope effects are cumulative (first rule of the geometric mean) 0

0

‘nðs=s Þf ðD2 O=H2 OÞ ¼ 2‘nðs=s Þf ðHDO=H2 OÞ

d. Equivalent isomers have the same isotope chemistry 0

0

0

‘nðs=s Þf ðC6 H4 D2ðoÞ Þ ¼ ‘nðs=s Þf ðC6 H4 D2ðmÞ Þ ¼ ‘nðs=s Þf ðC6 H4 D2ðpÞ Þ

4. The heavy isotopomer concentrates in the chemical species with the strongest bonding (largest vibrational force constants). Rule (2) was also formulated through numerical computation for kinetic isotope effects.62,63

B. STATISTICAL M ECHANICAL P ERTURBATION T HEORY Singh and Wolfsberg59 developed a perturbation method for the dissection of the isotopic reduced partition function ratio from the diagonal elements of the F and G matrix elements plus the sum of corrections from the off diagonal elements. The Hamiltonian of the molecule is written as a sum of diagonal and off diagonal elements. H ¼ H0 þ H1 X 2 ðgii p2þ 2H0 ¼ i fii qi Þ

ð1:34Þ ð1:35Þ

i

H1 ¼

X i; i,j

ðgij pi pj þ qi qj Þ

ð1:36Þ

where the qi s are the internal coordinates and the pi s are the conjugate momenta. With this division of the total Hamiltonian and Schwinger perturbation theory Singh and Wolfsberg obtain 0

0

‘nðs=s Þf ¼ ‘nðs=s Þf0 þ CORR

where 0

‘nðs=s Þf0 ¼

X i

‘n cðui0 Þ

ð1:37Þ

ð1:38Þ

14

Isotope Effects in Chemistry and Biology 0

0

‘n cðui0 Þ ¼ ‘nðui0 =ui0 Þ þ dui0 =2 þ ‘n 1 2 expð2ui0 Þ =½1 2 expð2ui0 Þ

ð1:39Þ

and ui0 ¼ ð"=kTÞðgii fii Þ1=2

ð1:40Þ

‘n cðui0 Þ is just the reduced partition function ratio of an uncoupled internal coordinate. CORR is a

function of both diagonal and off-diagonal F and G matrix elements.59 A comparison will be given of the results for RFPR from calculations through Equation 1.37 along with calculations using finite polynomials with exact calculations.64

C. POLYNOMIAL E XPANSIONS OF THE R EDUCED PARTITION F UNCTION R ATIO In this section I present an alternate to the Singh –Wolfsberg correction to the diagonal element approximation through the use of finite polynomials. We expand ‘n cðui Þ; the exact reduced partition function ratio of an oscillator, in an infinite series of u 2j and obtain the convergent, infinite series58 XX 0 ‘nðs=s Þf ¼ Aj dðu2j ½u0i , 2p ð1:41Þ i Þ j

i

2j 02j dðu2j i Þ ¼ u i 2 ui

Aj ¼ ð21Þ2jþ1 ½B2j21 =2jð2j!Þ ð"=kTÞ2j

ð1:42Þ

B2j21 are the Bernoulli numbers ðB1 ¼ 1=6; B3 ¼ 1=30; B5 ¼ 1=42 etc.). The numerical coefficients of the first three terms are 1/24, 2 1/2880, and 1/181440. Through Equation 1.41 we have extended the validity of Equation 1.33 to values of u0i , 2p: This corresponds to a maximum frequency of 1300 cm21 at 300 K. We can now remove this restriction through a method suggested . by T. Ishida.17 Instead of expanding ‘nðs=s0 Þf in an infinite series, we choose a finite polynomial for the expansion. This expansion replaces Equation 1.41 with XX 0 ‘nðs=s Þf ¼ Wj Aj dðu2j ½u0i , 1 ð1:43Þ i Þ j

i

The modulating coefficients, Wj ; are isotope independent and have values less than one. Equation 1.43 is similar in structure to Equation 1.41. Again it leads to the first order rules of isotope chemistry but without the restriction that u , 2p: Through Equation 1.43 we have now extended the first order rules of isotope chemistry to all molecules at all temperatures. The best values of the modulating coefficients are the WINIMAX coefficients.65 These are MINIMAX coefficients66 weighted appropriately for use in isotope chemistry. The WINIMAX coefficients supercede the prior FOP coefficients.17,67 Equation 1.43 can be used to derive the correction terms which appear in the Wolfsberg method and form the basis of the WIMPER method.64,68 o Xn 0 0 ‘nðs=s Þf ¼ ‘nðs=s Þf0 þ Wj Aj ð"=kTÞ2j d TrðH j Þ 2 TrðH j Þ0 ð1:44Þ j¼1

The terms d½TrðH j Þ 2 TrðH j Þ0 are the sums of the differences of powers of the eigenvalues of the normal coordinates and the internal coordinates, respectively. In Table 1.2 and Table 1.3 we give examples of the approximation of the reduced partition function ratios of a number of molecules by both the Singh –Wolfsberg and the WIMPER (2) method. The Singh – Wolfsberg method converges more rapidly than does WIMPER.

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

15

TABLE 1.2 Approximation of Reduced Partition Function Ratios by Perturbation Methods (D/H) at 300 K Isotopomer Pair HDO/H2O CH3D/CH4 HDCO/H2CO C2H3D/C2H4 C2H5D/C2H6 C6H5D/C6H6 out of plane

Method S-W W(2) S-W W(2) W(2) S-W W(2) S-W W(2) W(2)

ln(s/s 0 )f0

ln(s/s 0 )f, 2nd Order

ln(s/s 0 )f, Exact

2.5990 2.5812 2.4957 2.4789 2.2426 2.3066 2.3099 2.4406 2.3935 0.2349

2.6012 2.5821 2.4899 2.4825 2.2261 2.3399 2.2882 2.4123 2.3830 0.2514

2.6012 2.5829 2.4723 2.4926 2.1997 2.3399 2.3204 2.4177 2.3645 0.2656

S-W;59 W(2).64,68 The differences between the S-W and W(2) values for ‘nðs=s0 Þf0 and ‘nðs=s0 Þf ; respectively, are due to the different F matrices used by these authors.

TABLE 1.3 Approximation of Reduced Partition Function Ratios by Perturbation Methods (13C/12C and 18O/16O) at 300 K Isotopomer Pair 16 H18 2 O/H2 O 13

CH4/12CH4

12 H13 2 CO/H2 CO 18 H2C O/H2C16O 13 CCH4/12C2H4 13

CCH6/12C2H6

13

CC5H6/12C6H6 out of plane

Method S-W W(2) S-W W(2) W(2) W(2) S-W W(2) S-W W(2) W(2)

ln(s/s 0 )f0

ln(s/s 0 )f, 2nd Order

ln(s/s 0 )f, Exact

0.0677 0.0673 0.1313 0.1304 0.1580 0.0887 0.1379 0.1440 0.1447 0.1422 0.0339

0.0649 0.0643 0.1134 0.1134 0.1398 0.0893 0.1266 0.1251 0.1270 0.1314 0.0177

0.0648 0.0643 0.1126 0.1113 0.1369 0.0904 0.1299 0.1253 0.1280 0.1253 0.0183

See notes to Table 1.1.

The differences in the values of ‘nðs=s0 Þf0 between the two methods is the result of the usage of different F matrices by these authors. Corresponding differences also exist for the exact values of ‘nðs=s0 Þf : The zero order approximation, ‘nðs=s0 Þf0 ; to the exact values is within 1– 2% for deuterium substitution except for ring structures. On the other hand, heavy atom ‘nðs=s0 Þf0 values differ from the exact values by about 12%, again except for ring structures. The excellence of the diagonal element approximation in the case of deuterium –hydrogen substitution is a consequence of the fact that hydrogen is generally an end atom. The largest corrections to the diagonal element approximation come from the stretch –bend interaction. For this interaction the dgij matrix element is zero. Therefore there is no contribution to the correction at the j ¼ 1 level in the WIMPER method. The difference in this interaction for end and central atom substitution can be seen by

16

Isotope Effects in Chemistry and Biology

comparison of the 13C and 18O effects in formaldehyde, CH2CO, in Table 1.2. The 13C isotopomer has an 11% correction to ‘nðs=s0 Þf0 ; whereas the 18O correction is only 2%. The difference is primarily due to the fact that the 13C isotopomer involves a central atom substitution, whereas the 18 O isotopomer involves an end atom. An additional reason for the better approximation to ‘nðs=s0 Þf at the zero order for hydrogen – deuterium comes from the fact that all atoms have essentially infinite weight compared with that of the hydrogen atom, and to a lesser extent with the deuterium atom. This leads to small coupling of the motion of the hydrogen atom with the motions of any of the other atoms in the molecule.69 In Table 1.4 and Table 1.5 we tabulate the corrections at the j ¼ 1 and j ¼ 2 levels for each type of interaction for deuterium and heavy atom substitution in water and the out of plane vibrations of benzene. In Table 1.6 we show that the ratio ‘nðs=s0 Þf0 =‘nðs=s0 Þfexact deviates from unity by more than 2% only when there is a contribution to ‘nðs=s0 Þfexact at the level j ¼ 1: The j ¼ 1 values are identically zero for end atom substitution. The j ¼ 1 corrections for ring atom substitution are significant, cf. 13C substitution in benzene, Table 1.5. In that case the zero order approximation, ‘nðs=s0 Þf0 ; is 1.85 times the exact value. Yet the final value after correction at the j ¼ 2 level is within 3% of the exact value! This shows the power of the WIMPER (2) method to analyze all the interactions of the internal coordinates in the calculation of the reduced partition function ratio. Skaron and Wolfsberg72 have shown, without solutions to the vibrational problem, from Equation 1.37 how ‘nðs=s0 Þf changes with a change in force constant. A new value of ‘nðs=s0 Þf is calculated directly from the changed force constant and compared with the original value of ‘nðs=s0 Þf : In principle, their method can also be used to derive the absolute contribution of each force constant, F matrix element, to the reduced partition function ratio. Additional information concerning the role of structure to isotope chemistry can be derived from consideration of the j ¼ 2 corrections to ‘nðs=s0 Þf0 : In Appendix 1A we list some of the properties of the eigenvalues which are relevant to the j ¼ 2 corrections. The properties of isotopomers at the j ¼ 1 have been discussed under the first order rules of isotope chemistry. Deviations from the rule of the mean occur at the level of j ¼ 2 and they are primarily due to

TABLE 1.4 Dissection of the Reduced Partition Function Ratios of Water at 300 K Deuterium Substitution (HDO/H2O) Coordinate

Force Constant

ln c(ui0)

OH stretch HOH bend OH £ HOH OH £ OH Total ‘nðs=s0 Þf exact

8.454 0.762 0.237 20.100

2.2200 0.3611 0 0 2.5811

j51 0 0 0 0 0

j52

ln c(ui)

0.0025 0.0005 20.0016 20.0006 0.0009

2.2225 0.3616 20.0016 20.0006 2.5819 2.5829

16 Oxygen Substitution (H18 2 O/H2 O)

OH stretch HOH bend OH £ HOH OH £ OH Total ‘nðs=s0 Þf exact

8.454 0.762 0.237 20.100

ln c(ui0)

j51

j52

ln c(ui)

0.0550 0.0123 0 0 0.0673

0 0 20.0055 0.0003 20.0052

0.0008 0 0.0015 20.0002 0.0021

0.0558 0.0123 20.0040 0.0001 0.0642 0.0643

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

17

TABLE 1.5 Dissection of the Reduced Partition Function Ratios of the Out of Plane Vibrations of Benzene at 300 K Deuterium Substitution (C6H5D/C6H6) Coordinate

Force Constant

ln c(ui0)

C2C –H op wag CC–CC torsion WAG £ TORS 3C’s WAG £ TORS 2C’s WAG £ WAG 2C’s TORS £ TORS 3C’s TORS £ TORS 2C’s TORS £ TORS 2 &C’s Total ‘nðs=s0 Þf exact

0.378 0.086 ^0.088 ^0.043 20.057 20.060 0.008 0.019

0.2346 0 0 0 0 0 0 0 0.2349

j51 0 0 0 0 0 0 0 0 0

j52

ln c(ui)

0.0132 0 0.0054 0.0010 20.0031 0 0 0 0.0165

0.2478 0 0.0054 0.0010 20.0031 0 0 0 0.2514 0.2656

13

C Substitution (13C C5H6/C6H6)

C2C –H op wag CC–CC torsion WAG £ TORS 3C’s WAG £ TORS 2C’s WAG £ WAG 2C’s TORS £ TORS 3C’s TORS £ TORS 2C’s TORS £ TORS 2 &C’s Total ‘nðs=s0 Þf exact

0.378 0.086 ^0.088 ^0.043 20.057 20.060 0.008 0.019

ln c(ui0)

j51

j52

ln c(ui)

0.0226 0.0113 0 0 0 0 0 0 0.0339

0 0 20.0299 20.0081 0.0045 0.0127 0.0008 20.0005 20.0203

0.0020 20.0008 0.0045 0.0011 20.0007 20.0015 20.0001 0.0001 0.0041

0.0246 0.0105 20.0256 20.0070 0.0038 0.0112 0.0007 20.0004 0.0177 0.0183

See Ref. 68.

TABLE 1.6 Total Correction to the Zero Order Approximation of the Reduced Partition function Ratio Isotopomers

ln(s/s 0 )f0/ln(s/s 0 )f Exact

HDO/H2O 16 H18 2 O/H2 O HDCO/H2CO 12 H13 2 CO/H2 CO H2C18O/H2C16O C6H5D/C6H6 planar 13 CC5H6/C6H6 planar C6H5D/C6H6 out of plane 13 CC5H6/C6H6 out of plane

0.999 1.047 1.020 1.112 0.981 0.980 1.295 0.884 1.852

See Ref. 68.

Correction at j 5 1 None Yes None Yes None None Yes None Yes

18

Isotope Effects in Chemistry and Biology

70 the G0 F 0 terms relating to the bending vibrations. be seen from P 2 PThis0 2can readily P the2 trace 2 2 2 0 þ 2ð ¼ ð Þa 2 In Cartesian coordinates d l m m m m of the H matrix. 2 Þ i i j mj aji : The i i i ii i P identically for isotopic disproportionation reactions. first term, P i ðm0i 2 2 m2i Þa2ii ; cancels P P P P The sums, i ðm0i 2 2 m2i Þa2ii and i ðm0i 2 mi Þ j mj a2ji . The term i ðm0i 2 mi Þ j mj a2ji is larger for the isotopomer pair D2O/HDO than for HDO/H2O. Since the sign of A2, in Equation 1.41 and Equation 1.43, is negative ‘n Kqm # ‘n KCl in the harmonic oscillator approximation. Wolfsberg and coworkers38,47 – 50,71 have shown that anharmonic corrections and corrections to the Born –Oppenheimer approximation do not affect this conclusion. In Appendix 1A, I also give a summary of the structure of the j ¼ 2 corrections. For complete details of the individual matrix elements and their application to a variety of isotopomeric molecules consult the original literature.68 Section III and Section IV of this chapter summarize some of the significant advances in the theory of equilibrium isotope effects since the Bigeleisen –Mayer publication. These include the corrections to the harmonic oscillator and Born – Oppenheimer approximations by Max Wolfsberg and coworkers.47 – 49 A second correction to the Born – Oppenheimer approximation is the shift in the electronic energy levels due to the nuclear size and shape.54 Major advances have resulted from the ability to calculate frequencies of large isotopomers through electronic computers. In addition, we now have the capability of calculating these force fields by quantum mechanics. For both of these developments we are indebted to Max Wolfsberg. Our detailed understanding of the role of molecular structure has been advanced by the first and second order rules of isotope chemistry.60,70 The role of individual forces within a molecule to its isotope chemistry has been elucidated through the perturbation theories of Max Wolfsberg59,72 and those developed by Takanobu Ishida and Myung W. Lee64,68 with whom I experienced a joyful and fruitful collaboration.

V. KINETIC ISOTOPE EFFECTS The availability of gram quantities of heavy water in the early 1930s made possible the study of the effects of isotopic substitution on the rates of chemical reactions. Initial studies were directed at testing the recent developments in the theory of reaction rates. In 1931, Eyring and Polanyi73 introduced an empirical method for the quantum mechanical calculation of the energy surface of a three atom system. Shortly thereafter modifications to the classical transition state theory74 to include quantum statistics were derived.75 – 77 In the transition state theory one assumes a rapid reversible equilibrium between the reactants and an energy rich complex of the reacting species — the transition state. The rate of the reaction is simply the rate at which the transition state complex decomposes into products. The rate is Rate ¼ c‡n‡ ¼ kcA cB

ð1:45Þ

k ¼ c‡n‡=cA cB ¼ ðkT=hÞK‡

ð1:46Þ

The rate constant is

where c‡ is the concentration of activated complexes, cA cB is the product of the concentrations of reactants in the rate determining step, n‡ is the frequency with which activated complexes decompose into products, k is the rate constant and K‡ is the equilibrium constant between reactant and transition state molecules. Equation 1.46 is conventionally corrected for a transmission coefficient, k; and for the fact that some reaction occurs by tunneling through, rather than motion over, the energy barrier to the reaction.

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

19

The equilibrium constant, K‡; can be written as the quotient of the partition function of the transition state, Q‡; and the product of the partition functions of the reactants. K‡ ¼ Q‡=QA QB …

ð1:47Þ

K‡ can be calculated, in principle, from Equation 1.47. The partition function of the transition state, Q‡; differs in one important way from the conventional partition function of a stable molecule. Of the 3n coordinates in the transition state, 3 are allocated to the translation of the center of mass, 3 (2 in the case of linear transition states) are allocated to the rotation of the transition state, one belongs to the reaction coordinate, n‡; which is an imaginary frequency. This leaves 3n 2 7 ð3n 2 6 in the case of linear transition states), to the internal vibrations. Early applications of Equation 1.46 were made to the absolute and isotopic reactions of the type H þ H2. For these reactions Farkas and Wigner78 gave the explicit formula for the rate constant, k k ¼ ð1=2ÞkðM‡=Ma Mm Þ3=2 ðI‡=Im Þ"2 ð2p=kTÞ1=2 expð2Q=RTÞðNL =1000Þ sinh bnm =½ðsinh bnd Þ2 sinh bns ð1 þ ðbn‡Þ2 =6Þ

ð1:48Þ

In Equation 1.48 k is the transmission coefficient, M‡ is the molecular weight of the H3 complex, Ma is the atomic weight of the hydrogen atom, Mm is the molecular weight of the hydrogen molecule, I‡=Im is the ratio of the moments of inertia of the H3 complex and the hydrogen molecule, Q is the activation energy measured from the bottom of the potential energy surface (PES) of the H2 molecule to the minimum in the PES of the activated complex, NL is Loschmidt’s number, and the hyperbolic functions ðsinhbxÞ21 ¼ 2ðexpð2u=2ÞÞ=ð1 2 expð2uÞÞ; bx ¼ u=2kT: nm ; nd ; and ns are the stretching frequency of H2, the doubly degenerate bending frequency of H3, and its stretching frequency, respectively. They recognized that there were terms both in Equation 1.47 and the ratio of rate constants, k1 =k2 ; that cancelled. Neither Farkas and Wigner nor subsequently Hirschfelder, Eyring, and Topley79 were able to obtain agreement between calculations based on Equation 1.48 and the Eyring– Polanyi PES. Whether this was a problem with the PES or the quantum statistical transition state theory or both was a problem which awaited future solution. Heavy atom kinetic isotope effect studies became possible after WWII with the ready availability of radioactive isotopes and isotope ratio mass spectrometers for the measurement of stable isotope ratios. An early study was the 14C effect in the decarboxylation of malonic acid by Yankwich and Calvin.80 14C singly labeled in the carboxyl group of malonic acid, CH3CH14 2 COOH, can decompose in two ways: to give either 12CO2 or 14CO2. For the ratio of the rate constants to give 12CO2 compared with 14CO2, k4 =k3 ; they reported 1.12 ^ 0.03 at 1508C. For the bromine derivative, with substitution at the methylene position, they reported a ratio of 1.41 ^ 0.08 at 1158C These isotope effects appeared large to me in comparison with what was known about equilibrium isotope effects for carbon, typically 1.03 for 12C vs. 13C. This would translate to 1.06 for 12C vs. 14C. I decided to study the transition state theory of reaction rates. It became immediately obvious that the method used by Pelzer and Wigner and by Eyring could be greatly simplified by the introduction of the concept of the reduced partition function ratio. Starting with Equation 1.46 and Equation 1.47 one finds30 for the ratio of rate constants of light to heavy isotopomers, kL =kH ; kL =kH ¼ ðnL =nH Þ‡ð f =f ‡Þ

ð1:49Þ

apart from corrections due to the transmission coefficient and tunneling. The partition function ratio, f ; has been defined in Equation 1.16. The partition function ratio of the transition state, f ‡; is s 0 =s times the conventionally defined reduced partition function ratio, ðQ=Q0 Þ‡qm =ðQ=Q0 Þ‡Cl : Since the translation, rotation and reaction coordinates of the transition state are treated classically, f ‡ is the

20

Isotope Effects in Chemistry and Biology

product of ð3n 2 7Þ reduced vibrational partition function ratios, Equation 1.21. Each vibrational reduced partition function ratio consists of three factors: the classical correction, ðui =u0i Þ‡=2, the zero point energy term, dui ‡=2; and the Boltzmann excitation term, {½1 2 expð2u0i Þ =½1 2 expð2ui Þ ‡: The product of the ratio of the frequencies of the crossing the barrier, ðnH =nL Þ‡; when combined with the product ðui =u0i Þ‡ of the ð3n 2 7Þ real vibrations is abbreviated MMI. An important feature of kinetic isotope effect and its applications is the fact that the absolute height of the potential barrier of the transition state from the reactants does not enter into the kinetic isotope effect. The absolute value of the height of the barrier is what causes the major problem in the accurate theoretical calculation of absolute rate constants. Equation 1.49 brings to the study of kinetic isotope effects all that we have learned about equilibrium isotope effects. There are the first order rules of isotope chemistry spelled out in Section IV; there is the ability to correlate kinetic isotope effects with molecular structure. It greatly simplifies our understanding of the origin of kinetic isotope effects and the design and analysis of experiments. A complete exposition of the theory and the design of kinetic isotope effect experiments is given in the chapter in Advances in Chemical Physics by Bigeleisen and Wolfsberg30. Equation 1.49 is properly referred to as the Bigeleisen – Wolfsberg equation. More recent expositions of the theory of kinetic isotope effects include the monographs by Melander and Saunders32 and Willi.81 The latter texts include summaries of the major applications of the kinetic isotope effect to the study of reaction mechanisms. The chapter by Max Wolfsberg in this volume gives details about the calculation of kinetic isotope effects from Equation 1.49 by high speed computers. An early application of Equation 1.49 was to calculate the maximum kinetic isotope effects for all elements in the Periodic Table.82 Apart from the statistical factors ðsL =sH Þ and ðsL =sH Þ‡ which appear in the ratio ðf =f ‡Þ; each of these quantities in f and f ‡ is positive and equal to or greater than unity. The maximum kinetic isotope effect in any reaction occurs when f ‡ is equal to unity. This corresponds to a transition state of an assembly of free atoms, with no bonding whatsoever. Inasmuch as there are extensive tables of the reduced partition function ratio, tabulated as f (cf. Ref. 18), and these values are not significantly different for different compounds of a given element, it was possible to set upper limits for kinetic isotope effects for all elements. This amounted to a factor of 1.5 for 12C vs. 14C at 300 K. This cast doubt on the validity of the Yankwich – Calvin experiments on the decarboxylation of malonic acid. In the relative modes for the decarboxylation of labeled malonic acid, f is unity, since both paths involve the same substrate. I carried out a simplistic calculation, using the Slater coordinate to calculate ðnL =nH Þ‡ and neglecting the factor f ‡ for the isomeric transition states, to obtain a theoretical ratio for k4 =k3 of 1.038 for 12C vs. 14C.83 The experiments on the 13C and 14C kinetic isotope effects in the decarboxylation of malonic acid were repeated in several laboratories.84 – 89 A summary of some of the pertinent results along with the initial findings of Yankwich and Calvin is given in Table 1.7. Both Lindsay, Bourns, and Thode85 and Yankwich and Promislow87 found commercial samples of malonic acid were not homogeneous in their isotopic distribution of 13C between the methylene and carboxyl groups. They made appropriate correction for this inhomogeneity. The inhomogeneity explains the discrepancy between the Bigeleisen – Friedman experiment84 and the later results.85,87 Table 1.7 also includes a comparison of the experimental results with theoretical calculations,30,83,90 which include revisions to the use of the Slater coordinate, which was subsequently shown to be physically unrealistic.30,91 Although my back-of-the-envelope calculation of k4 =k3 83 is incorrect as a result of using the Slater coordinate and neglecting of the contribution from the partition function ratio of the vibrational terms in the activated complex, fortuitously it turned out to be of the correct order of magnitude. The results in Table 1.7 and subsequent experiments on kinetic isotope effects involving the isotopomers of carbon and nitrogen showed the power of the reduced partition function ratio to predict even small kinetic isotope effects. The later experiments on malonic acid also confirm the theoretical prediction that the 14C isotope effect is 1.9 times that of the 13C effect. I now return to the question of the failure of the transition state theory to calculate the rate constant for a simple reaction of an atom with a diatomic molecule, H þ H2 and its isotopomers.

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

21

TABLE 1.7 Intramolecular Isotope Effect in the Decarboxylation of Malonic Acids Label

T (8C)

14

C 14 C Br 13 C 14 C 14 C 13 C 13 C 13 C 13 C 14 C 14 C 14 C Br

150 115 138 148 138 138 138 140 140 140 140 118

13

138 140 133

C 13 C 13 C 14 13 C/ C (experimental)

100 1

H

Authors

Experiments 14 ^ 3 No 41 ^ 8 No 2.0 ^ 0.1 No 6^2 No 8.7 2 10.5 No 2.0 No 2.6 ^ 0.4 Yes 2.99 ^ 0.05 Yes 2.73 ^ 0.05 Yes 5.47 ^ 0.36 No 5.77 ^ 0.20 No 6.46 ^ 0.24 No

Yankwich & Calvin (1949) Yankwich & Calvin (1949) Bigeleisen & Friedman (1949) Roe & Hellman (1951) Yankwich, Stivers & Nystrom (1951) Lindsay, Bourns & Thode (1952) Lindsay, Bourns & Thode (1952) Yankwich & Promislow (1954) Yankwich & Nystrom (1954) Yankwich & Nystrom (1954) Grigg (1956) Grigg (1956)

Theory 2.0 2.5 1.84 2.0 ^ 0.1

Bigeleisen (1949) Bigeleisen & Wolfsberg (1958) Stern & Wolfsberg (1963) Yankwich, Promislow & Nystrom (1954)

14

C/13C ¼ 1.9 C/13C ¼ 1.9 14 13 C/ C ¼ 1.9 14

1 ¼ ðk4 =k3 Þ 2 1; H ¼ correction for homogeneity.

In 1951, W. M. Jones added to the kinetic data on the reaction of chlorine atoms with the isotopomers H2 and D2 data for the reaction with HT as a function of temperature.92 Plots of the relative rate constants for each of these isotopic reactions as a function of 1=T had positive slopes. Contrary to the experimental findings, calculations for this reaction,92,93 similar to the ones for the H þ H2 reaction, gave a negative temperature coefficient for both the isotopomers HD and HT. Jones entertained a suggestion by Magee94 that the transition state Cl – H –H was bent rather than linear. Wolfsberg proposed to examine whether this could account for the discrepancy between theory and experiment. He approached me about the possibility of collaboration on this study. I welcomed the opportunity to collaborate with Max, an expert in quantum chemistry. Bigeleisen and Wolfsberg95 showed that a simple change in the geometry per se could not remove the discrepancy between the theoretical and experimental signs of the activation energy difference in the kinetic isotope effect in the H2/HT þ Cl reaction. A bent transition state differs from a linear one in the moments of inertia and the number of bending vibrations. The rotation of the transition state complex is classical and, therefore, does not contribute to the kinetic isotope effect. Bending vibrations have low frequencies and as such make a minor contribution to the reduced partition function ratio. Given the known properties of the isotopomers of H2, only the magnitude of the real stretching frequency of the H2Cl transition state can affect the activation energy of the kinetic isotope effect of this reaction. On this basis Bigeleisen and Wolfsberg were able to construct linear and bent transition states, which were in reasonable agreement with all the known kinetic isotope effects. The properties of their linear transition state is given in Table 1.8. Additional experimental data were necessary to validate the result. The crucial additional experimental data were the kinetic isotope effect of H2/HD as a function of temperature. These experiments were carried out by Fritz S. Klein, a guest at Brookhaven National Laboratory (BNL) from the Weizmann Institute for Science, in collaboration with Jacob Bigeleisen. Their experiments, as well as those of Jones on H2/HT, did not measure the branching

22

Isotope Effects in Chemistry and Biology

TABLE 1.8 ˚ 21 ) and Vibrational Frequencies (in cm21) of the H2Cl Force Constants (in mdyne A Transition State

Year f (H–Cl) f (H–H) f12 ns nb nL

WTE

BW

1936 LEP 3.69 20.30 0 2489 439 720i

1955 EMP 1.86 20.48 0 1460 (200) 720i

BKWW 1959 LEPS 1.56 0.57 1.50 1348 727 1497i

SPK

ALTG

BW2

1973 LEPS 1.77 0.45 1.46 1357 707 1483i

1996 EMP

2000 ab initio

1358 581 1520i

1360 540 1294i

WTE, Wheeler, A., Topley, B., and Eyring, H., J. Chem. Phys., 4, 178, 1936; BW, Bigeleisen, J. and Wolfsberg, M., J. Chem. Phys., 23, 1535, 1955; BKWW, Bigeleisen, J., Klein, F. S., Weston, R. E., Jr., and Wolfsberg, M., J. Chem. Phys., 30, 1340, 1959; SPK, Stern, M. J., Persky, A., and Klein, F. S., J. Chem. Phys., 58, 5697, 1973; ALTG, Allison, T. C., Lynch, G. C., Truhlar, D. G., and Gordon, M. S., J. Phys. Chem., 100, 13575, 1996; BW2, Bian, W. and Werner, H-J., J. Chem. Phys., 112, 220, 2000.

ratio between HX and XH reacting withPCl to give H and X atoms, respectively. They measured the total kinetic isotope effect, kðH2 Þ= kðHXÞ þ kðXHÞ: At this time Ralph Weston at BNL was exploring the Sato96 modification of the Eyring –Polanyi method of calculating the PES of a three atom systems. He mapped out the PES for the H2Cl transition state as a function of the Sato parameter. The PES obtained by the four man collaboration,97 which represented the best fit to all the experimental data, is give in Table 1.8. Additional experiments were carried out by Bar-Yaacov, Persky, and Klein, Persky and Klein, and by Persky. Stern, Persky, and Klein98 developed additional PESs, both empirical and of the Sato type. A comparison of their results for both an empirical model and a Sato type potential is given in Figure 1.4. The potential parameters for their Sato like potential (LEPS) is given in Table 1.8. Their Sato like PS does not differ significantly from the BKWW surface. Their calculated kinetic isotope effects include tunneling within both the Wigner and Eckart approximations. The average root mean square deviation between either the BKWW or SPK calculated kinetic isotope effects and experiment is less than 15% over the temperature range 245– 445 K It is characteristic of all the PESs subsequent to the WTE surface that they have much smaller H – Cl binding force constants and a small attractive rather than repulsive H – H force constant compared with the WTE surface. Further, the later force fields involve significant potential energy coupling between the H – H and H –Cl stretching coordinates. The results of BKWW and SPK along with the experimental data constituted the best quantitative validation of the transition state theory at the time. Subsequent to Weston’s 1979 review99 of the H2 þ Cl reaction, important new theoretical calculations of the PES of this reaction100,101 have been published along with theoretical values of the absolute rate and the kinetic isotope effect102,103 and experimental data.104 The experimental data for some of the isotopic reactions cover the range from 250 to 3000 K. The theoretical calculations102,103 use quantum dynamics rather than transition state theory. Earlier calculations using quantum dynamics102 with the G3 surface100 differ insignificantly from variational transition state theory with multidimensional tunneling. The theoretical BW surface, designated as BW2,101 given in Table 1.8, is a modified ab initio quantum calculation. It includes an empirical correction to the electron correlation energy to bring the computation of BW1, the ab initio calculation, into quantitative agreement with the dissociation energies of H2 and HCl. A comparison

Theoretical Basis of Isotope Effects from an Autobiographical Perspective 50

23

HHCL TTCL

40 30

HHCL OTCL+TDCL

k1 / k 2

20

HHCL ODCL

10

HHCL HTOL+THCL

5

HHCL HDDL+DHCL

2

3.1

3.3

3.5 3.7 103 /(T°K)

3.9

1.8

4.1

DHCL HDCL

1.5 1.2

2.2

2.4

2.6

2.8

3.0

3.2

3.4

103 /(T°K)

FIGURE 1.4 Comparison of calculated and experimental isotope effects in the H2 þ Cl reaction. Solid lines, empirical PES with Eckart tunneling; dashed lines, empirical Sato PES with Wigner tunneling; points, experimental. (Reproduced from Stern, M. J., Persky, A., and Klein, F. S., J. Chem. Phys., 58, 5697– 5706, 1973. With permission.)

of the theoretical calculations of the absolute rate with the experimental values of the H2 þ Cl reaction is given in Figure 1.5. Also included in Figure 1.5 are calculations based on the G3 and BW1 surfaces. The agreement of calculations from the BW2 surface with the experimental data is excellent. Comparison of the BW2 with the BKWW and SPK surfaces illustrates the utility of deriving PESs from kinetic isotope effects. Recently Michael et al.105 have reported new experimental data for the reactions of D þ H2 and H þ D2 in the temperature range 1150 –21008 K. The absolute rate constants for these reactions now span 8 orders of magnitude! Arrhenius plots of the logarithms of the rate constants vs. 1=T are not linear due to the Boltzmann excitation terms, tunneling and the fact that quantum dynamics are necessary to describe the low temperature rates. A new theoretical calculation of the PES for this important reaction106 is claimed to be accurate to within ^ 10 cal mol21. Theoretical calculations based on this surface which used quantum dynamics rather than transition state theory107,108 are compared with experiment in Figure 1.6. The solid lines are, respectively, the experimental D þ H2 and H þ D2 reactions. The dotted lines, which are hardly distinguishable from the solid lines, are the theoretical calculations. The excellent agreement between theory and experiment brings to a close 75 years of experimental and theoretical work on this reaction, an important milestone in the history and development of gas phase kinetics. The authors of the recent works point out that the new PES for H3 “ranks with earlier solved problems in molecular quantum mechanics, the short

24

Isotope Effects in Chemistry and Biology 1θ−11

k [cm°/s]

1θ−12

1θ−13

surface I surface II

1θ−14

G3 (Ref.1) Miller and Gordon Kumaran et al. Kita and Stedman Lee et al.

1θ−15

Westenberg and de Haas

1

2

3 1000 /T [1/K ]

4

FIGURE 1.5 Plot of the absolute rate constant, k; as a function of temperature for the H2 þ Cl reaction. Solid line, calculated from the BW2 potential energy surface. (Reproduced from Manthe, U., Bian, W., and Werne, H-J, Chem. Phys. Lett., 313, 647– 654, 1999. With permission.)

1E-10 1E-11

k /cm3 molecule−1 s−1)

1E-12 1E-13 1E-14 1E-15 1E-16 1E-17 1E-18 1E-19

0

10

20

30 40 10000 K /T

50

60

FIGURE 1.6 Plot of the absolute rate constants for D þ H2 (top) and H þ D2 (bottom) reactions. Solid lines, experimental; dashed lines, theoretical. (Reproduced from Michael, J. V., Su, M-C., and Sutherland, J. W., J. Phys. Chem., 108, 432– 437, 2004. With permission.)

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

25

list which includes the electronic spectra of H2 and He and the vibrational – rotational spectra of H2, þ 108 Hþ 2 and H3 .”

VI. CONDENSED MATTER ISOTOPE EFFECTS The first theoretical treatment of isotope effects, published in 1919,3,4 dealt with the difference in vapor pressures of monatomic isotopomers in a solid with a Debye frequency distribution. The partial separation of the neon isotopes by fractional distillation in 19315 was the first demonstration of an isotope effect on the physical or chemical properties of matter. In the design of their experiment to search for a heavy isotope of hydrogen, Urey, Brickwedde, and Murphy6 made an estimate of the enrichment of an isotope of mass 2 above natural abundance that could be achieved by a Rayleigh distillation of liquid hydrogen. For the calculation of the elementary separation factor, the vapor pressure ratio, they assumed a Debye solid at the triple point. They neglected effects from rotation and internal vibration consistent with the known thermodynamic and spectroscopic properties of liquid and solid hydrogen. It is interesting to call attention to the fact that K. F. Herzfeld pointed out to them (see footnote 16 of Ref. 6) that the asymmetry in a molecule like HD would perturb the rotational energy states of the condensed phase. I first became aware of this insight by Herzfeld in writing this chapter. Progress on the development of a general theory of condensed matter isotope effects (CMIE) started in the late 1950s. During the 1930s and 1940s experimental results on vapor pressure isotope effects (VPIE) and liquid vapor fractionation factors (LVFF) were reported. These results are summarized elsewhere.34,109 CMIE are an order of magnitude smaller than chemical exchange fractionation factors. For instance the logarithm of the vapor pressure ratio of H2O to HDO is 0.07 at 258C. The logarithm of the chemical exchange factor for the reaction HD þ H2O ¼ HDO þ H2 is 1.59, also at 258C. The intermolecular forces, which are largely responsible for the VPIE, are much weaker than the intramolecular forces, which determine the chemical fractionation. Generally, the light isotope has a higher vapor pressure than the heavier one. This is due to its larger zero point energy and smaller heat of vaporization. However, for deuterocarbons the general rule is that the deutero isotopomer has a higher vapor pressure than the protio isotopomer. This was explained by Topley and Eyring110 as due to the red shift of the C – H vibrations of the molecule on condensation. Later it was found that at low temperatures the VPIE is normal, a topic to which we will return. The chapter by W. Alexander Van Hook on CMIE111 in this volume presents a concise summary of the theory of VPIE, nonideal behavior of isotopic mixtures and other interesting thermodynamic and spectroscopic properties of isotopomers. Earlier reviews of the experimental data and theory of CMIE are given in Refs. 34,35,109– 114. The thermodynamic properties related to the CMIE of isotopomers are all related to the free energy differences between the isotopomers in the condensed and vapor phases. These free energy differences are directly related to VPIE. In this chapter I add an historical perspective to the earlier reviews and the chapter by Van Hook. I became interested in the subject of CMIE in the early 1950s in connection with the possibility of large scale production of heavy water by low temperature distillation of liquid hydrogen. The vapor pressures of solid and liquid H2, HD, D2 and T2 were known.115,116 de Boer117 suggested that the VPIE of simple liquids could be explained by a quantum parameter Lp ¼ h=s ðm1Þ1=2 ; where s is the range of the intermolecular force, m is the molecular weight and 1 is the depth of the intermolecular potential. The VPIE was proportional to Lp2 ; in accordance with the work of Herzfeld and Teller.118 The latter follows from the formulation of the VPIE in terms of the Wigner first quantum correction to the classical partition function. Although the de Boer method was successful in predicting the properties of 3He from 4He, it failed for all the isotopomers of H2.119 In particular the de Boer theory, like all theories in which the VPIE depends on the molecular weights and moments of inertia, predicts that the vapor pressure of HD is related to the vapor pressures of H2 and D2 by the relation PðHDÞ3 ¼ PðH2 Þ £ PðD2 Þ2 in contrast to the experimental

26

Isotope Effects in Chemistry and Biology

finding PðHDÞ2 ø ½PðH2 Þ £ PðD2 Þ : The de Boer method also failed to correlate the vapor pressures of the homonuclear diatomic isotopomers H2, D2 and T2. In support of the de Boer method was the claim by Libby and Barter120 that HT and D2 had the same vapor pressures. The lack of a general theory to explain the vapor pressures of isotopomers was summarized by Urey in his compendium “Thermodynamic Properties of Isotopic Substances”18 with the following: “The differences in the vapor pressures of other compounds of protium and deuterium” (other than H2) “have been observed but no satisfactory theory with regard to these differences exists. With the exception of the hydrogens and the neons, none of these vapor pressure differences can be satisfactorily related to the translational or sound vibrations of the solids and liquids. The isotopic compounds of the hydrogens differ in heats of fusion, vaporization, molar volumes, heats of solution and many other ways. The differences in the hydrogen compounds have prompted the study of similar differences between the isotopic compounds of other elements, and though the differences are very much smaller, they show similarities to the differences in the hydrogen compounds.” Concurrent with my work on the heavy water problem, I turned my attention to a reconsideration of isotope effects with small quantum corrections. I realized there was more in Equation 1.33 than was recognized in the original Bigeleisen – Mayer paper.15 The first result was the derivation of the rule of the geometric mean.57 In that paper I showed that within the first quantum correction, the rule of the geometric mean also applied to VPIE. It was a small step to question the validity of the Libby – Barter results on HT. Consider the hypothetical liquids H2, HT and D2 at high enough temperatures such that the first quantum correction is applicable. The mass P effect for all motions according to Equation 1.33 depends on ð1=m0i 2 1=mi Þ where the sum runs over all atoms in the molecule. Within the first quantum correction the mass effect is not a function of the molecular weight, the moment of inertia and the reduced mass. These terms are appropriate for the use of independent translation, rotation and internal coordinates. From the sums P ð1=m0i 2 1=mi Þ; H2 2 HT ¼ ð1 2 13 Þ ¼ 2=3 and H2 2 D2 ¼ 2ð1 2 12 Þ ¼ 1 it is obvious that HT behaves as a light isotopomer compared with D2. It should have a higher vapor pressure than D2. An effect that exists in the first quantum approximation must be manifest at lower temperatures. I, therefore, decided to test this prediction by repeating the Libby –Barter experiment. At this time I was on the staff of the Chemistry Department of BNL. We did have facilities for the production of liquid hydrogen, but it would have taken a year or more to assemble and fabricate the equipment necessary to carry out the experiment. I decided it would be advantageous to carry out the experiment at Los Alamos where they had an outstanding cryogenic laboratory and facilities for handling tritium at any level within the same laboratory. I approached R. D. Fowler, Director of the CMF Division, and E. F. Hammel, Jr., Group Leader of the Cryogenic Section of the CMF Division, in the spring of 1955 about the possibility of doing the experiment at Los Alamos. They supported my proposal; Eugene C. Kerr of the cryogenic group volunteered to join me in the experiments. Approval of the experiment did not require any additional review either by Los Alamos management or AEC Division of Research. I did require approval from BNL management for me to continue as a BNL employee while working at Los Alamos. My entire family have fond memories of our stays in Los Alamos in the summers of 1955 and 1956. The experiments on the LVFF of HT in H2 were carried out in the summer of 1955. Care was taken in the experiments to achieve isotopic equilibrium between the liquid and vapor phases. To achieve this, the vapor phase was bubbled through the liquid. A countercurrent heat exchanger minimized temperature changes due to the gas recirculation. Equilibrium was approached from both enriched and depleted gas phases compared with the liquid. We found the distribution

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

27

coefficient (LVFF) was 2.00 ^ 0.01 at 20 K121 in good agreement with the theoretical calculation R ¼ 2:03:119 The known vapor pressure ratio PðH2 Þ=PðD2 Þ is 2.89 at 20 K. A similar experiment on the LVFF of DT in D2 was carried out in the summer of 1956, where the value of R ¼ ðXDT ÞL =XðDTÞV where X is the mol fraction of DT, at 20 K is 1.303.122 In both our HT and DT experiments the mol fraction of tritium was approximately 1028. The values reported by Libby and Barter from their Rayleigh distillation experiments are 3.0 ^ 0.06 and 2.1 ^ 0.05, respectively. The fractionation factors measured by Bigeleisen and Kerr121,122 have been confirmed by experiments related to the design of fuel cycles for fusion reactors and the decontamination of the heavy water moderator in power reactors. A plot of the experimental data of ‘n R vs. 1=T 2 for the solution of HT in H2 is given in Figure 1.7.122 For comparison I show a value for ‘n R for a dilute solution of D2 in H2 calculated from the vapor pressures of the pure substances after correction for nonideal solution behavior. The major correction comes from the gas imperfection. The deviation of the behavior of the heteronuclear diatomic molecule, HT, from that of a homonuclear diatomic molecule was suggested by Herzfeld.6 The internal vibration in liquid hydrogen differs imperceptibly from that in the gas. The rotation in both the liquid and the gas are free and do not contribute to the LVFF. The difference between the heteronuclear and the homonuclear comes from the coupling of the rotation with the translation in the liquid. Consider two neighboring HT molecules in the liquid. The translational motion of each molecule can be described in terms of its center of gravity. The interaction energy between the two molecules is described by the mutual distance of the center of force, which is measured from the geometric 1.40

1.20

1.00

0.80

0.60

0.40

0.20

0

10 Te (H2)

20

30 4 2 10 / T

40

50

FIGURE 1.7 Fractionation factor of HT between solution in liquid H2 vs. vapor as a function of temperature. R ¼ XHTðliquidÞ =XHTðvaporÞ : Upper dotted line is calculated for a nonideal solution of D2 in H2(liquid). (Reproduced from Bigeleisen, J. and Kerr, E. C., J. Chem. Phys., 39, 763– 768, 1963. With permission.)

28

Isotope Effects in Chemistry and Biology

centers of the two molecules. Thus, in the heteronuclear molecule, there is a torque exerted on the molecule from the fact that these two centers are not coincident. A quantum mechanical calculation of this perturbation was first carried out by A. Babloyantz in Prigogine’s laboratory.123 She calculated the zero point energies for a cell model of harmonic oscillators and a smooth intermolecular potential. Theory shows that the effective mass, Meff ; of an heteronuclear diatomic molecule is related to the mass, M; of the homonuclear molecule by the relation ð1=Meff Þ1=2 ¼ ð1=MÞ1=2 {1 þ a½ðm1 2 m2 Þ2 =m1 m2 }

ð1:50Þ

m1 and m2 are the atomic masses of the heteronuclear molecule. The values of a calculated by Babloyantz were of the correct sign but large compared with values derived from the experimental data. The use of an anharmonic cell model gave good agreement with the value of a derived from the experimental data.124 In considering the theory of CMIE, I concluded that the theoretical work prior to 1955 could be simplified and significant insight brought to understanding the basic origins of these isotope effects by the introduction of the reduced partition function ratio for the condensed phase.33 My first effort was to review the data of Keesom and Haantjes125 on the vapor pressures of the neon isotopes. In the preface to their paper they cite the derivation by Otto Stern of the VPIE for a monatomic substance within the first quantum approximation. The result is ‘nðP 0 =PÞ is proportional to 1=T 2 : To my surprise they never made such a correlation. They plotted ‘nðP 0 =PÞ vs. 1=T as one would expect from the van’t Hoff equation. When I made the 1=T 2 plot I found that the data conformed to 0

2

‘nðP =PÞ ¼ A=T þ C

ð1:51Þ

where C was zero for the liquid, but finite for the solid. An extrapolation of the data for the solid should give C equal to zero at 1=T 2 ¼ 0: Here was an experimental result by an established and well respected scientist from the famous Kammerling-Onnes Laboratory at the University of Leiden that was in disagreement with the most fundamental theoretical results of Lindemann, Stern, and Herzfeld and Teller. I decided that a reinvestigation of the VPIE of the neon isotopes, 20 Ne/22Ne, was called for. I designed a cryostat based on the basic designs of Giauque and Egan126 and Johnston et al.127 The measurements would consist of differential vapor pressure measurements of enriched samples along with the absolute vapor pressure of a sample of natural abundance. My design was based on the fact that there would be small quantities of enriched isotopes available to me; temperature equilibration between samples was of utmost importance as well as temperature stability during the time of measurements. At the Amsterdam Symposium on Isotope Separation in April 1957 I was approached by Jules Gueron, head of the physical chemistry section of CEA at Saclay, France, about the possibility of one of his scientists working with me at Brookhaven. Gueron had as his objective that Etienne Roth work on a problem that would be suitable for a Ph.D. thesis. Etienne was a graduate of the Ecole Polytechnique; his graduate studies were interrupted by WWII. He spent the war years in Montreal as a member of the French team working on the Canadian atomic energy program. Etienne joined me in 1958; we assembled the cryostat and we made differential measurements between a sample of 99.8% 20Ne and one of 72.2% 22Ne. The results were extrapolated to pure isotopic composition with the assumption of Raoult’s Law. Keesom and Haantjes had shown it to be applicable to mixtures of 20Ne and 22Ne. There was curvature in a plot of ‘n½ðP20 NeÞ=ðP22 NeÞ vs. 1=T 2 ; the data extrapolated to ‘nðP20 NeÞ=ðP22 NeÞsolid equal to zero at 1=T 2 equal to zero.128,129 Although our measurements were made with a Model “T” cryostat and differential pressures were measured on millimeter ruled graph paper with oil as the manometric fluid, they had much greater precision than the measurements of Keesom and Haantjes. The data on liquid neon129 were in quantitative agreement with the LVFF determined by Boato et al.,130 after the latter were corrected for gas imperfections.131 Direct confirmation of our work on the VPIE of the liquid was made

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

29

by Furukawa,132 who had highly enriched samples and more precise manometric equipment. The triple point pressure of 20Ne measured by Furukawa agreed within 0.02 mm Hg with that measured by Bigeleisen and Roth. The one measured by Clusius et al.133 differed by 0.20 mm Hg. The zero point energy of solid neon determined from VPIE was in excellent agreement with the theoretical calculation of Bernardes.134 The curvature in the 1=T 2 plot was due to the fact that the lattice vibrations were anharmonic and there were appreciable corrections to the 1=T 2 term from the 1=T 4 term. Etienne returned to Paris at the end of 1959, wrote up his work and presented his thesis to the Faculty of Sciences of the University of Paris on 22 March, 1960. I did not see the thesis prior to the examination nor was I invited to the final examination. Was this because I did not have academic rank? Later I learned that it is not just academic rank that matters in France. It was also a question of where. In the 1960s Etienne sent a junior member of his staff, Marc Dupuis, to the United States for graduate work in theoretical chemistry. He successfully completed his degree with Prof. Lars Onsager at Yale. When Dupuis returned to France and applied for a faculty position, he was asked about his educational credentials. He had a degree from the Ecole Polytechnique, which was an accepted credential. His degree with Onsager was not recognized! Was there anyone in all of France then active in the area of the physical sciences of stature comparable to Onsager? This bureaucratic chauvinism has not diminished the friendship that exists between Etienne and me. The molecular theory of CMIE33 was supported by the work on the isotopomers of hydrogen.121,122,124 The theory also predicted differences in vapor pressures of isotopic isomers. The first system we studied were the isomers 15N14N16O and 14N15N16O. These were shown to differ in their vapor pressures135 due to the hindered rotation in liquid N2O. The next set of isotopic isomers studied were cis-, trans-, and gem dideuteroethylene. The oil differential manometer in the “Model T” apparatus of Bigeleisen and Roth was replaced by a system of interconnected mercury manometers so that the difference between the vapor pressure of C2H4 and trans-dideuteroethyelene and either cis-or gem-dideuteroethylene could be measured simultaneously.136 The order of the vapor pressures were trans . cis ø gem . C2H4. At 120 K the cis isomer has a higher vapor pressure than the gem isomer. The order of these two is reversed at 170 K. The coefficients A and B in the A – B equation, ‘nðP 0 =PÞ ¼ A=T 2 þ B=T; (see Equation 35 of Ref. 111), are different for all three dideuteroethylenes. The differences in the A terms is due to the different moment of inertia of the three isotopomers. That all three liquid dideuteroethylenes have higher vapor pressures than C2H4 in the above temperature range is due to the B terms which have negative values for all three dideutero-isomers. These are due to the red shift of the C – H vibrations in the liquid compared with the gas. In order to dissect the A terms into their translational and rotational components Marvin 137 The Stern and Alexander Van Hook measured the LVFF of 13CH12 2 CH2 and the VPIE of C2H3D. A terms of these two isotopomers should differ only in their rotational components. Marvin and Aleck convinced me that their data was inconsistent with a value of Bðd 2 2Þ=Bðd 2 1Þ equal to 2. I should have realized this from the fact that the B values for the three d 2 2 isotopomers were different. Once I accepted their analysis, I realized that there was vibrational –rotational coupling in the liquid, which is absent in the ideal gas, and this was responsible for the deviation of the Bðd 2 2Þ=Bðd 2 1Þ ratio from 2. I was making preparation to leave for a year’s sabbatical in Switzerland and I suggested that Marvin and Aleck pursue this problem with Max Wolfsberg, who was independently working on the general problem of the effects of interactions on VPIE.138 Stern, Van Hook and Wolfsberg139 developed the general theory of the perturbation of all modes, including the internal vibrations, in a condensed phase due to the fact that the translation and rotation are mass dependent coordinates. The use of different coordinates for different isotopomers leads to differences in their potential energy values, contrary to the Born– Oppenheimer theorem. They calculated the differences of the internal vibrations between the liquid and vapor, the B values, for each of the three dideuteroethylenes and the deviation of the ratios of these B values to that of the mono-deutero-in quantitative agreement with experiment. One would think that this closed the chapter on the CMIE of the isotopomers of ethylene. When I returned from sabbatical leave in 1963, I realized that further progress in the experimental

30

Isotope Effects in Chemistry and Biology

determination of VPIE would require major improvements in the operation of the “Model T ” and the accuracy that could be attained with that apparatus. The differences between the isomers of dideuteroethylene were 1022%, which was the limit of the measurements. It was a time when major advances in high vacuum techniques, cryogenic equipment and pressure transducers came through the space program. In collaboration with Frank Brooks of the Cryogenic Division of the Accelerator Department of BNL, I designed the “Rolls Royce” cryostat for VPIE to cover the temperature range 4 –300 K. Together with the new differential pressure measuring equipment, a precision of 1023% in VPIE was achieved.140 This was an order of magnitude better than the state of the art at the time. I decided to test this new equipment against our former results with the dideuteroethylenes. We had the original samples through which Ribnikar and Van Hook had determined the differences between the three dideutero-isomers. To our surprise, Bodo Ribnikar, who had returned from Belgrade for his second appointment at BNL, found that the trans isomer had the smallest vapor pressure of three isomers! We had to resolve this discrepancy before we could proceed with any other work. Attempts to purify the samples by bulb-to-bulb condensation did not change the result. We decide to start from scratch and synthesize new samples. By this time we had adopted the method of low temperature gas chromatography introduced by Van Hook141 for the chemical purification of small samples with high recovery. The new samples reproduced the original data of Ribnikar and Van Hook quantitatively. It was apparent that the stored sample of the transdideuteroethylene had picked up an impurity with a vapor pressure smaller than the ethylene. This was then confirmed by gas chromatographic analysis. The journey with the VPIE of the isotopomers was concluded with the complete redetermination of all the data with the “Rolls Royce” equipment.142 The new data showed that whereas the A – B equation accurately fitted the Ribnikar –Van Hook data; the new data showed that there was curvature in a plot of T ‘nð fc =fg Þ vs 1=T; Figure 1.8. Not surprisingly, the translation and rotation in the liquid were not harmonic. This was treated in the final analysis by Van Hook’s method141 of the use of temperature dependent F matrices for all coordinates, internal as well as external. The results of this final analysis,

3.0

Series 1−9 (1967)

2.0

+ B, R and VH (1961)

Series 23−44 (1967)

trans− C2H2D2 / C2H4 T In (fc /fg)

1.0

0 ++

−1.0 −2.0

−3.0

+

+

+

++

500

6

+

+ ++ ++

100

7

+

+

++ + ++

10

8 103 /T

1

9

0.1

10

FIGURE 1.8 Plot of T ‘nð fc =fg Þ of trans-ethylene-d2 as a function of temperature. Series 1 – 9 (1967), Ref. 142; BR & VH (1961), Ref. 136. (Reproduced from Bigeleisen, J., Fuks, S., Ribnikar, S. V., and Yato, Y., J. Chem. Phys. 66, 1689– 1700, 1977. With permission.)

Theoretical Basis of Isotope Effects from an Autobiographical Perspective

31

reproduced in Table 4.7 of Van Hook’s chapter, were in quantitative agreement with experiment. I succeeded in reaching the end of this journey only through “the inspiration from teachings of the late Claude W. Dukenfield.”142 I decided that my next experimental study of VPIE should be directed at the isotopomers of argon. There were earlier measurements on the liquid that covered the narrow range 84 –88 K143 and by fractionation between the vapor and the condensed phase.130,144 I had already shown that isotopic equilibrium was not reached between the solid and the vapor in LVFF experiments.129 The “Rolls Royce” was ideally suited for further investigation of the CMIE in argon. This was important in order to relate the CMIE in liquid and solid argon with the extensive theoretical and experimental literature on the structure and thermodynamic properties of liquid and solid argon. Our first results were disappointing. We could not reproduce the absolute triple point pressure of normal argon. Our problem was solved as a result of a discussion with Bob Sherman of the Los Alamos Laboratory. For the absolute pressure in this and subsequent experiments we used a Texas Instrument Bourdon gauge, which we calibrated against a mercury manometer. We had an early production gauge. Bob pointed out that the early gauges had a periodic error in the screw, which turned the quartz capillary. There were large errors from this periodic error, which were superimposed on a general exponential correction. These could only be detected by very close incremental pressure calibrations. We then recalibrated the quartz Bourdon gauge against a calibrated dead weight gauge and found the absolute triple point pressure of normal argon in agreement within ^ 0.02 K of the accepted literature value. During the course of the measurements with the “Rolls Royce” cryostat the temperature difference between the isotopic samples drifted by less than 1024K. Our argon measurements spanned the range from 62.8 K in the solid to 101 K in the liquid, where the absolute vapor pressure of liquid argon is 3.6 atmospheres.145 This was the limit to which we were able to make absolute vapor pressure measurements with the necessary precision. Our VPIE measurements on the liquid were in excellent agreement with those of Clusius et al.,143 which covered a very limited temperature range. The value of the mean second derivative of the intermolecular potential, k72 U p l; in liquid argon agreed well with calculations based on molecular dynamics. Values of k72 U p l calculated from experimental values of the radial distribution function were in poor agreement. It was desirable to measure CMIE of liquid argon up to the critical point. For this purpose a cryostat was built at the University of Rochester, to where I had relocated starting in the fall of the 1968 –1969 academic year. The new cryostat was similar in design to the “Rolls Royce” cryostat. The sample container in the VPIE cryostat was replaced by a container for liquid and a vapor recirculation system similar in design to the one used at Los Alamos for the HT and DT work. Measurements were made of ‘n a for the isotopes 36Ar/40Ar from the triple point to within 0.1 K of the critical temperature, 150 K.146 In the temperature range 84– 100 K, which overlapped the VPIE measurements,145 the LVFF measurements of T ‘n a were systematically 5% lower than those derived from VPIE data. This difference was more than 10 times the precision and accuracy claimed for the VPIE data and 5 times that of the LVFF data. We were prescient when we pointed out in the original paper “We are at a loss to explain this small discrepancy, amounting to 3 £ 1024 in ‘n a; other than to suggest that there may be isotope effects on mixing not considered to date. We do not rule out a systematic experimental artifact which has escaped us.” An artifact related to the effect of recirculation of the vapor was suggested by me in a note I prepared for publication. The referee kindly advised me informally to withdraw my note from publication. He included a preprint147 of his revision of the Prigogine theory of the isotope effect on mixing.148 The revised theory quantitatively explained the difference between the VPIE and LVFF factors. This nonideal behavior on mixing liquid argon isotopes was subsequently directly confirmed by Rebelo and coworkers.149 The scaling law exponent for k72 U p ll 2 k72 U p lv was shown to be equal to that of the density law exponent of the density difference between liquid and vapor.150 The correlation of the CMIE of the condensed rare gases with the structure of the liquids is given in Section II.C.4 of the chapter by

32

Isotope Effects in Chemistry and Biology

W. Alexander Van Hook and references therein. Using integral equation theory Lopes et al.151 have shown the values of k72 U p ll for all the rare gases can be calculated in good agreement with the values derived from VPIE and LVFF data over the entire liquid range. They show that k72 U p ll ; as are thermodynamic properties, is a scalable quantity for the liquid rare gases. Although the Lennard-Jones 12-6 potential suffices to calculate the energy, pressure and mean value of the second derivative of the intermolecular potential, it fails to reproduce the VPIE of solid neon and argon. For these simple solids the VPIE calculated from a 13-6 potential152 reproduces the experimental data.145 Although ab initio calculations of the potential energies of polyatomic molecules suffice to calculate both equilibrium and kinetic isotope effects, such calculations have not even been attempted for condensed phase isotope effects. The condensed phase isotope effects are an order of magnitude smaller than the gas phase ones. The intermolecular potential near the minimum of the intermolecular potential is much more anharmonic than that of the intramolecular potential. There is the interplay of effects, usually of opposite sign, from the internal and external modes. Finally, there are the contributions from coupling between external modes amongst themselves and between these modes with the internal modes. To date almost all calculations of condensed phase isotope effects are carried out within the harmonic oscillator approximation of a cell model, with one molecule per unit cell. Anharmonic corrections have been applied through the use of temperature dependent F matrices. The F matrices are constructed in part from spectroscopic data combined with experimental CMIE data. Progress toward a more general approach than the single molecule per unit cell was made by Myung Lee’s successful calculation of the lattice modes of the isotopomers of CO2 consistent with the structure of the solid, 4 molecules per unit cell.153 The calculations of the VPIE of solid 13C16O2 and 12C18O2 compared with 12C16O2 based on 4 molecules per unit cell differ insignificantly from those with one molecule per unit cell, which agree with experiment.154 Extension of the method was made to the isotopomers of N2O.155A more general treatment of the anharmonic effects on VPIE and LVFF was made by relating the temperature dependence of both the A and B terms in Equation 1.51 to the linear difference between the densities of the liquid and vapor over most of the entire temperature of the liquid.146,150,156

ACKNOWLEDGMENTS My research on the theory of isotope effects began in November 1943 as a classified project at the SAM Laboratory of Columbia University under Contract W 7405-eng-50 with the Manhattan Project. The Argonne National Laboratory kindly served as a conduit for the declassification and open publication of this work in 1947. This chapter summarizes what I have learned over a period of 60 years about the theoretical basis of isotope chemistry. The learning process was greatly enhanced by collaboration with the following colleagues in chronological order: Maria Goeppert Mayera Max Wolfsberg Etienne Roth Marvin J. Sterna James T. Phillips Frederic Mandel Anthony M. Popowicz a

Lewis Friedman Eugene C. Kerr Slobodan V. Ribnikar Takanobu Ishida Carl U. Linderstrom-Langa Yumio Yato Luis P.N. Rebelo

Ralph E. Weston, Jr, Fritz S. Kleina W. Alexander Van Hook William Spindela Myung W. Leea Masahiro Kotaka

Deceased.

From July 1948 through January 1987 my research was supported for the most part by the USAEC and its successors ERDA and DOE. Additional support came from an unrestricted grant from ACS – PRF, and fellowships from the National Science Foundation and the John Simon Guggenheim Foundation. Institutional financial support came from the University of Chicago,

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the University of Rochester and the State of New York, Stony Brook. I had the responsibility and pleasure of mentoring a total of three graduate students during my career. For the conduct of the experiments cited in this chapter of which I am one of the authors, I had the able assistance of the late John Densienski from 1966 to 1968 at Brookhaven National Laboratory and William Watson at the University of Rochester from 1968 to1978. In the preparation of this manuscript I have received assistance from Prof. Amnon Kohen and Todd Fleischmann in the compilation of the bibliography.

APPENDIX 1A. SOME PROPERTIES OF THE G AND H MATRICES AND OF THE EIGENVALUES OF ISOTOPOMERS X X li ¼ mi aii ;

li ¼ 4p2 n2i ; X ij

dl i ¼

X ij

i

dli ¼

X i

ðm0i 2 mi Þaii ;

X ij

li ¼

X ij

fij gij ;

fij dgij

li0 ¼ 4p2 n2i0 ¼ hii0 ¼ gii fii ;

dli0 ¼ ðm0i 2 mi Þfii ; lH 2 lIl ¼ 0;

H ¼ FG; X

Classification of the G0 F 0 G0 F 2 G1 F 1 G2 F 0 G1 F 2 G2 F 2

X

P

dl2i ¼

X i

dli0 ¼ ð1=m0i 2 1=mi Þaii

X j li ¼ TrlHj l

ðm02i 2 m2i Þa2ii þ 2ðm0i 2 mi Þ

X j

mj a2ji

ð1:52Þ

dl2i in terms expressed in internal coordinates.68,70

¼ g2ii fii2 statistical mechanical correction and kinetic energy coupling. ¼ F matrix coupling of internal coordinates. ¼ correction to the diagonal element approximation of the H matrix. ¼ kinetic energy coupling of internal coordinates. ¼ correction to the diagonal element approximation of the H matrix. ¼ correction to the diagonal element approximation of the H matrix due to the interactions of three, four coordinates.

The superscript is the order of the off diagonal term in the j ¼ 2 correction to the diagonal element approximation. For a complete enumeration of the G and F matrix elements in each of these terms see Appendix A and Equation 4 of Ref. 68.

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Theoretical Basis of Isotope Effects from an Autobiographical Perspective

39

134 Bernardes, N., Theory of solid neon, argon, krypton, and xenon at 0 K, Phys. Rev., 112, 1534– 1539, 1958. 135 Bigeleisen, J. and Ribnikar, S. V., Structural effects in the vapor pressures of isotopic molecules. O18 and N15 substitution in N2O, J. Chem. Phys., 35, 1297– 1305, 1961. 136 Bigeleisen, J., Ribnikar, S. V., Slobodan, V., and Van Hook, W., Molecular geometry and the vapor pressure of isotopic molecules. Equivalent isomers cis-gem-, and trans-dideuterioethylenes, J. Chem. Phys., 38, 489–496, 1963. 137 Bigeleisen, J., Stern, M. J., and Van Hook, W., Molecular geometry and the vapor pressure of isotopic molecules. C2H3D and C12H2 – C13H2, J. Chem. Phys., 38, 497–505, 1963. 138 Wolfsberg, M., Isotope effects on intermolecular interactions and isotopic vapor pressure differences, J. Chim. Phys., 60, 15 – 22, 1963. 139 Stern, M. J., Van Hook, W., and Wolfsberg, M., Isotope effects on internal frequencies in the condensed phase resulting from interactions with the hindered translations and rotations. The vapor pressures of the isotopic ethylenes, J. Chem. Phys., 39, 3179– 3196, 1963. 140 Bigeleisen, J., Brooks, F. P., Ishida, T., and Ribnikar, S. V., Cryostat for thermal measurements between 2 to 300 K, Rev. Sci. Instrum., 39, 353–356, 1968. 141 Van Hook, W., Vapor pressures of the deuterated ethanes, J. Chem. Phys., 44, 234– 251, 1966. 142 Bigeleisen, J., Fuks, S., Ribnikar, S. V., and Yato, Y., Vapor pressures of the isotopic ethylenes. V. Solid and liquid ethylene-d1, ethylene-d2 (cis, trans, and gem), ethylene-d3, and ethylene-d4, J. Chem. Phys., 66, 1689– 1700, 1977. 143 Clusius, K., Schleich, K., and Vogelmann, M., Low-temperature research. XL. The vapor pressures of Ar36 and Ar40 between the melting and boiling points, Helv. Chim. Acta, 46, 1705– 1714, 1963. 144 Boato, G., Casanova, G., and Vallauri, M. E., Vapor pressure of isotopic liquids II. Ne and Ar above the boiling point, Nuovo Cimento, 16, 505– 520, 1960. 145 Lee, M. W., Fuks, S., and Bigeleisen, J., Vapor pressures of argon-36 and argon-40 intermolecular forces in solid and liquid argon, J. Chem. Phys., 53, 4066– 4075, 1970. 146 Phillips, J. T., Linderstrom-Lang, C. U., and Bigeleisen, J., Liquid– vapor argon isotope fractionation from the triple point to the critical point. Mean Laplacian of the intermolecular potential in liquid argon, J. Chem. Phys., 56, 5053– 5062, 1972. 147 Singh, R. R. and Van Hook, W., Excess free energy in solutions of isotopic isomers. I. Monatomic species. II. Polyatomic species, J. Chem. Phys., 86, 2969– 2975, 1987. 148 Prigogine, I., Bellemans, A., and Mathot, V., Molecular Theory of Solutions, North Holland Publishing Co., Amsterdam, 1957. 149 Rebelo, L. P. N., Dias, F. A., Lopes, J. N. C., and Calado, J. C. G., Nunes da Ponte M. Evidence for nonideality in the fundamental liquid mixture (36Ar þ 40Ar), J. Chem. Phys., 113, 8706– 8716, 2000. 150 Lee, M. W. and Bigeleisen, J., Calculation of the mean force constants of the rare gases and the rectilinear law of mean force, J. Chem. Phys., 67, 5634– 5638, 1977. 151 Lopes, J. N. C., Padua, A. A. H., Rebelo, L. P. N., and Bigeleisen, J., Calculation of VPIE in the rare gases and their mixtures using an integral equation theory, J. Chem. Phys., 118, 5028–5037, 2003. 152 Klein, M. L., Blizard, W., and Goldman, V. V., Calculation of the vapor-pressure ratio of the isotopes of solid neon and argon, J. Chem. Phys., 52, 1633–1635, 1970. 153 Lee, M. W., Calculation of the lattice modes of the isotopic carbon dioxide molecules and their reduced partition function ratios, J. Chem. Phys., 62, 2094– 2097, 1975. 154 Bilkadi, Z., Lee, M. W., and Bigeleisen, J., Phase equilibrium isotope effects in molecular solids and liquids. Vapor pressures of the isotopic carbon dioxide molecules, J. Chem. Phys., 62, 2087– 2093, 1975. 155 Yato, Y., Lee, M. W., and Bigeleisen, J., Phase equilibrium isotope effects in molecular solids and liquids. Vapor pressures of the isotopic nitrous oxide molecules, J. Chem. Phys., 63, 1555– 1563, 1975. 156 Popowicz, A. M., Lu, T. H., and Bigeleisen, J., Temperature dependence of the liquid– vapor isotopic fractionation factors in methane-d3-methane and fluoromethane-d3-fluoromethane, Z. Naturforsch, 46, 60 – 68, 1991.

2

Enrichment of Isotopes Takanobu Ishida and Yasuhiko Fujii

CONTENTS I. II.

III.

IV.

V.

VI.

Overview ............................................................................................................................ 42 A. Separation Factor, Material Balance, and Cascade of Separation Stages ................ 42 Enrichment Processes......................................................................................................... 44 A. Enrichment Processes Based on Steady State Phenomena of Reversible Processes.............................................................................................. 44 1. Distillation ........................................................................................................... 50 2. Chemical Exchange............................................................................................. 50 3. Gas Centrifugation .............................................................................................. 53 B. Enrichment Processes Based on Nonsteady State Phenomena of Reversible Processes.............................................................................................. 53 C. Enrichment Based on Irreversible Processes............................................................. 53 1. Laser Isotope Separation ..................................................................................... 53 2. Gaseous Diffusion ............................................................................................... 54 3. Thermal Diffusion ............................................................................................... 55 4. Electrolysis .......................................................................................................... 55 5. Electromagnetic Method: Calutron..................................................................... 56 Separation Cascade ............................................................................................................ 56 A. Ideal Cascade: Thermodynamic Efficiency and No-Mixing..................................... 56 B. Product-End Refluxer................................................................................................. 58 C. McCabe –Thiele Diagram for Square Cascade ......................................................... 61 1. Case of Total Reflux ........................................................................................... 63 2. Case of Minimum Reflux Ratio.......................................................................... 63 D. Separative Capacity for Close-Separation, Ideal Cascade ........................................ 64 E. HETP (Height Equivalent of Theoretical Plate) ....................................................... 65 Startup of Isotope Enrichment Cascade ............................................................................ 66 A. Time-Dependence of Enrichment Profile along the Length of Cascade during Startup.......................................................................................... 66 B. Rate of Attainment of Steady-State Profile vs. Holdups .......................................... 67 Empirical Determination of HETP and Separation Factor a............................................ 67 A. By Use of Analytic Solution of Material Balance Equation under Transient Condition.................................................................................................... 67 B. From Graphical Solution of Material Balance Equation under the Condition of Zero Time-Dependence at All Stages............................................ 69 Miscellaneous Other Considerations ................................................................................. 69 A. Possible Needs of Chemical Waste Disposal............................................................ 70 B. Possibility of Failure to Achieve a High Target Enrichment ................................... 70 C. Possible Explosion of Working Material .................................................................. 71 D. Consideration of Supply for the Feed ....................................................................... 72 41

42

Isotope Effects in Chemistry and Biology

VII.

Enrichment by Nonsteady State Phenomena Involving Reversible Process .................... 72 A. Ion Exchange Isotope Separation .............................................................................. 72 B. Chromatographic Isotope Separation......................................................................... 74 C. Nonsteady-State Enrichment...................................................................................... 75 1. Enrichment Profile............................................................................................... 75 2. HETP ................................................................................................................... 77 D. Isotope Separation by Ion Exchange ......................................................................... 78 1. Boron Isotope Separation .................................................................................... 78 2. Nitrogen Isotope Separation................................................................................ 79 3. Uranium Isotope Separation................................................................................ 81 VIII. Concluding Remarks............................................................................................................... 82 Acknowledgments .......................................................................................................................... 83 References....................................................................................................................................... 83

I. OVERVIEW Similarity in the chemical and physical properties of isotopomers is the basis of isotopic tracer techniques, while isotope effect studies take advantage of the slight differences, and isotope separation works against the similarities. Isotope separation is thus one of the practical applications of the theories of isotope effects (IE), which suggest which features of molecules would cause an effective separation of isotopes. The theoretical basis of isotope effects is presented by Jacob Bigeleisen1 in the present publication and in other excellent reviews, of which Refs. 2– 18 are classic examples. Regarding the distinction between the terms isotopomers and isotopologues, “isotopomer” will be used exclusively in this chapter, because the discussions of this chapter will be equally applicable for both. The distinction becomes important only when discussion involves mechanics of interactions of molecular structures, molecular forces, and external forces.

A. SEPARATION FACTOR, M ATERIAL B ALANCE, AND C ASCADE OF S EPARATION S TAGES Because one pass of isotopic material through an isotope separation unit would not achieve a sufficiently high enrichment, many such units must be interconnected one after another to effect a practically useful joint multiplicative result. Such a system of interconnected units is called a cascade of separation stages. A separating unit for any process can be envisioned as a black box (Figure 2.1) that, in the simplest scheme, separates a single feed stream into a product stream which is somewhat enriched in the desired isotope, and a waste stream, which is somewhat depleted in that component. In Figure 2.1, F, P, and W are the total molar flows of the molecules containing the desired isotope in the feed, the product, and the waste streams, respectively, and x represents the mole fraction of the desired isotope or, more precisely, the average atom fraction of the desired isotope in the form of exchangeable state in each stream. For each stage, material conservation requires that the total amount of material fed into the stage be equal to the amounts leaving in the forms of product and waste streams (Figure 2.1) and that the quantities of the desired isotope in the two exit streams be equal to the moles of the isotope that have entered the stage: Total molecular balance: F ¼PþW

ð2:1Þ

Fxf ¼ Pyp þ Wzw

ð2:2Þ

Total isotopic balance:

Enrichment of Isotopes

43 Depleted Stream W moles Atom fraction = x w

Feed Stream

Separation Stage

F moles Atom fraction = x f

P moles Atom fraction = x p Enriched Stream

FIGURE 2.1 An isotope separation stage.

The degree of separation achieved by one passage through such a unit is expressed by the singlestage (or elementary) separation factor, a, defined as the abundance ratio of the desired isotope in the product stream divided by the abundance ratio in the waste stream: Elementary separation factor:

a;

xp =ð1 2 xp Þ xw =ð1 2 xw Þ

ð2:3Þ

If a is unity, there is no separation.1 With exceptions for hydrogen isotope exchanges and laser isotope separation (LIS) and a few other notable exceptions, the order of magnitude of ða 2 1Þ of all isotope separation processes are several percent at the most. The separations for which the magnitude of a 2 1 is very small compared to unity is called the close separation, for which ln a < a 2 1: The isotope separation processes that have been studied have some merits and some disadvantages, and evaluating them for a specific isotope depends on several factors. Two most important factors are: (a) whether isotopes of light mass (e.g., deuterium), intermediate mass (e.g., nitrogen), or heavy mass (e.g., uranium) elements are to be separated, and (b) whether the quantities needed are grams per day (as in research) or multiple tons per year (as for use in power reactors). The choice depends on properties of the element, the degree of enrichment needed, and the scale and continuity of the demand; even for a given isotope, there is usually no one best method. For large scale applications, availability of feed materials, capital costs, environmental cleanliness (vide infra) and energy requirements may be the overriding considerations, while for laboratory needs simplicity of operation and/or versatility may be most important. For a large-scale production of a highly enriched isotope, the net content of the desired isotope in the feed becomes a very important issue: the number of moles of the desired isotope produced by a plant must be contained in the feed material. Since the natural abundance of desired isotope is usually very low, a large quantity of the feed material for the isotope plant must be abundantly available. The problems of feed supply and considerations regarding the processes that do not provide high throughputs or require large energy inputs or chemical refluxing will be discussed in Sections III and VI of the present chapter.

44

Isotope Effects in Chemistry and Biology

II. ENRICHMENT PROCESSES Most isotope enrichment processes fall into one of the following three categories: (1) The steady state phenomena of reversible processes: examples: distillation, chemical exchange, gaseous centrifugation. (2) The nonsteady state phenomena of reversible processes: examples: ion exchange separation of isotopes of heavy elements. (3) Irreversible processes: examples: laser isotope separation, gaseous diffusion, thermal diffusion, electromagnetic separation, electrolysis, ionic migration. Examples of applications of these processes are tabulated in Table 2.1. The separation factors listed in Table 2.1 are for the ambient conditions as quoted in the references cited in the last column of Table 2.1.

A. ENRICHMENT P ROCESSES B ASED ON S TEADY S TATE P HENOMENA OF R EVERSIBLE P ROCESSES Distillation can be regarded as an exchange process between liquid and its vapor. Except for the associated refluxer, the same theory of separation cascade can be used for both processes. In a distillation column a liquid flows downward and a gas stream flows upward, and a refluxer converts the liquid into the gas at the bottom and the gas is converted into liquid at the top. Isotope exchange takes place between the liquid and the gas in the column. When the isotope effect is normal, that is, when the vapor pressure of the heavier isotopomer is lower than that of the lighter isotopomer, the heavier isotope enriches in the liquid. In relatively rare cases of inverse isotope effects at the temperature of operation, in which the direction of net migration of isotopes between two phases are reversed, the heavier isotope moves toward the gas. This occurs when an increase in the effect of intramolecular forces on the reduced partition function ratio associated with vaporization outweighs the effect of intermolecular forces in the liquid upon the reduced partition function ratio (Ref. 1 and later in this section). The equilibrium constant K of an exchange reaction AX0 þ BX ¼ AX þ BX0

ð2:4Þ

is K¼

QðAXÞQðBX0 Þ QðAX0 ÞQðBXÞ

ð2:5Þ

where AX0 refers to a lighter isotopomer and AX refers to the heavier isotopomer, A and B are the polyatomic entities of two chemical species to which the isotopes X0 and X are attached, and Q is the molar partition function. For the ideal gas, Q ¼ qN0 =N0 !; where q is the molecular partition function and N0 is the Avogadro number. When isotopic atoms distribute themselves between two chemical species, the preference of an isotope for associating themselves with one chemical species rather than the other is determined by the chemical environments provided by the two species competing for the isotopic atom in question. The chemical environments for a gaseous molecule are provided by the intramolecular (vibrational) forces, while those for a condensed phase or in a solution are provided by the intermolecular forces as well as the intramolecular vibrational forces.1

6

Lithium Li ¼ 7.56% 7 Li ¼ 92.44%

Hydrogen H ¼ 99.985% D ¼ 0.015%

Element

Water (‘) vs. hydrogen sulfide (g)

Chemical exchange

Li-amalgam vs. LiBr in DMF Li-amalgam vs. Liþ (C2H5OH soln) Li- amalgam vs. LiCl (in DMF, THF, DMSO, etc.)

Ammonia (‘) vs. dihydrogen (g)

Chemical exchange Chemical exchange Chemical exchange

Water (‘) vs. dihydrogen (g) Water (‘) vs. dihydrogen (g) Water (‘) þ dihydrogen, and electrlysis of water Water

Chemical exchange Chemical exchange Chemical exchange þ Electrolysis Chemical exchange þ Distillation Chemical exchange

LiCl (fused)

Water (‘) vs. dihydrogen (g)

Chemical exchange

Ion-migration

Water (‘)

Electrolysis

LiCl(aq), LiOH(aq)

Water (‘)

Distillation

Electrolysis

Dihydrogen (g)

Working Substances

Distillation

Method

TABLE 2.1 Examples of Isotope Enrichment

Self-contained heavy water production by bithermal process For a, see Table 2.2 Self-contained heavy water production in bithermal process Both isotopes; laboratory scale a ¼ 1:055 6 Li enrichment; laboratory scale. a ¼ 1:0075 6 Li enrichment. a ¼ 1:05 6 Li enrichment. a ¼ 1:025 6 Li enrichment. Lab scale a ¼ 1:04 , 1:058 @ T ¼ 2 8 , 858C

Production of D2O: final enrichment a ¼ 3.61 @ Ttriple;1.81@ nbp Production D2O, final enrichment a ¼ 1.120 @ Ttriple;1.026 @ nbp Production: final enrichment of output from other process, or partial enrichment for input of other processes. a < 3 , 10 (See text) Self-contained heavy water production by bithermal (dual temperature) process Development of hyrophobic Pt-based catalyst Production: India, Argentina, and others The CECE process: production of heavy water. For a, see Table 2.2 Detritiation

Remarks

29 29 30

26–28

24,25

8,9,12

8,9,12

23

22

21

8,9,11,12

8,9,20

continued

8,9,11– 13,20

8,9,11– 13,59

Reference

Enrichment of Isotopes 45

Nitrogen 14 N ¼ 99.63% 15 N ¼ 0.37%

Carbon 12 C ¼ 98.89% 13 C ¼ 1.11%

Boron 10 B ¼ 19.61% 11 B ¼ 80.39%

Element

Method

(NHþ 4 ) vs. NH3(g) 15

Chemical exchange

Thermal diffusion

N2(g) þ 14N15N(g) þ 14N15N(g)

HNO3 (aq) vs. NO (g)

Chemical exchange

Distillation

HCN(g) vs. CN2 (aq) Bicarbonate(aq) vs. CO2(g) Carbamate (in organic solution) vs. CO2(g) NO

Li(s) vs. Li (glymes, propylene carbonate) Li(s) vs. LiCl(aq) BF3(g) vs. BF3(CH3)2O(‘) BF3(g) vs. BF3 –Et2O(‘) BF3(g) vs. BF3-ethers (‘) BF3 CO

þ

Working Substances

Chemical exchange Chemical exchange Chemical exchange

Chemical exchange Exchange distillation Exchange distillation Exchange distillation Distillation Distillation

Chemical exchange

TABLE 2.1 Continued

Enriched 15N production For a, see Table 2.4 Enriched 15N production: Nitrox process, a ¼ 1:055 (258C) Laboratory scale. Column overall separation < 1.22 , 1.79 Laboratory scale, Clusius –Dickel Column

Labolatory scale. a < 1:0075 , 1:02

32 33–35 36 37,38 39 8,9,40

K ¼ 1.046 ^ 0.013 @ 296.6K Mainly 10B: laboratory scale. a ¼ 1:016 Pilot scale. a ¼ 1.026 @ 738C Laboratory scale. a ¼ 1:016 Laboratory scale for 10B. a ¼ 1:0075 Laboratory scale for 12C and 13 C a ¼ 1:011 for (12/13) a ¼ 1:008 for (16/18) Laboratory scale a ¼ 1:026

47

44

45,70,71

43

41 53 42

31

Reference

K ¼ 1.030 ^ 0.005 @ 296.6K

Remarks

46 Isotope Effects in Chemistry and Biology

CO NO Water O2 Water (‘) vs. CO2 (g) Sulfite(aq) vs. SO2(g)

UF6(g) UF6(g) UF6(g) UCl4(g) U atoms or UF6(g)

Distillation

Distillation

Distillation þ Electrolysis Thermal diffusion Chemical exchange Chemical exchange

Gas diffusion

Centrifuge

Thermal diffusion

Electro-magnetic

Laser

Production of enriched 235U. For a and other data, see Ref. 8 Production of enriched 235U. For a and other data, see Ref. 8 Laboratory scale. For a and other data, see Ref. 8 Production of 235U (1945–1946) For all data, see Ref. 8 Production (Discontinued in U.S. and France) a ¼ 6:20 , 6:24

Laboratory scale. a ¼ 1:044 Pilot scale for 34S a34 ¼ 1:012

By-product of CO-distillation for 13C a ¼ 1:0038 By-product of NO distillation for 15N. For a, see Table 2.4 Laboratory scale, for 18O and 17O

8,9,60–65

8,9,16–18,57,69

8,9,15–18

8,9,14–18,58,66–68

8,9,14–18

8,9,48–50 51,52 53,54 55,56

43

40

Ttr ; Triple point; nbp, Normal boiling point of the liquid. The separation factors without any indication of temperatures are the values reported by the authors of the Reference, which is mostly at ambient temperatures.

Sulfur32S ¼ 95.0% 34 S ¼ 4.22% Uranium

Oxygen 16 O ¼ 99.76% 17 O ¼ 0.037% 18 O ¼ 0.204%

Enrichment of Isotopes 47

48

Isotope Effects in Chemistry and Biology

In the Boltzmann distribution, relative attractiveness of a quantum state of a molecule is expressed by the molecular partition function q¼

AllX states

e21j =kT

ð2:6Þ

j

where the relative weight of a state is the negative exponential of the reduced energy, the energy relative to the thermal energy, kT. The greater the q-value or Q-value, the greater the effective capacity of the molecule or the molar ensemble of the molecules. Equation 2.6 simply states that K is the ratio of the relative capacities of AX compared to AX0 and the relative capacities of BX compared to BX0 . If AX provides higher molecular forces than BX, it follows that 1,

QBX Q , AX QBX0 QAX0

ð2:7Þ

and, therefore K.1

ð2:8Þ

The heavier isotope X enriches in AX rather than in BX. Classically1 QAX Q0AX0

!

s0 ¼ s Cl

Isotopic Yatoms i

mi m0i

!3=2 ð2:9Þ

where s and s0 are the symmetry numbers of the isotopomers, AX and AX0 , respectively, and mi and atomic masses in the isotopomer molecules except the m0i are the masses of the isotopic atoms. All Q isotopic atom(s) cancel out for the product in Equation 2.9. The equilibrium constant, Kclassical ; according to the classical statistical mechanics is, from Equation 2.9 QAX QAX0 QBX QBX0

KCl ¼

Cl

¼

Cl

s0 s s0 s

AX=AX0

ð2:10Þ

BX=BX0

Thus, KCl is independent of temperature and it is only the ratio of the symmetry number ratios. Quantum mechanically1 Qqm < ðQTranslation ÞCl ðQRotation ÞCl ðQVibration Þqm

ð2:11Þ

and Qqm QCl

AX

¼

AX 2uj =2 Y e =ð1 2 e2uj =2 Þ ð1=uj Þ j

ð2:12Þ

where ðQqm =QCl ÞAX is the reduced partition function of the isotopomer AX, and

Kqm ¼

QAX QAX0 QBX QBX0

qm qm

¼

Qqm QCl Qqm QCl

AX AX0

Qqm QCl Qqm QCl

BX0 BX

QAX QBX0 QAX0 QBX

Cl

¼

s f s0 s f s0

AX=AX0 BX=BX0

KCl

ð2:13Þ

Enrichment of Isotopes

49

where s f s0

AX=AX0

Qqm QCl Qqm QCl

;

AX

ð2:14Þ

AX0

in which ðs=s0 Þf of the AX/AX0 pair is the isotopic ratio (AX/AX0 ) of Bigeleisen – Mayer’s2 reduced partition function ratio (RPFR), Qqm =QCl : The separation factor for the exchange reaction Equation 2.4 is the ratio of QðAX=AX0 Þqm over and above its finite classical limit QðAX=AX0 ÞCl in comparison to QðBX=BX0 Þqm over and above its classical limit QðBX=BX0 ÞCl ; that is s 0 0 f ðAX=AX Þ Kqm s ¼ a¼ s KCl f ðBX=BX0 Þ s0 ln a ¼ ln

s AX f s0 AX0

2 ln

s s0

Qqm QCl

2ln

Qqm QCl

¼ ln

AX

ð2:15Þ

BX BX0 AX0

2 ln

Qqm QCl

BX

2ln

Qqm QCl

BX0

ð2:16Þ

For an ideal gas, each reduced partition function (RPF), Qqm =QCl ; consists of contributions of intramolecular motions, while for a condensed phase molecule each RPF consists of contributions of inter- and intra-molecular motions. The intermolecular motions such as translation and libration and low-frequency vibrations such as a bond-torsion and a wagging contribute a 1=T 2 terms of the form 1 h 24 k

2

W1 T2

Low-energy motions X

ðnj02 2 n2j Þ;

or /

X j

ðuj02 2 u2j Þ

to the RPF, where nj 0 and nj are the frequencies (sec21) for the lighter and heavier isotopomers, and u ; hn=kT: The coefficient W1 is the modulating coefficient for the first order term generated by the orthogonal polynomial approximation19 of the reduced partition function, Equation 2.14. The contributions to the RPF of the higher energy-motions such as the bond-stretching and most of the bond-angle-bending vibrations are, at ordinary temperatures, of the form of the zero-point energy approximation of RPF, that is 1 h 1 2 k T

High-energy motions X

ðn 0j 2 nj Þ;

or /

X j

ðu0j 2 uj Þ

The motions that contribute the terms of the form u=2 at low temperatures tend to move toward the terms of the form u2 =24 at higher temperatures, so that effective contributions at intermediate temperatures may take other forms of temperature-dependence. The words, high-energy and low-energy are only relative to the thermal energy at the temperature of the operating system in question.

50

Isotope Effects in Chemistry and Biology

1. Distillation Examples of distillation systems are found in Table 2.1. Except for the hydrogen isotopes, the separation factor, a, of distillation is generally small primarily due to two reasons. First, the separation factor for isotopic distillation is s Low-energy f X 1 h 2 W1 motions s0 condensed ðnj02 2 n2j Þcondensed ln a < ln ¼ 2 s 24 k T f s0 vapor 1 ½dðZPEÞcondensed 2 dðZPEÞvapor þ 2

ð2:17Þ

where dðZPEÞ ; ðZPEÞ0 2 ðZPEÞ; which represents the isotope shift in the zero-point energy. And, dðZPEÞcondensed 2 dðZPEÞvapor ¼ ½ðZPEÞ0condensed 2 ðZPEÞ0vapor 2 ½ðZPEÞcondensed 2 ðZPEÞvapor , which is the isotopic difference in the ZPE-shift upon condensation of the vapor. This term is usually negative and tends to partially cancel the positive first term of Equation 2.17. Secondly, the distillation is usually carried out near the normal boiling point of the liquid for practical reasons and, T being in the denominator in Equation 2.17, these relatively high temperatures for the distillate used in the isotope separation reduces both terms of Equation 2.17. For heavy water production, the deuterium enrichment system is built as a parasitic plant adjacent to a chemical plant, e.g., a plant consisting of a hydroelectric water-electrolysis plant producing large quantities of dihydrogen, plus a dihydrogen distillation plant that uses the hydroelectric dihydrogen as its feed. Another distillation process that is notable for yielding a relatively high a for distillation of an isotope outside heavy water is that of nitric oxide for 15N production. The separation factor is high (Ref. 39 and Table 2.1 of this chapter) because, unlike other distillations, NO in its liquid state exists primarily as a dimer (NO)2, in which two NO molecules are bound to each other by a weak bond between approximate mid-points of the two NO molecules.76 2. Chemical Exchange Two chemicals flow countercurrently. The heavier isotope usually enriches in the stream of the chemical in which stronger molecular forces are exerted on the isotopic atom than in the other chemical.1 For an exchange reaction AX0 ð‘; or aqÞ þ BXðgÞ ¼ AXð‘; or aqÞ þ BX0 ðgÞ 1 h ln a < 24 k þ

2

1 T2

Low-energy motions X

2

AX;AX0

AX;AX0

0

2 2 0 2 nj;AX Þ ðnj;AX

High-energy X 6 motions

1 h 16 2 k T4

ð2:4aÞ

ðnj;AX0 2 nj;AX Þ 2

High-energy motions X BX;BX0

3 7 ðnj;BX0 2 nj;BX Þ7 5

ð2:17aÞ

Because the chemical exchange processes take advantage of relatively large differences in the isotopic differences in the vibrational ZPE’s between two chemical species, the separation factors of chemical exchange processes are usually greater than those of distillation at comparable temperatures. Because a is large, the chemical exchange requires a smaller number of separation stages, and needs shorter process columns and requires lower flow rates of the material through the stages. The chemical process, however, usually needs chemical refluxing of enriched product

Enrichment of Isotopes

51

(cf. Section III.A). Feed of a third chemical to the refluxer is needed and by-products of the reflux chemical reaction must be disposed of. The amounts of the refluxing chemical and the by-products increase proportionately to the amount of refluxing required, but the latter increases very rapidly with the amount of enriched isotope withdrawn and with the level of enrichment of the product (cf.: Section III.C). In the early days of isotope chemistry, i.e., in the early 1930s, H.C. Urey and his students (and, later, the students’ collaborators) investigated many chemical exchange reactions that formed the bases of the contemporary isotope exchange processes. They have been included in Table 2.1. For the chemical exchange reactions given in Table 2.1, the compound in which the heavier isotope enriches is listed first under the column “working substances”, and the compound depleted in the heavy isotope is listed second, except for the processes for uranium. Some of the exchanging species listed in Table 2.1 may appear to be violating the rule that the heavier isotope enriches in the species that provides a stronger molecular force around the isotopic atom, but all of these cases in fact exemplify the subtlety of the rule. For example, take the exchange distillation involving BF3(g) vs. BF3 anisole (‘). (Do not be bothered by the word “distillation” in “exchange distillation,” because the word simply refers to a special type of product-end reflux and has nothing to do with the direction of isotope enrichment. The word will be explained in Section III.B). In the reaction, Donor 11 BF3 ð‘Þ þ 10 BFðgÞ ¼ Donor 10 BF3 ð‘Þ þ 11 BF3 ðgÞ 11

B enriches in the gaseous BF3, while the common chemist’s intuition anticipates that 11B will go to the BF3-donor complex. The key for the apparently unusual behavior of boron is that the fluorine atom is so small. When the BF3 molecule is formed with fluorine, one of the three electron pairs on one of the fluorine atoms is repulsed from the F-atom and occupies one of two vacant orbitals of boron perpendicular to the B – F bond, thus forming a p bond. The resonance energy of BF2· · ·F has been calculated to be 48 kcal/mol. When the Lewis base forms a coordination bond with BF3, the coordination energy is smaller than the resonance energy of the p bond.37,38 The chemical exchange process is of major importance for the isotopes of light elements, especially for heavy water production. Table 2.2 is an excerpt of a tabulation by Benedict, Pigford, and Levi8 on the separation factors of hydrogen exchange. From the viewpoint of the separation factor and its temperature-dependence, the exchange between water and dihydrogen is the best of all. (For the importance of T-dependence of a, see below.) However, this exchange reaction between liquid water and dihydrogen gas requires a catalyst, unlike others listed in Table 2.2. Even platinum catalysts readily lose their activity for cleaving the hydrogen – hydrogen bond of

TABLE 2.2 Separation Factors of H/D Exchange Reactions Involving Watera Separation Factor a

Ratio

Exchange Reaction

a/K

08C

258C

508C

1008C

1258C

2008C

a25/a125

H2O(‘Þ þ NH2D(g) ¼ HDO þ NH3 H2O(‘Þ þ HDS(g) ¼ HDO þ H2S H2O(‘Þ þ DCl(g) ¼ HDO þ HCl H2O(‘) þ HD(g) ¼ HDO þ H2

3/2 1 1/2 1

1.02 2.60 2.87 4.53

1.00 2.37 2.51 3.81

1.00 2.19 — 3.30

0.99 1.94 — 2.65

0.99 1.84 1.88 2.43

0.99 1.64 — 1.99

1.01 1.24 1.34 1.57

a

Excerpt from Table 13.17 of Benedict, Pigford, and Levi (Ref. 8).

52

Isotope Effects in Chemistry and Biology

dihydrogen, because the platinum surfaces are too hydrophilic: the surface is quickly covered by water molecules after the surface is exposed to water leaving no room for dihydrogen. The problem was solved by an invention of “hydrophobic platinum catalysts”21,22 in which platinum is deposited on porous polytetrafluoroethylene or on high surface area carbon bonded to a variety of column packings using Teflon. All water (‘) — dihydrogen exchange processes listed in Table 2.1, including the process for detritiation, are dependent on hydrophobic platinum catalysts. Removal of tritium from heavy water is an isotope separation process and detritiation is, for instance, needed to keep the level of tritium in the heavy water coolant of nuclear power reactors under a safety limit. The problem associated with the needs for chemical reflux becomes acute in large-scale isotope plants such as the ones for heavy water production. It has been solved by an ingenious method called the dual-temperature (or, bithermal) process. The principle of the bithermal process is schematically illustrated by Figure 2.2. Figure 2.2a is the scheme of an ordinary exchange process. In Figure 2.2b which is a block diagram of bithermal process, the cold tower operates like the exchange column of Figure 2.2a in which deuterium is enriched in the water stream but, instead of the chemical product-refluxer of (A), the hot tower provides an enriched hydrogen sulfide, because a lower separation factor at the higher temperature (cf.: Table 2.2) for the reaction, H2O(‘) þ HDS(g) ¼ HDO(‘) þ H2S(g), generates a requisite highly enriched hydrogen sulfide. Product refluxing is thus done physically rather than chemically. The effective separation factor of a dualtemperature process8 is

aeff ¼

aC aH

ð2:18Þ

where aC and aH are the separation factors at the temperatures of the cold and hot towers, respectively. Table 2.2 illustrates that the water –hydrogen system is best with regard to the high value of aeff : See Ref. 8 for a detailed analysis of the dual-temperature principle.

Feed (Natural water)

Feed

Depleted Hydrogen Sulfide

Depleted (Waste)

(Natural water)

Depleted Hydrogen Sulfide

Cold Tower 25 °C a = 2.37

Exchange Column

Product

Product

Hot Tower

Enriched Hydrogen Sulfide

(Heavy Water)

125 °C a = 1.84

Refluxing Chemical Product Refluxer

(a)

D2S

(Heavy Water)

Chemical Waste

FIGURE 2.2 Dual temperature process.

(b)

Waste Depleted Water

Depleted Hydrogen Sulfide

Enrichment of Isotopes

53

3. Gas Centrifugation8,9,15,16,18,66 – 68 The gas centrifuge is based on the dependence of equilibrium population of gaseous molecules under the influence of position-dependent centrifugal force. When a gas of molar mass M is placed in a uniform gravitational field g, the partial pressure at altitude h is, according to the Boltzmann distribution law, pðhÞ ¼ pð0Þexpð2Mgh=RTÞ: Similarly, when a gas of mass M and density r is placed in a centrifuge of radius r rotating with an angular velocity v, the pressure gradient of the gas at distance r from the axis is dp=dr ¼ M rv2 r ¼ ðMp=RTÞv2 r; the ratio of the partial pressure at the axis ð p0 Þ to that at the periphery ð pÞ is p0 =p ¼ expðM v2 r 2 =2RTÞ; and the separation factor between the molar masses M 0 and M is "

ðM 0 2 MÞv2 r 2 a ¼ exp 2RT

#

The separation factor thus depends on the square of the peripheral velocity and on the difference in the masses. This last dependence is an important advantage of the centrifugal method in comparison with other isotope separation processes, especially of the isotopes of heavy elements, because in most other methods a depends on ðM 0 2 MÞ=M 0 M rather than on ðM 0 2 MÞ: When gaseous uranium hexafluoride spins in a centrifugal cylinder at a high rotational speed, the heavier 238UF6 molecules move preferentially toward the periphery. The gas enclosed is subject to centrifugal acceleration thousands of times greater than the gravity of the earth. At a peripheral speed of above 500 m/sec, the 235UF6 content at the center of the cylinder could be as much as 18% greater than at the periphery. A system of rotating baffles and stationary scoops induces longitudinal countercurrent gas flow with light gas flowing upward near the axis and heavier gas flowing downward near the periphery. Only seven stages would be needed in the ideal cascade to produce 3% 235U product and waste at 0.25%. The separation factor being dependent on high spin speed, the centrifuge process depends on high strength cylinder material and design of the bearings.

B. ENRICHMENT P ROCESSES B ASED ON N ONSTEADY S TATE P HENOMENA OF R EVERSIBLE P ROCESSES The ion-exchange separation of uranium isotopes is an example of the processes in this category. In the column, a solution of uranium compounds flows down a stationary column packed with ion exchangers, and the adsorption bands of uranium isotope migrate down and, in the process, isotope fractionation takes place (Ref. 1 and Section VII in the present chapter). Except for the Asahi plant, ion-exchange methods have not left the environment of laboratories. Besides accomplishing a separation of uranium isotopes, however, it has led to the discovery of a new type of nonBorn – Oppenheimer isotope effect, the effects due to differences in the shape and size of the isotopic atoms of uranium (and other heavy elements), which cause isotopic irregularities of the extranuclear electronic energies, resulting in reversals of orders of magnitudes of separation factor vs. mass number.

C. ENRICHMENT B ASED ON I RREVERSIBLE P ROCESSES 1. Laser Isotope Separation8,9,15 – 18,60 – 65 Figure 2.3 is a cross-sectional view depicting an atomic vapor laser isotope separation (AVLIS) module. It takes advantage of small differences in absorption spectra of isotopic species. By using sufficiently monochromatic light of an appropriate wavelength, a particular isotopic species is preferentially excited to an upper energy level. Referring to Figure 2.3, atoms of uranium heated in a high vacuum chamber vaporize and flow upward through a region between a pair of negatively charged plates and get illuminated by a laser having an appropriate energy to excite 235U but not

54

Isotope Effects in Chemistry and Biology Tails Collection Surface

Depleted Uranium Flow

Laser - illuminated area

Electromagnetic (Plasma) ion Extraction Structure Ion deflector plates

Product Collector plates

Magnetic Field electron bea m rgy ne

hig h

e

Uranium vapor flow

U

Water - cold crucible

FIGURE 2.3 Cross-sectional view of an atomic vapor laser isotope separation module for (Redrawn based on the references listed under Ref. 8.)

235

U enrichment.

238

U. Then, at least one other laser beam having energies sufficient to ionize the excited 235U but insufficient to ionize unexcited 238U is used, and 235U ions are collected by a bank of negatively charged plates. Since even the largest energy difference between the atoms of 235U and 238U corresponds to a difference of the order of 1 in 50,000, the excitation laser must be tuned more closely than this. Lasers generate pulses of photons at a fixed repetition rate. The lasers for the LIS must have a sufficiently rapid repetition rate to catch all the 235U atoms passing between the collection plates leading to ionization of the 235U atoms. Thus, LIS requires a high-power, rapid-repetition, finely tunable laser. The method has been used successfully for laboratory scale separation of isotopes as heavy as uranium and, among others, those as light as hydrogen, boron, chlorine, sulfur, and bromine. 2. Gaseous Diffusion8,9,15 – 18

Although this process is usually called gaseous diffusion, the process is based on the physical phenomenon of molecular effusion. The difference is that effusion is a phenomenon of individual molecules passing through a tiny hole on a wall of a vessel just large enough to allow such molecular passage without permitting continuous mass flow of the gas. The rate of passage of two gaseous isotopomers is inversely proportional to the square root of their molecular mass (Graham’s law). For example, sffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffi mð238 UF6 Þ 352 a¼ ¼ ¼ 1:00429 235 349 mð UF6 Þ The unit stage of a gaseous diffusion cascade consists of a compressor, a cooling chamber, and a converter containing a porous diffusion barrier. See Ref. 8 for detailed description of theory

Enrichment of Isotopes

55

and equipment of the U.S. plant that began its operation in 1945 and others built in European countries since then. 3. Thermal Diffusion51,52 When a mixture of fluid (usually a gas mixture) is subjected to a temperature gradient, a small diffusion current is induced, with one component transported in the direction of heat flow and the other component in the opposite direction. The phenomenon is called thermal diffusion. The theory of the thermal diffusion phenomenon is detailed by Hirschfelder, Curtiss, and Bird.46 The basic design of thermal diffusion apparatus that multiplies the separation effect of thermal diffusion was first developed by Clusius and Dickel.51,52 The Clusius – Dickel column is a vertical tube externally cooled and internally (along the tube axis) heated, and a (gaseous isotopic) mixture is confined in a narrow annular space between the two walls. When a temperature difference is maintained between the two walls, not only is the lighter isotope concentrated along the hotter inner wall while the heavier isotope is concentrated along the cooler outer wall, but also a convection current is induced so that the lighter-isotope-enriched gas near the axis flows upward and the heavier-isotope-enriched gas near the outer wall flows downward. The convection flow multiplies the effect of the thermal diffusion. A Clusius –Dickel-type column with a narrow annular space (e.g., a few millimeters wide) can produce an height equivalent of theoretical plate (HETP) (cf. Section III.E) of the order of a centimeter, so that a single Clusius –Dickel column can yield thousands of stages. The design of the thermal diffusion column is simple and has no moving parts and is extremely convenient for a small-scale operation, and the same Clusius– Dickel column can be easily converted for other isotope separation purposes. However, the throughput (i.e., the magnitude of interstage flows) are small, and the energy consumption is high. 4. Electrolysis8,9,11 – 13 When a portion of an aqueous electrolyte solution, e.g., KOH(aq), is electrolyzed, the deuterium content of the residual water becomes higher than the D-content of the feed water and higher than the D-abundance of dihydrogen generated at the cathode. If electrolysis is continued, the residual water becomes more and more enriched in deuterium. If desired, the richer gas may be burned, and the water condensed and returned to an appropriate stage of the cascade of electrolytic cells. This is the basis of the electrolytic production of heavy water. All of the heavy water produced until 1943 was made by electrolysis. The largest of the plants in this period were: The Norsk Hydro Co at Rjukan, Norway (1.5 metric tons per year), which used cells equipped with steel cathodes and steel diaphragms between the cathodes and anodes; the Trail, British Columbia, plant, which used a chemical exchange process to an enrichment of a few percent in deuterium (6 metric tons of 2.37% D2O), which was then further enriched by electrolysis to 99.7%; Manhattan District plant at Morgantown, W. Va, which used cells equipped with steel cathodes and diaphragms. The Trail BC plant was initially an electrolytic production plant. It was later converted to an electrolysis and water – dihydrogen exchange scheme by SAM scientists. At Savannah River, S. Carolina, batch electrolysis was used for a final enrichment from 90 to 99.87% D. Other electrolysis plants are at Ems, Switzerland, and Nangal, India. The separation factor ranged between about 3 and about 13: a ¼ 5:3 (, 608C) at Rjukan, a ¼ 3 (60 , 708C) for the primary plant at Trail, and a ¼ 8 (238C) for the secondary plant at Trail, and a ¼ 6:0 , 8:2 (408C) at Morgantown. The separation factor is influenced by the cell design and operating conditions that are known to increase the irreversibility of the electrolysis process; for example, a is lower at higher temperature, at lower voltage, and in the presence of diaphragms, and with poisoning of the surfaces of the electrodes. Also, electrode materials that have catalytic properties for the reactions, 2H ! H2 and H2(g) þ HDO(‘) ¼ HD(g) þ H2O(‘),

56

Isotope Effects in Chemistry and Biology

reduce the separation factor. It is plausible that the overvoltage for the dihydrogen generation at the cathode is higher for the dideuterium generation than for dihydrogen generation and that this is the cause of H/D separation in electrolysis. (The lowest electrode voltage usable for the enrichment purposes is about 1.7 V, which is to be compared with V ¼ 1:23 V for the reversible electrolysis of water.) 5. Electromagnetic Method: Calutron57,69 The process is essentially that of a huge-scale mass spectrometer and dates back to World War II Manhattan District Project for enriching 235U in kilogram quantities. Since the end of the War the electromagnetic separators, called Calutron for California University Cyclotron because they were originally developed at the University of California by E.O. Lawrence, have been used for separating an amazing variety of isotopes. The Calutron was used for separating relatively small quantities of high enrichment isotopes of practically every element, including the target isotopes for use in the production of a variety of radio-pharmaceuticals by bombardment in a particleaccelerator. Unless the target material is isotopically pure, bombardment might produce radioisotopes of undesirable properties not fit, for instance, for injection in human patients. A workshop organized by the National Research Council, Washington, DC, issued a report,72 a wish-list (as of early eighties) of the research communities in various scientific fields for the need of enriched isotopes that could be separated especially by taking advantage of the versatility of Calutron.

III. SEPARATION CASCADE This section primarily deals with the cascade theory of steady state of cascade of reversible processes.

A. IDEAL C ASCADE : T HERMODYNAMIC E FFICIENCY AND N O -MIXING To achieve a useful enrichment, it is necessary to construct a stack of separation stages in which the product stream from a lower stage is fed to the next higher stage, whose product is in turn fed to the stage above, and so on. Here, the “lower stages” are the stages containing less enriched material than those in the “higher stages.” Such a stack of interconnected stages is called a separation cascade. Thermodynamically, the most efficient cascade is the one in which the streams containing different isotopic enrichments are not allowed to mix: such mixing would achieve an entropy of mixing, a process that will counteract the very purpose of isotope separation. At every stage of a cascade called an ideal cascade, two streams, i.e., the product stream from the stage below and the waste stream from the stage above converging into the stage in question, have the same isotopic enrichment. The ideal cascade that satisfies Equation 2.1 to Equation 2.3 at every stage necessarily have flow rates that must change with stage numbers as depicted in Figure 2.4.66 The interstage flows are at their highest at the feed point, because the flow of the low isotopic abundance material at this point must supply, at least, all the requirement for the target isotopes in the stages nearer the more enriched product end. Stages below the feed point in Figure 2.4, called the stripping section of the cascade, work to squeeze out some of the residual isotope left out by the stages above the feed, called the enriching section. The stripping section is useful especially in cases where the feed material is valuable. The feed material must be conserved in the most practical way where it has already been enriched to some extent and/or when the natural resources of the feed material are limited, e.g., uranium. The ideal cascade calls for a smooth decrease in the flow rates at every stage from the feed point to the product end (Figure 2.4). The ideal cascade is the most efficient cascade but such

Enrichment of Isotopes

57 Product (Enriched)

} A Stage Separating Units

} A Stage

Feed Natural Abundance

Tails (Waste) (Depleted)

FIGURE 2.4 Schematic representation of relative flow rates of an ideal cascade. The width of each stage is meant to be proportional to the interstage flow rate in the stage in a cascade of discrete separating units such as centrifuges in which each rectangle represents a discrete unit. For a continuous equipment such as a packed column, the number of the rectangles in the horizontal direction in the figure is proportional to the crosssectional flow rate. (Adapted from Ref. 66; “The Gas Centrifuge,” by Donald R. Olander, Copyright Scientific American, August, 1978.)

a smooth scale-down of flow rates requires changes of design of stage equipment from stage to stage, a highly expensive endeavor. Such a cascade is thus feasible only for a large-scale plant for enriching highly valuable products such as enriched uranium. For the isotopes of light elements, say, for those approximately up to sulfur in the periodic table, a cascade in which the interstage flow rates are constant throughout is more practical. It is called the square cascade. Sometimes, as a compromise between an ideal cascade and a square cascade, two or more straight columns of different diameters are interconnected, one on top of another, that crudely simulate the shape of an ideal cascade. This arrangement is called the tapered cascade. These cascades still require an often expensive product refluxer (vide infra) at every junction of the columns of different diameters. Of course, one loses the ideality, but the square design offers much simplified equipment and lower capital costs than an ideal cascade.

58

Isotope Effects in Chemistry and Biology

B. PRODUCT-END R EFLUXER Let us suppose that the plant is for enriching an isotope I1 of an element and that the plant works on two chemical streams A and B in a square cascade of an equilibrium process. Further, the Isotope I1 concentrates in the A-stream, which flows down, and an Isotope I2 concentrates in the B-stream, which flows up. The feed for every stage, i, is a mixture of the enriched stream from the stage “above” (i.e., stage number i 2 1), and the I1-depleted stream from the stage “below” (stage number i þ 1). The net feed to stage i is an average composition of the two streams. Here again, the word “below” is meant to be the next higher stage containing the isotope I1 in a higher enrichment, and the word “above” is meant to be the next lower stage containing the isotope in a lower enrichment. This common usage of the words, “above” and “below,” has come from the facts that (a) the target isotope is usually heavier than the other isotope, e.g., deuterium and tritium in hydrogen, 13C and 14C in carbon, 15N in nitrogen, and 17O and 18O in oxygen, and (b) the heavier isotope usually enriches in a molecule (i.e., in the A-stream in the present notation) that exerts higher molecular forces on the isotopic atom in question than the other molecule does,1 and (c) such a molecule is usually in a liquid state or in a liquid solution that physically flows downward in the cascade column, while the other molecule is usually gaseous that flows upward in the column (Figure 2.5). That is, the stage “below” is the higher stage, enrichment-wise, and the stage “above” is the lower stage, enrichment-wise. As one follows further and further downward into the more and more enriched stages, one finally comes to the “product” end where the enrichment has become sufficiently high to be called your product. But then a question arises as to what is the source of the highly enriched material B for the next-to-the-product-end stage, the stage that is one stage away from the satisfactorily high enrichment, i.e., the stage number n in Figure 2.5? Because you are the one producing the high enrichment product, you cannot expect to find from elsewhere a source of a sufficiently highly enriched stream of the material B to feed this highest stage. The only solution to this bootstrapping problem is to sacrifice a portion of your own high-enrichment product (still in the form of Chemical A) and convert it into Chemical B. Such a device is called the product-end refluxer. For example, in a process called Nitrox process for the enrichment of 15N, the isotope exchange reaction is H14 NO3 ðaqÞ þ 15 NOðgÞ ¼ H15 NO3 ðaqÞ þ 14 NOðgÞ In a packed column for this process, nitric acid (of about 10 M) flows down the column countercurrently against a gas stream of nitric oxide. Nitrogen-15 enriches in nitric acid with a separation factor a ¼ 1:055 (258C). At the bottom of the column nitric acid can be made to contain a highly enriched (, 99.9%) 15N. During the total reflux period, 100% (vide infra) of the nitric acid is reduced to nitric oxide using sulfur dioxide 3SO2 ðgÞ þ 2HNO3 ðaqÞ þ 2H2 Oð‘Þ ¼ 3H2 SO4 ð‘Þ þ 2NOðgÞ When the reflux conversion is complete, the nitric oxide thus produced is as highly enriched as the product nitric acid and does not contain any higher nitrogen oxides or nitrogen oxyacids. The gas then flows up the exchange column as it gradually gets depleted by the transfer of 15N to the nitric acid stream. It sounds like we have gone to great lengths in which 15N has been transported back and forth between the two chemicals without making any gain, but we actually do gain. One should remember that isotope separation takes advantage of a slight difference for our purpose just as we do in isotope effect research. Effects of the tiny differences will become clear in the sections that follow. Before leaving this section, it is extremely important to emphasize that the refluxing process should not tolerate an incomplete chemical conversion and any physical leaks around

Enrichment of Isotopes

59

Point A

L moles / hr

G moles / hr

yf

xf =x1

yf

x1

Stage 1 Point B

y1

x2

Stage 2 y2

x3

Stage 3 y3

x4

yi −1

xi

Envelope E

Point C

Stage i yi

xi +1

Stage i+1 yi +1

xi +2

yn−1

xn

Stage n Point P

yn = y p

xp

yp

xp

Product-end Refluxer

FIGURE 2.5 Schematics of a square cascade. The mole fractions xi and yi are in isotope exchange equilibrium with each other in the streams leaving the stage-i; xiþ1 and yi are the mole fractions of the desired isotope in the streams at the same horizontal levels along the cascade, i.e., xiþ1 enters the stage i and yi leaves the stage i. The Points A, B, C, and P correspond to the Points A, B, C, and P in Figure 2.7.

the refluxer: incomplete conversion will be a potential source of parasitic isotope fractionation and difficulty in achieving the target enrichment, and leaks will offset the material balance and be a major cause of inability to reach the target level of enrichment. After a successful buildup of isotope concentration profile along the length of the separation column (vide infra) in the startup phase of the operation, one would start withdrawing the enriched product from the product end, always with a small and controlled rate of withdrawal. Even with extreme care, the withdrawal will inevitably lower the enrichment level just achieved prior to the beginning of the product withdrawal, as will be explained later in the present chapter. The higher the withdrawal rate, the more depressed

60

Isotope Effects in Chemistry and Biology

the enrichment of the withdrawn product will become. If there is a leak from around the product refluxer, the enrichment will never reach an expected level, because your cascade cannot distinguish between a purposeful withdrawal for the extraction of the product and an unintentional withdrawal called a leak. The cascade system responds to a leak in the same manner as to an intentional withdrawal. There is another chemical engineering consideration concerning the choices of a type of refluxer. The issue becomes a major one when considering the cost of large-scale enrichment. There are three schemes of refluxing method commonly used in the reversible isotope separation processes, as depicted in Figure 2.6. Distillation requires only addition or removal of heat for the product and waste end refluxing. Chemical exchange needs a chemical to convert the enriched product, e.g., from HNO3 to NO (Figure 2.6) using SO2. The refluxer also produces a chemical waste, e.g., H2SO4 (at a concentration about 10 M and saturated with sulfur dioxide) in the case of the Nitrox process. Exchange distillation (Figure 2.6) is an excellent compromise that takes advantage of a relatively high a of chemical exchange method and a clean and inexpensive thermal refluxing of distillation. In the example of exchange distillation shown in Figure 2.6, 13C/12C are exchanged between CO2 and a nonaqueous solution of carbamate, RR0 – NCOOH, and 13C is enriched in the carbamate. In its product refluxer, heat is applied to decompose the enriched carbamate to carbon dioxide, which is sent back up the separation column, and the amine in an organic solvent is pumped to the top of the column, where it is recombined with CO2 emerging from the low-enrichment end of the column. The chemical exchange processes usually provide a higher single stage separation factor than distillation processes operating at the same temperature, but the distillations require only thermal reflux. However, the chemical exchange needs a chemical to effect the refluxing conversion and produces chemical wastes. This is one of the most significant issues especially in a large-scale production, because the amount of reflux reaction needed is easily many

Feed NO

Depleted CO2 Heat

Condenser

Boiler

Enriched CO2

Amine CO2

Thermal Decomposer Heat

Heat DISTILLATION

EXCHANGE DISTILLATION

FIGURE 2.6 Comparison of reflux methods.

H2O, O2 Depleted HNO3

Depleted NO Feed HNO3

Packed Column

Packed Column

Carbamate

Waste Refluxer

CO2

Carbamate

Enriched NO

Heat

Thermal Recombiner

Feed CO2

Packed Column

Depleted NO

HNO3

Enriched HNO3

NO

Product Refluxer

SO2 CHEMICAL EXCHANGE

Waste H2SO4

Enrichment of Isotopes

61

thousands of times the number of moles of enriched product withdrawn (cf.: Sections III.C and VI.A).

C. MC CABE – THIELE D IAGRAM FOR S QUARE C ASCADE Although mathematical theory and design formula that describe the square and ideal cascades have been derived8,9 from Equation 2.1 to Equation 2.3, the principle generated from Equation 2.1 to Equation 2.3 can be best illustrated using a McCabe – Thiele diagram. For the flow scheme of Figure 2.5, which shows a two-isotope system involving two countercurrent streams, a McCabe – Thiele diagram, Figure 2.7, is constructed using two lines in the x ¼ ½0; 1 , y ¼ ½0; 1 space, where x and y represent the mole fractions of the desired isotopes in streams g and ‘; respectively. For simplicity in this discussion of the diagram the stripping section and the refluxers will not be included. The ‘-stream having a molar flow rate L (cf. Figure 2.5) and a composition of the feed stage yf enters Stage 1 and meets the g-stream having a molar flow rate G and a composition x2 : After an isotopic exchange equilibrium is reached in Stage 1, it produces the atom fraction y1 in the ‘-stream and atom fraction xf ( ¼ x1 ) in the g-stream. The stream compositions are (xf ; yf ), i.e., Point A of Figure 2.5 and Figure 2.7, just above Stage 1 and (x2 ; y1 ), which is Point B, just below Stage 1. Similarly, the column composition at Point C is (x3 ; y2 ). The line A –P is the locus of the points representing the compositions of the working fluids on the same horizontal planes between the successive stages (cf.: Figure 2.5). The line is straight, because the isotope balance in the Envelope E in Figure 2.5 is yf L þ xiþ1 G ¼ yi L þ xf G

or

yi ¼ yf þ

G ðx 2 xf Þ L iþ1

ð2:19Þ

where ðxf ; yf Þ is a fixed point and G and L are constant in a square cascade. The straight line, A through P, in Figure 2.7 is called the operating line. 1 yp

P

y

Equilibrium Line

Operating Line y2

C

y1

B

yf

A 0

x2 0

xf

x3 X

xp

1

FIGURE 2.7 Basic McCabe– Thiele diagram for a square cascade. See Figure 2.5 for the flow scheme notations.

62

Isotope Effects in Chemistry and Biology

The points representing the equilibrium composition in Figure 2.7 are given by an equivalent of Equation 2.3 which, in terms of the notation of Figure 2.5 and Figure 2.7, y 12y a¼ x 12x

ð2:20Þ

or y¼

ax 1 þ ða 2 1Þx

ð2:21Þ

This line is called the equilibrium line. The line is concave upward when a . 1: As has been shown above, one separation stage corresponds to a step consisting of a vertical line and a horizontal line drawn between the equilibrium line and the operating line. The diagram illustrated by Figure 2.7 is called the McCabe – Thiele diagram. The operating line ends at its highenrichment end on the main diagonal in the McCabe – Thiele diagram, which corresponds to a perfect product-end refluxing; “perfect” in the sense that the refluxer does not change the isotopic abundance during the reflux process and does not have any leaks. The number of stages needed for a given separation can be graphically determined by counting the number of steps between the operating and equilibrium lines between Points A and P of the McCabe – Thiele diagram. As is evident from Equation 2.21, the smaller the magnitude of la 2 1l; the closer the equilibrium line is to the operating line, and the greater the number of steps required to accomplish a given separation. Mathematically, the number of stages required for a given set of terminal conditions and flow rates can be obtained from the material balance and equilibrium equations. For instance, for the ideal cascade without stripping section, the overall separation S is xp 1 2 xp S¼ xw 1 2 xw

ð2:22Þ

where xp and xw are the mole fractions in the product and waste streams, respectively. Also, S ¼ an

ð2:23Þ

where n is the total number of stages in the cascade. In the case of close separation in which a is very small compared to unity, i.e., lal ! 1; we have

an < eða21Þn ¼ e1n

ð2:24Þ

1;a21

ð2:25Þ

where

Here, a is the overall separation of a stage between two streams leaving the stage. This a is the same as the a used by Benedict,1 Bigeleisen,4 and many others, which is the “heads-to-tails” separation factor. This is in contrast to the a used by Karl Cohen, which is “heads-to-feed” ratio. Because of the eminence of Cohen’s book,14 there has sometimes been minor confusion among the latter day researchers in the use of the word, separation factor, despite the general trend that favors the heads-to-tails terminology.

Enrichment of Isotopes

63

When yp ð¼ xp ) and yf are given, the slope of the operating line of the square cascade is (cf.: Figure 2.5 and Figure 2.7) Slope ¼

yp 2 yf 1 G ¼12 ¼ yp 2 xf R L

ð2:26Þ

where R ¼ Reflux ratio ;

L P

ð2:27Þ

1. Case of Total Reflux For a given rate of product enrichment, i.e., when Point P is fixed, fewer stages are required when the slope of the operating line is steeper (cf.: Figure 2.7). The steepest slope is the case in which the feed point is on the main diagonal of the McCabe – Thiele diagram. This corresponds to Slope ¼ 1 and R ! 1; called the case of total reflux: The entire product is refluxed, and there would be no product withdrawn. Then nmin ¼

lnbxp =ð1 2 xp Þc 2 ln½xw =ð1 2 xw Þ ln a

ð2:28Þ

2. Case of Minimum Reflux Ratio Another extreme case corresponds to the smallest slope of the operating line, that is, the case of minimum reflux ratio. For the square cascade it occurs when the feed point A (Figure 2.7) is on the equilibrium line. Mathematically, from the equilibrium condition, Equation 2.21 y2x¼

yða 2 1Þð1 2 yÞ y þ að1 2 yÞ

and the material balance between the feed point and the product point yf L ¼ Pxp þ ðL 2 PÞÞxf Combining the two equations Rmin ¼

L P

min

¼

yp b1 þ ða 2 1Þð1 2 yf Þc 2 yf ða 2 1Þyf ð1 2 yf Þ

ð2:29Þ

A cascade operating on the minimum reflux ratio will produce a (fictitious) maximum product rate but will require infinite number of stages, since it would take an infinite number of steps to move the operation point away from the equilibrium line. Table 2.3 illustrates examples of the number of stages and the need for large amounts of reflux reaction in the Nitrox process. Note that the ratio of the amount of the enriched product to that of the reflux chemicals is huge. For an ideal cascade in which the interstage flow rates change from one stage to the next, the slope of the operating line changes from stage to stage, so that the condition for the minimum slope must be applied on the individual segments of the operating line. As the reflux ratio decreases, say, from that of the total reflux to smaller values, the increase in the enrichment between the stages decreases, and the difference in the enrichment between the successive stages decreases toward an eventual zero. In this situation there would be no enhancement of enrichment and the reflux ratio

64

Isotope Effects in Chemistry and Biology

TABLE 2.3 An Example of Parameters of Two-Section Tapered Cascade: Case of Nitrox Process Producing 100 kg 15N per year (5 18.26 mol 15N per day) Flow Rates (mol N/day)f

Enrichment

Number of Stagesd

Reflux Ratio

HNO3

NO

Reflux

SO2

90.0%a —

92 (84) 137 (125)

11.470 (10.428) 411.6 (343)

94,004 7,538

93,985 7,520

86,465 7,520

99.0%b —

92 (84) 188 (171)

11.388 (10.352) 456.0 (380)

103,428 8,349

103,410 8,331

95,079 8,331

99.9%c —

90 (82) 246 (224)

11.165 (10.150) 468.6 (426)

104,370 8,580

104,350 8,561

95,790 8,561

129,698 11,280 140,978 142,619 12,497 155,116 143,685 12,842 156,527

e

Product

18.26 18.26 18.26 18.26 18.26 18.26

a Atom fraction of depleted stream out of cascade ¼ 0.00037. A stripping section has been included in these calculations but not shown in the table. Taper point atom fraction ¼ 0.05. k1 ¼ 1:10; k2 ¼ 1:20: b Atom fraction of depleted stream out of cascade ¼ 0.00037. A stripping section has been included in these calculations but not shown in the table. Taper point atom fraction ¼ 0.05. k1 ¼ 1:10; k2 ¼ 1:20: c Atom fraction of depleted stream out of cascade ¼ 0.00037. A stripping section has been included in these calculations but not shown in the table. Taper point atom fraction ¼ 0.045, k1 ¼ 1:10; k2 ¼ 1:10: d In each enrichment entry the first number (e.g., 92) is the number of stages used in the calculation. The number in the parentheses (e.g., 84) in each entry is the minimum number of stages. The first and second lines for each enrichment entry are for the Section 1 and Section 2 of the two-section enrichment cascade, respectively. e The reflux ratio used in the Section 1 calculation (e.g., 11.470) is k1 times the minimum reflux ratio, which is enclosed in the parentheses (e.g., 10.428). The reflux ratio used in the Section 2 calculation (e.g., 411.6) is k2 times the minimum reflux ratio, which is enclosed in the parentheses (e.g., 343). f All flow rates are in moles of N per day, except for SO2, which is 1.5 times the rate of product-end reflux.

is at its minimum. The minimum reflux ratio for the ideal cascade is necessarily a function of stage number8 Liþ1 P

min

¼

ðyp 2 yi Þ½1 þ ða 2 1Þð1 2 yi Þ ða 2 1Þyi ð1 2 yi Þ

ð2:30Þ

In both square and ideal cascades the minimum reflux ratio and thus the flow rates at any stage in the cascade are inversely proportional to ða 2 1Þ: Therefore, the number of separation units, e.g., the compression –cooling – diffusion units at a given stage in a gaseous diffusion plant, is inversely proportional to ða 2 1Þ: Because the minimum number of stages, nmin ; is also inversely proportional to ða 2 1Þ; the minimum total size of the plant and thus the initial construction cost is inversely proportional to ða 2 1Þ2 .

D. SEPARATIVE C APACITY FOR C LOSE -SEPARATION, I DEAL C ASCADE The sum of the total flow rates of the “heads” (enriched) streams, J, and the sum of the total flow rates of the “tails” (depleted) streams, K, for a close-separation ideal cascade is8,14 JþK ¼

8 D ða 2 1Þ2

ð2:31Þ

Enrichment of Isotopes

65

where D is called the separative capacity or separative duty and defined by D ; Wð2xw 2 1Þln

xp xw xf 2 Fð2xf 2 1Þln þ Pð2xp 2 1Þln 1 2 xp 1 2 xw 1 2 xf

ð2:32Þ

in which xw ; xp ; and xf are the isotopic atom fractions in the waste, product and feed streams for the cascade, respectively, and W, P, and F are the molar flow rates for the waste, product, and feed streams of the cascade, respectively. Note that they are for the whole plant and not for an individual stage. Equation 2.31 is obtained by summing up the material balance relations on all (i.e., feed, heads, and tails) streams of all stages. The factor, 8=ða 2 1Þ2 ; is a measure of the relative ease or difficulty of the separative process or the quality of the separation phenomenon in question, while the separative capacity, D, is a measure of the quantity of the actual separation being done. Many important characteristics of a plant are proportional to the separative capacity. When predicting an effect of changing a design and operating conditions, the most dependable, first hand, prediction can be made on the basis of changes in the separative capacity.

E. HETP (HEIGHT E QUIVALENT OF T HEORETICAL P LATE) If a distillation column consists of a series of discrete bubble-cup plates and if the cup-plate stage were designed to let the liquid and vapor streams entering it reside for a sufficiently long period of time so that the streams would reach an equilibrium with each other before they leave the stage, then each of such stages will be equivalent to a fully bona-fide separation stage. In a packed column for distillation or isotope exchange system, one can imagine a (hopefully short) vertical section of the column to be designated as an equivalent of one stage in which an isotope exchange equilibrium is attained. Whatever the actual two-phase kinetics is, it is conceivable to expect that an equilibrium would be reached sooner or later so that after such a period the vertical section of the column becomes an equivalent of one stage. The two-way streams continue to flow while they exchange toward the equilibrium, so that, the slower the two-phase exchange kinetics is, the higher the vertical thickness of the column it would take to achieve one equilibrium. This is the height equivalent of theoretical plate (HETP). The HETP multiplied by the number of stages equals the height of the separation cascade: Height of cascade ¼ ðHETPÞn

ð2:33Þ

The HETP of a packed column is influenced by the rate of isotope exchange, the flow rates per cross-sectional area and nature of the packing. The rate of exchange depends on the chemical kinetics and the rates of diffusion of the isotope-exchanging chemicals through two fluid layers, especially through the liquid. The primary aim of the packing is to provide an increased interphase area and a thin liquid layer without losing its own wet surface area while maintaining a smallest possible holdup. The higher the column height, the higher the pressure drop through the column, and the higher will be the holdup of the desired isotope by the column, causing among other things a higher capital cost of construction and longer startup transition period longer (cf.: Section IV of present chapter). This period is not only a waste of time because it does not produce the enriched isotope but also because, in case of an operational accident that requires a shut-down of the cascade, all efforts and costs expended up to that point will become wasted. And, the longer the startup period, the more probable that an accident may occur. Depending on the scale of production, holdups, and target enrichment, the transition period may be days, weeks, or months (cf.: Section IV of present chapter).

66

Isotope Effects in Chemistry and Biology

IV. STARTUP OF ISOTOPE ENRICHMENT CASCADE A. TIME -DEPENDENCE OF E NRICHMENT P ROFILE OF C ASCADE DURING S TARTUP

ALONG THE

L ENGTH

We will consider a startup of a simple case of a square, close separation cascade involving two isotopes and two streams, one stream of chemical ‘ and another of g. The plant’s feed point may be taken as Stage 0 and the abundance of the desired isotope at the feed point is kept constant (usually at the natural abundance) at all times. The stage number, s, increases from 1 to n, n being the highest stage which may be identified with the product-end refluxer. The desired isotope is assumed to enrich in the stream of ‘: The magnitude of the holdup, H moles of the desired isotope, in the refluxer depends on the design of the reflux process. Similarly, the holdup, h moles of the desired isotope per stage per unit flow, or the average process time per stage, also depends on the design of the separation process and the stage component such as the column packing material. Prior to a startup of the process, the separation column is flooded with the substance ‘ of the isotopic composition of the feed. Thus, at time t ¼ 0; the mole fraction of the desired isotope, N, which corresponds to the notations x and y in Figure 2.5 and Figure 2.7, is N0 throughout the cascade, where N0 is the enrichment of the feed material. The cascade operation is then started by initiating the input flow at the feed stage and the total reflux operation at the product-end. As the stream of the material ‘ flows toward the higher stages countercurrently against the stream of the material g, the target isotope moves toward the ‘-stream and the other isotope moves from the ‘-stream into the g-stream in the isotope exchange. When the front end of the ‘-stream first reaches the product end, however, the enrichment in that stream at that time will not be anywhere near the target enrichment level, because the level in every stage has just started with the very low enrichment of the feed material. As the ‘-steam comes into the product-end refluxer, it is (supposedly) 100% converted to the material of the g-stream in the refluxer, which then begins its journey up toward the plant feed point. This g-stream is somewhat more enriched than the original flooding material and provides a source of somewhat higher levels of enrichment in the ‘-stream flowing down. As plant operation under total reflux continues, the isotope concentration profile along the column slowly builds up. At any time during the startup period, the enrichment becomes higher with the higher stage gaining more in the enrichment than the lower stage. Thus, N is a function of the stage number, s, and the time, t, during the startup period: At t ¼ 0;

Nðs; t ¼ 0Þ ¼ N0

At s ¼ 0 ðat feed pointÞ;

½all s

Nð0; tÞ ¼ N0

At s ¼ n ðat product endÞ; P ¼ Product withdrawal rate ¼ 0

ð2:34aÞ ½all t ½all t , t1

ð2:34bÞ ð2:34cÞ

where t1 is the time it takes for the cascade to asymptotically attain a satisfactorily high enrichment profile, at which time the withdrawal of the enriched product may begin. Since the enrichment at every stage in the cascade will drop when the product is withdrawn (cf.: Section III.D, this chapter), the total reflux operation must be continued for a while longer even after the enrichment at the refluxer has reached a satisfactory level, in order to provide a buffer against such a drop. The time-dependent material balance between the product stage and the sth stage in a square, close separation cascade with a constant flow rate L,14 leads to a partial differential equation:

l

›Nðs; tÞ ›2 Nðs; tÞ › 2 1 {CNðs; tÞ þ Nðs; tÞ½1 2 Nðs; tÞ } ¼ 2 ›t ›s ›s

ð2:35Þ

Enrichment of Isotopes

67

where 1;a21

ð2:25Þ

l ; 2h

ð2:36Þ

2P 1L

ð2:37Þ

and

C;

in which P is the production rate at the product-end, and L the total interstage flow rate at stage s. L is independent of s at all times in a square cascade and of t during the total reflux operation. When the mole fraction of the target isotope is small compared to unity as in most of the isotope separation processes, Equation 2.35 becomes linear:

l

›N ›2 N ›N ¼ 2 1ð1 þ CÞ ›t ›s ›s2

ð2:38Þ

The solution of Equation 2.38 is in the form N 2 N0 ¼ 1 2 A1 e2B1 t 2 A2 e2B2 t 2 · · · N1 2 N0

ð2:39Þ

where N1 ¼ e1n is the overall separation at steady state, t ; t=ln2 is the reduced time, or the time measured in the units of average process time ðlnÞ divided by the number of stages ðnÞ; and ðN 2 N0 Þ=ðN1 2 N0 Þ represents the fractional equilibrium attainment. Cohen14 tabulated A1 and B1 for the cases of K=ln ¼ H=hLN ¼ 0.1 (0.1) 0.5 and 1n between 0.1 (0.1) 1.2, where K ; 2H=L in which H is the product-refluxer holdup in moles. Wieck and Ishida73 extended the solution of Equation 2.38 to include the second term of the solution, Equation 2.39.

B. RATE OF ATTAINMENT OF S TEADY-STATE P ROFILE VS. H OLDUPS As a more direct application of Equation 2.39, calculated reduced time, t0:95 ; the time required for the achievement of 95% of the steady-state enrichment has been plotted in Figure 2.8. It clearly shows an increasing trend of the transient time with increasing holdup in the product refluxer, K=ln; and the overall separation, 1n: The actual time it takes is ln2 t0:95 : Figure 2.9 illustrates an approach of the overall separation toward a steady state in a laboratory scale experiment74 on an exchange of nitrogen isotopes between liquid N2O3 – N2O4 mixture and their vapor phase under pressured conditions. The HETP for this system is 1.07 cm, while for the Nitrox process under comparable conditions70,71 (with 10 M HNO3 at 1.6 ml cm22 min through a 2.5 cm dia £ 95 cm long column packed with Helipak #3013) is HETP ¼ 2.8 cm.

V. EMPIRICAL DETERMINATION OF HETP AND SEPARATION FACTOR a A. BY U SE OF A NALYTIC S OLUTION OF M ATERIAL B ALANCE E QUATION UNDER T RANSIENT C ONDITION The method is an application of Equation 2.39 assuming first that the approximation up to the first transient term, A1 e2B1 t ; is sufficient. A plot of ln½ðN1 2 NÞ=ðN1 2 N0 Þ against t would be a straight line whose intercept and slope are ln A1 and 2B1 ; respectively. Both A1 and B1 are functions of 1 and n. This information, combined with experimental data on 1nð¼ ln N1 Þ leads to the first approximation for 1 and n. The second transient term of Equation 2.39 may then be added

68

Isotope Effects in Chemistry and Biology 10 9 8 7 6

l = 0.9 kn

5

t 0.95

4 0.5 0.4 0.3 0.2

3

2

0.1 0.0

1

0

1.0 en

2.0

FIGURE 2.8 The 95% equilibrium-attainment time, t0:95 : See the text for the notations; K=ln ¼ Ratio of product refluxer holdup and total separative stage holdup, 1n ¼ Overall separation of the cascade. (Reprinted with permission from Ref. 73. Courtesy of Marcel Dekker Inc.)

Overall Separation, S(t)

4.0

3.0

2.0

1.0

0

24

48

72 96 Time (hours)

120

144

FIGURE 2.9 Overall separation as a function of time in a laboratory experiment under the following conditions. Points up to 96 h were taken under total reflux. Points thereafter were observed at the product withdrawal rate, P ¼ 0.207 m mol N/min. (Reprinted from Ref. 74 by courtesy of Marcel Dekker Inc.) Exchange system: Exchange of 15N and 14N between liquid N2O3 – N2O4 mixture and their vapor phase under pressured condition. 15 NOðgÞ þ 14 N2 O3 ð‘Þ ¼ 14 NOðgÞ þ 15 N14 NO3 ð‘Þ Liquid composition: A mixture of the major component, N2O3(‘), and the minor component, N2O4(‘). Gaseous component: major component: NO(g). Minor components: NO, N2O3, NO2, N2O4. Product-end reflux: reduction of NOx by SO2(g), similar to the reflux for the Nitrox process. Other operating conditions: T ¼ þ 15.08C, total pressure ¼ 4.08 atm, liquid flow rate ¼ 0.48 ml/cm2 min, liquid molar flow rate ¼ 17.4 mmol Nitrogen per min, column packing ¼ Podbielniak SS Helipak No. 3013. Parameters obtained: aeffective ¼ 1:030; HETP ¼ 1.07. Minimum required parameters: nmin ¼ 345; min molar flow rate ¼ 9.31 mmol N/cm2 min, min column height ¼ 369 cm, min column cross-section ¼ 24.8 cm2.

Enrichment of Isotopes

69

to ðN1 2 NÞ=ðN1 2 N0 Þ to correct, especially, for possible deviations from a straight line for small t. The natural logarithm of the resulting quantity would then be plotted against t, and a second approximation for 1 and n would be obtained accordingly. Then, the HETP is equal to the total height of all separation stages divided by the number of stages, n.

B. FROM G RAPHICAL S OLUTION OF M ATERIAL B ALANCE E QUATION UNDER THE C ONDITION OF Z ERO T IME -D EPENDENCE AT A LL S TAGES Another method for emipirical determination of a and HETP that has been used by many researchers in laboratory scales is based on a formula obtained by integrating Equation 2.38 over the stages, s, under the condition of steady state, that is, when ›N=›t ¼ 0: The method does not rely on the tables of coefficients such as these14,73 for the parameters A1 ; A2 ; B1 , and B2 (cf. Equation 2.39). The integration yields P 1 L ln n¼ P P 1þ 1þ L L1 2 P L1 Sp 12

ð2:40Þ

where n ¼ number of theoretical plates, P ¼ product withdrawal rate (moles of desired isotope per unit time), L ¼ total interstage flow rate of the ‘ and g streams (moles per unit time), N0 ¼ mole fraction of the target isotope in the feed, Sp ¼ Np =N0 ¼ overall separation when the production rate is P moles per unit time. Set P=L1 ; g

ð2:41Þ

and rewrite Equation 2.40. Then, 2

13

0

6 B 1 þ g C7 1 C7 ¼ B exp6 4n1@ P A5 1þg 12 2g L Sp and en1 ¼ S0 ¼ the separation at total reflux. Thus, 2

S0

1þg P 12 L

¼

1þg 2g Sp

ð2:42Þ

Given the experimental values of P, L, S0 ; and Sp ; each side of Equation 2.42 is a function of g. Let the left-hand side of Equation 2.42 be FðgÞ; and let the right-hand side be GðgÞ: Then, the value of g that satisfies Equation 2.42 is obtained as an intersect of the plot of FðgÞ vs. g and the plot of GðgÞ vs. g. The corresponding 1 is equal to P=gL; and n is obtained from S0 ¼ ð1 þ 1Þn :

VI. MISCELLANEOUS OTHER CONSIDERATIONS The single stage separation factor as a function of temperature, pressure, etc. and the HETP are not the only factors that one should take into consideration for design of an isotope enrichment plant and evaluation of experimental results. There are numerous “other” considerations, which depend on the particular isotope separation in a particular production scale. In the following, four examples will be discussed to illustrate varieties of “other” consideration.

70

Isotope Effects in Chemistry and Biology

A. POSSIBLE N EEDS OF C HEMICAL WASTE D ISPOSAL As illustrated in Table 2.3, the refluxer of the Nitrox process that produces 100 kg of 99.9% enriched 15N per year (18.26 mol 15N per day) needs 156,527 mol of sulfur dioxide per day and produces the equivalent number of moles of sulfuric acid per day. This sulfuric acid is produced at about 10 M and saturated with sulfur dioxide and it is not marketable as is. It must be treated as an industrial waste: a large-scale Nitrox plant must be run either with a large waste-disposal cost or with a sulfur-resource recycling system attached to it.

B. POSSIBILITY OF FAILURE TO ACHIEVE A H IGH TARGET E NRICHMENT One of the first things to suspect when one becomes aware that an expected target enrichment is not being attained after a prolonged total reflux period is a possible leak around the high-enrichment end of the plant such as the product refluxer, as previously explained. However, under special circumstances, there could be other more fundamental reasons for an apparent difficulty in reaching a design enrichment level. For example, the cryogenic distillation of nitric oxide cannot attain a high 15N enrichment by a single-pass distillation process, because nitrogen and oxygen isotopes in NO do not undergo exchanges at cryogenic temperatures, although they do at ambient temperatures. At ambient temperatures, the isotopes 14N, 15N, 16O, 17O, and 18O are distributed in accordance with the completely random distribution; when the isotopes are randomly distributed, i.e., when the distribution of isotopomers is in the classical limit of the Boltzmann distribution, the mole fractions of the isotopomers are the terms of the binomial expansion of the equality such as ½ð1 2 xÞ þ x ½ð1 2 y1 Þ þ y1 þ y2 ¼ 1

ð2:43Þ

where x, y1 ; y2 are the atom fractions of 15N, 17O, and 18O, respectively, (cf. Refs. 43,76; and Table 2.4). The isotopes are locked in this distribution in the feed material, which is cooled and distilled without further isotope exchange. As the distillation continues, the less volatile isotopomers become enriched near the boiler (the product-end refluxer) and the more volatile isotopomers get enriched near the top of the column in accordance with the relative volatilities (Table 2.4). The separation factors here are approximately equal to the vapor presure ratios.1

TABLE 2.4 Mole Fraction of the Isotopomers of Nitric Oxide at Room Temperature and Separation Factors of Distillation at 121K Separation Factor Isotopic NO 14–16 14–17 14–18 15–16 15–17 15–18 a

Mole Fractiona

At Natural Abundance

McInteerb

Bigeleisenc

ð1 2 xÞð1 2 y1 2 y2 Þ ð1 2 xÞy1 ð1 2 xÞy2 xð1 2 y1 2 y2 Þ xy1 xy2

0.99390 0.00037 0.00203 0.00369 0.00000137 0.00000755

1.000 1.019 1.027 1.037 1.046 1.064

1.000 1.019 1.028 1.040 1.049 1.069

x ¼ Natural abundance of 15N ¼ 0.37%; y1 ¼ Natural abundance of 17O ¼ 0.037%; y2 ¼ Natural abundance of O ¼ 0.204%; For the mole fraction formula, see Equation 2.43. b Fitted to column operation data at 121K on the basis of an assumption of equal difference in ða 2 1Þ for the oxygen series Ref. 39. c From measurements of isotopic vapor pressures at 120K. Ref. 72. 18

Enrichment of Isotopes

71 Product end

Feed point for Column 2

100 90 80 14–16

Mole Percent

70 60

15–16

50 40

14–18

30 15–18

20 10

14–17

0

15–17

0

100 200 Stage number

300

FIGURE 2.10 Isotopomer distribution profile along the second column of a two-section tapered cascade for the cryogenic distillation of nitric oxide at 121K under 1 atm. Composition of the feed NO ¼ Random distribution as in Table 2.4. (Reprinted with permission from Ref. 43. Copyright (1965) American Chemical Society).

However, as illustrated in Figure 2.10 (Ref. 43), the mole fractions of the isotopomers that were present in the feed in low concentrations, e.g., xy2 ¼ 0:00000755 for 15 –18, cannot build up a high concentration profile near the refluxer because, having started with low abundance, the buildup of 15– 18 is blocked by the isotopomers such as 14– 18 that are available in abundance. A compromise solution to circumvent the problem is (i) withdrawing the enriched product in a manner as if it is the final product, warming it up to the ambient temperature outside the cyrogenic column, accumulating it, and using it as the feed for a separate distillation column, or (ii) running the distillation column with an isotope exchange catalyst.

C. POSSIBLE E XPLOSION OF W ORKING M ATERIAL For example, in the cryogenic distillation of nitric oxide, solid deposits may accumulate in the parts of the distillation column causing clog-up of the gas flow, which may lead to an explosion. The solid deposit could be the higher oxides of nitrogen, all of which have higher melting points than nitric oxide. The most prominent culprit is the disproportionation of NO, 3NO ! NO2 þ N2O. The rate of this disproportionation does not significantly change with temperature.77 The accumulation of the higher oxides with time at ambient temperature is illustrated in Table 2.5. A careful purification of initial feed material, nitric oxide, as soon prior to the feeding as possible, is of primary importance. If the purified NO and enriched NO must be held in storage tanks, it is highly recommended that the storage pressure be kept as low as practical. The partial pressure of NO below 5 atm is advisable.78

72

Isotope Effects in Chemistry and Biology

TABLE 2.5 Calculated Accumulation of NO2 and N2O through Disproportionation of NO at 298K Mole Fractions of NO2 and N2O Time (h)

P0 5 50

P0 5 10

P0 5 5

P0 5 2

P0 5 1

24 48 72 240 30 days 1 year

0.0010 0.0019 0.0029 0.0094 0.0264 0.1689

0 0 0.0001 0.0004 0.0011 0.0124

0 0 0 0.0001 0.0003 0.0032

0 0 0 0 0 0.0001

0 0 0 0 0 0.0001

P0 ¼ Initial pressure of pure NO in atm.

D. CONSIDERATION OF S UPPLY FOR

THE

F EED

The feed material for an isotope separation plant is a chemical that is usually not available inexpensively and in abundance, and the desired isotope contained in the typical feed is usually a minor component of the feed. Therefore, a large supply of the feed material is needed especially for a large-scale plant. For instance, if the atom fraction of the target isotope in a feed is, say, 0.0001 (i.e., 0.01%), the absolutely minimum amount of the feed required to produce 1 mol of 100% enriched target isotope is 10,000 mol of the feed material, which is without any regard to the separation factors and other cascade realities such as the target isotopes lost with the deleted (waste) stream. A common solution of the problem is that of a parasitic isotope separation plant, parasitic in the sense of setting up the isotope plant adjacent to a chemicals plant that produces the feed chemical in a sufficiently large quantity. Use a product of an appropriate chemicals plant as the feed for the isotope plant, extract the desired isotope to produce an enriched isotope, and return the isotope-depleted chemical to the chemicals plant. The isotopically depleted chemical will pass as the ordinary chemical, and the amount not returned by the isotope plant will be a small fraction of the amount borrowed.

VII. ENRICHMENT BY NONSTEADY STATE PHENOMENA INVOLVING REVERSIBLE PROCESS A. ION E XCHANGE I SOTOPE S EPARATION The isotope effect in ion exchange process was first observed by Taylor and Urey. They found that the isotopic abundance ratios of lithium, potassium, and nitrogen were changed when lithium, potassium, and ammonium ions were eluted from inorganic ion exchanger “zeolite” packed in a stainless steel pipe.79 Isotope effects of lithium were intensively studied by Lee, Begun, and Drury by using organic cation exchange resin.80 – 82 They determined single stage separation factors of lithium isotopes between the phases of an aqueous solution and synthetic cation exchange resin at different experimental conditions of temperature, concentration of eluent, crosslinking and functional groups of resin, etc. Ion exchange was applied for nitrogen isotope separation by Spedding et al. They were successful in enrichment of 15N and obtained highly enriched 15N starting from natural abundance of 0.366% by ion exchange chromatography.83 Since then several works have been carried out on nitrogen isotope separation by cation exchange resins. The observed single stage separation factor of the NH3 – NHþ 4 system is in the range 1.020 , 1.025. The experimentally

Enrichment of Isotopes

73

determined isotope separation factor arises from the equilibrium constant of the following isotopic exchange reaction 14

NH4 – R þ 15 NH3 H2 O ¼ 15 NH4 – R þ 14 NH3 H2 O

ð2:R:1Þ

where R represents a cation exchange resin. In general, isotope exchange in the ion exchange resin system is expressed as M – R þ M0 L ¼ M0 – R þ ML

ð2:R:2Þ

where M and M0 represent heavy and light isotope ion, respectively, and L is the ligand used for the elution of the isotopic species. In many cases of isotopic metal ions, light isotopes are enriched in the resin phase, while the heavy isotopes are enriched in the complex species in the aqueous phase. Recently studied examples are copper, zinc, vanadium, and gadolinium.84 – 86 The cations of these elements make strong complexes with organic acids and chelating reagents, such as malate and EDTA, while they are in the form of hydrated ions in the resin phase. As expected from the theory of isotope effects, the heavy isotopes are fractionated in the complex species in the solution phase. In the case of lithium isotopes, M is Liþ ion, L is a water molecule or a group of water molecules and ML represents fully hydrated Liþ ion in the aqueous phase. Due to the dehydration in ion exchange resin, light isotope 6Li is enriched in the resin phase and 7Li is enriched in the aqueous phase. On the other hand in the case of the above-mentioned NH3 –NHþ 4 system, the direction of the fractionation of nitrogen isotopes becomes opposite; the heavy isotope is enriched in the resin phase due to the formation of NHþ 4 , for which the RPFR in the resin phase is greater than that of NH3 in the aqueous phase. This tendency of heavy isotope enrichment in the ion exchange resin also occurs among the heavy alkali metals such as Rb: heavy isotope 87Rb is fractionated in the cation exchange resin, while light isotope 85Rb is fractionated in the aqueous solution. Isotope effects in pure ion exchange are closely related to the hydration states of the ions. Isotope effects provide a tool for the study of inorganic solution chemistry. The separation factor of the system is defined by the following equation and the deviation from unity is expressed in terms of 1,

a ¼ ½M0 – R ½ML =½M – R ½M0 L ¼ 1 þ 1

ð2:44Þ

The separation factor is experimentally determined by measuring the isotopic abundance ratios in each phase as

a ¼ {½M0 =½M }re ={½M0 =½M }aq ¼ {½M =½M0 }aq ={½M =½M0 }re

ð2:45Þ

where subscripts re and aq represent the resin and aqueous solution, respectively. The isotopic fractionation in an ion exchange system is depicted in Figure 2.11. It should be noted that the pure AX + BY = BX + AY K>1

Isotope Exchange Equilibrium Ion Exchange Equilibrium

Isotopic Fractionation

MY

MX [A] [B] resin Resin Phase

FIGURE 2.11 Isotopic fractionation in ion exchange resin system.

<

[A] [B] sol. Solution Phase

74

Isotope Effects in Chemistry and Biology

ion exchange processes show only very small isotope effects which are usually negligible, while the observed isotope effect appearing in the ion exchange process is not so negligible and is attributed to the difference in the states of complex formation between the resin and the solution phase.

B. CHROMATOGRAPHIC I SOTOPE S EPARATION Ion exchange is a promising process for isotope separation. The single stage equilibrium shows isotope fractionation, but the value of the separation factor, 1, is so small that the isotope enrichment is not practically realized by a single stage process. The single stage effect of isotope fractionation is multiplied by the chromatographic operation that uses a column packed with ion exchange resin. There are two types of chromatographic systems applicable for the separation; elution chromatography and displacement chromatography (Figure 2.12). Elution chromatography is used as an analytical tool to separate elements and molecules of which chemical nature or adsorption properties are close to each other. A small amount of mixed sample involving different chemical species is injected into the column and the components eluted down using an eluent. During the elution, chemical components are detected as a series of separated peaks. Gas chromatography, ion chromatography, and high performance liquid chromatography are typical examples. To detect distinctive peaks, separation factors larger than approximately 1.1 are generally required, which means that the retention times of concerned species are different by 10% from each other. The separation factors of isotopes are much smaller than 1.1 except for the case of hydrogen: the separation factors of heavy elements are extremely small, i.e., la 2 1l being usually smaller than 0.001. Elution chromatography is not useful in such cases of small separation factors. In such cases displacement type chromatography is used for isotope separation. The adsorption band moves down through the resin bed packed in the ion exchange column, repeating adsorption at the front boundary and the elution at the rear boundary. The enrichment is accumulated at the band boundary region. When the boundary region is eluted out of the column, the separation factor is calculated according to the following equation by using the analytical data of the concentration profile and Eluent

Chromatogram

Eluent

Chromatogram

A,B,C A,B,C C B A C

B A

C C

B

A

A

(a)

concentration

(b)

B

concentration

FIGURE 2.12 Illustration of chromatographic models: (a) elution-type chromatography, (b) displacementtype chromatography.

Enrichment of Isotopes

75

the isotope abundance ratios of eluted band: 1¼a21¼

X

qi ðRi 2 R0 Þ=QR0 ð1 2 R0 Þ

ð2:46Þ

R ¼ ½M0 ={½M0 þ ½M } Where q is the amount of total isotope in the ith fraction, R is the atom fraction of the isotope concerned, Q is the total ion exchange capacity for the isotopic atoms, subscript i is the fraction number and 0 refers to the original feed. The isotope separation factor is also calculated by using isotopic ratios of one isotopic pair, r ¼ ½M0 =½M ; which is directly measured by the isotope ratiomass spectrometry: 1¼

X

qi ðri 2 r0 Þ=Qr0 ð1 þ ri Þ

ð2:47Þ

The isotope separation factor is defined for a pair of isotopes. However, Equation 2.47 indicates that the separation factor can be calculated as long as an isotopic ratio of any isotopic pairs is available. Thus the separation factors can be determined for any multiple isotopic mixture as long as the experimental data on the isotope ratio is available. Isotope enrichment by displacement chromatography was successfully realized in the case of NH3 – NHþ 4 exchange system. In the chromatographic process of cation exchange resin, an ammonia molecule in the form of NH3H2O (or NH4OH) in aqueous phase is adsorbed in Hþ type cation exchange resin packed in a column and forms an ammonium adsorption band. A sharp band boundary is created between H-type zone and the ammonium zone since the equilibrium constant of the following reaction is quite large ðK ¼ 109 Þ: H – R þ NH4 OH Y NH4 – R þ H2 O

ð2:R:3Þ

The ammonium adsorption band is eluted by alkali solutions such as NaOH. At the rear end of the ammonium band, another sharp band boundary is also created due to the large equilibrium constant ðK ¼ 105 Þ of the following reaction: NH4 – R þ NaOH ¼ Na – R þ NH4 OH

ð2:R:4Þ

The eluted ammonia is re-adsorbed at the front adsorption boundary according to above-mentioned Reaction (2.R.3). During migration of the ammonia molecule through the ammonium adsorption band, the isotopic exchange reaction takes place. The equilibrium constant of exchange reaction (2.R.1) is larger than unity, which leads to the fact that the heavy isotope 15N is fractionated in the resin phase. The fractionated 15N in the resin phase is accumulated at the rear band boundary in the course of the ammonium band elution. The system of displacement type chromatography of 15 N enrichment is illustrated in Figure 2.13.

C. NONSTEADY-STATE E NRICHMENT 1. Enrichment Profile The isotope effects in chemical reactions of ionic isotopomers such as adsorption, complex formation, oxidation– reduction reaction, etc. can be studied by ion exchange chromatography. Even in the cases of small separation factors of ln a < 1025 ; deviations in the isotopic abundance ratios are observed in the front and the rear boundary regions of an adsorption band in displacement chromatography. During the course of migration, isotopic accumulation takes place at the band boundary regions. The isotopic enrichment profile in the band changes with the migration time. The enrichment is presumed to proceed in the nonsteady state manner. Theoretical study on

76

Isotope Effects in Chemistry and Biology Eluent : NaOH

+

Na type

15N/14N

Rear +

Ratio

+

NH4 •R + NaOH ← → Na •R + NH3 + H2 O 15N 14 15

14

is concentrated.

N

N

+ 15 15 + 14 NH4 •R+ NH3 ← → Na4 •R+ NH3

NH4+ type Front NH3 + R•H+ ← → +

Length of migration band

Plateat

Na4+•R

14N

is concentrated.

+ H2O

H type R represents exchanger of resin. Effluent

FIGURE 2.13 Schematic model of 15N-enrichment by ion exchange.

the nonsteady state enrichment is important for the analysis of the isotope effects. The calculation of the separation factors has been mentioned in an earlier section. The present section discusses the isotopic accumulation at the band boundary and Height Equivalent to the Theoretical Plate (HETP) of the nonsteady-state enrichment process. The first systematic treatment of the mathematical analysis was made by Gleuckauf on the chromatographic separation process of isotopes.87 He derived the formulae for the isotopic profiles of elution and displacement chromatography and an empirical formula for the highly enriched 15N profile experimentally obtained by ion exchange chromatography. Since then, a number of research works have been reported on the analysis of chromatographic enrichment. To derive the fundamental equation for the enrichment process, usually the mass balance is taken for isotopes concerned. Therefore, the derived equation consists of the terms of concentrations of isotopes. This treatment faces difficulties in describing the enrichment profile when the system involves attainment of a high enrichment. Another approach was proposed: a mathematical treatment was developed which describes the changes in the isotopic ratios with time and position rather than the changes in the concentrations of individual isotopes.88 The treatment of individual isotopes is based on the concept that the enrichment phenomenon is described by using the chemical potential of each component or the concentration as the most fundamental property. In contrast the second approach was based on the concept that the enrichment proceeds based on the relative chemical potential between the isotopes. In this case all equations that describe isotope enrichment and the concentration of the isotope R are replaced by the isotopic ratio r. The net enrichment at any segment is expressed as the difference between the enrichment flows of 1r and the diffusion flow of DV ðdr=dxÞ and the net enrichment is equal to 1r0; where dr=dx at the starting point of migration, x ¼ 0; is zero. On the basis of this concept,

Enrichment of Isotopes

77

the fundamental equation becomes 1r 2 DV ðdr=dxÞ ¼ 1r0

ð2:48Þ

where DV is a constant including the terms of diffusion coefficient and velocity of migration. By integrating Equation 2.48 in the range 0 # x # L; the following equation is derived r 2 r0 ¼ ðrL 2 r0 Þexp kðx 2 LÞ k ¼ 1=DV

ð2:49Þ

where L is the migration distance of the front boundary, rL is the isotopic ratio at x ¼ L; and k is the slope coefficient indicating the steepness of the enrichment curve. Equation 2.49 clearly suggests that the plotting of lnðr 2 r0 Þ vs. ðx 2 LÞ in the front boundary region gives a straight line with a slope of k, in spite of the nonsteady-state occurring in the ion-exchange column. The developed enrichment profile is described by using isotopic atom fraction R. Since the isotopic ratio is expressed as r ¼ R=ð1 2 RÞ for the two-isotope system, Equation 2.49 is converted to R ¼ 1 2 ð1 2 R0 Þ={1 þ ðexp 1kR0 L 2 1Þexp kðx 2 LÞ}

ð2:50Þ

Equation 2.50 is an S-shaped function with a symmetric center at R ¼ ð1 þ R0 Þ=2: In the case where the enrichment is not so developed, the four-factor product 1kR0 L is sufficiently smaller than unity, i.e., 1kR0 L ! 1; and Equation 2.50 is simplified as R ¼ R0 þ 1kR0 Lð1 2 R0 Þexp kðx 2 LÞ

ð2:51Þ

The maximum enrichment obtained at the front boundary, RL ; is RL ¼ 1 2 ð1 2 R0 Þ=expð1kR0 LÞ

ð2:52Þ

The required length of migration L to obtain a product enrichment RL is calculated using Equation 2.52 and the known value of k or HETP mentioned later. 2. HETP Determination of HETP from the experimental results of a nonsteady-state enrichment was quite difficult. Reliable HETP was experimentally determined by a very long distance chromatography with a narrow band width, in which steady state isotope enrichment profile is developed. Such experiments need time and resources.89 A more convenient method applicable for the nonsteadystate isotope separation by displacement chromatography has been derived. The HETP of the separation medium is defined as a length, H, which gives an increase in isotopic enrichment by 1r: Hðdr=dxÞ ¼ 1r

ð2:53Þ

H ¼ 1=ðd ln r=dxÞ

ð2:54Þ

or

1. In the steady state enrichment, d ln r=dx is equal to the slope kS in the entire range of the enrichment curve. Then HETP is expressed by H ¼ 1=kS

ð2:55Þ

78

Isotope Effects in Chemistry and Biology

2. In the nonsteady-state enrichment, the steady state model is applicable only at the extreme front boundary at x ¼ L; Based on the above-mentioned definitions H ¼ 1=ðd ln r=dxÞx¼L ¼ 1rL =kðrL 2 r0 Þ

ð2:56Þ

H ¼ {1 þ R0 =ðexp 1kR0 L 2 1Þ}ð1=kÞ

ð2:57Þ

In the region where enrichment is not so high, i.e., 1kR0 L ! 1; the above equation is simplified to H ¼ 1=k þ 1=k2 L

ð2:58Þ

When L is small, the second term in the right hand side becomes major. At a very long migration, on the other hand, the second term becomes negligible and expressed by the first term which is in the same form as Equation 2.55.

D. ISOTOPE S EPARATION BY I ON E XCHANGE Ion exchange is a sophisticated separation technique applicable for both fine analysis, such as ion chromatography, and also large scale industries, such as metallurgy, chemical purification, water treatments, etc. If compared with similar techniques of solvent extraction, ion exchange would be recognized as a more compact separation system and application for smaller plant and process would be appropriate. This is due to the small HETP expected for an ion exchange process using small particle resins. Since isotope separation requires huge number of separation stages, small HETP is a very attractive characteristic. Furthermore, efforts have been made to improve in the kinetics. Very rapid ion exchange reactions are realized by using highly porous small particle resins. The ion exchange process could be reconsidered as a promising technique for isotope separation. 1. Boron Isotope Separation Natural boron consists of 10B (19.84%) and 11B (80.16%). Boron-10 has a large neutron absorption cross-section and is used as a neutron absorber or controller material of nuclear reactors. Application has been extended to medical uses. An enriched boron compound has been produced for neutron capture therapy for melanotic cancer and brain tumors. Low cost production of enriched 10 B will expand the application of boron isotopes in various fields, such as medical treatment and high technology industries. Boron isotope separation by ion exchange was first reported by Makishima et al.90 They used strong base anion exchange resin, Amberlite CG-400-I, 100– 200 mesh. Kakihana et al. developed a weak base resin system where boric acid solution is charged into the column and boric acid into the resin.91 The adsorbed boron is eluted by pure water eluent. The observed separation factors by strong base resin are around 1.02 and those observed by weak base are around 1.01. Isotope enrichment by ion exchange is based on the following reaction in which the following isotope fractionation takes place 10

11 10 2 BðOHÞ3 þ 11 BðOHÞ2 4 – R ¼ BðOHÞ3 þ BðOHÞ4 – R

ð2:R:5Þ

where – R represents the resin phase. The chemical form of boric acid B(OH)3 in the pH lower than 6 is trigonal planer, and that of B(OH)2 4 is tetrahedral in the range higher than pH 11. These forms correspond to the electronic configuration of sp2 (trigonal) and sp3 (tetrahedral). Due to

Enrichment of Isotopes

79 100

1.00 202 m 302 m 402 m 502 m 602 m 754 m

0.2

−200

−100

0 0

Distance x −l / cm (a) The isotopic accumulation profiles at different migration distances. The solid curve were calculated.

orig

B

1

B

B

x−l

B

10 10 ( 11 ( −( 11 (

0.4

Atomic Fraction of B-10

0.6

−300

10

0.8

202 m 302 m 402 m 502 m 620 m 754 m

0.1

0.01

0.001 −300

−200

−100

0

Distance x−l / cm (b) Plots of measured isotopic ratios at different migration distances vs. the inner band distance. The k's indicated are the slopes of line in cm−1.

FIGURE 2.14 Chromatographic enrichment of 10B. (a) Observed isotope abundance in atom fraction vs. distance from boundary, x 2 L; (b) plots by isotopic ratio, r 2 r0 vs. distance from boundary, x 2 L:

the drastic change in the molecular structures, large isotope effects are expected for the boric acid ion exchange system. Aida conducted a long-distance chromatography using porous weak base resin Diaion WA21.92 The isotope enrichment developed in the boundary region is presented in Figure 2.14(a), plotted in the isotopic atom fraction. The experimental work is an appropriate example to verify the theorem described in the earlier section for nonsteady-state enrichment by ion exchange. According to the theorem, Equation 2.51, the experimental data are converted to lnðr 2 r0 Þ and plotted in Figure 2.14(b). From the straight lines in Figure 2.14(b), the slope coefficient k can be calculated for each migration distance and HETP is obtained by Equation 2.57. The determined value of HETP is 1.85 ^ 0.2 mm in the whole range of migration distance. It should be noted that a constant HETP is obtained in a nonsteady-state enrichment, in despite of the migration length. 2. Nitrogen Isotope Separation As mentioned in an earlier section, Spedding was successful in the enrichment of 15N by ion exchange using cation exchange resin Dowex 50Wx12.83 Since then several works have been reported by Urgell et al.,93 Park et al.,94 Kruglov et al.,95 and Ohtsuka et al.96 Urgell et al. enriched 15 N from the natural abundance of 0.366 to 25% at the rear boundary of the adsorption band after the migration distance of 81 m. Ohwaki conducted experimental work on kinetic properties of isotope separation by ion exchange.97,98 He measured HETP (Height Equivalent to the Theoretical Plate) in the wide range of band velocity by using 1 m column packed with high porous strong acid type cation exchange resin TITECH-H2, cross linking 20%, 80– 180 m. The adsorption band velocity was varied from 6.3 £ 1023 cm/min (0.38 cm/h) to 12.5 cm/min (7.5 m/h). The separation factor, which is a thermodynamic property of isotopic equilibrium constant, was observed to be constant in the range of the band velocity. The value of HETP, which is the kinetic factor, was observed to be changeable

80

Isotope Effects in Chemistry and Biology

101

102

Reynolds number 103

104

HETP /cm

10−1

10−2

10−2

10−1 100 Band velocity UB /cm/min

101

Plots of HETP against band velocity and Reynolds number

FIGURE 2.15 Band velocity dependence of HETP in

15

N enrichment.

with a minimum value of 0.12 mm at the band velocity of 0.6 cm/min (36 cm/h), as seen in Figure 2.15. The velocity dependence of HETP is quite different in the lower velocity side and the higher velocity side of the minimum point at the velocity of 1 cm/min (60 cm/h). In the lower velocity region, HETP is inversely proportional to the band velocity, UB ; and expressed by H ¼ a=UB þ b

ð2:59Þ

where a and b are constants. In the higher band velocity region, on the other hand, HETP is proportional to the square root of the band velocity and expressed as H ¼ cðUB Þ1=2 þ d

ð2:60Þ

where c and d are constants. The observed flow rate dependence of HETP shows a very similar pattern to that of gas chromatography. The reason for the increase in HETP in the lower band velocity region is seen to correspond to the molecular diffusion of gas chromatography. However, molecular diffusion is not the case of liquid chromatography. Estimation of the Reynolds number of the ion exchange eluent suggests the flow pattern is changed from laminar flow to turbulent flow at both sides of the minimum HETP point. For the analysis of liquid chromatography, Giddings introduced a concept of the reduced plate height99 h ¼ H=dp

ð2:61Þ

v ¼ dp UB =Dm

ð2:62Þ

and the reduced velocity

where dp is the particle diameter, Dm is the diffusion coefficient of species in the mobile phase. Ohwaki calculated values of h and v from the work of Spedding, Kruglov, Park and plotted in

Enrichment of Isotopes

81

Reduced plate height h

102

101

100 100 102 Reduced velocity ν Plots of reduced plate height vs. reduced velocity

HETPs of gel-type resigns: ( ) Spedding et al.1: ( ) Park and Michales 2: ( ) Urgels et al.[15]. HETPs of macroreticular resin: (e) Karuglov et al. 3. HETPs of microreticular resins: (O) present work; ( ) our previous work 13 Open symbols, single column operations; solod symbols, multi-column operations

FIGURE 2.16 Plots of reduced HETP vs. reduced band velocity.

a figure (Figure 2.16) with his experimental values.98 The reported data were obtained by using different techniques under quite different conditions. In spite of such differences, the plotted points show a systematic tendency concerning the flow rate dependence of HETP. 3. Uranium Isotope Separation Isotope effects in uranium complex formation (or, ligand exchange reaction) and U(IV) – U(VI) redox reaction (or, electron exchange reaction) have been intensively studied by using cation exchange resin and anion exchange resin, respectively. Isotope effects in uranyl (UO2þ 2 ) ligand exchange reaction represented by the following reaction have been studied with various ligands, Cl2, acetate, lactate, glycolate, citrate, and malate100 235

UO2 – R þ 238 UO2 L ¼ 238 UO2 – R þ 235 UO2 L

ð2:R:6Þ

The isotope separation coefficient expressed by 1ð¼ a 2 1Þ; has been determined to be 1 ¼ 0:75 , 2:2 £ 1024 for carboxyl ligands; where a corresponds to the equilibrium constant of the above reaction. Since the separation coefficient 1 is, theoretically, inversely proportional to the square of isotopic mass, it was questionable whether it would be possible to attain a detectably significant separation coefficient due to the large mass of uranium. Therefore, it was rather surprising to see such values of separation coefficients of uranium. In addition the results were strange; the light isotope is enriched not in the resin phase but in the complex species in the solution phase. To elucidate the mechanism of this isotope effect, IR spectra of these uranyl complexes were examined and the IR spectra clearly explained that the UyO bond is loosened by the complex formation with carboxyl ligand; the asymmetric OyUyO stretching frequency is shifted to a lower frequency by the complex formation.

82

Isotope Effects in Chemistry and Biology

Uranium isotope separation based on the U(IV) – U(VI) exchange reaction was studied for the industrial purpose of enriched uranium production. The ion exchange uranium enrichment is regarded as a nuclear proliferation-resistant process due to the fact that the production is limited to low enrichment uranium for commercial reactor use. Since the separation medium consists of water and organic materials, i.e., light atoms of H, C, O, these atoms become the neutron moderator and, therefore, the separation plant itself reaches criticality and the chain reaction of nuclear fission begins in the plant when the plant should reach a production of highly enriched uranium for military use. In a laboratory-scale work, 3% enriched uranium was obtained by a two step enrichment; the first step being for enrichment from the natural 0.71 to 1.7%, and the second step being for enrichment from 1.7 to 3%.101 In the project, efforts were made to develop a high speed, durable anion exchange resin usable under industrial process conditions. The thus developed resin has shown outstanding characteristics. The resin may be useful for other isotope separation by anion exchange chromatography. The basis of this redox exchange process is in the isotope fractionation in the following reaction 238

UO2 Cln – R þ 235 U4þ ¼ 235 UO2 Cln – R þ 238 U4þ

ð2:R:7Þ

in which uranium-235 is enriched in the resin phase. The separation coefficient of the reaction is 1 ¼ 0:0006 , 0:001; which is much larger than the values of ligand exchange systems. Uranium enrichment by anion exchange was surely successful, but the scientific reason as to why light isotope 235U is enriched in uranyl species, has been unexplained for a long time. To elucidate the mechanism of the enrichment, experimental work was conducted on uranium enrichment by anion exchange redox chromatography based on U(IV)– U(VI) exchange isotope effects by using different isotopes of 232U, 233U, 234U, 235U, 236U, and 238U. After a long distance chromatography, isotope abundances of all isotopes involved were analyzed and correlation among the isotopes was examined. The results were unbelievable. The even-mass isotopes show the normal mass dependence, that is, the separation is proportional to the mass difference in the isotope pair, while the odd-mass isotopes, e.g., 233U and 235U, show a deviation from the regularity exhibited among the even-mass isotopes. A kind of odd– even effect was observed.102 The phenomena was elucidated as the nuclear size effects on the chemical isotope fractionation which was formulated by Bigeleisen103 and Fujii,102 independently and practically simultaneously, and called “the nuclear size and shape effects” on chemical isotope fractionation. The nuclear mass and nuclear shape of the isotopes of extremely large, multinucleon isotopes such as those of uranium are not isotopeindependent, which leads to isotope-dependent interactions of the nuclear charge with the extranuclear electrons that have relatively high densities in the neighborhood of the nucleus.1 This is a new type of nonBorn –Oppenheimer isotope effect and an example of contribution of isotope enrichment subfield to the knowledge and understanding of isotope effects. The interests of many isotope separation researchers are being directed toward discoveries of similar isotope dependences in other heavy elements.

VIII. CONCLUDING REMARKS A variety of methods has been developed for separating isotopes. Some methods such as gas centrifuge are directly based on mass differences while others such as thermal diffusion, chemical exchange, distillation and photochemical methods result from less obvious mass-dependent changes in atomic and molecular properties. Although the principles underlying isotope separation are relatively few, a large variety of processes has been considered using a principle. All individual processes have some merit, but evaluation depends on several factors; whether isotopes of light-weight (e.g., nitrogen), heavy-weight elements (e.g., uranium), or elements in between, are to be separated; and whether the

Enrichment of Isotopes

83

quantities needed are grams (as in research) or tons (as for use in power reactors). The choice depends on the properties of the element, the degree of separation needed, and the scale and continuity of the demand. Even for a given isotope, there is no one best method. For large-scale applications availability of feed materials, capital costs, continued stability of demands, and power requirements are the overriding considerations, while for laboratory needs simplicity of operation and versatility of equipment and process may be primary. Ready availability of enriched isotopes will benefit all fields of science and technology. The high costs of enriched isotopes are partly due to the high unit costs of construction and operation of enrichment plant, and partly due to reluctance of the isotope separator of venturing into a new plant in the face of uncertainty of the demands for their product. The principle of supply and demand is operating in the isotope enrichment field as in other fields. A worldwide assurance for a level of demand by the users would help in reducing the cost. Because the enrichment of a partially enriched material, e.g., from 50% enrichment to near 100%, is surprisingly inexpensive compared to the cost of enrichment starting with a natural abundance material (cf.: Section III), a scheme of enriched material-recycling networks in which the “wastes” of chemical syntheses for labeled compounds and final wastes of other uses are collected and chemically decomposed to yield a feed material for an isotope separation plant would significantly reduce the net cost of isotopes.

ACKNOWLEDGMENTS One of the authors, TI, would like to express his deepest acknowledgement to his mentors, Professors Manson Benedict, Jacob Bigeleisen, and William Spindel. He would also like to express his appreciation to Prof. Jacob Bigeleisen, who has read this manuscript and has given many useful suggestions.

REFERENCES 1 Bigeleisen, J., Theoretical basis of isotope effects: an historical perspective, In Isotope Effects in Chemistry and Biology, Kohen, A. and Limbach, H.-H., Eds., Marcel Dekker, New York, 2004. 2 Bigeleisen, J. and Mayer, M. G., Calculation of equilibrium constant for isotope exchange reaction, J. Chem. Phys., 15, 261– 267, 1947. 3 London, H., Ed., Separation of Isotopes, George Newnes Ltd, London, 1961. 4 Kistemaker, J., Bigeleisen, J., and Nier, A. O. C., Eds., Proceedings of The International Symposium on Isotope Separation, North-Holland Publishing, Amsterdam, 1958. 5 Bigeleisen, J., Chemistry of isotopes, Science, 147, 463– 471, 1965. 6 Bigeleisen, J., Isotope separation practice, In Isotope Effects in Chemical Processes, Adv. Chem. Ser., 89, Gould, R. F., Ed., American Chemical Society, pp. 1 – 24, 1969. 7 Spindel, W., Isotope separation processes, In Isotopes and Chemical Principles, ACS Symp. Ser., 11, Rock, P. A., Ed., American Chemical Society, pp. 77 – 100, 1975. 8 Janes, G. S., Forsen, H. K., and Levy, R. H., Research and development prospects for the atomic uranium laser isotope separation process, In Developments in Uranium Enrichment, Benedict, M.,Ed., AIChE Symposium Series 169, pp. 62 – 68; Benedict, M., Pigford, T., and Levi, H. W., Nuclear Chemical Engineering, 2nd ed., McGraw-Hill, New York, 1981. 9 Villani, S., Isotope Separation, American Nuclear Society, Hinsdale, IL, 1979. 10 Spindel, W. and Ishida, T., Isotope separation, In Encyclopedia of Physics, 2nd ed., Lerner, R. G. and Trigg, G. L., Eds., VCH Publication, New York, pp. 573– 579, 1991. 11 Murphy, G. M., Ed., Production of heavy water, Nuclear Science and Energy Series, III, 4F, McGrawHill, New York, 1955. 12 Benedict, M., Survey of heavy water production processes, In Proceedings of International Conference on the Peaceful Uses of Atomic Energy, Geneva, United Nations, Vol. 8, pp. 377– 405, 1956. 13 Rae, H. K., Ed., Separation of hydrogen isotopes, ACS Symp. Ser., American Chemical Society, 1978.

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14 Cohen, K., The Theory of Isotope Separation as Applied to the Large-Scale Production of 235U, McGraw-Hill, New York, 1951. 15 Benedict, M., Berman, A. S., Bigeleisen, J., Powell, J. E., Shacter, J., and Vanstrum, P. R., Report of Uranium Isotope Separation Review AD HOC Committee, ORO-694, US AEC, 1972. 16 Benedict, M., Ed., Developments in Uranium Enrichment, AIChE Symposium Series No. 169, Vol. 73, American Institute of Chemical Engineering, New York, 1977. 17 Uranium Enrichment: Topics in Applied Physics Series, Vol. 35, Villani, S., Ed., Springer, Berlin, 1979. 18 Benedict, M., Uranium enrichment — past, present and future, In Diamond Jubilee Historical/Review Volume, AIChE Symposium Series No. 235, Vol. 80, Resen, L., Ed., American Institute of Chemical Engineering, New York, pp. 149–156, 1983. 19 Bigeleisen, B. and Ishida, T., Application of finite orthogonals to the thermal functions of harmonic oscillators. I. Reduced partition function of isotopic molecules, J. Chem. Phys., 48, 1311– 1330, 1968; Ishida, T. and Spindel, W., Theoretical analysis of chemical isotope fractionation by orthogonal polynomial methods, In Isotope Effects in Chemical Processes, Adv. Chem. Ser, Vol. 89, American Chemical Society, pp. 192– 247, 1969. 20 Bebbington, W. P. and Thayer, V. R., Concentration of Heavy Water by Distillation and Electrolysis, Proceedings of International Conference on the Peaceful Uses of Atomic Energy, Geneva, United Nations, Vol. 4, pp. 527– 533, 1958. 21 Butler, J. P., Rolston, J. H., and Stevens, W. H., Novel catalysts for isotopic exchange between hydrogen and liquid water, In Separation of Hydrogen Isotopes, ACS Symp. Ser., Vol. 68, Rae, H. K., Ed., American Chemical Society, pp. 93 – 109, 1978. 22 Hammerli, M., Stevens, H., and Butler, J. P., Combined electrolysis catalytic exchange (CECE) process for hydrogen isotope separation, In Separation of Hydrogen Isotopes, ACS Symp. Ser., Vol. 68, Rae, H. K., Ed., American Chemical Society, pp. 119–125, 1978. 23 Pautrot, P. H. and Damiani, M., Operating experience with the tritium and hydrogen extraction plant at the Laue – Langevin institute, In Separation of Hydrogen Isotopes, ACS Symp. Ser., Vol. 68, Rae, H. K., Ed., American Chemical Society, pp. 163– 170, 1978. 24 Taylor, T. I. and Urey, H. C., The electrolytic and chemical exchange methods for the separation of the lithium isotopes, J. Chem. Phys., 5, 597– 598, 1937. 25 Johnston, H. L. and Hutchison, A., Efficiency of the electrolytic separation of lithium isotopes, J. Chem. Phys., 8, 869– 877, 1940. 26 Klemm, A., Hintenberger, M., and Hoernes, P., The enrichment of the heavy isotopes of Li and K by electrolytic ion migration in fused chloride, Z. Naturforsch., 2a, 245– 249, 1947. 27 Klemm, A., Concentration of 6Li by electrolytic transference in molten LiCl, Z. Naturforsch., 6a, 512, 1951. 28 Klemm, A., Ionenwanderung in Geschmoltzenen Salzen: Dauerversuche, Temperaturund Konzentrations-Effekt, In Proceedings of The International Symposium on Isotope Separation, Kistemaker, J., Bigeleisen, J., and Nier, A. O. C., Eds., North-Holland Publishing, Amsterdam, pp. 275– 282, 1958. 29 Perret, L., Rozand, L., and Saito, E., Investigation of the Separation Coefficient of Certain Processes Involving the Isotopes of Lithium, Proceedings of International Conference on the Peaceful Uses of Atomic Energy, Geneva, United Nations, Vol. 4, pp. 595– 601, 1958. 30 Palko, A. A., Drury, J. S., and Begun, G. M., Lithium isotope separation factors of some two-phase equilibrium systems, J. Chem. Phys., 64, 1828– 1835, 1976. 31 Singh, G., Hall, J. C., and Rock, P. A., Thermodynamics of lithium isotope exchange reactions. II. Electrochemical investigations in diglyme and propylene carbonate, J. Chem. Phys., 56, 1855– 1862, 1972. 32 Singh, G. and Rock, P. A., Thermodynamics of lithium-isotope-exchange reactions. III. Electrochemical studies of exchange between isotopic metals and aqueous ions, J. Chem. Phys., 57, 5556– 5561, 1972. 33 Edmunds, A. O. and Loveless, F. C., Production of boron-10 and other stable isotopes, Proceedings of International Conference on the Peaceful Uses of Atomic Energy, Geneva, United Nations, Vol. 4, 1958, pp. 576– 584. 34 Miller, G. T., Kralik, R. J., Belmore, E. A., and Drury, J. S., Production of boron-10, Proceedings of International Conference on the Peaceful Uses of Atomic Energy, Geneva, United Nations, Vol. 4, pp. 585– 594, 1958.

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35 Kilpatrick, M., Separation of boron isotopes, National Nuclear Energy Series, Vol. III-5, McGraw-Hill, New York, 1952. 36 McIlroy, R. W. and Pummery, F. C. W., A pilot plant for the production of highly enriched boron isotopes, In Proceedings of The International Symposium on Isotope Separation, Kistemaker, J., Bigeleisen, J., and Nier, A. O. C., Eds., North-Holland Publishing, Amsterdam, pp. 178– 198, 1958. 37 Palko, A. A. and Drury, J. S., Separation of boron isotopes. VIII. BF3 addition compounds of dimethyl ether, dimethyl sulfide, dimethyl selenide, dimethyl telluride, dibutyl ether, and ethyl formate, J. Chem. Phys., 47, 2561– 2566, 1967. 38 Palko, A. A. and Drury, J. S., The chemical fractionation of boron isotopes, In Isotope Effects in Chemical Processes, Adv. Chem. Ser, 89, Gould, R. F., Ed., American Chemical Society, pp. 40– 56, 1969. 39 Nettley, P. T., Cartwright, D. K., and Kronberger, H., The production of 10B by low-temperature distillation of boron trifluoride, In Proceedings of The International Symposium on Isotope Separation, Kistemaker, J., Bigeleisen, J., and Nier, A. O. C., Eds., North-Holland Publishing, Amsterdam, pp. 385– 407, 1958. 40 London, H., Isotope separation by fractional distillation, In Proceedings of The International Symposium on Isotope Separation, Kistemaker, J., Bigeleisen, J., and Nier, A. O. C., Eds., North-Holland Publishing, Amsterdam, pp. 319– 335, 1958. 41 Hutchison, C. A., Stewart, D. W., and Urey, H. C., The concentration of 13C, J. Chem. Phys., 8, 532– 537, 1940. 42 Ghate, M. R. and Taylor, T. I., Production of 13C by chemical exchange reaction between amine carbamate and carbon dioxide in a solvent – carrier system, Sep. Sci., 10, 547– 569, 1975. 43 McInteer, B. B. and Potter, R. M., Nitric oxide distillation plant for isotope separation, Ind. Eng. Chem. Proc. Des. Level, 4, 35 – 42, 1965. 44 Thode, H. G. and Urey, H. C., The further concentration of 15N, J. Chem. Phys., 7, 34 – 39, 1939. 45 Taylor, T. I. and Spindel, W., Preparation of highly enriched nitrogen-15 by chemical exchange of NO with HNO3, In Proceedings of The International Symposium on Isotope Separation, Kistemaker, J., Bigeleisen, J., and Nier, A. O. C., Eds., North-Holland Publishing, Amsterdam, pp. 158– 164, 1958. 46 Hirschfelder, J. O., Curtiss, C. F., and Bird, R. B., Molecular Theory of Gases and Liquids, Wiley, New York, 1954. 47 Clusius, K., Das Trennrohs IV. Reindarstellung des Schweren Stickstoffs 15N, Helv. Chim. Acta, 33, 2134– 2152, 1950. 48 Dostrovsky, I. and Raviv, A., Separation of the heavy isotopes of oxygen by distillation, In Proceedings of The International Symposium on Isotope Separation, Kistemaker, J., Bigeleisen, J., and Nier, A. O. C., Eds., North-Holland Publishing, Amsterdam, pp. 336– 349, 1958. 49 Dostrovsky, I., Production and distribution of the heavy isotopes of oxygen, Proceedings of International Conference on the Peaceful Uses of Atomic Energy, Geneva, United Nations, Vol. 4, pp. 605–607, 1958. 50 Lewis, G. N. and Cornish, R. E., Separation of the isotopic forms of water by fractional distillation, J. Am. Chem. Soc., 55, 2616– 2617, 1933. 51 Clusius, K. and Dickel, G., Neues Verfahren zur Gasentmischung und Isotopentrennung, Naturwiss, 26, 546, 1938. 52 Clusius, K., Expose´ Ge´ne´ral. Diffusion thermique, J. Chim. Phys., 60, 163– 169, 1963. 53 Reid, A. F. and Urey, H. C., The use of the exchange between carbon dioxide, carbonic acid, bicarbonate ion, and water for isotopic concentration, J. Chem. Phys., 11, 403– 412, 1943. 54 Kirschenbaum, I., Physical Properties of Heavy Water, McGraw Hill, New York, 1951. 55 Stewart, D. W. and Cohen, K., The further concentration of 34S, J. Chem. Phys., 8, 904– 907, 1940. 56 Andreev, B. M. and Polevoi, A. S., Separation of sulfur isotopes by physicochemical methods, Russ. Chem. Rev., 52, 213– 228, 1983. 57 Koch, J., Ed., Electromagnetic Isotope Separators and Application of Magnetically Enriched Isotope, Interscience, New York, 1958. 58 Olander, D. R., Technical Basis of Gas Centrifuge: Advances in Nuclear Science and Technology, Vol. 6, Academic Press, New York, pp. 105– 174, 1972. 59 Lewis, G. N. and MacDonald, R. T., Concentration of 2H isotope, J. Chem. Phys., 1, 341– 344, 1933. 60 Zare, R. N., Laser separation of isotopes, Sci. Am., 236(2), 86 –98, 1977. 61 Letokhov, V. S. and Moore, C. B., Laser isotope separation, In Chemical and Biochemical Applications of Lasers, Vol. III, Moore, C. B., Ed., Academic Press, New York, pp. 1 – 165, 1977.

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62 Paisner, J. A. and Solarz, R. W., Resonance photoionization spectroscopy, In Laser Spectroscopy and Its Applications, Radziemski, L. J., Solarz, R. W., and Paisner, J. A., Eds., Marcel Dekker, New York, pp. 175– 260, 1987. 63 Lyman, J. L., Laser-induced molecular dissociation: applications in isotope separation and related processes, In Laser Spectroscopy and Its Applications, Radziemski, L. J., Solarz, R. W., and Paisner, J. A., Eds., Marcel Dekker, New York, pp. 417– 505, 1987. 64 Becker, F. S. and Kompa, K. L., The practical and physical aspect of uranium isotope separation with lasers, Nucl. Technol., 58, 329– 353, 1982. 65 Paisner, J. A., Atomic vapor laser isotope separation, Appl. Phys. B, 46, 253– 260, 1988. 66 Olander, D. R., The gas centrifuge, Sci. Am., 239(8), 37 – 43, 1978. 67 Whitley, S., Reviews of the gas centrifuge until 1962. Part I: principles of separation physics, Rev. Mod. Phys., 56, 41 – 66, 1984. 68 Whitley, S., Review of the gas centrifuge until 1962. Part II: principles of high-speed rotation, Rev. Mod. Phys., 56, 67 – 97, 1984. 69 Love, L. O., Electromagnetic separation of isotopes at oak ridge, Science, 182, 343– 352, 1973. 70 Spindel, W. and Taylor, T. I., Separation of nitrogen isotopes by chemical exchange between NO and HNO3, J. Chem. Phys., 23, 981– 982, 1955. 71 Spindel, W. and Taylor, T. I., Preparation of 99.8% nitrogen-15 by chemical exchange, J. Chem. Phys., 24, 626– 627, 1956. 72 Office of Chemistry and Chemical Technology. Separated Isotopes: Vital Tools for Science and Medicine, National Research Council, Washington, DC, 1982. 73 Wieck, W. and Ishida, T., Kinetics of a square cascade of close-separation stages under total reflux, Sep. Sci., 12, 587– 605, 1977. 74 Principe, P., Spindel, W., and Ishida, T., Nitrogen-15 fractionation by countercurrent exchange between Liquid N2O3 –N2O4 mixture and their vapor phases under pressured condition, Sep. Sci. Technol., 20, 489– 511, 1985. 75 Monse, E. U., Spindel, W., Kauder, L. N., and Taylor, T. I., Enrichment of nitrogen-15 by chemical exchange of NO with liquid N2O3, J. Chem. Phys., 32, 1557– 1566, 1960. 76 Eshelman, D. M., Torre, F. J., and Bigeleisen, J., Temperature dependences of the isotopic liquid– vapor fractionation factor for nitric oxide, J. Chem. Phys., 60, 420– 426, 1974. 77 Melia, T. P., Decomposition of nitric oxide at elevated pressures, J. Inorg. Nucl. Chem., 27, 95 – 98, 1965. 78 Tsukahara, H. and Ishida, T., Gas phase disproportionation of nitric oxide at elevated pressures, Free Radical Res., 37(2), 171– 177, 2003. 79 Taylor, T. I. and Urey, H. C., Fractionation of lithium and potassium isotopes by chemical exchange, J. Chem. Phys., 6, 429– 438, 1938. 80 Lee, D. A. and Begun, G. M., Enrichment of lithium isotopes by ion-exchange chromatography. I. The influence of the degree of crosslinking on the separation factor, J. Am. Chem. Soc., 81, 2332– 2335, 1959. 81 Lee, D. A., Enrichment of lithium isotopes by ion-exchange chromatography. The influence of exchange functional group on separation factor, J. Chem. Eng. Data, 6, 565– 566, 1961. 82 Lee, D. A. and Drury, J. S., The enrichment of lithium isotopes by ion exchange chromatography. The influence of eluent concentration on the separation factor, J. Inorg. Nucl. Chem., 27, 1405– 1407, 1965. 83 Spedding, F. H., Powell, J. E., and Svec, H. J., Laboratory method for separating nitrogen isotopes by ion exchange, J. Am. Chem. Soc., 77, 6125– 6132, 1955. 84 Ban, Y., Aida, M., Nomura, M., and Fujii, Y., Zinc isotope separation by ligand exchange chromatography using cation exchange resin, J. Ion Exch., 13, 46 – 51, 2002. 85 Ismail, I. M., Matin, MD. A., Nomura, M., Begum, S., Aida, M., and Fujii, Y., Isotope effects in Cu(II) ligand-exchange systems by ion exchange chromatography, J. Ion Exch., 13, 40 – 45, 2002. 86 Ismail, I. M., Fukami, A., Nomura, M., and Fujii, Y., Anomaly of 155Gd and 157Gd isotope effects in ligand exchange reactions observed by ion exchange chromatography, Anal. Chem., 72, 2841– 2845, 2000. 87 Glueckauf, E., Isotope separation by chromatographic methods, In Separation of Isotopes, London, H., Ed., George Newnes Ltd., London, pp. 209– 248, 1961. 88 Fujii, Y., Aida, M., and Okamoto, M., A Theoretical study of isotope separation by displacement chromatography, Sep. Sci. Technol., 20(5/6), 377–392, 1985. 89 Hagiwara, Z. and Takakura, Y., Enrichment of stable isotopes (III), enrichment of 6Li and 7Li by ion exchange column, J. Nucl. Sci. Technol., 6, 326– 332, 1969.

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90 Yoneda, Y., Uchijima, T., and Makishima, S., Separation of boron isotopes by ion exchange, J. Phys. Chem., 63, 2075, 1959. 91 Kakihana, H., Kotaka, M., Satoh, S., and Nomura, M., Bull. Chem. Soc. Jpn., 50, 158, 1977. 92 Aida, M., Fujii, Y., and Okamoto, M., Chromatographic enrichment of 10B by using weak-base anionexchange resin, Sep. Sci. Technol., 21(6/7), 643– 654, 1986. 93 Urgell, M. M., Iglesias, J., Casas, J., Saviron, J. M., and Quintanilla, M., Third U.N. International Conference on the Peaceful Uses of Atomic Energy, 1964, A/CONF28/P/491. 94 Park, W. K. and Michaels, E. D., Sep. Sci. Technol., 23, 1875, 1988. 95 Kruglov, A. V., Andreev, B. M., and Pojidaev, Y. E., Sep. Sci. Technol., 31, 471, 1996. 96 Ohtsuka, H., Ohwaki, M., Nomura, M., Okamoto, M., and Fujii, Y., J. Nucl. Sci. Technol., 32, 1001, 1995. 97 Ohwaki, M., Fujii, Y., Morita, K., and Takeda, K., Nitrogen isotope separation using porous microreticular cation-exchange resin, Sep. Sci. Technol., 33, 19 – 31, 1998. 98 Ohwaki, M., Fujii, Y., and Hasegawa, M., Flow-rate dependence of the height equivalent to a theoretical plate in nitrogen isotope separation by displacement chromatography, J. Chromatogr. A, 793, 223– 230, 1998. 99 Giddings, J. C., Dynamics of Chromatography, Part 1, Dekker, New York, 1965. 100 Kim, H. Y., Kakihana, M., Aida, M., Kogure, K., Nomura, M., Fujii, Y., and Okamoto, M., Uranium isotope effects in some ion exchange systems involving uranyl – carboxylate complexes, J. Chem. Phys., 81, 6266– 6271, 1984. 101 Onitsuka, H., Takeda, K., and Miyake, T., Operation and success of the ACEP semi-commercial plant, Bull. Res. Lab. Nucl. Reactors, 187– 191, 1992, Special Issue. 102 Nomura, M., Higuchi, N., and Fujii, Y., Mass dependence of uranium isotope effects in the U(IV) –U(VI) exchange reaction, J. Am. Chem. Soc., 118, 9127– 9130, 1996. 103 Bigeleisen, J., J. Am. Chem. Soc., 118, 3676– 3680, 1996.

3

Comments on Selected Topics in Isotope Theoretical Chemistry Max Wolfsberg

CONTENTS I. II.

Introduction ........................................................................................................................ 89 Born –Oppenheimer Approximation and Molecular Vibrations/Potential Energy Surfaces ................................................................................................................. 90 A. Born – Oppenheimer Approximation.......................................................................... 90 B. The Adiabatic Correction to the Born – Oppenheimer Approximation..................... 91 C. Molecular Vibrations/Potential Energy Surfaces ...................................................... 96 1. General................................................................................................................. 96 2. The Determination of Harmonic Force Constants in Valence Coordinates.......................................................................................................... 97 3. The Determination of Harmonic Force Constants in Cartesian Displacement Coordinates................................................................................... 98 D. Two Important Equalities for Harmonic Frequencies of Isotopomers ..................... 99 III. The Statistical Mechanics of Equilibrium Isotope Effects in the Gas Phase ................. 100 A. Equilibrium Constants.............................................................................................. 100 B. Rate Constants.......................................................................................................... 102 C. The Symmetry Number in Isotope Chemistry ........................................................ 104 IV. Numerical Calculations of Isotope Effects ...................................................................... 109 A. “Early” Calculations ................................................................................................ 109 B. Isotope Effect Calculations Coupled with A Priori Calculation of Electronic Structures ................................................................................................ 110 1. Some General Considerations of Electronic Structure Calculations................ 110 2. The Program THERMISTP............................................................................... 112 References..................................................................................................................................... 115

I. INTRODUCTION This chapter traces the principles underlying the understanding of the theory of isotope effects from quantum mechanics to the calculation of isotope effects on equilibrium constants and on rate constants (within the transition state theory often associated with Henry Eyring1 and commonly referred to as TST). In order to keep this chapter from becoming too lengthy, it became necessary to restrict this discussion to some specific topics in isotope effect theory; the title of this chapter arises from this requirement. There are many topics which are missing. The missing topics include isotope effects on molecular dipole moments and polarizabilities and isotope effects on rates within theories beyond TST, including the successful applications of variational transition state theory, especially by D.G. Truhlar (e.g., Ref. 2). Explicit consideration here is limited to molecular systems in the 89

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ideal gas phase. W.A. Van Hook discusses the theory of condensed phase isotope effects in this volume (W.A. Van Hook, Chapter 4 in this volume). In writing this chapter, the author has been aware of some overlap of purpose with the chapter in this volume by his colleague Jacob Bigeleisen (J. Bigeleisen, Chapter 1 in this volume) and he has tried to avoid overlap of content. In the following, the discussion in Section II starts with the Born –Oppenheimer approximation which leads to the concept of isotope independent potential energy surfaces for molecular vibrations; this isotope independence is basic to understanding isotope effects and has often been referred to as the First Law of Isotopics by the author. Correction to the Born– Oppenheimer approximation is briefly discussed; some new material on such corrections for muonium chemistry is briefly mentioned. There is a brief discussion of vibrational potential energy surfaces and of the theory of molecular vibrations. In Section III, the statistical mechanics of equilibrium isotope effects in the gas phase is discussed, as well as the development of rate constant isotope effects within TST. This section includes a new extended discussion on symmetry numbers and the relationship of symmetry numbers to isotope effects. Finally, in Section IV, numerical calculations of reduced isotopic partition function ratios are discussed. A new computer program THERMISTP is introduced and applied. It is to be noted that this chapter is not a review article. While the author has many references to his own work, there is no attempt to survey the field.

II. BORN –OPPENHEIMER APPROXIMATION AND MOLECULAR VIBRATIONS/POTENTIAL ENERGY SURFACES A. BORN – OPPENHEIMER A PPROXIMATION Consider a system of atomic nuclei and electrons, which can form a stable molecule. The quantum mechanical Schro¨dinger equation of this system for the wave functions Cmol and corresponding energy eigenvalues Emol ^ mol Cmol ¼ Emol Cmol H

ð3:1Þ

contains an energy operator H^ mol that consists of potential energy terms and kinetic energy terms. If one forgets about electron spin, the potential energy terms are all electrostatic terms resulting from the attractions between nuclei and electrons and from the repulsions between particles of like charge (nucleus –nucleus repulsions as well as electron – electron repulsions). There is a kinetic energy term for each particle (electrons and nuclei). Isotopomers from the point of view of Equation 3.l differ only in the nuclear mass factors that enter into the kinetic energies of the respective nuclei. It was first demonstrated by Born and Oppenheimer3 that the problem of Equation 3.1 could be much simplified largely because the masses of the individual nuclei are much larger than that of an electron. The simplification leads to the result that the single Schro¨dinger equation, Equation 3.1, is replaced by two Schro¨dinger equations: an electronic Schro¨dinger equation and a nuclear motion equation. The electronic Schro¨dinger equation contains the kinetic energy terms for the individual electrons and the various aforementioned electrostatic potential energy terms. In the solution of the electronic Schro¨dinger equation, the nuclei are considered fixed; the wave function, which depends on the electronic, coordinates ri , Celec , and corresponding electronic energy Eelec are calculated as a function of the nuclear geometry (the collective coordinates defining the nuclear geometry will be designated as S here). Thus H^ elec celec ðS; ri Þ ¼ ðT^ elec þ Velec ðS; ri ÞÞcelec ðS; ri Þ ¼ Eelec ðSÞcelec ðS; ri Þ

ð3:2Þ

where T^ elec is the operator for the kinetic energy of the electrons. Note that Celec and Eelec are both independent of nuclear masses; the nuclear mass independence of Eelec leads to the First Law of Isotopics. The nuclear motion Schro¨dinger equation contains a kinetic energy term consisting

Comments on Selected Topics in Isotope Theoretical Chemistry

91

of the kinetic energies of the nuclei and a potential energy term which is just Eelec ðSÞ: Thus, H^ nuc cnuc ¼ ðT^ nuc þ Eelec ðSÞÞcnuc ¼ Emol cnuc

ð3:3Þ

Moreover, the total wave-function Cmol is given by

Cmol ¼ cnuc ðSÞcelec ðS; ri Þ

ð3:4Þ

Equation 3.3, the equation for nuclear motion, may be further simplified for a stable molecule. This is standard textbook material. The translational motion of the center of mass may be separated and what remains then is the rotational – vibrational problem. The separation of the center-of-mass motion is exact; there is no approximation made. In the rigid rotor approximation (rotor corresponding to the equilibrium geometry of the stable molecule), one completely separates the rotational motion of the molecule and is then left with the vibrational problem which contains the vibrational kinetic energy of the atoms together with Eelec ðSÞ which becomes the (isotope independent) potential energy for the molecular vibrations. Thus, Equation 3.3 is replaced by three equations H^ trans ctrans ¼ T^ trans ctrans ¼ Etrans ctrans

ð3:5Þ

H^ rot crot ¼ T^ rot crot ¼ Erot crot

ð3:6Þ

^ vib cvib ¼ ðT^ vib þ Eelec ðSÞÞcvib ¼ Evib cvib H

ð3:7Þ

As a consequence of these three equations, the molecular wave functions Cmol become a quadruple product of an electronic, a translational, a rotational and a vibrational wave function. Correspondingly, the molecular energy Emol is a sum Emol ¼ Etrans þ Erot þ Evib

ð3:8Þ

Note that the electronic energy appears in the vibrational problem. For isotope effect calculations, it is usually sufficient to consider only the ground electronic state and the energy surface called Eelec ðSÞ corresponding to this state. Within the Born – Oppenheimer approximation, the minimum of this surface is taken as the zero of energy, which is the same for all the isotopomers.

B. THE A DIABATIC C ORRECTION TO THE B ORN – OPPENHEIMER A PPROXIMATION For very accurate calculations, one can develop corrections4 to the Born – Oppenheimer approximation, which has been outlined above. The mathematical development of this procedure is beyond the scope of this chapter. The development involves, among other matters, the investigation of terms which arise from the action of the kinetic energy/momentum operators of the nuclei on the electronic wave function and which contain inverse nuclear masses; these terms are ignored in the Born – Oppenheimer (BO) approximation. In order to correct the Born– Oppenheimer approximation for the missing operators, one starts with the Born– Oppenheimer approximation as the first approximation to Celec and expands the molecular wave function Cmol in G terms of the Born – Oppenheimer states now designated as celec X G G Cmol ¼ F ðSÞcelec ðS; ri Þ ð3:9Þ G

Thus, when one considers a full correction to the BO approximation, the molecular wave function no longer corresponds to a single Born – Oppenheimer electronic state but a linear combination of such states. To the extent that such mixing is important, the concept of a molecular system being in

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a given electronic state loses its meaning. Such mixing clearly occurs in nonadiabatic reactions where a system “jumps” from one electronic state to another (usually as a result of “curve crossing”). Also, such mixing occurs during the nonradiative decay of a molecule in an excited electronic state. Ordinarily, however, mixing of electronic states need not be considered unless two electronic states of a molecule are close lying in energy. Ordinarily, one can thus forget about the mixing of electronic states implied by Equation 3.9. In this situation, one can make a first order correction to the Born – Oppenheimer approximation by averaging the terms ignored in the Born – Oppenheimer electronic Hamiltonian operator over the electronic wave function of the state considered and replacing Equation 3.3 by ðT^ nuc þ Eelec ðSÞ þ CðSÞÞcnuc ¼ Emol cnuc

ð3:10Þ

where CðSÞ is the “perturbation” averaged over the electronic state. C is often referred to as the Adiabatic Correction. Generally CðSÞ will depend on nuclear configuration. In practice, this author has usually evaluated CðSÞ only at the equilibrium internuclear configuration (and, unless otherwise indicated, C refers to this value). As already pointed out, the operators that are averaged over the electronic wave function contain the nuclear mass. Thus, in Equation 3.10, the potential function for nuclear motion is no longer independent of isotopic substitution as it is in the BO approximation. To the extent that the configuration dependence of C is important, the shape of the potential energy surface on which the nuclear motion takes place is isotope dependent and both force constants (vide infra) and equilibrium geometries of molecules do depend on isotopic substitution. In limited theoretical work and in many deductions from experimental work, it has been shown that for most purposes this isotope dependence of the shape of the potential energy surface is sufficiently small so that it can be ignored. Thus, for a stable molecular system here, C will be considered to have been evaluated at the nuclear configuration corresponding to the minimum energy and it will be considered as an additive constant to Eelec ðSÞ, which forms the potential energy surface for the nuclear motion of the molecule in the BO approximation. Thus, with the introduction of the adiabatic correction C, the zero of energy of a molecular system becomes isotope dependent. This isotope dependence will introduce previously nonexistent isotope effects into the thermochemistry of isotopes. Consider an isotopic exchange equilibrium involving isotopes X1 and X2 , RX1 þ SX2 ¼ RX2 þ SX1

ð3:11Þ

The energy change DU of this reaction from the zeroes of molecular energy of the reactants to the corresponding zeroes of the products will be zero in the BO approximation, but in the adiabatic approximation it will be given by DUBOELE ¼ CðRX2 Þ þ CðSX1 Þ 2 CðRX1 Þ 2 CðSX2 Þ ¼ CðSX1 Þ 2 CðSX2 Þ 2 ½CðRX1 Þ 2 CðRX2 Þ ¼ DCðSXÞ 2 DCðRXÞ ¼ DDC

ð3:12Þ

where DC refers to the difference between the X1 isotopomer and the X2 isotopomer (with 1 always referring to light isotope and 2 to heavy isotope). Note that, if one wants to consider isotope effects on chemical equilibria, it is sufficient to consider isotopic exchange reactions because any isotope effect on the thermodynamics of a reaction can be re-expressed as an isotope effect on an exchange reaction (vide infra). Note also that, if the energy change DU from reactants to products is negative as written, then the contribution of this term to the equilibrium constant of the reaction will be to

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93

increase K: Thus, the contribution to K is given by a multiplicative factor KBOELE KBOELE ¼ expð2DUBOELE =kTÞ

ð3:13Þ

with k the Boltzmann constant and T the absolute temperature. In previous work,5,6 C values and corresponding DC values have been obtained for a small number of diatomic molecules, mostly hydrides. Testing did reveal that the values of DC change as the quality of the electronic structure calculation is changed. The most reliable values of DC should be available with the calculations of wave functions obtained close to the variational limit which, for many-electron systems, implies high electron correlation (vide infra). When highly correlated calculations are not available, there are indications that it is “best” to obtain DDC values (Equation 3.12) from DC values obtained with similar types of wave functions. For the majority of systems for which DC values are available, the DC’s correspond to no correlation and at best to near Hartree –Fock limit values (vide infra). There are only two molecular systems for which highly correlated values are available, molecular hydrogen and the diatomic molecule lithium hydride.6 In dealing with DC values, it helps to know that it can be shown that the nuclear mass dependence of C can be expressed simply as,7 C¼

X bi i mi

ð3:14Þ

where the sum is over all the nuclei in the molecule and the bi ’s are constants for the particular molecule. Thus, once one calculates DC for a D for H isotopic substitution, one can also evaluate DC values for all other isotopes of hydrogen. The equivalent of an adiabatic correction and corresponding DC value also exists for atomic systems. Thus, the so-called Rydberg correction for finite nuclear mass in consideration of the hydrogen atom spectrum is basically an adiabatic correction, which leads to an increase of the energy of the ground state of the hydrogenic atom. For infinite nuclear mass, the electron kinetic energy operator of the hydrogen atom after removal of the motion of the center of mass is written "2 2 7 T^ 1 elec ¼ 2 2me i

ð3:15Þ

where 1 refers to infinite nuclear mass, " has its usual meaning ("/2p), and me is the mass of the electron. For finite nuclear mass this term is replaced by "2 2 "2 2 "2 2 7 7 7 T^ M ¼ 2 2 ¼ 2 elec 2m i 2me i 2M i

ð3:16Þ

with

m¼

me M me þ M

where M is nuclear mass and m is the so-called reduced electron mass.When M is infinite compared to me ; m ¼ me : The ground state energy of the hydrogen atom is M E1s ¼2

e4 m ¼ 2RM 8h2

ð3:17Þ

With m ¼ me , one obtains the infinite mass Rydberg, R1 , while with M ¼ proton mass, one obtains the proton Rydberg. The difference is the equivalent of the adiabatic correction for an atom; in fact,

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similar type terms do, of course, appear in the evaluation of C and DC for molecules. The kinetic energy of the electron in the hydrogen atom is found by averaging the kinetic energy operator over the ground state wave function. It can be shown that this average kinetic energy is just the negative of the ground state energy (namely one Rydberg). From the second equality in Equation 3.16, one sees that the nuclear mass correction to the kinetic energy (which is also the correction to the Hamiltonian operator) can be written ! "2 2 me "2 2 7 ¼ 2 7 ð3:18Þ 2 2M i M 2me i One obtains the energy correction from this term by averaging this term over the ground state 1s wave function of the hydrogen atom for the case m ¼ me : One knows in classical mechanics that, in a two-particle system, both particles must move with equal but opposite momentum to keep the center of mass at rest. Equation 3.18 is just the operator for the kinetic energy of a particle of mass M with the same average squared momentum as that of the electron. So, the extra energy of the system is just the energy of the nuclear motion necessary to keep the center of mass at rest. This exercise has been carried out to give the reader some feeling of how the adiabatic correction in molecules arises because, as pointed out above, there are some terms in the molecular C value calculation which are similar to what has just been considered. If one takes the average value of the correction of Equation 3.18 one obtains me 1 m ðTelec Þ ¼ e R1 M M

ð3:19Þ

One obtains a similar result for the nuclear mass correction from Equation 3.17 by expanding m, 0 1 ! me M M 1 me m2e B C m¼ ¼ me ¼ me @ m A ¼ me 1 2 M þ M 2 þ · · · me þ M me þ M 1þ e M so that M E1s ¼ 2R1 þ R1

me me 2 R1 M M

2

þ· · ·

ð3:20Þ

This is the same result as Equation 3.19 except for the presence of additional terms involving higher powers of me =M: These additional terms come about because Equation 3.19 arises from a first order approximation (which is good enough). The atomic nuclear correction here will be regarded as a C factor for the hydrogen(ic) atom. It has a nuclear mass dependence of the same type as the molecular C factors Catom ¼

b M

ð3:21Þ

There is no electron correlation problem for the hydrogenic atom. So C values and DC values for hydrogen are accurate (except for higher order perturbations). Accurate C values also exist for H2 and of course for Hþ. The gas phase hydrogen ion is the trivial case with C equal to zero. Accurate C values and b values for H2, LiH, H, H2, and Hþ taken from the work of Bardo et al.6 are given in Table 3.1. From the DCðH – XÞ values in Table 3.1, one can calculate DDC values for corresponding H/D isotopic exchange reactions and KBOELE values, correction factors to the equilibrium constant calculated from statistical mechanics in the BO approximation. If one omits the Hþ system from

Comments on Selected Topics in Isotope Theoretical Chemistry

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TABLE 3.1 Accurate Values (cm21) of Adiabatic Corrections (C values) from Bardo et al.6 and DC Values for H – D and Mu – H Isotope Effectsa Molecule

C(HX)

C(DX)

DC(H –D)b

bH(cm21 amu)

DC(Mu–H)c

H2 LiH H2 H Hþ

114.59 198.34 67.01 59.77 0

85.96 163.82 33.52 29.90 0

28.63 34.52 33.49 29.87 0

57.71 69.58 67.50 60.20 0

451.5 544.4 528.1 471.0 0

a

Nuclear masses in atomic mass units: mH ¼ 1.007276, mD ¼ 2.013451, mu ¼ 0.113428. The H, D nuclear masses are the ones used in Ref. 6. The muon mass is from Ref. 9. b DC(H–D) ¼ C(HX) 2 C(DX). c DC(m–H) ¼ C(m–X) 2 C(H– X).

consideration, the largest magnitude DDC value from Table 3.1 is 5.7 cm21 which gives rise to a KBOELE value in the room temperature region that differs from unity by less than 3%. As noted, there are more tabulations of DCðH – DÞ values evaluated for wave functions at the Hartree – Fock level and even more at considerably lower levels of accuracy.5 If one looks through the tabulations of DC values, one finds magnitudes of DDC as high as 22 cm21 for the isotopic H, D exchange between H2 and BH. Such a DDC value yields KBOELE differing from unity by 10% at room temperature. In work at UC Irvine, only one isotopic exchange equilibrium was found where it made sense to compare calculation with experiment to see if the influence of KBOELE could be detected. In order to make such a comparison, gas phase equilibrium was needed involving fairly small molecules over a large temperature range with high accuracy. Gas phase was needed so that the data for theoretical calculations of equilibrium constants would be available, including many corrections that are not usually included in such calculations. Such equilibrium was found in the H/D isotopic exchange reaction between water and molecular hydrogen H2 OðgÞ þ HDðgÞ ¼ HDOðgÞ þ H2 ðgÞ

ð3:22Þ

This exchange is used for deuterium enrichment and has been measured by a number of workers over a temperature range from 280 to 588 K. Bardo and Wolfsberg8 carried out refined calculations by statistical mechanics (vide infra) and then evaluated DDC for this reaction with wave functions near the Hartree –Fock limit level. They found DDC ¼ 3:8 cm21. This corresponds to KBOELE ¼ expð25:5=TÞ

ð3:23Þ

This adiabatic correction to the equilibrium is not large. At room temperature KBOELE differs from unity by 2%. The plot obtained by Bardo and Wolfsberg clearly shows that KBOELE puts the calculated value of the exchange equilibrium in almost perfect agreement with the least squares fit to the experimental data. J. Bigeleisen (Chapter 1 in this volume) demonstrates the excellent agreement of the calculation with the experimental data. In the earlier work on DC and KBOELE ; at room temperature, there were also a couple of calculations of KBOELE for heavy atom exchange reactions involving 14N, 15N and 6Li, 7Li. In both cases studied, KBOELE differed from unity by about 0.001. Thus, as expected, Born– Oppenheimer corrections for heavy atom isotope effects will tend to be negligible. For H, D isotope effects, it would probably be reasonable to state that KBOELE at room temperature will usually differ from

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Isotope Effects in Chemistry and Biology

unity by less than 0.1 and, in fact, in the one case where comparison has been made with experiment the deviation was of the order of 2% at 300 K. The question now arises whether KBOELE values may be larger than has been indicated here. This author has asked himself this question when he learned about measurements of muonium (Mu), hydrogen isotope effects.10 Mu is a very light unstable isotope of hydrogen consisting of a nucleus, which is a positive muon (m), and an electron. The nuclear mass of Mu is 0.113 atomic mass units.9 It is then easy to calculate 21 21 21 DCðMu; HÞ=DCðH; DÞ ¼ ðm21 Mu 2 mH Þ=ðmH 2 mD Þ ¼ 15:77

ð3:24Þ

The resulting DC values for Mu, H isotope effects are listed in Table 3.1. It is seen that the larger DC values obtained for Mu, H effects lead to a DDC value for the LiH, H2 isotopic exchange reaction involving Mu, H of 93 cm21. This DDC value leads to a deviation of KBOELE from unity at room temperature of about 0.4. A DDC value of 22 cm21 has been mentioned for H, D exchange for a reaction involving H2 and BH. For a Mu, H exchange this would yield KBOELE ¼ expð500=TÞ, indeed a large factor compared to unity at room temperature. While Mu, H isotope effects are expected to be much larger than H, D isotope effects, it would seem that the answer to the question about Mu, H isotope effects and the Born – Oppenheimer approximation is that one cannot obviously ignore the possibility of KBOELE effects in considering thermodynamics involving Mu,H equilibria. There is no question the KBOELE will be more important in Mu,H isotope effect considerations than in H,D isotope effects.

C. MOLECULAR V IBRATIONS/ POTENTIAL E NERGY S URFACES 1. General A stable molecule is a molecule for which EðSÞ in Equation 3.3 and Equation 3.7 has a minimum occurring at a geometry which is referred to as the equilibrium geometry of the molecule in a particular electronic state (usually here the ground state). For a stable molecule, the potential energy surface for vibrational motion Eelec (which will often be referred to as V here) is expanded in a Taylor’s series about the equilibrium nuclear configuration. The coordinates favored by chemists tend to be displacements of bond lengths Dr, displacements of valence bond angles Da and displacements of torsion angles Dt, all the displacements being measured by their respective deviations from the values in the equilibrium configuration. Torsion angles are defined by the positions of four atoms 1, 2, 3, 4. By convention t is restricted 2p , t # p; t is the angle between the 1 2 3 plane and the 2 3 4 plane; for the sign convention for t, please see Wilson, Decius, and Cross (WDC).11 These three types of displacement coordinates and others are described in WDC.11 Since there is a minimum, the lowest terms in the expansion will be quadratic; the higher order terms will be cubic, quartic, etc. The quadratic terms are often referred to as the harmonic terms and the collection of harmonic terms is referred to as the harmonic potential. The higher order terms are referred to as the anharmonic potential. The anharmonic potential is usually restricted to cubic and quartic terms since the thermal properties of molecules are not expected to depend strongly on large vibrational displacements of the molecule from its equilibrium configuration at room temperature. In very accurate calculations of isotope effects, anharmonic corrections need to be considered. The discussion here will be mainly in the harmonic approximation. Now the solution of Equation 3.7 is considered. The values of the vibrational energy levels are needed in the harmonic approximation. It is interesting that the difficulties in solving this quantum mechanical problem are all in the solution of the classical mechanical problem. Once the classical mechanical problem is solved to obtain the so-called normal mode frequencies ni of the molecular system, the quantum mechanical problem is trivially simple.

Comments on Selected Topics in Isotope Theoretical Chemistry

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The solution of the classical problem with the potential for vibration expressed in harmonic displacements from equilibrium is well described in WDC.11 The method is known as the GF matrix technique. The elements of the F matrix fij describe the potential in terms of valence displacement coordinates (stretches, bend angles, torsion angles, etc.), 2V ¼ 2Eelec ¼

X i#j

fij si sj

ð3:25Þ

The elements of the G matrix gij are related to the vibrational kinetic energy expressed in terms of momenta pi which are conjugate to the coordinates si , X 2T ¼ 2Tvib ¼ gij pi pj ð3:26Þ i#j

The reader need not worry about the meaning of the term “momentum conjugate to a coordinate.” What is important is that WDC shows exactly how to calculate gij if the equilibrium geometry of the molecule is known as well as the atomic masses of the atoms. The method of constructing G matrix elements is very simple. The process of solving the classical mechanical problem of the vibrations is continued by multiplying the G matrix by the F matrix following the law of matrix multiplication to form the GF matrix. The GF matrix must then be diagonalized and the diagonalization procedure yields eigenvalues li related to the normal mode vibrational 2 2 frequencies P ni by li ¼ 4p ni and corresponding eigenvectors Qi related to the coordinates si by Qi ¼ cij sj : The trouble with the mathematical process described is the following. Diagonalization of symmetric matrices B with elements bij ¼ bji is one of the most frequently used processes in applied mathematics and many computer programs exist to diagonalize symmetric matrices. Unfortunately, the GF matrix is not a symmetric matrix. There are alternative ways of finding the eigenvalues and eigenvectors corresponding to the GF matrix which only involve diagonalization of symmetric matrices (actually these usually involve two diagonalizations, see for example Ref. 12). For an N-atomic nonlinear molecule, there will be 3N 2 6 normal frequencies ni and corresponding 3N 2 6 normal coordinates. For a complete solution, there should be 3N 2 6 coordinates si : If more than 3N 2 6 coordinates are used, there is a redundancy and there will be a zero frequency corresponding to each redundancy. (For some discussion on redundancy, see WDC.11) The classical problem then becomes a problem of 3N 2 6 independent harmonic oscillators, each corresponding to one of the normal coordinates and vibrating with the corresponding frequency. The 3N 2 6 oscillators are independent and noninteracting and their phases and amplitudes have to be set by initial or other conditions. The quantum mechanics of the vibrations of the nonlinear molecule is completely analogous — 3N 2 6 harmonic oscillators with frequencies ni , the energy of the individual oscillators being given by the usual vibrational energy level formula 1 i ¼ ni þ

1 hni 2

ni ¼ 0; 1; 2; …

ð3:27Þ

The total energy of the vibrational system is given by the collection of 3N 2 6 vibrational quantum numbers ni : When dealing with a linear molecule, 3N 2 6 above must be replaced by 3N 2 5: 2. The Determination of Harmonic Force Constants in Valence Coordinates How does one determine the force constants fij of the previous section? The idea of force constants corresponding to valence bond stretches, to angles between two valence bonds on the same atom, to torsion angles often created by four successive valence bonds, and similar

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Isotope Effects in Chemistry and Biology

coordinates relating to bonds11 arose early in the twentieth century and it proved to be a qualitatively useful idea. It was anticipated that diagonal stretching force constants fii are “large” by comparison to bending force constants and that torsion force constants would be even smaller although the meaning of “large” and “small” is not entirely clear since units are not necessarily the same. Also, it was anticipated that diagonal force constants fii would be larger than offdiagonal force constants ( fij with i – j). It was also anticipated that force constants involving similar bonds (say C – H bond-stretch in hydrocarbons) in different molecules would be the same. Until 1970 or so, the only practical method for determining force constants of a molecule was from “experimental” observation of normal frequencies in infrared and Raman spectra of molecules. There are methods of “correcting” observed fundamental frequencies for anharmonicity and such correction was frequently carried out. Then it becomes a matter of using the methods of the previous section to find those force constants that give the “best” fit to observed frequencies. Without digital computers this was a tedious process. These calculations could only be carried out for small molecules. There are many examples of the results of such calculations in the classic book by Herzberg.13 These calculations did indeed lead to agreement with what had been anticipated about valence force constants. Most of the early (and also later) calculations by this author and Stern (e.g., Refs. 14 ,15) on isotope effects in which digital computers were used to calculate frequencies of polyatomic molecules in model calculations are based on diagonal force constants from force constant tables (in Refs. 11 ,13), often with off-diagonal force constants omitted. It should, however, be pointed out that such simple force fields will at best only give qualitative agreement with observed vibrational spectra. 3. The Determination of Harmonic Force Constants in Cartesian Displacement Coordinates The quadratic potential function of the previous section in valence displacement coordinates can be readily transformed into a potential function in Cartesian displacement coordinates; this procedure leads to the concept of Cartesian force constants. Compact formulas are given in Chapter 4 of WDC11 in connection with the evaluation of the elements of the G matrix. If the Cartesian displacements from the equilibrium configuration are designated as ji with three Cartesian displacements per atom (say i ¼ 1; 2; 3 are x; y; z coordinates of atom 1, i ¼ 4; 5; 6 refer to atom 2, etc.), the vibrational potential becomes 2V ¼

3N X i#j

aij ji jj

ð3:28Þ

It can then be shown11 that the eigenvalues and eigenvectors that result from the diagonalization of the matrix A0 , with elements a0ij ¼ aij =ðmi mj Þ1=2

ð3:29Þ

are the harmonic vibrational frequencies of the molecule and the eigenvectors correspond to the normal coordinates expressed in terms of so-called mass weighted Cartesian displacement 1=2 coordinates, i.e., qi ¼ ðxi 2 x0i Þmi : Here mi refers to the mass of the atom to which the Cartesian coordinate refers. It is of some utility that this matrix is symmetric. Note that for an N-atomic molecule, there are 3N Cartesian displacement coordinates. Thus the matrix A0 has dimension 3N £ 3N: Correspondingly, the matrix has 3N eigenvalues and 3N eigenvectors. For a nonlinear polyatomic molecule six of these frequencies are zero (for a linear molecule, there will be five zeroes). The eigenvectors corresponding to the zero frequencies correspond to molecular translations and rotations. If the matrix diagonalization yields fewer than six (respectively, five,

Comments on Selected Topics in Isotope Theoretical Chemistry

99

for a linear molecule) zero frequencies, this is a danger signal which points to an error either in the evaluation of the force constant matrix elements or to an error in the diagonalization procedure. It is usually not sensible to carry out the transformation to Cartesian displacement coordinates once the force constants are known for valence coordinates. Force constants in valence coordinates are very useful since one can look at the magnitude of the force constants and decide if this is a reasonable force field (diagonal force constants larger than off-diagonal, etc.) There are no such apparent niceties known for Cartesian force constants. Yet, Cartesian force constants have been introduced here for a reason. The force constants are obtained from a Taylor’s expansion of Eelec ðSÞ: While the discussion of valence force constants mentions only experimental data on vibrational spectra as a source of force constants, Eelec ðSÞ can be obtained from a priori calculations directly from electron structure packages which are widely available. There are a number of such packages available; reference here will be made only to the so-called GAUSSIAN package, in particular the latest version GAUSSIAN 03,16 also referred to as g03. These derivatives are generally taken with respect to Cartesian displacements of the atomic nuclei. While the transformation from valence coordinates to Cartesian displacement coordinates is straightforward, the inverse of this transformation is more complicated since it involves explicit consideration of molecular translations and rotations. Hence, the computer program employed for the calculation of equilibrium isotope effects from theoretically evaluated force constants does use as input from GAUSSIAN 03 the Cartesian coordinates of the atoms in their respective equilibrium configuration and the Cartesian force constants. This program will be further discussed in Section IV.B.2.

D. TWO I MPORTANT E QUALITIES FOR H ARMONIC F REQUENCIES OF I SOTOPOMERS There are two very important equalities11 involving the harmonic vibrational frequencies of isotopic isomers (isotopomers) in the Born– Oppenheimer approximation (isotope independent force constants). In the present context, these equalities are exact both for normal molecules and for transition states (vide infra). Deviations represent errors in the calculation either because of round-off errors or true errors made in the theory. The first is the Teller – Redlich product rule. Refer to the two isotopomers as 1 and 2, respectively; then Y n1i ¼ n2i i

M1 M2

3=2

I1x I1y I1z I2x I2y I2z

!1=2

N Y m2k m1k k¼1

3=2

ð3:30Þ

where the isotopic frequency ratio product extends over 3N 2 6 vibrations for a nonlinear (3N 2 5 for a linear) N-atomic molecule, the isotopic mass ratio product extends over the atomic masses m in the molecule, M refers to molecular mass, and there are three principal moments of inertia Ix ; Iy ; Iz ; for a nonlinear molecule while there are only two such moments for a linear molecule. The second rule is the sum rule. It derives from the well-known fact that the sum of the eigenvalues li of a matrix equals the sum of the diagonal elements of the matrix. Applied to the problem in Cartesian force constants for the case of two isotopomers which differ by one isotopic substitution of a mass m2 for a mass m1 this rule takes the simple form ! X 2 X 2 1 1 2 2 ð3:31Þ 4p n1i 2 n2i ¼ a m1 m2 i i where a is the sum of the three diagonal Cartesian force constants at the position of isotopic substitution. Such a rule can also be stated for valence force constants, but the rule is particularly “attractive” in Cartesian force constants.

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Isotope Effects in Chemistry and Biology

III. THE STATISTICAL MECHANICS OF EQUILIBRIUM ISOTOPE EFFECTS IN THE GAS PHASE A. EQUILIBRIUM C ONSTANTS Thermodynamic equilibrium constants in the ideal gas phase can be formulated in terms of canonical molecular partition functions17 (constant T; V) for ideal gas phase molecules, Q: In quantum mechanics, Q is a sum over states X 21 =kT Q¼ e i ð3:32Þ i

where the sum is over all energy states 1i of the system, k is the Boltzmann constant, and T the absolute temperature. It is well known and simple to show that, when a Hamiltonian Schro¨dinger equation such as Equation 3.3 can be “broken up” into separate Schro¨dinger equations for translation, vibration, and rotation, the partition function can be written as a product of partition functions. Thus, Q can be written, Q ¼ Qtrans Qrot Qvib

ð3:33Þ

Moreover, since the vibrational Schro¨dinger Equation 3.7 can be broken up in the harmonic approximation into separate equations for each normal mode of vibration, Y Qvib ¼ qvib i ð3:34Þ i

where q refers to the partition function for one normal mode and the product over i extends over the molecular normal modes. It has already been mentioned that all molecules will be considered to be in their respective ground electronic states. Thus, the electronic state enters into the partition function expression only through its possible degeneracy but that degeneracy will cancel when partition function ratios are calculated for isotopomers. Consider now the isotope effect on an equilibrium constant, with subscripts 1 and 2 referring to two isotopomers, where, by usual convention, 2 will here again refer to the heavier isotopomer (note, however, that Bigeleisen (Chapter 1 in this volume) uses just the opposite convention) A1 þ B ¼ C1 þ D

K1

ð3:35Þ

A2 þ B ¼ C2 þ D

K2

ð3:36Þ

The equilibria 3.35 and 3.36 differ by an isotopic substitution in a reactant and in a product. The ratio K1 =K2 is termed an isotope effect. Now subtract the above two reactions in the usual chemical manner. One obtains the isotopic exchange equilibrium A1 þ C2 ¼ A2 þ C1

K3 ¼ K1 =K2

ð3:37Þ

Thus, the equilibrium constant of an isotopic exchange reaction K3 can be considered to be the isotope effect on a reaction K1 =K2 :17 K3 can be written straightforwardly in terms of a ratio of isotopic partition function ratios as18 K3 ¼

QA2 QC1 QA2 =QA1 ¼ QA1 QC2 QC2 =QC1

ð3:38Þ

Here the isotopic partition function ratio is always taken as heavy/light (when appropriate). Note that, from its definition, K3 is independent of concentration units17 so that Kp ¼ Kc : Also note that gas phase equilibrium constants refer to the ideal gas at 1 atm and that the partition

Comments on Selected Topics in Isotope Theoretical Chemistry

101

function for NA molecules of A is written as QNAA =NA !; the factor NA ! arises from the use of Boltzmann statistics17 for indistinguishable molecules. It is usually appropriate in the room temperature region and above to work with vibrations treated quantum mechanically. However, the molecular translations and rotations may usually be treated by classical statistical mechanics.17 If one substitutes the classical partition functions for the rotations and the translations and the quantum partition functions for the normal mode vibrations into Equation 3.38, one obtains the often seen expression for K3 in terms of vibrational frequencies of the molecular species, products of moments of inertia, molecular masses, and symmetry numbers. If one substitutes the Teller – Redlich product rule, one can obtain the isotope effect just in terms of normal mode vibrational frequencies and symmetry numbers. The last result is more pleasing because of its simplicity but the two results are equivalent and, as stated previously, any failure of the Teller – Redlich product rule in a calculation must be regarded as a sign to proceed with caution. Bigeleisen (Chapter 1 in this volume) handles the problem of the isotopic partition function ratios in the elegant manner devised by himself and M.G. Mayer,18 recognizing that the classical isotopic partition function ratio Q2 =Q1 can be written in terms of a symmetry number ratio s1 =s2 (vide infra) multiplied by an isotopic ratio of atomic masses. When an isotope effect on an equilibrium constant is evaluated, the atomic mass factor always drops out. Thus, the classical mechanical value of the isotope effect (high temperature value) will depend only on symmetry numbers. This then leads to the formulation of the isotope effect on an equilibrium constant in terms of the reduced isotopic partition function ratios of Bigeleisen and Mayer, ðs2 =s1 Þf ðA2 =A1 Þ: The reduced isotopic partition function ratio is equal to the ratio of the quantum mechanical value of QA2 =QA1 divided by the corresponding classical value. Since classical partition functions are used for translations and the rotations even in the quantum formulation, this quantity is expressed in terms of the normal mode vibrational frequencies of A2 and A1. The equilibrium constant is then expressed as (Bigeleisen, Chapter 1 in this volume) K3 ¼

sA =sA ðsA2 =sA1 Þf ðA2 =A1 Þ f ðA2 =A1 Þ ¼ 1 2 sC1 =sC2 ðsC2 =sC1 Þf ðC2 =C1 Þ f ðC2 =C1 Þ

ð3:39Þ

where ðsA2 =sA1 Þf ðA2 =A1 Þ ¼

QA2 =QA1 ¼ ðQA2 =QA1 Þclassical

QA2 =QA1 N Y ðsA1 =sA2 Þ ðm2i =m1i Þ3=2

Y u2i 1 2 e2u1i ðu 2u Þ=2 1i 2i ¼ 2u2i e 1 2 e u 1i i

ð3:40Þ

Here, m refers to atomic mass and the mass ratio product is taken over all the atoms in the isotopomers, the frequency product term is taken over all 3N 2 6 normal mode frequencies n of the N-atomic nonlinear molecule (3N 2 5 for a linear molecule), u ¼ hn=kT: T is the absolute temperature. Since the frequencies of the heavier isotopomer2 are equal to or less than those of the light one, the reduced partition function ratio is equal to or larger than unity; it decreases with temperature, approaching unity as the temperature increases (Bigeleisen, Chapter 1 in this volume). While the following statement follows directly from the statement above, it is repeated for emphasis. The expression 3.39 for K3 in terms of symmetry numbers and normal vibrational frequencies of the molecules A2, A1, C2, C1 is exactly the same expression in terms of frequencies and symmetry numbers that could have been derived from Equation 3.38 from the appropriate quantum and classical partition functions (in the harmonic approximation with no rotational – vibrational interaction) with application of the Teller – Redlich product rule. The symmetry

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number factor in front of the ratio of reduced partition function ratios is the classical mechanical value of the isotope effect. As is usually stated in the literature18 and is demonstrated in detail in a following Section III.C of this chapter, this symmetry number factor does not lead to equilibrium isotopic enrichment in either the A or C molecular species. The high temperature form of this equation, originally stated by Bigeleisen and Mayer,18 is found by expanding ðs2 =s1 Þf and keeping the first nonvanishing term in T 22 : Thus, ðsA2 =sA1 Þf ðA2 =A1 Þ ¼ 1 þ

1 h 24 kT

2X i

ðn21i 2 n22i Þ ¼ 1 þ

1 " 24 kT

2

aA

1 1 2 m1 m2

ð3:41Þ

and K3 for a single atom (say, H for D) isotope exchange equilibrium becomes, at high temperature ! 1 " 2 1 1 K3 ¼ gðsÞ 1 þ 2 ðaA 2 aC Þ ð3:42Þ 24 kT m1 m2 Here aA is equal to the sum of the three diagonal Cartesian displacement force constants at the position of isotopic substitution in molecule A; the meaning of aC is similar except the reference is to the C molecule; m refers to atomic mass. As noted, Equation 3.41 is a high temperature equation. Yet, the author likes this equation as a qualitative indicator even at lower temperatures. It predicts that the equilibrium, aside from the symmetry number factor gðsÞ which does not lead to isotopic enrichment, favors the heavy atom in that molecule where the position of isotopic substitution has the larger force constant. In other words, with aA . aC , A2 is the favored molecule rather than A1. One can obviously reach this same conclusion with the usual zero-point energy argument, which will not be repeated here. Also note that Equation 3.41 correctly indicates that a high force constant at the position of isotopic substitution tends to yield high ðs2 =s1 Þf values. Both of the above equations indicate that heavy atom isotope effects (12C, 13C) will tend to be smaller than H, D effects.

B. RATE C ONSTANTS This discussion of isotope effects on rate constants will be very brief so as to keep overlap with the discussion of Bigeleisen (Chapter 1 in this volume) to a minimum. The discussion is based on TST. Partition functions are needed for isotopic transition states as well as for reactants. In TST, the position of the transition state is isotope independent just as it is for a stable molecule. The transition state corresponds to a maximum in the potential energy surface along the “reaction coordinate” from reactant(s) to product(s), which, in the BO approximation, is Eelec ðSÞ; the electronic energy of the system as a function of nuclear configuration. When a reaction is considered, one is interested in the motion of atomic nuclei in a system which may contain several molecular species, not only in regions in configuration space which lie close to the energy minima of the individual molecule(s), but also in those regions in configuration space which connect stable molecules. Within the BO approximation, the potential energy for the motion of the atomic nuclei is still the electronic energy Eelec ðSÞ as a function of nuclear configuration S: Generally, for a chemical reaction to take place, which is from one set of stable molecules that are reactants to another set of stable molecules that correspond to products, the system has to pass through a maximum in Eelec ðSÞ: As already noted the transition state corresponds to the highest energy in the potential surface along the lowest energy pathway from reactants to products. In TST, the position of the transition state is a saddlepoint in Eelec : The transition state corresponds to a maximum in the direction of the “reaction coordinate”. With respect to motion in the 3N ‡ 2 7 other “directions” of the transition state, there is bound motion (vibration). From the electronic Schro¨dinger equation, one can determine the second derivatives of Eelec ðSÞ with respect to Cartesian displacements from the transition state position and then use this transition state force field for the calculation of the normal mode frequencies ni‡ of the transition state (the symbol ‡ is used to designate a transition state). For a nonlinear N ‡ -atom transition state, the appropriate

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diagonalization of the matrix derived from the force constant matrix will yield 3N ‡ 2 7 real frequencies which corresponds to real bound vibrational motion and one frequency which is imaginary (or maybe zero) which does not correspond to a real vibration and which is often designated as n‡L (for a linear transition state, there will be 3N ‡ 2 6 real frequencies). In standard transition state theory, one assumes equilibrium between reactants and transition state so that the rate constant contains the equilibrium constant K ‡ between reactants and transition state. The transition state is taken to have three translational degrees of freedom like a normal molecule and also three (two in a linear transition state) rotational degrees of freedom with corresponding moments of inertia. It has 3N ‡ 2 7 (3N ‡ 2 6 linear) bound vibrations corresponding to the real frequencies discussed above. The motion along the reaction coordinate is handled as a translation. The combination of the rate of decomposition arising from this translation and the number of translational decomposition states, gives rise to a universal factor kT=h so that the rate constant kr for a bimolecular reaction of A with B is given by kr ¼

kT ‡ K h

ð3:43Þ

where d½product ¼ kr ½A ½B dt

ð3:44Þ

and K‡ ¼

Q0 ‡ ½A 2 B‡ ¼ A2B ½A ½B QA QB

ð3:45Þ

Note that a subscripted k, kr, here and below k1 and k2, is being used to designate a rate constant while k without a subscript refers to the Boltzmann constant. The prime on the transition state partition function indicates a missing vibrational degree of freedom corresponding to the imaginary frequency n‡L : The transition state is designated as A 2 B‡. If A1 and A2 are isotopomers, then the isotope effect on the rate constant is QA =QA k1 ¼ 0 ‡2 0 ‡1 k2 Q2 =Q1

ð3:46Þ

One now proceeds to calculate partition function ratios just as for ordinary equilibria, assuming classical rotations, classical motion of the center of mass and 3N ‡ 2 7 harmonic quantum vibrations for the transition state. When these partition functions are substituted into Equation 3.46, one obtains an equation for k1 =k2 which contains isotopic ratios of molecular masses, isotopic ratios of products of moments of inertia, and ratios of quantum partition functions with 3N 2 6 vibrations in the reactant but only 3N ‡ 2 7 vibrations in the transition state. In this expression, the imaginary frequency in the transition state does not appear. If one now introduces the Teller – Redlich product rule, one must remember that this rule applies to all the eigenvalues of the appropriate matrix diagonalization including the imaginary frequency in the transition state which is usually called n‡L : One then obtains, with omission of symmetry number ratios (Ref. 19 and Bigeleisen, Chapter 1 in this volume), k1 n‡ f ðA2 =A1 Þ ¼ 1L k2 n‡2L f 0‡ ð‡ 2 =‡ 1 Þ

ð3:47Þ

The form of ðs2 =s1 Þf in terms of normal mode frequencies is exactly the same as the reduced partition function ratio in Equation 3.40, except that for the transition state, there are only 3N ‡ 2 7 frequencies (3N ‡ 2 6 for a linear transition state).

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When one carries out TST calculations, one often19 adds, as a correction for quantum tunneling T, the so-called Wigner tunneling correction (consideration of more sophisticated tunneling correction necessary for large tunneling is beyond the scope of this Chapter) that is expressed in terms of the absolute value of the imaginary frequency. It has the form T1 ¼ 1 þ

2

1 h 24 kT

ln‡1L l2

ð3:48Þ

Remember now that n‡1L is pure imaginary (or zero) so that ln‡1L l2 ¼ 2n‡2 1L and that k1 k2

corrected

¼

k1 T 1 k2 T 2

ð3:49Þ

Again, the author is interested in the high temperature form of this equation since he finds that it gives him a good qualitative insight into kinetic isotope effects. Proceeding as for ordinary equilibrium isotope effects in Equation 3.42, one finds a very similar expression for ðk1 =k2 Þcorr k1 k2

( n1L 1 " ¼ gðsÞ ‡ 1 þ 24 kT n2L corr

2

1 1 ðaA 2 a‡ Þ 2 m1 m2

) ð3:50Þ

Here aA and a‡ are the respective sum of diagonal Cartesian force constants at the position of isotopic substitution in reactant A and in transition state and m1 and m2 are the respective atomic masses of the light isotope and the heavy isotope. As is appropriate, we have applied the sum rule to all the frequencies in the transition state including the zero or imaginary frequency n‡L : It is to be noted that the “pleasing” result is only possible if the Wigner tunnel correction is included. The high temperature equation leads to the result that, if the force constant at the position of isotopic substitution in the transition state is less than in the reactant, then k1 =k2 is larger than unity. On the other hand, if the force constants in the transition state are larger than those in the reactant, then the nonclassical part (for the classical part, see below) of the isotope effect can be less than unity and thus lead to what is called an inverse isotope effect. This result, again, can be derived from zeropoint energy arguments. However, the author prefers this derivation. It is to be noted that the high temperature limit (classical limit) of an isotope effect on an equilibrium constant is from Equation 3.42 equal to gðsÞ; the appropriate symmetry number factor, while the infinite temperature limit of a rate isotope effect is from Equation 3.50 equal to n‡1L =n‡2L times the appropriate symmetry number factor gðsÞ: If one works on isotope effects, one knows that these symmetry number factors are often pretty obvious and do not lead to isotope enrichment in one species with respect to another. In the next section, it is shown explicitly that one does not actually need to know symmetry numbers to calculate gðsÞ and that indeed gðsÞ does not lead to isotopic enrichment. Note, however, that one needs to know gðsÞ to compare with experiment and it will be shown how to calculate the factor easily.

C. THE S YMMETRY N UMBER IN I SOTOPE C HEMISTRY As already noted, the quantization of energy levels only very significantly affects the vibrational partition functions of molecules at room temperature and above. For the translational and rotational partition functions, one usually replaces summation over energy levels by integration over phase space (vibrational/rotational) momenta and coordinates, p’s and q’s, because the spacing between energy levels is small compared to kT:17,20 Thus, the molecular rotational partitional function Qrot

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105

is given by Qrot ¼

1 1 ð 2Hrot =kT e dp:dq::dpf dqf hf s

ð3:51Þ

Here s is the symmetry number, h is Planck’s constant, and f is the number of rotational coordinates, 3 for a general polyatomic molecule and 2 for a linear molecule, Hrot is the rotational kinetic energy, k and T have their usual meanings. Because of the factor h f , this partition function is often referred to as semi-classical; this factor makes it possible to compare a phase space integration which yields a phase space volume with a sum over states and to ensure that the classical and quantum partition functions agree at high temperatures, as they must from the Correspondence Principle. For a detailed discussion of the origin of h f , please refer to a statistical mechanics text.20 The interest is now in the calculation of the ratio of symmetry numbers of two isotopomers s2 =s1 : In quantum mechanics one has to take into account the indistinguishability of particles. Thus, the electrons in molecules are indistinguishable; the general consideration of this property leads to the two types of statistics applicable to fundamental particles, Bose –Einstein and Fermi – Dirac, and eventually to the Pauli Exclusion Principle which is the basis of the periodic table of the elements. If one has NA atoms of A in the molecule, there are NA ! permutations of these A atoms which all correspond to one configuration and by carrying out the phase space integration over all of space, one is overcounting indistinguishable configurations by a factor NA !; consequently one divides the phase space 20 integral by NQ A !: Thus, one divides the phase space integral in the molecular partition function by the product i Ni ! for all nuclei which appear multiple times in the chemical formula of the molecule. However, further thought show that this is overdoing it. One should only divide the phase space integral by factors Ni ! for atoms which are chemically equivalent. Thus, as pointed out in the book by Mayer and Mayer20 for the molecule HCOOH, which contains two oxygen atoms and two hydrogen atoms, the molecular partition function requires no division. Permutation of the oxygens and/or the hydrogens is a nonfeasible permutation, which corresponds to a new molecular minimum in coordinate space; the permuted configuration should be counted because it corresponds to a different energy minimum. On the other hand, the three hydrogens in CH3Cl and the four hydrogens in methane are indeed chemically equivalent and division by N! is necessary. However, in these molecules a new problem arises.20 Look at CH3Cl. There are six permutations of the three hydrogen atoms. Study of these permutations shows that there are two types of molecules with numbered hydrogen atoms, three for which the numbering of the hydrogen atoms 1, 2, 3 will be clockwise looking down the CCl axis and three counterclockwise. The two types cannot be converted into each other and, in fact, if you substitute three different hydrogen isotopes, the phenomenon gives rise to different optically active forms of isotopically substituted methylchloride. These two forms correspond to different minima. Thus, for methylchloride, we must divide the classical phase space integral by 3! first and then must multiply by a factor L ¼ 2 for the fact that there are two distinct minima. A similar situation occurs for methane, where there are again two minima and one must divide by 4! (arising from the N! term) and multiply by L ¼ 2: It is sufficient to multiply the rotational partition function by two. This is the classical equivalent of doubling all energy levels. With the definition of the symmetry number s as the quantity by which the classical phase space must be divided to take proper account of the permutation of equivalent nuclei, one has s ¼ N!=L, where L refers to the “different” atom numbered minima of the molecule. An alternative way of determining the symmetry number of a molecule is to look at the point group of the molecule. The various symmetry elements of the molecular point group correspond to permutations of chemically equivalent nuclei (for certain point groups one may have to include not only permutation but also permutation inversion operations). Be that as it may, the rotation operations of the point group will correspond to the feasible permutation operations of indistinguishable nuclei. Indeed, this understanding leads to the statement20 that the symmetry number of a molecule is equal to the number of rotations in the point group (including the identity E as one of the operations).

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Determination of symmetry numbers of molecules requires either knowledge of the molecular point group of the molecule or an analysis of the number of distinct minima of the molecule (based on numbering of atoms). Many chemists are reluctant to do either of these two analyses. The point here is that, in isotope chemistry, one does not need the symmetry number s of molecules but only the ratio of the symmetry Q numbers of two isotopomers, s2 =s1 : In the definition of symmetry numbers, the factor of i Ni !, the products of N! for the individual nuclei of the molecules is very easy to determine. The difficulty is the determination of L: Now remember that L is found from the different distinct configurations of a molecule based on numbering the atoms. L is the same for all isotopomers of a molecule. Thus L cancels out when one calculates s2 =s1 for two isotopomers. One then obtains for the H, D isotopomers of methane, with CH4 as molecule 1, the following results Molecule 2

CH3 D

CH2 D2

CHD3

CD4

s2 =s1

3!1 1 ¼ 4! 4

2!2! 1 ¼ 4! 6

3!1! 1 ¼ 4! 4

4! ¼1 4!

So far only rigid molecules have been considered, i.e., molecules which are confined to one potential well. There are so-called nonrigid molecules; detailed discussion is beyond the scope of this presentation. However, at very low temperature the NH3 molecule is “stuck” in an equilibrium structure corresponding to a symmetric triangular pyramid. There is, however, a second minimum and the molecule will invert from one structure to the other already at temperatures well below room temperature. If one wants to take into account this inversion motion, one must treat ammonia as a nonrigid molecule. In order to do this, one must use the average state between the two pyramids, which corresponds to the triangular planar ammonia molecule. In order to calculate isotope ratios for the ammonia isotopomers as nonrigid molecules, one only needs to know that, for the nonrigid molecule, the three protons are indistinguishable just as they are for the rigid molecule. Consequently, the ratios of symmetry numbers for nonrigid ammonias are exactly the same as for rigid ammonias. However, the symmetry numbers themselves are different for NH3, sðrigidÞ ¼ 3, sðnonrigidÞ ¼ 6: Since symmetry number ratios of isotopomers have now been calculated, we now investigate what can be said about isotope effects in the classical limit, i.e., at high temperatures. Consider isotope exchange reactions involving two isotopes (for convenience called H and D here) and two molecular species with the all protio formulas RHm and SHp, where RHm has m chemically equivalent protium nuclei and SHp is similarly defined. The general exchange reaction involving the exchange of q hydrogen atoms in the R species for q deuterium atoms can be written RHm2n Dn þ SHp2b Db ¼ RHm2n2q Dnþq þ SHp2bþq Db2q

ð3:52Þ

In the system at equilibrium, there will be a number of such equilibria with 0 # n # m; 0 # b # p, and appropriate values of q: In order to simplify the notation, the above molecular formulas are replaced by A1, A2, B1, B2 with 1 referring to light species and 2 to heavy species so that the equilibrium becomes A1 þ B2 ¼ A2 þ B1

ð3:53Þ

with A1 ¼ RHm2n Dn , etc. One can then write for the equilibrium constant of the gas phase reaction K¼

QðA2 Þ=QðA1 Þ ½A2 =½A1 ¼ QðB2 Þ=QðB1 Þ ½B2 =B1

ð3:54Þ

where the Q’s refer to gas phase molecular partition functions and [ ] refer to molecular mole numbers or to isotopic mole fractions of A and B molecules, respectively, (the total numbers of A and of B molecules are constant). As Bigeleisen (Chapter 1 in this volume) has emphasized, one can show that in the classical limit (high temperature in the quantum world), K just becomes

Comments on Selected Topics in Isotope Theoretical Chemistry

107

the appropriate ratio of symmetry numbers (Equation 3.39). K¼

sðA1 Þ=sðA2 Þ sðB1 Þ=sðB2 Þ

ð3:55Þ

Use will now be made of the isotopic symmetry number ratios, which have been derived here sA 1 wA2 cm;nþq sðRHm2n Dn Þ ðm 2 nÞ!n! ¼ ¼ ¼ ¼ cm;n sA 2 sðRHm2n2q Dnþq Þ ðm 2 n 2 qÞ!ðn þ qÞ! wA1

ð3:56Þ

Here cm;nþq and cm;n are the so-called binomial coefficients cm;nþq ¼

m! ðm 2 n 2 qÞ!ðn þ qÞ!

cm;n ¼

m! ðm 2 nÞ!n!

ð3:57Þ ð3:58Þ

cm;nþq is the number of ways of distributing m numbered objects into two boxes such that the first box contains m 2 n 2 q of the objects and the second box contains n þ q objects. It is also equal to the number of ways of forming RHm2n2q Dnþq from H atoms and D atoms. The binomial coefficient is really a weight factor and it is referred to here as w: For example, w for H2O equals unity while, for HDO, w equals two. The binomial coefficient has been introduced here into the isotopic symmetry number ratio because of its property as a weight factor. Similarly, cm;n is the weight factor for molecule RHm2 n Dn : Note that the contribution of the classical isotopic partition function ratio QA2 =QA1 to K is equal to the inverse ratio of symmetry numbers sA1 =sA2 , which is simply given by the ratio of the corresponding weight factors wA2 =wA1 : The equality of such a symmetry number ratio to a ratio of binomial coefficients has already been noted in the referenced literature.18 In the cases considered here, there are two isotopes of an element. For two isotopes, the classical isotopic partition function ratio contribution can always be written in terms of a corresponding ratio of binomial coefficients. If three or more isotopes had been considered, the symmetry number ratio would have been written in terms of multinomial coefficients. In either case, the classical isotopic partition function ratio contribution would have been equal to a ratio of corresponding statistical weights of the molecules. For sðB1 Þ=sðB2 Þ in Equation 3.55, one obtains similarly s B1 s B2 ¼

cp;b cp;b2q

¼

wB2 wB1

ð3:59Þ

Thus, the classical value of the equilibrium constant for the gas phase reaction is given by K¼

QðA2 Þ QðA1 Þ

wA2 QðB2 Þ ¼ QðB1 Þ wA1

wB2 wB1

ð3:60Þ

It will now be demonstrated that this value of K demands that, at equilibrium, there is no isotopic fractionation between the A and the B molecules, i.e., the mole fraction of H atoms, xH , is the same in both molecular species. Consider the isotopic exchange equilibria between RHm (and its isotopomers) and hydrogen atoms (H and D). Note that the symmetry number of H(D) is equal to unity. Then the equilibrium constant for the gas phase reaction RHm þ nD ¼ RHm2n Dn þ nH

ð3:61Þ

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Isotope Effects in Chemistry and Biology

in the classical limit can be written K¼ ¼

½RHm2n Dn ½H ½RHm ½D n

n

cm;n sðRHm Þ ¼ sðRHm2n Dn Þ cm;0

ð3:62Þ ð3:63Þ

With H and D concentrations in mole fractions, x equal to mole fraction of hydrogen atom and y the corresponding quantity for deuterium, one obtains ½RHm2n Dn ¼

cm;n ½RHm ðy=xÞn cm;0

ð3:64Þ

By definition cm;0 equals unity. One can now write an expression for the sum of the concentrations of all the m isotopomers of RHm , then factor ½RHm out of the sum, and then multiply and divide by xm , to obtain sum ¼

½RHm xm

ð3:65Þ

The terms in the sum, starting with n ¼ 0 for RHm , correspond to relative concentrations of the isotopomers RHm2n Dn : One, of course, recognizes the series following the summation sign in Equation 3.65 as the binomial expansion of ðx þ yÞm ðx þ yÞm ¼ xm þ mxm21 y þ · · · ¼

m X n¼0

m X m! xm2n yn ¼ cm;n xm2n yn ðm 2 nÞ!n! n¼0

ð3:66Þ

The symbols x and y represent mole fractions and add to unity so that the concentrations represented by terms in the summation in Equation 3.66, cm;n xm2n yn , are the isotopic mole fractions of the RHm2n Dn isotopomers. The binomial expansion clearly indicates that the equilibrium concentrations of the isotopomers in equilibrium with hydrogen and deuterium atoms merely reflect the number of ways in which the various isotopomers can be formed from the atoms; there is no isotopic discrimination. It is clear that the overall mole fractions of D and H in the RHm isotopomers are exactly the same as the mole fractions of H and D in the atoms with which they are in equilibrium. This can be explicitly shown mathematically by using the methods of Mayer and Mayer,20 Appendix A VI.5. The lengthy details are omitted. Returning now to the nomenclature of the equilibrium of Equation 3.52 and Equation 3.53, one can show from Equation 3.66 for the R type molecules A2 in equilibrium with H, D gas (mole fraction H equal to x) that wA ½A2 ðm 2 nÞ!n! xm2n2q ð1 2 xÞnþq cm;nþq 2q x ð1 2 xÞq ¼ 2 x2q ð1 2 xÞq ¼ ¼ ½A1 ðm 2 n 2 qÞ!ðn þ qÞ! xm2n ð1 2 xÞn cm;n wA1

ð3:67Þ

Similarly, one can show for the S type molecules B2 and B1 in equilibrium with H, D gas (mole fraction H equal to z) wB ½B2 ¼ 2 z2q ð1 2 zÞq wB1 ½B1

ð3:68Þ

If one substitutes these expressions into Equation 3.54, one finds that agreement with Equation 3.60 requires that x must be equal to z: Clearly, in the classical limit, equilibrium between A and B

Comments on Selected Topics in Isotope Theoretical Chemistry

109

isotopomers requires such concentrations that the respective mole fractions of H and D in the A isotopomers and in the B isotopomers are the same. Thus, there is no isotopic fractionation between different molecular species from the symmetry number factors which determine the isotope exchange equilibrium constants in the classical limit (high temperature). A review of the above discussion on a two isotope system (e.g., H, D) might well lead the reader to the conclusion that the final result (no isotope fractionation between different molecular entities) was assured when it was shown that the ratio of the symmetry numbers of the two isotopomers may be given as a ratio of binomial coefficients, Equation 3.56 and Equation 3.59, which are also referred to as weight factors here. It might be argued that it was unnecessary to go through the exercise of showing that there is no isotopic fractionation between a pool of hydrogen/deuterium atoms and the isotopomers of RHm molecules in the classical (high temperature) limit. In that spirit, it is pointed out again that in a situation with more than two isotopes (e.g., H, D, T) the symmetry number ratios will correspond to ratios of multinomial coefficients (e.g., the trinomial coefficients of the expansion ðx þ y þ zÞm ). Thus, the conclusion about symmetry numbers and high temperature (classical) isotopic fractionation clearly is also applicable to systems dealing with more than two isotopes of an element.

IV. NUMERICAL CALCULATIONS OF ISOTOPE EFFECTS A. “EARLY ” C ALCULATIONS Prior to about 1962, calculations of isotope effects discussed in Section III, K1 =K2 and k1 =k2 were usually restricted to systems of 4 or fewer atoms because of difficulty in diagonalizing the matrices necessary for obtaining frequencies of larger molecules from force fields. The work by Bigeleisen and Wolfsberg21 on isotope effects in the gas phase reaction between molecular hydrogen and chlorine atoms is typical of such work. In 1962, the author’s then colleague J.A. Ibers arranged for him to obtain from J.H. Schachtschneider the computer programs which he and R.G. Snyder12 were using to calculate harmonic vibrational frequencies of larger molecules corresponding to given equilibrium geometries and given force fields in valence coordinates. (The author was/is very grateful to Ibers and Schachtschneider.) Soon afterwards, the author wrote his first computer program (FORTRAN) thermo, as a subroutine to the program received from Schachtschneider. Thermo converts calculated frequencies of isotopomers into ðs2 =s1 Þf values and then into calculated values of isotope effects (Equation 3.39 or Equation 3.47). Among the first applications of this new program resource was the development of a cell model approach for studying vapor pressure isotope effects,22 an approach that is still in use today. Further use of the new program was a study with M.J. Stern on whether the idea suggested by the high temperature equations (Equation 3.41 and Equation 3.50) that equilibrium (rate) isotope effects reflect force constant changes at the position of isotopic substitution between reactant and product (transition state) is a correct one even at room temperature.15,23 It was shown that such isotope effects are indeed very small unless force constants do change at the position of isotopic substitution. Stern and Wolfsberg also tested a number of approximation procedures for calculating equilibrium (rate) isotope effects; in particular the cut-off procedure was tested.14,24 Since it had been shown that isotope effects are probes for force constant changes at the position of isotopic substitution, the question was asked whether one can “cut-off”, in a calculation involving large molecules, parts of the molecule distant from the reaction center. Subject to certain conditions, such a cut-off procedure yields valid results. As already pointed out in Section II.C.2, most of the model calculations above were carried out with valence force fields, the diagonal force constants of which were taken from tables such as those in Refs. 11,13. Off-diagonal force constants were usually taken to be zero unless the effects of off-diagonal force constants were being tested. Equilibrium geometries were taken from tables

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of bond distances and bond angles and from “chemical intuition”. For transition states, the geometries and force constants away from the supposed reaction center were taken as for normal molecules. If a bond was broken in a reaction, then in the transition state the stretching force constant was often chosen to be zero or slightly negative (with equilibrium geometry corresponding to an elongated bond). Such a choice would ensure an imaginary (or zero) frequency n‡L in the transition state. As is required by the BO approximation, isotopomers were always chosen to have the same force constants and geometries. The point is that these were model calculations carried out to study the “nature” of isotope effects. Reproduction of experimentally observed spectra or rate/ equilibrium isotope effects was not usually attempted. Some calculations, of course, were carried out where reproduction of experimental values was attempted. In these calculations, use of molecular parameters, which reproduce experimental vibrational spectra, and use of experimental data on equilibrium structures (e.g., Ref. 8) was attempted. In the latter calculations, corrections for anharmonic vibrations and for rotational – vibrational interaction and even the KBOELE factor from the correction to the BO approximation were introduced. While the author has been interested in anharmonicity and while anharmonic effects on isotope effects are usually larger than the effect of the KBOELE factor, anharmonicity is not discussed further in this chapter.

B. ISOTOPE E FFECT C ALCULATIONS C OUPLED WITH A P RIORI C ALCULATION OF E LECTRONIC S TRUCTURES 1. Some General Considerations of Electronic Structure Calculations Electronic structure calculations and Cartesian coordinate force fields have already been introduced in Section II.C.3. Over the years the methodologies used by the electronic structure computer packages have grown more sophisticated while the computational capabilities of digital computers have increased “astronomically”. Computations that would have been considered almost impossible a few years ago except for a “supercomputer” can today be carried out on a PC. Consequently, electronic structure packages have become a working tool for many chemists (e.g., Ref. 25). A detailed discussion of the technologies available for a priori calculations of electronic energies of molecules and transition states is beyond the scope of this chapter. For a recent look at the situation, the reader is referred to the literature (e.g., Ref. 26). For isotope effects considered here, the main interest is in the energy of the ground electronic state of the molecular system as a function of nuclear configuration. There are two chief types of calculations in the a priori computer packages: (1) the (true) a priori methods based on molecular orbital theory some of which make use of the variational principle of quantum mechanics; in this methodology there is usually a systematic way of improving the calculation; (2) density functional calculations (DFT) which are based on a theorem proved by Kohn and Hohenberg27 in which one does not need to know the electronic wave function to calculate the electronic energy but only the electron density. (While this author includes DFT under the broad heading of a priori calculations, he hesitates to regard these calculations as presently carried out as true a priori calculations (vide infra).) Among a priori methods, the lowest level is the Hartree – Fock method (designated HF), which corresponds to the straightforward molecular orbital method of individual electron orbitals obtained by considering the problem of an electron subject to the field created by all the other electrons and the atomic nuclei. The method leads to the concept of a self-consistent field (SCF). To go beyond the Hartree – Fock method, one considers configuration interaction (CI) which for a molecule is equivalent to saying that the ground electronic state of the helium atom is not adequately described by the electron configuration (1s)2 but that mixing is required in the ground state with configurations like (1s) (2s), (2s)2, etc. Configuration interaction is necessary because a given electron “sees” not only the average field created by the other electrons. When considering CI, one therefore says that one is dealing with electron correlation. The first improvement to Hartree – Fock

Comments on Selected Topics in Isotope Theoretical Chemistry

111

methods is to consider configuration interaction by perturbation theory; the acronym MP (for Moeller Plesset) is often attached to such methods. MP2 and MP4 refer to increasing orders of perturbation theory. There are a number of methods which go beyond just perturbation theory. Full configuration interaction calculations are impractical because too many configurations are possible. Therefore, methods have been developed to choose those configurations which are most important; these methods have acronyms like CCSD, CISD, etc. Reference should be made to the literature (e.g., Ref. 26). Almost all basis sets used in calculations today are gaussian (G) basis sets because of the ease of evaluating necessary integrals with such basis sets. The simplest meaningful basis set is referred to as STO3G. Bigger basis sets tend to involve more numbers in the acronym and tend to lead to “convergence” (better results) for the chosen method. For HF calculations, one generally has to use basis sets which are larger than ST03G in order to converge to the Hartree – Fock limit. One will note names like 3-21G, 6-31G, 6-311G as acronyms for “better and larger” basis sets. There are some very superior basis sets which start with the acronym (cc). The reader is again referred to the literature. DFT is the “new kid on the block.” While the electronic wave function of an n-electron molecule is a function of the 3n Cartesian coordinates of the electrons, the electron density is a function of only 3 Cartesian coordinates. Thus, the DFT approach which only requires the electron density is a much simpler approach. The “fly in the ointment” is that one does not know how to calculate the electronic energy from the density, i.e., one does not know the density functional. A number of density functionals have been developed over the years. None of these is “exact.” When energies and properties are compared with experimental results, some functionals are better than others, but not consistently. For a priori calculations, one often knows from the Variational Principle of quantum mechanics that better calculations yield lower and lower energies for the system; this Principle has no meaning for DFT. Thus, a lower energy does not mean that the density functional is better. Better density functionals are judged to be better by the fact that they yield better agreement with experiment. The “best” density functionals today are probably furnished by the so-called hybrid methods which are admixtures of various density functionals chosen in such a way as to give “best” agreement with experimental results for a set of molecular systems. A widely used DFT method is that designated by the acronym B3LYP. Again reference should be made to the literature (e.g., Ref. 26). The Kohn – Sham (KS) theorems27 represented a major advance in DFT. These theorems enabled one to cast the computational problem of DFT into a mathematical form very closely related to a Hartree –Fock calculation. This development gradually led to the inclusion of DFT capability into most electronic structure calculation packages (e.g., GAUSSIAN16). The DFT methods also require basis functions and, in fact, the same basis functions are available for DFT as for a priori calculations. The use of larger basis sets should again converge the given method, designated by the functional used. However, while convergence in a priori theory means better wave function, there is no such guarantee in DFT. The attraction of DFT is cost in terms of computer time; one hopes with DFT to obtain molecular parameters of a quality obtainable with pure a priori methods considering configurational interaction at the cost of a much cheaper Hartree –Fock calculation. In most (all) the electronic structure packages, the user can request that electronic structure calculations be carried out as a function of nuclear configuration in such a way as to find a minimum energy configuration corresponding to a stable molecule. Moreover, for the stable molecule, the program can be requested to find the second derivatives with respect to Cartesian displacements from the molecular minimum. These second derivatives are then used to calculate the harmonic vibrational frequencies. By a somewhat more sophisticated algorithm, one can also determine the geometries, electronic energies, and the harmonic force constants of transition states. There is a large literature of applications of these techniques, especially to organic molecules and to the study of the reactions of organic molecules.

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With respect to isotope effects, it has already been noted in Section II.C.3 that the 3N £ 3N Cartesian force constant matrix for an N-atomic molecule yields six zero frequencies (five for a linear molecule) and 3N 2 6 (3N 2 5 for a linear molecule) nonzero frequencies which correspond to the normal vibrations of the molecule. In a transition state, one of the “normal” frequencies will be imaginary and is designated n‡L (Equation 3.47). In practice, all six (respectively, five) of the “small” frequencies (which are referred to here as the zero frequencies) are rarely calculated equal to zero. This is probably the result of the fact that one does not find the exact stationary point that corresponds to the desired equilibrium configuration of the molecule (or to the transition state). The determination of the force constants corresponding to second derivatives assumes, of course, that the first derivatives are zero. To the extent that the first derivatives are in error, there are errors in the force constant matrix. The GAUSSIAN program16 specifies that one should request a very tight (vtight) optimization in finding the normal mode vibrational frequencies of molecules and in addition specify Int ¼ ultrafine in DFT calculations. While these options slow down the computation, the author does always use them if possible. If these options are not employed, the calculations will be deemed to have been carried out in the default mode. In addition, the Teller – Redlich product rule should always be tested for the vibrations which are nonzero. The author regards deviations of more than 0.01% from the Teller – Redlich product rule (equality) as a danger signal in an isotope effect calculation (see later discussion on this point). 2. The Program THERMISTP In using electronic structure calculations in order to calculate equilibrium (rate) isotope effects, the author originally used successive electronic structure calculations to compute force constants and frequencies for various isotopomers and then substituted these frequencies into the formulas for ðs2 =s1 Þf or ðs2 =s1 Þf 0‡ to evaluate isotope effects by the methods of Equation 3.39 or Equation 3.47. Doing the electronic structure calculations more than once is, of course, very wasteful since the Cartesian force constants are independent of isotopic substitution. When Martin Saunders suggested joint work on equilibrium isotope effects measured by NMR methods, he also suggested that the computer program thermo be revised to accept force constant output and equilibrium geometry output from GAUSSIAN computations and then to use these data to calculate ðs2 =s1 Þf and ðs2 =s1 Þf 0‡ for isotopomer pairs. This revision was carried out at Yale University largely through the programming efforts of Keith Laidig. The new program called Quiver was based on the older program thermo and it was available, on request, from Yale University. The important output of the program is ðs2 =s1 Þf for stable molecules and ðs2 =s1 Þf 0‡ for transition states. The program assumes that all frequencies larger than 50 cm21 are real frequencies. It assumes that frequencies less than 50 cm21 are the translational/rotational frequencies and, in a transition state, the imaginary frequency n‡L since the program designates an imaginary frequency as a negative frequency. The program does not check whether these assumptions are met by the calculation but the user can look at the frequencies in the output. The program was used very successfully in two separate studies.28,29 The author has recently re-written Quiver. The new program is THERMISTP. The program requires the same input as Quiver. The output includes ðs2 =s1 Þf and, for transition states, ðs2 =s1 Þf 0‡ as well as ðn‡1L =n‡2L Þ ðs2 =s1 Þf 0‡ : The program is more user friendly in that the output explicitly notes the values of real frequencies, translation/rotation frequencies as well as n‡L and prods the user to change the program if the assumptions on the values are wrong. Finally, the new program tests the Teller – Redlich product rule. Quiver also tests the Teller – Redlich product rule but because of an error in the calculation of moments of inertia, Quiver usually indicates that the Teller – Redlich product rule does “not work.” Thus, the test is useless in Quiver. THERMISTP will be fully described in a separate publication by Saunders and Wolfsberg30 and the program will be available from either author. Shiner and his students31,32 have fitted force fields to reproduce observed fundamental frequencies for a given molecule and available isotopomers. They have used these force fields to

Comments on Selected Topics in Isotope Theoretical Chemistry

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obtain reduced isotopic partition function ratios for single H – D substitutions in about 100 different stable molecular species. It has been shown33 that force fields so obtained lead to ðs2 =s1 Þf values which include some anharmonicity correction. Shiner has used these ðs2 =s1 Þf values quite successfully in explaining observed kinetic isotope effects in terms of arguing that he expects a given transition state to behave very much like a given stable molecule and he then predicts rate isotope effects on the basis of equilibrium isotope effects calculated from reduced partition function ratios. Shiner’s table is the only extensive table of its kind. It is now proposed to compare his partition function ratios to those calculated with THERMISTP. Keep in mind that the values calculated by THERMISTP correspond to the harmonic approximation. A particular interest here is to use different electronic structure methods and different basis sets for the THERMISTP calculations and to compare results; this means is also being used to acquaint the reader with some of the aspects of calculating reduced isotopic partition function ratios based on force fields derived from electronic structure calculations. In Table 3.2, results of ðs2 =s1 Þf at 298.15 K are presented for seven different gas phase molecules and for six different electronic structure calculation methods (terminology is discussed in the previous section here). The electronic structure program used is GAUSSIAN, g03. The results are compared with the ðs2 =s1 Þf values reported by Shiner and Neumann.32 It is noted that the ðs2 =s1 Þf values calculated with THERMISTP do depend on the type of calculation. The first four columns of calculations are what have been called here a priori calculation while the last two are DFT calculations. The first four columns correspond in the order listed to increasing accuracy of wave functions. The CCSD calculation corresponds to a fair amount of configuration interaction and the basis set is fairly large. The two DFT calculations, the B3LYP calculations, correspond to what is usually regarded as a fair basis set calculation for DFT and the second one corresponds to a larger basis set (cc-type). The author would be inclined to “like” the larger basis set more. The RHF/ ST03G calculations lead to very high values compared to Shiner and also compared to the other values. This result is expected because it is well known that RHF/ST03G frequencies are much too high (i.e., the force constants are too large). As might have been expected, the CCSD calculation most closely parallels the Shiner values among the a priori calculation. Among the two DFT calculations, the larger basis set leads to slightly better results (B3LYP/cc-pVDZ) when compared to Shiner values. Remember now that the ratio of ðs2 =s1 Þf values for two molecular species in TABLE 3.2 Values of (s2/s1)f for Single D/H Substitutions in Ideal Gas Phase Molecules. Comparison of Shiner32 Spectroscopic Values and Values Computed by THERMISTP with Force Constants Obtained with the Program g0316 System

Shinera

HF STO3G

HF 631 1 G*b

MP2 ccpVDZ

CCSD ccpVDZ

B3LYP 631G*

B3LYP ccpVDZ

H2S C2H2 H2CO C2H4 C2H6 NH3 H2O

5.402 8.248 9.122 10.37 11.23 12.20 12.69

9.765 15.62 17.40 20.09 23.97 23.41 20.27

6.853 11.62 13.02 14.28 15.09 17.22 17.04

6.176 9.293 10.67 12.07 13.10 14.43 13.86

5.986 9.213 10.56 11.81 12.79 14.17 13.81

5.820 9.284 10.35 11.94 12.87 14.02 12.90

5.622 9.422 9.740 11.54 12.24 13.46 12.82

The different calculations are indicated by the gaussian acronyms in the headings, T ¼ 298.15 K. Shiner and co-workers list values relative to C2H2. Prof. Shiner has kindly informed the author that the absolute value for C2H2 is 8.248. This table lists the absolute values derived from this value. b 631 þ G* means 631þ þG**. a

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Table 3.2 gives the value of the equilibrium constant for the single H/D isotope exchange between the two species except for the symmetry number factor (Equation 3.39). However also note that, for the hydrogen atom, the Schro¨dinger equation of the system is replaced by an electronic Schro¨dinger equation and a nuclear motion Schro¨dinger equation for the translational motion of the center of mass. Since the translational motion is treated classically in statistical mechanics, the reduced isotopic partition function ratio ðs2 =s1 Þf for D/H atom requires no calculation and equals unity (independent of temperature). Thus, one can add another row to Table 3.2, labeled H atom and the reduced partition function ratio would be unity for each column of Table 3.2. Therefore, the ðs2 =s1 Þf value for each molecular species is the D, H exchange equilibrium constant for that molecular species with atomic hydrogen. Clearly, for exchange equilibrium with the atomic hydrogen species, the ST03G values in Table 3.2 compare badly with the one obtained with the Shiner values. However, it is true that for exchange equilibria between the species actually listed in Table 3.2, even the ST03G values lead to results that are often not too different from the Shiner values. The other values do tend to compare better with Shiner values. If one is interested in studying ratios of ðs2 =s1 Þf values obtained by the various methods with the corresponding ratios obtained with the Shiner values, one should look at the correlation between the logarithm of Shiner’s values and the logarithm of the corresponding values obtained from THERMISTP with the various methods. Table 3.2 will be expanded by Saunders and Wolfsberg30 to include more of the systems studied by Shiner; the study of this correlation is deferred to this publication. For the present, one concludes from Table 3.2 that, in calculating isotope effects from ratios of calculated ðs2 =s1 Þf values, one should stick to one method of calculation, e.g., do not mix HF/ ST03G with CCSD/cc-pVDZ. High-level calculations tend to give better results than low-level results, but even low level results can often be useful. Also, it is very impressive that high-level results seem to be converging to a lower value which still tends to be larger than the Shiner value. Further study will be interesting. One must, however, bear in mind that the Shiner value does contain an “anharmonicity correction” and is therefore expected to be lower than a calculated harmonic value. It appears reasonable to say that one can today calculate ðs2 =s1 Þf values for many gas phase molecules, which are fairly close to the exact harmonic values. As already noted, the Teller – Redlich product rule is tested in all calculations with THERMISTP. The test consists of dividing the right-hand-side of Equation 3.30 by the left-handside of the equation; the result is referred to as TRTEST. While the explicit form of Q2 =Q1 evaluated directly from moments of inertia and molecular masses without using the Teller – Redlich product rule has not been shown in this chapter, it is straightforward to demonstrate that the division of the calculated value of ðs2 =s1 Þf by TRTEST will yield the reduced isotopic partition function ratio that would have been calculated if the Teller –Redlich product rule had not been applied. This statement is correct both for normal molecules as well as for transition states (with n‡L included in the frequency product). As has been stated, TRTEST should be unity and therefore calculated isotope effects should have the same value independent of whether the Teller – Redlich product rule is employed or not. However the six (five for a linear molecule) rotational/translational frequencies calculated from the diagonalization of the appropriate Cartesian force constant matrix should also be zero. In practice, they are not all equal to zero, but in the present context these six frequencies (five for a linear molecule) will be referred to as zero frequencies. This result means that the Cartesian force field is faulty because the optimization procedure in the electronic structure program did not lead to the exact geometry of the molecule (transition state). Then the nonzero (the 3N 2 6 or 3N 2 5) larger frequencies do not correspond to a correct molecular force field either. If a THERMISTP calculation is based on an electronic structure calculation in the default mode of optimization, the zero frequencies can often be as large as 50 or 60 cm21 in absolute value (they may be imaginary). If the electronic structure calculation of the force field with the g03 program is based on the recommended mode (vtight, etc.) for the optimization procedure, then these zero frequencies tend to be much closer to zero; often they are all less than 1 cm21 in absolute value (but not always). One expects the large

Comments on Selected Topics in Isotope Theoretical Chemistry

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frequencies also to be “improved” then. As for the TRTEST, the recommended mode for the systems in Table 3.2 always leads to values which deviate from unity by less than one part in ten thousand. What surprised the author is that the TRTEST for the default mode calculations (and these calculations were carried out on a small subset of Table 3.2 systems) also usually gave TRTEST values deviating from unity by less than one part in ten thousand but some larger deviations were also observed, one as large as six parts in a thousand. On the other hand, one can also compare the ðs2 =s1 Þf values obtained with default mode and with recommended mode calculations. There are systems where the ðs2 =s1 Þf value changes by less than one part in ten thousand in going from one mode to the other but there are other cases where the vtight mode yields a value differing by several parts in a thousand from that obtained in the default mode although TRTEST in the default mode differs from unity by less than one part in ten thousand. Thus, there are three criteria by which one can judge the quality of the computed reduced isotopic partition function ratio (or the isotope effects calculated by taking appropriate ratios of these partition function ratios): the magnitudes of the so-called zero frequencies, the deviation of TRTEST from unity, and the behavior of the reduced isotopic partition function ratio when one compares default mode values with recommended mode values. From the rather limited sample that has been studied here, one concludes that, for H, D isotope effects, if an accuracy better than one part in a thousand is desired, then use the recommended mode (and be aware of the three criteria by which the quality of the calculation can be judged). Schaad and co-workers34 have recently studied the kinetic deuterium isotope effect in the isomerization of formamide by internal rotation. They calculated the kinetic H, D isotope effect (corresponding to a ratio of reduced isotopic partition function ratios) for the ND2 compound and obtained with default mode computations a value of TRTEST which differs from unity by 3 parts in one thousand. For a recommended mode (vtight) calculation, they obtained a value of TRTEST, which differs from unity by less than one part in a hundred thousand. They reached a convincing conclusion that the value obtained for an isotope effect from reduced isotopic partition function ratios obtained with use of the Teller – Redlich product rule will be more accurate than one obtained without use of the rule. This conclusion can be shown to be in agreement with a very general rule that the author has found valid for many sorts of approximation procedures in calculating isotope effects — if possible, try to use equations which lead to the correct high temperature limit of the isotope effect. Calculations done in connection with work on the molecules of Table 3.2 and some other molecules, tend to confirm the conclusion of Schaad and coworkers. However, in the recommended mode, which is still being advocated here, there is very little difference between isotope effects reported with or without use of the Teller –Redlich product rule. The length of this discussion on TRTEST reflects the worries of the author when he first noted the large values of the “zero” frequencies that resulted from some of the GAUSSIAN calculations; however, in the end, even default mode isotopic partition function ratios are not “that bad.” The Table 3.2 calculations have been used here to illustrate a use of Cartesian force constants generated by quantum mechanical calculations to evaluate isotope effects. It would be a most serious omission not to call attention to the extensive literature that has been created over the past few years which reports on such calculations and compares with experimental work (e.g., Refs. 32,35). It is an impressive collection of works amply demonstrating the power of isotope effects to determine/confirm mechanism. In closing this chapter, it appears appropriate to point out that the progress in this field of isotope applications could never have been imagined in 1957 when J. Bigeleisen and the author wrote their chapter in volume 1 of Advances in Chemical Physics.19

REFERENCES 1 Glasstone, S., Laidler, K. J., and Eyring, H., The Theory of Rate Processes, McGraw-Hill, New York, 1941.

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2 Pu, J. Z. and Truhlar, D. G., Tests of potential energy surfaces for H þ CH4 $ CH3 þ H2: deuterium and muonium kinetic isotope effects for the forward and reverse reaction, J. Chem. Phys., 117, 10675– 10687, 2002. 3 Born, M. and Oppenheimer, R., Zur quantentheorie der molekeln, Ann. Phys. (Leipz), 84, 457– 484, 1927. 4 Van Vleck, J. H., On the isotope corrections in molecular spectra, J. Chem. Phys., 4, 327– 338, 1936. 5 Kleinman, L. I. and Wolfsberg, M., Correction to the Born – Oppenheimer approximation and electronic effects on isotope exchange equilibria, II, J. Chem. Phys., 60, 4749– 4754, 1974. 6 Bardo, R. D., Kleinman, L. I., Raczkowski, A. W., and Wolfsberg, M., The effects of electron correlation on the adiabatic correction and on equilibrium constants of isotopic exchange reactions, J. Chem. Phys., 69, 1106– 1111, 1978. 7 Bardo, R. D. and Wolfsberg, M., The nuclear mass dependence of the adiabatic correction to the Born – Oppenheimer approximation, J. Chem. Phys., 62, 4555 –4558, 1975. 8 Bardo, R. D. and Wolfsberg, M., A theoretical calculation of the equilibrium constant for the isotopic exchange reaction between H2O and HD, J. Phys. Chem., 80, 1068– 1071, 1965. 9 Mohr, P. J. and Taylor, B. N., CODATA recommended values of the fundamental physical constants 1998, Rev. Mod. Phys., 72, 351– 495, 2000. 10 Lossack, A. M., Roduner, E., and Bartels, D. M., Solvation and kinetic isotope effects in H and D abstraction reactions from formate ions by D, H, and Mu atoms in aqueous solution, Phys. Chem. Phys., 3, 2031– 2037, 2001. 11 Wilson, E. B., Decius, J. C., and Cross, P. C., Molecular Vibrations, McGraw-Hill, New York, 1955. 12 Schachtschneider, J. H. and Snyder, R. G., Vibrational analysis of the n-paraffins II normal coordinate calculations, Spectrochim. Acta, 219, 117– 168, 1963. 13 Herzberg, G., Molecular Spectra and Molecular Structure II, Infrared and Raman Spectra of Polyatomic Molecules, Van Nostrand, New York, 1945. 14 Wolfsberg, M. and Stern, M. J., Validity of some approximation procedures used in the theoretical calculation of isotope effects, Pure Appl. Chem., 8, 225– 243, 1964. 15 Wolfsberg, M. and Stern, M. J., Secondary isotope effects as probes for force constant changes, Pure Appl. Chem., 8, 325– 338, 1964. 16 Frisch, M. J. et al. Gaussian 03, Gaussian Inc., Pittsburgh, 2003. 17 McQuarrie, D. A., Statistical Mechanics, Harper Collins, New York, 1976. 18 Bigeleisen, J. and Mayer, M. G., Calculation of equilibrium constants for isotopic exchange reactions, J. Chem. Phys., 15, 261– 267, 1947; Bigeleisen, J., The Significance of the Product and Sum Rules to Isotope Fractionation Processes, Proceedings of the International Symposium on Isotope Separation, North-Holland Publishing Co., Amsterdam, pp. 121– 157, 1958. 19 Bigeleisen, J., The relative reaction velocities of isotopic molecules, J. Phys. Chem., 17, 675– 678, 1949; Bigeleisen, J. and Wolfsberg, M., Theoretical and experimental aspects of isotope effects in chemical kinetics, In Advances in Chemical Physics, Vol. I, Prigogine, I., Ed., Interscience, New York, pp. 15 – 76, 1958. 20 Mayer, J. E. and Mayer, M. G., Statistical Mechanics, 2nd ed., Wiley, New York, 1977. 21 Bigeleisen, J. and Wolfsberg, M., Semi-empirical study of the H2Cl transition complex through the use of hydrogen isotope effects, J. Chem. Phys., 23, 793– 795, 1955. 22 Stern, M. J., Van Hook, W. A., and Wolfsberg, M., Isotope effects on internal frequencies in the condensed phase resulting from interactions with the hindered translations and rotations — the vapor pressures of the isotopic ethylenes, J. Chem. Phys., 39, 3179– 3196, 1963. 23 Stern, M. J. and Wolfsberg, M., On the absence of isotope effects in the absence of force constant changes, J. Chem. Phys., 45, 2618– 2629, 1966. 24 Stern, M. J. and Wolfsberg, M., A simplified procedure for the theoretical calculation of isotope effects involving large molecules, J. Chem. Phys., 45, 4105– 4124, 1966. 25 Leach, A. G. and Houk, K. N., Diels – Alder and ene reactions of singlet oxygen, nitroso compounds and triazolinediones: transition states and mechanisms from contemporary theory, Chem. Commun., 1234– 1255, 2002. 26 Jensen, F., Introduction to Computational Chemistry, Wiley, New York, 2002. 27 Parr, R. G. and Yang, W., Density Functional Theory of Atoms and Molecules, Oxford University Press, Oxford, 1989.

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28 Saunders, M., Laidig, K. E., and Wolfsberg, M., Theoretical calculation of equilibrium isotope effects using ab initio force constants: application to NMR isotopic perturbation studies, J. Am. Chem. Soc., 111, 8989– 8994, 1989. 29 Saunders, M., Cline, G. W., and Wolfsberg, M., Calculation of equilibrium isotope effects in a conformationally mobile carbocation, Z. Naturforsch., 44a, 480– 484, 1989. 30 Wolfsberg, M. and Saunders, M., to be published. 31 Hartshorn, S. R. and Shiner, V. J., Calculation of H/D, 12C/13C, and 12C/14C fractionation factors from valence force fields derived for a series of simple organic molecules, J. Am. Chem. Soc., 94, 9002– 9012, 1972. 32 Shiner, V. J. and Neumann, T. E., Protium – deuterium fractionation factors for organic molecules calculated from vibrational force fields, Z. Naturforsch., 44a, 337–354, 1989. 33 Goodson, D., Sarpal, S., Bopp, P., and Wolfsberg, M., The influence on isotope effect calculations of the method of obtaining force constants from vibrational data, J. Phys. Chem., 86, 659– 663, 1982. 34 Schaad, L. J., Bytautas, L., and Houk, K. N., Ab initio test of the usefulness of the Redlich – Teller product rule in computing kinetic isotope effects, Can. J. Chem., 77, 875– 878, 1999. 35 Singleton, D. A., Merrigan, S. R., Beno, B. R., and Houk, K. N., Isotope effects for Lewis acid catalyzed Diels – Alder reactions. The experimental transition state, Tetrahedron Lett., 40, 5817– 5821, 1999.

4

Condensed Matter Isotope Effects W. Alexander Van Hook

CONTENTS I. II.

Introduction ...................................................................................................................... 120 The Vapor Pressure Isotope Effect in Liquids and Solutions ......................................... 120 A. Measurements on Separated Isotopes...................................................................... 120 1. The Vapor Phase ............................................................................................... 121 2. The Condensed Phase ....................................................................................... 122 3. The VPIE ........................................................................................................... 123 B. Fractionation Factors................................................................................................ 123 C. Relation of VPIE to Condensed Phase Molecular Properties and Vibrational Dynamics ....................................................................................... 124 1. Application to Polyatomics ............................................................................... 125 2. What Happens When Molecules Condense? A Simplified Physical Picture ................................................................................................. 125 3. VPIEs in Monatomic and Polyatomic Systems. Approximate Vibrational Analysis.......................................................................................... 127 4. Monatomic Systems Continued. Accurate Calculations of VPIE.................... 128 5. Polyatomic Systems in First Approximation: The Cell Model........................ 129 6. Spectroscopic vs. Thermodynamic Precision ................................................... 130 7. A Further Approximation. The AB Equation................................................... 130 III. Illustrations. Representative Effects, Especially H/D Effects ......................................... 131 IV. Further Remarks on Connections with Spectroscopy ..................................................... 134 A. Anharmonicity.......................................................................................................... 134 B. The Dielectric Effect................................................................................................ 134 1. Example, VPIE of Carbon Disulfide ................................................................ 134 V. The Molar Volume Isotope Effect................................................................................... 136 VI. Excess Free Energies in Solutions of Isotopes. The Relation between VPIE, the Liquid Vapor Fractionation Factor, a, and RPFR .................................................... 138 VII. Anharmonicity.................................................................................................................. 139 VIII. Some Examples ..................................................................................................................... 140 A. Ethylene.................................................................................................................... 140 B. Benzene .................................................................................................................... 141 C. Water ........................................................................................................................ 142 IX. Solute and Solvent IEs in Polymer –Polymer and Polymer Solvent Mixtures............... 145 A. Demixing of Polymer – Polymer Isotopomer Solutions .......................................... 145 B. Demixing in Polymer –Solvent Systems ................................................................. 145 X. Conclusion........................................................................................................................ 148 References..................................................................................................................................... 148

119

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NOMENCLATURE CMIE EOS IE LVFF MVIE PES RPFR VCIE VPIE ZPE

Condensed matter isotope effect Equation of state Isotope effect Liquid vapor fractionation factor Molar volume isotope effect Potential energy surface Reduced partition function ratio Virial coefficient isotope effect Vapor pressure isotope effect Zero point energy

I. INTRODUCTION In this chapter we discuss isotope effects (IEs) on condensed phase physical properties like vapor pressure, molar volume, compressibility, expansivity, heats of vaporization or solution, and so forth, for liquids, solids and solutions. Some properties (for example the vapor pressure isotope effect or VPIE) are of great practical interest because they form the basis for isotope separation (e.g., distillation, solvent extraction, etc.).1 Of equal interest, however, is the fact that the sign and magnitude of condensed matter isotope effects (CMIEs) are closely related to the intermolecular forces which cause condensation. Therefore CMIEs serve as useful probes to test ideas about the nature of condensation and the motions of molecules in condensed phases. In other words there are both good practical and fundamental theoretical reasons which justify studies of CMIEs. A number of previous relevant reviews are available.1 – 8

II. THE VAPOR PRESSURE ISOTOPE EFFECT IN LIQUIDS AND SOLUTIONS We begin with the vapor pressure because it is related to the free energy of the condensed phase in a simple fashion. To illustrate, consider the measurements schematized in Figure 4.1. In Figure 4.1a the vapor pressures of two liquids or solutions of different isotopic composition are compared by difference, and at the same time the vapor pressure of one sample (conveniently the one containing the more common isotope) is measured. In the limit the experiment involves a difference measurement between pure separated isotopes. It differs in an important fashion from fractionation measurements (like distillation) carried out on mixtures of isotopes (Figure 4.1b) where the isotope concentrations in the two equilibrating phases are separately determined to define the liquid vapor fractionation factor (LVFF). In Figure 4.1a liquid and vapor phases of each sample are in equilibrium at the same temperature, and the pressure is the equilibrium vapor pressure P0 ; or P ¼ P0 2 DP: By convention the prime refers to the lighter isotopic molecule. If solutions are being studied it is necessary to determine condensed and vapor phase solute and solvent mole fractions for each sample, either by analysis or from materials balance.9 It is convenient to express VPIE using the logarithmic vapor pressure ratio, VPIE ¼ lnðP0 =PÞ:

A. MEASUREMENTS ON S EPARATED I SOTOPES When each liquid is in equilibrium with its vapor (Figure 4.1a), the thermodynamics of equilibrium requires mi 0 ðcÞ ¼ mi 0 ðvÞ and mi ðcÞ ¼ mi ðvÞ; so Dmi ðcÞ ¼ mi 0 ðcÞ 2 mi ðcÞ ¼ Dmi ðvÞ ¼ mi 0 ðvÞ 2 mi ðvÞ: Here “D” refers to an isotopic difference, light – heavy, and c and v refer to the condensed and vapor phases, respectively. For solutions, the mole fraction of the ith component in the condensed phase is designated xi ; in the vapor yi : The ms are partial molar free energies (chemical potentials), the total

Condensed Matter Isotope Effects

121

P DP

A'

A

(a)

T

To analysis

y, y' x, x' T

(b)

FIGURE 4.1 The determination of (a) vapor pressure differences between isotopomer samples and liquid vapor fractionation factors (b), compared. A and A 0 are sample containers for pure isotopomers, P and DP are pressure and differential pressure transducers, x, x 0 , y and y 0 are isotopomer concentrations in condensed and vapor phase, respectively. T ¼ thermometer. From Jancso, G., Rebelo, L. P. N., and Van Hook W. A., Chem. Soc. Rev., 257– 264, 1994. With permission.

free energies of the condensed and vapor phases are G0 ðcÞ ¼

X

X xi mi ðcÞ

ð4:1aÞ

X X 0 0 yi mi ðvÞ and GðvÞ ¼ yi mi ðvÞ

ð4:1bÞ

x0i m0i ðcÞ and GðcÞ ¼

and G0 ðvÞ ¼ 1. The Vapor Phase With the hypothetical ideal vapor at unit pressure selected as the standard state (i.e., the ideal gas at unit fugacity, fio ¼ 1Þ; the partial molar free energy of each vapor component is written as

mi ðvÞ ¼ moi ðvÞ þ RT lnðfi =fio Þ ¼ moi ðvÞ þ RT ln Pi þ RT lnð fi =ð yi PÞÞ

ð4:2Þ

P is the total pressure, and Pi the partial pressure Pi , yi P: The fugacity coefficient fi =yi P is defined in terms of the PVT properties of the gas mixture ð RT lnð fi =ð yi PÞÞ ¼ ðVi ðvÞ 2 RT=PÞdP

ð4:3Þ

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The integral extends from 0 to P, and Vi ðvÞ is the partial molar vapor volume for the ith component. For a one component system, yi ¼ 1; and Vi ðvÞ ¼ VðvÞ the total vapor volume. At low enough pressures Vi ðvÞ is given by the virial equation of state, which for one component is PVðvÞ=ðRTÞ ¼ 1 þ BP þ CP2 þ …

ð4:4Þ

RT lnð f=PÞ ¼ BP þ CP2 =2 þ …

ð4:5Þ

B and C are second and third virial coefficients in the pressure expansion. If the vapor phase is a mixture the situation gets more complicated. Most authors proceed by writing the virial coefficients as composition weighted sums. Illustrating, we find for the second virial coefficient of a two component vapor mixture BðmixÞ ¼ y21 B11 þ 2y1 y2 B12 þ y22 B22

ð4:6Þ

h12 ¼ B12 2 ðB11 þ B22 Þ=2

ð4:7Þ

and define a deviation function

This leads to expressions for the partial molar free energy of each vapor component. For a two component system

m1 ðvÞ ¼ mo1 ðvÞ þ RT lnðP1 Þ þ ðB11 þ 2y22 h12 ÞP þ …

ð4:8aÞ

m2 ðvÞ ¼ mo2 ðvÞ þ RT lnðP2 Þ þ ðB22 þ 2y21 h12 ÞP þ …

ð4:8bÞ

Higher order terms involve third, fourth, and higher virial coefficients and are usually neglected. Generalization to more components is straightforward but tedious. 2. The Condensed Phase For the condensed phase we employ the Raoult’s law reference state (i.e., the free energy of the pure liquid), except when attention is focused on the solute component in highly dilute solutions. Then the Henry’s law (infinite dilution) reference is more convenient. With Raoult’s law the free energy of the ith component is

mi ðcÞ ¼ moi ðcÞ þ RT ln xi þ RT ln gi þ

ð

Vi ðcÞdP

ð4:9Þ

Here the activity is ai ¼ gi xi ; ln gi ¼ 0 when xi ¼ 1; and Poi is the vapor pressure of the pure component reference. Also P is the total vapor pressure, and Vi ðcÞ the partial molar volume of the ith component in the condensed phase. For Henry’s law (only suitable for use with components of low concentration), ln gpi ¼ 0 when xi ¼ 0; and

mi ðcÞ ¼ mpi ðcÞ þ RT ln xi þ RT ln gpi þ

ð

Vj1 ðcÞdP

ð4:10Þ

Superscript “o” specifies the Raoult’s law, and superscript “ p ” the Henry’s law, reference. In Equation 4.10, Ppj is the pure solvent vapor pressure, and Vj1 ðcÞ is the partial molar volume of the solute at infinite dilution. The integrals in Equation 4.9 and Equation 4.10 extend from Poi or Poj to the total pressure P and correct for the change in partial molar free energy on compression (or expansion) from the standard state pressure to the total pressure. They can be neglected except at high pressure.

Condensed Matter Isotope Effects

123

For either reference state one defines the excess free energy as a difference between real and ideal solutions. For two components (neglecting the last terms in Equation 4.9 and Equation 4.10) GE ¼ RTðx1 ln g1 þ x2 ln g2 Þ

ð4:11Þ

but remember the activity coefficients are connected by the Gibbs –Duhem equation and may show an isotope effect, x1 dðln g1 Þ ¼ 2x2 dðln g2 Þ

ð4:12Þ

3. The VPIE The VPIE is obtained by equating partial molar free energies in condensed and vapor phases, remembering mi 0 ðcÞ 2 mi ðcÞ ¼ mi 0 ðvÞ 2 mi ðvÞ and using Equation 4.8a, Equation 4.8b and Equation 4.9. For the ith component in a two component (i, j) solution RT lnðP 0i =Pi Þ ¼ ðmoi ðcÞ 2 moi ðvÞÞ0 2 ðmoi ðcÞ 2 moi ðvÞÞ þ RT lnðx 0i g 0i =xi gi Þ ð ð þ V 0i dP 2 Vi dP 2 ðB 0ii þ 2y2j h0ij ÞP 0 þ ðBii þ 2y2j hij ÞP

ð4:13Þ

The integrals extend from Poi 0 or Poi to P; respectively. For the one component case Equation 4.13 simplifies; xi ¼ xi 0 ¼ 1; gi ¼ gi 0 ¼ 1; Bii ¼ B; Bii 0 ¼ B0 ; hij ¼ hij 0 ¼ 0; Poi 0 ¼ P0 ; and Poi ¼ P: Introducing upper case delta, D, to represent the isotopic difference (light – heavy), and lower case delta, d, for the phase difference (vapor – condensed), we obtain lnðP 0 =PÞ ¼ VPIE ¼ 2dDðmo Þ=RT þ ðBP 2 B 0 P 0 Þ=RT

ð4:14Þ

Equation 4.14 is a simple and important result which equates VPIE to the difference on phase change of the isotopic difference of standard state free energies, plus a small correction for vapor phase nonideality, here approximated through terms involving the second virial coefficient. It is thus limited to relatively low pressure. As T and P increase third and higher virial corrections may be needed, and at higher pressures yet the virial expansion must be abandoned for a more accurate equation of state. To make comparisons with statistical thermodynamic model calculations (most often carried out in the ðT; V; NÞ ensemble) it is useful to rewrite Equation 4.14 using Helmholtz free energy differences, dDA o, because Ao (and hence dD A o) is connected to the ðT; V; NÞ partition function, Q, in a particularly simple fashion; A ¼ 2RT ln Q (and dD A ¼ 2RT dD ln Q). From A ¼ m 2 PV we obtain dDðmo Þ ¼ dD Ao 2 ðP0 V 0 2 PVÞc where V 0 and V are condensed phase molar volumes, (in the vapor ðP0 Vv 0 Þo ¼ ðPVv Þo ¼ RTÞ so lnðP 0 =PÞ ¼ VPIE ¼ 2dDAo =RT 2 ðB0 P 0 2 BPÞ=RT þ ðP 0 V 0 2 PVÞc =RT

ð4:15Þ

B. FRACTIONATION FACTORS Now consider the phase equilibrium of a single sample containing at least two isotopic isomers (isotopomers or isotopologues) with isotopic analyses carried out on each phase after equilibrium is established (Figure 4.1b). [In this chapter we prefer the use of the older terms, “isotopomer” or “isotopic molecule” rather than “isotopologue”. In each case the terms refer to molecules of the same compound that only differ in isotopic compostion.8] Labeling the isotopomers in the two

124

Isotope Effects in Chemistry and Biology

component case as “1” and “2”, the fractionation factor, a12 ¼ a21 21 ; is defined

a12 ¼ ð y1 =y2 Þ=ðx1 =x2 Þ

ð4:16Þ

Here x and y denote mole fractions (for liquid/vapor fractionation y refers to vapor, and x to liquid). To connect a12 with thermodynamic properties consider Equation 4.8a, Equation 4.8b and Equation 4.9. Collecting terms, denoting the lighter isotope as “1” and remembering dDm ¼ 0, one finds RT ln a21 ¼ 2 dDmo þ ðB11 2 B22 ÞP þ ð y22 2 y21 Þ2h12 P ð ð 2 V1 d P þ V2 d P 2 RT lnðg1 =g2 Þ

ð4:17Þ

The integrals are from Po1 or Po2 to P, respectively. Equation 4.17 is of general validity for P and T not too large, but usually in fractionation experiments one is interested in the special case where one isotopomer is at an extremely dilute concentration. Designating the trace isotope as “2” and recognizing x1 ¼ y1 , 1; g1 ¼ 1; and P , Po1 ; Equation 4.17 simplifies RT ln a21 ¼ 2dDmo þ ðB11 2 B22 ÞP þ 2h12 P þ

ð

V2 dP þ RT lnðg2 Þ

ð4:18Þ

Introducing the Helmholtz free energy, neglecting the pressure dependence of V2 over the narrow range Po2 to Po1 ; writing DV o ¼ V1o 2 V2o ; and V2E ¼ V2 2 V2o ðV2E is the excess partial molar volume of component 2 at high dilution in component 1, it is often assumed V2E , DV o Þ; and neglecting h12 ; RT ln a21 ¼ 2dD Ao þ ðB22 2 B11 ÞP 2 RT lnðg2 Þ þ ðPo1 V1o ÞðPo2 V2E =Po1 V1o 2 DV o =V1o Þ ð4:19Þ The last term involves molar volume isotope effects (MVIEs) and excess volumes in solutions of isotopomers, and is very small (except in the critical region). It is usually neglected. Also the second term on the right can be neglected when the virial coefficient isotope effect (VCIE) and/or P is small. The result is particularly simple RT lna21 ¼ 2dD Ao 2 RT lnðg2 Þ

ð4:20Þ

RT lnðg2 Þ is the partial molar excess free energy of isotopomer 2, the solute, now at high dilution in isotopomer 1, the solvent. This term may be significantly different from zero and should not be neglected.

C. RELATION OF VPIE TO C ONDENSED P HASE M OLECULAR P ROPERTIES AND V IBRATIONAL DYNAMICS To make the connection with molecular structure and dynamics we introduced the standard state Helmholtz free energy difference, dD Ao ; to obtain Equation 4.15 and Equation 4.20. This is done because molecular properties are conveniently described using standard state canonical partition functions for the condensed and vapor phases, Qoc and Qov ; remember Ao ¼ 2RT ln Qo : The Q’s are 3nN dimensional, n is the number of atoms per molecule and N is Avogadro’s number. For convenience, from this point on we drop superscript “o’s” on the Q,s. Those “o’s” specified standard state conditions, and are now to be implicitly understood. For VPIE and a21 ; respectively, we obtain, lnðP0 =PÞ ¼ VPIE ¼ lnðQ0v Qc =Qv Q 0c Þ 2 ðB0 P 0 2 BPÞ=RT þ ðP 0 V 0 2 PVÞc =RT RT ln a ¼ lnðQ0v Qc =Qv Q0c Þ 2 RT lnðgÞ

ð4:21Þ ð4:22Þ

Condensed Matter Isotope Effects

125

Because partition functions are in principle calculable from molecular properties, Equation 4.21 and Equation 4.22 are important. They connect VPIE and ln(a), both measurable properties, with basic theoretical ideas. It remains true, of course, that condensed phase Qs are often complicated and difficult to evaluate. Except for particularly simple systems (e.g., monatomic isotopomers) it becomes necessary to introduce approximations to make further progress. We will return to Equation 4.21 and 4.22 in later sections which treat simple monatomic systems but continue here with the development of an approximate practical treatment for polyatomic molecules. The treatment is based on the one Bigeleisen developed in his influential paper on condensed phase isotope effects.10 To simplify the initial discussion we focus on Equation 4.21 and defer discussion of excess free energies in isotopomer solutions (i.e., the lnðgÞ term in Equation 4.22). 1. Application to Polyatomics In order to treat complicated polyatomic systems it is convenient to define an average molecular partition function, lnkQl ¼ ðln QÞ=N; for an assembly of N molecules. In the dilute vapor reference state this introduces no difficulty as there is no effect of intermolecular interaction and lnkQl ¼ ðln QÞ=N ¼ lnðqÞ rigorously11 (q is the microcanonical partition function). In the condensed phase, however, there are important contributions from intermolecular interactions and the canonical Qs are no longer strictly factorable. Be that as it may, the result on formal substitution is superficially the same as Equation 4.21, lnðP 0 =PÞ ¼ VPIE ¼ lnðkQ0v lkQc l=kQv lkQ0c lÞ 2 ðB 0 P 0 2 BPÞ=RT þ ðP0 V 0 2 PVÞc =RT

ð4:23Þ

Still, a nagging point of concern is that in the end one must deal with the accuracy of the approximation leading from Equation 4.21 to Equation 4.23. The equations are of fundamental importance. They relate the molecular energy states of the two isotopomers in condensed and vapor phase to VPIE using canonical partition functions. The correction terms account for the difference between the Gibbs and Helmholtz free energies of the condensed phase, and for vapor nonideality. The comparison is between separated isotopomers at a common temperature, each existing at a different equilibrium volume, V 0 or V; and different pressure, P0 or P: It is interesting that since the condensed phase Qs are sensitive functions of volume, Q ¼ QðT; V; NÞ; the rigorous use of Equation 4.23 requires detailed knowledge of the volume dependence of the partition function, and thus MVIE, since the comparisons are made at V 0 and V: This point is further developed in a later section. Reasoning from the idea that the potential energy surfaces (PES) of both isotopomers are identical (Born – Oppenheimer approximation), Bigeleisen and Mayer12 showed the Q ratio in Equation 4.23 is equivalent to the ratio of reduced partition functions (RPFRs) in the two phases, ðs=s0 Þfi ¼ ðkQl . =kQ 0 lÞqm =ðkQl=kQ 0 lÞcl (s and s 0 are symmetry numbers, qm ¼ quantum mechanical, cl ¼ classical) lnðP 0 =PÞ ¼ VPIE ¼ lnðfc =fv Þ 2 ðB 0 P 0 2 BPÞ=RT þ ðP 0 V 0 2 PVÞc =RT

ð4:24Þ

2. What Happens When Molecules Condense? A Simplified Physical Picture Our interest is in the connection between the intermolecular forces that cause condensation and the thermodynamics of CMIEs. To set the stage for later refinement we begin with a simple model. Of the 3n coordinates required to describe an n-atom molecule, 3 are reserved for the motions of the center of mass, 3 to describe angular motion (rotation, hindered rotation, or libration) about the axes used to define the principal moments of inertia (2 if the molecule is linear, 0 if monatomic), and the remaining 3n 2 6 (3n 2 5; if linear, 3n 2 3 ¼ 0; if monatomic) to describe atom to atom displacements (vibrations). In some cases it may not be possible to separate translation cleanly from

126

Isotope Effects in Chemistry and Biology

rotation and vibration, but when the separation can be made it is a great convenience. Elementary treatments assume kQl ¼ ðkQTRANS lkQROT lkQVIB lÞ

ð4:25Þ

To begin, consider the energy of an isolated pair of rotationally –vibrationally averaged molecules as their center to center distance, R, changes.13,14 In the transfer from dilute gas reference to the imperfect gas (dimer), or to the condensed phase important changes occur in all degrees of freedom. This is schematized in Figure 4.2 which shows the shifts in intermolecular potential energy, and in the internal vibrational potential energy of a single vibrational mode, as the molecules approach each other along RINTERMOL ¼ R12 : A similar diagram applies for each internal mode. In Figure 4.2 the upper path depicts transfer between dilute gas and the complexed (dimerized) vapor which lies at the bottom of the upper well. It is this path which describes the VCIE.14 The lower path refers to condensation and accounts for the VPIE. Thus, in the first case uðR12 Þ represents the pair intermolecular potential energy, in the second it represents the average potential that a single molecule feels when embedded in the field of ðN 2 1Þ molecules. The interaction is with c nearest neighbors (and ðN 2 c 2 1Þ more distant neighbors), and this increase in number accounts for the significantly deeper and sharper well for condensation. During the transfer the intermolecular PES shifts to lower energy, and, of at least equal importance, the curvature in the vibrational dimension, r, is perturbed due to coupling between internal and external degrees of freedom. Increased curvature on condensation corresponds to a blue shift (shift to higher frequency), decreased curvature to a red shift (shift to lower frequency), and many examples of such frequency shifts, albeit small, have been spectroscopically identified. External motions are also quantized. These correspond to blue shifts because ð›EPOT =›R12 Þ ¼ 0 in the ideal gas reference state, and is necessarily positive in the condensed phase. To sum up, within the precision of the Born – Oppenheimer approximation properly calculated PESs are isotope independent, but quantization of

Red energy, u(R12 or r)/ε

5

a

0

b

–5

–10

0.01 r ∆r / de –0.01 litu p am or llat i c Os 0.00

2 olecula 1. r dista nc

Interm

1.6 e, R 12 /R

12, m

2.0

in, dim

er

FIGURE 4.2 Schematic projection of the potential energy surface describing intermolecular interaction. The LJ potential energy of interaction is plotted in one plane, the shift in the position of the minimum and curvature for an internal mode, such as a CH stretch, is plotted in the other. The heavy upper curve, a, represents a “gas– gas” dimer interaction, the lower heavy curve, b, represents condensation. The lighter parabolic curves show the internal vibration in dilute gas (at the right), gas dimer, and condensed phase. The first few vibrational levels are shown. From Van Hook, W. A., Rebelo, L. P. N., and Wolfsberg, M., J. Phys. Chem. 105A, 9284– 9297, 2001. With permission.

Condensed Matter Isotope Effects

127

the kinetic energies of the various motions on the surface, vibrational energy for example, is isotope dependent because of isotopic differences in mass and mass distribution. Figure 4.2 illustrates the truism that the intermolecular interactions accounting for VCIE (upper curve) and VPIE (lower curve) differ not in kind, only in degree. The well depth for gas-gas interaction is available from analysis of the virial coefficient of the parent isotopomer, and that for the condensed phase can be obtained by combining measured energies of vaporization with the zero point energies of the condensed and ideal vapor phases. A detailed discussion is available.14 3. VPIEs in Monatomic and Polyatomic Systems. Approximate Vibrational Analysis The picture introduced above is that of a freely translating gas which drops into the potential well defined by its interaction with a large number of neighbors. We begin by considering the VPIE for monatomics in crude approximation, calculating the first quantum correction on the classical LJ potential well using the Wigner15 high temperature approximation (appropriate because the level spacing in the quantized intermolecular well is small compared to the thermal energy) lnð fc =fv Þ ¼ ð1=24Þðh=kTÞ2 k72 Ulðm0 2 mÞ

ð4:26Þ

k72Ul is the mean square force defining the motion in question and m 0 and m are reduced masses. In the harmonic approximation we obtain, lnð fc =fv Þ ¼ 3ð1=24Þðhc=kTÞ2 n 0TR 2 ð1 2 m0 =mÞ

ð4:27Þ

where n TR0 is the harmonically approximated lattice translational frequency, and the factor 3 is due to the assumption of isotropic motion. IEs calculated with Equation 4.27 are compared with experiment for several monatomic liquids at their melting and boiling points in Table 4.1. The sign and magnitude of VPIE, and its temperature dependence, are predicted within reasonable, albeit crude, approximation. In a later section we outline more sophisticated calculations in quantitative agreement with experiment. Now compare the VPIEs for inert gases (Table 4.1), small (a few tenths percent or less) and “normal” ðP 0 . PÞ; with the hydrocarbon/deuterocarbon effects also in Table 4.1 (methane and benzene as typical examples). The H/D effects are larger (a few percent) and inverse ðP 0 , PÞ: For these isotopomer pairs the small positive effect arising from quantization in the LJ potential well is overridden by a much larger negative contribution coming from zero point energy (ZPE) shifts in the internal modes. The origin of such effects was illustrated in Figure 4.2. In Figure 4.2 the intramolecular coordinate refers to an appropriate combination of bond extensions or angle TABLE 4.1 Prediction of VPIEs for Two Rare Gases and Nitrogen using a Crude Oscillator Model (Equation 4.27). Comparisons with experiment at the melting point, TM, and boiling point, TB, and with experimental VPIEs for two hydrocarbons Calc. VPIE, (Equation 4.27) System Ar/40Ar Kr/84Kr 14 N2/15N2 CH4/CD4 C6H6/C6D6

36

80

21

0

n (cm 47 32 48 — —

)

At TM 0.0068 0.00093 0.010 — —

At TB (K) 0.0063 0.00086 0.007 — —

Experimental VPIE3 At TM 0.0062 0.0010 0.006 20.014 20.028

At TB 0.0055 0.0008 0.003 20.027 20.025

128

Isotope Effects in Chemistry and Biology

deformations following normal coordinate theory.16 The shape of the potential surface changes on condensation due to intermolecular forces operating in the condensed phase. Most often the curvature of the well at its minimum is less in the condensed phase than in the gas; the frequency is red-shifted. For many internal modes excitation to upper vibrational states is negligible because the potential wells for internal modes are very deep. For example the potential well for the CH stretch in methane is more than 50 times deeper than the LJ well for condensation. The zeroth vibrational level in that potential well lies about 20 RTB above the minimum, the first excited level about 60 RTB, and thermal excitation is inconsequential (TB is the boiling temperature). Thus, the contribution of this mode to the energy is accurately described using the ZPE approximation. The relative widths of the wells are important because width determines amplitude. At the zeroth level the vibrational width is about 1% of the LJ width at RTB. The CH bond vibrates many times (35 or so), but with small amplitude, during the time the molecule is executing one relatively slow motion in the LJ potential well. This leisurely sampling of the intermolecular potential by the internal degree of freedom accounts for the success of perturbation developments. It clearly follows from the discussion that proper description of VPIE must take account of the details of molecular structure, because generally each of 3n modes is perturbed on condensation and contributes to VPIE. Before developing a formalism to accomplish that description, we will return to monatomic systems to illustrate the application of accurate theory. 4. Monatomic Systems Continued. Accurate Calculations of VPIE Lopes, Padua, Rebelo and Bigeleisen,17 and (independently) Chialvo and Horita,18 have recently described extensive and accurate calculations of VPIEs of the rare gases, mixtures of rare gases, and rare gas isotopomer mixtures. The calculations are in good agreement with each other and with the experimental data. Those data, many of them due to Bigeleisen and coworkers, have been thoroughly reviewed.3,6,19,20 To begin, Lopes et al.17 remind us that the mean intermolecular potential energy, kUl, and mean force constant k72 Ul are obtained from the pair correlation function gðrÞ of the fluid phase, ð kUl ¼ 4pr gðrÞuðrÞr 2 dr

ð4:28Þ

ð k72 Ul ¼ 4pr gðrÞ›=›rðr 2 ›uðrÞ=›rÞdr

ð4:29Þ

where uðrÞ is the intermolecular potential, r the intermolecular distance, r the number density, and the integrals extend ð0 , r , 1Þ: They applied integral equation theory21 to extend the calculation of k72 Ul across a broad range of density (using the reduced Lennard– Jones potential). Their numerical methods constitute an efficient alternative to computer simulation methods (much more lengthy and expensive), with the added benefit that the solution converges to higher numerical precision than that allowed by statistical error in the simulations. Density dependence was treated by introducing Gruneisen constants, GT ¼ 2½1=2 ½› ln k72 Ul=› ln V T and GOB ¼ 2½1=2 £ ½› lnk72 Ul=› ln V OB employed along the isothermal (subscript “T”) or orthobaric (subscript “OB”) paths, respectively. Figure 4.3a and Figure 4.3b compare calculated RPFRs and experimental results. The calculations correctly predict rare gas VPIEs and thermodynamic activities in isotopomer mixtures, using a simple two-parameter corresponding states principle, that is, from nothing more than a knowledge of their Lennard –Jones parameters and masses. The agreement with experiment is remarkably good, especially considering that the VPIEs span several orders of magnitude. At any specific reduced temperature ðT p ¼ kT=1Þ; RPFR scales approximately as ½ðDm=m2 Þ=ðs 2 1Þ ; and the excess free energies, GE ; and activity coefficient, ln g1 ; in isotopomer mixtures scale as ½ðDm=m3 Þ=ðs4 1Þ and ½ðDm=m3 Þ=ðs4 12 Þ ; respectively. The temperature dependence of the Gruneisen parameters GT and GOB is shown in Figure 4.3c. Although

Condensed Matter Isotope Effects

129

1200

1.E-01

1.35 Ne

1.05

Ar

Γr or Γ1

In(f/fgo)

T *In(f/fgo)*

1.E-02

1.E-03 Kr

200

(a)

0.7

1/T *

1.5

1.E-05

(b)

0.75

Xe

1.E-04

0.6

T*

1.3

0.45

(c)

0.6

T*

1.3

FIGURE 4.3 Theory and experiment compared for VPIEs of rare gases. (a). T p lnð fc =fg Þ ¼ T p lnð fc =fv Þ vs. 1=T p (points ¼ experiment, line ¼ calculation, triangles ¼ 20Ne/22Ne, rhombs ¼ 36Ar/40Ar, squares ¼ 80 Kr/84Kr, circle ¼ 130Xe/136Xe). (b) lnð fc =fg Þ vs. T p ; same symbols. (c) Temperature dependence of Gruneisen constants for the LJ fluid along the saturation curve (circles ¼ liquid, triangles ¼ vapor, open ¼ isothermal, filled ¼ isobaric). From Lopes, J. N. C., Padua, A. A. H., Rebelo, L. P. N., and Bigeleisen, J. J., Chem. Phys., 118, 5028– 5037, 2003. With permission.

the LJ potential is only approximate, it turns out to be an excellent choice to represent IEs in rare gas systems, and has the decided advantage of being able to describe the calculated thermodynamic properties with a single master (reduced) equation. Theory and experiment are in satisfactory agreement. 5. Polyatomic Systems in First Approximation: The Cell Model With harmonic oscillator partition functions to describe both internal and external modes the logarithmic Q ratios, lnðQv 0 Qc =Qv Qc 0 Þ ¼ lnðQc =Qc 0 Þ þ lnðQv 0 =Qv Þ; in Equation 4.21 to Equation 4.24 can be written, # " X ½ui expð2ui =2Þ=ð1 2 expð2ui ÞÞ c 0 ð4:30aÞ lnðQc =Q c Þ ¼ ½ui 0 expð2u0i =2Þ=ð1 2 expð2u0i ÞÞ c 3n and "

lnðQ0v =Qv Þ

X ½ui expð2ui =2Þ=ð1 2 expð2ui ÞÞ ¼2 ½u0i expð2u0i =2Þ=ð1 2 expð2u0i ÞÞ

# v v

ð4:30bÞ ð3n26Þint

Notice in the condensed phase the sum is over all 3n frequencies, but in the vapor phase the 6 external frequencies have null value and do not contribute, the sum is over the remaining 3n 2 6 internal frequencies. For rare gases the harmonic assumption is highly approximate, and no doubt this is also true for the lattice modes of structured molecules. However as size, mass, and molecular complexity increase, the relative contribution of the external modes becomes less and less important relative to internal contributions, and is therefore easier to approximate. Numerical evaluation of Equation 4.30a and Equation 4.30b implies the availability of a self consistent set of 3n condensed phase and 3n 2 6 vapor phase frequencies for both members of the isotopomer pair. These are best obtained from the known molecular geometry, masses, and mass distributions using an isotope independent force field consistent with the spectroscopically

130

Isotope Effects in Chemistry and Biology

observed gas phase frequencies, and the frequency shifts on condensation. It is dangerous to substitute directly observed frequencies and frequency shifts into Equation 4.30a and Equation 4.30b because spectroscopic experimental error accumulates and may result in large errors on calculated VPIEs. The recommended and now standard procedure is based on the FG matrix treatment of vibrational dynamics.16,22 Computer programs to obtain RPFR and VPIE from information on molecular geometry, atomic masses, and force constants are available. 6. Spectroscopic vs. Thermodynamic Precision To illustrate application of Equation 4.30a and Equation 4.30b consider the contribution of a single high energy vibrational mode to VPIE, say a CH vs. a CD stretch at room temperature (300 K). Using the typical CH value of 3000 cm21, shifting 10 cm21 to the red on condensation, the contribution to RPFR is lnðRPFRi Þ ¼ 2ð1=2Þðhc=ðkTÞÞðnv 2 nc Þð1 2 ðGi =G0i Þ1=2 Þ ¼ 20:0064

ð4:31Þ

where the ratio of G matrix elements has been calculated using a “diatomic approximation,” ðGi =Gi 0 Þ ¼ ½ð1=12Þ þ ð1=2Þ =½ð1=12Þ þ ð1=1Þ : While in the gas phase one can measure the frequency of each isotopomer to high precision, say ^ 0.05 cm21 or better, this is impossible in the liquid because of the inherent broadening caused by intermolecular forces. Except in special cases band centers cannot be located to better than , ^ 0.5 cm21, that limit being imposed by the nature of the liquid state. It is in no sense “a spectroscopic problem” to be eliminated by improved instrumentation or refined technique. Of course there is an identical uncertainy for each isotopomer, so

d lnðRPFRi ÞSPEC , ½2 £ 0:052 þ 2 £ 0:52

1=2

=½10ð1 2 1=21=2 Þ , 0:25

ð4:32Þ

VPIE measurements are routinely carried with a precision d lnðP0 =PÞ , ^1 £ 1024 or better in favorable cases. In the example spectroscopic uncertainty is much more than that. Of course real molecules have many vibrational frequencies and VPIE is the sum of all contributions. It becomes clear that the best physical understanding will be achieved from an interactive approach using both spectroscopic and thermodynamic information. In that process isotope independent force fields are employed to calculate gas frequencies and gas-to-liquid frequency shifts, which, on the one hand, are consistent with spectroscopic measurements, and on the other with measured IEs. 7. A Further Approximation. The AB Equation It is often useful to have an approximate relation for calculation of VPIEs when there is not enough force constant information to define the complete problem, or when a quick, albeit inexact, estimate is desired. The AB approximation introduced below accomplishes that, and in addition sometimes furnishes more physical insight than do detailed, but very complicated, calculations based on Equation 4.30a and Equation 4.30b. The equation takes advantage of the fact that often the 3n condensed phase normal modes factor in two groups; the first containing the high frequencies, ui q 1 (most often the internal modes, ui ¼ hcni =kT), where the zero point (low temperature) approximation is appropriate, and X lnð fc =fv ÞB ¼ B=T ¼ ð1=2Þðhc=ðkTÞÞ ½ðn 0i;c 2 ni;c Þ 2 ðn 0i;v 2 ni;v Þ

high freqs

ð4:33Þ

The second group contains the low frequencies which are treated in the high temperature approximation, accounting for excitation into upper levels by expanding lnðs=s0 Þf in even

Condensed Matter Isotope Effects

131

powers of u23,24 lnðs=s0 Þfi ¼

X

½ð21Þjþ1 b2j21 du2j i =ðð2jÞð2jÞ!Þ

low freqs

ð4:34Þ

2j 0 Here du2j i ¼ ui 2j 2 ui and the bs are the Bernoulli numbers ðb1 ¼ 1=6; b2 ¼ 1=30; …Þ: Reorganizing and dropping higher order terms we have from Equation 4.33 and Equation 4.34

lnð fc =fv Þ ¼ A=T 2 þ B=T

ð4:35Þ

with B defined in Equation 4.33 and A ¼ ð1=24Þðhc=kÞ2

X 02 2 ½ðn i;c 2 n2i;c Þ 2 ðn 0i;v2 2 ni;v Þ

low freqs

ð4:36Þ

with the additional simplification, should the A frequencies be limited to hindered translations and rotations, that ni;g 0 ¼ ni;g ¼ 0: Still, in many molecules there are low lying internal modes (often internal rotations or skeletal bends), and in those cases both terms contribute in Equation 4.36. A is usually positive, in the direction of a normal VPIE ðP0 . PÞ: The B term results from the intermolecular interaction which causes shifts in internal force constants on condensation. An increased force constant leads in the direction of a normal VPIE, a decrease towards an inverse effect ðP0 , PÞ: Notice the different temperature dependences. At low enough temperature A=T 2 must predominate. The IE will be normal and fall off as 1=T 2 : At intermediate temperatures the B term, which can be positive, but more often is negative, may dominate. This accounts for the commonly observed crossover to inverse IEs. At still higher temperatures, both terms decay to zero. The overall temperature dependence of VPIE is thus quite complicated. The discussion above has been under the assumption of temperature independent force constants. In real liquids we expect small temperature dependence of frequencies and force constants due to anharmonicities, lattice expansion, etc. The incorporation of these effects into VPIE formalism using pseudoharmonic lattice theory is treated in a later section.

III. ILLUSTRATIONS. REPRESENTATIVE EFFECTS, ESPECIALLY H/D EFFECTS Figure 4.4 shows VPIEs per atom D for a representative sampling of organic compounds and for water. For now we restrict attention to H/D because these are the largest IEs commonly studied, and because they have been analyzed in greatest detail. The effects range from 10% per D normal for H2O/HOD at 273 K (the freezing point, the effect in ice is even larger), to 1.6% inverse for CH3CCH/CH2DCCH at 160 K. The temperature coefficients also vary widely (from › lnðP0 =PÞ=›ð1=TÞ ¼ 2110 K for HOD at 273 K, to 1.3 K for CH2DCCH at 160 K). D for H substitution at atoms participating in hydrogen bonds in the condensed phase results in large normal VPIEs with large negative temperature coefficients (e.g., HOD/H2O, CH3CCH /CH3CCD). Large contributions from the hindered lattice modes characteristic of associated species far outweigh the redshift in internal frequencies, even though strongly associated molecules usually show larger redshifts in internal modes than do nonassociated ones. For example in the condensation of water there is a net redshift of , 280 cm21 in the internal OH stretching frequencies, but that is more than compensated by the appearance of the 3 (blue-shifted) librational (each in excess of 500 cm21) and hindered translational (, 160 cm21) modes.25,26 Similar large effects exist for deuteration on amino27 and alcoholic28,29 hydrogen. On the other hand, substitution at methyl, methylene, or phenyl, is different (much smaller) even when those sites are contained in molecules with NH2, OH, etc. associated functional groups; this is because the isotopically substituted site is not associated.

132

Isotope Effects in Chemistry and Biology 2 CD3CCH/3

1

CH2DCH3

CDH2CHCH2

CHDCH2

102 In (PD/PH)

0

CH3CDCH2

−1 −2

CH3CCD

−3 −4 −5

CH3ND2/2

HOD 2

3

4

5 6 103/T, in °K−1

7

8

FIGURE 4.4 Representative vapor pressure isotope effects for monodeutero hydrocarbons and for HOD. From Van Hook, W. A., Isotopenpraxis, 4, 161– 169, 1968. With permission.

Since VPIE is determined by intermolecular forces, significant changes should occur when one varies the condensed phase environment. This is nicely exemplified in studies of two component systems. Dutta-Choudhury, Miljevic and Van Hook30 and others31,32 have studied CMIEs for H2O/D2O dissolved in benzene, cyclohexane, and CCl4, comparing them with effects for the pure waters. Isotopic ratios of thermodynamic activity (from VPIEs of pure waters or Henry’s law coefficients for dilute solutions in hydrocarbon solvent) are shown in Figure 4.5. In the hydrocarbon solvents water is monomeric, or nearly so. For H2O/D2O in benzene the IE is about 5% inverse (lower curve) compared with the VPIE of the pure waters varying between 12 and 20% normal over the temperature range of the experiment. The IE on the free energy of transfer, DdGTRANS ¼ RT½lnðKH =KD Þ 2 lnðPH =PD Þ ; is dramatic; it is consequent to large changes in both internal and lattice frequencies on transfer from the hydrogen bonded to the nonpolar medium (KH and KD are the Henry’s law constants for H2O and D2O, respectively). The same effect can be studied spectroscopically (see Table 4.2, and the calculated line in Figure 4.5a). Figure 4.5b shows IE data on reduced transfer free energy, this time for C6H6 or C6D6 between H2O and D2O. A comparison of “1” and “2” in the figure shows ln[a(C6H6)/a(C6D6)] increases markedly (from , 22:5 to 6%) on transfer from the pure hydrocarbon liquid to the Henry’s law reference state, H2O. Moreover there is a solvent IE on this process. Comparison of “1” to “2” with “1” to “3” shows the benzene transfer free energy to the Henry’s law standard state in D2O lies about 1.5% below that in H2O. Other transfer free energies included in the figure explore interesting solute and solvent isotope dependences. In other systems large IEs on transfer free energies have been reported by Wolff and Hopfner27 for amine/hexane solutions, and Wolff et al.33 and Kooner and Van Hook34 for alcohol solutions. The first authors reported ln½aðCH3 NH2 Þ=aðCH3 ND2 Þ for (0 , x(amine) , 1). The IE falls smoothly from a large normal effect with a large temperature coefficient at x(amine) ¼ 1, to a small inverse effect with a small temperature coefficient at x(amine) ¼ 0. The effect is the one expected as the strong directionally dependent hydrogen bonding between amino groups is progressively diminished on dilution with hexane. The authors present a thermodynamic analysis. The connection between IEs on gas solubility,35 infinite dilution Henry’s law constants, and transfer free energy IEs, imply that gas-liquid chromatography is a convenient way to study such effects. That in fact is the case, and several laboratories36 – 38 have reported on the use of chromatography for isotope separation and on the interpretation of the separation factors in terms of the transfer free energy IEs.

4

3 8

−4

−.08

4

−.12

5

fe c

−.04

Ef

0

ts

1

(a)

280

300

320

T, in °K

6 -B

Bh

−8

d

6

102 In (PH/PD)

.04

8

7

6

WATER

2

BENZ. SOLN.

.08 12

9

W at er Ef fe ct s

16

TMU SOLN.

.12

UREA SOLN.

20

AQ SOLN.

133

BENZENE

Condensed Matter Isotope Effects

(b)

FIGURE 4.5 The effect of environment on IEs on thermodynamic activity in the H2O/D2O and C6H6/C6D6 systems. (a) VPIE’s for H2O/D2O in the pure liquids (upper curve) compared with their thermodynamic activity ratio at high dilution in C6H6 (lower curve).31 From Van Hook, WA., Isotopenpraxis, 4, 161– 169, 1968. With permission. (b) Comparison of IEs on thermodynamic activities, condensed to gas phase at 306 K. Dutta-Choudhury, M. K., Miljevic, N., and Van Hook, W. A., J. Phys. Chem., 86, 1711– 1721, 1982. With permission. (TMU ¼ tetramethylurea). Code 1 2 3 4 5 6 7 8 9

Solutes

Solvent(s)

C6H6/C6D6 C6H6/C6D6 C6H6/C6D6 C6H6/C6H6 C6D6/C6D6 C6H6/C6D6 C6H6/C6D6 H2O/D2O H2O/D2O

C6H6/C6D6 H2O/H2O D2O/D2O H2O/D2O H2O/D2O Urea/H2O TMU/H2O C6H6/C6H6 H2O/D2O

Remarks VPIE of separated benzene isotopomers C6H6 or C6D6 at high dilution in H2O C6H6 or C6D6 at high dilution in D2O C6H6 at high dilution in H2O or D2O C6D6 at high dilution in H2O or D2O C6H6 or C6D6 at high dilution in conc. urea/H2O soln. C6H6 or C6D6 at high dilution in conc. TMU/H2O soln. H2O or D2O at high dilution in C6H6 VPIE of separated water isotopomers

TABLE 4.2 Frequency Shifts of H2O on Transfer from the Vapor; (a) to Infinite Dilution in Benzene, and, (b) to Liquid H2O, (Room Temperature)30,31,84 Frequency 1 2 3 Librations Hindered transl Dn ¼ ðnCOND -nVAPOR Þ/(cm21).

Solvent 5 Benzene 262 8 274 ,200 Low

Solvent 5 Water 2207 50 2126 ,500 ,160

134

Isotope Effects in Chemistry and Biology

IV. FURTHER REMARKS ON CONNECTIONS WITH SPECTROSCOPY A. ANHARMONICITY In a previous section we discussed the ZPE contribution of a single high energy oscillator to VPIE, (that is to the B term). lnð fc =fv Þi ¼ ðZPEc 2 ZPEv Þ0 2 ðZPEc 2 ZPEv Þ , ðhc=2kTÞ½ðnc 2 nv Þ0 2 ðnc 2 nv Þ

ð4:37Þ

In making the connection with optical measurements it is important to recall that spectroscopic experiments do not measure ZPE itself, but rather the energy difference between the zeroth and first levels. With the curvature of PES thoroughly understood it is possible to calculate ZPE from dEð0; 1Þ: For the harmonic oscillator the result is the well known ZPE ¼ ð1=2Þhcn0 – 1 which is the result employed in the derivations above. For anharmonic surfaces small corrections (refinements) are necessary, and may be important,39 but they do not change the thrust of the development. Even for an anharmonic potential well there is a straightforward relationship between spectroscopic frequency measurements and ZPE. Wolfsberg and coworkers40 have developed the theory of anharmonic corrections to IEs on equilibrium properties.

B. THE D IELECTRIC E FFECT In applying the oscillator model to CMIEs there is an additional complication to consider when comparing thermodynamic and spectroscopic IEs. The very intermolecular interaction which we describe in terms of an effective well depth and shape, and in this way account for CMIE, also affects band shapes, centers of gravity, spectral intensities, and related spectroscopic properties. For high intensity IR bands, shape corrections for condensed phase effects are substantial; in the spectroscopic literature they are referred to as “dielectric corrections”. Dielectric corrections take account of the difference between the effective electric field of a light wave acting on a molecule in a condensed medium, and the average electric field in that medium. The two are different because of the absorption of part of the radiation. Neglect of the dielectric corrections affecting condensed phase band shapes and frequency assignments can lead to large errors in CMIEs derived from spectroscopic measurements. This was pointed out by Jancso and Van Hook41 for 13C/12C and 34 32 S/ S VPIEs in carbon disulfide.42 Wolfsberg and coworkers43,44 have provided a theoretical analysis of dielectric effect IEs with a prescription for calculation of dielectric corrections. For IR intense bands these can amount to 10 cm21 or more. Table 4.3 gives a few values. Bearing in mind that typical spectroscopic precision in the condensed phase is , ^ 0.5 to ^ 1.0 cm21, the corrections are significant. They cannot be ignored when making accurate comparisons between spectroscopic and thermodynamic CMIEs for molecules with intense IR bands. 1. Example, VPIE of Carbon Disulfide41 CS2 is a symmetric linear molecule with carbon at the center. There are five external (3 hindered translations and 2 librations) and four internal modes, the symmetric and antisymmetric stretches and the doubly degenerate bend. The spectrum is available in both phases (Table 4.4). The carbon VPIE is large and negative (2 0.0018 at 250 K, 2 0.0013 at 325 K), the sulfur effect is positive and much smaller (0.0003 at 250 K, 0.0002 at 325 K).42 Because carbon lies at both center of mass and center of symmetry there is no IE on the symmetric stretch nor on libration. The 13C/12C effect is a sum of IEs for hindered translation, degenerate bending, and assymmetric stretching, but the isotope dependence of translation and bending are both small. These contributions do not explain the unusually large carbon VPIE, nor do they rationalize the fact that it is inverse. The predominant effect is from n3 ; and its phase frequency shift determined from VPIE is 13 cm21 as compared to the spectroscopically observed (uncorrected) value of 27 cm21. The difference is far outside

Condensed Matter Isotope Effects

135

TABLE 4.3 Some Larger Dielectric Shifts Calculated for Various Frequencies of Pure Liquids Substance CO CO2 CS2 SO2

CH4 CF4 C6H6

(CH3)2CO

Frequency

cm21

n2 n3 n2 n3 n2 n1 n3 n4 n3 n4 n3 n4 n13 n12 n3 n17

2143 667 2349 397 1535 518 1151 1362 1306 3019 632 1283 673 1486 3080 1742 1218

Dielectric shift, Dn (cm21)

103 Intensity (cm2mol21)

1.3 2.6 11.6 0.4 10.3 4.0 1.6 10.9 1.3 1.2 0.7 29.7 2.5 0.2 0.4 2.0 1.3

2.7 6.2 27.4 1.4 36.8 5.2 2.1 14.1 2.7 2.3 1.9 80.1 13.1 0.9 2.0 8.7 5.6

Selected from Maessen, B. and Wolfsberg, M., Z. Naturforsch. 38a, 191–195, 1983. With permission.

the combined thermodynamic and spectroscopic experimental uncertainty, which is no more than , 2 cm21. The authors concluded that the appropriate phase frequency shift to employ in thermodynamic calculation is the one corrected for the dielectric effect. That correction had been variously estimated by spectroscopists to lie between 10 and 15 cm21 in good agreement with the thermodynamic calculation, and the later theoretical evaluation from the Wolfsberg group43,44 (Table 4.3 and Table 4.4). In the past many analyses of CMIEs depended heavily on spectroscopic measurements of ZPE differences to give limits on parameter assignments (force constant shifts). It is therefore a matter of some importance to determine the magnitude of dielectric corrections. Fortunately dielectric corrections are larger than typical spectroscopic uncertainties in phase frequency shifts only for more intense IR bands.

TABLE 4.4 Calculated and Observed Frequency Shifts on Condensation for CS2 following Jancso and Van Hook41 Frequency

n1 n2 n3 ntr nlibr a b c

Degeneracy 1 2 1 3 2

Observed. After spectroscopic correction for dielectric effect. Theoretical dielectric correction (Table 4.3) ¼ 10.3 cm21.

(ng 2 nI)OBS (cm21) 5.2 5.5 27.1a (10 to 15)b,c 252 270

(ng 2 nI)CALC (cm21) 5.2 5.5 13.1 252 270

136

Isotope Effects in Chemistry and Biology

V. THE MOLAR VOLUME ISOTOPE EFFECT Because volume is a function of T, P, and isotope substitution it is important to develop as clear an understanding of the MVIE as possible. The experimental data prior to 1970 or so were reviewed by Rabinovich,2 and later by Jancso and Van Hook.3 Around that time MVIE was discussed by Bartell and Roskos45 who used a mechanical oscillator model, considered liquids not too far above their melting points, and concluded that a major part of MVIE for larger molecules (benzene or cyclohexane for example) is due to IEs on mean square amplitudes of internal modes. Using Van der Waals radii for molecules of interest, and recognizing that ˚ (CH . CD), they rationalized properly averaged CH/CD bond lengths differ by about 0.005 A MVIEs for some hydrocarbons in terms of this “steric” or “core-size” effect. They argued that the steric contribution to MVIE is a natural consequence of the isotope independent intramolecular potential surface and is consistent with the Born – Oppenheiner approximation. Bartell and Roskos45 also applied the mechanical model to external modes finding a very small effect. The Bartell model predicts little or no temperature coefficient for MVIE and is only applicable to liquids near the melting point. Bigeleisen, Menes and Dorfmuller46 took a different approach when rationalizing MVIEs of deuteroethylenes where the relative contribution of the external modes is large. Their approach was straightforward. Since from thermodynamics ðd m=d PÞT ¼ V; they proceeded by differentiating Equation 4.35. The lattice contribution to the condensed phase MVIE is then DVðtr; libÞ ¼ 2kGðtrÞRAðtrÞ=T þ 2kGðlibÞRAðlibÞ=T

ð4:38Þ

where k is the isothermal compressibility, and G(tr or lib) ¼ 2› ln nðtr or libÞ=› ln V are Gruneisen constants for hindered translation or libration with characteristic frequency n(tr or lib). Application of Equation 4.38 to benzene47 gave results in reasonable agreement with Bartell’s estimate of the lattice contribution to MVIE. Van Hook47 pointed out that the temperature dependence of MVIE can be understood using a model which includes the steric and lattice effects described above, but also takes account of second order effects introduced by thermal expansion. In benzene, for example, the (orthobaric) liquid expands by more than eight percent between the triple and boiling points, then by another factor of three or so between the boiling point and the critical temperature (TC). The MVIE, small and positive around room temperature (, 0.2%), falls with increasing T, crossing to the negative quadrant at ð1 2 T=TC Þ , 0.1, thereafter changing exponentially as TC is approached (, 21:5% at T=TC ¼ 0:998). The departure from the first order steric-lattice harmonic model is dramatic. Isotope effects on thermal expansivity and compressibility become increasingly important, finally dominating in the near critical region.48 Very recently Van Hook, Wolfsberg and Rebelo49 have developed a corresponding states based description of molar density IEs, MrIE ¼ 1/MVIE. Commonly encountered equations of state (EOS) are based on classical analysis. They do not account for quantization effects and of themselves cannot rationalize IEs on PVT properties. Even so, the physical properties of isotopomers are almost the same, and it would be surprising if isotopomers are not in some sense in corresponding states. The authors assumed both isotopomers are described by cubic EOSs of identical form, and questioned if MrIE and VPIE can be rationalized in terms of the observed IEs on critical properties, judging success by the quality of fit over the range ð0:5 , TR ¼ T=TC , 1Þ: The method reduces the prediction of PVT IEs (i.e., the contribution of quantization) to the prediction of IEs on critical properties (or for an extended EOS, to IEs on critical properties and acentric factor9). Writing the generic Van der Waals EOS as P ¼ RT=ðV 2 b1 Þ 2 u=½VðV þ b2 Þða21Þ

ð4:39Þ

Condensed Matter Isotope Effects

137

f* MρlE =f* In[ρ'/ρ]

0.4 0.2 0.0 −0.2 −0.4 −0.6

0.03

0.1 1-T/T'CR

0.3

1

FIGURE 4.6 Molecular density IEs multiplied by a factor, f, for 3He/4He, H2/D2, C6H6/C6D6 and CH4/CD4 (points) compared with corresponding states based calculations.49 Note the ten-fold difference between helium ðf ¼ 1Þ or hydrogen ðf ¼ 1Þ; and benzene ðf ¼ 10Þ or methane ðf ¼ 10Þ: In each case the points are experimental, the lines show the corresponding states calculation.

the modification the authors found most convenient to employ set b1 ¼ b, b2 ¼ 0, and u ¼ a=T b : The reduced EOS is, PR ¼ ½4a=ða2 2 1Þ ½TR =ðVR 2 ða 2 1Þ=ða þ 1ÞÞ 2 ½ða þ 1Þ=ða 2 1Þ =½VRa TRb

ð4:40Þ

By fitting Equation 4.40 to the PVT properties of the parent isotopomers, a and b were empirically correlated with the system specific Pitzer acentric factors, v, to good precision; a ¼ 1:912 2 0:791v þ 0:969v2 ; b ¼ 2:44 2 1:06a: The acentric factors are tabulated.9 MrIE correlates very well with critical property IEs alone, no IE on v need be introduced to rationalize MrIE, but the treatment of VPIE with this approach does require introduction of an IE on the acentric factor or its equivalent. To find MrIEs Equation 4.40 was solved for the parent isotopomers at fixed TR to define a master equation, rR ¼ rR(TR, PR, v). Using the observed critical temperature IE, lnðTR 0 =TR Þ was calculated at convenient temperature intervals and lnðrR 0 =rR Þ deduced from the master equation. MrIE was recovered from the observed IE on critical density, MrIE ¼ ln(r0 /r) ¼ ln(rR0 /rR) þ ln(rC0 /rC). The results are encouraging. Calculated MrIEs for 3He/4He, H2/D2, CH4/CD4, and C6H6/C6D6 are compared with experiment in Figure 4.6. Critical property IEs for these (and some other) isotopomer pairs are shown in Table 4.5. Figure 4.6 demonstrates that the method TABLE 4.5 Critical Property IEs for Some Isotopomer Pairs2,49,86,87 Pair T 0 C (K) lnðTC 0 =TC Þ P 0 C (MPa) lnðP0C =PC Þ 103 rC 0 (mol/m3) lnðr0C =rC Þ va a

3

He/4He

H2/D2

H2O/D2O

CH4/CD4

C6H6/C6D6

3.34 20.443 0.121 20.658 13.8 20.231 20.387

33.25 20.143 1.322 20.250 15.1 20.104 20.22

647.10 0.0050 22.06 0.0180 17.9 0.0056 0.344

190.52 0.0070 4.70 20.0086 10.1 20.0081 0.008

562.10 0.0025 4.92 — 3.9 — 0.212

Pitzer acentric factor for parent isotopomer.9

138

Isotope Effects in Chemistry and Biology

straightforwardly correlates orthobaric MrIEs with IEs on critical temperature and critical density over a wide temperature range (0.5 , TR , 1) Unfortunately reliable critical property IEs are not available for very many systems. A necessary step for wider application of the method will be the theoretical calculation,17 experimental observation,2 or empirical correlation of critical property IEs (for example with the VPIE in the near critical region49).

VI. EXCESS FREE ENERGIES IN SOLUTIONS OF ISOTOPES. THE RELATION BETWEEN VPIE, THE LIQUID VAPOR FRACTIONATION FACTOR, a, AND RPFR solutions. Typical A number of authors50 – 55 have considered excess free energies in isotopomer 0 approaches expand the free energy about the equilibrium volumes, V o or V o, to obtain an expression for the contribution of DV to the excess free energy AE ðVÞ ¼ xx0 ðV 0 2 VÞ½ðd A=dVÞ 2 ðd A0 =d VÞ þ ð1=2Þxx0 ðV 0 2 VÞ2 ½xðd2 A0 =d V 2 Þ þ x0 ðd2 A=d V 2 Þ þ …

ð4:41Þ

Here x and x 0 are isotopomer mole fractions. Remembering x 0 ¼ 1 2 x; differentiating to obtain partial molar free energies (recalling mE ðVÞ ¼ AE ðVÞ 2 x 0 ðd AE ðVÞ=d x 0 Þ and mE0 ðVÞ ¼ AE ðVÞ þ x 0 ðd AE ðVÞ=d x 0 Þ one finds expressions for the excess partial molar free energies, mE ðVÞ and mE0 ðVÞ:52 In the high dilution limit, a case of special practical interest, the excess chemical potential of the trace isotopomer, say the unprimed one, is

m1;E ¼ ðV o0 2 V o0 Þ½ðd A=d VÞ 2 ðd A0 =d VÞ þ ð1=2ÞðV o0 2 V o Þ2 ½ðd2 A0 =d V 2 Þ ; x0 ¼ 1

ð4:42Þ

To this point the development is sound but difficulties arise as simplifications are introduced. Early authors continued by discarding the first term in Equation 4.42 since V o , V o0 and ½ðd A=d VÞT ¼ P , ðd A0 =d VÞT ¼ P0 , and identified ðd2 A=d V 2 ÞT with 1=ðkVÞ; k is the isothermal compressibility. It then follows m1;E ¼ ðV o0 2 V o Þ2 =ð2kV o Þ: Since RPFR ¼ 2dD Ao =RT; at this level of approximation one finds lnðgÞ ¼ VðDV=VÞ2 =ð2k RTÞ: Finally, restricting application to low enough pressure to neglect compressibility IEs lnðaÞ ¼ RPFR 2 ðV o0 2 V o Þ2 =ð2kRTV o Þ

ð4:43Þ

Equation 4.43 was long used to convert liquid vapor fractionation factors (LVFFs ¼ a) to the theoretically more interesting RPFRs. Still, Jancso and Van Hook52 argued that identification of lnðgÞ with VðDV=VÞ2 =ð2kRTÞ seriously underestimates the contribution of internal modes (i.e., the high frequency motions), because these modes make negligible contribution to k; but do make an important contribution to the free energy. They suggested the free energy of mixing be calculated following Prigogine50 in a two step process — (1) compressing or dilating each of the separated components to the molar volume of the solution, and (2) mixing at constant volume, assuming the free energy change in step (2) is zero. In terms of an oscillator model the free energy change in the first step (1) is ð ðX ½ðd lnðQi Þ=d ni Þðd ni =d VÞdV 2RT lnðgÞ ¼ 2 ðd A=dVÞdV ¼ 2RT

3n freqs

ð4:44Þ

The integral is from V to V 0 : With Equation 4.44 the disadvantages of the slowly converging Taylor series in Equation 4.41 are avoided and the contributions of internal modes properly evaluated.

Condensed Matter Isotope Effects

139

TABLE 4.6 Excess IEs in Some Isotopomer Solutions System 36

Ar/40Ar

36

Ar/40Ar in Kr CH4/CD4 HCl/DCl H2S/D2S NH3/ND3 C6H6/C6D6 c-C6H12/c-C6D12 H2O/D2O 16 H18 2 O/H2 O

Polybutadiene-h/-d Polystyrene-h/-d

T (K)

x0

(104 lng)a

84 90 118 100 170 190 230 298 298 305 278 –363 305 296 300

0 0.5 0 0.5 0.5 0.5 0.5 0.5 0.5 1 0.5 1

23

G E (J/mol)

0.015

H E (J/mol)

0.12

24

28 4 9 4

0.57 0.66 20.92 25.9 0.58 1.08

1.3 2.1

0b

0b

1.1 3.1

Ref. 4, 58, 58 4, 58, 4, 59 4 4 4, 89 4, 66 4, 66 4, 56 90 4, 56 4, 54, 4, 54,

88 88

79 79

a

Activity coefficient of the more dilute species 36Ar, HDO, or H18 2 O. But recall the complication introduced by the disproportionation equilibrium H2O þ D2O ¼ 2 HDO and its IE, lnðPHOH =PDOD Þ=lnðPHOH =PHOD Þ ¼ 1:92 ^ 0:02:72 b

With this approach apparent differences between lnð fc =fv Þ obtained via VPIE or LVFF were successfully rationalized for a number of systems,52,56 and the excess free energies in concentrated solutions of isotopomers, one in the other, successfully interpreted.54,57 – 59 Examples are shown in Table 4.6.

VII. ANHARMONICITY A realistic look at potential functions for intermolecular or intramolecular modes shows the harmonic approximation is no better than “rough-and-ready.” For external modes the best fit harmonic potential is approximately coincident with more reasonable potentials (like LJ-6/12) only over narrow ranges. Thus it is no surprise that harmonic calculations only agree with experiment over narrow temperature ranges. The harmonic model does not properly account for thermal expansion of the lattice, and for the frequency shifts associated with expansion. These contributions to lattice anharmonicity can be treated empirically using volume dependent lattice force constants following the pseudo-harmonic approximation.60 In the Gruneisen notation, Gi ¼ 2d ln ni =d ln V ¼ 2ð1=2Þd lnð fi Þ=d lnV: Gi is the Gruneisen constant for the frequency of interest ni ; and fi its harmonic force constant. The temperature dependence arises only indirectly, i.e., through the temperature dependence of the volume. Anharmonic corrections for lattice modes are important. For example in the C2H6/C2D6 system it was found that the lattice constants decrease , 8% between 120 and 200 K,60 and for C2H4/C2D4 , 10% (trans) to 20% (libration) between the triple point (104 K) and 175 K.61 Anharmonic contributions to RPFR can also be significant for internal modes. As the molecule of interest gets larger and/or heavier, either by accumulating mass in the form of heavier atoms, or by adding more atoms, the relative contribution of the external degrees of freedom to RPFR gets smaller. Massive molecules have small hindered translational and librational frequencies, n 2(tr) , 1/M and n 2(lib) , 1/I, so A in Equation 4.35 drops off rapidly with mass. If at constant mass one increases the number of bonds, the change in M and I is modest, but the number of internal oscillators increases rapidly ðd NðnÞ , 3d n; N the number of oscillators, n the number of atoms).

140

Isotope Effects in Chemistry and Biology

Thus the relative contribution of B grows at the expense of A. In the early eighties Jancso and coworkers62,63 and Van Hook47 using different lines of thought, demonstrated the contribution of internal anharmonicity to RPFR. The Hungarian group made a careful comparison of spectroscopic and VPIE data in bromoform and chloroform. Since A is small (both CHCl3 and CHBr3 are relatively massive), and well defined (via comparisons of carbon, hydrogen, and halogen IEs), B(VPIE) is well defined. Phase frequency shifts and dielectric corrections for these molecules are established with good precision. The authors62 found B from harmonic spectroscopic analysis to be inconsistent with the thermodynamic (VPIE) value. They attributed the difference to anharmonicity which they treated using a Birge –Sponer formalism. Van Hook’s treatment was different. For VPIE in C6H6/C6D6 he noted an inconsistency between the harmonically calculated and thermodynamically measured IE on the enthalpy of vaporization which exceeded any possible contribution from pseudo-harmonic lattice modes. The difference was assigned to anharmonicity in the CH/CD stretching modes and evaluated using Wolfsberg’s G0 formalism.64 To illustrate, we follow the Jancso, Jakli and Fetzer62 treatment of CHCl3/CDCl3. Most of the anharmonic effect is due to the CH/CD stretch since HCCl bending makes only a small contribution to VPIE, and all other modes contribute negligibly. lnð fc =fv ÞCH=CD ¼ ðhc=kTÞ{½ðv0c 2 vc Þ=2 2 ðv0v 2 vv Þ=2 þ ½ðX 0c 2 Xc Þ=4 2 ðX 0v 2 Xv Þ=4 }

ð4:45Þ

The vs are harmonic frequencies, X the anharmonicity constant (which is mass dependent, X 0 =X ¼ m=m0 ; m the vibrational reduced mass). The “best fit” harmonic approximation, on the other hand, gives lnð fc =fv ÞCH=CD ¼ ðhc=kTÞ{½ððn 0c 2 nc Þ=2Þ 2 ððn 0v 2 nv Þ=2Þ }

ð4:46Þ

where ni is the fundamental frequency, ni ¼ vi þ 2Xi : The anharmonicity correction is thus

d lnð fc =fv Þ ¼ ð3=4Þðhc=kTÞ½1 2 m0 =m dX 0

ð4:47Þ

where dX 0 is the shift of the anharmonicity constant on condensation. The authors deduced a 4% decrease in X 0 for the CH stretching mode of CHCl3 around room temperature. The uncertainty in this thermodynamic estimate ðd dX 0 ¼ 22:6 ^ 0:6 cm21)VPIE is less than the spectroscopic uncertainty ðd dX 0 ¼ 21:3 ^ 2 cm21)SPEC. The results indicate that intermolecular forces in liquid chloroform make the CH stretch more nearly harmonic than in the vapor.

VIII. SOME EXAMPLES To illustrate the principles developed above we now discuss CMIEs for some thoroughly studied systems.

A. ETHYLENE VPIEs for C2DH3, cis-, trans-, and gem-C2D2H2, C2D3H and C2D4 and for 12C13CH4 have been reported and thoroughly analyzed by the Bigeleisen group61,65 (Figure 4.7). The liquid vapor effects are inverse and show a large temperature coefficient. There is a large discontinuity on crystallization. The effect for C2D4 changes sign, increases to 0.0376 in the crystal at 104 K and increases even further as temperature falls. Ethylene is the first system subjected to detailed molecular vibrational analysis using the harmonic cell model22 (later refined to take account of lattice anharmonicity61). Agreement with experiment is excellent (Figure 4.7). The differences between cis-, trans-, and gem-C2H2D2 were shown to be principally due to hindered rotation in the liquid, but in addition superposed on that

Condensed Matter Isotope Effects

141

800 d4 400

d3 trans-d2

T 2 In(fc/fg)

d1

13

CH2=12CH2

0 d1 −400

trans-d2 d3

−800

d4 −1200

80

100

120

140 T

160

180

200

FIGURE 4.7 Plot of T 2 lnð fc =fg Þ ¼ T 2 lnð fc =fv Þ for some isotopically substituted ethylenes. The solid lines are theoretically calculated from an isotope independent force field. From Bigeleisen, J., Fuks, S., Ribnikar, S. V., and Yato, Y., J. Chem. Phys., 66, 1689– 1700, 1977. With permission.

effect is a ZPE contribution due to coupling of hindered rotation with certain internal modes. An interesting feature of IE calculations using comprehensive vibrational analysis is that contributions from individual frequencies can be sorted out and separately considered. Although the net IE is a complicated linear combination of all 3n motions it can be instructive to consider the contribution of individual terms. This is exemplified for the deuteroethylenes in Table 4.7 which shows the most important contributions to VPIE are from the librational and CH/CD stretching modes.

B. BENZENE High precision VPIEs for C6H5D, ortho-, meta-, and para-C6H4D2 and C6D6 are available between the normal melting and boiling points, together with the excess free energy of mixing (small and positive) for C6H6/C6D6,66 the 13C/12C effect at the boiling temperature,67 VPIEs and MVIEs for C6H6/C6D6 across the entire coexistence range to the critical points,68 and isothermal compressibility IEs between 288 and 313 K.69 The pseudo-harmonic analysis of Jancso and Van Hook57 showed good agreement with spectroscopic information, including subtle deviations from the law of the mean in the series D1, D2,…, D6, and for the IE between equivalent isomers ortho-, meta, and para-C6H4D2. For liquid benzenes the A=T 2 contribution is relatively small. At the triple point hindered translation and rotation contribute about 3 and 5% of total VPIE, respectively, at the boiling point only 2 and 3%. This is not enough to rationalize the anharmonic contribution to VPIE which Jancso and Van Hook suggest is best done by an appeal to anharmonicity in the CH/CD stretching modes using volume dependent force constants. With ðd fCH =d VÞ obtained from VPIE one finds mE ¼ 1:8 J/mol for the low temperature isotopomer mixture (0.5 C6H6 þ 0.5 C6D6) in good agreement with experiment, 2.1 ^ 0.3 J/mol.4,66 The agreement was taken as strong support for the approach described in Section VI.

142

Isotope Effects in Chemistry and Biology

TABLE 4.7 The Contribution of Various Motions to T 2 ln( fc/fv) in the Ethylene/Deuteroethylene System at 104 K d1 Hindered-tr Libration C ¼ C stretch C –H stretch CH2 bend CH2 wag (in pl) (out of pl) Torsion Nonclassical-rot(vap) Total Experiment

62 433 53 2411 1268 21222 86 2133 4 142 145

trans-d2 84 801 123 2814 1497 21425 395 2423 8 247 249

cis-d2

gem-d2

92 830 106 2803 1065 2981 2186 131 8 262 262

111 833 72 2749 293 2205 20 2114 8 269 267

d3 134 1120 122 1255 21103 21162 161 2206 11 333 334

d4 153 1408 113 21102 578 2743 50 264 13 406 407

Bigeleisen et al.61

Excess volumes in C6H6/C6D6 solutions have been measured at 298 and 318 K by DuttaChoudhury, Dessauges and Van Hook70 (Figure 4.8). The effects are small, at the minimum they only amount to 0.006 cm3 /mol (compared to V(C 6H 6) , 90 cm 3/mol and DV(C 6H 6/ C6D6) , 0.18 cm3/mol). The interpretation of the complicated shape of excess MVIE is based on the assumption that the properties of the guest molecule are determined by the host lattice. Consider the addition of an infinitesimal quantity of (smaller) C6D6 to C6H6 host. If the partial molar volume of the solute is that of the cavity determined by the host size, the total volume of the solution is greater than the ideal solution, V E is positive. Similarly at the other extreme, C6H6 as guest in C6D6, the deviation is negative. Obviously the properties of the solution do not change discontinuously, and one concludes the (V E, x) plot contains both a maximum and a minimum. The contribution of this effect, properly labeled as “steric”, is expected to be symmetric. Superposed on the steric effect is an energetic effect of sign and magnitude estimated from corresponding states theory. The sum gives reasonable agreement with experiment (Figure 4.8).70

C. WATER The principal features of water VPIEs are illustrated in Figure 4.9 which compares VPIEs for the various isotopomers from well down in the solid vapor part of the diagram to near 400 K. For 18 H2O/D2O and H16 2 O/H2 O high precision data from VPIE or LVFF, respectively, are available all the way to the critical point, a total range well in excess of 400 K. The experimental work on which the diagram is based has been often reviewed.2,3,71 VPIEs in the solid, and in the liquid not too far from the melting point, are normal and unusually large. At 273.15 K for H2O/D2O, lnðP0 =PÞSOLID ¼ 0:242 and lnðP0 =PÞLIQ ¼ 0:204: These large effects are consequent to large vibrational frequency shifts which occur on the condensation of a freely rotating vapor molecule to the librationally hindered and hydrogen-bonded condensed phase. Although the OH/OD stretching modes red shift significantly on condensation, that change is more than compensated by the very large and isotope sensitive blue shift in external frequencies. The temperature coefficient of VPIE is also unusually large. The VPIE (normal at low temperature) falls steeply from 0.204 at the triple point to 0.051 at the boiling point, continues to drop as it crosses into the inverse VPIE region at 494 K, and still falls, albeit more slowly, all the way to the critical point, 647.3 K, where it is 2 0.024. Within experimental error RPFR is zero at

143

0.6

109 VE, m3/mol

Condensed Matter Isotope Effects

a

4

b

2 0

×

c

2 4

×

d

6

×

× 0.2

0.4

0.6

C6H6

0.8

FIGURE 4.8 Excess volumes of C6H6/C6D6 solutions; circles ¼ exp. points, 298 K, Xs ¼ exp. points 318 K. Heavy solid line ¼ least squares fit. Lines “a”, “b”, “c” ¼ contributions according to semiempirical theory. Line “d” ¼ calculated total effect. From Dutta-Choudhury, M. K., Dessauges, G., and Van Hook, W. A., J. Phys. Chem., 86, 4068– 4075, 1982. With permission.

40

TOT DOT

DOD

36 32

102 In (PH2O /Px)

28

HOT

24 20

HOD

16 12 8 4

H2O78 H2O77

0 2.5

3.0

3.5

4.0

4.5

103/T, in °K−1

FIGURE 4.9 VPIEs of Waters and Ices. The points are experimental from various sources.84 The lines are calculated using an isotope independent harmonic force field consistent with spectroscopic information. From Van Hook, W. A., J. Phys. Chem., 72, 1234 –1244, 1968; Van Hook, W. A., Isotopenpraxis, 4, 161–169, 1968. With permission.

144

Isotope Effects in Chemistry and Biology

the critical point.71 The interpretation of these temperature dependences is straightforwardly connected with the loss of hydrogen bonding as temperature increases. At high enough T, Hbonding has essentially disappeared, and the librational frequencies have also all but vanished. Near TC the most important contribution to IE is from the red-shifted OH/OD stretches. Thus, in this region the interpretation is similar to that of a hydrocarbon or other Van-der-Waals bonded fluid. In treating mixtures of isotopes in aqueous systems it is necessary to consider the disproportionation, H2O þ D2O ¼ 2 HOD, while recognizing that a pure sample of HOD cannot be obtained in bulk. Van Hook72 considered numerous literature data and concluded ln½PðH2 OÞ=PðD2 OÞ =ln½PðH2 OÞ=PðHODÞ ¼ 1:910 ^ 02 (273 , T/K , 473) and is essentially independent of HOD concentration. In first approximation one would expect the value 2 predicted by the law of the geometric mean, but that law does not properly account for the interactions between internal and external modes, large and important in water because of hydrogen bonding. IEs on some of the other properties of water are shown in Table 4.8. Many properties (like the enthalpies of phase change, triple points, etc.) are closely related to VP and can be interpreted in similar terms. Molar volume effects are interesting. The high-temperature behavior of MVIE has been interpreted in Section V using corresponding states. In the low temperature liquids the molar volumes are in the order TOT . DOD . HOH . HH18O. These inverse effects have been ascribed to librational motion in the hydrogen bonded condensed phase.73 The center of mass and center of interaction (about which the molecule librates) do not coincide and coupling must be taken into account. The center of mass moves away from the oxygen progressively in the series 18 OHH, 16ODD, 16OTT and this explains the ordering in the low temperature liquids. At higher temperatures MVIE falls off to more negative values due to ever increasing isothermal compressibility (unbounded at TC). Near TC this should fix the order of isobaric molar volumes as TTO . DDO . HH18O . HH16O and the ordering of critical temperatures as HH16O . DDO . TTO. Numerous model calculations of aqueous IEs have been reported3,25,26,74 but are not summarized here due to lack of space, except to remark that VPIEs can be reasonably well correlated using simple harmonic cell models (see Figure 4.9). Such calculations show the importance of the librational hydrogen bonded modes and the stretch-libration interaction in determining VPIE for D or T substitution. Neither does space permit here a discussion of the large and interesting literature which exists on aqueous solvent isotope effects, including the effects of

TABLE 4.8 IEs on Some of the Thermodynamic Properties of Isotopic Waters3

t(triple) (8C) (t(boil) 2 100) (8C) VPIE crossover (c.o.) (8C) Critical Properties

HOH

DOD

TOT

H18 2 O

H17 2 O

0.0 0.0

3.82 1.42 220.9

4.49 1.51 190

0.38 0.15 no c.o.

0.21 0.08 no c.o.

67 46

38 25

See Table 4.5 Relative Enthalpy (Lx 2 LH2O)

258C (J mol21) 1008C (J mol21) dHFUS(J mol21), 08C (J mol21), 3.828C Cp/(J mol21 K21), 258C Cv/(J mol21 K21), 258C 106 kT (atm21), 258C

— — 6010 — 75.3 74.5 45.3

1750 810 5960 6280 84.3 83.7 46.6

1960 1210

Condensed Matter Isotope Effects

145

both electrolyte75,76 and nonelectrolyte77 solutes on LVFFs and VPIEs in H2O/HOD and H16 2 O/ O mixtures or separated H O and D O solvents, respectively. H18 2 2 2

IX. SOLUTE AND SOLVENT IES IN POLYMER– POLYMER AND POLYMER SOLVENT MIXTURES We end this chapter with a brief description of recent work on IEs in polymer systems. The motivation for such studies is partly due to interest in the unusually large IEs found in polymer systems, and partly to the general scientific and economic interest in polymers. Because CMIEs often follow 1/T or 1/T 2 dependences, one way to enhance the magnitude of IEs is to lower the temperature, but that soon reaches its natural limit. A more effective method is to note that many IEs scale with the number of isotopically substituted atoms; following that path one soon arrives at the study of polymer IEs.

A. DEMIXING OF P OLYMER – POLYMER I SOTOPOMER S OLUTIONS A natural consequence of molecular weight scaling for IEs and excess IEs is the prediction of isotopomer demixing for polymer – polymer solutions. Demixing thermodynamics was first applied to polymers by Buckingham and Hentschel78 with later refinements by Bates and coworkers,79 and Singh and Van Hook.54 For the simplest polymer solutions the free energy of mixing can be described with a one term Flory – Huggins equation of state, AE , GE ¼ c1 c2 x: The cs are volume fractions (subscripts 1 and 2 equal “H” and “D”, respectively), but for solutions of isotopomers of equal polymerization, mole fractions would serve. As the Flory– Huggins interaction parameter, x, increases to its critical limit x ¼ 2RTC ; the solution demixes at an upper critical solution temperature TC. Singh and Van Hook54 used the oscillator model to find

x ¼ ðN Gi ri =2ÞðDV=VÞðu0 2 uÞ

ð4:48Þ

N is the number of monomers per polymer, ri the number of H/D substituted bonds per monomer, Gi the Gruneisen coefficient for the effective frequency (assumed to be the CH/CD stretch), DV=V the MVIE and u ¼ hcn=kT: For polymers the relative contribution of the external modes to the excess free energy is negligible. The conditions for phase separation are obtained by differentiating the expression for total free energy (not shown here) and setting the second and third derivatives to zero. This gives f1 ¼ f2 ¼ 0:5 and ðN Gi ri =2ÞðDV=VÞu0 ð1 2 m0 =mÞ ¼ 2: The ms are reduced masses for CH/CD oscillators. The excess free energy per oscillator is small but the effects are cumulative. As the number of isotopically substituted bonds increases, the excess free energy becomes large enough to cause phase separation. For polybutadiene the critical polymerization number at room temperature is , 1.4 £ 103 in reasonable agreement with the demixing data reported by Bates and coworkers79 obtained from neutron diffraction. That experiment and its theoretical analysis offer a powerful verification of the general ideas brought forward in this review.

B. DEMIXING IN P OLYMER – S OLVENT S YSTEMS Polymer – solvent demixing diagrams (Figure 4.10a,b) generally show both upper and lower consolute branches which are functions of chain length, pressure, and H/D isotope substitution. In Figure 4.10a the upper and lower branches approach each other as polymerization number, r, increases but the natural limit, r ¼ 1; is reached before they join. A polymer/solvent mixture of this type is known as a u solution (example, polystyrene/cyclohexane). In Figure 4.10b, on the other hand, the two branches meet at a double critical (hypercritical) point at finite r. The behavior is typical of solutions in “poor” solvents (example, polystyrene/acetone). Figure 4.10a,b plot demixing curves, TDEMIX ¼ Tðc; rÞP;H=D; but equally well might have shown TDEMIX ¼ TðH=D; rÞP;c

146

Isotope Effects in Chemistry and Biology

T Xhyp

T

X

X

−1

y

(a)

T

T

X X hyp X −1

(b)

y

FIGURE 4.10 Schematics for upper and lower consolute temperature demixing in (T, C, X) space (T ¼ temperature, C ¼ volume fraction, X ¼ another variable such as 1/MW1/2, P, H/D substitution on polymer or solvent, etc.). The upper and lower branches of the two phase regions are shaded. Skewing in the (T, C ) plane is understood in terms of the Flory– Huggins theory. (a) For solutions in theta (u) solvents (like methylcyclohexane), X ¼ 1/MW1/2, there exists a range of temperature where the polymer of infinite MW at critical concentration is soluble. (b) For solutions in poor solvents (like acetone) where the upper and lower branches join at a hypercritical temperature at finite MW. The polymer of MW . MW(hypercitical) and critical concentration is insoluble at all temperatures. From Imre, A. and Van Hook, W. A., J. Phys. Chem. Ref. Data, 25, 637– 661, 1996. With permission.

TDEMIX ¼ TðH=D; PÞr;c or TDEMIX ¼ Tðc; PÞr;H=D ; that is they could have emphasized the isotope or pressure dependence of demixing at constant concentration (most likely the critical concentration) and polymerization number. Detailed investigations of demixing as a function of all these variables have been reported in considerable detail by groups in Tennessee, Warsaw, Budapest and Lisbon. The studies extend to investigations over the wide ranges of solute and solvent H/D substitution, polymer molecular weight (, 102.5 , MW , 107), and pressure (2 1 , P/MPa , 200). The resulting data have been systematically interpreted using a continuous mean field thermodynamic description which accounts for the dependence of the phase equilibria on pressure, temperature, polymerization number, and polydispersity.80 Along the critical line multidimensional scaling permits economical representation of the data. The isotope effects are large as is expected for polymer systems. Examples are shown in Figure 4.11 and Figure 4.12. Figure 4.11 shows the dependence of the upper critical demixing temperature of polystyrene/ acetone in (CH3)2CO/(CD3)2CO mixtures, yD ¼ n[(CD3)2CO]/[n(CD3)2CO þ n(CH3)2CO], at

Condensed Matter Isotope Effects

147

400

Tc /k

350

300

250 0.00

0.20

0.40

0.60 yD

0.80

1.00

FIGURE 4.11 Critical loci for some [XCRIT(PS) þ (1 2 XCRIT)((1 2 yD)(CH3)2CO þ yD (CD3)2CO)] solutions at P ¼ 0.5 MPa at several MWs (PS: polystyrene). Filled squares MW ¼ 22,000, open circles MW ¼ 13,500, open squares MW ¼ 7500.4,80 From Luszczyk, M., Rebelo, L. P. N., and Van Hook, W. A., Macromolecules, 28, 745– 767, 1995. With permission.

several molecular weights.80,81 Thus PS-22,000 is barely soluble in (CH3)2CO (the hypercritical MW for polystyrene-H at yD ¼ 0.05, is 22,000), (but an increase in pressure increases the hypercritical MW). In pure deuterated acetone, yD ¼ 1, the hypercritical MW has dropped to 13,500 (deuteration operates in the same direction as a drop in pressure, i.e., toward “poorer” 6 (a)

5

Pressure (MPa)

4 3 2 1 0 −1 320

340

360

380 400 420 Temperature (K)

440

460

FIGURE 4.12 Critical UCST and LCST cloud point loci for solutions of polystyrene (MW ¼ 22,100) in propionitrile as a function of pressure at several solvent deuterations, [(1 2 yD) CH3CH2CN þ ( yD) CH3CD2CN]. Open circles yD ¼ 0, filled circles yD ¼ 0.48, filled squares yD ¼ 0.64, filled triangles yD ¼ 0.75. From Luszczyk, M. and Van Hook, W. A., Macromolecules, 29, 6612– 6620, 1996. With permission.

148

Isotope Effects in Chemistry and Biology

solvent). At P ¼ 0.1 MPa, then, the solvent IE on hypercritical MW, ln[MW(yD ¼ 0)/ MW(yD ¼ 1)] , 0.49, a remarkably large IE, and at constant MW the solvent isotope effect on the upper critical temperature is lnðTC 0 =TC Þ13;500 , 20:22 and lnðTC 0 =TC Þ7500 , 20:09: The effect of critical enhancement is obvious. The coupling of pressure and isotope effects on demixing is interesting. A good example is shown in Figure 4.12 using the polystyrene-22100/propionitrile system.82 Here critical demixing in the (P, T) projection is plotted as a function of deuteration on the methylene group of the solvent, CH3CH2CN/CH3CD2CN. The hypercritical demixing loci move from (P ¼ 2 1 MPa, T ¼ 383 K) at yD ¼ 0 to (P ¼ 3.7 MPa, T ¼ 404 K) at yD ¼ 0.75, once again evidencing remarkably large IEs. The Lisbon/Warsaw collaborating groups continue to publish detailed experimental studies and theoretical analysis on isotope and pressure dependences on liquid – liquid demixing in interesting systems.83 Clearly, such phenomena are well worth careful study. They furnish a sensitive probe of solute/solvent intermolecular forces.

X. CONCLUSION Isotope effects on the thermodynamic and excess thermodynamic properties of condensed phases have been discussed. The effects can be understood using the statistical theory of isotope effects based on the notion that PESs which describe condensed and vapor phases are isotope independent. That granted, the interpretation of the observed effects has provided and is providing a powerful tool to aid the understanding of the nature of intermolecular interaction in the condensed phase. The excess effects are small, but they are additive, and for long chain H/D substituted polymers total effects can be, and are, genuinely large. Theoretical analysis indicates that CMIEs, including excess CMIEs, are vibrational in origin. Proper consideration of the volume dependence of vibrational properties of the component molecules is important.

REFERENCES 1 Van Hook, W. A., Isotope separation, Handbook of Nuclear Chemistry, Vol. 5, Vertes, A., Nagy, S., and Klencsar, Z., Eds., pp. 177– 212, 2003. 2 Rabinovich, I. B., Influence of Isotopy on the Physicochemical Properties of Liquids, Consultants Bureau, New York, 1970. 3 Jancso, G. and Van Hook, W. A., Condensed phase isotope effects, Chem. Rev., 74, 689– 750, 1974. 4 Jancso, G., Rebelo, L. P. N., and Van Hook, W. A., Isotope effects in solution thermodynamics: excess properties in solutions of isotopomers, Chem. Rev., 93, 2645– 2660, 1993. 5 Jancso, G., Rebelo, L. P. N., and Van Hook, W. A., Nonideality in isotopic mixtures, Chem. Soc. Rev., 257– 264, 1994. 6 Bigeleisen, J., Lee, M. W., and Mandel, F., Equilibrium isotope effects, Ann. Rev. Phys. Chem., 407– 440, 1973. 7 Imre, A. and Van Hook, W. A., Liquid– liquid equilibria in polystyrene solutions: the effect of pressure on solvent quality, Recent Res. Devel. Polym. Sci. Transworld Trivanaddam, 2, 539– 546, 1998. 8 Jancso, G., Isotope effects, Handbook of Nuclear Chemistry, 4, Vertes, A., Nagy, S., and Klencsar, Z., Eds., pp. 85 – 116, 2003. 9 Van Ness, H. C. and Abbott, M. M., Classical Thermodynamics of Solution, McGraw-Hill, New York, 1982. 10 Bigeleisen, J., Statistical mechanics of isotope effects on the thermodynamic properties of condensed systems, J. Chem. Phys., 34, 1485– 1493, 1961. 11 Fowler, R. and Guggenheim, E. A., Statistical Thermodynamics, Cambridge, Cambridge, 1939. 12 Bigeleisen, J. and Mayer, M. G., Calculation of equilibrium constants for isotopic exchange reactions, J. Chem. Phys., 15, 261– 267, 1947. 13 Moelwyn-Hughes, E. A., Physical Chemistry, Pergamon, New York, 1957, Chap.VII.

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14 Van Hook, W. A., Rebelo, L. P. N., and Wolfsberg, M., An interpretation of the vapor phase second virial coefficient isotope effect. Correlation of virial coefficient and vapor pressure isotope effects, J. Phys. Chem., 105A, 9284– 9297, 2001. 15 Wigner, E., On the quantum correction to thermodynamic equilibrium, Phys. Rev., 40, 749– 759, 1932. 16 Wilson, E. B., Decius, J. C., and Cross, P. S., Molecular vibrations, McGraw-Hill, New York, 1955. 17 Lopes, J. N. C., Padua, A. A. H., Rebelo, L. P. N., and Bigeleisen, J., Calculation of vapor pressure isotope effects in the rare gases and their mixtures using an integral equation theory, J. Chem. Phys., 118, 5028– 5037, 2003. 18 Chialvo, A. A. and Horita, J., Isotopic effect on phase equilibria of atomic fluids and their mixtures: a direct comparison between molecular simulation and experiment, J. Chem. Phys., 119, 4458– 4467, 2003. 19 Bigeleisen, J., Lee, M. W., and Mandel, F., Mean square force in simple liquids and solids from isotope effect studies, Acc. Chem. Res., 8, 179– 184, 1975. 20 Lee, M. W. and Bigeleisen, J., Calculation of the mean force constants of the rare gases and the rectilinear law of mean force, J. Chem. Phys., 67, 5634– 5638, 1977. 21 Duh, D. M. and Henderson, D., Integral equation theory for Lennard– Jones fluids: the bridge function and applications to pure fluids and mixtures, J. Chem. Phys., 104, 6742–6754, 1996. 22 Stern, M. J., Van Hook, W. A., and Wolfsberg, M., Isotope effects on internal frequencies in the condensed phase resulting from interactions with the hindered translations and rotations. The vapor pressures of the isotopic ethylenes, J. Chem. Phys., 39, 3179– 3196, 1963. 23 Bigeleisen, J., The significance of product and sum rules to isotope effect fractionation processes, Proceedings of the International Symposium on Isotope Separation, Kistemaker, J., Bigeleisen, J., and Nier, A. O. C., Eds., North Holland, Amsterdam, pp. 121– 157, 1958. 24 Bigeleisen, J., Ishida, T., and Lee, M. W., Correlation of the isotope chemistry of hydrogen, carbon and oxygen with molecular forces by the WIMPER (2) method, J. Chem. Phys., 74, 1799– 1816, 1981. 25 Van Hook, W. A., Vapor pressures of the isotopic waters and ices, J. Phys. Chem., 72, 1234– 1244, 1968. 26 Kresge, A. J., O’Ferrall, R. M., and Koeppel, G. W., Vibrational analyses of liquid water and the hydronium ion in aqueous solution, J. Am. Chem. Soc., 93, 1 – 9, 1971, Solvent isotope effects upon proton transfer from the hydronium ion. J. Am. Chem. Soc., 93, 9 – 20, 1971. 27 Wolff, H. and Hopfner, A., Wasserstoffbruckenassoziation und Dampfdruck-Isotopieeffekt von N-deuteriertem Methyl- und Athylamin, Ber. Buns. Ges. Phys. Chem., 69, 710– 716, 1965. 28 Kiss, I., Jakli, G., Jancso, G., and Illy, H., Vapor pressures of some deuterated alcohols and amines, J. Chem. Phys., 47, 4851– 4852, 1967. 29 Borowitz, J. L. and Klein, F. S., Vapor pressure isotope effects in methanol, J. Phys. Chem., 75, 1815– 1820, 1971. 30 Dutta-Choudhury, M. K., Miljevic, N., and Van Hook, W. A., Isotope effects in aqueous solution. The hydrophobic interaction. Some thermodynamic properties of benzene/water and toluene/water solutions and their isotope effects, J. Phys. Chem., 86, 1711– 1721, 1982. 31 Moule, D. C., The deuterium isotope effect and the molecular dynamics of water in benzene, Can. J. Chem., 44, 3009– 3015, 1966. 32 Goldman, S. and Backx, P., Water water – d2 solubility isotope effects. An estimate of the extent of nonclassical rotational behavior of water, when dissolved in benzene or carbon tetrachloride, J. Phys. Chem., 85, 2975– 2979, 1981. 33 Wolff, H., Bauer, D., Gotz, R., Landeck, H., Schiller, D., and Schimpf, L., Association and vapor pressure isotope effect of variously deuterated methanols in n-hexane, J. Phys. Chem., 80, 131– 138, 1976. 34 Kooner, Z. and Van Hook, W. A., Activities of H2O, D2O, CH3OH and CH3OD in hexane, decane, and hexadecane, Fluid Phase Eq., 27, 81 – 92, 1986. 35 Jancso, G., Interpretation of isotope effects on the solubility of gases, Nukleonika, 47S, 53 – 57, 2002. 36 Van Hook, W. A., Isotope separation by gas chromatography, Adv. Chem. Ser., 89, 99 –118, 1969. 37 Cartoni, G. P., Liberti, A., and Pela, A., Gas chromatographic separation of polar isotopic molecules, Anal. Chem., 39, 1618– 1622, 1967.

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38 Matucha, M., Jockish, W., Verner, P., and Anders, G., Isotope effect in gas-liquid chromatography of labeled compounds, J. Chromatogr., 588, 251– 258, 1991. 39 Jancso, G. and Van Hook, W. A., The effect of condensation on vibrational anharmonicity as determined by the vapor pressure isotope effect, J. Mol. Struct., 48, 107– 113, 1978. 40 Wolfsberg, M., Theoretical evaluation of experimentally observed isotope effects, Acc. Chem. Res., 5, 225– 233, 1972. 41 Jancso, G. and Van Hook, W. A., 13C and 32,34S Isotope effects on the vapour pressure of liquid carbon disulfide, Can. J. Chem., 55, 3371– 3376, 1977. Jancso, G., and Van Hook, W. A., The effect of intermolecular interaction on the asymmetric stretching vibration of CS2, Chem. Phys. Lett., 48, 481– 482, 1977. 42 Betts, R. H. and Buchannon, W. D., 13C and 32,34S isotope effects on the vapour pressure of liquid carbon disulfide, Can. J. Chem., 54, 3007– 3011, 1976. 43 Maessen, B. and Wolfsberg, M., Estimation of dielectric shifts in infra-red spectra of pure liquids for use in the evaluation of vapor pressure isotope effects, Z. Naturforsch., 38a, 191– 195, 1983. 44 Warner, J. W. and Wolfsberg, M., Dielectric effects on the spectra of condensed phases, J. Chem. Phys., 78, 1722–1730, 1983. 45 Bartell, L. S. and Roskos, R. R., Isotope effects on molar volume and surface tension: simple theoretical model and experimental data for hydrocarbons, J. Chem. Phys., 44, 457– 463, 1966. 46 Bigeleisen, J., Menes, F., and Dorfmuller, T., Molal volumes of the isotopic homologs of ethylene, J. Chem. Phys., 53, 2869– 2878, 1970. 47 Van Hook, W. A., Isotope effects in condensed phases; the benzene example. Influence of anharmonicity. Harmonic and anharmonic potential surfaces and their isotope independence. Molar volume isotope effects in isotopic benzene, J. Chem. Phys., 83, 4107– 4117, 1985. 48 Matsuo, S. and Van Hook, W. A., Isothermal compressibility of benzene, deuteriobenzene (C6D6), cyclohexane and deuteriocyclohexane (C6D12) and their mixtures from 0.1 to 35 MPa at 288, 298 and 313 K, J. Phys. Chem., 88, 1032– 1040, 1984. 49 Van Hook, W. A., Rebelo, L. P. N., and Wolfsberg, M., Correlation of isotope effects on PVT properties of fluids using corresponding states. Critical point shifts on isotopic substitution. J. Am. Chem. Soc., submitted. 50 Prigogine, I., The molecular theory of solutions, North Holland, Amsterdam, 1957. 51 Bigeleisen, J., Quantum effects in liquid hydrogen, J. Chem. Phys., 39, 769– 777, 1963. 52 Jancso, G. and Van Hook, W. A., The excess thermodynamic properties of solutions of isotopic isomers, one in the other, Physica, 91A, 619– 624, 1978. 53 Singh, R. R. and Van Hook, W. A., Excess free energies in solutions of isotopic isomers. I. Monatomic species. II. Polyatomic species, J. Chem. Phys., 86, 2969– 2975, 1987. 54 Singh, R. R. and Van Hook, W. A., Excess free energies in solutions of isotopic isomers. III. Solutions of deuterated and protiated polymers, Macromolecules, 20, 1855– 1859, 1987. 55 Calado, J. C. G., Dieters, U. K., Lopes, J. N. C., and Rebelo, L. P. N., The excess free energy of nuclidic liquid mixtures, Ber. Buns. Phys. Chem., 99, 721– 729, 1995. 56 Jakli, G. and Van Hook, W. A., D/H and 18O/16O fractionation factors between vapor and liquid water, Geochem. J., 15, 47 – 50, 1981. 57 Jancso, G. and Van Hook, W. A., Vapor pressure isotope effects in benzene – cyclohexane systems. III. Theoretical analysis, J. Chem. Phys., 68, 3191– 3202, 1978. 58 Calado, J. C. G., Diaz, F. A., Lopes, J. N. C., Nunes da Ponte, M., and Rebelo, L. P. N., Nonideality of an “ideal” liquid mixture. (36Ar þ 40Ar), Phys. Chem. Chem. Phys., 2, 1095– 1097, 2000. 59 Calado, J. C. G., Jancso, G., Lopes, J. N. C., Marko, L., Nunes da Ponte, M., Rebelo, L. P. N., and Staveley, L. A. K., The excess thermodynamic properties of liquid (CH4 þ CD4), J. Chem. Phys., 100, 4582– 4590, 1994. 60 Van Hook, W. A., Vapor pressures of the deuterated ethanes, J. Chem. Phys., 44, 234– 251, 1966. 61 Ishida, T. and Bigeleisen, J., Vapor pressure of the isotopic ethylenes. IV. Liquid ethylene-d3 and d4, J. Chem. Phys., 49, 5498– 5509, 1968; Bigeleisen, J., Fuks, S., Ribnikar, S. V., and Yato, Y., Vapor pressures of the isotopic ethylenes. V. Solid and liquid ethylene – d1, ethylene– d2 (cis, trans, and gem), ethylene– d3, and ethylene– d4, J. Chem. Phys., 66, 1689 –1700, 1977. 62 Jancso, G., Jakli, G., and Fetzer, C., Vapour pressure isotope effects of chloroform, Z. Naturforsch., 38a, 184– 190, 1982.

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151

63 Jancso, G., Jakli, G., and Juhasz, P., Deuterium vapour pressure isotope effect in bromoform, Acta Chim. Sci. Hung., 114, 133– 148, 1983. Toth, K., and Jancso, G., Application of vapor pressure isotope effect to the determination of vibrational anharmonicity in the liquid phase, Z. Naturforsch., 44a, 355– 358, 1989. 64 Wolfsberg, M., Correction to the effect of anharmonicity on isotopic exchange equilibria, Adv. Chem. Ser., 89, 185– 191, 1969. 65 Bigeleisen, J., Ribnikar, S. V., and Van Hook, W. A., Molecular geometry and the vapor pressure of isotopic molecules. The equivalent isomers cis-, gem- and trans-dideuteroethylene, J. Chem. Phys., 38, 489– 496, 1963; Bigeleisen, J., Stern, M. J., and Van Hook, W. A., Molecular geometry and the vapor pressure of isotopic molecules, C2H3D and 12CH2 ¼ 13CH2, J. Chem. Phys., 38, 497– 504, 1963. 66 Jakli, G., Tzias, P., and Van Hook, W. A., Vapor pressure isotope effects in the benzene(B)cyclohexane(C) system from 5 to 80 degrees C. I. The pure liquids B-d0, B-d1, ortho-, meta-, and paraB-d2, B-d6, C-d0 and C-d12. II. Excess free energies and isotope effects on excess free energies in the solutions B-h6/B-d6, C-h12/C-d12, B-h6/C-h12, B-d6/C-h12, and B-h6/C-d12, J. Chem. Phys., 68, 3177– 3190, 1978. 67 Narten, A. and Kuhn, W., Genaue Bestimmung kleiner Dampfdruckunterschiede isotoper verbindung II. Der 13C/12C-Isotopieeffekt in Tetrachlorkohlenstoff und in Benzol, Helv. Chim. Acta, 44, 1474– 1479, 1961. 68 Kooner, Z. and Van Hook, W. A., Isotope effects of simple fluids. Pressure-volume-temperature properties of benzene/deuterobenzene, acetone/acetone-d3 and methanol/methanol-d1 from 0.1 MPa and 298 K to critical conditions or to 570 K and 6 MPa, whichever is less, J. Phys. Chem., 92, 6414– 6426, 1988. 69 Matsuo, S. and Van Hook, W. A., Isothermal compressibilities of benzene-h6, benzene-d6, cyclohexane-h12, cyclohexane-d12 and their mixtures from 0 to 30 MPa at 288, 298 and 313 K, J. Phys. Chem., 88, 1032– 1040, 1984. 70 Dutta-Choudhury, M. K., Dessauges, G., and Van Hook, W. A., Excess volumes in the solutions benzene-h6/benzene-d6, cyclohexane-h12/benzene-h6, cyclohexane-h12/benzene-d6 and benzene-h6/ benzene-d6/water, J. Phys. Chem., 86, 4068 –4075, 1982. 71 Horita, J. and Wesolowski, D. J., Liquid vapor fractionation of oxygen and hydrogen isotopes of water from the freezing to the critical temperature, Geochem. Cosmochem. Acta, 58, 3425 –3437, 1994; Levelt Sengers, J. M. H., Straub, J., Watanabe, K., and Hill, P. G., Assessment of critical parameter values for H2O and D2O, J. Phys. Chem. Ref. Data, 14, 193– 207, 1985; Wagner, W. and Kruse, A., Properties of Water and Steam, Springer Verlag, Berlin, 1997; Harvey, A. H. and Lemmon, E. W., Correlation for the vapor pressure of heavy water from the triple point to the critical point, J. Phys. Chem. Ref. Data, 31, 173–182, 2002. 72 Van Hook, W. A., Vapor pressure isotope effect in aqueous systems. III. Vapor pressure of HOD (2 60 to 200 deg C), J. Phys. Chem., 76, 3040– 3043, 1972. 73 Dutta Choudhury, M. K. and Van Hook, W. A., Isotope effects in aqueous systems. 11. Excess volumes in water/water – d2 mixtures. The apparent molal volume of sodium chloride/water/water –d2 mixtures, J. Phys. Chem., 84, 2735– 2740, 1980. 74 Jancso, G. and Bopp, P., The dependence of the internal vibrational frequencies of liquid water on central force potentials, Z. Naturforsch., 38a, 206– 213, 1983. 75 See for exampleFriedman, H. L. and Krishnan, C. V., Thermodynamics if ionic hydration, Water. A Comprehensive Treatise, Vol. 3, Franks, F., Ed., Plenum, New York, pp. 1 – 118, Chap.1, 1973. 76 Horita, J., Cole, D. R., and Wesolowski, D. J., The activity composition relationship of oxygen and hydrogen isotopes in aqueous salt solutions. III. Liquid vapor equilibrations of NaCl solutions to 350 C, Geochem. Cosmochem. Acta, 59, 1139– 1151, 1995. 77 See for exampleJakli, G. and Van Hook, W. A., Isotope effects in aqueous systems. Excess thermodynamic properties of 1,3-dimethylkureasolutions in H2O and D2O, J. Chem. Eng. Data, 42, 1274– 1279, 1997, Excess thermodynamic properties of H2O and D2O solutions of tetramethylurea, an azeotropic system. Vapor pressures, excess vapor pressures, and vapor pressure isotope effects. J. Chem. Eng. Data, 46, 777– 781, 2001, and references therein. 78 Buckingham, A. D. and Hentschel, H. G. E., Partial miscibility of protonated and deuterated high polymers, J. Polym. Sci. Polym. Phys., 18, 853–861, 1980.

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79 Bates, F. S., Wignall, G. D., and Kohler, W. C., Critical behavior of binary liquid mixtures of deuterated and protonated polymers, Phys. Rev. Lett., 55, 2425– 2428, 1985. Bates, F. S., and Wignall, G. D., Nonideal mixing in binary blends of perdeuterated and protonated polystyrenes, Macromolecules, 19, 932– 934, 1986. 80 Luszczyk, M., Rebelo, L. P. N., and Van Hook, W. A., Isotope and pressure dependence of liquid– liquid equilibria in polymer solutions. 5. Measurements of solute and solvent isotope effects in polystyrene– acetone and polystyrene –methylcyclopentane. A continuous polydisperse thermodynamic interpretation of demixing measurements in polystyrene-acetone and polystyrene-methylcyclopentane solutions, Macromolecules, 28, 745– 767, 1995. 81 Szydlowski, J. and Van Hook, W. A., Isotope and pressure effects on liquid– liquid equilibria in polymer solutions. H/D solvent isotope effects in acetone-polystyrene solutions, Macromolecules, 24, 4883– 4891, 1991. 82 Luszczyk, M. and Van Hook, W. A., Isotope and pressure effects on liquid – liquid equilibria in polymer solutions. 7. Solute and solvent H/D isotope effects in polystyrene-propionitrile solutions, Macromolecules, 29, 6612– 6620, 1996. 83 Rebelo, L. P. N., Visak, Z. P., de Sousa, H. C., Szydlowki, J., de Azevedo, R., Ramos, A. M., Najdanovic-Visak, V., Nunes da Ponte, M., and Klein, J., Double critical phenomena in (water þ polyacrylamide) solutions, Macromolecules, 35, 1887 – 1895, 2002; Najdanovic-Visak, V., Esperanca, J. M. S. S., Rebelo, L. P. N., Nunes da Ponte, M., Guedes, H. J. R., Seddon, K. R., de Sousa, H. C., and Szydlowski, J., Pressure, isotope and water co-solvent effects in liquid– liquid equilibria of (ionic liquid þ alcohol) systems, J. Phys. Chem., 107B, 12797– 12807, 2003. 84 Van Hook, W. A., Condensed phase isotope effects, Isotopenpraxis, 4, 161– 169, 1968. 85 Imre, A. and Van Hook, W. A., Liquid– liquid demixing from solutions of polystyrene. 1. A review. 2. Improved correlation with solvent properties, J. Phys. Chem. Ref. Data, 25, 637– 661, 1996. 86 Landolt, H. and Bornstein, R., Zahlenwerte und Funktionen, IV, 316 ff, 1967, 1971; IV; 632 ff. 87 Grigor, A. F. and Steele, W. A., Physical properties of fluid CH4 and CD4: Experimental, J. Chem. Phys., 48, 1032–1037, 1968. 88 Lee, M. W., Eshelman, D. M., and Bigeleisen, J., Vapor pressures of isotopic krypton mixtures. Intermolecular forces in solid and liquid krypton, J. Chem. Phys., 56, 4585– 4592, 1972. 89 Lopes, J. N. C., Rebelo, L. P., and Calado, J. C. G., Deviations from ideal behavior in isotopic mixtures of ammonia, J. Chem. Phys., 115, 5546– 5553, 2001. 90 Jancso, G. and Jakli, G., Vapor pressure and ideality of the equimolar mixture of H2O and D2O, Aust. J. Chem., 33, 2357– 2362, 1980.

5

Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes Janet E. Del Bene

CONTENTS I. II.

Introduction ...................................................................................................................... 153 Hydrogen Bond Types ..................................................................................................... 154 A. Traditional ................................................................................................................ 154 B. Ion-Pair..................................................................................................................... 154 C. Proton-Shared........................................................................................................... 154 III. X – H Stretching Bands in the IR Spectra of Complexes with X – H – Y Hydrogen Bonds....................................................................................... 154 A. Anharmonicity Effects ............................................................................................. 154 B. Matrix Effects........................................................................................................... 158 C. Deuterium Substitution Effects on Proton-Stretching Frequencies ........................ 162 IV. Two-Bond Spin – Spin Coupling Constants across Hydrogen Bonds ............................. 165 A. Anharmonicity and Field Effects ............................................................................. 166 B. Isotopic Substitution Effects on Zero-Point Motion and Thermal Vibrational Averaging of Coupling Constants ......................................... 170 V. Concluding Remarks........................................................................................................ 171 References..................................................................................................................................... 172

I. INTRODUCTION During the last thirty years of the twentieth century, dramatic progress was made in ab initio studies of hydrogen-bonded complexes.1 At the turn of the century, the minimum level of theory required to obtained reliable structures and binding energies had been identified. Detailed studies of these complexes at very high levels of theory had also been carried out which demonstrated that both the structures and binding energies of complexes could be computed to an accuracy comparable to that of experimental data.2 – 5 Moreover, if the anharmonicity correction was not usually large, the shift of the X –H stretching band upon formation of an X – H· · ·Y hydrogen bond could be reliably estimated, and computed spectra could be used to assist in the assignment of bands in the experimental IR spectra of hydrogen-bonded complexes. Despite the tremendous work of many investigators and the remarkable insights that have been gained, there are still some very fundamental and challenging theoretical and experimental problems related to hydrogen-bonded complexes that have not been fully investigated. These include the unusually large shift to low frequency of the proton-stretching X – H band in the infrared (IR) spectrum of some complexes as a result of increased anharmonicity of the X – H stretch; environmental effects on the X –H stretching band; the NMR property of two-bond X –Y spin –spin 153

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coupling across a hydrogen bond; and the effect of isotopic substitution on both IR and NMR properties. In this chapter, the results of ab initio studies carried out in this laboratory that address some of these problems will be presented.

II. HYDROGEN BOND TYPES Because IR and NMR properties of hydrogen-bonded complexes can vary with hydrogen bond type, it is advantageous to begin by giving the following general classification of hydrogen bonds.

A. TRADITIONAL A traditional hydrogen bond has a normal (as opposed to short) X –Y distance, and a slightly elongated X –H bond relative to the monomer. The X – H stretching band in the infrared spectrum is shifted downfield relative to the monomer, and has a significantly increased intensity. Such hydrogen bonds are designated X –H· · ·Y. Traditional hydrogen bonds are by far the most common. FH:NH3, ClH:NH3, BrH:NH3, and ClH:pyridine are stabilized by traditional hydrogen bonds in the gas-phase.6 – 8

B. ION -PAIR An ion-pair hydrogen forms as a result of proton transfer from X to Y, yielding an X2· · ·þH – Y bond. The X – Y distance in such a complex is similar to that in the corresponding complex with a traditional hydrogen bond. The Y – Hþ distance is slightly elongated relative to the corresponding cation, and an intense Y – Hþ stretching band appears in the IR spectrum at a frequency that is lower than that of the isolated cation. The BrH:N(CH3)3 complex has an ion-pair hydrogen bond in the gas phase.7

C. PROTON -S HARED If the X –H proton is not completely transferred to Y, then a proton-shared X· · ·H· · ·Y hydrogen bond forms. In a proton-shared hydrogen bond the X – Y distance is short, and the frequency shift of the proton-stretching band is unusually large, giving rise to a very intense low-frequency band. Other strong bands may also appear at relatively low frequencies, depending on the nature of the proton donor and proton acceptor moieties. The gas-phase structure of ClH:N(CH3)3 is stabilized by a proton-shared hydrogen bond.7 In summary, three types of hydrogen bonds have been identified experimentally from microwave spectroscopy.6 – 8 Their existence can be inferred from experimental IR spectral data,9 and from the temperature-dependence observed experimentally for spin – spin coupling constants across hydrogen bonds. (See Chapter 7.) From a theoretical point of view, the identification of hydrogenbond type has been a key factor in gaining insight into both IR and NMR properties of complexes.

III. X–H STRETCHING BANDS IN THE IR SPECTRA OF COMPLEXES WITH X–H–Y HYDROGEN BONDS A. ANHARMONICITY E FFECTS In most hydrogen-bonded complexes, the frequency shift of the X –H proton-stretching band upon formation of a traditional X – H· · ·Y hydrogen bond can be estimated reliably at second-order Møller – Plesset perturbation theory with the 6-31 þ G(d,p) basis set [MP2/6-31 þ G(d,p)]. The shift can be obtained by computing the harmonic vibrational spectra of the proton-donor monomer and the complex, and then taking the difference between the frequencies of the two X –H stretching bands. Figure 5.1 shows the experimental Ar matrix10 and the computed gas-phase spectrum of FH:NH3. These spectra are similar, although the bands in the computed spectrum are found at

FH:NH3

a

875

3041

−1697

I

6

b

−1093 −916

3 −1679

A

1250

500

BrH:NH3 −1163

a

−729

−1689

0

−2006

I

2000

2750

−800

0 3500 2500

155

−1195 −1054

1750

−3296

Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes

.5

b

A

0 2200

1700

1200 n

−591

−1147

.25

700

200

FIGURE 5.1 Computed harmonic spectra (a) and experimental Ar matrix spectra (b) of FH:NH3 and BrH:NH3.

higher frequencies than the corresponding experimental bands. This is primarily a result of neglecting the anharmonicity of vibrational motion. Nevertheless, it is apparent that the strong bands in the experimental spectrum could be assigned from the computed spectrum. This implies that for this and many other similar complexes that are stabilized by traditional hydrogen bonds, the anharmonicity correction for the proton-stretching vibration arises primarily in the monomer, and is only slightly increased in the complex. There are some complexes for which the harmonic approximation fails miserably, even when the complex is stabilized by a traditional hydrogen bond. This is evident by comparing the experimental and the computed harmonic spectra of BrH:NH3,11,12 also shown in Figure 5.1. Other

156

Isotope Effects in Chemistry and Biology

examples of the inadequacy of the harmonic approximation are found for complexes of the hydrogen halides HCl and HBr with nitrogen bases such as NH3, N(CH3)3, and pyridine and its derivatives.9,13 – 16 An interesting example is the ClH:pyridine complex for which the computed harmonic IR spectrum bears little resemblance to the spectrum observed experimentally, despite the fact that the computed gas-phase structure of the ClH:pyridine complex and the computed proton affinity of pyridine agree with experimental gas-phase data.17 An early semi-empirical attempt to resolve this problem used a combination of experimental Ar and N2 matrix data, derived HCl distances, and effective anharmonic force constants for the Cl – H and N –H stretches and the coupling between them.16 With these parameters, it was possible to simulate the entire anharmonic spectrum of this complex and reproduce the frequencies and intensities observed experimentally for ClH:pyridine in both Ar and N2 matrices. While this study provided insight into the anharmonicity of the X –H stretch and the influence of the matrix, such an approach is limited insofar as both the computed spectra and the experimental spectra for a closely related series of complexes must be available. A more general approach for investigating anharmonicity effects was initiated in studies of complexes of hydrogen halides (HX, for X ¼ F, Cl, Br) with NH3.18,19 These complexes were chosen because they are structurally similar but have dramatically different experimental IR spectra in Ar and N2 matrices. Two-dimensional potential energy surfaces in the X –H and N –H coordinates were generated for these complexes, and model two-dimensional vibrational problems were solved to obtain anharmonic eigenvalues and eigenvectors for the dimer- and protonstretching modes. The choice of these complexes was indeed serendipitous. Although all three complexes have equilibrium structures stabilized by traditional hydrogen bonds, they are prototypical of three distinctly different types of complexes with respect to the nature of their potential energy surfaces, and the shape and location on those surfaces of wavefunctions for the ground state and lower-energy vibrational states. Figure 5.2 shows the potential surface for FH:NH3 with the square of the ground-state wavefunction superimposed on it. Figure 5.3 and Figure 5.4 show the potential surfaces for ClH:NH3, BrH:NH3, ClH:N(CH3)3, and BrH:N(CH3)3.15 Superimposed on these surfaces are the square of the wavefunctions for the ground ðv ¼ 0Þ state (Figure 5.3) and first-excited ðv ¼ 1Þ state (Figure 5.4) of the proton-stretching mode. From Figure 5.2 it is apparent that the FH:NH3 complex is confined within the potential well surrounding the equilibrium structure in the ground state. This is also the case for the v ¼ 1 state of the proton-stretching mode. As a result, the anharmonicity correction for the F –H stretching

2.5

2.0

H−F distance (bohr)

3.0

1.5 2.0

2.5

3.0 3.5 N−H distance (bohr)

4.0

4.5

FIGURE 5.2 Square of the ground state vibrational wavefunction superimposed on the potential energy surface of FH:NH3 at zero-field strength. Contour values are at 0.0005, 0.001, 0.002, 0.003, 0.005, 0.01, 0.02, 0.03, 0.04, and 0.05 au above the global minimum. (Source: Reprinted with permission from Del Bene, J. E. and Jordan, M. J. T., J. Chem. Phys. 108, 3205, Copyright 1998, American Institute of Physics.)

Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes

157 4.5

ClH:NH3

BrH:NH3

4.0

3.0 2.5

(a)

(b)

2.0

ClH:N(CH3)3

BrH:N(CH3)3

4.0

X−H distance (bohr)

3.5

3.5 3.0 2.5

(c) 1.5

(d) 2.0

2.5

3.0

3.5

1.5

2.0

2.5

3.0

3.5

2.0 4.0

N−H distance (bohr) FIGURE 5.3 Square of the ground state vibrational wavefunctions for (a) ClH:NH3, (b) BrH:NH3, (c) ClH:N(CH3)3, and (d) BrH:N(CH3)3 superimposed on the potential energy surfaces at zero field strength. Contour values are at 0.0005, 0.001, 0.002, 0.003, 0.005, 0.01, 0.02, and 0.03 au above the global minimum. (Source: Reprinted with permission from Jordan, M. J. T. and Del Bene, J. E., J. Am. Chem. Soc. 122, 2101, Copyright 2000, American Chemical Society.)

frequency is not significantly greater in the complex than it is in the monomer, and as noted above, the computed shift of the F– H stretching band evaluated from harmonic frequencies is in acceptable agreement with the shift obtained from experimental Ar matrix data. The situations for the ClH:NH3 and BrH:NH3 complexes are quite different. The computed gasphase harmonic proton-stretching frequency of ClH:NH3 is 2289 cm21, which is significantly greater than the experimental Ar matrix value of 1371 cm21.13 (The computed results for HCl and its complexes are MP2/aug0 -cc-pVDZ results. See Ref. 19.) Since the difference between the computed harmonic and the experimental anharmonic gas-phase Cl –H stretching frequency is only 146 cm21, the shift of the Cl –H stretching band upon complex formation is significantly underestimated. That the anharmonicity correction in the complex must be large can be inferred from Figure 5.3 and Figure 5.4. In the ground state, the ClH:NH3 complex is essentially localized in the region of the surface surrounding the equilibrium structure which has a traditional hydrogen bond, and the proton-shared region, although sampled, is not readily accessible in this state. However, the proton-shared region is accessible in the v ¼ 1 state. The potential surface in this region is relatively flat, and this introduces a large anharmonicity into the proton-stretching vibration and coupling between proton- and dimer-stretching modes. The failure of the harmonic approximation for the BrH:NH3 complex is even more dramatic, as evident from Figure 5.1. Although the equilibrium structure of this complex is also stabilized by a traditional hydrogen bond, Figure 5.3 and Figure 5.4 show that the proton-shared region of the potential surface is accessible in both the v ¼ 0 and v ¼ 1 states of the proton-stretching mode. This introduces a very large anharmonicity correction, and leads to significant coupling between

158

Isotope Effects in Chemistry and Biology 4.5 ClH:NH3

BrH:NH3

4.0

3.0 2.5

(a)

(b)

2.0

ClH:N(CH3)3

BrH:N(CH3)3

4.0

X−H distance (bohr)

3.5

3.5 3.0 2.5

(c) 1.5

(d) 2.0

2.5

3.0

3.5

1.5

2.0

2.5

3.0

3.5

2.0 4.0

N−H distance (bohr) FIGURE 5.4 Square of the wavefunctions for the v ¼ 1 state of the proton-stretching vibration for (a) ClH:NH3, (b) BrH:NH3, (c) ClH:N(CH3)3, and (d) BrH:N(CH3)3 superimposed on the zero-field potential energy surfaces. Contour values are the same as those in Figure 5.3. (Source: Reprinted with permission from Jordan, M. J. T. and Del Bene, J. E., J. Am. Chem. Soc. 122, 2101, Copyright 2000, American Chemical Society.)

dimer- and proton-stretching vibrations. Thus, while the experimental Ar matrix proton-stretching band is found at 728 cm21,11 the computed harmonic proton-stretching band has a frequency of 2289 cm21. The two-dimensional gas-phase anharmonic frequency is 908 cm21. These data, as well as corresponding frequency data and surfaces for ClH:N(CH3)3 and BrH:N(CH3)314,15 clearly demonstrate the importance of an anharmonic treatment of the proton-stretching vibration in such complexes.

B. MATRIX E FFECTS An examination of Table 5.1 shows that the computed two-dimensional anharmonic gas-phase (zero-field) proton-stretching frequencies for ClH:NH3 and BrH:NH3 are higher than the experimental Ar matrix values. Since it is well-known that the matrix influences the frequency of the proton-stretching vibration,20 the matrix effect was mimicked by applying external electric fields along the hydrogen-bonding axis,12,15,18 with field strengths consistent with Onsager’s model for dipolar fluids.21 Since the hydrogen-bonded complexes of the hydrogen halides with nitrogen bases such as NH3, N(CH3)3, and pyridine have large dipole moments, these complexes can polarize adjacent matrix atoms or molecules, which in turn, can further polarize the complex. This suggests that the matrix preferentially stabilizes more polar structures. This effect can be seen in Figure 5.5, which shows one-dimensional proton-stretching curves for ClH:NH3 as a function of field strength. The curves have been drawn so that the equilibrium structure on each curve is found at zero displacement along the normal coordinate vector. At zero field the minimum in the curve is the complex with a traditional hydrogen bond, and the shoulder that appears at higher energy

Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes

159

TABLE 5.1 Anharmonic Two-Dimensional Proton-Stretching Frequencies (cm21) as a Function of Field Strength for Complexes of HCl and HBr with NH3 and N(CH3)3 and Deuterated Analogues Field

ClH:NH3

ClD:ND3

0.0000 0.0010 0.0025 0.0040 0.0055 0.0100 0.0150 Experimental Ar matrixa Experimental N2 matrixa

1567 1415 1219 1067 936 1137 1781 1371 720

1263 1136 932 758 654 886 1419 1113 550

Field

BrH:NH3

BrD:NH3

0.0000 0.0005 0.0010 0.0025 0.0040 0.0080 0.0120 Experimental Ar matrixb Experimental N2 matrixb

908 865 833 809 954 1544 2032 728 1386

633 570 518 536 758 1247 1601

Field

ClH:N(CH3)3

ClD:N(CH3)3

0.0000 0.0010 0.0040 Experimental Ar matrixc Experimental N2 matrixc

1134 1221 1478 1486 1615

811 896 1118 1244 1344

Field

BrH:N(CH3)3

BrD:N(CH3)3

0.0000 0.0010 0.0040 Experimental Ar matrixb Experimental N2 matrixb

1595 1683 1933 1660 1890, 1782

1207 1274 1455

a b c

Ref. 13. Ref. 14. Ref. 22.

corresponds to a proton-shared structure. As the external field increases, the energy difference between the minimum and the shoulder decreases as the complex acquires increased proton-shared character. At a field strength of 0.0055 au, the curve exhibits a broad, flat minimum, and corresponds to a complex with a quasi-symmetric proton-shared hydrogen bond. (A quasisymmetric proton-shared hydrogen bond is one in which the proton is shared equally between the hydrogen-bonded atoms, meaning that the forces exerted on H by the two hydrogen-bonded atoms are equal.) The lowest curve in Figure 5.5 corresponds to a field strength of 0.0100 au, and indicates

160

Isotope Effects in Chemistry and Biology −516.63 −516.64

Energy, a.u.

−516.65 −516.66 −516.67 −516.68 −516.69 −516.70 −516.71

−1.5

−1.0 −0.5 0.0 0.5 1.0 Normal Coordinate Displacement, √amu bohr

1.5

FIGURE 5.5 One-dimensional curves along the normal coordinate for the proton-stretching mode obtained from the two-dimensional surfaces as a function of field strength. The curve at highest energy (solid) is the zero-field curve. The remaining curves are obtained from surfaces with external fields of 0.0010, 0.0025, 0.0040, 0.0055, and 0.0100 au. The minima on these curves occur at lower energy as the field strength increases. (Source: Reprinted with permission from Jordan, M. J. T. and Del Bene, J. E., J. Am. Chem. Soc. 122, 2101, Copyright 2000, American Chemical Society.)

that the equilibrium structure is now on the ion-pair side of quasi-symmetric. At higher fields, the ion-pair character of the hydrogen bond increases. The effect of external electric fields on the two-dimensional potential energy surface of ClH:NH3 is illustrated in Figure 5.6. As the field strength increases, the minimum moves along the surface in a valley connecting the traditional, proton-shared, and ion-pair regions, and the square of the ground-state vibrational wavefunction follows the minimum. At a field of 0.0055 au, the minimum in the potential surface is very broad and flat, and the wavefunction is delocalized and centered in this valley. As the field strength increases, the wavefunction moves toward the ion-pair region, and becomes more localized. The changes in the potential surfaces and wavefunctions as a function of field strength are accompanied by changes in two-dimensional anharmonic proton-stretching frequencies, as well as changes in equilibrium and expectation values of Cl – N and Cl – H distances, as seen from Table 5.1 and Table 5.2. The variation in the proton-stretching frequency and the expectation value of the Cl – N distance for ClH:NH3 is illustrated graphically in Figure 5.7, which also shows the values of the experimental proton-stretching frequencies in Ar and N2 matrices.13 It is apparent that a weak field of between 0.0010 and 0.0025 au is sufficient to reduce the computed two-dimensional anharmonic Cl – H stretching frequency and bring it into agreement with the experimental Ar matrix value. Further increasing the field strength decreases the proton-stretching frequency until it exhibits its minimum value at a field of 0.0055 au, as the hydrogen bond approaches a quasi-symmetric proton-shared bond. The computed anharmonic proton-stretching frequency at this field strength is 936 cm21, which is about 200 cm21 higher than the N2 matrix value. Further increasing the field leads to an increase in the frequency for the proton-stretching vibration, which now corresponds to a perturbed N – H stretch. Thus, these data suggest that the ClH:NH3 complex is stabilized by a traditional hydrogen bond in an Ar matrix and a protonshared hydrogen bond in an N2 matrix. Figure 5.7 also shows that the expectation value of the Cl – N distance changes with field strength in a manner similar to the proton-stretching frequency.

Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes

161 4.5

0.0010 au field

0.0025 au field

4.0 3.5 3.0 2.5

(b)

2.0

0.0040 au field

0.0055 au field

4.0 3.5 3.0 2.5

(c)

(d)

Cl−H distance (bohr)

(a)

2.0

0.0100 au field

0.0150 au field

4.0 3.5 3.0 2.5

(e) 1.5

(f) 2.0

2.5

3.0

3.5

1.5

2.0

2.5

3.0

3.5

2.0 4.0

N−H distance (bohr) FIGURE 5.6 Square of the ground state vibrational wavefunctions for ClH:NH3 superimposed on the potential surfaces at field strengths of 0.0010, 0.0025, 0.0040, 0.0055, 0.0100, and 0.0150 au. Contour values are the same as those in Figure 5.3. (Source: Reprinted with permission from Jordan, M. J. T. and Del Bene, J. E., J. Am. Chem. Soc. 122, 2101, Copyright 2000, American Chemical Society.)

It initially decreases as the field increases, exhibits its minimum value for the near quasisymmetric proton-shared hydrogen bond, and then increases as the ion-pair character increases. The shortening of the Cl –N distance in a proton-shared hydrogen bond facilitates proton transfer from Cl to N. The proton-stretching frequencies and the expectation values of the X – N distances as a function of field strength for BrH:NH3, ClH:N(CH3)3, and BrH:N(CH3)3 are also reported in Table 5.1 and Table 5.2, and plotted in Figure 5.8 to Figure 5.10, respectively. The experimental values of the proton-stretching frequencies in both Ar and N2 matrices are indicated.14,22 Although the BrH:NH3 complex is stabilized by a traditional hydrogen bond in the gas-phase, the hydrogen bond approaches a quasi-symmetric proton-shared bond in an Ar matrix and has ion-pair character in an N2 matrix. In contrast, ClH:N(CH3)3 and BrH:N(CH3)3 have proton-shared and ion-pair hydrogen bonds, respectively, in the gas phase, and ion-pair hydrogen bonds in Ar and N2 matrices.

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Isotope Effects in Chemistry and Biology

TABLE 5.2 Equilibrium (Re) and Ground-State Expectation Values (R0) of X –N and X – H Distances (A˚) as a Function of Field Strength for HCl, DCl, HBr, and DBr with NH3 and N(CH3)3 ClH:NH3

ClD:NH3

Cl–N

Cl–H

Cl–N

Cl–D

Field

Re

R0

Re

R0

R0

R0

0.0000 0.0010 0.0025 0.0040 0.0055 0.0100 0.0150

3.080 3.056 3.019 2.975 2.832 2.896 3.004

3.016 2.986 2.941 2.905 2.870 2.898 2.988

1.341 1.349 1.363 1.383 1.575 1.766 1.917

1.392 1.411 1.448 1.495 1.546 1.692 1.857

3.041 3.011 2.960 2.909 2.878 2.895 2.996

1.374 1.389 1.421 1.473 1.538 1.710 1.878

BrH:NH3

Br–N Field

Re

0.0000 0.0005 0.0010 0.0025 0.0040 0.0080 0.0120

3.247 3.224 2.976 2.993 3.009 3.068 3.152

Br–H R0

Re

3.053 1.462 3.037 1.469 3.027 1.778 3.016 1.830 3.021 1.868 3.006 1.965 3.142 2.075 ClH:N(CH3)3 Cl–N

Field

Re

0.0000 0.0010 0.0040

2.825 2.836 2.872

R0 1.623 1.651 1.676 1.740 1.794 1.910 2.027 Cl–H

R0

Re

R0

2.850 1.658 2.854 1.685 2.879 1.752 BrH:N(CH3)3 Br–N

1.615 1.640 1.706 Br–H

BrD:NH3

Br–N

Br–D

R0

R0

3.075 1.588 3.040 1.633 3.018 1.672 3.015 1.741 3.015 1.812 3.067 1.928 3.147 2.044 ClD:N(CH3)3 Cl–N

Cl–D

R0

R0

2.843 1.624 2.849 1.651 2.877 1.719 BrD:N(CH3)3 Br–N

Br–D

Field

Re

R0

Re

R0

R0

R0

0.0000 0.0010 0.0040

2.983 2.993 3.027

2.991 3.000 3.016

1.870 1.887 1.937

1.830 1.849 1.888

2.989 2.999 3.031

1.842 1.861 1.915

From these data it is easy to understand how the change from an Ar to an N2 matrix may appear to have disparate effects on proton-stretching frequencies, decreasing frequencies for some complexes and increasing them for others.

C. DEUTERIUM S UBSTITUTION E FFECTS ON P ROTON -S TRETCHING F REQUENCIES It is well-known that the effect of deuteration is to lower proton-stretching frequencies. This is evident from the data of Table 5.1,23 and is illustrated in Figure 5.11 where the Cl – H and Cl –D stretching frequencies for ClH:NH3 and ClD:NH3 are plotted as a function of field strength.

Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes 1800

ClH:NH3

3.05

1600 Ar matrix

3.00

1400

2.95

1200

2.90

1000

2.85 2.80

N2 matrix 0

0.002 0.004 0.006 0.008 0.010 0.012 0.014 Field Strength, a.u.

nproton, cm−1

N−Cl distance, Angstrom

3.10

163

800 600

FIGURE 5.7 Expectation value of the Cl– N distance (solid line) in the ground vibrational state and the proton-stretching frequency (dashed line) in ClH:NH3 as a function of field strength. Experimental Ar and N2 matrix values are indicated. (Source: Reprinted with permission from Jordan, M. J. T. and Del Bene, J. E., J. Am. Chem. Soc. 122, 2101, Copyright 2000, American Chemical Society.)

Experimental data are available for the fully-deuterated isotopomer ClD:ND3.13 Once again, the computed anharmonic frequencies at field strengths of 0.0010 and 0.0025 au bracket the experimental frequency measured in an Ar matrix. The lowest Cl – D stretching frequency is found for the near quasi-symmetric proton-shared complex at a field strength of 0.0055 au. At this field strength, the computed anharmonic frequency is approximately 100 cm21 higher than the experimental Cl – D stretching frequency obtained in an N2 matrix. Unfortunately, experimental data are not available for deuterated isotopomers of BrH:NH3, but the computed anharmonic Br– D frequencies show a similar variation with field strength as the Br –H stretching frequencies. Both ClH:N(CH3)3 and ClD:N(CH3)3 have zero-field, gas-phase

3.08 N−Br distance, Angstrom

1600

BrH:NH3

1500 N2 matrix

3.07

1400

3.06

1300

3.05

1200

3.04

1100

3.03

1000

3.02

900

3.01 3.00

Ar matrix

0

nproton, cm−1

3.09

800

700 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 Field Strength, a.u.

FIGURE 5.8 Expectation value of the Br – N distance (solid) in the ground vibrational state and the protonstretching frequency (dashed) in BrH:NH3 as a function of field strength. Experimental Ar and N2 matrix values are indicated.

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Isotope Effects in Chemistry and Biology

2.88 N−Cl distance, Angstrom

1700

ClH:N(CH3)3

N2 matrix Ar matrix

2.87

1600 1500

2.86

1400

2.85

1300

2.84

1200

2.83

1100

2.82

0

0.001 0.002 0.003 Field Strength, a.u.

nproton, cm−1

2.89

1000 0.004

FIGURE 5.9 Expectation value of the Cl – N distance (solid) in the ground vibrational state and the protonstretching frequency (dashed) in ClH:N(CH3)3 as a function of field strength. Experimental Ar and N2 matrix values are indicated.

frequencies that are lower than the corresponding experimental Ar matrix frequencies. However, the computed anharmonic frequencies for these two complexes increase with increasing field strength, and approach the Ar matrix values at a field strength of 0.0040 au when the hydrogen bond has ion-pair character. The experimental frequencies in N2 are even higher,22 suggesting that the ion-pair character is greater in this more polarizable medium. The two-dimensional anharmonic frequencies for both BrH:N(CH3)3 and BrD:N(CH3)3 correspond to perturbed N – H and N –D stretches, respectively, even in the gas-phase, and these frequencies increase with increasing field strength as the ion-pair character increases. Experimental data are not available for the deuterated isotopomers of BrH:N(CH3)3. However, it is expected that BrD:N(CH3)3 and BrD:N(CD3)3 would also exhibit strong high-frequency N – D stretching bands in both Ar and N2 matrices.

2000

BrH:N(CH3)3

3.01

} N2 matrix

1800

3

1700

2.99

Ar matrix 1600

2.98 2.97

1900 nproton, cm−1

N−Br distance, Angstom

3.02

0

0.001 0.002 0.003 Field Strength, a.u.

1500 0.004

FIGURE 5.10 Expectation value of the Br– N distance (solid) in the ground vibrational state and the protonstretching frequency (dashed) in BrH:N(CH3)3 as a function of field strength. Experimental Ar and N2 matrix values are indicated.

Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes

165

1800 1600

n(cm−1)

1400 1200 1000 800 600 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 Field strength (au)

FIGURE 5.11 Cl – H (solid line) and Cl– D (dashed line) stretching frequencies in ClH:NH3 and ClD:NH3 as a function of field strength. (Source: Reprinted with permission from Bevitt, J. et al., J. Phys. Chem. A, 105, 3371, Copyright 2001, American Chemical Society.)

Harmonic analysis gives a value for the ratio n(D)/n(H) of 0.71. What is this ratio when the computed X –D and X – H frequencies are anharmonic? Comparing ClH:NH3 with ClD:NH3, and ClH:ND3 with ClD:ND3 shows that n(D)/n(H) is greater than 0.71 at all field strengths except 0.0055 au. At this field strength the hydrogen bond approaches a quasi-symmetric proton-shared hydrogen bond, and n(D)/n(H) approaches the harmonic value. This is a surprising result, given that a quasi-symmetric hydrogen bond is considered to be very anharmonic by other measures. The ratio n(D)/n(H) for BrH:NH3 and BrD:NH3 is slightly less than 0.71 at zero field, decreases to a minimum value of 0.62 at a field of 0.0010 au when the frequencies either have or are near their minimum values for proton-shared hydrogen bonds, and then subsequently increases to a maximum value of 0.81 at a field of 0.0080 au. The ratio n(D)/n(H) for ClH:N(CH3)3 and ClD:N(CH3)3, and for BrH:N(CH3)3 and BrD:N(CH3)3, is always slightly greater than the harmonic ratio of 0.71.

IV. TWO-BOND SPIN– SPIN COUPLING CONSTANTS ACROSS HYDROGEN BONDS In the mid-1990s, the first experimental measurements of two-bond spin – spin coupling constants across hydrogen bonds were reported.24 This NMR property has subsequently been investigated by both theorists and experimentalists, and gives promise of providing new information about hydrogen bond type and intermolecular distances.25,26 Studies of spin – spin coupling constants carried out in this laboratory have employed the equation-of-motion coupled cluster singles and doubles (EOM-CCSD) method in the CI-like approximation, with the Ahlrichs (qzp, qz2p) basis set. Two-bond X – Y coupling constants (2hJX – Y) across X – H – Y hydrogen bonds have been systematically investigated for various series of complexes stabilized by C– H – N, N – H –N, N – H –O, F – H –N, O – H –O, and Cl –H –N hydrogen bonds.27 – 37 For all of these complexes, the total coupling constant (2hJX – Y) is determined solely by the Fermi-contact (FC) term, which is more than an order of magnitude greater than the paramagnetic spin –orbit, diamagnetic spin – orbit, and spin – dipole terms. These studies have also established that the value of 2hJX – Y decreases (in an absolute sense if 2hJX – Y is negative) quadratically with increasing X – Y distance. Most importantly,

166

Isotope Effects in Chemistry and Biology

this level of theory has been found to give good agreement with experimental two-, three-, and fourbond coupling constants across hydrogen bonds when these are available.31,34,38 – 40 Although coupling constants involving Cl have not been measured experimentally because of signal broadening due to its large nuclear quadrupole moment, ClH:NH3 has proven to be an ideal model system for theoretical studies of coupling constants. First, it is possible to change hydrogenbond type in this complex from traditional, to proton-shared, to ion-pair by application of an external electric field. Second, relatively high-level ab initio calculations are feasible for this complex, thereby permitting investigations of anharmonicity, environmental, isotopic substitution, zero-point motion, and thermal vibrational averaging effects on coupling constants for different types of hydrogen bonds using the same theoretical methods on the same chemical system. The ClH:NH3 complex has in fact become our workhorse system, and has yielded a rich harvest of general information about two-bond spin –spin coupling constants across hydrogen bonds.

A. ANHARMONICITY AND F IELD E FFECTS In our studies it has been customary to fully optimize the structures of a related series of complexes, and then compute 2hJX – Y for each equilibrium structure. Curves have been generated illustrating the distance-dependence of 2hJX – Y for fixed X and Y by plotting equilibrium values of 2hJX – Y against equilibrium X – Y distances. This is illustrated in Figure 5.12 for complexes with N – H –N and N – Hþ –N hydrogen bonds.31 Each point on this graph represents the value of 2hJN – N at the equilibrium N – N distance for a particular complex. As is evident from Figure 5.12, 2hJN – N decreases quadratically with increasing N –N distance. This curve and other similar curves can be used to extract intermolecular distances from experimentally measured coupling constants. It should be noted, however, that experimental measurements of coupling constants are performed on complexes at their ground-state geometries in solvents. Table 5.1 demonstrates that the expectation value of the X –Y distance can be either longer or shorter than the equilibrium distance depending on hydrogen bond type. How does 2hJX – Y vary with hydrogen bond type? How can the effects of zero-point motion and thermal vibrational averaging be taken into account, and how should curves such as the one shown in Figure 5.12 be interpreted? 18 16

2hJ

N-N

(Hz)

14 12 10 8 6 4 2 2.6

2.7

2.8

2.9 N−N (Å)

3.0

3.1

3.2

FIGURE 5.12 N– N spin– spin coupling constants versus the N– N distance for equilibrium structures of complexes stabilized by N – H– N and N –Hþ – N hydrogen bonds. (Source: Reprinted with permission from Del Bene, J. E. et al., Magn. Reson. Chem., 39, S109, Copyright 2001, John Wiley & Sons Ltd.)

6

10

5

8

4

6

3

4

2

2

1

0

0

Coupling Constant

12

167

−0.05

−100

−0.10

−200

−0.15

−300

−0.20

−400

−0.25

−500

−0.30

−600

Proton Stretching Frequency

Cl−N Distance

Chemical Shift

Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 Field Strength, a.u.

FIGURE 5.13 Vibrational and NMR properties of the ClH:NH3 complex as a function of the strength of an external electric field imposed along the Cl– H – N direction. These plots have been made relative to a value of 0.0 for the property at zero field. The upper plot shows the proton chemical shift (dppm, solid line) and the Cl – N spin– spin coupling constant (Hz, dashed line). The lower plot shows the proton-stretching frequency ˚ , solid line). (Source: Reprinted with permission (cm21, dashed line) and the equilibrium Cl – N distance (A from Del Bene, J. E. and Jordan, M. J. T., J. Am. Chem. Soc., 122, 4794, Copyright 2000, American Chemical Society.)

Since the X –Y distance varies with hydrogen bond type, the X –Y spin –spin coupling constants should also depend on hydrogen bond type. Proton-shared hydrogen bonds, which have short intermolecular distances, also have coupling constants that are significantly greater in magnitude than complexes stabilized by traditional hydrogen bonds. Indeed, as the degree of proton-shared character increases, the coupling constant also increases.36 The variation of 2hJCl – N for ClH:NH3 as a function of field strength and therefore changing hydrogen bond type can be seen in Figure 5.13. Also shown in this figure are the field-dependence of the equilibrium Cl –N distance, the anharmonic proton-stretching frequency, and the chemical shift of the hydrogen-bonded proton.27,33 Each of these curves has an extremum value for the proton-shared hydrogen bond. Thus, these properties, particularly the proton-stretching frequency and the spin – spin coupling constant across the hydrogen bond, can be used to provide information about intermolecular distances, and are fingerprints of hydrogen bond type. A similar observation concerning the relationship between spin – spin coupling constants and hydrogen bond type has been made by

Isotope Effects in Chemistry and Biology

−3 Hz

0 Hz

3 Hz

4.5 4.0 3.5 3.0 2.5

1.5

2.0

2.5 3.0 N−H distance (bohr)

Cl−H distance (bohr)

168

2.0 4.0

3.5

FIGURE 5.14 2hJCl – N surface at zero-field. Contours are in increments of 3 Hz, and values of selected contours are indicated. The equilibrium geometry is marked (p), and the geometry corresponding to the expectation values of the Cl – N and Cl– H distances in the ground vibrational state is marked (£). (Source: Reprinted with permission from Del Bene, J. E. and Jordan, M. J. T., J. Phys. Chem. A, 106, 5385, Copyright 2002, American Chemical Society.)

Limbach and co-workers on the basis of their experimental studies of the temperature dependence of coupling constants. (See Chapter 7.) To further investigate spin –spin coupling constants across hydrogen bonds, global spin –spin coupling constant surfaces (more specifically Fermi-contact term surfaces) have been generated in the Cl – H and N – H coordinates for ClH:NH3 at field strengths of 0.0000, 0.0055, and 0.0150 au. At these field strengths the complexes are stabilized by traditional, near quasi-symmetric protonshared, and ion-pair hydrogen bonds, respectively.33 These surfaces are illustrated in Figure 5.14 to Figure 5.16, respectively. It is apparent that the contours on the surface at a field strength of 0.0055 are more closely spaced then they are on the other two surfaces. This implies that coupling constants for proton-shared hydrogen bonds are most sensitive to changes in intermolecular distances.

4.5

−6 Hz

4.0

−6 Hz −9 Hz

3.5 −3 Hz

3.0 2.5

1.5

2.0

2.5 3.0 N−H distance (bohr)

3.5

Cl−H distance (bohr)

−3 Hz

2.0 4.0

FIGURE 5.15 2hJCl – N surface at a field of 0.0055 au. Contours are in increments of 3 Hz, and values of selected contours are indicated. Equilibrium and ground-state geometries are marked as in Figure 5.14. (Source: Reprinted with permission from Del Bene, J. E. and Jordan, M. J. T., J. Phys. Chem. A, 106, 5385, Copyright 2002, American Chemical Society.)

Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes

4.0

−6 Hz −9 Hz

3.5 −6 Hz

3.0 2.5

1.5

2.0

2.5 3.0 N−H distance (bohr)

3.5

Cl−H distance (bohr)

4.5

−9 Hz

−3 Hz

169

2.0 4.0

FIGURE 5.16 2hJCl – N surface at a field of 0.0150 au. Contours are in increments of 3 Hz, and values of selected contours are indicated. Equilibrium and ground-state geometries are marked as in Figure 5.14. (Source: Reprinted with permission from Del Bene, J. E. and Jordan, M. J. T., J. Phys. Chem. A, 106, 5385, Copyright 2002, American Chemical Society.)

The equilibrium geometry on each surface has been indicated by an asterisk, and the geometry corresponding to the ground-state expectation values of the N –H and Cl –H distances has been indicated by an “x” in Figure 5.14 to Figure 5.16. Using these indicators, it is easy to compare equilibrium and expectation values of 2hJCl – N. At zero field when the hydrogen bond is traditional, 2h JCl – N for the equilibrium structure (2 5.9 Hz) is less than the ground-state expectation value (2 7.5 Hz). (Since Cl – N coupling constants are negative, comparisons are made in terms of absolute values.) This is consistent with the expectation value of the ground-state Cl – N distance, which is shorter than the equilibrium distance. Similarly, for the ion-pair hydrogen bond, the ground-state expectation value of 2hJCl – N is 2 7.8 Hz, slightly greater than the equilibrium value of 2 7.1 Hz, once again reflecting the shorter ground-state Cl –N distance. In contrast, when the hydrogen bond is proton-shared, 2hJCl – N at equilibrium is 2 12.2 Hz, whereas the expectation value of 2hJCl – N is 2 10.7 Hz. This reflects the longer ground-state Cl –N distance compared to the equilibrium distance. The generation of coupling constant surfaces and the calculation of expectation values of coupling constants from these surfaces using anharmonic wavefunctions is a demanding computational task. However, since the difference between these two values can be significant, it would be advantageous to estimate the ground-state value in a more computationally tractable way. One possibility is to perform a single-point calculation to obtain 2hJCl – N using the groundstate geometry. As is evident from Table 5.3, this approach provides a good approximation to the expectation value of 2hJCl – N when the complex has a traditional or ion-pair hydrogen bond. However, when the hydrogen bond is proton-shared, the value of 2hJCl – N from the single-point calculation (2 11.5 Hz) overestimates the expectation value (2 10.7 Hz), but is still an improvement over the equilibrium value (2 12.2 Hz). It is most probably the diffuseness of the ground-state wavefunction for a complex with a proton-shared hydrogen bond which cannot be captured by a single-point calculation. How general are these results? Table 5.4 compares equilibrium and ground-state values of two bond spin –spin coupling constants for a set of complexes (CNH:NCH, FH:NH3, FD:NH3, ClH:NH3, and ClD:NH3) that are stabilized by traditional hydrogen bonds. The ground-state expectation value of 2hJX – Y is always greater (in an absolute sense) than the equilibrium value. In contrast, the ground-state values of the F– F coupling constants for the anions FHF21 and FDF21 that are stabilized by symmetric proton-shared hydrogen bonds, are less than the value computed

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Isotope Effects in Chemistry and Biology

TABLE 5.3 ClH:NH3 and ClD:NH3 Equilibrium and Expectation Values of Ground-State Cl –N Spin – Spin Coupling Constants (Hz) as a Function of Field Strength (au) and Temperature (K)a Cl:NH3

ClD:NH3

Field

Eq.b

G.S.c

k2hJCl – Nld

k2hJCl – Nle

G.S.c

k2hJCl – Nld

k2hJCl – Nle

0.0000 0.0055 0.0150

25.88 212.19 27.05

27.46 211.51 27.83

27.52 210.68 27.81

27.23 210.48 27.68

26.83 211.34 27.54

26.95 210.90 27.59

26.75 210.53 27.46

a

Data quoted at field strengths other than 0.0000 au are implicit values computed from the zero-field FC surface using the anharmonic vibrational wavefunctions corresponding to the non-zero field surface. The difference between implicit and explicit values is discussed in detail in Ref. 33. While the magnitudes of implicit and explicit 2hJCl – N are different, the trends observed are the same. b 2h JCl – N evaluated at the equilibrium geometry. c 2h JCl – N evaluated by a single-point calculation at the ground-state expectation values of the Cl–N and Cl–H or Cl–D distances. d 2h JCl – N evaluated as an expectation value from the Fermi-contact surface using the corresponding two-dimensional anharmonic wavefunctions. e Vibrationally-averaged value of 2hJCl – N at 298K.

for the equilibrium structure, as seen in Table 5.4.38 It is the ground-state value of 2hJF – F for FHF21 which is in agreement with the value estimated experimentally.41 This suggests that curves such as the one shown in Figure 5.12 should be interpreted in terms of ground-state X – Y distances, even though they have been obtained from data for the equilibrium structures of hydrogen-bonded complexes.

B. ISOTOPIC S UBSTITUTION E FFECTS ON Z ERO - P OINT M OTION AND T HERMAL V IBRATIONAL AVERAGING OF C OUPLING C ONSTANTS A deuterated isotopomer sits lower in the potential well than the normal isotopomer. This implies that sampling of the potential surface in the ground vibrational state (zero-point motion) will be more restricted in a complex with an X – D – Y hydrogen bond than in the corresponding complex

TABLE 5.4 Effects of Zero-Point Motion and Thermal Vibrational Averaging on X – Y Spin –Spin Coupling Constants (2hJX – Y) Across X– H – Y and X –D –Y Hydrogen Bonds Complex CNH:NCH FH:NH3 FD:NH3 ClH:NH3 ClD:NH3 FHF21 FDF21 a

Ref. 41.

2h

JX – Y,eq

6.4 243.8 243.8 25.9 25.9 254.4 254.4

k2hJX – Ylgs

k2hJX – Yl298

7.1 248.1 246.6 27.5 27.0 212.7 (,220)a 223.1

6.8 247.2 245.7 27.2 26.8 212.7 223.1

Anharmonicities, Isotopes, and IR and NMR Properties of Hydrogen-Bonded Complexes

171

with an X – H – Y bond. As a result, the expectation value of the two-bond spin –spin coupling constant for the deuterated isotopomer should be closer to the equilibrium value than the expectation value for the normal isotopomer. That this is the case is evident from Table 5.4, which shows that the ground-state expectation value of the X –Y spin –spin coupling constant in the deuterated isotopomer always lies between the equilibrium and ground-state expectation values of the normal isotopomer. Unlike the effect of deuteration on the IR spectra of complexes where X –D stretching bands always have lower frequencies than corresponding X – H stretching bands, the effect of deuteration on the X – Y coupling constant depends on hydrogen bond type. It is apparent from Table 5.3 and Table 5.4 that deuteration decreases the expectation value of 2hJX – Y for complexes with traditional and ion-pair hydrogen bonds. However, if the hydrogen bond is proton-shared, the effect of deuteration is to increase the expectation value of 2hJX – Y. If such differences could be measured experimentally, then the effect of deuteration on 2hJX – Y could be useful for confirming the presence or absence of proton-shared hydrogen bonds. Experimental measurements of NMR coupling constants are done at finite temperatures, whereas computed expectation values of spin –spin coupling constants are 0K values. Since dimer vibrational states in hydrogen-bonded complexes may have relatively low frequencies, and since the X – Y distance increases as the energy of the dimer vibrational state increases, thermal population of excited dimer vibrational modes at finite temperatures should lead to a decrease in the expectation values of coupling constants. This effect is illustrated in Table 5.3 and Table 5.4 for all complexes except FHF21 and FDF21, although the effect is relatively small.42 Thermal vibrational averaging has no effect on 2hJF – F for FHF21 and FDF21 since dimer- and proton-stretching modes have relatively high frequencies.

V. CONCLUDING REMARKS It is now possible to compute accurate equilibrium structures and binding energies of hydrogenbonded complexes. In the future, the techniques for doing this will be applied to larger and more sophisticated complexes, and the error bars on computed structures and binding energies will be reduced. However, the calculation of accurate binding enthalpies remains a major challenge because these depend on vibrational frequencies. Since the harmonic approximation fails to capture anharmonicity effects particularly for complexes with proton-shared hydrogen bonds or those that sample the proton-shared region of the potential surface in the ground vibrational state, the computed enthalpies for these complexes can be significantly in error. An anharmonic treatment in at least two-dimensions is absolutely necessary if accurate binding enthalpies are to be obtained, and reliable assignments made of IR spectral bands. Such treatments are also required for reliable estimates of deuterium substitution effects on vibrational frequencies. Anharmonic frequencies computed from three- or higher-dimensional surfaces may be required in some cases, and efforts to obtain these should be pursued. For many complexes stabilized by either traditional or ion-pair hydrogen bonds, there is only a small difference between computed equilibrium and ground-state values of 2hJX – Y. However, if the complex has a proton-shared hydrogen bond, or if a complex with a traditional or ion-pair hydrogen bond can access the proton-shared region of the surface in the ground vibrational state, then the difference between equilibrium and ground-state values of 2hJX – Y can be significant. Once again, obtaining reliable estimates of coupling constants for such complexes requires that expectation values be computed from coupling constant surfaces using anharmonic wavefunctions, and this is a daunting computational task. There is hope even here, however, since the use of external fields can alter structures of hydrogen-bonded complexes, and coupling constants can be computed for equilibrium structures as a function of field strength. If the equilibrium values of these coupling constants are viewed as ground-state

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values, then insight into the variation of coupling constants as a function of temperature and solvent ordering can be obtained and used to assist in the interpretation of experimental data.34,43,44 Another interesting prediction resulting from these studies is that substitution of D for the hydrogen-bonded H decreases X – Y coupling constants of traditional or ion-pair hydrogen bonds, but increases 2hJX – Y for proton-shared hydrogen bonds. If this effect could be measured experimentally, then a valuable tool would be available to confirm the presence or absence of proton-shared hydrogen bonds. Future theoretical work must expand on what has already been done, with the aim of producing reliable results which can be used to assist in the interpretation of experimental data, and to predict 2hJX – Y for systems that cannot be measured experimentally. It is hoped that future work will lead to a better understanding of the IR and NMR properties of normal and deuterated isotopomers of hydrogen-bonded complexes, and provide greater insight into the factors that determine the signs and magnitudes of spin – spin coupling constants across hydrogen bonds.45

REFERENCES 1 Del Bene, J. E. and Jordan, M. J. T., What a difference a decade makes: progress in ab initio studies of the hydrogen bond, J. Mol. Struct. (Theochem), 573, 11 – 23, 2001. 2 Del Bene, J. E. and Shavitt, I., The quest for reliability in calculated properties of hydrogen-bonded complexes. In Molecular Interactions: From van der Waals to Strongly-Bound Complexes, Scheiner, S., Ed., Wiley, Susex, Chichester, New York, pp. 157– 179, 1997. 3 Del Bene, J. E., Hydrogen bonding: 1, In The Encyclopedia of Computational Chemistry, Vol. 2, Schleyer, P. v. R., Allinger, N. L., Clark, T., Gasteiger, J., Kollman, P. A., Schaefer, H. F. III, and Schreiner, P. R., Eds., Wiley, Chichester, pp. 1263– 1271, 1998. 4 Feyereisen, M. W., Feller, D., and Dixon, D. A., Hydrogen bond energy of the water dimer, J. Phys. Chem., 100, 2993–2997, 1996. 5 Dunning, T. H. Jr., Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen, J. Chem. Phys., 90, 1007– 1023, 1989. 6 Howard, N. W. and Legon, A. C., An investigation of the hydrogen-bonded dimer H3NAAAHBr by pulsed-nozzle, Fourier-transform microwave spectroscopy of ammonium bromide vapor, J. Chem. Phys., 86, 6722–6730, 1987. 7 Legon, A. C., The nature of ammonium and methylammonium halides in the vapour phase: hydrogen bonding versus proton transfer, Chem. Soc. Rev., 22, 153– 163, 1993. 8 Cooke, S. A., Corlett, G. K., Lister, D. G., and Legon, A. C., Is pyridinium hydrochloride a simple hydrogen-bonded complex C5H5NAAAHCl or an ion pair C5H5NHþAAACl2 in the gas phase? An answer from its rotational spectrum, J. Chem. Soc., Faraday Trans., 94, 837– 841, 1998. 9 Szczepaniak, K., Chabrier, P., Person, W. B., and Del Bene, J. E., Ab initio theoretical and matrix isolation experimental studies of hydrogen bonding IV. The HBr:pyridine complex, J. Mol. Struct., 367– 386, 436– 437, 1997. 10 Johnson, G. L. and Andrews, L., Matrix infrared spectrum of the ammonia-hydrogen fluoride hydrogen-bonded complex, J. Am. Chem. Soc., 104, 3043– 3047, 1982, Experimental spectrum shown courtesy of Person, W. B. and Szczepaniak, K. 11 Szczepaniak, K., Chabrier, P. and Person, W.B., private communication. 12 Del Bene, J. E., Jordan, M. J. T., Gill, P. M. W., and Buckingham, A. D., An ab initio study of anharmonicity and matrix effects on the hydrogen-bonded BrH:NH3 complex 3, Mol. Phys., 92, 429– 439, 1997, The experimental Ar matrix spectrum of BrH:NH3 was generously provided by Person, W. B. and Szczepaniak, K. 13 Barnes, A. J., Beech, T. R., and Mielke, Z., Strongly hydrogen-bonded molecular complexes studied by matrix-isolation vibrational spectroscopy. Part 1. The ammonia-hydrogen chloride complex, J. Chem. Soc. Faraday Trans. 2, 80, 455– 463, 1984.

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14 Barnes, A. J. and Wright, M. P., Strongly hydrogen-bonded molecular complexes studied by matrixisolation vibrational spectroscopy. Part 3. Ammonia-hydrogen bromide and amine-hydrogen bromide complexes, J. Chem. Soc. Faraday Trans. 2, 82, 153– 164, 1986. 15 Jordan, M. J. T. and Del Bene, J. E., Unraveling environmental effects on hydrogen-bonded complexes: matrix effects on the structures and proton-stretching frequencies of hydogen-halide complexes with ammonia and trimethylamine, J. Am. Chem. Soc., 122, 2101– 2115, 2000. 16 Szczepaniak, K., Chabrier, P., Person, W. B., and Del Bene, J. E., Experimental infrared spectra of matrix isolated complexes of HCl with 4-substituted pyridines. Evaluation of anharmonicity and matrix effects using data from ab initio calculations, J. Mol. Struct., 520, 1 – 18, 2000. 17 Del Bene, J. E., Person, W. B., and Szczepaniak, K., Ab initio theoretical and matrix isolation experimental studies of hydrogen bonding: vibrational consequences of proton position in 1:1 complexes of HCl and 4-X-pyridines, Chem. Phys. Letters, 247, 89 – 94, 1995. 18 Del Bene, J. E. and Jordan, M. J. T., Vibrational spectroscopy of the hydrogen bond: an ab initio quantum-chemical perspective, Int. Rev. Phys. Chem., 18, 119–162, 1999. 19 Del Bene, J. E. and Jordan, M. J. T., A comparitive study of anharmonicity and matrix effects on the complexes XH:NH3, X=F, Cl, and Br, J. Chem. Phys., 108, 3205– 3212, 1998. 20 Jacox, M. E., Vibrational and Electronic Energy Levels of Polyatomic Transient Molecules [Washington, D.C]: American Chemical Society; New York: American Institute of Physics, for the National Institute of Standards and Technology, J. Phys. Chem. Ref. Data, 1994, Monograph No. 3. 21 Onsager, L., Electric moments of molecules in liquids, J. Am. Chem. Soc., 58, 1486– 1493, 1936. 22 Barnes, A. J., Kuzniarski, N. S., and Mielke, Z., Strongly hydrogen-bonded molecular complexes studied by matrix-isolation vibrational spectroscopy. Part 2. Amine-hydrogen chloride complexes, J. Chem. Soc. Faraday Trans. 2, 80, 465–476, 1984. 23 Bevitt, J., Chapman, K., Crittenden, D., Jordan, M. J. T., and Del Bene, J. E., An ab initio study of anharmonicity and field effects in hydrogen-bonded complexes of the deuterated analogues of HCl and HBr with NH3 and N(CH3)3, J. Phys. Chem. A, 105, 3371– 3378, 2001. 24 Dingley, A. J. and Grzesiek, S. J., Direct observation of hydrogen bonds in nucleic acid base pairs by internucleotide 2JNN couplings, J. Am. Chem. Soc., 120, 8293– 8297, 1998. 25 Elguero, J. and Alkorta, I., Review on DFT and ab initio calculations of scalar coupling constants, Int. J. Mol. Sci., 4, 64 – 92, 2003. 26 Limbach, H.-H., ed. Special issue on NMR Spectroscopy of Hydrogen-Bonded Systems, Magn. Reson. Chem. 39, S1-S213, 2001. 27 Del Bene, J. E. and Jordan, M. J. T., Vibrational spectroscopic and NMR properties of hydrogen-boned complexes: do they tell us the same thing?, J. Am. Chem. Soc., 122, 4794 –4797, 2000. 28 Del Bene, J. E. and Bartlett, R. J., N – N spin-spin coupling constants [2hJ(15N– 15N) across N – HAAAN hydrogen bonds in neutral complexes: to what extent does the bonding at the nitrogens influence 2h JN – N?, J. Am. Chem. Soc., 122, 10480 –10481, 2000. 29 Del Bene, J. E., Perera, S. A., and Bartlett, R. J., What parameters determine N –N and O– O coupling constants (2hJX – X) across X –Hþ – X hydrogen bonds?, J. Phys. Chem. A, 105, 930– 934, 2001. 30 Chapman, K., Crittenden, D., Bevitt, J., Jordan, M. J. T., and Del Bene, J. E., Relating environmental effects and structures , IR, and NMR properties of hydrogen-bonded complexes: ClH:pyridine, J. Phys. Chem. A, 105, 5442– 5449, 2001. 31 Del Bene, J. E., Perera, S. A., and Bartlett, R. J., 15N,15N spin-spin coupling constants across N– H– N and N – Hþ – N hydrogen bonds: can coupling constants provide reliable estimates of N– N distances in biomolecules?, Magn. Reson. Chem., 39, S109– S114, 2001. 32 Toh, J., Jordan, M. J. T., Husowitz, B. C., and Del Bene, J. E., Can proton-shared or ion-pair N– H– N hydrogen bonds be produced in uncharged complexes? A systematic ab initio study of the structures and selected NMR and IR properties of complexes with N– H – N hydrogen bonds, J. Phys. Chem. A, 105, 10906– 10914, 2001. 33 Del Bene, J. E. and Jordan, M. J. T., To what extent do external fields and vibrational and isotopic effects influence NMR coupling constants across hydrogen bonds? Two-bond Cl – N spin-spin coupling constants (2hJCl – N) in model ClH:NH3 complexes, J. Phys. Chem. A, 106, 5385– 5392, 2002. 34 Del Bene, J. E., Bartlett, R. J., and Elguero, J., Interpreting 2hJ(F,N), 1hJ(H,N) and 1J(F,H) in the hydrogen-bonded FH – collidine complex, Magn. Reson. Chem., 40, 767– 771, 2002.

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35 Del Bene, J. E., Perera, S. A., Bartlett, R. J., Ya´n˜ez, M., Mo´, O., Elguero, J., and Alkorta, I., Two-bond 19 F– 15N spin-spin coupling constants (2hJF – N) across F-H000N hydrogen bonds, J. Phys. Chem. A, 107, 3121– 3125, 2003. 36 Del Bene, J. E., Perera, S. A., Bartlett, R. J., Ya´n˜ez, M., Mo´, O., Elguero, J., and Alkorta, I., Two-bond 15 N!19F spin!spin coupling constants (2hJN – F) across N!Hþ…F hydrogen bonds, J. Phys. Chem. A, 107, 3126– 3131, 2003. 37 Del Bene, J. E., Perera, S. A., Bartlett, R. J., Mo´, O., Ya´n˜ez, M., Elguero, J., and Alkorta, I., Two-bond 13 C!15N spin!spin coupling constants (2hJC – N) acoss C!H!N hydrogen bonds, J. Phys. Chem. A, 107, 3222– 3227, 2003. 38 Del Bene, J. E., Jordan, M. J. T., Perera, S. A., and Bartlett, R. J., Vibrational effects on the FIF spin!spin coupling constant (2hJF – F) in FHF2 and FDF2, J. Phys. Chem. A, 105, 8399– 8402, 2001. 39 Del Bene, J. E., Perera, S. A., Bartlett, R. J., Alkorta, I., and Elguero, J., 4hJ(31P– 31P) coupling constants through N– Hþ – N hydrogen bonds: a comparison of computed ab inito and experimental data, J. Phys. Chem. A, 104, 7165– 7166, 2000. 40 Del Bene, J. E., Perera, S. A., Bartlett, R. J., Elguero, J., Alkorta, I., Lo´pez-Leonardo, C., and Alajarin, J., 3hJ(15N!31P) spin!spin coupling constants across N!H000O!P hydrogen bonds, J. Am. Chem. Soc., 124, 6393, – 6397, 2002. 41 Benedict, H., Shenderovich, I. G., Malkina, O. L., Malkin, V. G., Denisov, G. S., Golubev, N. S., and Limbach, H-H., Nuclear scalar spin!spin couplings and geometries of hydrogen bonds, J. Am. Chem. Soc., 122, 1979–1988, 2000. 42 Jordan, M. J. T., Toh, J. S-S., and Del Bene, J. E., Vibrational averaging of NMR properties for an N– H – N hydrogen bond, Chem. Phys. Lett., 346, 288– 292, 2001. 43 Shenderovich, I. G., Burtsev, A. P., Denisov, G. S., Golubev, N. S., and Limbach, H-H., Influence of the temperature-dependent dielectric constant on the H/D isotope effects on the NMR chemical shifts and the hydrogen bond geometry of the collidine– HF complex in CDF3/CDClF2 solution, Magn. Reson. Chem., 39, S91 – S99, 2001. 44 Golubev, N. S., Shenderovich, I. G., Smirnov, S. N., Denisov, G. S., and Limbach, H-H., Nuclear scalar spin-spin coupling reveals noval properties of low-barrier hydrogen bonds in a polar environment, Chem. Eur. J., 5, 492– 497, 1999. 45 Del Bene, J. E. and Elguero, J., What determines the sign of the Fermi-contact contribution to the NMR spin-spin coupling constant?, Chem. Phys. Lett., 382, 100– 105, 2003.

6

Isotope Effects on Hydrogen-Bond Symmetrization in Ice and Strong Acids at High Pressure Katsutoshi Aoki

CONTENTS I.

Introduction ...................................................................................................................... 175 A. Hydrogen-Bond Symmetrization ............................................................................. 175 B. Candidate Compounds and Promising Probe .......................................................... 176 II. Hydrogen-Bond Symmetrization in Ice........................................................................... 177 A. Crystal Structure ...................................................................................................... 177 B. Infrared Absorption Study ....................................................................................... 178 1. Symmetrization in Ice VIII ............................................................................... 179 2. Symmetrization in Ice VII ................................................................................ 181 3. Phase Diagram and Isotope Effect.................................................................... 182 III. Hydrogen-Bond Symmetrization in Hydrogen Chloride ................................................ 183 A. Crystal Structure ...................................................................................................... 183 B. Raman Scattering Study........................................................................................... 185 1. Symmetrization in HCl ..................................................................................... 185 2. Symmetrization in DCl ..................................................................................... 187 3. Isotope Effect on Stretching Vibration and Symmetrization ........................... 189 IV. Summary .......................................................................................................................... 190 References..................................................................................................................................... 191

I. INTRODUCTION A. HYDROGEN -BOND S YMMETRIZATION The hydrogen bond is an intermolecular force connecting adjacent molecules rather tightly, often producing the specific structure and property of hydrogen-bonded molecular solids. In contrast to the isotropic nature of coulomb and van der Waals interactions, the hydrogen bond has a strong directional feature and hence plays a dominant role in molecular arrangement. The hydrogen atom loses its electron to another atom in an adjacent molecule and the bare proton strongly attracts the proton-sharing neighboring molecules. In addition, the proton can move along the hydrogenbonding axis by thermal activation or tunneling motion, allowing a variety of phase transitions with proton ordering or disordering. These characteristics of the hydrogen bond are responsible for the helical conformation of DNA and the dielectric property of ice, etc. 175

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(a)

(b)

(c)

(d)

FIGURE 6.1 Variation of hydrogen-bonding potential and proton distribution on compression. The potential is converted from double (a) to single minimum shape (d) via transition states (b and c) as the hydrogenbonding distance decreases.

The symmetrization of the hydrogen bond is intimately related to the quantum motion of proton tunneling and has been one of the major subjects of high pressure chemistry. We consider here the variation of hydrogen bonding potential with pressure to recall the essential aspect of symmetrization process. The energy potential for the proton motion along the hydrogen bonding axis can be described as a double-minimum potential with an energy barrier on the midpoint (Figure 6.1). Although the potential is drawn here as a symmetric shape for convenience of explanation (Figure 6.1a), it should be deformed in an actual system to an asymmetric shape with the proton localized in the lower minimum. As the bond length decreases on applying pressure, the potential barrier is gradually depressed and eventually smeared out (Figure 6.1d); the potential would converge from double-minimum to single-minimum shape at sufficiently high pressure. The proton initially located at the lower minimum will move to eventually occupy the midpoint symmetric position. An intermediate or transition state characterized by enhanced tunneling motion may appear in the process of symmetrization (Figure 6.1b,c). The proton moves frequently by tunneling between the two minima separated by the significantly depressed potential barrier and on average occupies the two minima equally (Figure 6.1b). The transition state with the bimodal proton distribution arises from tunneling motion and hence can be characterized as quantum-mechanical symmetrization. The proton centering is completed on further compression when a well-defined single minimum potential is realized as shown in Figure 6.1d. This state can be characterized as classical symmetrization. The transition from the quantum-mechanical to classical symmetrization state progresses continuously as expected from the gradual potential conversion and hence definite separation of the two states can hardly be made. The transition to the quantum-mechanically symmetrized state is driven by tunneling motion and is therefore expected to be influenced by isotope substitution. In other words, the observation of the isotope effect provides experimental support for the model of the symmetrization process. Tunneling probability is governed by the shape of the potential barrier (width and height) and the mass of the tunneling particle. We can assume that the essential shape of the hydrogen-bonding potential is not affected by proton– deuteron substitution. This assumption readily leads to the prediction that the quantum-mechanical symmetrization occurs earlier for the proton system than for the deuteron one. pffiffiFirst, the ground state or zero-point energy of proton vibration is located at energy higher by 2 than that of deuteron vibration within a harmonic approximation and consequently the effective height of the potential barrier is reduced for proton tunneling. Second, the tunneling probability is inversely proportional topparticle mass and hence the lighter proton has ffiffi higher tunneling probability again by a factor of 2: Both factors will accelerate the tunneling motion of the proton and result in earlier transition to the quantum-mechanically symmetrized state on compression.

B. CANDIDATE C OMPOUNDS AND P ROMISING P ROBE Ice and hydrogen halide are appropriate candidates for the high pressure study of hydrogen-bond symmetrization. Ice is a prototype of hydrogen bonded systems and its bond symmetrization has been intensively theoretically investigated over half a century.1 – 9 The latest results obtained by first-principle molecular dynamics successfully demonstrated the symmetrization process and

Isotope Effects on Hydrogen-Bond Symmetrization in Ice and Strong Acids at High Pressure

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the isotope effects on it7; on compression the OH distance lengthens faster than the OD distance due to zero point fluctuation and consequently hydrogen bond symmetrization occurs in H2O ice at a pressure much lower than the symmetrization pressure in D2O ice. Estimated symmetrization pressures ranging from 35 to 80 GPa are accessible using the current high-pressure technique. Hydrogen halide forms the simplest hydrogen-bonded system in its condensed state. In contrast to the three dimensional hydrogen-bonded network of ice, hydrogen-bonded molecules form onedimensional zigzag chains in solid hydrogen halide. The simplicity of the molecular and crystal structures is of great help in interpreting the experimental results and hence clarifying the symmetrization process. Vibrational spectroscopy is a promising tool for investigating the hydrogen bonding state and hence detecting the symmetrization process. As expected from the potential deformation drawn in Figure 6.1, the vibrational frequency related to the proton stretching motion along the hydrogenbonding axis decreases initially on compression and turns to increase on further compression beyond the symmetrization pressure. The symmetrization would thus be detected as a turnaround from softening to hardening of the proton stretching frequency. Dramatic change in spectral feature is another signal proving the symmetrization. Proton centering implies dissociation of constituted molecules. The peaks related to the molecular vibrations of water in ice, for instance, should disappear and those related to the lattice vibration of the atomic crystal appear instead. In this chapter we review hydrogen-bond symmetrization in ice and hydrogen chloride investigated by infrared absorption and Raman scattering at pressures up to 120 GPa and temperatures down to 10 K. The symmetrization process is shown to be described in terms of the softening of the proton stretching vibrations and the mode conversion from the molecular to lattice vibrations at the transition point. It is also shown from the observed isotope effects that quantum mechanical symmetrization is driven by tunneling motion; the symmetrization pressure is shifted to higher pressure by deuteration.

II. HYDROGEN-BOND SYMMETRIZATION IN ICE A. CRYSTAL S TRUCTURE Ice exhibits more than ten solid phases including amorphous and metastable states in its pressure – temperature phase diagram.10 At pressures above 2 GPa, however, there exist only two known molecular phases, ice VII and ice VIII. These ices consist of two interpenetrating hydrogen bonded lattices as described in more detail below and are often called dense ices. Ice VIII is the low temperature phase of ice VII. Ice VII transforms to ice VIII in association with proton ordering on cooling. The dense ices VII and VIII have been predicted to transform into ice X with symmetrized hydrogen bond on compression. The crystal structures of the molecular phases of ices VII (Pn3m) and VIII (I41/amd) and the atomic phase of ice X (Pn3m) are drawn in Figure 6.2. Each water molecule is connected by hydrogen bonding to four neighboring tetrahedrally arranged molecules. They have essentially the bcc configuration of oxygen atoms and are distinguished by proton ordering and location. The ice-VII structure, for example, can be converted into the ice-X structure simply by displacing the hydrogen atoms to the midpoints of the hydrogen bonds. As a result, four hydrogen atoms form a tetrahedron with one oxygen atom at the center in ice X. In ice VIII the oxygen sublattice is slightly distorted from the bcc lattice owing to the dipole – dipole interaction between the orientationally ordered water molecules. Four distorted bcc sublattices build up a tetragonal unit cell in ice VIII. The transition from ice VII (and hence ice VIII) to ice X has been suggested from the equation of state measured by high pressure x-ray diffraction.11 – 16 At room temperature, ice VII is found to retain the bcc sublattice of oxygen atoms while the hydrogen bond length (oxygen– oxygen distance) continuously decreases to 0.22 nm at 128 GPa. This value is shorter than the distance of 0.23 –0.24 nm, at which the hydrogen bonding potential is expected to be sufficiently deformed to

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X

VII

VIII

FIGURE 6.2 Crystal structures of ices VII, VIII, and X. Ices VII and VIII are molecular crystals consisting of two interpenetrating hydrogen-bonded sublattices of water molecules. Ice X is atomic crystal consisting of regularly aligned hydrogen and oxygen atoms. Thick lines indicate unit lattices.

allow proton centering or bond symetrization. The x-ray diffraction measurement is unable to determine the proton position, providing no experimental indication for the symmetrization. However, it is expected that ice VII transforms to ice X at a certain point in the pressure range measured. The transition to ice X is also expected to take place at a corresponding pressure region for ice VIII having the crystal structure very close to that of ice VII.

B. INFRARED A BSORPTION S TUDY Infrared absorption measurement reveals that the hydrogen-bond is symmetrized at pressures of 60 –70 GPa for ice VII at room temperature and ice VIII below 100 K.17 – 19 The spectra show complicated patterns particularly near the transition pressure. The turnaround predicted for the proton stretching vibration is not definitely observed. The stretching vibration is distorted owing to anharmonic deformation of the hydrogen-bonding potential, showing significant broadening in the peak shape. In addition, the stretching mode causes sequential mixing with another vibrational mode in the course of softening; the perturbed frequency prevents us from tracing the softening process and hence detecting the softening-hardening turnaround. Infrared absorption no longer seems a promising probe to detect hydrogen-bond symmetrization. However, careful analysis of the observed spectra enables us to extract the essential features of the symmetrization process. We first present the hydrogen-bond symmetrization observed for the low-temperature phase, ice VIII. Ice VIII has an ordered structure and the observed vibrational peaks can definitely be assigned on the basis of fundamental vibrational analysis. After showing the prospect of the symmetrization process for ice VIII, we present the results of the high-temperature phase, ice VII. Ice VII has a crystal structure consisting of orientationally disordered molecules, which does not allow definite peak assignment. The observed spectral changes can be interpreted by analogy with those of ice VIII.

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179

Phase study of ices VII, VIII and X has also been made by Carnegie and Paris groups, providing experimental results consistent with those presented in the following sections.20 – 26 1. Symmetrization in Ice VIII The crystal structure of ice VIII belongs to space group I41/amd and point group D19 4h. Group theory analysis of the vibrational modes predicts six IR-active modes27: two stretching modes (symmetric stretching vibration n1 ðA2u Þ and asymmetric one n3 ðEu Þ), one bending mode ðn2 ðA2u ÞÞ, two rotational modes (here termed as high- and low-frequency rotational modes nR ðEu Þ and nR0 ðEu Þ, respectively), and one IR-active translational mode ðn T ðEu ÞÞ: The transition from ice VIII to ice X with symmetrized hydrogen bonds is observed by infrared absorption measurement at temperatures below 100 K.19 Figure 6.3 shows a representative spectral set for a 60 K isothermal run in a pressure range 8 to 90 GPa. The spectrum of ice VIII at 8.6 GPa exhibits three molecular peaks, which are related to the asymmetric n3 and symmetric n1 stretching vibrations, and to the bending n2 vibration, observed at approximately 3300, 3150, and 1500 cm21, respectively. The peak arising from the fundamental rotational vibrations and its overtone and combination bands are observed clearly at higher pressures, around 20 –40 GPa. Each molecular vibrational mode shows characteristic pressure behavior. The stretching peaks rapidly shift to low frequency with increasing pressure, whereas the rotational peaks shift to high frequency. In contrast, the bending peak is insensitive to pressure, staying at approximately the same position for all pressures up to 50 GPa. A significant spectral change with the ice VIII –X transition is observed at approximately 60 GPa. The molecular vibrational peaks broaden significantly and the spectral profile becomes featureless. No signal of the transition seems to be detected. However, the spectra show distinct changes in the wavenumber range of 1100– 1300 cm21; the rotational peak nR at approximately 1100 cm21 disappears above 62.5 GPa, while a new peak at approximately 1250 cm21 appears at

1.0

H2O VIII-X 60 K 89.2

nD

73.6

Absorbance

66.7 62.5 60.1 57.3 54.6 50.3 41.8 33.6 25.5

2nR 2nR' n2 + nR

nR'

nR 1000

n2

n3

n1

2000 3000 Wavenumber (cm−1)

15.7 8.6 GPa 4000

FIGURE 6.3 Infrared absorption spectra of H2O ice measured up to 90 GPa at 60 K.19 Peak assignment is given in text. Disappearance of the n R and n R0 rotational peaks and n2 bending peak takes place around 60 GPa, indicating dissociation of water molecule with bond symmetrization. The asymmetric peak newly appearing at approximately 1150 cm21 is assigned to the distortional lattice mode of ice X.

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the slightly lower pressure of 60.1 GPa as indicated with arrows in Figure 6.3. The newly appearing peak grows in intensity with further increase in pressure, showing an asymmetric shape even at 89.2 GPa. This peak can be assigned to the lattice mode n D of ice X. Ice X is predicted to have another infrared-active lattice vibration ðn T Þ, which still remains below 650 cm21 out of the low frequency limit of the measurement even at 90 GPa. According to the correlation diagram for the vibrational modes of ice VIII and ice X, the rotational modes (nR and nR0 ) of ice VIII are converted into the distortional lattice mode (nD ) of ice X in association with the hydrogen-bond symmetrization.27 The transition pressure at 60 K can hence be determined to be 62.5 GPa at which the peak conversion is completed. A significantly broadened band, which extends over the whole frequency region measured, may involve the vibrational peaks of several molecular species such as neutral and ionized water molecules. Theoretical calculation predicts ionization of water molecules as a result of enhanced proton tunneling.4,9 In such a mixed state, each molecular species would vibrate with short lifetime in a potential largely deformed from an ideal harmonic potential. Several peaks related to the vibrational motions of these molecular species probably merge into the broadened band. The symmetrization process is described by a soft-mode analysis of the proton stretching vibration. The peak frequencies are plotted as a function of pressure in Figure 6.4. The n1,3 stretching modes cause mixing with the n2 þ nR;R0 combination and 2nR0 and 2nR overtones, while their frequencies decrease with increasing pressure from approximately 3300 cm21 at 5 GPa to about 1000 cm21 at 60 GPa. The perturbation is strongly enhanced around 30, 40, and 48 GPa. The unperturbed frequencies are obtained by following a fitting procedure18,21 and presented with solid lines. The stretching frequency may be described as a function of pressure by a phenomenological form vOH ¼ ðv20 2 aPÞ1=2 , with a critical pressure Pc of 67.5 GPa for vOH ¼ 0.19,24 The soft-mode analysis is based on a harmonic approximation and hence is not appropriate for exactly locating the ice VIII – X transition point. Actually it gives a critical or transition pressure of 67.5 GPa, higher by 5 GPa than that determined from the vibrational mode conversion. The symmetrization process, however, is shown to be interpreted in terms of the softening of the proton stretching vibration, 3500

n1

Frequency (cm−1)

3000 2500

H2O

n3

60 K

n2 + nR,R'

2000 2nR' n2 1500 2nR 1000 nR'

500 0

0

20

nD

nR VIII 40 60 Pressure (GPa)

X 80

100

FIGURE 6.4 Variation of peak frequencies with pressure measured for H2O ice at 60 K.19 The frequencies of the overtone and combination modes are significantly modified as a result of vibrational mode mixing with the softening stretch modes. Solid lines are inferred bare frequencies fitted by a formula v ¼ ðv20 2 aPÞ1=2 for the stretching mode and by a second-order polynomials of pressure for the other vibrational modes. The transition from ice VIII to X takes place at about 60 GPa with disappearance of some molecular vibrational peaks and appearance of the distortional lattice mode n D :

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181

which arises from the continuous deformation of the hydrogen-bonding potential from double to single minimum shape. 2. Symmetrization in Ice VII Disordering in molecular orientation in ice VII destroys the normal selection rules for infrared absorption, making it hard to distinctly assign the observed peaks to the predicted vibrational modes. The similarity of the spectra of ice VII and VIII, which have essentially the same shortrange order in molecular arrangement and can be converted to each other by introducing a slight distortion of the primitive or sublattice cells, allows reasonable peak assignment for ice VII on the basis of that made for ice VIII. The spectra of H2O and D2O ices measured across the symmetrization pressure are shown in Figure 6.5 and Figure 6.6, respectively.18 The spectrum of H2O at 56.6 GPa is featureless, showing smoothly sloping hills with peak tops around 1300 and 1000 cm21. With a further increase in pressure, the absorption peak located at 1300 cm21 develops to a very sharp and intense peak. Another peak, which should have moved to high frequency with pressure, eventually exhibits a whole peak shape around 900 m21. The spectrum at 120 GPa shows two peaks alone at about 900 and 1400 cm21. A very similar change can be seen in the spectra measured for D2O ice as shown in Figure 6.6. Two peaks at about 800 and 1100 cm21 in a 141-GPa spectrum correspond to those in the 120-GPa spectrum of H2O ice. Ice X has a highly symmetric structure with symmetrized hydrogen bonds, showing two IR-active lattice modes: the distortional twisting of the tetrahedron of hydrogen atoms around a resting oxygen atom, nD , and the translational motion of oxygen and hydrogen atoms in the opposite direction, nT (Figure 6.7). The spectra of H2O ice above 101 GPa and D2O ice above 124 GPa exhibit two IR absorption peaks in agreement with the prediction, providing clear evidence for the transition to ice X with classically symmetrized hydrogen bond. The high and low

νT

H2O

νD

Absorbance

120 GPa 101 79.3 67.6 62.0 56.6 52.7 44.3 33.3 1000

2000 3000 4000 Wavenumber (cm−1)

5000

FIGURE 6.5 Infrared absorption spectra of H2O ice measured up to 120 GPa at 298 K.18 The spectrum at 33.3 GPa shows Fermi resonance between n1;3 and n2 þ n R as observed for ice VIII. At pressures above 52.7 GPa, the peak observed around 1200 cm21 is assigned to the distortional lattice mode ðn D Þ of ice X. The peak additionally observed around 900 cm21 in a 120-GPa spectrum is assigned to the translation lattice mode ðn T Þ of ice X.

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νT

D2O

νD

Absorbance

141GPa 124 104 93 80.8 74.4 70.5 58.2 37.5 1000

2000 3000 Wavenumber (cm−1)

4000

FIGURE 6.6 Infrared absorption spectra of D2O ice measured up to 141 GPa at 298 K.18 The peaks around 800 and 1000 cm21 observed in a 141-GPa spectrum is assigned to the distortional lattice ðn D Þ and translational lattice ðn T Þ modes of ice X, respectively. The transition from ice VII to X takes place around 70 GPa.

n1

n2

nT

n3

nR

nD

molecular vibrations in ices VII and VIII

lattice vibrations in ice X

FIGURE 6.7 Infrared active molecular and lattice vibrational modes in ice. Symmetric stretching mode n1 , bending mode n2 , asymmetric stretching mode n3 , and rotational mode n R in ices VII and VIII. Translational lattice mode n T and distortional lattice mode n D in ice X.

frequency peaks can be assigned as the n D distortional and n T translational mode, respectively. The frequency ratios of vH2 O =vD2 O at a corresponding pressure of 120 GPa are very close to 1.41 derived from ðmH =mD Þ1=2 for both the n D and n T modes; these peaks are shown to be related to the lattice vibrations dominated by proton or deuteron motions in agreement with the mode analysis. 3. Phase Diagram and Isotope Effect The phase diagram determined for H2O and D2O ices VII, VIII and X by IR measurement is shown in Figure 6.8.19 As shown in the phase diagram of H2O ice (Figure 6.8a), the VII – VIII transition with proton ordering shows unusual behavior. The transition temperature is insensitive to pressure and located at 273 K at low pressures. It begins to decrease at about 12 GPa and develop into

Isotope Effects on Hydrogen-Bond Symmetrization in Ice and Strong Acids at High Pressure

183

300 VII

Temperature (K)

250

VII

H2O

D2O

200 150 VIII

X

VIII

100

X

50 0

(a)

0

20

40 60 Pressure (GPa)

80

100

0

(b)

20

40 60 Pressure (GPa)

80

100

FIGURE 6.8 (a) Phase diagram of H2O ice at pressures above 2 GPa and temperatures below 298 K.19 (b) Phase diagram of D2O ice at pressures above 2 GPa and temperatures below 298 K.19 The phases appearing below 2 GPa are omitted. Solid triangles O and P represent transition points determined by cooling and warming runs, respectively. Solid squares represent those determined by isotherm runs. The previously reported data are also presented by open circle22 and square.25

the VIII –X boundary line approximately at 60 GPa. At temperatures below 100 K, ice VIII transforms into ice X without passing through the ice VII region. The triple point among ices VII, VIII and X can be located at about 110 K and 60 GPa. The transition pressure from the molecular VII or VIII to atomic phase X tends to increase slightly with decreasing temperature; the pressure of 58 GPa at 298 K shifts to 63 GPa at 40 K, giving a temperature coefficient DP/DT ¼ 0.02 GPa/ K. The phase diagram of D2O ice is quite similar to that of H2O (Figure 6.8b). The VII – VIII phase boundaries of H2O and D2O ices overlap at low pressures below 20 GPa, showing gradual deviation from each other with further increase in pressure. The boundary between the molecular and atomic phases becomes slightly inclined with a coefficient of DP/DT ¼ 0.03 GPa/ K and apparently shifts to higher pressure: 66 GPa at 298 K and 78 GPa at 20 K by 15 –25% increase in pressure compared to the corresponding transition pressures of H2O ice. This is in agreement with the theoretical prediction that the low zero-point energy and small tunneling probability of deuteron pushes up the transition pressure of the symmetrization. The triple point of VII, VIII and X phases is located at 100 K and 74 GPa.

III. HYDROGEN-BOND SYMMETRIZATION IN HYDROGEN CHLORIDE A. CRYSTAL S TRUCTURE Hydrogen halide, HX (X ¼ F, Cl, Br, I), is a diatomic molecule forming a hydrogen bond in its condensed state. Three crystalline phases are known to exist at low temperature and ambient pressure.28 – 30 The lowest temperature phase (phase III) has an orthorhombic Cmc21 structure consisting of planar zigzag chains of hydrogen-bonded molecules. Temperature elevation gives rise to transition from phase III to II and further to phase I. Phase II has an orthorhombic Cmca structure in which the halogen atoms are located essentially in the same positions as those in phase III but the H atoms are in the twofold disordered positions around halogen atoms. In phase I, halogen atoms construct a face-centered-cubic lattice (Fm3m) with completely disordered protons occupying one of twelve equivalent sites. The sequential transformation from the ordered (Cmc21)

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FIGURE 6.9 Crystal structures of hydrogen halide. Left: molecular arrangement in the bc plane of the orthorhombic Cmc21 structure. The proton displacements associated with a symmetric ns ðA1 Þ and an asymmetric stretching vibration ns ðB2 Þ are presented with arrows along the upper and lower zigzag chains, respectively. Right: the hydrogen bonds are symmetrized by putting the protons (small spheres) at the midpoints between the halogen neighbors (large spheres). The crystal structure changes into the orthorhombic Cmcm structure with proton centering.

to disordered structure (Fm3m) via the partially disordered one (Cmca) is interpreted as step-bystep breakage of hydrogen bonding by thermally activated vibrations. Structural transition with proton ordering takes place at high pressure as well.31 – 33 The fully disordered phase I transforms to the ordered phase III at 19 GPa in HCl and at 13 GPa in HBr at room temperature. The hydrogen-bond symmetrization is hence expected to occur in phase III when the proton located close to one halogen atom is pushed out to the midpoint between the neighboring halogen atoms at sufficiently high pressure. Proton centering changes the crystal structure from the Cmc21 to CmCm orthorhombic lattice (Figure 6.9). The high pressure phase thus appearing in association with hydrogen bond symmetrization is denoted as phase IV hereafter. Proton stretching vibration plays a key role in the symmetrization. The proton displacements associated with a symmetric ns(A1) and an antisymmetric stretching vibration ns(B2) are presented with arrows along the upper and lower zigzag chains, respectively. These stretching modes are expected to show initially softening behavior with pressure and then hardening behavior after the symmetrization. A factor group analysis predicts three translational lattice modes with Raman activity for both Cmc21 and Cmcm structures and additionally four librational and two stretching modes for the Cmc21 structure.33 Approximate vibrational motions are drawn for the fundamental vibrational modes in Figure 6.10. The symmetrized phase with the Cmcm structure is hence identified by Raman spectra showing just three lattice peaks. translation libration stretch

translation

A1

Ag

B2

B3g

A2 B1

+ + − −

+ − + − + −

Cmc21

B1g

+

−

+− Cmcm

FIGURE 6.10 Raman active vibrational modes for the Cmc21 (phase III) and Cmcm (phase IV) structures of hydrogen halide. Open and solid circles represent halogen and hydrogen atoms, respectively. Allows indicate atomic displacement in the plane. þ and 2 indicate upward and downward atomic displacement out of the plane.

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B. RAMAN S CATTERING S TUDY Hydrogen-bond symmetrization is observed for HCl, DCl and HBr for a pressure range from 40 to 60 GPa at room temperature.31 – 33 The Raman spectra clearly show the softening behavior of the proton stretching vibration and the spectral change associated with the symmetrization. The isotope effect is observed for DCl; the symmetrization pressure shifts slightly to higher pressure by substitution of proton with deuteron. The symmetrized Cmcm structure seems unstable for HBr. The dissociation of HBr molecule to H2 and Br2 molecules is probably induced by formation of Br– Br bonding between neighboring molecules.34,35 1. Symmetrization in HCl Figure 6.11 shows Raman spectra measured for HCl up to 60 GPa at 298 K. The transition from the proton-disordered cubic (Fm3m) to proton-ordered orthorhombic phase (Cmc21) takes place at 19 GPa with a split in the stretching peak and an appearance of lattice mode peaks. The stretching peaks rapidly shift to low frequency. In particular, the shift of the ns(A1) symmetric stretching peak is remarkable, falling towards zero frequency with increasing pressure. The stretching peak reaches the lattice vibrational region around 1000 cm21 above 40 GPa. The translational and librational lattice peaks are enhanced in intensity and more distinctly observed in the frequency region below 500 cm21 and between 700 and 1000 cm21, respectively, also above 40 GPa. The variation of Raman frequencies with pressure is plotted in Figure 6.12. The ns ðA1 Þ stretching mode shows Fermi resonance when the frequency approaches those of the librational modes with the same A1 symmetry. The resonance behavior is clearly seen between the ns ðA1 Þ and nL ðA1 Þ þ nL ðA2 Þ combination modes for a pressure span from 30 to 40 GPa. We can examine a soft mode analysis for the ns ðA1 Þ mode. The frequencies are fitted with v ¼ ðv20 2 apÞ1=2 ; v0 is the frequency at ambient pressure and a the pressure coefficient.24,31 The frequencies apparently perturbed by the resonance effect are excluded in the fitting procedure. Extrapolation of the

HCl

Intensity

48.2 42.8 stretch 32.3 21.1 18.6 0

500 1000 1500 2000 2500 3000 3500 Raman shift (cm−1)

FIGURE 6.11 Raman spectra measured for molecular phase of solid HCl up to 48 GPa at 298 K.31 Inserted numerical values represent pressures in GPa. The phase transition from the proton-disordered cubic to protonordered orthorhombic structure occurs at about 19 GPa with splitting in the stretching peak located around 2300 cm21 and appearance of lattice peaks. Also presents the position of the symmetric stretching peak showing rapid shift to low frequency with pressure.

186

Isotope Effects in Chemistry and Biology 3000 HCl

Raman shift (cm−1)

2500

2000

1500

I

III

IV

1000

500

0

0

10

20

30 40 50 Pressure (GPa)

60

70

FIGURE 6.12 Variation of Raman peak frequencies with pressure measured for solid HCl at 298 K.31 The proton-disordered cubic phase (Fm3m, phase I) transforms to the proton-ordered orthorhombic phase (Cmc21, phase III) at about 19 GPa and further to the symmetric phase (Cmcm, phase IV) at about 51 GPa. The soft mode analysis of the stretching frequency provides an extrapolated pressure of 51 GPa at which the frequency falls into zero (solid curve).

frequency to zero using the optimized parameters v0 ¼ 2858 cm21 and a ¼ 1.55 £ 105 (cm21)2/ GPa provides an estimate of 51 GPa for the symmetrization. The definite sign of hydrogen-bond symmetrization is the spectral changes in the lattice vibrational region (Figure 6.13). The translational lattice modes of the Cmc21 orthorhombic

HCl

Intensity

54 GPa 52 GPa 50 GPa 45 GPa 0

200

400 600 800 Raman shift (cm−1)

1000

1200

FIGURE 6.13 Raman spectral change associated with hydrogen-bond symmetrization in solid HCl measured at pressures of 45 – 54 GPa at 298 K.31 The stretching peak, which is barely observed around 900 cm21 at 45 GPa and 750 cm21 at 50 GPa, disappears at 52 GPa. The three lattice peaks appear above 52 GPa in agreement with the vibrational mode analysis for the Cmcm structure with bond symmetrization.

Isotope Effects on Hydrogen-Bond Symmetrization in Ice and Strong Acids at High Pressure

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structure are replaced with new ones and the stretching peak having shifted to about 750 cm21 disappeared. The Raman spectra measured above 52 GPa show three peaks in the frequency region of lattice vibration. This is in agreement with the lattice mode prediction for the Cmcm structure. The hydrogen-bond symmetrization pressure is hence determined to 51 GPa in agreement with that estimated from the soft mode analysis. 2. Symmetrization in DCl Typical Raman spectra of DCl measured in the pressure range up to 47 GPa are shown in Figure 6.14. The spectral change associated with the transition from the Fm3m cubic to the Cmc21 orthorhombic structure is clearly displayed in a spectrum measured at 19.7 GPa. The broadened stretching peak splits into intense, well resolved peaks at approximately 1640 and 1800 cm21. In addition to the peak splitting, three translational and four librational lattice peaks appear in the frequency regions around 250 cm21 and of 400 –800 cm21, respectively. The spectral change corresponds well with that observed for HCl at the same pressure of 19 GPa. The split peaks at 1640 and 1800 cm21 are assigned as the A1 symmetric and the B2 antisymmetric stretching mode. The lattice vibrational peaks, which are indistinct in the 19.7-GPa spectrum, become intense and sharp peaks with increasing pressure. In a 35.7-GPa spectrum the Raman peaks located at 174, 211 and 321 cm21 are assigned to A1 ; A2 ; and B1 translational modes, and those at 413, 523, 596 and 725 cm21 to A1 ; A2 ; B1 ; and B2 librational modes, respectively. Dramatic spectral changes arising from hydrogen-bond symmetrization are observed in the lattice region at pressures of 50 –60 GPa (Figure 6.15). The A1 stretching peak located at approximately 660 cm21 at 52.6 GPa gradually disappears during a pressure increase up to 58 GPa, whereas a new peak grows at 420 cm21. A change in the spectral feature is also observed in the region of 200 –300 cm21; the two lattice peaks merge into a single peak at about 58.5 GPa.

DCl 47.3 43.5 Intensity

stretch

38.9 37.4 35.7 19.7 17.2 1.8 GPa

0

1000 2000 Raman shift (cm−1)

3000

FIGURE 6.14 Raman spectra measured for molecular phase of solid DCl up to 47 GPa at 298 K.33 Inserted numerical values represent pressures in GPa. The phase transition from the proton-disordered cubic to protonordered orthorhombic structure occurs at about 19 GPa with splitting in the stretching peak located around 1800 cm21 and appearance of lattice peaks. Also presents the position of the symmetric stretching peak showing rapid shift to low frequency with pressure.

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Isotope Effects in Chemistry and Biology

DCl

66.6 Intensity

58.5 56.3 54.4 52.6 51.8 50.2 47.3 200

400

600

800

1000

Raman shift (cm−1)

FIGURE 6.15 Raman spectral change associated with hydrogen-bond symmetrization in solid DCl measured at pressures of 47 –66 GPa at 298 K.33 The stretching peak disappears at about 56 GPa. The three lattice peaks observed in the top spectrum are attributed to the translational lattice modes of the Cmcm structure with bond symmetrization.

One peak located around 500 cm21 remains unchanged, showing a continuous shift to high frequency, the peak at 500 cm21 shows a shift and does not remain unchanged. Eventually, three lattice peaks remain at pressures above 60 GPa and the other peaks originally related to the molecular motions such as stretching and rotational vibrations disappear completely. The Cmc21 structure transforms to the Cmcm one with proton centering. 2500 DCl

Raman shift (cm−1)

2000

1500 I

III

IV

1000

500

0

0

10

20

30 40 50 60 Pressure (GPa)

70

80

FIGURE 6.16 Variation of Raman peak frequencies with pressure measured for solid DCl at 298 K.33 The proton-disordered cubic phase (Fm3m, phase I) transforms to the proton-ordered orthorhombic phase (Cmc21, phase III) at about 19 GPa and further to the symmetric phase (Cmcm, phase IV) at about 56 GPa.

Isotope Effects on Hydrogen-Bond Symmetrization in Ice and Strong Acids at High Pressure

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The phase transitions and the vibrational mode mixings are more clearly demonstrated in the frequency vs. pressure plots (Figure 6.16). The stretching frequencies show an abrupt drop by roughly 100 cm21 in association with I– III transition at 19.0 GPa. The A1 symmetric and B2 antisymmetric stretching frequencies continue to decrease, whereas the frequencies of the librational and translational modes are found to increase. Mixing of the stretching modes occurs around 35 GPa, at which the A1 stretching frequency almost reaches the overtone frequency of A1 libration. They should have crossed each other at 35 GPa, avoiding crossing as a result of mode mixing and separating rapidly on further compression. An anomalous change in frequency is also observed for the fundamental A1 libration. Its frequency increases monotonically up to 42 GPa and then turns to decrease above it. The symmetrization pressure is determined to be 56 GPa. 3. Isotope Effect on Stretching Vibration and Symmetrization The isotope effects on the stretching vibration are clearly seen in Figure 6.17. The frequencies of proton vibration are located below those of deuteron vibration over the whole pressure range measured. Careful investigation reveals that frequency difference becomes large as the pressure pffiffiffiffi pffiffiffiffi increases about 90 cm21 in frequency (0.95 in the ratio of mH nHCl = mD nDCl ) at 20 GPa and about p p ffiffiffiffi ffiffiffiffi 165 cm21 (0.76 in mH nHCl = mD nDCl Þ at 50 GPa. The isotope substitution may influence the stretching vibration through both anharmonicity and tunneling; the former would be dominant at low pressure and the latter at high pressure. At low pressures roughly below 40 GPa, proton (deuteron) oscillates at the bottom of one minimum, approximated with a quadratic form. The actual potential curve would deviate downward from the ideal quadratic or harmonic curve as the vibrational energy rises apart from the bottom toward the barrier top. This anharmonicity leads to significant lowering of the first excitation level of proton vibration compared to that of deuteron vibration, since the former has an energy level about 2 as high as the latter. The anharmonicity in

stretch mode

Frequency (cm−1)

1500

DCl 1000

HCl

500 20

30 40 50 Pressure (GPa)

60

FIGURE 6.17 Variation of proton stretching frequency with pressure.33 Open and solid circles represent the frequencies of HCl and DCl, respectively.pffiffiThose of DCl are corrected for the atomic mass difference: the observed frequencies are multiplied by 2: The axis of the ordinate on the right gives their observed frequencies.

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Isotope Effects in Chemistry and Biology

a

b

c

D

Energy

H a

1' b c

1

0'

0 Pressure

FIGURE 6.18 Pressure variation of hydrogen bonding potential and proton (deuteron) vibrational state drawn schematically on the basis of calculated results.33 The thick and thin curves represent the energy levels of proton and deuteron vibrations, respectively. The deuteron vibrational energies are corrected for the mass pffiffi difference by a factor of 2 to allow immediate comparison with those of proton vibration.

the potential thus explains the frequency difference between HCl and DCl in the low-pressure region. The increased difference in the stretching frequency at higher pressure can be explained in terms of tunneling effects.33 As displayed in Figure 6.18, the first excitation and ground states show successive energy splitting owing to the tunneling motion of the proton (deuteron). The S-shape variation of the frequency with pressure (see again the solid curves in Figure 6.17) can be qualitatively explained in terms of tunneling effects. The tunneling splitting will take place earlier in proton vibration than in deuteron vibration as shown for the first excitation state (see top panel b) and the ground state (see top panel c). The earlier 0 –00 splitting in proton vibration, for example, results in a faster decrease in the 00 –1 excitation energy and hence in the Raman stretching frequency. This is exactly what we see in the frequency – pressure relations at pressures around 45 GPa (Figure 6.17); the stretching frequency of HCl falls slightly earlier than that of DCl to produce a large frequency difference of about 160 cm21 at 50 GPa. The symmetrization pressure increases from 51 GPa in HCl to 56 GPa in DCl. The pressure shift of 5 GPa is rather small compared with 12 GPa observed for ice, showing, however, that the symmetrization is driven by tunneling motion.

IV. SUMMARY 1. Hydrogen-bond symmetrization is observed for ice at pressures of 60– 70 GPa and hydrogen chloride at pressures of 51 –56 GPa. The symmetrization pressure of ice is in agreement with the prediction of the latest theoretical study. 2. The symmetrization process is described in terms of the softening of proton stretching vibration. The hydrogen-bonding potential is converted from the usual double minimum shape to a single minimum shape with increasing pressure.

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3. Symmetrization is confirmed by spectral change showing disappearance of the molecular vibrational peaks and appearance of the lattice vibrational peaks. Protons are situated at the center of hydrogen bonds to form atomic crystals. 4. Isotope effect is found in the symmetrization pressure. Deuteron substitution pushes the transition pressure by 10 GPa in ice and 5 GPa in hydrogen chloride. 5. The proton stretching vibration exhibits mode mixing with other vibrational modes in the course of softening. The potential surface changes singularly along the hydrogen bonding axis with pressure.

REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

Reid, C., J. Chem. Phys., 30, 182, 1959. Holzapfel, W. B., J. Chem. Phys., 56, 712, 1972. Stinllinger, F. H. and Schweizer, K. S., J. Phys. Chem., 87, 4281, 1983. Schweizer, K. S. and Stinllinger, F. H., J. Chem. Phys., 80, 1230, 1984. Lee, C., Vanderbilt, D., Laasonen, K., Car, R., and Parrinello, M., Phys. Rev. Lett., 69, 462, 1992. Lee, C., Vanderbilt, D., Laasonen, K., Car, R., and Parrinello, M., Phys. Rev. B, 47, 4863, 1993. Benoit, M., Marx, D., and Parrinello, M., Nature, 392, 258, 1998, London. Johannsen, P. G., Phys. J.: Condens. Matter, 10, 2241, 1998. Benoit, M., Marx, D., and Parrinello, M., Solid State Ionics, 125, 23, 1999. Petrenko, V. F. and Whiteworth, R. W., Physics of Ice, Oxford University Press, Oxford, 1999. Hemley, R. J., Jephcoat, A. P., Mao, H. K., Zha, C. S., Finger, L. W., and Cox, D. E., Nature, 330, 737, 1987. Hama, J. and Suito, K., Phys. Lett. A, 187, 346, 1994. Wolanin, E., Pruzan, Ph., Chervin, J. C., Canny, B., Gauthier, M., Hausermann, D., and Hanfland, M., Phys. Rev. B, 56, 5781, 1997. Holzapfel, W. B., Physica B, 265, 113, 1999. Loubeyre, P., LeToullec, R., Wolanin, E., Hanfland, M., and Hausermann, D., Nature, 397, 503, 1999, London. Benoit, M., Romero, Aldo H., and Marx, Dominik, Phys. Rev. Lett., 89, 145– 501, 2002. Aoki, K., Yamawaki, H., Sakashita, M., and Fujihisa, H., Phys. Rev. B, 54, 15673, 1996. Song, M., Yamawaki, H., Fujihisa, H., Sakashita, M., and Aoki, K., Phys. Rev. B, 60, 12644, 1999. Song, M., Yamawaki, H., Fujihisa, H., Sakashita, M., and Aoki, K., Phys. Rev. B, 68, 014106, 2003. Goncharov, F., Struzhkin, V. V., Somayazulu, M. S., Hemley, R. J., and Mao, H. K., Science, 273, 218, 1996. Struzhkin, V. V., Goncharov, A. F., Hemley, R. J., and Mao, H. K., Phys. Rev. Lett., 78, 4446, 1997. Goncharov, A. F., Struzhkin, V. V., Mao, H. K., and Hemley, R. J., Phys. Rev. Lett., 83, 1998, 1999. Pruzan, Ph., Chervin, J. C., and Canny, B., J. Chem. Phys. B, 99, 9842, 1993. Pruzan, Ph., J. Mol. Struct., 322, 279, 1994. Pruzan, Ph., Wolanin, E., Gauthier, M., Chervin, J. C., Canny, B., Hausermann, D., and Hanfland, M., J. Phys. Chem. B, 101, 6230, 1997. Pruzan, Ph., Chervin, J. C., Wolanin, E., Canny, B., Gauthier, M., and Hanfland, M., J. Raman Spectrosc., 34, 591, 2003. Hirsch, K. R. and Holzapfel, W. B., J. Chem. Phys., 84, 2771, 1986. Ikram, A., Torrie, B. H., and Powell, B. M., Mol. Phys., 79, 1037, 1993. Vesel, J. E. and Torrie, B. H., Can. J. Phys., 55, 592, 1977. Anderson, A., Torrie, B. H., and Tse, W. S., J. Raman Spectrosc., 10, 148, 1981. Aoki, K., Katoh, E., Yamawaki, H., Sakashita, M., and Fujihisa, H., Physica B, 265, 83, 1999. Katoh, E., Yamawaki, H., Fujihisa, H., Sakashita, M., and Aoki, K., Phys. Rev. B, 59, 11244, 1999. Katoh, E., Yamawaki, H., Fujihisa, H., Sakashita, M., and Aoki, K., Phys. Rev. B, 61, 119, 2000. Ikeda, T., Sprik, M., Terakura, K., and Parrinello, M., Phys. Rev. Lett., 81, 4416, 1998. Ikeda, T., Sprik, M., Terakura, K., and Parrinello, M., J. Chem. Phys., 111, 1595, 1999.

7

Hydrogen Bond Isotope Effects Studied by NMR Hans-Heinrich Limbach, Gleb S. Denisov, and Nikolai S. Golubev

CONTENTS I. II.

Introduction ...................................................................................................................... 193 Theoretical Section .......................................................................................................... 194 A. The Crystallographic View of Hydrogen Bonded Systems .................................... 194 B. Origin of Hydrogen Bond Isotope Effects............................................................... 196 1. Influence of the Hydron Potential..................................................................... 196 2. Effects of the Environment ............................................................................... 197 C. Inclusion of Quantum Corrections in Hydrogen Bond Correlations ...................... 198 D. H/D Isotope Effects on NMR Parameters and Hydrogen Bond Geometries: The Point Approximation.................................................................... 202 E. H/D Isotopic Fractionation, Hydrogen Bond Geometries and NMR Parameters............................................................................................... 204 III. Applications...................................................................................................................... 205 A. H/D Isotope Effects in Strong NHN Hydrogen Bonds ........................................... 205 B. H/D Isotope Effects in OHN Hydrogen Bonded Pyridine –Acid and Collidine –Acid Complexes .............................................................................. 210 1. Low-Temperature NMR Spectroscopy of Pyridine –Acid Complexes Dissolved in Liquefied Freon Mixtures ............................................................ 210 2. Geometric Hydrogen Bond Correlations of OHN Hydrogen Bonded Complexes............................................................................................ 211 3. H/D Isotope Effects on the NMR Parameters of Pyridine – Acid and Collidine –Acid Complexes ....................................................................... 213 4. H/D Isotopic Fractionation and NMR Parameters of Pyridine – Acid Complexes ........................................................................... 217 C. Temperature-Induced Solvent H/D Isotope Effects on NMR Chemical Shifts of FHN Hydrogen Bonds.............................................................. 217 D. H/D Isotope Effects on the NMR Parameters and Geometries of Coupled Hydrogen Bonds ................................................................................... 222 IV. Conclusions ...................................................................................................................... 226 Acknowledgments ........................................................................................................................ 227 References..................................................................................................................................... 227

I. INTRODUCTION For a long time, Nuclear Magnetic Resonance (NMR) spectroscopy has been used to study the kinetics of hydrogen transfer in condensed phases. As NMR is traditionally regarded as a “slow” 193

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Isotope Effects in Chemistry and Biology

kinetic method, systems exhibiting relatively large barriers for the proton transfer had been studied, giving rise to kinetic H/D isotope effects.1,2 In the last years, the dynamic range of NMR has been increased into the micro- and nanosecond time scale.2b However, a long time ago it was also recognized that NMR is an excellent tool for the study of low-barrier hydrogen bonds, as isotope effects on NMR parameters can be observed which give interesting insights into the type of hydrogen bonds in solution. In past years, two developments in NMR have especially promoted the understanding of H/D isotope effects on NMR parameters and hydrogen bond geometries in liquids and solids. The first is dipolar solid-state NMR from which new insights into geometric isotope effects can be obtained directly. This method is also a link between the world of hydrogen bond geometries and NMR parameters obtained in solution or in biomolecular systems. The second development is the combination of solid-state NMR methods with low-temperature liquid-state NMR methods3 using liquefied deuterated gases such as freons as NMR solvents. The possibility to perform liquid-state NMR measurements at low temperatures allows one to observe different hydrogen bonded environments, protonated and deuterated, in the slow NMR hydrogen bond exchange regime. The scope of this chapter is, therefore, to present examples of how liquid- and solid-state NMR can be used as a tool in order to determine the geometries of protonated and deuterated intermolecular hydrogen bonded systems in solids and liquids. Firstly, we will describe geometric correlations that provide a basis for the classification of hydrogen bonded systems. The advantage and the limits of these correlations are discussed. Then, the correlations are applied to describe geometric hydrogen bond isotope effects of single as well as of multiple hydrogen bonded systems. In a series of acid –base complexes we will deal with the determination of solvent induced temperature effects on hydrogen bond geometries and of H/D isotopic fractionation factors. Finally, we will discuss the occurrence of H/D isotope effects on the geometries of coupled hydrogen bonds. At this point we would like to caution the reader: this review is written with the eyes of an experimental chemist who uses the simplest concepts available to describe and interpret experimental results. A description of the approximations used and their justification is beyond the scope of this review. Finally, in this book, the chapters of Hansen and Perrin review the NMR research field of H/D isotope effects of hydrogen bonded systems in solution, with special emphasis on intramolecular hydrogen bonds. The chapter of Del Bene deals with high-level quantummechanical calculations of strong protonated and deuterated intermolecular hydrogen bonds. In this chapter, we will use the notion “hydron” L ¼ H, D as a general term for mobile hydrogen isotopes such as the proton and the deuteron.

II. THEORETICAL SECTION A. THE C RYSTALLOGRAPHIC V IEW OF H YDROGEN B ONDED S YSTEMS The easiest way to classify hydrogen bonded systems is the geometric information available by x-ray and neutron diffraction crystallography.4 Therefore, we will review in this section recent results of this field, especially the phenomenon of hydrogen bond correlations. To any hydrogen bond A – H· · ·B one can normally associate two distances, the A· · ·H distance r1 ; rAH for the diatomic unit AH, and the H· · ·B distance r2 ; rHB for the diatomic unit HB. According to Pauling,5 one can associate to these distances so-called valence bond orders or bond valences, which are nothing else than the “exponential distances” p1 ¼ exp{ 2 ðr1 2 r1o Þ=b1 }; p2 ¼ exp{ 2 ðr2 2 r2o Þ=b2 }; with p1 þ p2 ¼ 1

ð7:1Þ

where b1 and b2 are parameters describing the decrease of the bond valences of the AH and the HB unit with the corresponding distances. ro1 and ro2 are the equilibrium distances of the fictive nonhydrogen bonded diatomic molecules AH and HB. If one assumes that the total valence for

Hydrogen Bond Isotope Effects Studied by NMR

195

hydrogen is unity, it follows that when one distance is varied, the other automatically adjusts, leading to an ensemble of allowed r1 and r2 values called the hydrogen bond correlation curve. The hydrogen bond angle does not appear in Equation 7.1. This correlation implies also a correlation between the hydrogen bond coordinates q1 ¼ 1=2ðr1 2 r2 Þ and q2 ¼ r1 þ r2 : For a linear hydrogen bond, q1 represents the distance of H from the hydrogen bond center and q2 the distance between atoms A and B. We note that correlations of the type of Equation 7.1 have been used a long time ago in the context of describing the “bond energy bond order conservation” reaction pathway of the H2 þ H reaction.6 A typical geometric hydrogen bond correlation according to Equation 7.1 is depicted in Figure 7.1a. When H is transferred from one heavy atom to the other, q1 increases from negative values to positive values. q2 goes through a minimum which is located at q1 ¼ 0 for AHA and near 0 for AHB systems. This correlation means that both proton transfer and hydrogen bonding coordinates can be reduced to a single coordinate s representing the pathway along the correlation curve, with s ¼ 0 at q1 ¼ 0: In Figure 7.1b are depicted the corresponding valence bond orders.

q2 = r1 + r2 /Å

3.6

AH + B

3.2

E(q1)

r1 r2 A·····H−B

F H

A

r1 r2 A··H··B

3.8

D H

s

2.4

(a) −0.8

−0.4

1.0 0.8 0.6 0.4 0.2 0.0 −0.2

p1

0.0 p1+ p2

0.4

0.8

D H B

p2

E

D H D

C

p1p2

−0.8

−0.4

0.0 0.4 q1 = ½(r1 − r2) /Å

0.8

(c)

G

L

H

M

I

N

K

O 0.0 q1 = ½(r1 − r2) /Å

(e)

H D

H D

D

(b)

(d)

A + HB

r1 r2 A−H·····B

0.0 q1 = ½(r1 − r2) /Å

0.0 q1 = ½(r1 − r2) /Å

FIGURE 7.1 (a) Correlation of the hydrogen bond length q2 ¼ r1 þ r2 with the proton-transfer coordinate q1 ¼ 1/2(r1 2 r2). The variable s represents the pathway along the correlation curve, with s ¼ 0 at q1 ¼ 0. (b) Evolution of the valence bond orders defined in Equation 7.1. (c) One-dimensional potential energy curves (schematically) for the proton motion in hydrogen bonds. (d) Vibrationally and solvent averaged onedimensional hydron density distribution functions of quasisymmetric hydrogen bonds of different strength. (e) Vibrationally and solvent averaged one-dimensional hydron density distribution functions of nonsymmetric hydrogen bonds of different strength. For further explanation see text.

196

Isotope Effects in Chemistry and Biology

They are unity in one and zero in the other limit. The product p1 p2 is zero in both limits, but exhibits a maximum of 0.25 at q1 ¼ 0: The sum of both bond orders is unity. In order to establish the parameters of Equation 7.1 used to calculate the solid line in Figure 7.1a it is necessary to know the position of the proton in the hydrogen bond. As this is a difficult task for x-ray diffraction it is not surprising that hydrogen bond correlations have been developed only recently after a large number of low-temperature neutron diffraction data have become available. Thus, parameters of Equation 7.1 have been derived empirically by Steiner et al.7 from lowtemperature neutron diffraction data for OHO, NHN, and OHN hydrogen bonds, mainly of weak and medium strength. These correlations seem also to hold for strong hydrogen bonds, as verified by dipolar solid-state NMR8 and by theoretical calculations.8 – 10 The advantage of the correlation is that the two phenomena of hydrogen bond formation and proton transfer are linked together in a single pathway. From the crystallographic view, where proton donors and acceptors are held together not only by hydrogen bonds, there is no clear border between the free molecules and the hydrogen bonds. The usual van der Waals cutoff criterion of ˚ for A· · ·B heavy atom distances of hydrogen bonds AHB seems to be too restrictive.7a about 3.6 A Also, it is not easy to define the border line between weak, medium, and strong and short hydrogen bonds.

B. ORIGIN OF H YDROGEN B OND I SOTOPE E FFECTS 1. Influence of the Hydron Potential The above correlation implies that the crystallographic view of atoms in defined positions is valid. From a quantum-mechanical standpoint, atomic positions represent observables, which are averaged over the vibrational wave functions. Because of the anharmonicity of the proton potential of single- or double-well hydrogen bonded systems the averaged proton positions in hydrogen bonds do not coincide with the equilibrium positions. A general quantum-mechanical treatment of single-well hydrogen bonds has been proposed by Sokolov et al.11 As a detailed description of all types of hydrogen bonds is beyond the scope of this chapter, we only sketch some typical one-dimensional potentials in Figure 7.1c for a sequence of configurations where the equilibrium position of H is shifted from A to B. The squared wave functions of the lowest vibrational states are included, where the vertical bars correspond to the average hydron positions kq1 lL ; qL1 ; L ¼ H; D: In principle, the one-dimensional potential curves represent averages over various zero-point motions. Therefore, they are slightly different for H and for D as indicated schematically. Note that this does not represent a deviation from the Born –Oppenheimer approximation. A, B and E, F represent configurations with asymmetric single-well potentials. A and F exhibit already a considerable Morse-type proton potential, with different average positions of H and D because of their different zero-point energy and hence different ground-state wave functions. This anharmonic effect gives rise to “geometric” H/D isotope effects on the average hydron positions. The effect is increased for the stronger hydrogen bonds B and E. We note that if other modes are taken into account, it could be that, for example, configuration pairs such as A and F, or B and E, or A and E constitute stable “tautomers” which can interconvert with each other in terms of a chemical rate process characterized by forward and backward rate constants. As the two forms of a pair are separated by a barrier, this situation is often not distinguished from a true double-well situation such as C, where the vibrational wave function is extended over both potential wells and exhibits two maxima indicated by arrows. The lowest vibrational levels occur in pairs, separated by a tunnel splitting, not depicted in Figure 7.1c. Such a situation is found for proton transfer in small molecular systems in the gas phase.12 For weak and medium hydrogen bonds, solid state interactions generally lower the symmetry and lead to proton localization as shown directly by various spectroscopic techniques.13

Hydrogen Bond Isotope Effects Studied by NMR

197

Nevertheless, when the barrier is low or even negligible, situations such as C or D can exist for small values of q1 even in condensed matter, although they may be difficult to detect. When the two maxima of the squared ground-state wave functions are well separated, two half protons can be localized by neutron diffraction, and for each position the hydrogen bond correlation may be fulfilled. In the case of C the proton density in the hydrogen bond center is, however, already substantial, and when the usual ellipsoid is used to fit the proton locations average H and D positions qL1 ¼ 0 may result not only for D but also for C. As both configurations exhibit different heavy atom distances, the hydrogen bond correlation will be no longer fulfilled. 2. Effects of the Environment In polyatomic molecules, the atoms distant from the hydrogen bond provide a “bath,” i.e., an intramolecular environment which influences the hydron potential for which examples were given in Figure 7.1c. There are two main phenomena that generally occur. The first concerns the nature of hydron transfer in a double-well potential. When the system is small and isolated, a double oscillator is realized with delocalized vibrational states split by hydron tunneling, for example the double proton transfer in formic acid dimer in the gas phase.14 Increasing the size of the system and incorporation into the solid state leads to an incoherent double-proton transfer in solid benzoic acid dimer.15 For this phenomenon also the term “tautomerism” has been used, characterized by rate constants and kinetic H/D isotope effects. If the system is asymmetric, it will also be characterized by an equilibrium constant and equilibrium isotope effects or isotopic fractionation between the tautomers. Generally, H is enriched in the tautomer exhibiting the smaller zero-point energy (see chapter of J. Bigeleisen). Not only chemical substitution lowers the symmetry, but also asymmetric isotope labeling, for example 17O or 18O substitution in an otherwise symmetric OHO hydrogen bond. This case will be addressed below as the “quasisymmetric” case. An equilibrium between Case A and F in Figure 7.1c would correspond to such a situation. Such systems are described in the chapter of Perrin who used the technique of isotopic perturbation in order to elucidate the symmetry of hydrogen bonds in solution.16 A detailed description of these phenomena is beyond the scope of this chapter. The second phenomenon is concerned with the influence of intermolecular interactions on the potential of the hydron in the hydrogen bonded system. As compared to the gas phase, the potential may be modified in the solid state. For example, a symmetric double well can become slightly asymmetric. In glassy state, a distribution of different sites is observed, where the double wells exhibit a different asymmetry.13b In the liquid state, the distribution of different sites will convert slowly in the IR timescale, but fast within the NMR time scale. The resulting vibrationally and solvent averaged hydron density distribution for the quasisymmetric case is depicted schematically in Figure 7.1d as a function of the hydrogen coordinate q1 (Figure 7.1d, Case G). Clearly, two tautomers with different geometries are well defined. The width of the peaks is caused mainly by intermolecular interactions. As has been derived qualitatively by a discussion of IR bands, the width of the distribution can be substantial in solution, i.e., much larger than in the crystalline state.17 As discussed above, the position of both peaks is different for H and for D, giving rise to an intrinsic H/D isotope effect on the hydrogen bond geometries, and hence also to modified spectroscopic parameters. D is located farther away from the hydrogen bond center as compared to H. Because of isotopic fractionation, a modified intensity distribution of the proton and the deuteron probabilities are realized as indicated schematically by the dashed line for D. When the barrier of the hydron motion in the quasisymmetric hydrogen bonded system becomes smaller, the hydron density peaks move towards each other according to the hydrogen bond correlation of Figure 7.1a, and eventually merge. Here the case of “easily polarizable” hydrogen bonds is reached, a phenomenon studied in many papers by Zundel et al.18 Hynes et al. (see chapter of Kiefer and Hynes) have studied the dynamics of hydron transfer under these

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Isotope Effects in Chemistry and Biology

conditions. In this regime, the width of the hydron density distribution increases strongly. In Case K, the deuteron is more confined in the hydrogen bond center than the proton; however, the average H and D density can still be different as the system is only quasisymmetric. By contrast, in Case I, D is still farer away from the hydrogen bond center than H. The asymmetric case is depicted in Figure 7.1e. In Case L only a single configuration is realized, characterized by an intrinsic geometric H/D isotope effect. When the hydron is shifted through the hydrogen bond center, similar configurations of the type M or N can be realized, which are similar to those of I and K. In these cases, it is no longer possible to distinguish intrinsic and equilibrium isotope effects: isotopic substitution does not only shift the position and the intensities of the maxima of the hydron density distribution functions but the whole functions themselves. Therefore, some of us have proposed to use the term hydrogen bond isotope effects19 instead of intrinsic and equilibrium isotope effects in the case of strong hydrogen bonds. A scenario arising from ab initio calculations of various acid-base complexes AHB in the presence of electric fields of Zundel et al.,18 Del Bene et al.,20 and Ramos et al.,10a which explains the distribution of hydrogen bond geometries in solution is illustrated in Figure 7.2 which is adapted from Ref. 21. The individual solvent dipoles create a temperature-dependent effective electric field at the solute site. This field is large at low temperatures where the dipoles are ordered and small at high temperatures where they are disordered. As a consequence, the temperaturedependent electric field induces in the solute a dipole moment m that is small at high temperatures, but increases as the temperature drops. The dipole moment induced depends on the polarizability of the complex, which consists of two contributions. The electronic contribution is associated with a reorganization of the electronic cloud under the action of the electric field. The nuclear part which is also called nuclear or vibrational polarizability is associated with changes of the nuclear geometry. In a molecular complex A –H· · ·B exhibiting a relatively small permanent electric dipole moment the latter can only be enhanced by charge transfer from B to AH. The energy for the charge separation is provided by the electric field; this energy is minimized by a contraction of the hydrogen bond. In other words, a molecular complex A – H· · ·B contracts in the presence of an electric field. This hydrogen bond contraction is accompanied by a displacement of the proton to the hydrogen bond center. Once the proton has crossed the center, an increase of the dipole moment is achieved by a further displacement of the proton towards B, accompanied by an increase of the A· · ·B distance. However, the question is now whether the shortest quasisymmetric hydrogen bond configuration constitutes a transition state or the top of the barrier of a double-well potential as indicated in Figure 7.2a, or whether this state is a stationary stable state. For this purpose, it is necessary to consider the thermodynamics of these systems. Solvent reorientation around the solutes leads to a gain in energy but also requires a decrease of the entropy. Therefore, the zwitterionic forms will be more stable at low temperature as depicted in the free-energy diagrams of Figure 7.2. The conventional picture of proton transfer assumes that Ad2· · ·H· · ·Bdþ is a transition state as illustrated in Figure 7.2a. By contrast, Figure 7.2b represents the case of a continuous temperature-dependent distribution of stationary states exhibiting various geometries. Here, the quasisymmetric complex dominates at an intermediate “transition” temperature which is high for zwitterionic complexes and very low for molecular complexes. Whereas in the double-well situation of Figure 7.2a H/D substitution always leads to an increase of the A· · ·B distance, i.e., to a high-field shift of D as compared to H, the single-well situation of Figure 7.2b should lead to the opposite, i.e., a low-field shift of D as compared to H.

C. INCLUSION OF Q UANTUM C ORRECTIONS IN H YDROGEN B OND C ORRELATIONS In order to take into account zero-point stretching vibrations in the valence bond model, and hence derive a separate description of the hydrogen correlations for protonated and deuterated hydrogen

Hydrogen Bond Isotope Effects Studied by NMR

199 +

− +

−

μ

−

H A+ D A+

−

−

−

−

−

μ

+

− + B− B− +

+

−+

+

H D

−

+

μ

−+

−+

δ B B

−

B B

δ A A

+

+

H D

−

−

− −

+

A A

−

−

+

+−

+

+

−

−

+

+

+

+

+

free energy

high T

medium T

low T

−

δ A A

−

+

−

μ

δ

B B

−

− +

−+

− − + B− B

− +

μ

H A+ + D A

−+ − +

−

+

+

+

−

H D

−

−

+

+

B B

+

+

−

+

−

+

−

−

−

−

A H A D + μ

+

−+

−

+

reaction coordinate + +

−

+

(a)

free energy

high T

medium T

low T

(b)

reaction coordinate

FIGURE 7.2 Temperature-dependent free-energy reaction profiles of hydron transfer in a 1:1 acid – base complex. (a) The case where Ad2· · ·L· · ·Bdþ corresponds to a transition state. (b) The case where Ad2· · ·L· · ·Bdþ corresponds to a stationary state. Both exhibit different H/D isotope effects on the hydrogen bond geometries and hence on associated NMR parameters, in particular the primary isotope effect dðADBÞ 2 dðAHBÞ: (Source: From Golubev, N. S., Shenderovich, I. G., Smirnov, S. N., Denisov, G. S., and Limbach, H. H., Nuclear scalar spin – spin coupling reveals novel properties of low-barrier hydrogen bonds in a polar environment, Chem. Eur. J., 5, 492– 497, 1999. With permission.)

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Isotope Effects in Chemistry and Biology

bonds, Limbach et al.22 have proposed that Equation 7.1 is valid only for a “classical” system exhibiting an “equilibrium” geometry corresponding to a minimum of the potential energy surface. In other words, p1 and p2 are essentially equilibrium bond orders. A real hydrogen bond ALB, L ¼ H, D, experiencing anharmonic zero-point vibrations of H and D, is then characterized by the real average bond orders pL1 and pL2 . For the latter, Equation 7.1 may no longer be valid in the strong hydrogen bond regime, i.e., pL1 þ pL2 , 1

ð7:2Þ

Limbach et al.22 proposed the following empirical relations between the classical and real bond orders, which are justified by comparison with experimental data: pL1 ¼ p1 2 cL ðp1 p2 Þf ðp1 2 p2 Þ 2 d L ðp1 p2 Þg ; pL2 ¼ p2 þ cL ðp1 p2 Þf ðp1 2 p2 Þ 2 d L ðp1 p2 Þg ; L ¼ H; D

ð7:3Þ

The corresponding average distances are calculated from the real valence bond orders using Equation 7.1. The parameters c L and d L as well as the values of the powers f and g are empirical and have to be adjusted by comparison with experimental geometries. The term dL ðp1 p2 Þg is a correction term describing the flattening of the real correlation curves in the minimum. If d H ¼ d D ¼ 0, the classical or equilibrium correlation and the real correlation curves of AHB and ADB coincide. The “correlation” term cL ðp1 p2 Þf ðp1 2 p2 Þ indicates then by how much the geometry of ALB is shifted on the correlation curve as compared to the classical value. Equation 7.3 also allows one to calculate the so-called primary geometric hydrogen bond isotope effect ( primary GIE)8: Dq1 ; q1D 2 q1H

ð7:4Þ

and the secondary geometric hydrogen bond isotope effect (secondary GIE): Dq2 ; q2D 2 q2H

ð7:5Þ

The secondary effect has also been called the Ubbelohde effect, as it was observed by this author and coworkers for a number of hydrogen bonded systems.23 A negative value of Dq2 has also been called an inverse Ubbelohde effect. Generally, secondary effects can be observed quite easily by x-ray crystallography, as hydron positions do not need to be determined. The primary geometric isotope effects are, however, difficult to study by x-ray diffraction, and neutron diffraction is needed. Indirect spectroscopic methods such as IR, Raman or NMR24 – 26 can give more precise results after a suitable calibration of the spectroscopic parameters. Naturally, theoretical methods can also be used to calculate these geometric isotope effects.11 The performance of Equation 7.3 is depicted qualitatively for NHN bonds in Figure 7.3 for two arbitrary sets of parameters listed in Table 7.1. Figure 7.3a corresponds to a series of hydrogen bond configurations exhibiting a double well in the symmetric case as depicted by situation C in Figure 7.1c. At all geometries, Dq2 is positive, i.e., the hydrogen bond is widened upon deuteration. Dq1 is negative for negative values of qL1 but changes sign when the latter becomes positive. In other words, D is always farther away from the hydrogen bond center than H. The bond order sum pL1 þ pL2 predicted by Equation 7.3 is shown on top of Figure 7.3a for the protonated and the deuterated hydrogen bonds. The deviation from unity is well pronounced in the strong hydrogen bond region; the reduction is stronger for D than for H. This reduction of the bond order sum is responsible for the larger qL2 values as compared to the equilibrium geometries, where the bond order sum is always unity.

Hydrogen Bond Isotope Effects Studied by NMR

equilibrium (p1 + p2 = 1)

p1L + p2L

1.00

H 0.95

q2 = r1 + r2/Å

2.70

D r1 r2 A–L·······B

r1 r2 A·······L–B

2.65 2.60 A··L·L··B 2.55

(a)

201

−0.3

H D

equilibrium

−0.2

−0.1

0.0

0.1

0.2

0.3

equilibrium (p1 + p2 = 1)

p1L + p2L

1.00 D 0.95

q2 = r1 + r2/Å

2.70

r1 r2 A–L·······B

r1 r2 A·······L–B

2.65 2.60 2.55

−0.3

(b)

H

r1 r2 A··L··B H D −0.2

equilibrium −0.1 0.0 0.1 q1 = ½(r1 − r2) /Å

0.2

0.3

FIGURE 7.3 Geometric hydrogen bond correlations according to Equation 7.3 for systems with a strong hydrogen bond NHN, i.e., in the region around q1 ¼ 0. The solid line represents the equilibrium geometries (a) for systems with double-well proton potential at the symmetric midpoint (according to the series A –B – C– E– F in Figure 7.1c and b) for systems with single-well proton potential at the symmetric midpoint (according to the series A – B– D– E– F in Figure 7.1c). The bond order sum as a function of q1 is shown for H and D particles both for case (a) and (b). The parameters used to calculate the curves are included in Table 7.1. For further explanation see text. (Source: From Limbach, H. H., Pietrzak, M., Benedict, H., Tolstoy, P. M., Golubev, N. S., and Denisov, G. S., Empirical corrections for quantum effects in geometric hydrogen bond correlation, J. Mol. Struct., 706, 115– 119, 2004. With permission.).

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Isotope Effects in Chemistry and Biology

TABLE 7.1 Parameters of the Anharmonic Correction in Equation 7.3 for NHN Hydrogen Bonded Units Figure

System

bNH/A˚

˚ roNH/A

f

g

cH

dH

cD

dD

Figure 7.3a Figure 7.3b Figure 7.7 Figure 7.8 Figure 7.9

NHN NHN 1, 2, 3 3 1, 2

0.370 0.370 0.404 0.370 0.370

0.997 0.997 0.992 0.997 0.997

5 5 — 5 5

2 2 — 2 2

330 330 — 330 330

0.3 0.4 — 0.4 1.2

30 30 — 30 30

0.4 0.3 — 0.3 1.1

The behavior of a series of hydrogen bonded complexes exhibiting a single-well potential also in the symmetric configuration is illustrated in Figure 7.3b. In contrast to Figure 7.3a, in the symmetric case, Dq2 is negative, i.e., the hydrogen bond is shortened upon deuteration as is well established for ions such as FHF2.27 Again, the sum pL1 þ pL2 for H and D is included in Figure 7.3b. Now, the hydrogen bond shortens in the symmetric case after deuteration, whereas in the asymmetric cases the contrary is found. We note that only the values of the parameters d H and d D needed to be exchanged in order to describe the change of the effective correlation curves between Figure 7.3a and b. For a more detailed description of the origins of deuteration effects on strong hydrogen bonds we refer to the chapter of Del Bene (Chapter 5).

D. H/D ISOTOPE E FFECTS ON NMR PARAMETERS AND H YDROGEN B OND G EOMETRIES: T HE P OINT A PPROXIMATION It is now plausible that when the geometry of a series of hydrogen bonded complexes changes in a systematic way, also the associated NMR parameters will follow. In this section, we will therefore discuss the links between the world of hydrogen bond geometries and NMR parameters. To establish such links is an important task as it opens the possibility to obtain geometric information of selected nuclei in molecules in solution where the usual diffraction or other NMR methods fail. Correlations between NMR parameters and geometries of A – H· · ·B hydrogen bonds, for example proton chemical shifts or A· · ·B coupling constants, have been observed using high-level ab initio methods by Del Bene et al. For a description of this work we refer to their chapter. From an experimental standpoint, chemical shifts of heavy atom nuclei involved in H-bonding as well as proton chemical shifts can now be measured using high-resolution solid-state NMR techniques. The aim of such work is, therefore, to correlate experimentally the world of H-bond geometries and NMR parameters. Once such links are established, it is possible to correlate other NMR parameters measured in solution such as coupling constants across hydrogen bonds or H/D isotope effects on chemical shifts of remote atoms such as 13C with the chemical shifts and hence the hydrogen bond geometries which can be used for fine tuning of H-bond geometries in solution. However, in contrast to systems with intramolecular hydrogen bonds or biomolecules where the dissociation and formation is linked to other molecular processes, the determination of intrinsic NMR parameters of intermolecular hydrogen bonds in solution is very difficult. For this task, lowtemperature liquid-state NMR methods have been developed.3 The main task is then to establish correlations between NMR parameters and hydrogen bond geometries, i.e., the value of q1 which would imply also a correlation with q2 : It is desirable to have isotope-insensitive correlation functions which can be applied to both the protonated and the deuterated systems. In other words, for a hydrogen bond ALB, L ¼ H, D, the secondary H/D isotope effect on the chemical shifts of B DdðADBÞ ; dðADBÞ 2 dðAHBÞ

ð7:6Þ

Hydrogen Bond Isotope Effects Studied by NMR

203

energy

V

chemical shift d

A··L··B

d (ADB) d (AHB)

q1H q1D

H D

(a) d(ALB)

(b) 0 q1 = ½(r1 − r2)

FIGURE 7.4 (a) One-dimensional potential curve and squared ground-state wave function (schematically) of a symmetric single-well hydrogen bond ALB, L ¼ H, D. (b) Chemical shifts (schematically) of B and L as a function of q1. The average chemical shift of D is larger than for H because of the maximum of dðALBÞ and the narrower wave function of D as compared to H. By contrast, as dðALBÞ is a linear function of q1, dðAHBÞ ¼ dðADBÞ: (Source: From Limbach, H. H., Pietrzak, M., Sharif, S., Tolstoy, P. M., Shenderovich, I. G., Smirnov, S. N., Golubev, N. S., Denisov, G. S., NMR- parameters and geometries of OHN- and ODN hydrogen bonds of pyridine –acid complexes. Chem. Eur. J., 10, 5195– 5204, 2004. With permission.)

and the primary isotope effects on the hydron chemical shifts DdðADBÞ ; dðADBÞ 2 dðAHBÞ

ð7:7Þ

arise in this approximation only from isotope effects on the hydrogen bond geometries. Such correlations can be found for the case where the NMR parameters are in good approximation to a linear function of q1. This condition is normally met, except in the case of the shortest and strongest hydrogen bond in a given series of hydrogen bonds exhibiting a single-well potential as depicted in Figure 7.4. 15 D N is an almost linear Around qH 1 ¼ q1 ¼ 0, the chemical shift dðALBÞ of a heavy atom such as function of q1. This means that the values of dðAHBÞ and of dðADBÞ; averaged over the groundstate wave functions, are the same and in good approximation equal to the value at q1 ¼ 0. In other words, one can associate to each value of q1 also a unique value dðALBÞ: By contrast, dðALBÞ is a nonlinear function of q1 exhibiting a maximum around q1 ¼ 0. As D is more confined to the hydrogen bond center as compared to H, the average value dðADBÞ will be larger than the average value dðAHBÞ; because the mean square values kq21lH and kq21lD are different, whereas the mean values qL1 ¼ 0 are the same for both H and D. In other words, a correlation of dðALBÞ with the mean average values qL1 represents an approximation which might lead to a systematic error in the case of a symmetric single-well hydrogen bond. It is clear that a correct description can be found only if the nuclear wave functions and the chemical shielding surface are known for the particular environment studied. An example is the FHF2 anion, where H/D isotope effects on chemical shielding have been studied theoretically by Golubev et al.28 A theoretical analysis showed that averaging over all hydrogen bond vibrations including the bending vibrations needs to be taken into account. As such a case represents an exception at present, we will not take the breakdown of the correlation between dðAHBÞ and q1 for the symmetric single-well case into account in the following.

204

Isotope Effects in Chemistry and Biology

For a quantitative correlation of NMR parameters and geometries of hydrogen bonds of the type AHB, we therefore assume the validity of the following isotope-independent equations. For the chemical shifts of the base B it was assumed that29,30

dðALBÞ ¼ dðBÞ8pLAL þ dðLBÞ8pLLB þ 4dp ðALBÞpLAL pLLB ; L ¼ H; D

ð7:8Þ

1

for the H chemical shifts that

dðAHBÞ ¼ dðAHÞ8pHAH þ dðHBÞ8pHHB þ 4dp ðAHBÞpHAH pHHB

ð7:9Þ

and for scalar 1H – B couplings that 1

H 2 H p JðAHBÞ ¼1 JðHBÞ8pH HB 2 8J ðAHBÞðpAH Þ pHB

ð7:10Þ

where dðLBÞ8; dðHBÞ8 and 1 JðHBÞ8 represent the limiting B and 1H chemical shifts and 1H–B coupling constants of the fictive free HB, dðBÞ8 the chemical shift of the free base B, and dðAHÞ8 the 1H chemical shift of the free acid. dp ðALBÞ; dp ðAHBÞ and J p ðAHBÞ are excess terms describing the deviation of the parameters of the strongest AHB hydrogen bond from the average of the limiting values of the free forms. We note that all the NMR correlation curves exhibit a principal difference to the q1 vs. q2 curves: the values of the latter can increase to infinity as AH moves away from B or A from HB, but the NMR parameters of the molecular units do not change any more when they are separated.

E. H/D I SOTOPIC F RACTIONATION, H YDROGEN B OND G EOMETRIES AND NMR PARAMETERS In contrast to the case of hydron tautomerism in strong hydrogen bonds the phenomenon of isotopic H/D fractionation is well established between different hydrogen bonded systems AHB and XHY. This equilibrium can be expressed as AHB þ XDY Y ADB þ XHY

ð7:11Þ

The equilibrium constant of this reaction is given by K ¼ 1=F ¼

xADB xXHY < expð2DZPE=RTÞ xAHB xXDY

ð7:12Þ

where the inverse equilibrium constant F is also called the fractionation factor. xi are mole fractions or concentrations of the various isotopic species, R the gas constant, T the temperature and DZPE the zero-point energy difference of the hydrons in the hydrogen bonds. This quantity is given by DZPE ¼ ZPEðXHYÞ 2 ZPEðXDYÞ 2 ðZPEðAHBÞ 2 ZPEðADBÞÞ

ð7:13Þ

We note that Equation 7.11 does not describe H/D equilibrium isotope effects of tautomerism or, more generally, a different hydrogen or deuteron distribution within a given hydrogen bonded system, but isotopic fractionation between different systems. From a theoretical standpoint, fractionation factors are most often calculated in harmonic approximation, although anharmonic corrections are possible (see the chapters of Bigeleisen and of Wolfsberg). For hydrogen bonded systems, anharmonicities are, however, very important. Here, we conceive DZPE as an experimental quantity that includes all anharmonicities. Kreevoy and Liang31a have determined fractionation factors of the kind in Equation 7.11 for a series of hydrogen bonded anions exhibiting a different basicities. Harris et al.32 (see the chapter of Mildvan and references cited therein) have measured H/D isotopic fractionation factors of OHO and NHO hydrogen bonds in biomolecules in relation to the 1H chemical shifts of the hydrogen

Hydrogen Bond Isotope Effects Studied by NMR

205

bond protons and the O· · ·O and N· · ·O distances. Smirnov et al.29 have measured fractionation factors of OHN hydrogen bonded complexes. In principle, it would be desirable to express the values of ZPE(ALB), L ¼ H, D as a function of the corrected bond valences pL1 and pL2 from which DZPE could be calculated as a function of the hydrogen bond geometries. Unfortunately, general correlations for all isotopic sensitive vibrations and, therefore, of their ZPE values are not available at present. Therefore, we use the following approximation in which DZPE is expressed as a function of the classical bond valences p1 and p2 as has been proposed previously29: DZPE ¼ DZPEo ð4pOH pHN Þ2 þ DZPEðOHÞo pOH þ DZPEðHNÞo pHN

ð7:14Þ

where DZPE(OH)o and DZPE(HN)o represent the zero-point energy differences of the hypothetical free diatomic states OH and HN as compared to a reference complex XHY. DZPE8 represents the absolute drop of zero-point energy between the nonhydrogen bonded limiting states and the strongest hydrogen bonded quasisymmetric state. Equation 7.14 provides a link between the world of geometries, NMR parameters, and vibrations of hydrogen bonds.

III. APPLICATIONS A. H/D I SOTOPE E FFECTS IN S TRONG NHN H YDROGEN B ONDS In this section we discuss the case of a strong NHN hydrogen bond in polycrystalline 1 and 2 (Figure 7.5), studied experimentally in Ref. 8. In Figure 7.5a are depicted the high-resolution solidstate 15N NMR spectra of both compounds, before and after deuteration, obtained with 1H – 15N 1: K+ ≡ As(Ph)4+ 2: K+ ≡ N(n-propyl)4+

K+[(CO)5Cr-C≡N··· H···N≡C-Cr(CO)5]− B

B a

b

rNN = 2.56 Å

N··H··N 1-h

N2

N1

0

N··D··N 1-d

DND ~ rND−3

N2···H··N1 2-h 2-d 800

r1 ≡ N1...D = 1.274 Å r2 ≡ D...N2 = 1.274 Å

r1 ≡ N1...D = 1.13 Å r2 ≡ D...N2 = 1.50 Å

N2····D··N1

220 180 δ/ppm

−1

+1

600

400

200 0 −200 −400 −600 δ/ppm

FIGURE 7.5 (a) 9.12 MHz 15N {1H} CPMAS NMR spectra of compounds 1-h to 2-d at room temperature, 2 kHz spinning speed, 9 ms 908 pulses. No temperature dependence was observed. (b) Superposed experimental and calculated 9.12 MHz 15N {1H} CP NMR spectra of static solid powder samples of 1-h, 1-d, 2-h, 2-d at 190 K. For the simulation of spectrum (b) a H/D fraction of 35/65 had to be taken into account. (Source: From Benedict, H., Limbach, H. H., Wehlan, M., Fehlhammer, W. P., Golubev, N. S., and Janoschek, R., Solid state 15N NMR and theoretical studies of primary and secondary geometric H/D isotope effects on low-barrier NHN-hydrogen bonds, J. Am. Chem. Soc., 120, 2939– 2950, 1998. With permission.)

206

Isotope Effects in Chemistry and Biology

cross-polarization (CP) for intensity enhancement, and magic angle spinning (MAS) and 1H decoupling for resolution enhancement. For 1-h a single line is observed, indicating that both nitrogen atoms are equivalent. The line is shifted slightly to high field in 1-d. This situation is typical for a symmetrical potential somewhere between C and D in Figure 7.1c. By contrast, for 2 two signals are obtained, indicating the presence of different nitrogen atoms N1 and N2, exhibiting different nitrogen– hydrogen distances. Here, the symmetry of the hydrogen bond is lowered by the crystal field. The chemical shift difference is much larger for 2-d as compared to 2-h. This finding represents the first H/D isotope effect on chemical shifts in solid-state NMR spectra. The isotope effect is explained with a distance decrease of the shorter bond and a distance increase of the longer bond after deuteration, corresponding to a proton potential of C in Figure 7.1c. When the static powdered crystals are measured, the spectra of Figure 7.5b are obtained. In the case of 1-h a typical spectrum of an axially symmetric chemical shielding tensor is obtained. Here, crystallites where the molecular N· · ·N axis is parallel to the magnetic field B absorb at high and those perpendicular at low field. As there are more of the latter, the low-field intensity is stronger than the high-field intensity. After deuteration, a triplet splitting because of dipolar coupling of 15N with 2H arises. From the coupling constant DND the average inverse cubic distance rND can be obtained. The spectra of 2-d are more complex, giving rise to two nitrogen – deuteron distances. We note that in the case of 1 the distances r1 ; N1· · ·D ¼ r2 ; D· · ·N2 correspond to half the NN distance obtained by x-ray crystallography. From these data, a 15N-chemical shift – distance correlation could be obtained, which could be used to calculate the corresponding N· · ·H distances by extrapolation from the 15N chemical shifts of 1-h and 2-h. For comparison, the linear simplified system [CuN· · ·L· · ·NuC]2Liþ(3) was modeled theoretically,8 where the effects of the crystal field acting on the anion were generated by a variety of fixed C· · ·Li distances. In principle, a more-dimensional calculation as described in the chapter of Del Bene would be desirable. In Ref. 8, for calculating the dynamical corrections of geometries of 3 a procedure based on the crude adiabatic approximation was employed. The application of this approximation consisted in an iterative procedure for the calculation of the dynamically corrected interatomic distances. For this purpose, the potential energy function V(q1) for the collinear hydron motion was calculated pointwise at fixed optimized heavy atom positions. These calculations were performed at the MP2/6 2 31 þ G(d,p) level. Then, the one-dimensional Schro¨dinger equation for the anharmonic collinear hydron motion was solved separately for L ¼ H and D using known methods.33 For simplicity the masses of the heavy atoms were set to infinity. The expectation value qL1 ¼ kCoLlq1lCoLl, where CoL(q1) represents the anharmonic vibrational ground state wave function of the hydron L ¼ H, D, was used for calculating the distance r1 ; N1· · ·L. Finally, the heavy atom coordinates were reoptimized keeping the distance C· · ·Li and the dynamically corrected N1· · ·L distance constant. This procedure was repeated until self consistency was achieved. In this way the dynamically corrected vibrational wave functions of the proton and the deuteron and the corresponding energies were obtained. Using this information the zero-point energy differences DZPE between the protonated and the deuterated hydrogen bond (neglecting bending contributions) and the mean linear and cubic average distances N1· · ·L and N2· · ·L, L ¼ H, D, were calculated for the vibrational ground state. The results are depicted in Figure 7.6. Because of the approximation used, the average one-dimensional potential is different for the proton and the deuteron motion. When Liþ is moved from the right side of the molecule to infinity, the hydron L moves from the left nitrogen to the H-bond center. At the same time, the N· · ·N distance is reduced. The mirror process completing the L transfer is induced when Liþ is reapproaches from the left side. The effects on the ground-state wave function, the different H and D positions, and the zero-point energy changes are similar to those discussed in the previous section. The analysis of 1 to 3 in terms of Equation 7.1 is shown in Figure 7.7, where neutron diffraction data of strong NHN hydrogen bonds listed in the Cambridge Structural Database (CSD)34 are included as triangles. Figure 7.7 is taken from Ref. 22 and represents an improved version of

Hydrogen Bond Isotope Effects Studied by NMR 3 r1 r2 [C≡N–H·····N≡C]− Li

1000 cm−1

E

r1 r2 Li+ [C≡N·····H–N≡C]−

r1 r2 [C≡N···H···N≡C]−

7Å

5Å

207

9Å

∞

9Å

7Å

5Å

H D

0.5 Å

q1=½(r1−r2)

FIGURE 7.6 Calculated potential curves, vibrational ground-state energies and squares of wave functions for the proton and deuteron motion in 3-h and 3-d, respectively. The mean hydron positions kqL1 (L)l are indicated by vertical lines. (Source: From Benedict, H., Limbach, H. H., Wehlan, M., Fehlhammer, W. P., Golubev, N. S., and Janoschek, R., Solid state 15N NMR and theoretical studies of primary and secondary geometric H/D isotope effects on low-barrier NHN-hydrogen bonds, J. Am. Chem. Soc., 120, 2939– 2950, 1998. With permission.)

corresponding graph in Ref. 8. The calculated points for 3 are fairly well located on the correlation ˚ and roNH ¼ ro1 ¼ ro2 ¼ 0.992 A ˚ line, calculated with the parameters bNH ¼ b1 ¼ b2 ¼ 0.404 A proposed in Ref. 8. By contrast, the data points for 1 and 2 are systematically displaced. These deviations were not discussed in detail in Ref. 8. The correction parameters of Equation 7.1 are obtained according to Ref. 22 by comparison of the geometries of the homoconjugated anionic systems 1 to 3. In Figure 7.7 and Figure 7.8

1 : [(CO)5Cr-C≡N····H···N≡C-Cr(CO)5]− As(Ph)4+ 2 : [(CO)5Cr-C≡N····H···N≡C-Cr(CO)5]− N(n-propyl)4+ 3 : [C≡N····H···N≡C]− Li+

3.6

r1 r2 N··H·····N

q2 = r1 + r2 /Å

3.4

r1 r2 N·····H··N

3.2 3.0 2.8 2.6 2.4

CSD r1 r2 N··H··N

1h, 2h 1d, 2d 3h 3d

−0.8

−0.4

0.4 0.0 q1 = ½(r1 − r2) /Å

0.8

FIGURE 7.7 Correlation of the length q2 ¼ r1 þ r2 of NHN hydrogen bonds with the proton transfer coordinate q1 ¼ 1/2(r1 2 r2). CSD: neutron diffraction data of various NHN hydrogen bonds from the Cambridge Structural Database.34 The solid line was calculated in terms of Equation 7.1 with a single set ˚ and r o ¼ 0.992 A ˚ . (Source: From Limbach, H. H., Pietrzak, M., Benedict, H., of parameters b ¼ 0.404 A Tolstoy, P. M., Golubev, N. S., and Denisov, G. S., Empirical corrections for quantum effects in geometric hydrogen bond correlations, J. Mol. Struct., 706, 115– 119, 2004. With permission.)

208

Isotope Effects in Chemistry and Biology 3 : [C≡N····H···N≡C]¯ ······Li+ 3.6

q2 = r1 + r2/Å

3.4 3.2 3.0 2.8 2.6

(a)

2.4

3h 3d

+

+

+ 3-equilibrium

++ ++ +

+

+ ++++

+

+

equilibrium

−0.8

−0.4

0.0

0.4

0.8

−0.8

−0.4

0.0

0.4

0.8

−0.8

−0.4

0.0 q1 = ½(r1 − r2) /Å

0.4

0.8

0.06

Dq1/Å

0.04 0.02 0.00 −0.02 −0.04 −0.06

(b) 0.06

Dq2/Å

0.04 0.02 0.00 −0.02 −0.04 −0.06

(c)

FIGURE 7.8 (a) Hydrogen bond correlations adapted to 3h and 3d using the correction of Equation 7.3 and the parameters listed in Table 7.1. The solid line was fitted to the equilibrium geometries of 3 listed in Table 7.2 of Ref. 8, included as crosses in the graph. (b) Primary geometric isotope effects Dq1, (c) secondary geometric isotope effects Dq2. (Source: From Limbach, H. H., Pietrzak, M., Benedict, H., Tolstoy, P. M., Golubev, N. S., and Denisov, G. S., Empirical corrections for quantum effects in geometric hydrogen bond correlations, J. Mol. Struct., 706, 115–119, 2004. With permission.)

the geometric hydrogen bond correlations q2 and the primary and the secondary geometric isotope effects Dq1 and Dq2 are plotted as a function of q1. The dotted lines were calculated using Equation 7.3, where the parameters used are included in Table 7.1. The data points of Figure 7.5 are included separately in Figure 7.8a and Figure 7.9a; in addition, in Figure 7.8a are included the equilibrium distances of 3 as crosses, as calculated in Ref. 8 without dynamic correction.

Hydrogen Bond Isotope Effects Studied by NMR

209

1 : [(CO)5Cr-C≡N····H···N≡C-Cr(CO)5]¯ As(Ph)4+

2 : [(CO)5Cr-C≡N····H···N≡C-Cr(CO)5]¯ N(n-propyl)4+

3.6 q2 = r1 + r2 /Å

3.4 3.2 3.0 2.8 2.6 (a)

2.4

1h, 2h 1d, 2d

+

+

CSD −0.8

++ ++ +

−0.4

+

+ ++++

0.0

0.06

equilibrium 0.4

0.8

0.4

0.8

0.4

0.8

2

0.04 Dq1/Å

+ +

0.02 0.00

1

−0.02 −0.08 2

−0.06 (b)

−0.8

−0.4

0.0

0.06

Dq2 /Å

2

2

0.04 0.02 0.00 −0.02

1

−0.08 −0.06 (c)

−0.8

−0.4

0.0 q1 = ½(r1 − r2) /Å

FIGURE 7.9 (a) Hydrogen bond correlations adapted to the data of 1, 2 published in Ref. 8 and to the CSD data of Figure 7.1, using the correction of Equation 7.3 and the parameters listed in Table 7.1. (b) Primary geometric isotope effects Dq1, (c) secondary geometric isotope effects Dq2. (Source: From Limbach, H. H., Pietrzak, M., Benedict, H., Tolstoy, P. M., Golubev, N. S., and Denisov, G. S., Empirical corrections for quantum effects in geometric hydrogen bond correlations, J. Mol. Struct., 706, 115– 119, 2004. With permission.)

˚ and roNH ¼ 0.992 A ˚ of the NHN correlation curve were As the parameters bNH ¼ 0.404 A derived mainly for protonated systems, they could not be directly used for the calculation of the equilibrium geometry correlation. Therefore, in Figure 7.8a, the equilibrium geometries of ˚ and roNH ¼ 0.997 A ˚ reproduced these 3 were used for this purpose. The parameters bNH ¼ 0.370 A geometries in a satisfactory way, as indicated by the crosses and the solid line in Figure 7.8a. However, one can anticipate that these parameters might be subject to changes in the future.

210

Isotope Effects in Chemistry and Biology

The computational data of the model compound 3h and 3d need a relatively small correction for the anharmonic zero-point vibration, as indicated by the dotted line in Figure 7.9a; the primary GIE and the secondary GIE are well reproduced. A negative secondary isotope effect is predicted at q1 ¼ 0. However, the crude adiabatic approximation used for data points of system 3 does not predict any GIE at q1 ¼ 0, because, in contrast to the expectations of Figure 7.1c, it uses the same potential for H and D for the symmetric complex (Figure 7.6). A more-dimensional treatment would probably reveal a negative secondary GIE, found experimentally for 1 as depicted in Figure 7.9a and 9c. In the cases of 1 and 2, the correction terms (Table 7.1) are larger than for 3; Figure 7.9 illustrates that the GIE are well reproduced. However, the most important point is that the increase in the qL2 values of both compounds as compared to the values calculated from the equilibrium correlation is now well reproduced.

B. H/D I SOTOPE E FFECTS IN OHN H YDROGEN B ONDED P YRIDINE – ACID AND C OLLIDINE – A CID C OMPLEXES In this section, we will analyse in particular the relations between the NMR parameters and hydrogen bond geometries of 1:1 collidine (2,4,6-trimethylpyridine) – acid and pyridine– acid complexes in different environments. The former had been studied both by solid-state NMR35a as well as by low-temperature liquid-state NMR using a liquefied freon gas mixture CDF3/ CDF2Cl as NMR solvents.36 The pyridine – acid complexes in freons have been studied in several papers.30,19 1. Low-Temperature NMR Spectroscopy of Pyridine –Acid Complexes Dissolved in Liquefied Freon Mixtures As an example of solid-state NMR was already been discussed above, we will give here only an example of the liquid-state work. Figure 7.10 depicts some low temperature NMR spectra of samples of o-toluic acid (a –c) and of 2-thiophenecarboxylic acid (d – f) as proton donors AH in the presence of a small excess of pyridine-15N as base B in CDF3/CDF2Cl.19a The deuterium fractions in the mobile proton site was about 80%. Under the conditions employed, hydrogen bond exchange is slow in the NMR timescale. The signals of the H-bond protons are split into doublets by scalar coupling with 15N. The coupling constant JðOHNÞ between 15N and the hydrogen bond proton is relatively small (12 Hz) in the case of o-toluic acid as proton donor, indicating that the proton is preferentially localized near A, but is much larger (57 Hz) in the case of 2-thiophenecarboxylic acid, indicating that the proton is located near B. This result is plausible as 2-thiophenecarboxylic acid exhibits a greater acidity or proton donating power in comparison to o-toluic acid. The low field shift from 18.68 to 21.33 ppm indicates also a shortening of the OHN hydrogen bond. The 2H NMR spectra (Figure 7.10b and e) of the same samples reveal primary upfield isotope shifts DdðODNÞ ; dðODNÞ 2 dðOHNÞ which are negative in both cases. In other words, both hydrogen bonds are weakened by D-substitution. The 15N spectra give interesting additional information. Firstly, hydrogen bonding and proton transfer to pyridine leads to a high field shift of the pyridine 15N signal, where the chemical shift scale of Figure 7.10 refers to the absorption of free pyridine. These high field shifts are in accordance with the increase of JðOHNÞ: However, in the 80% deuterated samples two lines appear where the more intense line arises from the deuterated complexes ODN and the smaller line from OHN which are in slow exchange. We observe that the one-bond isotope effect across the hydrogen bond DdðODNÞ ; dðODNÞ2 dðOHNÞ; is negative in the case of o-toluic acid, but positive in the case of 2thiophenecarboxylic acid. Qualitatively, these effects can be explained in terms a widening of the hydrogen bond upon deuteration, and a larger shift of D away from the hydrogen bond center as compared to H.

Hydrogen Bond Isotope Effects Studied by NMR

211 O

B

15

a

B

N −30

b

N15 ≡ B

D A H

−40

−50

−60

H J =12 Hz

H 21

20

d

≡ B

f

B

H

HA ≡ H O

−50

B+ D

2H

J =57 Hz 1H

22

B+ 21

A

18

17 ppm

B+ D B+

−40

ppm

S

C

15N

−30

e

−80 D A

19 O

N15

CH3

−70 B

2

22

C

A

1

c

HA ≡ H O

−60

A−

A−

H −70

−80 ppm

A−

H

A− 20 19 δ/ppm

18

17 ppm

FIGURE 7.10 15N NMR (a,d), 2H NMR (b,e), 1H NMR (c,f) spectra (1H Larmor frequency 500.13 MHz) of solutions of pyridine-15N (B) and carboxylic acids (AL, L ¼ H, D) in a 1:2 mixture of CDF3/CDF2Cl at a deuterium fraction in the mobile proton sites xD ¼ 0.8. a– c: 125 K, CB ¼ 0.033 M, CAH þ CAD ¼ 0.028 M, d – f: 110 K, CB ¼ 0.045 M, CAH þ CAD ¼ 0.035 M. The 15N chemical shifts are referred to internal free pyridine, where d(CH3NO2) ¼ d(C5H5N) 2 69.2 ppm, and d(NH4Cl) ¼ d(CH3NO2) 2 353 ppm. (Source: From Smirnov, S. N., Golubev, N. S., Denisov, G. S., Benedict, H., Schah-Mohammedi, P., and Limbach, H. H., Hydrogen/deuterium isotope effects on the NMR chemical shifts and geometries of intermolecular low-barrier hydrogen bonded complexes, J. Am. Chem. Soc., 118, 4094 –4101, 1996. With permission.)

2. Geometric Hydrogen Bond Correlations of OHN Hydrogen Bonded Complexes In order to correlate NMR parameters with hydrogen bond geometries, Limbach et al.37 have recently introduced the empirical corrections of Equation 7.3 into the geometric correlation analysis of OHN hydrogen bonds studied previously by Steiner.7d Using his parameters, which are listed in the first row of Table 7.2, the classical or equilibrium correlation was obtained as depicted by the lower solid line in Figure 7.11a. The data points of the weak and medium hydrogen bonds from the CSD from which these parameters were derived are represented by filled circles in Figure 7.11a. The filled squares stem from data obtained by Lorente et al. for 1:1 hydrogen bonded complexes of collidine with various acids using a combination of dipolar solid-state NMR35a and x-ray crystallography.35b The filled triangles in Figure 7.11a stem from the low-temperature neutron

212

Isotope Effects in Chemistry and Biology 3.4 O-H

q 2 =r1+r2/Å

3.2

O

N

H-N

O H N

3.0 2.8

all OHN pyr-HA

2.6

col-HA

2.4

−0.5

(a)

0.0

0.5

q 1 = ½(r1 - r2) /Å 3.2

CH3

rND /Å

2.8

15N

col≡CH3

2.4

CH3

2.0

col-DA solid

1.6 1.2 0

−20

−40

(b)

−60

−80

−100

−120

−140

−120

−140

d(ODN)/ppm

d(OHN)/ppm

25 20

pyr ≡

15N

15

pyr-HA freon col-HA freon

10

[col-H-col]+ freon col-HA solid

5 0

(c)

pyr-HOSi 0

−20

−40

−60

−80

−100

d(OHN)/ppm 120 100

J (OHN)/Hz

80 60 40 20

pyr-HA freon

0

col-HA freon

−20 −40

(d)

[col-H-col]+ freon 0

−20

−40

−60

−80

d(OHN)/ppm

−100

−120

−140

Hydrogen Bond Isotope Effects Studied by NMR

213

TABLE 7.2 Parameters of the Geometric Hydrogen Bond Correlations of Pyridine – Acid and Collidine– Acid Complexes Systems

˚ a rOH8/A˚a bHN/A ˚ a rNH8/A ˚a bOH/A

Weak and medium strong OHN bonds 0.371 Pyridine–HA and collidine–HA in CDF3/CDF2Cl 0.371 Pyridine–HA and collidine–HA solid 0.371 a

0.942 0.942 0.942

0.385 0.385 0.385

0.992 0.992 0.992

f

g

cH

dH

0 0 0 0 5 2 360 0.7 5 2 360 0.7

cD

dD

0 0 50 0.6 110 0.6

Taken from Ref. 7d. Reproduced from Ref. 37.

diffraction data of the very strong OHN hydrogen bonds between 4-methylpyridine and pentachlorophenol and related compounds.38 As in the NHN case, systematic deviations from the equilibrium curve are observed. The dotted curve was calculated using the correction parameters listed in Table 7.2 by fitting the neutron diffraction data of the various OHN hydrogen bonds. 3. H/D Isotope Effects on the NMR Parameters of Pyridine –Acid and Collidine –Acid Complexes A quantitative analysis of all pyridine –acid and collidine –acid data available was performed by Limbach et al.37 and is depicted in Figure 7.11b to d and Figure 7.12. The dotted lines were calculated using the equations presented in the Theoretical section. For that purpose, A was identified with O, and B with N. In Figure 7.11b are plotted the ND distances rND of deuterated polycrystalline collidine –acid complexes as a function of their 15N chemical shifts dðODNÞ: The distances had been obtained previously from the dipolar 15N-D couplings, and the 15N chemical shifts by high-resolution solid-state NMR.35 The parameters of equations used to calculate the dotted lines are assembled in Table 7.3. For the case of collidine –acid complexes, the values of dðODNÞ are referenced to the 15N chemical shift of neat frozen collidine, resonating at 268 ppm,35 and those of the pyridine – acid complexes to neat frozen pyridine, resonating at 275 ppm with respect to solid 15NH4Cl.39 Small upfield solvent shifts DdðNÞ8 describing the difference of the 15N chemical shifts of the free bases dissolved in CDF3/CDF2Cl, arising from weak solute – solvent H-bond interactions were taken into account in Figure 7.11. In Figure 7.11c are depicted the 1H chemical shifts dðOHNÞ of the pyridine – and collidine – acid complexes obtained under various conditions as a function of the 15N chemical shifts dðOHNÞ: The filled symbols refer to the solid collidine – acid complexes with OHN hydrogen bonds, whereas

FIGURE 7.11 OHN hydrogen bond correlations. The parameters of the calculated curves are listed in Table 7.2 and Table 7.3. (a) Geometric OHN hydrogen bond correlations. OHN: neutron diffraction data in the Cambridge Structural Database as published by Steiner7d; pyr – HA: neutron diffraction data of the crystalline 4-methylpyridine-pentachlorophenol complex38; coll –HA solid: dipolar NMR of crystalline collidine – acid complexes.35 (b) ND distance — 15N chemical shift correlation for polycrystalline collidine– acid complexes obtained by dipolar solid-state NMR.35 (c) 1H – 15N chemical shift correlation of pyridine – acid and collidine – acid complexes. Open symbols: CDF3/CDF2Cl solution at low temperatures36,19a; [col – H –col]þ freon: Ref. 41; pyr– HOSi: pyridine in mesoporous silica.40 (d) Correlation between the H15N scalar couplings JðOHNÞ with the 15N chemical shifts of pyridine– acid and collidine– acid complexes in CDF3/CDF2Cl at low temperatures. (Source: From Limbach, H. H., Pietrzak, M., Sharif, S., Tolstoy, P. M., Shenderovich, I. G., Smirnov, S. N., Golubev, N. S., Denisov, G. S., NMR- parameters and geometries of OHN- and ODN hydrogen bonds of pyridine – acid complexes. Chem. Eur. J., 10, 5195– 5204, 2004. With permission.)

214

Isotope Effects in Chemistry and Biology O-L 12

O

L -N

[col-L-col] + freon

L=H,D

8

Dd (ODN)/ppm

O L N

N

col-LA solid

4 0 −4 −8

pyr-LA freon col-LA freon

−12 0

(a)

−20

−40

−60 −80 d(OHN)/ppm

−100

−120

−140

3

Dd(ODN)/ppm

2

[col-L-col] + freon

pyr-LA freon

1

freon

0 −1 −2 −3

(b)

solid 0

−20

−40

−60 −80 d(OHN)/ppm

−100

0.08 0.06

freon

Dq1/Å

0.04 0.02 0.00

solid

−0.02 −0.04 −0.06 −0.08

−0.5

(c) 0.08

0.0 q1 = ½(r1 - r2) /Å

0.5

freon

0.06

Dq2 /Å

0.04 0.02 0.00

−0.02

solid

−0.04 −0.06 −0.08

(d)

−0.5

0.0 q1 = ½(r1 - r2) /Å

0.5

−120

−140

Hydrogen Bond Isotope Effects Studied by NMR

215

in Figure 7.11b the corresponding complexes with ODN hydrogen bonds were depicted; in addition, a data point is included as solid triangle on the left side, characterizing the complex of pyridine with surface Si –OH groups of mesoporous silica of the MCM-41 and SBA-15 type.40 For this complex an 1H chemical shift of 10 ppm and a 15N chemical shift of 2 23 ppm with respect to frozen neat pyridine had been obtained. The open symbols refer to various pyridine – acid complexes19a and of collidine – acid36 complexes dissolved in CDF3/CDF2Cl mixtures (freon), as well as one data point referring to the homoconjugate cation of [collidine –H –collidine]þ reported in Ref. 41. The data indicate that the 1H chemical shifts are larger for freon solution as compared to the solid state as long as H is closer to oxygen or in the hydrogen bond center. In particular, the maximum 1H chemical shift is about 21.5 ppm for freon solution, but only about 19 ppm for the solid state. We found that we were able to reproduce these findings by using the same values of dðHNÞ8 ¼ 7 ppm for “free” collidinium and pyridinium, the same excess term dp ðOHNÞ ¼ 20 ppm; but different values of dðOHÞ8: For freon solution we had used previously a value of þ 2 ppm29 which we kept here, and for “free” oxygen acids in the solid-state values between 2 2 ppm42 and 2 4 ppm had been reported.43 We found that a slightly modified value of 2 3 ppm fitted better the solid-state data of Figure 7.11c. The calculated dotted lines reproduce now the experimental data in a very satisfactory way. Finally, Figure 7.11d illustrates how the scalar couplings 1 JðOHNÞ between 1H and 15N correlate with dðOHNÞ: The parameters of Equation 7.10 leading to the dotted line are included in Table 7.3. We come now to the problem of H/D isotope effects on NMR chemical shifts and geometries of the OHN hydrogen bonds studied. The secondary H/D isotope effect on the 15N chemical shifts DdðODNÞ ; dðODNÞ2 dðOHNÞ depicted in Figure 7.12a have been measured for pyridine – acid complexes in freon,19a and for collidine – acid complexes in the solid state35a and in freon,36 whereas the primary isotope effects on the hydron chemical shifts DdðODNÞ; dðODNÞ2 dðOHNÞ depicted in Figure 7.12b could be obtained only for the pyridine– acid complexes in freon.19a Because of the point approximation used, in order to calculate the dotted lines in Figure 7.12, it was only needed to adapt the values of the correction parameters c D and d D in Equation 7.3 listed in Table 7.2, as the NMR parameters in Table 7.3 were already known from Figure 7.10. Again, two data sets were obtained, one for the complexes in freon and the other for the solid state. Then, we calculated the corresponding ODN correlation curve q2 vs. q1. As these curves almost coincided with the dotted OHN curve of Figure 7.10a, it was not included in this graph. Instead, we calculated the isotope effects on the hydrogen bond geometries for the freon data which are included as dotted lines in Figure 7.12c and d. The values of DdðODNÞ change sign when H is moved across the hydrogen bond center; the absolute maximum and minimum values of DdðODNÞ are the same. By contrast, the absolute values of DdðODNÞ in two minima are unequal. This effect arises mainly from the smaller variation of the hydron chemical shifts with the distance from the hydrogen bond center when the proton is closer to nitrogen as compared to oxygen, i.e., because of the different slopes of the 1H vs. 15N curves depicted in Figure 7.11c. Thus, the graphs of Figure 7.12a and b represent the NMR analogs

FIGURE 7.12 (a) Secondary DdðODNÞ ; dðODNÞ2 dðOHNÞ and (b) primary DdðODNÞ ; dðODNÞ2 dðOHNÞ isotope effect on NMR chemical shifts of pyridine– acid and collidine – acid complexes around 130 K, dissolved in CDF3/CDF2Cl and solid collidine –acid complexes, data from Refs. 35a,36,41. (c) Primary geometric isotope effects Dq1 and (d) secondary geometric isotope effects Dq2 of pyridine – acid and collidine – acid complexes. The dotted curves in (a) to (d) were calculated using the parameters listed in Tables 7.2 and Table 7.3. (Source: From Limbach, H. H., Pietrzak, M., Sharif, S., Tolstoy, P. M., Shenderovich, I. G., Smirnov, S. N., Golubev, N. S., Denisov, G. S., NMR- parameters and geometries of OHN- and ODN hydrogen bonds of pyridine – acid complexes. Chem. Eur. J., 10, 5195– 5204, 2004. With permission.)

216

Isotope Effects in Chemistry and Biology

TABLE 7.3 Parameters of the NMR Hydrogen Bond Correlations of Pyridine– Acid and Collidine – Acid Complexes Systems

dðNÞ8=ppma

dðHNÞ8=ppm

DdðNÞ8=ppmb

dðOHÞ8=ppm

Pyridine–HA in CDF3/CDF2Cl Collidine–HA in CDF3/CDF2Cl Solid collidine –HA

0 0 0

126 126 126

24 28 0

2 2 23

dðHNÞ8=ppm

dp ðOHNÞ=ppm

JðHNÞ8=Hz

Jp ðOHNÞ=Hz

7 7 7

20 20 20

110 110 —

12.5 12.5 —

Pyridine–HA in CDF3/CDF2Cl Collidine–HA in CDF3/CDF2Cl Solid collidine –HA a b

1

With respect to neat frozen pyridine and collidine, resonating at 275 ppm and 268 ppm with respect to solid NH4Cl.35a DdðNÞ8 : 15N chemical shift difference between the free bases in the frozen solid state and in freon solution around 130 K.

of the corresponding graphs of the geometric isotope effects depicted in Figure 7.12c and d which exhibit only slight asymmetries because of the different parameters of OH and HN bonds in Table 7.2. The values observed for pyridine– and collidine – acid complexes in freon coincide within the margin of error. Moreover, in the zwitterionic regime, the values for the complexes in the liquid and the solid state are also very similar. By contrast, in the “molecular complex” regime where H is closer to oxygen, smaller absolute isotope effects are observed for the solids. Unfortunately, the origin of the different isotope effects in the liquid and the solid state could not be elucidated in this study. The parameter c D influences the curves in Figure 7.12a and c, and c D as well as d D influence the curves of Figure 7.12b and d. The liquid- and solid-state data are described by two different dotted curves of which the parameters are listed in Table 7.2. Besides the values of DdðODNÞ for the quasisymmetric pyridine– formic acid complex and the [collidine – H –collidine]þ cation, the agreement of the calculated curves with the experimental values is satisfactory. We note that the correction parameters c H, d H, c D and d D are very close to those found for the NHN hydrogen bonds22 described above, where geometric data could directly be analyzed. The agreement gives confidence that the corresponding graphs in Figure 7.12c and d of the geometric H/D isotope effects on the hydrogen bond are close to the reality. So far, we have not yet discussed the data points of the homoconjugate [collidine –H – collidine]þ cation represented by open triangles in Figure 7.11 and Figure 7.12. The data point in Figure 7.11c is located below the 15N – 1H correlation curve. As the OHN and NHN hydrogen bond geometric correlations are similar, this means that the corresponding data point in Figure 7.11a would be located around q1 ¼ 0, but the value of q2 would be larger than the minimum value. This effect could arise either from an intrinsic barrier between two degenerate potential wells of the kind depicted in type C of Figure 7.1c, or from a solvent barrier, where an asymmetric structure of the type B depicted in Figure 7.1c exchanges rapidly with the corresponding structure exhibiting a potential corresponding to the mirror image of B. These interpretations are supported by the observation of a negative primary H/D isotope effect DdðODNÞ (Figure 7.12b), indicating that deuteration increases the hydrogen bond length, a sign that H is not located in the hydrogen bond center. This finding is not in agreement with our treatment, which did not take into account two interconverting forms separated by a barrier. Thus, the correction parameters leading to the dotted

Hydrogen Bond Isotope Effects Studied by NMR

217

lines in Figure 7.12 predict a sign change of DdðODNÞ for the quasisymmetric complexes, as expected for systems with single well potentials in the strongest hydrogen bonds. Such a sign change had been observed for solid complexes exhibiting strong NHN hydrogen bonds,22 as well as for systems such as FHF2.41 Therefore, the calculated values of DdðODNÞ in Figure 7.12b deviate from the experimental ones found for freon solution. We did not attempt here to introduce a correction for this effect, in view of the fact discussed in the Theoretical section that the point approximation breaks down anyway in the case of the values of DdðODNÞ of very strong hydrogen bonds. The agreement between the experimental and calculated values in Figure 7.12b outside the region of the symmetric hydrogen bonds is better; here double wells will not be realized because of the large asymmetries of these complexes. In order to make further progress in this field, it will be necessary to measure the values of DdðODNÞ for the symmetric or quasisymmetric complexes not only for freon solution but also for the solid state. This task is, however, not easy in view of the large quadrupole coupling constant of D, and in view of the smaller gyromagnetic ratio of D as compared to H. 4. H/D Isotopic Fractionation and NMR Parameters of Pyridine – Acid Complexes Some time ago, Smirnov et al.19a performed a low-temperature NMR study of isotopic fractionation in the series of the 1:1 hydrogen bonded acid– pyridine complexes dissolved in CDF3/CDF2Cl according to Figure 7.13a. In particular, using the triphenylmethanol – pyridine complex as a reference the values of K had been determined around 130 K and correlated with the hydrogen bond geometries and 15N chemical shifts using Equation 7.14. Recently,37 the data analysis was improved by taking into account the empirical correction terms of Equation 7.3. Because of the close correlation with the geometric H/D isotope effects discussed in the previous section, we will discuss this case in more detail. In Figure 7.13b are depicted both the values of K as well as those of DZPE calculated according to Equation 7.12 as a function of the nitrogen chemical shift, serving again as a measure of q1. The K values are also plotted in Figure 7.13c as a function of the 1H NMR chemical shifts. A systematic correlation with both NMR parameters and hence the hydrogen bond geometries is observed. The dotted lines in Figure 7.13b were calculated neglecting differences of the zero-point energies in the free OH – and HN – acids using the following parameters: DZPEo ¼ 23:13 kJmol21 ; DZPEðOHÞo < DZPEðHNÞo < þ0:4 kJmol21

ð7:15Þ

with the other parameters of Table 7.2 and Table 7.3. Thus, a zero-point energy drop of about 3.5 kJmol21 between the free reference states and the strongest OHN hydrogen bond configuration is obtained. Without the correction, only a value of about 2.8 kJmol21 is obtained19a; this value corresponds to the difference between the ZPE in the limits and the effective ZPE in the minimum of the upper curve in Figure 7.13b. If the strongest OHN configuration were a classical transition state of a single H transfer reaction, one would expect a kinetic H/D isotope effect at 298 K of k H/ k D < exp(DZPEo/RT) < 3.5 for the new and 3.1 for the old value. Often larger values of k H/k D are assumed for this kind of reactions.44 The present analysis therefore supports the idea that larger kinetic isotope effects than k H/k D < 4 at 298 K will arise from tunneling contributions rather than from the loss of zero-point energy in the transition state.

C. TEMPERATURE-I NDUCED S OLVENT H/D I SOTOPE E FFECTS ON NMR C HEMICAL S HIFTS OF FHN H YDROGEN B ONDS In the previous sections we have described acid – base complexes in the solid state where dielectric constants are small, or in freon solvent mixtures around 130 K, where the dielectric constants are large.30b However, in the case of these mixtures, the dielectric constants decrease strongly when temperature is increased.

218

Isotope Effects in Chemistry and Biology

Ph3CO–D

N15

+ RO H N15

H D

K

H D

N15

Ph3CO–H

+ RO D N15

H D

(a)

H D

∆ZPE / kJmol−1

2

3.5 3.0 2.5 2.0 1.5 K 1.0 0.5 0.0

1 0

∆ZPE

−1 −2

K

−3 −4 −5

0

−20 −40 −60 −80 −100 −120 −140

(b)

δ(OHN)/ppm

3.5 3.0 2.5 2.0 K 1.5 1.0 0.5 0.0

(c)

0

5

10

15

20

25

δ(OHN)/ppm

FIGURE 7.13 (a) Isotopic fractionation between acid – pyridine complexes and the reference triphenylmethanol – pyridine complex studied in Ref. 29 around 130 K using CDF3/CDF2Cl as solvent. (b) Values of K (circles) and zero-point energy differences DZPE (squares) as a function of the 15N chemical shifts of hydrogen bonded pyridin – acid complexes. (c) Fractionation factors K as a function of the proton chemical shifts. The dotted lines were calculated as described in the text. (Source: From Limbach, H. H., Pietrzak, M., Sharif, S., Tolstoy, P. M., Shenderovich, I. G., Smirnov, S. N., Golubev, N. S., Denisov, G. S., NMR- parameters and geometries of OHN- and ODN hydrogen bonds of pyridine – acid complexes. Chem. Eur. J., 10, 5195– 5204, 2004. With permission.)

As a model system for this kind of problem Golubev et al.21 have studied various hydrogen bonded complexes of HF with collidine – 15N in CDF3/CDF2Cl (Figure 7.14 top). The chemical shifts and coupling constants were obtained again by low-temperature NMR using CDF3/CDF2Cl as solvent, in particular by the influence of temperature, as these solvent mixtures exhibit a strong increase of the dielectric constant 1 with decreasing temperature.30b The particular interest

Hydrogen Bond Isotope Effects Studied by NMR

219

F + H N H F Col(HF)2

q2=r1+r2 /Å

3.2

(a) JFH / Hz

1

+ N H

δ− δ+ F H N

F

− F −

H N

H

F

Col(HF)3 H + F

3.0 2.8

Col(HF)3

2.6 2.4

ColHF

200

Col(HF)2 [ColHFHCol]+

100 0

−100

1J NH / Hz

(b)

N

H +N

[CollHFCol]+

−

F H

F−

+ N H

δ+ δ− N H F ColHF

0 −50

−100

(d)

∆δ(FDN) )/ppm

2J / NF Hz

(c)

120 100 80 60 40 20 0 0.2 0 −0.2 −0.8

(e)

−0.4 0 0.4 q1=½ (r1 - r2) /Å

0.8

FIGURE 7.14 Hydrogen bond correlation q2 ¼ f(q1) for FHN systems. (a) Plot of the experimental coupling constants 1JFH (b), 1JHN (c) and 2JFN (d) and the primary isotope chemical shift effect DdðFDNÞ (e) as a function of q1. The solid lines are calculated according to the valence bond model as described in the text. (Source: From Shenderovich, I. G., Tolstoy, P. M., Golubev, N. S., Smirnov, S. N., Denisov, G. S., and Limbach, H. H., Low-temperature NMR studies of the structure and dynamics of a novel series of acid – base complexes of HF with collidine exhibiting scalar couplings across hydrogen bonds, J. Am. Chem. Soc., 125, 11710– 11720, 2003. With permission.)

in the FHN bond is that it resembles OHN bonds, but that 19F exhibits a spin 1/2, so that by 15N enrichment hydrogen bonds with three spins 1/2 can be realized. This allows one to measure coupling constants across hydrogen bonds in a similar way as for clusters of HF with F2.9 The NMR parameters of the FHN hydrogen bonds obtained have been linked to the corresponding hydrogen bond geometries in Ref. 30a as depicted in Figure 7.14a which shows the geometric FHN hydrogen bond correlation as a function of q1. Figure 7.14b to d refer to

220

Isotope Effects in Chemistry and Biology

the evolution of the experimental hydrogen bond coupling constants 1JFH, 1JHN and 2JFN. Furthermore, they are important NMR tools for the study of hydrogen bonds themselves, as well as for their theoretical description (see chapter of Del Bene). Finally, the primary isotope chemical shift effect DdðFDNÞ ; dðFDNÞ2 dðFHNÞ on the hydron chemical shifts of the complex collidine –HF is depicted in Figure 7.14e, but will be discussed in the following. In order to demonstrate the power of the method some typical multinuclear NMR signals of the collidine– HF and – DF complexes are depicted in Figure 7.15. The chemical shift of the bonding hydron (proton or deuteron) strongly depends on temperature and exhibits a maximum at the lowest reachable one, i.e., 103 K (Figure 7.15a and b). Primary isotope effects on the chemical shifts DdðFDNÞ ; dðFDNÞ2 dðFHNÞ are observed which are positive except at the highest temperature of 190 K where resolved signals could be obtained. A similar chemical shift dependence is observed for 19F (Figure 7.15a and b). However, it is interesting to note that the secondary isotope effect on J (FHN) c

J (FHN)

15N

155 K −60

CH3 δ+ 15N H(D)

CH3

CH3

−70 δ /ppm 1H

b 190 K

δ− F FDN 19F

FHN

2H

1H

170 K

2H

19F

FDN

1H

145 K

2H

19F

FHN −115

−120

−125

δ /ppm

a

1H

125 K

2H

FDN

19F

FDN

1H

103 K 21

FHN

FHN

2H

20

19 18 δ /ppm

19F

−105

−110 δ /ppm

FIGURE 7.15 1H, 2H, 15N and 19F NMR spectra of 1:1 complexes FHN and FDN between 15N –collidine and HF/DF in CDF3/CDClF2. The deuterium fraction: 55% (a and c) and 85% (b). (Source: From Shenderovich, I. G., Burtsev, A. P., Denisov, G. S., Golubev, N. S., and Limbach, H. H., Influence of the temperaturedependent dielectric constant on the H/D isotope effects on the NMR chemical shifts and the hydrogen bond geometry of the collidine-HF complex in CDF3/CDClF2 solution, Magn. Reson. Chem., 39, S91– S99, 2001. With permission.)

Hydrogen Bond Isotope Effects Studied by NMR

221

the 19F chemical shift, DdðFDNÞ ; dðFDNÞ2 dðFHNÞ, is positive at low temperatures, becomes immeasurably small at 170 K, and changes its sign at higher temperature. 15N NMR spectra of FHN/FDN mixtures were obtained only around 155 K (Figure 7.15c). The secondary isotope effect for the nitrogen nucleus, DdðFDNÞ ; dðFDNÞ2 dðFHNÞ; is negative in this case. The NHF complex does not only exhibit large scalar couplings JFH and JHN but also a large value of JFN around l96l Hz.21 The changes in the one- and two-bond coupling constants across the FHN hydrogen bond observed experimentally for FH· · ·collidine as a function of temperature have been supported by the results of an ab initio EOM – CCSD study of these coupling constants for FH – NH and FH – pyridine, which were used as models for FH –collidine.45 In Figure 7.16a we have plotted the secondary isotope effects on the fluorine chemical shifts DdðFDNÞ ; dðFDNÞ2 dðFHNÞ as a function of the dielectric constant which strongly increases with decreasing temperature. DdðFDNÞ is negative at high temperatures, indicating that D is closer to F than H. However, at higher dielectric constants the effect becomes opposite, i.e., D is now farther away from F than H. We expect that if we could increase the solvent polarity to a very high CH3 δ+ 15N

CH3

δ− ' ' H(D) ' ' F

CH3

(b)

0.3 ∆d(FDN)

ppm

0.2

0.0

eo

−0.2 10

15

20

25

30

35

40

35

40

(a) 3 ∆d(FDN)

ppm

2 1 0 −1 −2

eo 10

15

20

25

30

FIGURE 7.16 NMR parameters of the FHN and FDN complex dissolved in CDF3/CDClF2 mixtures as a function of the dielectric constant 1o. (a) Secondary, DdðFDNÞ ; dðFDNÞ 2 dðFHNÞ; H/D and (b) Primary, Dd(F DN) ; d(F DN) 2 d(F HN), H/D isotope effects on the NMR chemical shifts. (Source: From Shenderovich, I. G., Burtsev, A. P., Denisov, G. S., Golubev, N. S., and Limbach, H. H., Influence of the temperature-dependent dielectric constant on the H/D isotope effects on the NMR chemical shifts and the hydrogen bond geometry of the collidine-HF complex in CDF3/CDClF2 solution, Magn. Reson. Chem., 39, S91– S99, 2001. With permission.)

222

Isotope Effects in Chemistry and Biology

value DdðFDNÞ would again become smaller. As far as the corresponding nitrogen value is concerned, we obtained only a single value of DdðFDNÞ ; dðFDNÞ 2 dðFHNÞ ¼ 23:36 ppm at 155 K, indicating that D is closer to N than H. In Figure 7.16b are depicted the primary isotope effects, DdðFDNÞ ; dðFDNÞ2 dðFHNÞ as a function of the dielectric constant of the solvent mixture, which is strongly dependent on temperature. The values are the same as in Figure 7.14e, where they had been plotted as a function of q1. The values are found to be negative at high temperatures, i.e., low dielectric constants, indicating a situation where the deuteron is farther away from the hydrogen bond center than the proton. This finding is in agreement with the behavior of DdðFDNÞ; and expected for a situation prior to the formation of the quasisymmetric complex. When the dielectric constant is increased by lowering the temperature, DdðFDNÞ crosses zero at 1o < 15, goes through a maximum value of 0.27 ppm at 1o around 22 and then decreases again. This value is comparable to the value of 0.32 ppm found for FHF2.41 We estimate that this maximum is the best characterization of a chemically asymmetric hydrogen bonded complex, where the single well potential for the proton, averaged over all various solvent environments is the most symmetric one, although H does not need to be located directly in the H-bond center as indicated in Figure 7.14e. These results support the situation depicted in Figure 7.2b, where the increase of the solvent polarity symmetrizes a hydrogen bond rather than to lower the symmetry.

D. H/D I SOTOPE E FFECTS ON THE NMR PARAMETERS AND G EOMETRIES OF C OUPLED H YDROGEN B ONDS Up to now, we have dealt only with hydrogen bonded systems containing single hydrogen bonds. In this section, we will deal with systems where several hydrogen bonds are coupled. Naturally, these systems are still far away from complex systems like water exhibiting an infinite number of coupled hydrogen bonds. One can distinguish among the coupled hydrogen bonds cooperative and anticooperative bonds. In terms of hydrogen bond geometries this means that if the hydrogen coordinate s, defined in Figure 7.1a, of one hydrogen bond length is changed, the coordinate of a cooperatively coupled hydrogen bond is changed in the same and in an anticooperative coupled hydrogen bond in the opposite way as is illustrated in Figure 7.17. The two hydrogen bonds are uncoupled if a change of the geometry of one bond has no effect on the geometry of the other.

H H

B Y

s(XHY)

A X

cooperative coupling

noncoupled anticooperative coupling s(AHB)

FIGURE 7.17 Cooperative and anticooperative coupling of two hydrogen bonds. For further explanation see text.

Hydrogen Bond Isotope Effects Studied by NMR

223

A convenient way to determine these effects is NMR spectroscopy combined with hydrogen bond correlations which are able to link NMR parameters with hydrogen bond geometries. As an example, we refer to a recent paper of Tolstoy et al.43 of a low-temperature NMR study of acetic acid – acetate complexes in CDF3/CDF2Cl. Using appropriate chemical shift – geometry correlations similar to those discussed in the previous sections, the hydrogen bond geometries of the acetic acid dimer and of the acetate –acetic acid 1:2 complex (dihydrogen triacetate) were obtained. The distance changes after partial and full deuteration of the acetic acid dimer are depicted in Figure 7.18. Substitution of both H by D leads to a decrease of the shorter and an increase of the longer oxygen – hydrogen distances, where the overall average symmetry of the dimer is the same in the HH and the DD isotopologs. By contrast, the symmetry of the HD species is reduced as indicated in the geometry of the HD isotopolog depicted in Figure 7.18. Thus, the two hydrogen bonds in the cyclic dimer are cooperative. In contrast to acetic acid cyclic dimer, the hydrogen bonds in dihydrogen triacetate are anticooperative. This is manifested in a low-field shift of a given hydrogen bond proton signal upon deuteration of the neighboring bonds. The corresponding geometric change of this OHO group is indicated in Figure 7.18. Now, the shorter OH distance is lengthened and the longer H· · ·O distance shortened. Unfortunately, it was not possible to establish the geometric changes of the ODO group in a quantitative way, as the deuterium signal of the HD species could not be resolved. In a similar way, the NMR parameters and the hydrogen bond geometries of the isotopologs of [(FH)nF]2 clusters in CDF3/CDF2Cl have been determined by low-temperature NMR46 depicted schematically in Figure 7.19; [FHF]2 and [FDF]2 have been studied by Del Bene et al.47 (see also chapter by Del Bene) as well as recently by Golubev et al. using quantum-mechanical methods.28

r2=1.6101 Å

r1=1.0177 Å

O

H3C

H

O

H

O

O

O

r2=1.6101 Å r1=1.0177 Å r1=1.0163 Å r2=1.6151 Å

H3C

O H O

D

r2=1.6178 Å r1=1.0151 Å

H3C

O D O

O O

CH3

CH3 H

O O

O

CH3

CH3

CH3 O

r1=1.0158 Å

O H3C CH3 D O

r2=1.4865 Å r1=1.0593 Å

H

O

H3C

r2=1.6211 Å

O

D

O

r2=1.4766 Å r1=1.0634 Å

H

O

O O

O

CH3

r2=1.6211 Å r1=1.0151 Å

FIGURE 7.18 Hydrogen bond geometries of the three isotopologs of the cyclic dimer of acetic acid and of the two isotopologs of dihydrogen triacetate in CDF3/CDF2Cl obtained from 1H and 2H NMR chemicals shifts in Ref. 43. (Source: From Tolstoy, P. M., Schah-Mohammedi, P., Smirnov, S. N., Golubev, N. S., Denisov, G. S., and Limbach, H. H., Characterization of fluxional hydrogen bonded complexes of acetic acid and acetate by NMR: geometries, isotope and solvent effects, J. Am. Chem. Soc., 126, 5621– 5634, 2004. With permission.)

224

Isotope Effects in Chemistry and Biology

(a)

F D H

H

D

D∞h

D∞h

HD Cs

HH C2V

DD C2V

(b) HHH D3h

HHD C2V

HDD C2V

DDD D3h

(c) FIGURE 7.19 Hydrogen bond geometries of the isotopologs of [(FH)nF]2 clusters according to Ref. 46. (Source: From Shenderovich, I. G., Limbach, H. H., Smirnov, S. N., Tolstoy, P. M., Denisov, G. S., and Golubev, N. S., H/D isotope effects on the low-temperature NMR parameters and hydrogen bond geometries of (FH)2F2 and (FH)3F2anions in CDF3/CDF2Cl liquid solution, Phys. Chem. Chem. Phys., 4, 5488– 5497, 2002. With permission.)

˚ smaller for the latter,48 which would already lead to a small The F· · ·F distance is by 0.0054 A low-field shift of D as compared to H. On the other hand, as illustrated in Figure 7.19a, D is more confined in the hydrogen bond center than H because of the smaller amplitudes of the vibrational ground-state stretching and bending vibrations, which contributes to the low-field shift of D as compared to H. Calculations of isotope effects on the higher clusters are difficult and have not been performed up to date. Therefore, the NMR experiments give a first glance of what happens to the hydrogen bond geometries after deuteration. Especially useful are the coupling constants JFF which were correlated with calculated equilibrium values of q1 and q2. Figure 7.19b depicts the case of [FLFLF]2. Single deuteration of a given bond leads to an increase of the F· · ·F distance of that bond, and a displacement of D towards the external F atom. The remaining FHF bond is, however, shortened indicating an anticooperative coupling. When this bond is also deuterated, the both F· · ·F distances are the same and larger as compared to the nondeuterated isotopolog. The two deuterons are shifted somewhat back to the hydrogen bond center, as compared to D in the partially deuterated isotopolog. Similar effects, but less pronounced, are found for the [(FH)3F]2 cluster as depicted in Figure 7.19c. These changes were quantified as illustrated Figure 7.20 in further detail. Here, the solid lines correspond to “direct” or primary H/D isotope effects and broken lines to “vicinal” effects. We note that the isotope effects on q1 and on q2 calculated from the observed H/D isotope effects on the coupling constants JFF are similar for both clusters, which is not surprising in view of the correlation between both quantities which is linear in short intervals. The increase of the q2 values ˚ for deuterated (FD)2F2 and 0.0035 A ˚ for (FD)3F2. This after complete deuteration is 0.0022 A 2 change is opposite to the change in FLF . The corresponding increases of the distance q1 of D to ˚ and 0.0027 A ˚ . Most interestingly are the partially deuterated the H-bond center are 0.0032 A complexes. Single H/D substitution in a given hydrogen bond leads to a substantial increase of the F· · ·F distance and a substantial asymmetrization, i.e., increase of the distance of the deuteron from the H-bond center. On the other hand, we find also a significant decrease of the F· · ·F distance

Hydrogen Bond Isotope Effects Studied by NMR 155 JFF

100 FHFDF

150

JFF

FHFHF

90 FDFHF 0

1

1.30 2

(e)

m

2.435 q2 /Å 2.430

FDFHF

2.365 2.360

FDFDF

FHFHF

2.355

1

2

(f)

m

2.415

0.18

q1/Å

q1/Å

0.24

FDFHF

0.17

1

2

(g)

m

FDFDF

350

3.2

FDF(DF)2

FHF(DF)(HF)

FHF(DF)2

0

1

FDF(HF)2

0

m

2

3

FDF(DF)(HF) FDF(DF)2

1

FHF(DF)2 2

3 m

FDF(HF)2 FDF(DF)2 FHF(HF)2 FHF(DF)(HF)

428

FHF(DF)2 0

0.80

m

FDF(DF)(HF)

432

3.1

FHFDF

0.22

436

FHFHF

(d)

FHF(HF)2

440 J FL*

3.3 KFL

360

340

3

0.84

FDF(DF)(HF)

FHF(DF)(HF)

FDFHF

370

2

FHF(HF)2

FHFDF

380 J FL*

0.88

0.23

0.16 0

1

FDF(HF)2

FDFDF

FHFHF

(c)

0

2.420

KFF

FDF(DF)2 FDF(DF)(HF)

FDF(HF)2

2.425

FHFDF

0

85

0.92

FHF(DF)2

FHF(HF)2

1.35

2.370 q2 /Å

2.350 (b)

FHF(DF)(HF)

95

KFF

140 135

1.40

FDFDF

145

(a)

225

1

2

m

(h)

0

1

2

3 m

3.88 KFL 3.84 3.80 3.76

FIGURE 7.20 Coupling constants and hydrogen bond coordinates of (FL)2F2 and (FL)3F2 (L ¼ H,D) as a function of the number m of deuterons in the complexes. (a, e): scalar coupling constants JFF/Hz and reduced couplings KFF/1020 T2J21. (b,c and f,g) hydrogen bond coordinates, (d,h) fluorine– hydron couplings J*FL ¼ JFLgH/gL as well as the corresponding reduced couplings KFL ¼ gHgLJFL [1020 T2J21]. Full lines depict the direct isotope effect, and the vicinal isotope effects are denoted by dotted lines. For further explanation see text. Reproduced with permission from Ref. 46.

226

Isotope Effects in Chemistry and Biology

and a decrease of the hydron distance from H-bond center when single H/D substitution occurs in a neighboring H-bond. These single H/D effects are larger than the overall double HH/DD isotope effects. In the case of (FD)3F2 it seems that substitution of the last H by D exhibits no isotope effect. We come now to the, at first sight surprising, symmetry of the parallelograms in Figure 7.20 which indicate that for a given quantity, NMR parameter or hydrogen bond coordinate V, the following “sum rules” are valid: VH ðHHÞ þ VD ðDDÞ ¼ VH ðHDÞ þ VD ðDHÞ

ð7:16Þ

VH ðHHHÞ þ VD ðDDDÞ ¼ VH ðHDDÞ þ VD ðHHDÞ ¼ VH ðHHDÞ þ VD ðHDDÞ

ð7:17Þ

where VH refers to an FH· · ·F bond and VD to an FD· · ·F bond. In other words, the resulting effect of total (double or triple) deuteration corresponds approximately to the algebraic sum of the direct (or one-bond) and the vicinal (or secondary) isotope effects. Or, for every parameter, the average value over all isotopologs belonging to a definite local point group is constant. We note that the sum rules found in this paper for NMR parameters and H-bond geometries coincide with the wellknown sum rules for vibrational frequencies.49 We also note that sum rules have been previously established for other systems, e.g., the chemical shifts of methane H/D isotopologs50a and of ammonia,50b but deviations were observed for ammonium.50c The isotopic sum rules can facilitate considerably the assignment of spectral lines to given H/D isotopologs when scalar couplings across hydrogen bonds are absent, e.g., in the case of systems of hydrogen bonded complexes of the OHO type. The problem of line identification is especially important for isotopic modifications of hydrogen bonded systems, which cannot be studied separately but only as an equilibrium mixture. In addition, the rule provides grounds of the method of determination proposed in Ref. 51 of the composition of hydrogen bonded associates using the multiplicity of isotopic splitting. In particular, it excludes the possibility of occasional coincidence of several lines in such an isotopic multiplet. As the sum rules are valid for so many different quantities, they arise most probably, from a similar origin. On the other hand, the sum rules as well as the analogous product rules for vibrational frequencies were derived in harmonic approximation, whereas the isotope effects on the NMR parameters and on H-bond geometries are essentially caused by anharmonic effect. A consideration of this problem in terms of perturbation theory will be given elsewhere. These results of this section may be only semiquantitative and subject to systematic errors, but may motivate a theoretical chemist to study these ions in a quantitative way in the future, which justifies the use of the simple chemical shift – geometry correlations.

IV. CONCLUSIONS We conclude that liquid- and solid-state NMR spectroscopy constitutes a very important tool for the study of isotope effects of strong hydrogen bonds. The latter do not only play an important role with respect to the function of hydrogen bonds but also represent models for the transition states of high-barrier proton-transfer reactions. By introducing appropriate correction terms into the bond valence analysis a proper description of geometric hydrogen bond correlations both for protonated as well as for deuterated systems is achieved. By dipolar NMR the world of hydrogen bond geometries and NMR parameters are linked together. By low-temperature liquid-state NMR intrinsic NMR parameters of systems with single or coupled hydrogen bonds can be studied in the slow hydrogen bond exchange regime. This combination allows one to convert isotope effects on NMR parameters — which can be measured with a high precision — into isotope effects on hydrogen bond geometries.

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ACKNOWLEDGMENTS This research has been supported by the Deutsche Forschungsgemeinschaft, Bonn, the Fonds der Chemischen Industrie (Frankfurt) and the Russian Foundation of Basic Research, grant 03-0304009. We are indebted to Prof. Janet Del Bene, Youngstown State University, and Prof. Charles Perrin, University of California, San Diego, for helpful comments and proofreading of this manuscript.

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8

Isotope Effects and Symmetry of Hydrogen Bonds in Solution: Single- and Double-Well Potential Jonathan S. Lau and Charles L. Perrin

CONTENTS I.

Introduction ...................................................................................................................... 232 A. Single- and Double-Well H-Bonds.......................................................................... 232 B. Low-Barrier H-Bonds, Short, Strong H-Bonds, “Symmetric” H-Bonds ................ 232 1. Resonance-Assisted H-Bonds, Charge-Assisted H-Bonds ............................... 233 2. Sterically Enforced H-Bonds ............................................................................ 233 II. Computational Work........................................................................................................ 234 A. Energetic and Geometric Descriptions .................................................................... 234 B. Accounting for Solvation......................................................................................... 235 III. Methods of Observation................................................................................................... 236 A. Measurement of pKa ................................................................................................ 237 B. Fractionation Factors................................................................................................ 237 C. NMR Chemical Shifts.............................................................................................. 238 D. NMR Coupling Constants........................................................................................ 239 E. Infrared (IR) ............................................................................................................. 240 F. X-Ray and Neutron Diffraction ............................................................................... 240 IV. Current Work ................................................................................................................... 241 A. Intramolecular Systems............................................................................................ 241 1. Enol Tautomers of b-Dicarbonyls and Related Molecules.............................. 241 2. Proton Sponge ................................................................................................... 242 3. Dicarboxylic Acids............................................................................................ 243 4. Schiff Bases ....................................................................................................... 244 B. Intermolecular Systems............................................................................................ 245 1. Pyridine –Acid Complexes................................................................................ 245 2. Enzymes............................................................................................................. 246 V. Conclusion........................................................................................................................ 247 Acknowledgments ........................................................................................................................ 247 References..................................................................................................................................... 247

231

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Isotope Effects in Chemistry and Biology

I. INTRODUCTION Hydrogen bonds (H-bonds) create the stabilizing interactions between a donor, typically a heteroatom such as N or O bearing a hydrogen, and an acceptor, typically a heteroatom bearing a lone pair. A common convention is to represent the donor as A – H and the acceptor as B, so that the H-bond is represented as a dashed bond: A – H· · ·B.

A. SINGLE - AND D OUBLE - W ELL H - B ONDS For weak interactions, the potential describing an H-bond can be described as the superposition of the potential wells describing each species, A – H and H – B. The two species are not necessarily equally stable, but as their energies become equal, this becomes a symmetric double-well potential. As the distance between the heavy atoms decreases, the height of the barrier separating the wells decreases, leading to a single-well potential (Figure 8.1). Thus, there is a transition from an H-bond described by a tautomeric equilibrium between two structures, of unequal or equal energy, to a single structure represented by a resonance hybrid (Figure 8.2). The topic of short, strong, low-barrier H-bonds has been the subject of recent reviews.1,2 This continuing interest may be partly attributed to the invocation of such H-bonds in stabilizing intermediates or transition states that account for exceptional reactivity in certain enzymatic reactions.3 The purpose of this chapter is to discuss methods for distinguishing these -bonds and to present the evidence regarding the existence of single-well-potential H-bonds in solution. Among the most powerful methods are ones involving a study of the response to isotopic substitution.

B. LOW-BARRIER H-BONDS, S HORT, S TRONG H-BONDS, “S YMMETRIC ” H-BONDS Typical H-bonds represent weak interactions, resulting in , 20 kJ/mol of stabilization upon formation. These H-bonds are described by an asymmetric double-well potential, because one of the tautomers, either AH or HB, is much more stable than the other. In contrast, some H-bonds are unusually strong. These H-bonds are referred to as short, strong H-bonds (SSHBs) or as low-barrier H-bonds (LBHBs), the distinction being based on the criteria for recognizing them. They contribute up to 130 kJ/mol in stabilization and have shorter heavy-atom distances, less than 260 pm for the OHO system. As dOO approaches 240 pm, dOH is expected to increase from 100 to 120 pm, while dO0 H decreases from . 160 to 120 pm. In this case, the short, strong H-bond has become a single-well, symmetric H-bond. A prerequisite for such H-bonds is a

E

E

d(O–H)

E

d(O–H)

d(O–H)

FIGURE 8.1 Potential wells (asymmetric, symmetric double, single). A H B

A H B A H B

A H B

A H B

FIGURE 8.2 Tautomeric equilibrium (unbalanced or balanced) vs. resonance hybrid.

Isotope Effects and Symmetry of Hydrogen Bonds in Solution

O

H

233

O

O

H

O

FIGURE 8.3 Enol tautomer of benzoylacetone.

matching of the pKas of the two donor acids AH and HB. If there is to be a transition from a doublewell potential to a single-well potential, the two wells of the former must be of equal or nearly equal energy. Although a short –distance is a fundamental structural feature of H-bonding, it is not easy to distinguish whether an H-bond is of the single-well type or is of the more common double-well type. Computational methods do distinguish, but they require approximations. Most experimental methods are indirect. They provide parameters such as: energies (to be compared with the energy of some reference system that may be debatable), distances, NMR chemical shifts, and changes due to isotopic substitution. Certain values of these parameters are associated with a single-well H-bond, and they are consistent with such an H-bond, but they do not necessarily require one. 1. Resonance-Assisted H-Bonds, Charge-Assisted H-Bonds H-Bonding is generally ascribed to a stabilizing interaction between the partial positive charge on the hydrogen and the negatively charged lone pairs on the H-bond acceptor. This electrostatic interaction is enhanced because the hydrogen is so small that it allows an unusual proximity of positive and negative charges. Yet there is good reason to expect that short H-bonds are exceptionally strong, owing to some feature beyond mere electrostatics. This feature is a consequence of the lowering or disappearance of the barrier to H transfer as the A – B distance shortens and the H-bond becomes single well. One interpretation is to recognize that an H-bond can be represented as a resonance hybrid. If the two resonance forms are of identical energy, as in a H-bond where the A –H and B –H distances are the same, this corresponds to a maximum of resonance stabilization. Alternatively, such a H-bond is often viewed as possessing covalent character, which is the molecular-orbital counterpart to resonance that confers additional stabilization. Gilli et al. analyzed crystal structures of compounds containing apparently symmetric H-bonds.4 These could be grouped into three broad categories, namely resonance-assisted H-bonds, negative-charge-assisted H-bonds, and positive-charge-assisted H-bonds. The category of resonance-assisted H-bonds includes uncharged tautomeric systems, such as the enol of benzoylacetone (Figure 8.3),5 in which the H-bond donor and acceptor form a conjugated system. It has been proposed that this conjugation contributes to the strength of these H-bonds.6 Negativecharge-assisted H-bonds include systems such as 1,2-dicarboxylate monoanions (Example 1), whereas positive-charge-assisted H-bonds include systems such as the monoprotonated 1,8bis(dimethylamino)naphthalene (Example 2) (“proton sponge”). In these ionic cases it is proposed that the strength of the H-bond is nothing more than a result of electrostatic stabilization.7 δ− H δ − O O O

O R

R'

1

δ+

N

H

N

δ+

2

2. Sterically Enforced H-Bonds Certain structures possess an intramolecular H-bond in which the positions of the donor and acceptor enforce a short distance and perhaps a symmetric structure. This molecular conformation

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Isotope Effects in Chemistry and Biology

is suggested in derivatives of Kemp’s triacid (Example 3),8 2,2-dialkylmalonate monoanions (Example 4),9 and Schiff bases such as salicylidene imines (Example 5).10 It is proposed that the enforced geometries give rise to the strong H-bond. a,a0 -Dialkylsuccinate monoanions (Example 1) have also been suggested to possess such a geometry and a strong H-bond.11 For comparison, the parent hydrogen succinate anion is . 90% gauche in THF, owing to H-bonding, but only 66% in water, which corresponds to a statistical distribution of conformers.12 OH O

O

δ−

HO O N

O N O

O

H

O

OH

R1 R 2 3

N

R2 R1

O

O O

δ−

X

4

5

However, it is a fallacy, or perhaps a semantic mistake, to attribute the strength of an H-bond to such geometric constraints. If the constraint could be relaxed, the species would become even more stable. Therefore, the constraint acts to destabilize the H-bond, not stabilize it.

II. COMPUTATIONAL WORK A. ENERGETIC AND G EOMETRIC D ESCRIPTIONS High-level computations investigating H-bond strengths are ordinarily limited to relatively simple molecules. Since the potential-energy surface for H motion is highly sensitive to the heavy-atom separation, the geometry of the entire molecule must be calculated accurately in order to account for the H-bond. Moreover, the H motion is coupled to the motions of the rest of the molecule.13 An account of the application of computation to questions about H-bonds is in the chapter by Del Bene (this volume). Studies of the strong H-bond in the complex (Example 6) formed between formic acid and formate anion using ab initio and DFT calculations found good correspondence between calculated H-bond strengths and experimentally observed values, approaching 100 kJ/mol of stability as dOO approaches 243 pm.14 The stabilization decreases with increasing dOO, up to a 50 kJ/mol loss at 343 pm. Modeling showed a minimal effect on H-bond energy as the OHO angle deviates from linearity by up to 108, but there is a rapid decrease in stabilization beyond that angle. For example, a deviation of 308 decreases the strength of the H-bond by more than 20 kJ/mol, and a deviation of 408 weakens it by 33 kJ/mol. In contrast, only about a 1 kJ/mol variation is shown for deviations from linearity of less than 408 in the normal H-bond in the dimers of formamide and acetamide.15 O

H

δ−

O H

O 6

H O−

O

H

O

δ− O O

O 7

H O

8

Along with the formic acid –formate complex, the hydrogen maleate anion (Example 7) is a useful model system. Analysis of gas-phase hydrogen maleate anion at different levels of theory estimated the strength of the internal H-bond to be 60– 120 kJ/mol, with a barrier for proton transfer of 1 kJ/mol,16 confirming that this represents a symmetric low-barrier H-bond. 9-Hydroxyphenalen-1-one (Example 8), a model for the enol tautomer of b-diketones, was calculated at various levels of theory to have a 60 kJ/mol H-bond.17 The symmetric structure is

Isotope Effects and Symmetry of Hydrogen Bonds in Solution

235

approximately 10 kJ/mol less stable. These calculations indicate that this enol is a borderline case between a double- and a single-well, or low-barrier, H-bond. Calculations on acetylacetone found that the stabilization by the resonance-assisted H-bond of the cis-2-enol is , 60 kJ/mol but only , 25 kJ/mol for the cis-1-enol.18 The barrier for proton exchange was calculated to be , 10 kJ/mol, which is below the zero-point energy. In summary, many high-level and reliable calculations show that at short OO distances the H-bond can be of the single-well type, or else double-well but with the zero-point energy for H motion above the barrier between the two wells. In practice, these two possibilities are equivalent, since they cannot be distinguished experimentally.

B. ACCOUNTING FOR S OLVATION To use calculations to address the relations among H-bond length, strength, and symmetry (singlewell or low-barrier potential), it is essential to account for solvation. Although there is good reason, as described above, to expect such a relation, the most influential evidence is a graph presented by Hibbert and Emsley.19 They showed that four weak H-bonds (those in water dimer, methanol dimer, acetic acid dimer, and acetylacetone enol) have O – O distances . 250 pm and energies , 35 kJ/mol. In contrast, five other H-bonds, in (HCO2)2H2, (DMSO)2Hþ, (CH3OH)2Hþ, HOHOH2, and 3-(4-biphenylyl)acetylacetone enol, have O – O distances , 250 pm and energies . 100 kJ/mol. The graph shows a marked jump, almost a discontinuity, between these two groups. Yet it must be noted that the weak H-bonds are all neutrals, and the strong ones are gas-phase ions, except for the last one, where the H-bond strength was not measured directly but only inferred from correlations of IR frequencies, and where calculations indicate that its strength is not appreciably greater than that for acetylacetone itself.20 We interpret this dichotomy between ions and neutrals as arising because the energy of forming an H-bond involving an ion is greatly increased, simply from the ion – dipole attraction. Likewise the O –O distance is shortened, simply from ionic contraction accompanying that attraction. Therefore, the experimental relationship between H-bond length and strength in solution is tenuous. If short, low-barrier H-bonds with strengths of 100 kJ/mol are a feature only of gas-phase ions, then it is essential to account for solvation in assessing the strengths of H-bonds. As it is not often feasible to model all the individual solvent molecules, several approaches have been developed to account for solvation. These are employed to provide better correlation between computations and experimental results, which are almost always obtained in condensed phase. Since the strongest and shortest H-bonds are found in ions, it is particularly important to take adequate account of solvation. One approximation is to treat the solvent as a dielectric continuum. Solvation has been approximated by the Self-Consistent Isodensity Polarizable Continuum Method, in which the solute is treated as an electrostatic cavity and its interaction with the solvent is calculated as a function of the dielectric constant of the solvent. According to this method, the strength of a 115 kJ/mol H-bond in hydrogen maleate is reduced by , 25 kJ/mol upon solvation.21 Although the strength of the formic acid –formate H-bond approaches 100 kJ/mol, this decreases to , 25 kJ/mol in solvents with high dielectric constants.22 Proper modeling of solvation may require taking explicit account of the discrete nature of the solvent, which may not be adequately represented by a continuum dielectric. Thus, analyses of hydrogen phthalate anion (Example 9) in the gas phase or using cavity polarity methods indicate a symmetric hydrogen bond. Garcia-Viloca, Gonza´lez-Lafont, and Lluch have addressed the solvation of hydrogen phthalate anion through simulations.23 According to both B3LYP (DFT) and AM1-SRP (MO) calculations the structure of minimum energy has equal O –H distances of 119 pm, not only in the gas phase but also in continua of dielectric constant 4.81, to mimic chloroform, and 78.54, to mimic water. These media stabilize the anion by 142 and 180 kJ/mol, respectively. Moreover, the potential energy curve becomes much flatter, because asymmetric

236

Isotope Effects in Chemistry and Biology

structures are stabilized more than the symmetric one. To account for specific solvent – solute interactions, such as the formation of intermolecular hydrogen bonds, they performed combined QM/MM calculations, with the quantum mechanics treated by AM1-SRP and with the molecular mechanics treated by an AMBER force field in 379 molecules of TIP3P water. Now the structures of minimum energy are asymmetric, with one O –H distance shortened to 102 pm and the other lengthened to 173 pm, and the symmetric structure is 2.8 kJ/mol higher in energy. Although this barrier is so low that the proton might be expected to tunnel through it, making the potential surface effectively single well, the actual motion also involves the heavy atoms of the carboxyls and the solvent, so that the effective mass is much greater and tunneling is suppressed. A further molecular dynamics simulation found that only 1.6% of the structures are symmetric, whereas the most common structure has O – H distances of 117 and 178 pm. The asymmetry of the hydrogen bond of aqueous hydrogen phthalate anion is thus attributed to the low probability of solvent configurations with equivalent environments around both carboxyl group, and thus represents an entropic problem rather than an enthalpic one. To account for specific interactions with chloroform solvent, QM/MM calculations were performed with a force field describing 208 CHCl3 molecules. Here the structure of minimum energy remains symmetric, but the potential surface is exceedingly flat across ^ 20 pm. According to molecular dynamics simulation, only , 18% of the structures are symmetric, and most have one O –H distance , 116 pm and the other . 124 pm. Even though symmetric structures have lower energy than asymmetric ones, the probability that the proton lies within the symmetric region is quite low. This is attributed to the entropy term associated with the disorder of the solvent. A further QM/MM calculation with the inclusion of one Kþ cation found the structure of minimum energy to be asymmetric, with O – H distances of 103 and 166 pm. To reach the symmetric structure, the Kþ must also migrate to a position that is symmetric with respect to the proton. This migration has a high energy barrier of , 100 kJ/mol, because the Kþ stabilizes the asymmetric structures, with their more localized negative charge, more than it does the symmetric one. All in all, these calculations are in remarkable agreement with the experimental results of Perrin et al. described below, which found an asymmetric hydrogen bond in both aqueous solution and organic solvents, and with their interpretation in terms of solvent polarity, hydrogen bonding to solvent, the influence of the counterion, and the disorder of solvent molecules. O O− H O 9

O

Further modeling of the interaction of hydrogen phthalate anion (Example 9) with a tetraalkylammonium cation showed the strength of the H-bond to be reduced to 35 –55 kJ/mol.24 This result again indicates that interactions with the immediate environment can significantly affect the energy of the H-bond.

III. METHODS OF OBSERVATION Many experimental approaches have been used in the investigation of H-bonded systems. Analytical methods, such as measurements of pKa differences and of fractionation factors, can probe the strength of the H-bond. Spectroscopic methods, such as IR and NMR, are used to identify characteristics that have been correlated with strong and symmetric H-bonds. Direct determination of geometry may be performed by x-ray and neutron-diffraction studies, which may serve as a guide

Isotope Effects and Symmetry of Hydrogen Bonds in Solution

237

to geometries in solution. The nature of H-bonds in solution can be inferred by a combination of these methods.

A. MEASUREMENT OF P K a Dicarboxylic acids with a geometry favorable for the formation of an intramolecular H-bond exhibit a large difference between the first and second acid-dissociation constants, pKa1 and pKa2. This difference is a result of the increase in the acidity of the first proton and a decrease in the acidity of the second proton, compared to structurally similar acids (although it should be remembered that there is always a statistical contribution to DpKa of log104, or 0.60). The stability of the intramolecular H-bond formed after the first deprotonation increases the acidity of the first proton. The cost of breaking this H-bond increases the energy necessary for the second deprotonation, making the second proton less acidic. Examples of the correlations between pKa and H-bond strength are the isomeric pairs of maleic (DpKa ¼ 4.31) and fumaric acids (DpKa ¼ 1.36) and citraconic (DpKa ¼ 3.86) and mesaconic acids (DpKa ¼ 1.66). Measurements of the cis/trans equilibrium constants between fumaric (Example 10, R ¼ H) and maleic (Example 11) acids (Figure 8.4) or between mesaconic (Example 12, R ¼ CH3) and citraconic (Example 13) acids and their monoanions indicate that the strength of the intramolecular H-bond is 2 kJ/mol in water but , 20 kJ/mol in DMSO or chloroform.25 These results show that H-bond stabilities are greater in less polar solvents, although the dominant factor is probably the aprotic character, which leaves a carboxylate anion relatively unsolvated. Similarly, this difference in pKas has been observed in salicylic acid derivatives, which form an OHO H-bond.26 This system also provided comparisons of pKa1 with the pKa of the corresponding benzoic acid and of pKa2 with the pKa of the corresponding phenol. The largest DpKa corresponds to the derivative in which the pKas of the corresponding molecules became matched. Matched pKas do not necessarily result in the formation of a LBHB. A study of the intramolecular H-bonds in phthalic acid derivatives and of the intermolecular H-bonds formed between phenols and phenolates in DMSO showed a linear relation between increasing H-bond strength and decreasing DpKa, without any increase in strength near zero DpKa.27 This trend has been explained as resulting from simple electrostatic attraction, rather than from the formation of a low-barrier H-bond.

B. FRACTIONATION FACTORS Fractionation factors are defined as the equilibrium constant for a hydrogen exchange:

f ¼ ½R – D ½HOL =½R – H ½DOL

ð8:1Þ

where L ¼ H or D. If R –H is H-bonded, this represents the preference for deuterium in the H-bond of R – H, relative to that of water. Kreevoy and Liang measured fractionation factors for various H-bonds in acetonitrile, and found values of 0.30– 0.47 for several strong ones, but higher values of OH

δ− H δ− O O

O

O R

−

O

O 10, R=H 12, R=CH3

O R

11, R=H 13, R=CH3

FIGURE 8.4 Hydrogen maleate/fumarate or citraconate/measconate isomerization (Examples 10 – 13).

238

Isotope Effects in Chemistry and Biology

0.84 and 0.60 for hydrogen maleate and bifluoride anions.28 They also calculated the zero-point energies of isotopically substituted H-bonds as the shape of the potential well shifts from asymmetric to symmetric double well and further to single well. A fractionation factor of 0.22 was determined for the single-well potential. This result is a consequence of the flatness of the potential, which results in a low-frequency mode for H motion, and thus a low zero-point energy. Since the zero-point energy associated with deuterium is low regardless, the lowered frequency is more stabilizing for H, which prefers the H-bond position. This one-dimensional theoretical treatment has been further examined by calculating the potential of mean force along the H-bond coordinate, which includes factors such as intramolecular vibrations and proton tunneling.29 The potential calculated by this method does not require fitting to a harmonic expansion that is inappropriate for anharmonic potentials of low-barrier H-bonds. A fractionation factor of 0.58 for cis-3-hydroxycyclobutoxide (Example 14) was calculated by this method. OH

O−

14

Fractionation factors of 1 to 0.7 are typical for normal H-bonding, whereas values of 0.3 to 0.5 have been observed for strong H-bonds.30 Although these lower values are characteristic of strong H-bonds, they do not directly measure their strength or their geometry.

C. NMR C HEMICAL S HIFTS Downfield 1H NMR chemical shifts of 16– 20 ppm are considered to be indicative of strong H-bonding.19 Based on computational studies of hydrogen maleate (Example 7), hydrogen malonate (Example 15), and hydrogen oxalate (Example 16), Lluch and co-workers argue that although LBHB protons are strongly deshielded, a downfield chemical shift is not necessarily indicative of a LBHB.31 Experimental studies by Bru¨ck, McCoy, and Kilway of a number of dicarboxylic acid monoanions in nonpolar solvents showed uniformly downfield H-bond chemical shifts, but with little correlation to the reported DpKa of the diacid.32 The broadening of the downfield H-bonded proton NMR signal (16.01 ppm) of hydrogen 5,6-acenaphthenedicarboxylate (Example 17) at 250 K, the subsequent separation into two signals at 18.31 and 15.75 ppm, and the diminution of the intensity of the upfield peak below 230 K indicate that at higher temperatures, the intramolecular H-bond is in equilibrium with the intermolecular H-bond of the dimer, as was confirmed with calculated geometries.33 Therefore, the intramolecular H-bond is not so strong that the intermolecular one cannot compete. O

δ− δ− H O O

O − HO

15

OH

O

O

O

O

O−

16

17

O

A powerful method for probing the symmetry of hydrogen bonds is measurement of the isotope shift.34 This is the change of the chemical shift of a reporter nucleus X on replacing an atom N n bonds distant with a heavier isotope N0 . It is defined as n

D ¼ dX ðN0 Þ 2 dX ðNÞ

ð8:2Þ

although conventions differ and sometimes the order is reversed. With this convention the isotope shift is usually negative, insofar as a heavier isotope produces an upfield shift. For n ¼ 0 the reporter nucleus itself is subject to isotopic substitution, and both isotopes must be NMR active.

Isotope Effects and Symmetry of Hydrogen Bonds in Solution

239

Such an isotope shift is often called a primary one, and it is alternatively designated as PD rather than 0D. There are two contributions that add up to the observed isotope shift. One is an intrinsic one, designated as D0. It arises because bonds to a heavier isotope N0 are shorter, owing to a lesser influence of anharmonicity, which causes the vibrational wave function to sample larger separations. Consequently its reporter nucleus experiences a greater shielding by the electrons around the atom to which it is attached, and it is shifted upfield. This contribution can usually be measured independently in model species, and it can be subtracted from the observed isotope shift. The diagnostic contribution to an observed isotope shift is one that arises from isotopic perturbation of an equilibrium.35 If two equilibrating species are bonded differently to atom N, then replacing N by the isotopic N0 changes the relative zero-point energies of the two species and changes the position of the equilibrium. If the chemical shift of X is different in each of the two species, the average chemical shift is changed relative to that with N. This contribution is designated as De, and it is given by De ¼ ðD=2ÞðK 2 1Þ=ðK þ 1Þ

ð8:3Þ

where K is the equilibrium constant between the two species when N has been replaced by N0 and where D is the difference between the chemical shifts of X in the two species. Often D and K can be estimated from model systems. This method succeeds even if the equilibration between the two species is too fast for their individual signals to be detected. Indeed, it requires a rapid equilibration so that only an averaged chemical shift is measured. The key feature is that that average depends on the position of the equilibrium, as perturbed by the isotope. The essence of this method is that it makes it possible to distinguish a mixture of rapidly equilibrating tautomers from a single symmetric structure. The observation of a De is evidence for an equilibrium that is perturbed by the isotopic substitution, whereas only D0 is observed if there is a single species. It must be noted that the isotopic perturbation itself is not responsible for inducing a separation into two tautomers, or for converting a single-well potential into a double well. According to the Born –Oppenheimer approximation,36 the potential-energy surface controlling nuclear motion depends only on the electronic wave function and is independent of nuclear mass. Therefore, the method of isotopic perturbation of equilibrium can distinguish these. Perturbation of hydrogen exchange equilibrium by deuterium substitution in the H-bond has been reviewed by Dziembowska and Razwadowski,37 and isotope shifts are reviewed in detail by Hansen in another chapter (this volume). Substitution of the proton with either deuterium or tritium introduces an isotope effect on the chemical shift of the hydrogen itself.38,39 This is a primary isotope shift, defined as d(D)2 d(H) or d(T)2 d(H). The smaller vibrational amplitude of the heavier nuclei results in smaller excursions from the equilibrium position. In a single-well H-bond the equilibrium position is centered with respect to the heavy atoms. Lighter isotopes (H or D) deviate from this central position more, experiencing greater shielding from being closer to the heavy atoms. This results in a downfield shift for the heavier isotope (D or T), or a positive isotope shift. More generally, the deuterium isotope shift increases with the chemical shift of the H-bonded proton to a maximum upfield shift of 0.7 ppm at a chemical shift of 16 ppm, behavior associated with an anharmonic double-well potential. The isotope shift then decreases and becomes a downfield shift at a chemical shift of 19 ppm, corresponding to a single-well potential.40 The nature of the potential well is independent of solvent polarity, as this behavior is observed in CDF3/ CDF2Cl at low temperatures and consequently a higher dielectric constant.41

D. NMR C OUPLING C ONSTANTS The 1J(15N,D) dipolar coupling in the solid-state 15N NMR spectrum of 15N-labeled proton sponge was used to determine N – D bond lengths. Distances of 119 and 147 pm were obtained by this

240

Isotope Effects in Chemistry and Biology

method, which compare favorably with distances of 119 and 143 pm determined by crystallographic studies.42 The scalar coupling across an NHN H-bond, 2hJ, has been used as a measure of the H-bond length. The 2hJ(15N,15N) for normal NHN H-bonds is an exponential-type curve from 3 Hz at 315 pm to 17 Hz at a distance of 265 pm.43 By extrapolation a symmetric 256 pm NHN H-bond would have a maximum 2hJ(15N,15N) of 25 Hz.44

E. INFRARED (IR) The strength of H-bonds may be studied by IR spectroscopy inasmuch as the O – H bonds are lengthened by H-bonding, resulting in a decrease in the stretching frequencies. Since there is a huge hetereogeneity of environments, the decrease is variable, so that the band is broadened. Very broad IR bands centered at 2500 and 800 cm21, known as Speakman-Hadzi bands, are associated with symmetric H-bonds. Such bands have been observed in H-bonded complexes of 2 45 betaine derivatives [C5H5Nþ(CH2)nCO2 2 ] and the analogous carboxylates [C6H5(CH2)nCO2 ], 46 and hydrogen bis(4-nitrophenoxide) anions. Strong, although not necessarily symmetric, H-bonds also give rise to characteristically broad IR bands at 2500 cm21.47 Deuterium substitution of X –H bonds results in a decrease of vibrational frequencies typically by a factor of 1.4, owing to the increased mass of D, but in LBHBs this ratio is smaller.48 However, a ratio nOH/nOD approaching unity is observed in sodium hydrogen malate,49 and this is taken as evidence of an strong but asymmetric H-bond. Comparisons of solid-state IR spectra with geometries determined by x-ray diffraction confirm the association between IR absorptions attributed to symmetric H-bonds and the H-bond geometry.50,51 Isotope effects on the symmetry of the H-bonds of ice at high pressure are reviewed in the chapter by Aoki (this volume).

F. X- R AY AND N EUTRON D IFFRACTION Solving crystal structures of H-bonded systems by x-ray and neutron diffraction provides direct evidence for the geometry. Structures determined from neutron-diffraction data of the hydrogen maleate anion (Example 7) show a proton symmetrically displaced from the oxygens, consistent with a single-well potential.52 Geometries can be correlated with other properties describing the nature of the H-bond. 5,50 Dibromo-3-dimethylaminomethyl-2,20 -biphenol N-oxide (Example 18) possesses a 241.9 pm H-bond between the 2-phenol oxygen and the N-oxide oxygen and a 252.9-pm H-bond between the phenolic oxygens.53 The IR spectrum exhibits broad absorptions centered at 1000 and 2400 cm21 and appears to be due to the overlap of the spectra of both very short symmetric and asymmetric H bonds. The data indicate proton transfer from the phenol to the N-oxide to yield a symmetric H bond, but a strongly asymmetric H-bond between the phenolic oxygens. The crystal structures of H-bonded complexes of different acids and betaine [(CH3)3Nþ – CH2 – COO2] have been correlated with spectroscopic properties of the H-bond such as the 13C chemical

N+

O−

HO

HO

Br δ−

O

H

−O3S Br

18

O

δ−

SO3− 19

Isotope Effects and Symmetry of Hydrogen Bonds in Solution

241

shift of the carboxyl carbon and the CvO IR absorption.54 Such structures can be classified by the position of the proton in the H-bond, undergoing a transition from molecular complexes of the acid and the carboxylate of betaine to ion-pair complexes where the proton has been transferred to betaine, through a situation where the proton position is disordered, corresponding to the lowbarrier and single-well potentials. Crystal structures of 1,8-dihydroxynaphthalenes show an OHO H-bond length of 255 pm. The chemical shift of the H-bonded proton in sodium 4,5-dihydroxynaphthalene-2,7-disulfonate (Example 19) is 17.72 ppm.55 This deshielded chemical shift corresponds to a dOO of 250 pm based on an empirical fit, in good agreement with the x-ray structure of this compound. The increase upon deuteration of the distance between heavy atoms of an H-bond, known as the Ubbelohde effect,56 is observable by x-ray diffraction. In contrast, in single-well H-bonds the heavy-atom distance of deuterated H-bonds is the same or smaller, owing to the smaller vibrational amplitude of deuterium.57 However, in H-bonded complexes of hydrogen bis(4-nitrophenoxide) anion, which exhibit the Speakman-Hadzi IR bands, there is an increase in the distance between the heavy atoms upon deuteration, consistent with the Ubbelohde effect.56

IV. CURRENT WORK A. INTRAMOLECULAR S YSTEMS Systems containing intramolecular H-bonds provide good models for studying H-bonds because of the proximity of the donor and acceptor groups. Steric factors are then easier to predict and manipulate. Regardless of the intramolecularity there may still be an effect of counterions if charged species are involved. 1. Enol Tautomers of b-Dicarbonyls and Related Molecules The enol tautomers of b-diketones (Figure 8.5) are stabilized by H-bonding.4,58 The H-bond donor and acceptor are often conjugated in such systems, and the increased stability of the H-bond may be attributed to resonance stabilization. Neutron-diffraction studies indicate that the average dOO is 240 –250 pm.59 The short dOO does not guarantee a single-well potential because the OHO angle deviates from the optimum 1808 so that the effective distance along the OH bonds is longer. Downfield chemical shifts of the H-bonded proton of 15.3– 17.0 ppm and IR absorptions at 2566 –2675 cm21 47 are suggestive of the formation of LBHBs. The strength of the H-bond was calculated to range from 54 kJ/mol in acetylacetone to 80 kJ/mol in hexamethylacetylacetone.60 Isotopic perturbation of equilibrium was used in the Perrin lab to investigate 3-hydroxy-2phenylpropenal (Example 20) by deuteration of the aldehyde hydrogens.61 The phenyl group was simply for synthetic convenience, crystallinity, and fixed stereochemistry. The ability to detect a tautomeric equilibrium depends on the fact that the CH stretching frequency of an aldehyde is significantly lower than that of an enol ether. As a result of the zero-point energies, the monodeuterated substrate will preferentially be present as the tautomer with aldehydic CH and enolic CD, with an estimated equilibrium constant of 1.2 at 258C. Moreover, since the 13C signal of

O R

O

H H

O R'

H

R

O

O R'

H

FIGURE 8.5 Keto-enol tautomerism of b-diketones.

H

R

O

O R'

H

H

R

O R'

H

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Isotope Effects in Chemistry and Biology

an aldehyde is well downfield of an enol, the CH will be closer to the latter’s shift and thus , 1 ppm downfield of the average seen in the undeuterated substrate. This is opposite to the usual upfield isotope shift induced by deuterium and is thus diagnostic. Chemical shift differences, the sum of the equilibrium and intrinsic isotopic shifts, of 759, 753, and 560 ppb at the carbonyl carbon were observed in CDCl3, C6D6, and pyridine-d5, respectively. The magnitude and sign of this difference indicate that in solution 20 exists as two interconverting tautomers, rather than as a single structure bridged by a LBHB. OH

O X H X

H

H

X

20, X=CH 21, X=N

22, X=O 23, X=NHAr

S R

H

O R'

24

It should also be noted that many metal chelates of 3-hydroxy-2-phenylpropenal do not show such large isotope shifts, but only small intrinsic ones.62 Therefore, these are definitely symmetric with the metal shared equally between the two oxygens in a single-well potential. This result shows that the asymmetry of the H-bonds is not an artifact of the introduction of an isotope. A tautomeric equilibrium was claimed to cause a broadening of the 13C NMR signals of the carbonyl carbons in Example 20. The lack of similar peak broadening in 3-hydroxy-2-(4pyridyl)propenal (Example 21) was interpreted as evidence that this enol exists as a single symmetric structure.63 However, broadening requires “slow” chemical exchange of 13C nuclei between two different environments, whereas the H transfer that exchanges the carbonyl carbons proceeds by tunneling and is too fast to lead to broadening. A similar conjugated H-bond exists in 6-hydroxy-2-formylfulvene (Example 22) and N,N 0 diaryl-6-aminofulvene-2-aldimines (Example 23). Partial deuteration of the aldehyde hydrogens of Example 22 and of the imine hydrogens of Example 23 showed an isotope shift of þ 376 ppb at the unlabeled aldehyde carbon and þ 223 ppb at the unlabeled imine carbon.64 The downfield shift observed in each case is opposite to an upfield four-bond intrinsic isotope shift, and is evidence of a tautomeric equilibrium. The shape of the potential well for enol-enethiol tautomerism in the enolized b-thioxoketones thioacetylacetone, benzoylthioacetone, thiobenzoylacetone, and monothiodibenzoylmethane (Example 24, R,R0 ¼ CH3,Ph) was studied by Duus and Hansen.65 Upon deuteration of the H-bond a downfield primary isotope shift of 0.2 –0.7 ppm was observed. This contrasts with the intrinsic isotope shift, which is expected to be 0.9 ppm upfield. The downfield isotope shift confirms an equilibrium between OH· · ·SvC and SH· · ·OvC tautomers, in which the magnitude of the perturbation shift is larger than that of the intrinsic and produces a net downfield shift. Further support for an equilibrium comes from the large (up to 8 ppm) deuterium-induced isotope shifts at CO and CS carbons. 2. Proton Sponge 1,8-Bis(dimethylamino)naphthalene (Proton Spongew) has been extensively investigated for its unusually high basicity.66 There is a DpKa of , 7 between N,N-dimethylaniline (pKa , 5.1) and 1,8bis(dimethylamino)naphthalene (pKa , 12.1). 2,7-Dimethoxy-1,8-bis(dimethylamino)naphthalene is even more basic, with a pKa of 16.1 for its conjugate acid (Example 25). This further increase is attributed to a “buttressing” steric effect of the methoxy groups. The potential describing a simple NHN model system becomes single-well at dNN of 250 pm with an increase of the minimum energy

Isotope Effects and Symmetry of Hydrogen Bonds in Solution

243

of 11 kJ/mol.67 This increase in energy is equal to the height of the barrier of the double-well potential for the most stable geometry, with dNN of 272 pm.68

X

N δ+

H

N δ+

X 25, X=OCH3 26, X=Br

Computed geometries and deuterium isotope effects provide information about the shape of the H-bond potential well of bis(dimethylamino)naphthalenes.69 Optimized geometries calculated at the BPW91/6-31(d) level indicated an average dNN of 260 pm, in agreement with x-ray diffraction data. The sum of 1J(15N,H) for the two nitrogens, whether symmetrically substituted or unsymmetrically, is 62 Hz. The equilibrium constant for the two tautomers can be measured as the ratio of the two coupling constants. A 0.66 –0.75 ppm primary isotope shift is consistent with an asymmetric H-bond. Computational and x-ray diffraction studies of protonated 2,7-dibromo-1,8-bis(dimethylamino)naphthalene (Example 26) show a dNN of 254.7 pm and an H-bonded proton with two equilibrium positions.70 Deuteration resulted in a nH/nD ratio of 1.65 for the anharmonic NHN IR absorption at 500 cm21, corresponding to a barrier height , 0.7 kcal/mol for the H-bond potential well. Incremental substitution with a statistical mixture of 0– 4 CD3 groups of the N-methyls in protonated 1,8-bis(dimethylamino)naphthalene and in 25 by the Perrin lab probed the symmetry of the H-bonds.71 Perturbation of a tautomeric equilibrium of the H-bond for both species was most conclusive in the equilibrium isotope shifts of ^ 35 and ^ 47 ppb, respectively, at carbons 2 and 7, and nearly as large at the more distant carbons 4 and 5, significantly greater than any small (, 5 ppb) intrinsic isotope shift. The signs of the isotope shifts are uncertain, but positive values are more likely. The N-methyls of 25 exhibit the unusual feature of isotope shifts of 2 80 and 2 25 ppb from deuteration of the other N-methyls. The larger is from the adjacent methyl, but the smaller is from only one of the methyls on the other nitrogen, not both. It is not clear whether this latter is due to the trans methyl or to the cis, but the stereospecificity requires the cis/trans relationship to be preserved on the NMR timescale. 3. Dicarboxylic Acids Dicarboxylic acids with favorable geometry for forming intramolecular H-bonds exhibit large DpKas. The hydrogen maleate anion (Example 7) is a well-studied model system for investigations of symmetric H-bonds, with a chemical shift . 20 ppm for the H-bonded proton and a primary deuterium isotope shift of þ 0.03 ppm.38 Crystal structures show a symmetric H-bond, including at least one analog where the neutron-diffraction data are better fit with a centered hydrogen than with a structure that places half hydrogens at each oxygen, in a static or dynamic disorder.72 The chemical shift for the H-bonded proton of hydrogen cis-cyclohexane-1,2-dicarboxylate (Example 27) in aprotic organic solvents is 19.3 – 19.7 ppm, suggesting the presence of a LBHB. In contrast to trans-cyclohexane-1,2-dicarboxylate (Example 28), for which the fractionation factor is , 1.0, consistent with other evidence for normal H-bonding in this species, the fractionation factor for Example 27 is 0.69 in water and 0.52 in 9:1 acetone/water.73 These values are larger than typical for LBHBs but the decrease of the fractionation factor in less polar solvents is consistent with the behavior of strong H-bonds.74

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Isotope Effects in Chemistry and Biology

O−

HO

O

O O

O−

O 27

28

HO

Studies on a wide range of monoprotonated dicarboxylates were undertaken in the Perrin lab by mono-18O labeling of one carboxyl group.75 A short, symmetric H-bond would show only an intrinsic isotope shift. This was measured independently as an , 25 ppb upfield shift in the 13C NMR spectrum of the dicarboxylic acid or the carboxylate dianion.76 A larger isotope shift was uniformly observed in all carboxylate monoanions in water, a result of isotopic perturbation of a tautomeric equilibrium and indicative of asymmetric H-bonds in all systems studied. In organic solvents DMSO and THF, the equilibrium isotope shifts at the carboxyl carbon are positive but quite small, and hardly above the limit of resolution. Nevertheless, the carbon a to the carboxyl often shows a substantial positive equilibrium isotope shift, typically an order of magnitude greater than the intrinsic isotope shift, which was unresolvable at such distant carbons. Therefore, the H-bond in many monoanions of dicarboxylic acids is asymmetric in solution. Since hydrogen phthalate monoanion (Example 9) is symmetric in crystal structures,77 this was quite a surprising result when it was first obtained.75 At first it was thought to be a consequence of the high polarity of water, which would stabilize a localized negative charge better than a delocalized one. However, the H-bonds in many monoanions, across a broad range of solvents from THF and dichloromethane to water, are all asymmetric. To rationalize this general result, it was proposed that the disorder of solvation or of counterion association induces the asymmetry. Such a 2 breaking of the local symmetry due to solvation has been seen with I2 3 and NO3 , where resonance Raman spectra show nonzero intensities in transitions that are forbidden in the D1h or D3h symmetry of the isolated ion.78 These results contradict the symmetric H-bond that is inferred from the positive primary isotope shifts that were seen in maleate (Example 7) and phthalate (Example 9) monoanions.38 A further study found zero or negative values for monoanions of 1,2-cyclopentene-dicarboxylic, 3,4-furandicarboxylic, and 3,4,5,6-tetrahydrophthalic acids in four organic solvents.75 These results, along with the results of isotopic perturbation, are consistent with asymmetric H-bonds. For phthalate the primary isotope shift is not always positive but may be negative or zero, depending on solvent and temperature. Yet, by the method of isotopic perturbation, this anion is always asymmetric, and it was concluded that this method is more reliable. It must be acknowledged that the observation of a perturbation isotope shift does not require a double-well potential.71 What is required is an equilibrium between two tautomers, each of which must be asymmetric. Even if the potential is intrinsically single well, differential solvation could stabilize one tautomer more than the other. That would change the potential to a distorted one in which the H-bond at any instant is asymmetric. As the solvent reorganizes, the potential switches back and forth. Such a single-well potential would create an equilibrium between two tautomers, detectable through the ability of isotopic substitution to perturb the equilibrium. This sort of potential might reconcile the discrepancy between the asymmetry inferred from isotopic perturbation and the single well inferred from positive (downfield) primary isotope shifts. 4. Schiff Bases Strong H-bonding in salicylic acid and derivatives has been studied by DpK measurements.27 Salicylidene imines (Example 5) provide a model system for studying intramolecular H-bonds with the NHO motif. Variation of the N-alkyl group allows moderation of the imine pKa without affecting the phenolic pKa or introducing steric factors that would affect the H-bond length. The intramolecular H-bond is as short as 249.1 pm, with dOH of 133 pm and dNH of 122 pm and with an

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245

OHN angle of 1558.10 H-bonds in 5 and analogs have been studied by Hansen and coworkers by observing the effects of deuterium substitution at the H-bond on 15N and 13C NMR chemical shifts and coupling constants.79 The data are consistent with a mixture of OH and NH tautomers. The mole fractions of the two tautomers can be determined by comparison of the one-bond coupling constant 1J(15N,H) with the limiting values measured for the NH form. The isotope effect of deuteration on the 15N chemical shift decreases from 6 ppm, in parallel with 1J(15N,H). The deuterium isotope shift at the OH carbon appears as a distinctive S-shaped curve with a maximum of þ 600 ppb at a mole fraction of NH tautomer , 0.1 and with a minimum of 2 600 ppb at mole fraction , 0.9. The S-shaped curve arises because deuterium prefers the “deeper” potential-energy minimum, with its higher vibrational frequency, so that deuteration shifts the equilibrium constant further from unity. It is thus good evidence for the perturbation of an equilibrium even though the isotope shift vanishes when the equilibrium is exactly balanced, which would be equally consistent with a single-well H-bond.

B. INTERMOLECULAR S YSTEMS Intermolecular systems are of interest because of the generalizability of their results. They allow independent modification of the donor or acceptor, but they can be complicated by solvent effects and by the presence of competing H-bond donors or acceptors. Even in the crystalline state the location of a counterion can disrupt the symmetry of the H-bond between 2-methylimidazole and 2-methylimidazolium cation.80 1. Pyridine – Acid Complexes Complexes between pyridine and carboxylic acids (Example 29) have long been suspected to exhibit single-well H-bonds. This system is of interest because the OHN H-bond is a common motif found in biological systems. Experiments probed the role of matched pKas on the formation of LBHBs by moderating the DpKa by choice of the carboxylic acid or the pyridine.81 The chemical shift for the H-bonded proton in complexes of 2,6-dimethylpyridine is far downfield, at 16.6– 17.9 ppm, and 18.5 –19.7 ppm for 2,4,6-trimethylpyridine. O

O R O H 29

N

X

F

H

R

15N

O H 30

N

N

R'

31

Low-temperature NMR studies of complexes between pyridine-15N and a series of acids in CDClF2/CDF3 were performed by Limbach and coworkers.82 Acids with aqueous pKa’s ranging from 12.9 to , 1 were examined, with the maximum 1H chemical shift of 21.47 ppm corresponding to the complex with 2-furoic acid. The deuterium-induced isotope shift of this complex is 2 0.5 ppm for the H-bonded proton and 8.1 ppm for the pyridine nitrogen. The negative isotope shift of the H-bonded hydrogen and the behavior of the 15N isotope shift indicate that the H-bond is described by a double-well potential, in which deuteration increases dON. An exponential relationship was found to hold between the 15N NMR shift of 2,4,6-trimethylpyridine-15N and the H-bond length. The 1H and 15N chemical shifts are related by a quadratic curve with a maximum corresponding to the symmetric H-bond.83 In complexes of 2,4,6-trimethylpyridine with HF/DF (Example 30) the deuterium isotope effect on the F chemical shift is 2 0.2 ppm at 190 K, reaches a maximum of þ 0.27 ppm at 145 K, and decreases again at lower temperatures.84 This variation is consistent with the conversion of a neutral F –H· · ·N species, via a quasisymmetric F –H – N complex, to a zwitterionic 2F· · ·H –Nþ at low temperature, paralleling the variation of solvent polarity (dielectric constant), which favors

246

Isotope Effects in Chemistry and Biology

the zwitterion. There is also an inverse linear relationship between 1JFH and 1JNH, which does not extrapolate to the limiting values for HF and trimethylpyridinium ion. It was concluded that this is inconsistent with only two components, and that there must be a third, quasisymmetric component. The coupling constants can be used to determine q1, the deviation of the H from the center of the H-bond.85 At 150 K q1 ¼ 0; consistent with a quasisymmetric structure. Yet this value might be only a consequence of the principle of continuity, since the H is being transferred from F to N as the temperature decreases. Moreover, the relationship between 1JFH and 1JNH can be made consistent with a two-component mixture if coupling constants vary with the strength of the H-bonding interaction. This would lead to a curve that would extrapolate to the limiting values. Indeed, calculations on FH:pyridine produce 1JFH and 1JNH that can be fit to such a curve.86 The experimental data and the calculations are consistent with a proton-shared hydrogen bond, but neither the data nor the calculations require anything more than a continuous change from F –H· · ·N to 2F· · ·H – Nþ. By the principle of continuity there must be an intermediate stage where the equilibrium is balanced, but not necessarily one with a shared proton. Certainly the observed constancy of 2JNF across this temperature range does not require any structure fundamentally different from either of the extremes. Thus, we conclude that a quasisymmetric structure is possible, but it is difficult to distinguish this from a balanced equilibrium between F –H· · ·N and 2F· · ·H –Nþ. Perhaps the most convincing evidence for a quasisymmetric structure is the positive primary isotope shift.85 This occurs even with nonzero q1 ; so that the H-bond is not strictly symmetric. The distinction between quasisymmetric and symmetric may resolve the discrepancy between the symmetric H-bond inferred from the positive primary isotope shifts sometimes seen in dicarboxylate monoanions and the asymmetric one inferred from isotopic perturbation.75 Complexes of pyridines with dichloroacetic acid have been the subject of low-temperature NMR investigations in the Perrin lab.87 The H-bonded proton chemical shift approaches a maximum . 20 ppm in the complex with 3-picoline or the parent pyridine. The 18O-induced 13C isotope shift increases along with the 1H chemical shift, reaching a maximum of , 61 ppb. This is due to isotopic perturbation of an equilibrium between neutral Py· · ·HOCOCHCl2 and ion pair PyHþ· · ·– OCOCHCl2 complexes. However, at low temperatures the observed isotope shifts for the 3-picoline and parent complexes drop to 54 ppb, which is the intrinsic isotope shift. The absence of the equilibrium isotope shift observed in other complexes is consistent with the formation of a single-well H-bond, but this may be only a low-temperature phenomenon. The similar complexes of carboxylic acids and imidazoles (Example 31) show IR absorption bands at 1900 and 2500 cm21, which disappear on deuteration, indicating strong H-bonding.88 This was interpreted in terms of an LBHB in which the proton is free to move between N and O but the deuteron is more constrained. The 1H NMR chemical shift reaches a maximum of 17.88 ppm for the complex with 2,2-dichloropropionic acid. IR spectra of the complexes in chloroform also display carbonyl stretching frequencies at 1709 and 1673 cm21.89 These are due to H-bonded RCOOH and imidazolium, respectively, in a neutral or ion-pair complex. The latter is favored with stronger acids. The complex with 2,2-dichloropropionic acid is unusual, in that it shows both 1700 and 1647 cm21 bands, indicating a balanced proton-transfer equilibrium. According to solution calorimetry, the enthalpy of formation of the H-bond between 2,2-dichloropropionic acid and 1methylimidazole is 2 50 kJ/mol. This is comparable to the exothermicity of proton transfer from the stronger acids but , 25 kJ/mol greater than that for formation of ordinary H-bonds with weaker acids. The contribution of LBHB formation was therefore taken to be 50– 60 kJ/mol. 2. Enzymes Recent work proposing a role for strong H-bonding in enzyme-catalyzed reactions has stimulated a resurgence of interest in the nature of strong H-bonds. The potential role of LBHBs in stabilizing transition states has led to studies of enzymes such as ketosteroid isomerase, triosephosphate isomerase,90 phospholipase,91 and chymotrypsin.92 Because of the complexity of these systems,

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measurements of chemical shift and fractionation factors are the most common methods of investigation. The possible role of LBHBs in enzyme catalysis is reviewed in detail by Mildvan and Frey in other chapters (this volume). It is undeniable that there are various characteristics associated with LBHBs that sometimes appear in enzymatic systems. Among these are downfield 1H NMR chemical shifts, large primary isotope shifts, and unusual fractionation factors. What is still unclear is the role that LBHBs play in catalysis. Is there an energetic benefit from LBHB formation, and where does it come from? It is not simply from the reduction of zero-point energy, which contributes at most , 15 kJ/mol.1 Although LBHBs are unusually strong in the gas phase, there seems to be no great stabilization associated with them in solution, especially aqueous solution. Therefore, it has been proposed that the stabilization is due to the relief of strain in the enzymesubstrate complex.71 If this strain is relieved on passing to the transition state, the rate constant kcat can be increased. If so, it is not the LBHB per se that is responsible for the enhanced catalysis.

V. CONCLUSION Determination of the characteristics of H-bonds remains an active field of inquiry. The changes introduced by isotopic substitution can provide information that can be measured by a variety of methods. Although computational work predicts the existence of symmetric (single-well-potential) H-bonds and although potentially symmetric H-bonds have been observed in crystal structures, symmetric H-bonds in solution remain elusive.

ACKNOWLEDGMENTS This research was supported by NSF Grant CHE99-82103.

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47

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69 Grech, E., Klimkiewicz, J., Nowicka-Scheibe, J., Pietrzak, M., Schilf, W., Pozharski, A. F., Ozeryanskii, V. A., Bolvig, S., Abildgaard, J., and Hansen, P. E., Deuterium isotope effects on 15N, 13C and 1H chemical shifts of proton sponges, J. Mol. Struct., 615, 121– 140, 2002. 70 Bienko, A. J., Latajka, Z., Sawka-Dobrowolska, W., and Sobczyk, L., Low barrier hydrogen bond in protonated proton sponge. X-ray diffraction, infrared, and theoretical ab initio and density functional theory studies, J. Chem. Phys., 119, 4313– 4319, 2003. 71 Perrin, C. L. and Ohta, B. K., Symmetry of NHN hydrogen bonds in 1,8-bis(dimethylamino)naphthalene-Hþ and 2,7-dimethoxy-1,8-bis(dimethylamino)naphthalene-Hþ, J. Am. Chem. Soc., 123, 6520– 6526, 2001; Perrin, C. L. and Ohta, B. K., Symmetry of NHN hydrogen bonds in solution, J. Mol. Struct., 644, 1 – 12, 2003. 72 James, M. N. H. and Matsushima, M., Accurate dimensions of maleate mono-anion in a symmetric environment not dictated by crystallographic symmetry: imidazolium maleate, Acta Crystallogr. B, 32, 1708– 1713, 1976; Ellison, R. D. and Levy, H. A., A centered hydrogen bond in potassium hydrogen chloromaleate: a neutron-diffraction structure determination, Acta Crystallogr., 19, 260– 268, 1965. 73 Lin, J. and Frey, P. A., Strong hydrogen bonds in aqueous and aqueous-acetone solutions of dicarboxylic acids: activation energies for exchange and deuterium fractionation factors, J. Am. Chem. Soc., 122, 11258 –11259, 2000. 74 Markley, J. L. and Westler, W. M., Protonation-state dependence of hydrogen-bond strengths and exchange rates in a serine protease catalytic triad: bovine chymotrypsinogen A, Biochemistry, 35, 11092– 11097, 1996. 75 Perrin, C. L. and Nielson, J. B., Asymmetry of hydrogen bonds in solutions of monoanions of dicarboxylic acids, J. Am. Chem. Soc., 119, 12734– 12741, 1997; Perrin, C. L. and Thoburn, J. D., Symmetries of hydrogen bonds in monoanions of dicarboxylic acids, J. Am. Chem. Soc., 114, 8559– 8565, 1992; Perrin, C. L., Symmetries of hydrogen bonds in solution, Science, 266, 1665– 1668, 1994. 76 Rishavy, M. A. and Cleland, W. W., 13C, 15N, and 18O equilibrium isotope effects and fractionation factors, Can. J. Chem., 77, 967– 977, 1999. 77 Ku¨ppers, H., Kvick, A., and Olovsson, I., Hydrogen-bond studies. 142. Neutron-diffraction study of the 2 very short hydrogen-bonds in lithium hydrogen– phthalate – methanol, Acta Crystallogr. B, 37, 1203– 1207, 1981; Adiwidjaja, G. and Ku¨ppers, H., Lithium hydrogen phthalate – methanol, Acta Crystallogr. B, 34, 2003– 2005, 1978; Gonschorek, W., and Ku¨ppers, H., Crystal-structure of lithium hydrogen phthalate dihydrate, containing a very short hydrogen-bond, Acta Crystallogr. B, 31, 1068– 1072, 1975. 78 Johnson, A. E. and Myers, A. B., Solvent effects in the Raman spectra of the triiodide ion: observation of dynamic symmetry breaking and solvent degrees of freedom, J. Phys. Chem., 100, 7778– 7788, 1996; Waterland, M. R. and Kelley, A. M., Far-ultraviolet resonance Raman spectroscopy of nitrate ion in solution, J. Chem. Phys., 113, 6760– 6773, 2000. 79 Hansen, P. E., Sitkowski, J., Stefaniak, L., Rozwadowski, Z., and Dziembowska, T., One-bond deuterium isotope effects on N-15 chemical shifts in Schiff bases, Ber. Bunsen-Ges. Phys. Chem., 102, 410– 413, 1998; Rozwadowski, Z., Majewski, E., Dziembowska, T., and Hansen, P. E., Deuterium isotope effects on 13C chemical shifts of intramolecularly hydrogen-bonded Schiff bases, J. Chem. Soc. Perkin Trans., 2, 2809– 2817, 1999; Dziembowska, T., Rozwadowski, Z., Filarowski, A., and Hansen, P. E., NMR study of proton transfer equilibrium in Schiff bases derived from 2-hydroxy-1naphthaldehyde and 1-hydroxy-2-acetonaphthone. Deuterium isotope effects on 13C and 15N, Magn. Reson. Chem., 39, S67 – S80, 2001. 80 Song, X. J. and McDermott, A. E., Proton transfer dynamics and NH bond lengthening in NHN model systems: a solid-state NMR study, Magn. Reson. Chem., 39, S37– S43, 2001. 81 Dega-Szafran, Z., Dulewicz, E., and Szafran, M., Interaction of 2,6-dimethylpyridine with chloroacetic acid in some aprotic solvents: problem of stoichiometry, J. Mol. Struct., 177, 317– 325, 1988. Barczynski, P., Dega-Szafran, Z., and Szafran, M., Interaction of 2,4,6-trimethylpyridine with some halogenocarboxylic acids in benzene and dichloromethane problem of stoichiometry, J. Chem. Soc. Perkin Trans., 2, 901– 906, 1987. 82 Smirnov, S. N., Benedict, H., Golubev, N. S., Denisov, G. S., Kreevoy, M. M., Schowen, R. L., and Limbach, H. H., Exploring zero-point energies and hydrogen bond geometries along proton transfer

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83

84

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86 87 88 89 90 91 92

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9

NMR Studies of Isotope Effects of Compounds with Intramolecular Hydrogen Bonds Poul Erik Hansen

CONTENTS I. II. III. IV. V.

Introduction and Outline .................................................................................................. 253 Definitions ........................................................................................................................ 255 Theory .............................................................................................................................. 255 Experimental Conditions.................................................................................................. 258 Static Systems .................................................................................................................. 259 A. Resonance-Assisted Hydrogen Bonded Systems .................................................... 259 1. Transmission of Isotope Effects across Hydrogen Bonds ................................ 261 B. Non-RAHB Cases .................................................................................................... 262 C. Medium- and Long-Range Isotope Effects ............................................................. 264 D. Isotope Effects through Space ................................................................................. 264 E. nD15N(D) .................................................................................................................. 265 1. 1D15N(D)............................................................................................................ 265 2. 5D15N(D)............................................................................................................ 266 F. nD17O(D) .................................................................................................................. 266 G. nDYH(XH) (X,Y ¼ O or N) .................................................................................... 266 H. nD19F(XD) ................................................................................................................ 267 I. nD13C(18O)................................................................................................................ 267 J. Primary Isotope Effects............................................................................................ 267 K. Summary for Intrinsic Isotope Effects..................................................................... 268 1. RAHB Systems.................................................................................................. 268 2. Non-RAHB Systems ......................................................................................... 269 VI. Equilibrium Isotope Effects ............................................................................................. 269 A. General Findings ...................................................................................................... 269 B. Long-Range Effects in Equilibrium Systems .......................................................... 273 C. Identifying Equilibrium Systems ............................................................................. 274 VII. Calculations ...................................................................................................................... 275 References..................................................................................................................................... 275

I. INTRODUCTION AND OUTLINE The NMR parameters related to isotope effects are isotope effects on chemical shifts; isotope effects on coupling constants; and fractionation factors. Fractionation factors will be dealt with in detail in Chapter 7. Little information is available on isotope effects on coupling constants in intramolecularly hydrogen bonded systems. The common underlying feature is, in simple 253

254

Isotope Effects in Chemistry and Biology EHF (a.u.) 132.500

132.550

132.600

H D

132.630 s(N) ppm 260.0 250.0 240.0 D

H

230.0 0.8

1.0

1.2

1.4

1.6

1.8

2.0

A˚

FIGURE 9.1 Hydrogen bond potential and nuclear shielding surface.

Energy

terms, the different zero-point energies of the different isotopes, as illustrated for the H/D pair in Figure 9.1. Different zero-point energies result in differences in average bond length for the two isotopologues, depending on the asymmetry of the potential well. This difference in average bond length will ultimately lead to a difference in nuclear shielding (chemical shift), as illustrated in Figure 9.1 for a one-bond isotope effect. Isotope effects on chemical shifts in hydrogen bonded systems have great capability as a structural tool and lead to insight into the hydrogen bond potential and properties in particular. The intramolecular hydrogen bond can be of different types and can de divided according to the type of potential well1 (Figure 9.2) or according to the presence or absence of resonance assistance2,3 (Figure 9.3). An inherent property of the hydrogen bonded system is the possibility of proton transfer leading to equilibrium. The factors favoring proton transfer have been analyzed by Gilli et al.2,3 Isotope

(a)

(b)

(c)

FIGURE 9.2 Types of hydrogen bond potentials: (a) asymmetric; (b) symmetric double-well potential; (c) single-well potential.

NMR Studies of Isotope Effects

255

X'R

X

Y

− X

X'R H

Y

H Z+

Z

FIGURE 9.3 Resonance-assisted systems (RAHB).

effects on chemical shifts can be used advantageously to distinguish between systems that are with or without proton transfer (the latter are called static in this review). In order to distinguish and characterize such systems, a large number of parameters is required. Isotope effects on chemical shifts can often produce a large array of data in relation to hydrogen bonding. The isotope effects observed are most commonly due to the change of the H to a D at the hydrogen bonded (chelate) proton (see Figure 9.3). The secondary isotope effects (see below) at nuclei of the system nD13C(XD), nD15N(XD), n 17 D O(XD) can be studied, depending on the system. In the case that either X or Y are oxygens, n 13 D C(18O) can also be studied. Reviews dealing with this topic, either in general4 – 12 or more specifically13,14 exist. The latter provide a large amount of data.

II. DEFINITIONS The primary isotope effect is defined as pD1H(D) ¼ d(H) 2 d(D) or pD1H(T) ¼ d(H) 2 d(T) for the deuterium and tritium isotope effects, respectively. The definition for the secondary isotope effects on chemical shifts is as follows: nDX(Y) ¼ dX(Ylight) 2 dX(Yheavy). The isotope shifts will be referred to as either intrinsic or equilibrium. The intrinsic isotope shifts may also be called geometric as a consequence of their origin (see Section III). For equilibrium systems, part of the isotope effect is referred to as the equilibrium isotope effect. The majority of cases will have the isotope directly involved in the hydrogen bond as seen in Figure 9.4(a). However, the isotope may also occur as seen in Figure 9.4(b).

III. THEORY Two ways of analyzing the isotope will be described. Jameson uses the Born– Oppenheimer approximation and a ro-vibrational approach. Sokolov, Vener, and Savel’ev use an adiabatic separation of the “fast” subsystem (H and D) from the “slow” (heavy atoms, typically O).15 – 17 (D) RO

H

RO

H

(a)

Y

Y

C

C X

(b) FIGURE 9.4 Types of intramolecular hydrogen bonds.

R

H(D) X = Cor N Y = O, N or S

256

Isotope Effects in Chemistry and Biology

In Jameson’s18 ro-vibrational approach, the nuclear shielding ðsÞ can be expanded in a series as follows: ! ! X d2 s X ds X ds kDr l þ kDr Dr l þ kDaij l þ · · · ð9:1Þ k sl ¼ se þ dri e i dri dri e i j daij ij i ij ri is the average of the equilibrium distances, Dr the change in the bond length due to the stretching vibration, and Da the bond angle changes. The isotope effect is given as the difference in the chemical shielding for the light and heavier isotope (marked with *): ! i ds d2 s h p p 2 2 p ksl 2 ks l ¼ ½kDr1 l 2 kDr1 l þ kðDr Þ l 2 kðDr Þ l þ ··· ð9:2Þ 1 1 dr1 e dr12 kDrl2 is the mean square amplitude. On the basis of the Jameson theory, the one-bond deuterium isotope effect on a neighbor nucleus can be approximated by the expression:

s 2 sp ¼

X

ds drCH

e

½kDrCH l 2 kDrCD l

13

C

ð9:3Þ

rCH and rCD are the CH and CD bond length, respectively. For long-range deuterium isotope effects on a nucleus 13C remote from the substitution site, the system can be described as (13CAnH; A ¼ C, O, N). The shielding of the nucleus 13C change is in response to the shortening of the CA distance ðdsðCÞ=drCA Þe (primary electronic factor), as well as to the shortening of the AH bond, (ds (C)/drAH)e (secondary electronic factor), and also is a consequence of the change in the averaged distances AH, kDrAH l 2 kDrAD l (primary dynamic factor) and A – C (secondary dynamic factor). This leads to an expression for the isotope effects on 13C:

s 2 sp ¼

dsðCÞ drAH

e

½kDrAH l 2 kDrAD l þ

ds ðCÞ drAC

e

½kDrAC l 2 kDrAC lp þ · · ·

ð9:4Þ

It has been shown that the product of the secondary electronic factor (ds (C)/drAH)e and the primary dynamic factor kDrAH l 2 kDrAD l is more important than the second product of Equation 9.4. In general, the sign and magnitude of the isotope effect across two or more bonds reflect the sign and magnitude of the secondary electronic factor, ðds ðCÞ=drAH Þe : Based on the theory of Jameson and as demonstrated from the theoretical calculations, the intrinsic isotope effects can be understood as a change in the nuclear shielding upon deuteriation, multiplied by the change in the XH bond lengths upon deuteriation. For the o-hydroxyacetyl aromatics, a proportionality was found between the OH bond length and the change in the bond length upon deuteriation, DROH(D).19 The proportionality factor between ROH and DROH(D) will depend on the shape of the hydrogen bond potential, which is related to the strength of the hydrogen bond. The change in nuclear shielding upon deuteriation was found to be rather similar for similar carbons in different compounds. This change depends on the transmission pathway; bond order becomes an important feature. For hydrogen bonded compounds of similar type (e.g., o-hydroxybenzaldehydes, ketones and esters (Figure 9.5), one can expect to find isotope effects at a given position that are simply a multiple of the transmission factors given in Figure 9.6, modified to a smaller or larger extent by local substituent effects.20 The observation of the Ubbelohde effect in the solid state, a change in the heavy atom distance in going from a X-H· · ·X situation to a X-D· · ·X situation, X being O, N or S21 clearly indicates that

NMR Studies of Isotope Effects H

O

257 H3C

H (D)

O H

O

(D)

H3CO

0.18 ppm

0.28 ppm

X

H (D) O

O

0.23 ppm

O

X

X

FIGURE 9.5 o-Hydroxyacyl aromatics.

deuteriation may have an effect other than that of the average change of the OH/OD distance. Sokolov, et al.15 have set up an analysis using the adiabatic approximation and found a notable correlation between observed and experimental changes in heavy atom distances upon deuterium substitution. Unfortunately, only two experimental data sets exist for intramolecularly hydrogen bonded compounds, that of pyridine-2,3-dicarboxylic acid22 and bis(3-amino-3-methyl-2-butanone oximato) nickel(II) chloride.23 In the former compound the distance between O-2 and O-3 is ˚ . No change in this distance was found upon deuteriation of the chelate proton. 1.3995 A Measurements were taken at four different temperatures. For bis(3-amino-3-methyl-2-butanone ˚ an increase in the O· · ·O distance of 0.019 A ˚ oximato) nickel(II) chloride with a RO· · ·O of 2.439 A 23 was found. Neither of these compounds is, unfortunately, of RAHB type. From the graph of Sokolov et al.,15 it can be seen that the increase in the heavy atom distance for symmetrical ˚ , decreasing to zero for strong hydrogen bonds with RO· · ·O ¼ 2.42 A ˚. compounds is, at most, 0.02 A 17 Vener plotted primary isotope effects mainly from b-diketones deuterated at the OH position with ˚ ) and predicted a shape similar to that found experimentally, but Xe (Xe defined as RO· · ·O 2 2.4 A ˚ compared to experiment. with the maximum shifted from , 0.12 to , 0.08 A Another example concerns compounds like that of Figure 9.7; a 5D17O(XD) of 2 9 ppm ˚ was measured.24 The O· · ·O distance has, by neutron diffraction, been determined to be 2.42 A 17 (F. K. Larsen, personal communication). For similar compounds, the change in the O chemical ˚ . From the graph of Sokolov shift as a function of the O· · ·O distance was found to be 400 ppm/A 15 et al., the change in O· · ·O distance on deuteriation was found to be 0 ppm. In this particular case, this change contributes only very little to the isotope effect. Furthermore, the maximal change is ˚ . This corresponds to the distance estimated in predicted to be found for an O· · ·O distance of 2.52 A ˚ , which o-hydroxyacetophenone. The change in the O· · ·O distance is estimated to be 0.02 A translates into a change of 8 ppm in the 17O chemical shift, but for o-hydroxacetophenone, the isotope effect could not be measured, meaning that it is smaller than , 2 ppm. More experimental data are clearly needed before this effect on geometry and its influence on isotope effects on chemical shifts can be established clearly in intramolecularly hydrogen bonded cases.

O

X

H (D) O

(20)

X = H, OR or R

60

−30

10 25

FIGURE 9.6 Transmission coefficients. Printed with permission of MRC.

258

Isotope Effects in Chemistry and Biology −250

O

55 H

O

131 a

H (D) 17.09 O

186

O

722 a −35

a 0

ppm

−57

229 O

−45

H

O

57

FIGURE 9.7 Deuterium isotope effects of 1,3,5-triacetyl-2,4,6-trihydroxybenzene nD13C-2(OD), nDOH(OD) (in bold) and d(OH) (in italics).19

IV. EXPERIMENTAL CONDITIONS

165.398 164.957

ppm

174.341 174.155

Isotope effects on chemical shifts can be measured by two different techniques. The first one relates to compounds in which the isotope is in slow exchange with the surroundings on the NMR time scale. Examples are deuterium at the hydrogen bond bridge of compounds in dry and inert solvents like CDCl3, CD2Cl2, benzene-d6, toluene-d8 or deuterated freons. In these cases, the nuclei in question will appear as two lines, one due to the species with the light isotope and the other due to the one with the heavy isotope (see Figure 9.8). In this type of experiment, measurements are best done with both isotopic species in the same tube (one-tube experiment). For compounds in which the isotope is exchanging quickly with the surroundings on the NMR time scale, only averaged positions can be observed. The isotope content can be varied and the isotope effects plotted as a function of the isotope content, as illustrated in Figure 9.9, and the isotope effects obtained by extrapolation. This kind of experiment is typically performed in a proton donating solvent such as water or the alcohols and the isotope in question is deuterium. The degree of deuteriation is controlled by

ppm 174

FIGURE 9.8 Isotope splittings in

172 13

170

168

166

164

C NMR spectrum. The D-isotopologue is dominant.

NMR Studies of Isotope Effects

259

0

δ∗ [ppm]

−0.1 a4 = −5.43e-004

−0.2

a5 C4 C5 a6 C6 a7 C7 C3` a 3` C4` C2m a4` C3m α2m

−0.3

−0.4

α3m

0

10

20

= −2.68e-003 = −1.25e-003 = −2.92e-003 = −8.06e-004 = −1.51e-003 = −2.95e-004 = −9.83e-004 30

40

50 60 D2O [%]

70

80

90

100

FIGURE 9.9 Plot of measured deuterium isotope effects on chemical shifts vs. deuterium content of solvent. From Ref. 48 with permission from the Royal Society of Chemistry.

the mixture, H2O:D2O or ROH:ROD. The proton donating solvent should be in large excess and effects of fractionation (see Chapter 7) should be taken into account. It is essential that the amount of proton donating solvent is kept constant for the series of experiments to avoid the mixing-in of solvent effects. Isotope effects on 17O chemical shifts are best measured at conditions of low solvent viscosity. The broad nature of these resonances puts a lower limit of , 1 ppm on the magnitudes which can be measured. 1 15 D N(D) isotope effects can be difficult to measure due to long relaxation times — typical for secondary nitrogens as no protons are left on the nitrogen after deuteration. Furthermore, the onebond N –D coupling constant may either lead to a triplet or be washed out depending on conditions (e.g., temperature).25 This problem is avoided if an indirect detection is used, as demonstrated in proteins.26

V. STATIC SYSTEMS A. RESONANCE- A SSISTED H YDROGEN B ONDED S YSTEMS The resonance assistance is described in detail by Gilli et al.1 – 3 and is shown in Figure 9.3. The CvC bond can be isolated or part of a more conjugated system, often aromatic. The majority of the compounds investigated in this chapter are of that type and cover enamines Z ¼ N with YXX’R being COR,44 COOR,44 CSNR,27 NO2,28 – 31 C ¼ NO,32 enethioles with Z ¼ S and YXX0 R being COSR and CSSR.33,44 Planarity of the pseudo-aromatic six-membered ring hydrogen bonded system is clearly quite important. This type of hydrogen bonded system shows a nonlinear hydrogen bond as seen in Figure 9.3. The X-H· · ·Z angle is typically , 1508. The use of the X· · ·Z distance is obviously less optimal than for linear systems. However, as will be seen later, a good relation is found between the X· · ·Z distance and the other bond lengths of the hydrogen bonded system. This has been demonstrated in o-hydroxyacyl aromatics. What is more important is that these parameters also correlate well with the two-bond isotope effect, 2D13C(OD).19 The two-bond isotope effect on carbon chemical shifts, 2D13C-1(ZD), the four-bond 4D13C-1(ZD) and the one-bond 1D15N(D) isotope effects clearly vary as a function of hydrogen bonding25 as seen in Z- and E-enaminones (Figure 9.10).

260

Isotope Effects in Chemistry and Biology

0.06 ppm C2H5 C

O

(D) H

1.33 ppm C2H5

O

0

C2H5

N

C 0.27 ppm

C

H

0.10 ppm

C

C

H

H3C

(D)

C

NHC2H5 0.84 ppm

H3C 0.27 ppm

FIGURE 9.10 2D13C-2(ND) and 4D13CvO(ND) and 1D15N(D) (in italics) of enaminones.25

The two-bond isotope effect depends on four factors: (i) the double bond character of the intervening bond in the parent compound; (ii) the character of the acceptor RXY0 X; (iii) the acceptor ZH; and (iv) steric factors. The first of these factors is in accordance with the fact that the system is of RAHB type. This point is demonstrated for o-hydroxy aromatic esters,34 in which 2D13C(OD) isotope effects correlate well with the double-bond character of the CvC bond of the parent hydrocarbon, for salicylic acid ester the parent compound is benzene, etc. Without making plots, this can be seen by comparison of 2-hydroxyacetophenone (Figure 9.5) and 2-hydroxy-1acetylnaphthalene (Figure 9.12). The bond order is larger in the latter, as is the two-bond isotope effect.20 A further consequence of this is that bond “fixation” becomes important. This is clearly illustrated by 3- and 5-acyl-hydracetic acid35 or by a comparison of the data of the pair shown in Figure 9.11. In case A, the second hydrogen bond pair helps to localize the double bonds and, in a mutually optimal way, leads to strong hydrogen bonds and large deuterium isotope effects. In case B, the two hydrogen bonded systems are counteracting each other and no hydrogen bond strengthening is obtained. The ultimate case for bond fixation and strong hydrogen bonding is that of Figure 9.7. The reason bond order between the two intervening carbons is so important is of course that this is in the resonance form. Figure 9.3(b) results in a high bond order between the H or D and the carbon two bonds away. Steric compression is found in case A, in which the two pairs of hydrogen bonds are in the ring plane. This creates a strong steric interaction between a CH3 group of one system and the oxygen of the neighboring system.19 A comparison of the 2D13C(OD) of Figure 9.10(a) and (b) with those of Figure 9.7 reveals that the 2D13C(OD) isotope effects, as well as all other isotope effects, increase numerically with increasing steric interactions. The increase in the isotope effects relates very strongly to the increase in the OH bond length upon deuteration according to theoretical calculations19 (see Section VI). Steric compression effects are also seen in the enolic forms of acylindandiones36 and b-diketones. For the series 2,4-pentandione, 2,6-dimethyl-3,5-heptanedione and 2,2,6,6-tetramethyl-3,5-heptanedione, we find that the formal two-bond isotope effect increases from 0.639 to 0.745 ppm.37

−122

−50a

O 90

H(D) 14.79 ppm 68

O

129a

H(D) 13.09 ppm a O

O

309a

506 136 (a)

69a

79

O H

O

(b)

O

H

O

FIGURE 9.11 o-Hydroxyacyl aromatics, nD13C-2(OD) and d(OH) (in italics).19

NMR Studies of Isotope Effects

H H H

H3C

261

H O

H (D) O 0.66 ppm

(D) 0.1 ppm O H N C C2 C1 H CH3

t-bu −0.04 ppm (D) O H 2' H3C C N C C H CH3 t-bu

FIGURE 9.12 2D13C-2(OD) of 1-acetyl-2-hydroxynaphthalene20 and 3D13C-20 (ND) of enaminones.42

Steric compression effects of equilibrium systems like Schiff bases are also seen (Section VI.A). In addition to steric compression effects, one can also observe steric twist effects. Such effects are typically seen in systems like 1-acetyl-2-hydroxynaphthalene (Figure 9.12).20 They are formally of equilibrium type but, as steric compression and twist often go together, they are included here. In the twist cases, the steric interaction will push the acetyl group out of the ring plane. This is demonstrated by deuteration of the methyl group of the involved acetyl group (this is the type of hydrogen bond pattern shown in Figure 9.4(b). As the steric repulsion is less in the deuterated species, this will lead to less twist, leading to a stronger hydrogen bond and a high frequency shift of the OH resonance.10,38 A similar feature is not seen in compounds with steric compression.39,40 Deuteration at the OH position of e.g., 1-acetyl-2-hydroxynaphthalene (Figure 9.12) leads to a rather large two-bond isotope effect as the mechanism here is the opposite. Deuteration leads to a weaker hydrogen bond41 and, hence, to more twist, which in this case leads to a low frequency shift of the CvO carbon.38 A combined twist and steric compression is found in N-phenyl enaminones with substituents either at C-1 or at the C-20 ,C-60 position of the phenyl ring (Figure 9.12).42 The effects of the donor and acceptor group have been studied in a series of substituted enamines and o-hydroxyacyl aromatics (see Figure 9.5) leading to classification of acceptor groups according to their hydrogen bonding ability.9,33,43,44 For the donors, the OH is normally better than NH which again is better than SH as donors in intramolecular hydrogen bonds.43 1. Transmission of Isotope Effects across Hydrogen Bonds An extensive study of isotope effects of nitro-substituted 2-hydroxyacetophenones showed, for disubstituted compounds, very large 4D13CvO(OD) isotope effects. A principal component analysis revealed that 2D13(OD) depended on the parameters expected for a RAHB system, whereas for the 4D13CvO(OD) the following parameters were involved: RO· · ·O, RCvO, RC2 – O and RC1 – C7. This proves that the 4D13CvO(OD) isotope effect is transmitted via the hydrogen bond as the CvO bond length and the O· · ·O distance, as well as the C2 –O and the C1 – C7 bond length, come into play (see Figure 9.13(a)).45 As a consequence of the transmission across the hydrogen bond, 3,5-dinitro-2-hydroxyacetophenone 4D13CvO(OD) is rather large ¼ 0.30 ppm. In contrast, for o-hydroxyacyl aromatics with hydroxy or alkoxy substitutents at the 4- or 6-position lead to numerically very small 4D13CvO(OD) isotope effects because of the resonance scheme of Figure 9.13(b) being dominant.

262

Isotope Effects in Chemistry and Biology

O

R1 1 O (a)

O−

R1

H

O

O 2

O

N

+

O

O−

R1

H

O−

R1

H O+

O

(b)

O

OH

O R1

H

O N

N

O

O

R1

H

OR

H O

OR

OR

FIGURE 9.13 Resonance forms of nitro-, oxy-, and alkoxysubstituted o-hydroxyacyl aromatics.

Long range effects are found in highly conjugated systems like purpurogallin46 (Figure 9.14) or flavonoids.47,48 Most of the long-range effects are transmitted via the hydrogen bond. Effects across hydrogen bonds are also observed in Schiff bases, primarily at the proton transfer form. In this case, one can observe a nDN(D).49

B. NON-RAHB C ASES The proton sponges, such as DMANHþ (Figure 9.15), show some interesting features. The system is tautomeric and the barrier to proton transfer is rather small. First of all, deuteriation at the NH position leads to small deuterium isotope effects at the carbon resonances in the symmetrical DMANHþ. Likewise, the deuterium effects at the 15N chemical shifts are relatively small.50 For the symmetrical case, this deuteriation does not lift the symmetry. However, deuteriation at the CH3 groups does lift the symmetry (see Chapter 8). For deuteriation at the NH position of the symmetrical compounds, one observes averaged intrinsic isotope effects. For nonsymmetrical cases, see Section VI.A. Another system which is non-RAHB is that of pyridine and quinoline N-oxides (Figure 9.16). The system 8-hydroxyquinoline N-oxide shows strong hydrogen bonding (dOH varies from 15 –17.3 ppm) and relatively large isotope effects at carbons 2, 4, 5, 7 and 8. The only negative one is that at C-2. The positions showing large isotope effects are those conjugated with the OH group.51 This type of compound shows clearly that resonance assisted hydrogen bonding, as such, is not

O

HO 0.07

0

(D) H O

0.03

0.41

OH 0.03

0.18

0 0

0 0.09

−0.08

FIGURE 9.14 Long-range isotope effects of purpurogallin.46

OH

NMR Studies of Isotope Effects

263

(D) H CH3 CH3 + CH3 N 12 11 N CH3 Y

78 6 5

(a)

10

CH3 CH3

N

H

+

N

CH3 CH3

Y

1

9

(D)

2 3

4

(b)

X

X

FIGURE 9.15 DMANHþ.

a necessary condition for obtaining large isotope effects. The strong hydrogen bonding is caused by substituent effects and conjugation, combined with the negative charge at the oxygen. Conjugation throughout the system ensures transmission of the isotope effect (Figure 9.16). For the quinaldinic acids (dOH , 18 – 20 ppm) (Figure 9.16(b)), the situation is different. A small two-bond isotope effect due to deuteration at the COOH group is observed at the CvO carbon (0.11 ppm), whereas much larger isotope effects are seen at C-1, C-4, and C-8a.51 This strongly signals transmission via the hydrogen bond, as no conjugation via the carbonyl group can be established, hence leading to the small two-bond isotope effect. A small two-bond isotope effect at the carboxylic acid carbon is also found in citrinin.52 For the N-methyl quinoline-2-carboxyamide N-oxide (Figure 9.16(c)), a small two-bond isotope effect is seen, 0.11 ppm, due to deuteriation at the NH position. However, in this case the NH chemical shift is only 11.4 ppm, indicating a much weaker hydrogen bond and hence a rather normal NH chemical shift.53 – 56 In compounds like 16D and E, two-bond isotope effects at C-6 show that the effect is larger for E than for D, indicating that, in a non-RAHB case like the present, intramolecular hydrogen bonding does not lead to an increase of the two-bond isotope effects.57 An important group of compounds that can potentially make non-RAHB hydrogen bonds are the carbohydrates. For simple carbohydrates, deuterium isotope effects are seen over two bonds. These are typically , 0.1 ppm and can easily be measured by the so-called SIMPLE method, just adding D2O to a DMSO-d6 solution.58 – 60 Bosco et al.61 found from solvent studies and supported

X

O

(a)

H

O

N

N O

O

(b)

O

O H (D) O R

(d)

C

O

H

O

H

O

(D) H

H

O

N

C O

(c)

H

O

C H

N

R

O

O R

C

O

H

(e)

FIGURE 9.16 Quinoline N-oxides (a, b, and c) and 2,6-dihydroxy acylaromatics (d and e).

264

Isotope Effects in Chemistry and Biology

by studies of agarose that the two-bond deuterium isotope effects on carbon are increased at the OH groups when they are acting as an acceptor, whereas they are decreased when acting as a donor. This is nicely in line with the results from 2,6-dihydroxy acyl aromatics (see above) and also with suggestions made earlier by Davies et al.59

C. MEDIUM- AND L ONG- R ANGE I SOTOPE E FFECTS As shown in the theory section (Section III), all intrinsic isotope effects will depend on the change in the XH bond length upon deuteration. The same behavior, with local variations due to substituents, can therefore be expected (see Figure 9.6). In this text most emphasis has been placed on 2D13C(OD) and 4D13CvO(OD). This is of course not to indicate that all other isotope effects are not important or cannot be observed. Examples of isotope effects in benzene systems are shown in Figure 9.7 and Figure 9.11. In large conjugated systems, they may be transmitted over many bonds (see Figure 9.14). Isotope effects through single bonds fall off very quickly.

D. ISOTOPE E FFECTS THROUGH S PACE One of the early suggestions of the cause of isotope effects on chemical shifts goes back to Gutowsky,62 who suggested that the effect originated in the difference in electric field effects for XH and XD bonds. The rationale was that the XH and the XD bonds are not, on average, of equal length. For nonpolar cases such as CH bonds, this explanation was not very good (for a full account of these historic developments see Chapter 1). For peptide bonds an explanation in line with that of Gutowsky was suggested in order to explain 3D13CO(ND).55 Returning to long-range isotope effects, some of these are clearly very unusual and difficult to explain if they are supposed to go through bonds. The characteristic of an effect originating in electric fields is that the involved bond (e.g., the NH(D) bond) clearly must be highly polarized. The amide (peptide) bond is in that category; and the thioamide bond is even more so. A second prerequisite is a hydrogen bond that changes its length substantially upon deuteriation, that is, the potential of the hydrogen bond must be strongly asymmetric. Such a case is found for the compounds shown in Figure 9.17. As we assume that electric fields are important, the long-range effects of this type will depend both on the polarizability of the bond in which the observed nucleus is involved, and on the direction of that bond relative to the site of deuteriation. They will, of course, also depend on the 1/r 3, r being the distance between the site of deuteriation and the nuclei in question (Figure 9.18). Accordingly, they will be even more important over short distances. Examples of this can be found in enaminones, as seen in Figure 9.12. For a broader discussion of electric field effects, see Chapter 5.

1"

S

−60ppb N

H (D)

N

FIGURE 9.17 Long-range deuterium isotope effect in hydrogen bonded thioamide (Sosnicki, Jagodzinski, and Hansen, unpublished result).

NMR Studies of Isotope Effects

265 D H

O

N H

N H

H

H

FIGURE 9.18 Electric field effects in proteins.

E. 1.

n

D 15 N(D)

1

D15N(D)

The classical situation illustrating the importance of hydrogen bonding is the ketoenamines.9,25,44 They show (Figure 9.10) that hydrogen bonding leads to an increase of 1D15N(D) in the RAHB system. The value found in the non-hydrogen bonded E-isomer is close to that found in simple amides and amines , 0.65 ppm.53 A good correlation is found between 1D15N(D) and 2D13CO(ND) for enaminones.9 For non-RAHB systems, the symmetrical DMANHþs are an example (see Figure 9.15). The 1D15N(D) isotope effects are, in that case, rather small.50 This has been ascribed to the positive charge of the nitrogen and to the close presence of the nitrogen lone pair of the neighboring nitrogen.50 This trend is similar to that found in ammonium ions surrounded by water. In the ammonium case, it was found from calculations that the closer the OH2, the smaller the onebond isotope effect.63 This leads to the discussion of isotope effects in proteins in which the hydrogen bond donor is a CvONH and the acceptor is often a OvCNH group. Theoretical calculations again showed that a short NH(D)· · ·OvC distance and a directional (linear) approach led to smaller deuterium isotope effects. This was confirmed by experimental data for amino acid residues without side chains and with electronegative or charged atoms (Figure 9.19) (P. E. Hansen, Aa. E. Hansen, A. Liwang, A. Bax and J. Abildgaard, personal communication). For ammonium ions, the 1D15N(D) depended strongly on the nature of the counter ions.64 This is quite opposite to 1D15N(D) of the DMANHþ, which turned out to be rather insensitive to the nature of the counter ions.50

0.74

Aliphatic, Hydrogen bonded Glycine + 0.036 ppm, Hydrogen bonded Glycine + 0.036 ppm, Not Hydrogen bonded

0.70 0.68 0.66

1 15

∆ N(D) (Simulated) / ppm

Aliphatic, Not hydrogen bonded

0.72

0.64 0.62 0.60

0.62

0.64

0.66 0.68 0.70 0.72 ∆ N(D) (Experimental) / ppm

1 15

FIGURE 9.19 Plots of 1DN(D) experimental vs. predicted.

0.74

0.76

266

Isotope Effects in Chemistry and Biology

O

H

N C

R2

O

H

N C

R1

R2

O

molecular

N C

R1

R3

R3

H

R3

O

H

N C

R1

R2 R1

R3

keto

o -quinoid

R2

zwitterionic Proton-Transfer Form

Molecular Form

FIGURE 9.20 Resonance and tautomeric forms of Schiff bases.

2.

5

D15N(D)

Negative isotope effects have been found in nontautomeric Schiff bases on the molecular form (Figure 9.20).49 Although written as 5D15N(D), this is probably transmitted via the hydrogen bond and, hence, is a formal 1D15N(D). The negative sign is useful as it distinguishes this effect from the normal 1D15N(D). Negative effects may of course also be observed in equilibrium situations (see Section VI.A).

F.

n

D 17 O(D)

The 17O chemical shifts cover a very large range and would be, then, very suitable for studies of isotope effects were it not for the usually broad resonances. Small isotope effects in simple compounds like salicylaldehyde cannot be observed, at least not in one-tube experiments. 5 17 D O(OD) isotope effects formally over five bonds correlate well with nD13C(OD) for a series of o-hydroxyacyl aromatics (Figure 9.7 and Figure 9.11)65 and hence with other hydrogen bondrelated parameters19,66 (Section V.K).

G.

n

DYH(XH) (X , Y 5 O OR N )

Y and X are most often oxygen but can also be nitrogen. Isotope effects at one hydrogen bonded OH can, in some situations, be observed due to deuteriation at another position. These cases are divided into two groups. Figure 9.21 illustrates how deuteriation at OH-1 perturbs the chemical shift of OH-8, as both hydrogen bonds are competing for the same acceptor. Cases like this are found in 1,8-dihydroxyanthraquinones,67 2,20 -dihydroxybenzophenone,67 6-methyl-1,3,8-trihydroxyanthraquinone,68 and 8-hydroxy-2-quinolinecarboxyamide N-oxide.51 Figure 9.7 demonstrates how the effect in conjugated systems is transmitted through bonds. The effect at OH-4 is larger the stronger the hydrogen bond is at OH-1 (as given, e.g., by 2D13C(OD)). Such effects may be transmitted over several bonds, as seen in the hypericin anion.68,69 For such

(D) H

H O

FIGURE 9.21 Long-range isotope effects.

O

O

NMR Studies of Isotope Effects

267 O (D) H N N

t-bu

F 11 ppb

FIGURE 9.22 Long-range isotope effects on 19F chemical shifts.

long-range effects, it is important to make sure that one is not dealing with equilibrium isotope effects.14,70 A third kind of effect is found in tetrabutylammonium hydrogen disalicylate in CDF3 – CDF2Cl at 100– 120 K, in which the phenolic OH proton shows a negative isotope shift due to deuteriation of the central proton.71

H.

n

D 19 F(XD)

This type of isotope effect could seem strange in relation to hydrogen bonding. However, the fluorine chemical shift range is very large and fluorine chemical shifts are very sensitive to changes in electric fields or to structural changes. Isotope effects on fluorine chemical shifts have been monitored in a large series of compounds. Some rather long-range effects over formally ten bonds have been observed as seen in Figure 9.22.72 It is most likely that the effects are transmitted via the hydrogen bond (see also Section V.A.1).

I.

n

D 13 C( 18 O)

Oxygen chemical shifts on 13C chemical shifts are relatively small due to the small change in mass from 16 to 18. They have been shown to relate to bond order for simple compounds.18 For both double bonded and single bonded oxygens, they have been investigated in detail.73 For singlebonded cases, no clear correlations have been found. For double-bonded oxygens 1D13C(18O), isotope effects have been found to correlate both to the d13C74 in general, but not when looking at smaller groups of compounds75 and 17O chemical shifts.67 For intramolecularly hydrogen bonded cases 1D13C(18O) for CvO, groups clearly decrease with increasing hydrogen bond strength.67,76 This is in line with the RAHB scheme (Figure 9.3), in which the resonance form B becomes more important for strong hydrogen bonds. At the same time, a 2D13C(18O) isotope effect at the C-1 carbon can be found for compounds with strong hydrogen bonds.52,67 The nD13C(18O) isotope effect is also useful in studies of multiple hydrogen bonding in systems like that shown in Figure 9.21A. As both lone pairs of one carbonyl group are involved in hydrogen bonding simultaneously, the effect is less than twice that observed for single hydrogen bonding.38,65

J.

PRIMARY I SOTOPE E FFECTS

The primary isotope effects, pD1H(D) or pD1H(T) can act as very direct reporters of hydrogen bonding. Experimentally, deuterium resonances can be broad and, as a result, difficult to measure accurately. Primary tritium isotope effects are easy to measure because of the good sensitivity of tritium NMR and the sharp resonances.77 – 79 The main problem at present is to synthesize the compounds.

268

Isotope Effects in Chemistry and Biology

Primary deuterium and tritium isotope effects of intramolecularly hydrogen bonded compounds are found to correlate very well.79 The ratio is found to be very close to 1.4, as predicted theoretically.18 The primary isotope effects on 1H chemical shifts have from rather early on, been related to the shape of the hydrogen bond potential. The rule of thumb is: a very large and positive primary isotope effect is related to strong intramolecular hydrogen bonding, and small positive values to weak hydrogen bonding. In both cases a two-potential well, possibly symmetric, develops, as seen in Figure 9.2(b). Negative values, on the other hand, were ascribed to a single potential well (Figure 9.2(c)).77,78 For a discussion of symmetrical cases, see Chapter 8. For a discussion of equilibrium cases, see Section VI.A. As a consequence of dependence of hydrogen bond strength, a plot of dOH vs. pD1H(D) was found to be nearly linear.79 However, recently, a more extensive set of data and the inclusion of both RAHB and non-RAHB type compounds has made this relationship much less obvious.51 Primary isotope effects are shown to correlate quite well with 2D13C(OD) for RAHB systems.79 For non-RAHB systems, the primary isotope effect is probably a better measure than 2 13 D C(OD). An example is the symmetric DMANHþ (Figure 9.15) in which the 2D13C(OD) is small because of low bond order between H(D) and C-1, whereas pD1H(D) is large,50 reflecting the strong hydrogen bond.

K. SUMMARY FOR I NTRINSIC I SOTOPE E FFECTS 1. RAHB Systems

2.800 2.750 2.700 2.650 2.600 2.550 2.500 2.450 2.400 2.350 2.300

y = 0.5762x 2 − 0.8572x + 2.7237 R2 = 0.9402

0

0.1

0.2

0.3

0.4

0.5

0.6

2∆13C(OD)

0.7

0.8

0.9

1

ppm 0.3

y = −0.6802x + 1.8743 2 R = 0.9232

0.25 0.2 0.15

y = 0.5501x − 1.2974

0.1

2

R = 0.8598

2.650

2.600

2.550

2.500

2.450

2.400

0.05 0 2.350

RO…O in Å

FIGURE 9.23 Plots of 2D13C-2(OD) and 4D13C-2(OD) of nitro o-acylhydroxy aromatics.

4-bond isotope effects

RO...O Å

In summary, one finds from theoretical work as well as experiments80 that RO· · ·O is correlated to ROH, RCvO and 2D13C(OD) and to most of the other isotope effects but not to 4D13CvO(OD) (Figure 9.23).

NMR Studies of Isotope Effects

269

If we look at the isotope effects of 1,3,5-triacetyl-2,4,6-trihydroxybenzene (Figure 9.7) and similar compounds with strong hydrogen bonds,19,40 we find a number of features that will generally be valid to some extent. p 1 D H(D). The primary isotope effect is large and positive (2 0.42 ppm) indicating strong hydrogen bonding.14 n 13 D C(XD). All deuterium isotope effects on 13C chemical shifts can be observed. The twobond isotope effect is positive (0.72 ppm).19 The type is illustrated in Figure 9.4(a). 5 17 D O(OD) is as large as 2 9.0 ppm.65 1 13 D C(18O) is 39 ppb, which corresponds to strong hydrogen bonding. Strangely, no 2 13 D C(18O) isotope effect was observed at C-1, but a 2D13CH3(18O) of 8 ppb was found.65 n DOH(OH). Isotope effects of 55 and 57 ppb are seen.19 2. Non-RAHB Systems 2 13

D (XD) are small.

VI. EQUILIBRIUM ISOTOPE EFFECTS A. GENERAL F INDINGS

40

40

30

30 E, kcal / mol

E, kcal/ mol

For a correct analysis of isotope effects or in an analysis of the biologically active form of a compound or in order to understand reactivity of a compound, etc., it is essential to establish whether a compound is static or tautomeric. Isotope effects on chemical shifts can be very useful in this respect. For intramolecularly hydrogen bonded compounds with proton transfer, isotope substitution may lead to a change in the equilibrium constant due to a change in the zero-point energy of the two forms, provided the potential shapes are different (see Figure 9.24). A large number of equilibrium cases have been investigated by means of isotope substitution. The obvious isotope to use is deuterium, or tritium in rare cases, but also 18O has been used with great success by Perrin et al.13 (Chapter 8). It is advantageous to divide equilibrium cases into those in which the proton transfer leads to a situation indistinguishable from that of the original situation (symmetrical potential well) and asymmetrical cases. In the former category we find symmetrically substituted b-diketones and DMANHþs. In the latter group are the two just mentioned with unlike substituents, as well as Schiff bases,49,81 – 89 Mannich bases,90 – 92 o-hydroxyazo compounds,72,93 b-thioxoketones,9,94 – 96

20

10

10

0

20

H D 0.7

1.2

1.7

2.2

0

R(OH), Å

FIGURE 9.24 Double potential well with zero-point energies.

0.7

1.2

1.7 R(NH), Å

2.2

270

Isotope Effects in Chemistry and Biology

thioamides,43,97,98 acyl Meldrum acids99 and single cases like piroxicam,100 citrinin,52 usnic acid,20, rubazoic acids,11 2-hydroxy-1-nitrosophenol,11,101 1,8-dihydroxy-3,6-dimethyl-2-acetylnaphthalene and gossypol101,102 (Figure 9.25). For symmetrical cases like 6-hydroxy-2-formylfulvene and 6-aminofulvene-2-aldimines, see Chapter 8. This latter compound demonstrates how isotope effects on chemical shifts can be used to judge the presence of proton transfer.65 A characteristic pattern for equilibrium systems is, quite often, large isotope effects on chemical shifts, isotope effects of both signs and at positions far from the site of isotope substitution. The magnitudes of isotope effects can intuitively be seen to relate to the chemical shift difference between the observed nucleus in the two tautomers. A typical example is the b-thioxoketones in which the isotope effect at the CS carbon can be as large as 2 8 ppm.95 For this carbon, the chemical shift difference between a CvS and a CSH carbon is calculated as approximately 60 ppm.95 A simple approach to understand the deuterium isotope effects in equilibrium systems is to divide the effects into two parts, an intrinsic part and an equilibrium part:

34

DCmeasured ¼ DCint þ DCequi

ð9:5Þ

The intrinsic part is, of course, a weighted average of the intrinsic isotope effects for the two tautomers. This is illustrated for a Schiff base in Figure 9.26 for C-2 DCint ¼ XM DCint ðMÞ þ ð1 2 XM ÞDCint ðPTÞ XM being the mole fraction

H O

O

H

O

H

O

O

H

O

(b)

(a)

FIGURE 9.25 1,8-Dihydroxy-3,6-dimethyl-2-acetylnaphthalene.

O

H

O

N

H

N C

C

R

R Molecular Form

FIGURE 9.26 Isotope effect pathways for Schiff bases.

Proton Transfer Form

ð9:6Þ

NMR Studies of Isotope Effects

271 500 400 300

∆C1', ppb

200 100 −1500

−1000

−500

0 −100

0

500

1000

1500

−200 −300 −400 −500 ∆C2, ppb

FIGURE 9.27 Plot of D13C-2(XD) vs. D13C-10 (XD) of Schiff bases.

The equilibrium part is given as DCequi ¼ DXHðDÞ ðdCM 2 dCPT Þ

ð9:7Þ

where (dCM 2 dCPT) is the difference in chemical shifts between the nucleus in the two tautomeric forms; DXH(D) is the change in the equilibrium due to isotope substitution as seen in Figure 9.24. The former can in some cases be estimated by extrapolation or can be calculated (see Section VII). The latter can be calculated from the difference in vibrational frequencies, if these can be observed (see Chapter 8). This is, however, often difficult for strongly hydrogen bonded systems. For systems containing nitrogen, a comparison of 1J(N,H) and 1J(N,D) may be used.50 Another option is to calculate the difference by ab initio methods (Section VII). For symmetrical systems, no change in the equilibrium occurs if the isotope substitution takes place at a position that remains unaltered by proton transfer, typically the XH proton. In such a case, the observed isotope effect is the intrinsic contribution, as given in Equation 9.4. It may also be seen from Equation 9.5 and Equation 9.7 that if the equilibrium contribution dominates, plotting the observed equilibrium isotope effect for two nuclei (e.g., C-3 and C-10 of the Schiff base) should 0 0 result in a straight line with slope: (dC2M 2 dC2PT)/(dC1M 2 dC1PT ). This can be seen from Figure 9.27. Further support for such a simple scheme can be found in the analysis of symmetrical systems that can be made asymmetrical by substitution. One such case is the DMANHþs (Figure 9.15). Substitution at position 4 shifts the equilibrium depending on the substituent. From Equation 9.5 to Equation 9.7 and assuming that the hydrogen bonded system is not widely different for, say 4-picryl, 4-nitro and 4-bromo DMANHþ and DMANHþ itself, we can derive from Equation 9.5 and Equation 9.7 that the sum of isotope effects of nuclei at symmetrical positions in the asymmetrical compounds, like 4-picryl, 4-nitro and 4-bromo DMANHþ (e.g., N-1 and N-8 or C-2 and C-7), will be the same as that observed in DMANHþ itself. This is what is observed experimentally.50 Deuterium isotope effects on 1H and 13C chemical shifts have been studied in carbohydrates and polyols. Two schools on how to interpret this data exist, the static and the equilibrium. In the static way, the observed splittings are seen as intrinsic effects4,59,103 and as possibly long range effects.104 – 106 In the equilibrium school, the effects are ascribed to a flip-flop mechanism

272

Isotope Effects in Chemistry and Biology (D)

(D)

H

H

H

H O

O

O

O

CH

CH

CH

CH C H2

C H2

FIGURE 9.28 Flip-flop mechanism of carbohydrates.

(Figure 9.28), which gives rise to equilibrium isotope effects as originally suggested by Reuben.107 One piece of evidence for equilibrium is the finding that change of solvent may change the sign of the isotope effect.108,109 As a result of the equilibrium, a finding from these studies is that the OD group forms a stronger intermolecular hydrogen bond than the OH group. This is opposite to the RAHB systems, but in line with the predictions of Buckingham and Fan-Chen.110 More recently, this type of effect has been analysed in an elegant ISNOE (isotopomer selected NOE) experiment.111 – 113 This technique can provide distance constraints for structural calculations.113 Lewis and Schramm114 investigated in glucopyranose the effect of deuteriation of the OH groups on the anomeric equilibrium and showed that intramolecular hydrogen bonding played an important role in the isotopic perturbation of the anomeric equilibrium. A study of the isotope effect at the CO carbon chemical shift of b-diketones deuterated at the OH position revealed clearly that the isotope effect depends on the position of the equilibrium in an S-shaped way, as shown in Figure 9.29.95 Such a shape was also found for o-hydroxy Schiff bases as seen in Figure 9.29. This can be understood in the following way. For K ¼ 1; no perturbation of the equilibrium occurs. For K different from 1, the perturbation in energy depends on the difference in shape of the potential wells, as seen in Figure 9.24. The dependence of shape difference is complex and will depend on both the distance in heavy atoms and the difference in enthalpy (DH) for the two forms. Based on the finding that deuteriation leads to more of the stable form, Bordner et al.100 set up n∆C-2(D)

(ppb)

600 400 200 0 −200 −400 −600

0.0

0.2

0.4

FIGURE 9.29 Plot of 2DC(OD) vs. mole fraction.

0.6

0.8 1.0 mole fraction χ

NMR Studies of Isotope Effects

273

−0.20/−0.30 HC(H3C)

−0.34/−0.75 HC(H3C)2 O H(D)

/−0.02

O H(D) O

O 1.68/2.30

−0.10/−0.09 0.05/0.02

0.28/0.41 0.03/0.04

FIGURE 9.30 Equilibrium isotope effect of cyclic b-diketone at 300 and 220 K.37

an equation showing that the perturbation by deuteriation in a system like peroxicam or b-diketones is proportional to DG. This is not strictly true, but is a great simplification from the standpoint of analyzing the resulting expression. Making the above mentioned approximation, Bordner et al.100 finds an S-shaped relationship between the change in the mole fraction and equilibrium constant. S-shaped curves have also been observed for Schiff bases81,87,88 and b-thioxoketones.95 For compounds like the one seen in Figure 9.30, a dramatic temperature effect is found on the isotope effects. Temperature effects are also seen for intrinsic isotope effects, as judged from symmetrical b-diketones. For acetyl acetone, a change from 0.745 to 0.869 ppm is found with a change in temperature from 300 to 220 K in CD2Cl2.37 For the cyclic compound, the change is much larger. This is caused by the change in the equilibrium constant as the temperature is changed. This equilibrium isotope effect may lead to large effects as seen in Figure 9.30. For a group of similar compounds in which the chemical shift differences between a carbon in the two tautomers varies between compounds, the height of the maximum or minimum may vary as shown (e.g., in Schiff bases (Figure 9.29) and b-thioxoketones).85,95 For the Schiff bases, the value of the mole fraction for which the maximum or minimum is observed is also seen to vary somewhat.85 This is likely a consequence of plotting vs. the mole fraction or the equilibrium constant and is probably caused by differences in entropy for the different compounds, but this should be investigated further.

B. LONG-R ANGE E FFECTS IN E QUILIBRIUM S YSTEMS These can be of two kinds: One type is deuteriation of the transferred proton and observation of nuclei far from the site of deuteriation. The other type is isotope substitution far from the hydrogen bond. The latter type is illustrated in b-thioxoketones (Figure 9.31) in which the methyl group is deuterated in thioacetylacetone leading to a considerable effect at the XH proton. This effect is akin to that seen by deuteriation of one of the methyl groups of acetylacetone.115 In both cases, the CH3 groups next to the CvO or CvS group are in conjugation with a double bond and hence have a H S

O

C

C R

O

CH

R

S C

C R1

R = R1 = CH3 or PhenylX

FIGURE 9.31 b-Thioxoketones.

H

CH

R1

274

Isotope Effects in Chemistry and Biology

different vibrational frequency compared with methyl groups next to COH or CSH groups. This difference in vibrational frequency leads to a shift in the equilibrium upon deuteration. The XH proton is very suitable as a monitor because of the large chemical shift difference between the OH and the SH proton. For diphenylthioacetylacetone, deuteration at the phenyl ring closest to the oxygen again leads to equilibrium isotope effects whereas deuteriation at the phenyl ring closest to sulphur has no effect. Site-specific substitution of the phenyl ring close to oxygen showed that deuteration at the ortho position is larger than at the meta position which again gives an effect larger than at the para position.96 This pattern is typical for isotope effects due to deuteration on aromatic rings in general,116,117 but not for substitution in general. Observation of isotope effects far from the site of deuteration is a trademark of the equilibrium system and is, as such, the usual finding. As demonstrated for intrinsic isotope effects (Section V.F), 19 F is a very sensitive nuclei. Long-range effects can be monitored in equilibrium systems more advantageously if fluorine is present. An example is a para fluoro substituted b-thioxoketone (Figure 9.31, X ¼ p 2 F).72 Other examples are o-hydroxyazo compounds.72

C. IDENTIFYING E QUILIBRIUM S YSTEMS As the isotope effects measured in equilibrium system are a combination of intrinsic and equilibrium contributions, it is, of course, important to have a good description of the intrinsic part. For equilibrium systems like Schiff bases, this is the case for the molecular form (Figure 9.26) but only recently have data for the PT form become available.118 As the equilibrium part depends on a chemical shift difference, nuclei with large chemical shift ranges are clearly advantageous in the characterization of equilibrium systems. Isotope effects of 13C, 15N, 17O and, as already mentioned, 19 F chemical shifts are very useful; but primary isotope effects may also be included. Furthermore, as the intrinsic effects depend linearly on heavy atom distances, OH chemical shifts, and other parameters related to hydrogen bond strength, isotope effects from equilibrium systems will normally fall outside the correlation lines in plots of primary isotope effects vs. 2D13C-2(XD) þ 4 13 D C-2(XD).65 This is demonstrated very clearly for b-thioxoketones in Figure 9.32.

2∆C(XD)

+ 4∆C(XD)/ppm

2 1 0 49

−1 −2

53

−3 −4 −5 −1.0

43

49

54

52 54

−0.5

0.0

0.5

1.0

P∆(1H, 3H)/ppm

FIGURE 9.32 Plot of primary isotope effects vs. the sum of 2D13C-2(XD) þ 4D13C-2(XD). Static compounds are marked with squares; tautomeric compounds with crosses. 49 refers to a thioindandione,97 52 – 54 to b-thioxoketones. From Ref. 62 with permission from MRC.

NMR Studies of Isotope Effects

275

In equilibrium systems, the primary isotope effect also depends on the chemical shift difference between the protons in the two tautomers. Hence, both positive and negative primary isotope effects can be found. Negative ones are seen in b-thioxoketones.79

VII. CALCULATIONS Nuclear shieldings can be calculated very well at the present time using ab initio programs.119 For a more in-depth description see Chapter 8. This enables good estimates of shielding differences of nuclei in different tautomers, a parameter used to estimate the equilibrium contribution to isotope effects. This line of reasoning has been used to predict the equilibrium contribution to the isotope effects of DMNAHþs.50 Following Equation 9.4, the quantities needed for predicting the intrinsic isotope effects are the change in the XH bond length upon deuteriation and ds/dRXH. The latter quantity can be obtained from a plot of calculated nuclear shielding as a function of XH bond length. For the equilibrium XH distance, the nuclear shielding surface is linerar in most cases investigated. An example is seen in Figure 9.1. The change in the bond length can be obtained from the potential constructed by calculating geometries with fixed XH distances and either relaxing the parameters directly related to the hydrogen bond ring or relaxing the entire molecule. It should be remembered that in this treatment only the first term of the series expansion is retained (see Section III). The use of a one-dimensional potential for estimating the change in the bond length upon deuteriation is also very approximative. More sophisticated treatments using two-dimensional potentials are under way (J. Stare, personal communication.) From Equation 9.4, which defines intrinsic isotope effects, it is also seen that the ratio between isotope effects is equal to the ratio between the change in nuclear shielding. This was shown to be a good approximation for 13C chemical shifts of aromatic systems.19 For different nuclei within the same molecule, one obtains 5

D17 OðODÞ : 2 D13 C 2 2ðODÞ ¼ ds17 O=dROH : ds13 C 2 2=dROH

For 1,3,5-triacetyl-2,4,6-trihydroxybenzene (Figure 9.7), the calculated ratio of slopes is 17.6, whereas the experimentally determined ratio between isotope effects is 12.5.65

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32 Dziembowska, T., Rozwadowski, Z., Majewski, E., and Ambroziak, K., Deutrium isotope effects on 13 C NMR chemical shifts of some a-(2-hydroxyaryl)-N-phenylnitrones (Schiff base N-oxides), Magn. Reson. Chem., 39, 484– 488, 2001. 33 Duus, F. and Hansen, P. E., Determination of tautomeric phenotypes of b-thioxoester and characterization of the tautomeric enethiolic constituents by means of 13C NMR spectroscopy, Org. Magn. Reson., 22, 16 – 23, 1984. 34 Hansen, P. E., Deuterium isotope effects on the 13C nuclear shielding of intramolecularly hydrogenbonded systems, Magn. Reson. Chem., 24, 903– 910, 1986. 35 Hansen, P. E., Bolvig, S., and Kappe, T., Intramolecular hydrogen bonding and tautomerism of Acylpyran-2,4-diones, 2,4,6-triones and acylpyridinediones and benzannelated derivatives, deuterium isotope effects on 13C NMR chemical shifts, J. Chem. Soc. Perkin Trans., 2, 1901– 1907, 1995. 36 Liepins, E., Petrova, M. V., Gudriniece, E., Paulins, J., and Kuznetsov, S. L., Relationship between 1H, 13 C and 17O NMR chemical shifts and H/D isotope effects on 13C and 17O nuclear shieldings in intramolecular hydrogen-bonded systems, Magn. Reson. Chem., 27, 907– 917, 1989. 37 Hansen, P. E. and Bolvig, S., Deuterium-induced isotope effects on 13C chemical shifts as a probe for tautomerism in enolic b-diketones, Magn. Reson. Chem., 34, 467–478, 1996. 38 Hansen, P. E., Ibsen, S. N., Kristensen, T., and Bolvig, S., Deuterium and 18O isotope effects on 13C chemical shifts of sterically hindered and/or intramolecularly hydrogen bonded o-Hydroxy Acyl aromatics, Magn. Reson. Chem., 32, 399– 408, 1994. 39 Hansen, P. E., Intrinsic deuterium isotope effects on NMR chemical shifts of hydrogen bonded systems, Nukleonika, 47S, S37– S42, 2002. 40 Wozniak, K., Bolvig, S., and Hansen, P.E., Steric compression and twist in o-hydroxy acyl aromatics with intramolecular hydrogen bonding, J. Mol. Struct., Preview. 41 Hansen, P. E., Does an OH or an OD group form the stronger hydrogen bond to oxygen, In Interactions of Water in Ionic and Nonionic Hydrates, Kleeber, H., Ed., Springer Verlag, Berlin, pp. 287– 289, 1987. 42 Zheglova, D. K. h., Genow, D. G., Bolvig, S., and Hansen, P. E., Deuterium isotope effects on 13C chemical shifts of enaminones, Acta Chem. Scand., 51, 1016– 1023, 1997. 43 Hansen, P. E., Hydrogen bonding and tautomerism studied by isotope effects on chemical shifts, J. Mol. Struct., 321, 79 – 87, 1994. 44 Hansen, P. E., Bolvig, S., Duus, F., Petrova, M. V., Kawecki, R., Krajewski, S., and Kozerski, L., Deuterium isotope effects on 13C chemical shifts of intramolecularly hydrogen-bonded olefins, Magn. Reson. Chem., 33, 621– 631, 1995. 45 West-Nielsen, M., Dominak, P., Wozniak, K., and Hansen, P.E., Strong intramolecular hydrogen bonding involving nitro- and acetyl groups, deuterium isotope effects on chemical shifts, Eur. J. Org. Chem., submitted for publication. 46 Hansen, P. E., Long-range deuterium isotope effects on 13C chemical shifts of intramolecularly hydrogen-bonded compounds, purpurogallins, Magn. Reson. Chem., 31, 71 – 74, 1993. 47 Reuben, J., Intramolecular hydrogen bonding as reflected in the deuterium isotope effects on carbon-13 chemical shifts. Correlation with hydrogen bond energies, J. Am. Chem. Soc., 108, 1735– 1738, 1986. 48 Kozerski, L., Kamienski, B., Kawecki, R., Urbancy-Lipowska, Z., Bocian, W., Bednarek, E., Sitkowski, J., Zakrewska, K., Nielsen, K. T., and Hansen, P. E., Solution and solid state 13C NMR and X-ray studies of genistein complexes with amines, potential biological function of the C-7, C-5 and C-4’ OH groups, Org. Biomol. Chem., 1, 3578– 3585, 2003. 49 Hansen, P. E., Sitkowski, J., Stefanik, L., Rozwadowski, Z., and Dziembowska, T., One-bond deuterium isotope effects on 15N chemical shifts in schiff bases, Ber Bunsengesell Phys. Chem., 102, 410– 413, 1998. 50 Grech, E., Klimkiewicz, J., Nowicka-Scheibe, J., Pietrzak, M., Schilf, W., Pozharski, A. F., Ozeryanskii, V. A., Bolvig, S., Abildgaard, J., and Hansen, P. E., Deuterium isotope effects on 15N, 13C and 1H chemical shifts of proton sponges, J. Mol. Struct., 615, 121– 140, 2002. 51 Dziembowska, T., Rozwadowski, Z., and Hansen, P. E., Intramolecular hydrogen bonding in 8-quinolinol N-oxides, quinaldinic acid N-oxides and quinoline-2-carboxyamide N-oxide, deuterium isotope effects on 13C chemical shifts, J. Mol. Struct., 436– 37, 189– 199, 1997. 52 Hansen, P. E., Langga˚rd, M., and Bolvig, S., Isotope effects on chemical shifts in tautomeric systems with double proton transfer, citrinin, Pol. J. Chem., 72, 269– 276, 1998.

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53 Lycka, A. and Hansen, P. E., Deuterium isotope effects on 14N and 15N nuclear shielding in simple nitrogen containing compounds, Magn. Reson. Chem., 23, 973– 976, 1985. 54 Hansen, P. E. and Tu¨chsen, E., Deuterium isotope effects on carbonyl carbon chemical shifts of BPTI. Hydrogen bonding and structure determination in proteins, Acta Chem. Scand., 43, 710– 712, 1989. 55 Tu¨chsen, E. and Hansen, P. E., Hydrogen bonding monitored by deuterium isotope effects on carbonyl 13 C chemical shifts in BPTI: intra residue hydrogen bonds in antiparallel b-sheet, Int. J. Biol. Macromol., 13, 2 – 7, 1991. 56 Hansen, P. E., Kolonicny, A., and Lycka, A., Deuterium isotope effects on 13C nuclear shielding of amino and acetamido compounds, tautomerism and intramolecular hydrogen bonding, Magn. Reson. Chem., 30, 786– 795, 1992. 57 Hansen, P. E., Christoffersen, M., and Bolvig, S., Variable-temperature NMR studies of 2,6-dihydroxy acylaromatic compounds, deuterium isotope effects on chemical shifts, isotopic perturbation of equilibrium and barriers to rotation, Magn. Reson. Chem., 31, 893– 902, 1993. 58 Sandstro¨m, C., Baumann, H., and Kenne, L., NMR spectroscopy of hydroxy protons of 3,4-disubstituted methyl a-D -galactopyranoside in aqueous solution, J. Chem. Soc. Perkin Trans., 2, 809– 815, 1998. 59 Christofides, J. D. and Davies, D. B., Simple 1H NMR observation of cooperative hydrogen bonding in o-3’-sucrose derivatives, Magn. Reson. Chem., 23, 582– 585, 1985. 60 Everett, J. R., Hunt, E., and Tyler, J. W., Ketone-hemiacetal tautomerism in erythromycin A in nonaqueous solutions, an NMR spectroscopy study, J. Chem. Soc. Perkin. Trans., 2, 1481– 1487, 1991. 61 Bosco, M., Picoti, F., Radoicovich, A., and Rizzo, R., Hydroxyl group interactions in polysaccharides: a deuterium-induced differential isotopic shift 13C investigation, Biopolymers, 53, 272– 280, 2000. 62 Gutowsky, H. S., Isotope effects in high resolution NMR spectroscopy, J. Chem. Phys., 31, 1683– 1685, 1959. 63 Munch, M., Hansen, Aa. E., Hansen, P. E., and Bouman, T. D., Ab initio calculations of deuterium isotope effects on hydrogen and nitrogen nuclear magnetic shielding in the hydrated ammonium ion, Acta Chem. Scand., 46, 1065–1071, 1992. 64 Hansen, P. E. and Lycka, A., Reinvestigation of one-bond deuterium isotope effects on nitrogen and on proton nuclear shielding for the ammonium ion, Acta Chem. Scand., 43, 222– 232, 1989. 65 Bolvig, S., Hansen, P. E., Wemmer, D., and Williams, P., Deuterium isotope effects on 17O chemical shifts of intramolecularly hydrogen bonded systems, J. Mol. Struct., 509, 171– 181, 1999. 66 Kozerski, L., Kawecki, R., Krajewski, P., Kwicien, B., Boykin, D. W., Bolvig, S., and Hansen, P. E., 17 O chemical shifts and deuterium isotope effects on 13C chemical shifts of intramolecularly hydrogenbonded compounds, Magn. Reson. Chem., 36, 921– 928, 1998. 67 Hansen, P. E., Ibsen, S. N., Kristensen, T., and Bolvig, S., Deuterium and 18O isotope effects on 13C chemical shifts of sterically hindered and/or intramolecularly hydrogen-bonded o-hydroxy acyl aromatics, Magn. Reson. Chem., 32, 399– 408, 1994. 68 Etzlstorfer, C., Falk, H., Mayr, E., and Schwarzinger, S., Concerning the acidity and hydrogen bonding of hydroxypheanthroline quinones like fringelite D, hypericin and stentorin, Monatsh. Chem., 127, 1229– 1237, 1996. 69 Mazzini, S., Merlini, L., Mondelli, R., Nasini, G., Ragg, E., and Scaglioni, L., Deuterium isotope effect on 1H and 13C chemical shifts of intramolecularly hydrogen bonded perylenequinones, J. Chem. Soc. Perkin Trans., 2, 2013–2021, 1997. 70 Arnone, S., Merlini, L., Mondelli, R., Naini, G., Ragg, E., Scaglioni, L., and Weiss, U., NMR study of tautomerism in natural perylenequinones, J. Chem. Soc. Perkin Trans., 2, 1447– 1454, 1993. 71 Golubev, N. S., Smirnow, S. N., Schah-Mohameddi, P., Shenderovich, I. G., Denisov, G. S., Gindin, V. A., and Limbach, H. H., Study of acid – base interaction by means of low-temperature NMR spectra, structure of salicylic acid complexes, Russ. J. Gen. Chem., 67, 1082– 1087, 1997. 72 Hansen, P. E., Bolvig, S., Buvari-Barsza, A., and Lycka, A., Long-range intrinsic and equilibrium deuterium isotope effects on 19F chemical shifts, Acta Chem. Scand., 51, 881– 888, 1997. 73 Risley, J. M. and van Etten, R. L., Properties and chemical applications of 18O isotope shifts in 13C and 15 N nuclear magnetic resonance spectroscopy, NMR Basic Princ. Prog., 22, 81 – 168, 1990. 74 Everett, J. R., Oxygen-18 isotope effects on 13C NMR signals, Org. Magn. Reson., 2, 86 – 88, 1982. 75 Hansen, P. E., Nuclear magnetic resonance spectroscopy of CvC, CvO, CvN and NvN double bonds, In The Chemistry of Double-bonded Functional groups, Patai, S., Ed., Wiley, New York, 1989.

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76 Hansen, P. E., Nuclear Magnetic Resonance (NMR) of Acids and Acid Derivatives in Acid, Patai, S., Ed., Wiley, New York, 1992. 77 Altman, L. J., Laungani, D., Gunnarson, G., Wennersto¨m, H., and Forse´n, S., Proton, deuterium and tritium nuclear magnetic resonance of intramolecular hydrogen bonds, isotope effects and the shape of the potential energy function, J. Am. Chem. Soc., 100, 8264– 8266, 1978. 78 Gunnarson, G., Wennersto¨m, H., Egan, W., and Forse´n, S., Proton and deuterium NMR of hydrogen bonds: relationship between isotope effects and the hydrogen bond potential, Chem. Phys. Lett., 38, 96 – 99, 1976. 79 Bolvig, S., Hansen, P. E., Morimoto, H., Wemmer, D., and Williams, P., Primary tritium and deuterium isotope effects on chemical shifts of compounds having an intramolecular hydrogen bond, Magn. Reson. Chem., 38, 525– 535, 2000. 80 Steiner, T., C-H· · ·O hydrogen bonding in crystals, Crystallogr. Rev., 6, 1 – 57, 1996. 81 Rozwadowski, Z., Majewski, E., Dziembowska, T., and Hansen, P. E., Deuterium isotope effects on 13 C chemical shifts of intramolecularly hydrogen-bonded Schiff bases, J. Chem. Soc. Perkin Trans., 2, 2809– 2817, 1999. 82 Katritzky, A. R., Ghiviriga, I., Leeming, P., and Soti, F., Hydrogen bonding and tautomerism in anil of salicylaldehydes and related compounds, a study of deuterium isotope effects on 13C chemical shifts, Magn. Reson. Chem., 34, 518– 526, 1996. 83 Katritzky, A. R., Ghiviriga, I., Oniciu, D. C., O’Ferrall, R. A. M., and Walsh, S. M., Study of the enol– enamino tautomerism of a-heterocyclic ketones by deuterium effects on 13C chemical shifts, J. Chem. Soc. Perkin Trans., 2, 2605– 2608, 1997. 84 Katrizky, A. R. and Ghivriga, I., An NMR study of the tautomerism of 2-acylamonopyridines, J. Chem. Soc. Perkin Trans., 2, 1651– 1653, 1995. 85 Dziembowska, T., Rozwadowski, Z., Filarowski, A., and Hansen, P. E., NMR study of proton transfer equilibrium in Schiff bases derived for 2-hydroxy-1-naphthaldehyde and 1-hydroxy-2acetophenone, deuterium isotope effects on 13C and 15N chemical shifts, Magn. Reson. Chem., 39, S67– S80, 2001. 86 Rozwadowski, Z., Ambroziak, K., Dziembowska, T., and Kotfica, M., Interaction between the hydrogen bonds in unsymmetrical di-Schiff bases studied by means of deuterium isotope effects on C-13 Chemical shifts, J. Mol. Struct., 643, 93 –100, 2002. 87 Dziembowska, T., Ambroziak, K., Rozwadowski, Z., Schilff, W., and Kamienski, B., Deuterium isotope effects on 15N chemical shifts of double Schiff bases in the solid state and solution, Magn. Reson. Chem., 41, 135– 138, 2003. 88 Rozwadowksi, Z. and Dziembowska, T., Proton transfer equilibrium in Schiff bases derived from 5-nitrosalicylaldehyde, a study of deuterium isotope effects on 13C NMR chemical shifts, Magn. Reson. Chem., 37, 274– 278, 1999. 89 Ambroziak, K., Rozwadowski, Z., Dziembowska, T., and Bieg, B., Synthesis and spectroscopic study of Schiff bases derived from trans-1,2-diaminocyclohexane, deuterium isotope effects on 13C chemical shift, J. Mol. Struct., 615, 109– 120, 2002. 90 Rospenk, M., Koll, A., and Sobczyk, L., Proton transfer and secondary deuterium isotope effect in 13C NMR spectra of ortho-aminomethylphenols, Chem. Phys. Lett., 261, 283– 290, 1996. 91 Rospenk, M., Koll, A., and Sobczyk, L., Deuterium isotope effect on 13C NMR spectra of ortho mannich bases, J. Mol. Liq., 67, 63 – 69, 1995. 92 Sobczyk, L., NMR studies on hydrogen bonding and proton transfer in mannich bases, Appl. Magn. Reson, 18, 47 – 61, 2000. 93 Lycka, A. and Hansen, P. E., Deuterium isotope effects on 13C and 15N nuclear shielding in o-hydroxyazo dyes, Magn. Reson. Chem., 22, 569– 572, 1984. 94 Hansen, P. E., Duus, F., and Schmitt, P., Deuterium isotope effects on 13C nuclear shielding as a measure of tautomeric equilibria, Org. Magn. Reson., 18, 58 –61, 1982. 95 Andresen, B., Duus, F., Bolvig, S., and Hansen, P. E., Variable temperature 1H and 13C NMR spectroscopic investigation of the enol – enethiol tautomerism of b-thioxoketones. Isotope effects due to deuteron chelation, J. Mol. Struct., 552, 45 – 63, 2000. 96 Hansen, P. E., Skibsted, U., and Duus, F., Long-range deuterium isotope effects in tauomeric b-thioxoketones. A 1H and 13C NMR study, J. Phys. Org. Chem., 4, 225– 232, 1991.

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97 Hansen, P. E., Duus, F., Neumann, R., Wesolowska, A., Sosnicki, J. G., and Jagodzinski, T. S., Deuterium isotope effects of o-hydroxythioamides. 2DThiazolines and 5-Acyl-2-thiobarbituric Acids, Pol. J. Chem., 74, 409– 420, 2000. 98 Wesolowska, A., Jagodzinski, T. S., Sosnicki, J. G., and Hansen, P. E., Synthesis of the N-allylthioamide derivatives of cyclic Oxo- and Dioxo-acids and their cyclization to the derivatives of 4,5-dihydrothiazole, Pol. J Chem., 75, 387– 400, 2001. 99 Bolvig, S., Duus, F., and Hansen, P. E., Tautomerism of enolic triacetylmethane, 2-acyl1,3-cyclooalkandiones, 5-acyl meldrum’s acids and 5-acyl-1,3-demethylbarbituric acids studied by means of deuterium isotope effects on 13C chemical shifts, Magn. Reson. Chem., 36, 315– 324, 1998. 100 Bordner, J., Hammen, D. P., and Whipple, E. B., Deuterium isotope effects on carbon-13 NMR shifts and the tautomeric equilibrium in N-substituted pyridyl derivatives of piroxicam, J. Am. Chem. Soc., 111, 6572– 6578, 1989. 101 Hansen, P. E. and Bolvig, S., Deuterium isotope effects on 13C chemical shifts of o-hydroxyacyl aromatics, intramolecular hydrogen bonding, Magn. Reson. Chem., 35, 520– 528, 1997. 102 O’Brien, H. and Stipanovic, R. D., Carbon-13 magentic resonance of cotton terpenoids: carbon– proton long-range couplings, J. Org. Chem., 43, 1105– 1111, 1978. 103 Angyal, S. J. and Christofides, J. C., Intramolecualr hydrogen bonding in monosaccharides in dimethyl sulfoxide solution, J. Chem. Soc. Perkin Trans., 2, 1485– 1491, 1996. 104 Bock, K. and Lemieux, R. U., The conformational properties of sucrose in aqueous solution: intramolecular hydrogen-bonding, Carbohydr. Res., 100, 63 – 74, 1982. 105 Everett, J. R., Novel long-range 1H and 13C NMR, isotope effects transmitted via hydrogen bonds in a macrolide antibiotic: bafilomycin A1, J. Chem. Soc. Chem. Commun., 1878–1880, 1987. 106 Everett, J. R., Baker, G. H., and Dorgan, R. J. J., Nov long-range isotope effects in a macrolid antibiotic: bafilomycin A1, J. Chem. Soc. Perkin Trans., 2, 717–724, 1990. 107 Reuben, J., Fingerprints of molecular structure and hydrogen bonding effects in the 13C NMR spectra of monosaccharides partially deuterated hydroxyls, J. Am. Chem. Soc., 106, 6180– 6186, 1984. 108 Craig, B. N., Janssen, M. U., Wicerksham, B. M., Rabb, D. M., Chang, P. S., and O’Leary, D. J., Isotopic perturbation of intramolecular hydrogen bonds in rigid 1,3-diols: NMR studies reveal unusual large equilibrium isotope effects, J. Org. Chem., 61, 9610– 9613, 1996. 109 Vasquez, E. T. Jr., Bergset, J. M., Fierman, M. B., Nelson, A., Roth, J., Khan, S. I., and O’Leary, D. J., Using equilibrium isotope effects to detect intramolecular OH/OH hydrogen bonds: structural and solvent effects, J. Am. Chem. Soc., 124, 2931– 2938, 2002. 110 Buckingham, A. D. and Fan-Chen, L., Differences in the hydrogen and deuterium bonds, Int. Rev. Phys. Chem., 1, 253– 269, 1981. 111 Dabrowski, J., Kozar, T., Grosskurth, H., and Ninfant’ev, N. E., Conformational mobility of oligosaccharides; experimental evidence for the existence of an “Anti” conformer of the Galb1-3Glc b3-OMe disaccharide, J. Am. Chem. Soc., 117, 5334– 5339, 1995. 112 Kozar, T., Ninfant’ev, N. E., Grosskurth, H., Dabrowski, U., and Dabrowski, J., Conformational changes due to vicinal glycosylation: The branched a-L Rhap(1-2)[b-D-Galp(1-3)]-b-D-Glc1-OMe trisaccharides compared with its parent disaccharides, Biopolymers, 46, 417– 432, 1998. 113 Dabrowski, J., Grosskurth, H., Baust, C., and Ninfant’ev, N. E., Secondary H/D isotope effect on hydrogen-bonded hydroxyl groups as a tool for recognizing distance constraints in conformation analysis of oligosaccharides, J. Biomol. NMR, 12, 161– 172, 1998. 114 Lewis, B. E. and Schramm, V. L., J. Am. Chem. Soc., 123, 61327– 61336, 2001. 115 Saunders, M. and Handler, A., Private communication in Siehl, HU, Adv. Phys. Org. Chem., 23, 63 – 163, 1987. 116 Gardner Swain, C., Sheats, J. E., Gorenstein, D. G., and Harbison, K. G., Aromatic hydrogen isotope effects in reactions of benzenediazonium salts, J. Am. Chem. Soc., 97, 791– 795, 1975. 117 Streitwieser, A. Jr and Klein, H. S., Secondary isotope effects in solvolysis of various deuterated benzhydryl chlorides, J. Am. Chem. Soc., 86, 5170– 5173, 1964. 118 Ton, Q., Nguyen, K.P.P., and Hansen, P.E., Schiff Bases of Gossypol, A NMR and DFT Study. Magn. Reson. Chem., 43, 302–308, 2005. 119 Bu¨hl, M., Kaupp, M., Malkina, O. L., and Malkin, V. G., The DFT route to NMR chemical shifts, J. Comp. Chem., 20, 91– 105, 1999.

10

Vibrational Isotope Effects in Hydrogen Bonds Zofia Mielke and Lucjan Sobczyk

CONTENTS I. Introduction ...................................................................................................................... 281 A. Classical and Quantum Mechanical Calculations of Force Field and Vibrational Spectra in the Harmonic Approximation ...................................... 282 B. The Effect of Deuterium Substitution on the Vibrations Involving Hydrogen Motion ..................................................................................................... 284 C. The Isotopic Substitution, the Potential Energy Distribution, and the Frequency Isotopic Ratio (ISR).................................................................. 284 II. Sources of Anomalous H/D Isotope Effects in Hydrogen-Bonded Systems .................. 285 III. The Hydrogen Bond Effect on Anharmonicity of Protonic Vibrations.......................... 287 IV. Potential Energy Functions for the Proton-Stretching Vibrations .................................. 290 V. The Shape of the Potential and Evolution of IR Spectra of Hydrogen-Bonded Systems ......................................................................................... 292 VI. Frequency Isotopic Ratio (ISR) and Its Correlation with Other Parameters of Hydrogen Bonds .......................................................................................................... 294 VII. The Isotope Effect upon Other Spectroscopic Parameters of Hydrogen-Bonded Systems ......................................................................................... 296 VIII. Low-Barrier Hydrogen Bonds ......................................................................................... 298 References..................................................................................................................................... 301

I. INTRODUCTION Vibrational spectroscopy is a sensitive probe of the potential energy surface of the molecule which determines the dynamics of its nuclear motion. In theoretical treatment of vibrational spectra, the Born-Oppenheimer approximation is adopted — which assumes that an effective potential for the nuclear motion of the molecule is determined by its total electronic energy parametrized by the stationary nuclear coordinates. The Born-Oppenheimer approximation is valid only in the vicinity of a local minimum of this effective potential. The analysis of the vibrational spectra allows prediction of the structural and thermodynamical properties of the matter. Obtaining reliable information on the properties of the molecules often requires the use of isotopic data for the molecule under study. By far the most important example in vibrational spectroscopy is the substitution of hydrogen (H) with deuterium (D), although the heavy atom substitution has also been very successfully employed in many studies. In this section, we present the theoretical basis for the effect of isotopic H/D substitution on the infrared spectra of the regular systems in which the hydrogen bonding does not occur. In the whole chapter we limit our considerations to infrared spectra only as infrared spectroscopy is the most powerful tool in hydrogen bond research. 281

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A. CLASSICAL AND Q UANTUM M ECHANICAL C ALCULATIONS OF F ORCE F IELD AND V IBRATIONAL S PECTRA IN THE H ARMONIC A PPROXIMATION In a classical treatment of molecular vibrations,1 the relation between the potential energy of the molecule and its vibrational frequencies is obtained by solving the Lagrange vibrational equations. The variations of molecular configuration with vibrational motion are best described by a set of the internal coordinates R. The potential energy of the system in harmonic approximation is a quadratic function of coordinates and is described by the expression: 2V ¼

n X ij¼1

Fij Ri Rj

ð10:1Þ

where n is the number of internal coordinates. The coefficients Fij are the harmonic force constants and, as known, represent the second derivatives of V with respect to the vibrational coordinates in the equilibrium configuration: Fij ¼ ðd2 V=dRi dRj Þ

ð10:2Þ

Application of firmly established concepts of normal coordinates Q in solving the classical vibrational motion problem leads to vibrational secular equation1,2,3: GFL ¼ LL

ð10:3Þ

In the above equation, G is the symmetric matrix of the vibrational kinetic energy determined by the molecular geometry and atomic masses, F is the symmetric matrix of potential energy whose elements are harmonic force constants, L is a diagonal matrix of the positive eigenvalues lk of the matrix product GF, where lk ¼ 4p 2 c2 n 2k are frequency parameters. L is the eigenvector matrix of the GF product; the columns of the eigenvectors of L represent the transformation between the internal coordinates Rj and the normal coordinates Qk : R ¼ LQ

ð10:4Þ

The problem of evaluation of the matrix F elements, i.e., the harmonic force constants from the vibrational eigenvalues, and a given kinematic matrix G, is known as the inverse vibrational problem. The other essential result of solving the secular equation, in addition to matrix F, is matrix L. The coefficients Ljk quantify the relative contribution of each internal coordinate to the respective normal vibration in the molecule, i.e., describe the form of the normal vibrations. The potential energy distribution (PED), obtained from the L matrix, provides a convenient means for representing the contributions to the potential energy of the system from the internal coordinates. The commonly used definition for the PED contribution of internal coordinate j to the normal vibration k is described by expression4: cjk ¼ ðFjj L2jk =SFii L2ik Þ 100%

ð10:5Þ

The PED is particularly useful in understanding the origins of “group frequency” modes and in clarifying coupled vibrational modes of the same symmetry. The vibrational problem may also be solved in terms of 3N Cartesian displacement coordinates and symmetry coordinates that take advantage of the molecular symmetry in simplifying the secular determinant.1,2,3,4 The matrix L also plays an important role in interpretation of band intensities. The integrated absorption coefficient Ak of the infrared band corresponding to the fundamental mode Qk can be

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283

expressed as5,6: ~ Qk P ~ Qk Ak ¼ ðNa p=3c2 ÞP

ð10:6Þ

Where Na is the Avogadro number, c is the speed of light in vacuo and PQk are the derivatives of the molecular dipole moment with respect to the normal coordinate Qk : The above expression has been derived under conditions of mechanical harmonicity (the potential energy is a quadratic function, [See Equation 10.1]) and electrical harmonicity: the dipole moment P0 is a linear function of vibrational coordinates: P0 ¼

X k

PQk Qk

ð10:7Þ

The PQk values themselves do not elucidate particularly well the nature of intramolecular charge distribution. They are strictly related to the specific molecule, as they are strongly affected by intramolecular coupling and by the masses. The derivatives of the dipole moment with respect to internal coordinates PRj are another way of expressing Ak or PQk : PR are related to PQ by expression: PR ¼ PQ L21

ð10:8Þ

The PRj values have an apparent physical meaning, and have an advantage over PQk parameters in being free from dynamical effects (intramolecular coupling). They are isotopic invariant for apolar molecules; however, in the case of polar molecules, the rotational correction leads to different PR values for the molecule and its isotopic substituted derivative.6 Another way of expressing the Ak or PQk values is atomic polar tensors (APT). The elements of APT are derivatives of the molecular dipole moment with respect to Cartesian atomic displacement coordinates; one of each atom of the molecule is related to one atomic polar tensor ðPx Þ: APT are always isotopic invariant, both for the polar and apolar molecules. The infrared intensities can also be expressed in terms of the parameters that have a stronger degree of localization on bonds (EOP parameters) or atoms (ECCF parameters) than PR and Px. They are isotopic invariant and transferable between molecules. However, differently from PR and Px, the EOP and ECCF parameters are not unique, due to the fact that the number of these parameters exceeds the number of experimental data. Relying on the fact that the intensity parameters are isotopic invariant (which is strictly true for APT, EOP and ECCF parameters) the intensity values of one isotope can be calculated by using the intensity values of another. This fact is helpful when, in the spectra of one isotopic species, the strongly overlapping bands (or Fermi resonance) occur. A major difficulty in the inverse vibrational frequency problem is the high number of force constants defining the harmonic force field in relation to the number of observed vibrational frequencies in the infrared and Raman spectra. A satisfactory solution of the inverse vibrational problem can be obtained by using a set of experimental data that depend on the potential field.2,7 The frequencies of the isotopically substituted molecules are the most often used observables that serve as additional data in determining a set of force constants. Theoretical predictions of force constants from quantum chemistry calculations have proved extremely useful in deriving reliable force fields for many molecules.8,9 Ab initio calculations provide a complete set of force constants, from which a potential energy distribution may be determined for every normal coordinate. In most applications, the harmonic approximation is assumed in order to make the calculations more tractable. This assumption leads to abnormally large force constants for vibrations that are known to be appreciably anharmonic, e.g., for the stretching of bonds that involve the hydrogen atom. The prediction of the infrared intensities by quantum chemistry calculations is still considered not quite satisfactory, even when correlation effects are included.9 Even with the most sophisticated

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basis sets, the average error in the prediction of intensities is much larger than the average error in the prediction of frequencies. The calculated infrared spectrum has often been used in those cases where only a general shape of the spectrum is required. The comparison of the calculated spectrum with the experimental one supports, for example, the identification of the new, unknown species on the basis of its vibrational spectra.

B. THE E FFECT OF D EUTERIUM S UBSTITUTION ON I NVOLVING H YDROGEN M OTION

THE

V IBRATIONS

The most important effect in vibrational spectroscopy is the substitution of hydrogen (H) with deuterium (D). The frequency of the harmonic vibration (involving mainly pffiffi the motion of the hydrogen atom[s]) decreases after deuterium substitution by a factor of 1= 2; since the mass of deuterium is about twice that of the hydrogen atom mass. The band width pffiffi and the integrated band intensity of this vibration are also expected to decrease by a factor of 1= 2: The isotopic ratio of the two frequencies corresponding to the modes that involve hydrogen and deuterium motion (ISR) is pffiffi slightly less than 2; and usually ranges from 1.35 to 1.41. The value of the frequency isotopic ratio is often used in vibrational assignment as an indicator of the contribution of the hydrogen motion to the particular normal mode. However, a complete vibrational assignment for all normal modes of the molecule requires normal coordinate analysis, or quantum chemistry calculations that allow prediction of vibrational frequencies, the form of the normal vibrations, and PED for all isotopically substituted molecules. In recent times, quantum chemistry calculations of the vibrational spectra proved to be extremely useful in the analysis of experimental data. One of the most extensive initial applications of deuterium substitution was vibrational assignment for organic molecules, i.e., distinguishing the vibrations to which hydrogen atom(s) mainly contribute (CH3 rocks, CH2 wags, twist, rocks, CH in-plane and out-of-plane bends) from heavy atom vibrations, and determination of molecular force constants.10,11 The deuterium substitution enables identification of the methyl torsion vibration; this low intensity vibration occurs in the far-infrared region, where intense skeletal modes appear. Since methyl torsion involves essentially the motion of hydrogen atoms, the frequency of this vibration exhibits a frequency ratio of 1.35 –1.414 with deuterium substitution,10 which makes the identification of this mode quite definitive. The OH, SH, NH2 and PH2 torsions can also be identified with deuterium substitution, as numerous literature data prove. The isotopic substitution may affect the form of the normal modes (PED), increasing their intensity and enabling their identification. Due to this fact, the vibrational study of the N-d1 isotopologue of methylamine made possible the assignment of its two fundamentals ðn13 ; n14 Þ:12 Another example is nitrosoformic acid (HOC(O)NO), in which deuterium substitution changed the form of the normal modes, thus enabling the identification of an additional mode as compared to an unsubstituted molecule.13 This application requires support from normal coordinate analysis, or ab initio calculations of vibrational spectra. Excellent examples of the utility of deuterium substitution for identification and symmetry characterization of various molecules and intermediate species are the recent studies of numerous hydrides, and among them, the aluminium hydrides in low temperature matrices.14 The reaction of metal atoms with H2, H2 þ D2, HD and D2 led to formation of various isotopologues of metal hydrides, whose spectra allowed identification of the species formed under the conditions of the experiment and determination of their symmetry.

C. THE I SOTOPIC S UBSTITUTION, THE P OTENTIAL E NERGY D ISTRIBUTION, AND THE F REQUENCY I SOTOPIC R ATIO (ISR) The comparison of the experimental ISR value (determined by the frequencies of the band and its isotopic counterpart) with the theoretical ISR value calculated for a harmonic oscillator indicates whether the motion of the atom that is replaced by its isotope contributes to the potential energy

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285

of the normal mode to which the band corresponds. The close correspondence of the two ISR values, experimental and theoretical, indicates that the potential energy of the close-to-harmonic normal vibration is fully determined by the displacement coordinate involving the isotopically substituted atom. This is usually the case for high frequency stretching vibrations involving groups with light atoms. However, the potential energy of the normal mode characterized by the frequency from the “finger print region” is often a sum of contributions from more than one displacement coordinate; the isotopic substitution may change the distribution of the normal mode energy among the displacement coordinates. In such cases, the simple ISR value determined by the frequencies of the band and its isotopic counterpart is not very informative as far as the band assignment or anharmonicity of the mode are concerned. The potential energy distribution obtained from normal coordinate analysis or quantum chemistry methods allows for reliable band assignment of various isotopologues of the studied molecule. An example of the change of PED after isotopic substitution is phenol and phenol-OD molecules15 and para 4-fluorophenol.16 In phenol, the OH in plane bending vibration contributes to three normal modes, Q18(10%), Q20(45%), Q23(18%); in the phenol-OD molecule, it contributes essentially to one normal mode only, Q13(81%). The isotopic ratio of the frequencies corresponding to Q20 / Q13 modes in phenol and phenol-OD molecules equals 1.28; the low ISR value is due to the contribution of other displacement coordinates to the Q20 mode, not only the OH in plane bending. In para 4-fluorophenol, the situation is even more complicated; the OH in plane bending contributes to three normal modes, as in the case of unsubstituted phenol, and the OD in plane bending vibration is in Fermi resonance with a combination tone. Another example of varying PED with isotopic substitution is formohydroxamic acid, HCONHOH.17 The 1372 cm21 band in the spectrum of HCONHOH corresponds to a normal mode, to which contribute NOH in plane bending (50%), CN stretching (26%) and NH bending vibrations (10%). The counterpart of the 1372 cm21 band in the HCONDOD spectrum is observed at 944 cm21 and corresponds to a normal mode to which contribute NOD in plane bending (70%) and ND bending (25%) vibrations. The change in PED of the normal mode is responsible for high ISR value (1.45) of the corresponding frequencies.

II. SOURCES OF ANOMALOUS H/D ISOTOPE EFFECTS IN HYDROGEN-BONDED SYSTEMS In this section, we present a simple approach to understanding the anomalous isotope effects in infra-red spectra of hydrogen bonds, referring our analysis to the first review given by Sheppard.18 Anomalous isotope effects that are considered here may occur in hydrogen bonded systems in addition to isotope effects presented in the previous section. In our approach, we refer mainly to the experimental data and theoretical considerations that are modeling the hydrogen bonded systems. The analysis of similar topics based on the ab initio calculations is presented by J. Del Bene in this volume, p. 153. It seems that three main components of the anomalous H/D isotope effects can be distinguished: A. The increase of anharmonicity of the stretching n (AH) vibrations as compared to n (AD) ones is illustrated in Figure 10.1. The zero point energy of n (AH) oscillator is located above that of n (AD). Due to anharmonicity, the wave function peak for proton is shifted, compared to deuteron, towards the center of the AH· · ·B bridge. As a consequence, the stabilizing interaction for AH· · ·B is stronger than that for A – D· · ·B. The role of anharmonicity in hydrogen bond phenomena is of primary importance, as has been emphasized by Sandorfy.19 It will be analyzed in the next section. B. Some contribution to the bridge energy comes from the bending d(AH) and g(AH) vibrations, which are characterized by large amplitudes in-plane or out-of-plane. Obviously, the amplitudes for A –H· · ·B are larger than those for A –D· · ·B as shown in Scheme 10.1.

286

Isotope Effects in Chemistry and Biology

E

H E0(H)

D

E0(D)

Γ

FIGURE 10.1 Zero point energy levels for H and D and the wave function peaks in the case of anharmonic potential (dotted curve).

A

H

B

A

D

B

SCHEME 10.1

Hence, the AH deformation modes lead to a weakening of bridges in a higher degree than the AD modes. This is well-reflected in a negative Ubbelohde effect and in a positive effect in the hydrogen-bond stretching force constant ks ; determined from the rotational spectra in the gas phase.20 There are several known examples of weak hydrogen bonds, in which one observes an increase of interaction strength after deuteration. For instance, it was shown that H-bonded water dimers are ca 60 cm21 less stable than D-bonded dimers.21 The contribution of the deformation modes to the H/D isotope effects is discussed in detail by Sokolov and Savel’ev.22 One can conclude that, in analysis of the A – H stretching vibrations, it is necessary to take into account a modulation of the potential by low frequency deformation modes, as has been shown convincingly in Ref. 23. It might be worth emphasizing that the anharmonicity discussed in the next section and the bending modes have opposite effects on the relative strength of hydrogen bonds involving H or D. C. The third substantial contribution to the anomalous H/D isotope effect comes from tunneling that plays a role in strongly hydrogen-bonded systems characterized by the potential with two minima. The tunneling exerts particularly strong influence in the case of a double-minimum potential with a medium-high barrier for the proton transfer. Let us quote here the results of calculations by Matsushita and Matsubara24 who analysed the H/D isotope effect in dynamically symmetric OH· · ·O bonds in crystals by assuming the potential expressed by a double Morse function. The calculations demonstrated that the maximum isotope effect should appear for the ˚ . The results of calculations by Matsushita and OH· · ·O bridges of length approximately 2.50 A Matsubara are illustrated in Figure 10.2, where the plot of the difference between the proton and deuteron positions versus the bridge length is presented. One can distinguish a few bridge-length regions in which the H and D tunnelings are outlined in different degree. The analysis shown in Figure 10.2 is of particular importance in understanding the Ubbehlode effect and phase transitions in ferroic hydrogen bonded crystals. As will be shown later, the largest

Vibrational Isotope Effects in Hydrogen Bonds

287

∆ δ, Å 0.15 VI

V

IV

III

II

I

0.10

0.05

2.5

2.6

2.7 R0...0 /Å

FIGURE 10.2 Calculated by Matsushita and Matsubara24 Dd values (increase of the distance between the minima on deuteration) plotted versus R(0· · ·0); marked regions correspond to: no tunneling for both H and D (I), tunneling motion for H but not for D (II), tunneling both for H and D (III), highly significant proton tunneling (IV), highly significant deuteron motion and the proton wave function shows a single peak (V), single peaks for both H and D (VI).

isotope effects in infra-red spectra of OH· · ·O systems appear in the region close to the peak on the correlation curve presented in Figure 10.2. The correlation shown in Figure 10.2 is in strong agreement with experimental data gathered so far with respect to the Ubbelohde effect.25 One should mention that the three isotope effects considered above affect the energy of hydrogen bonding. Thus, deuteration leads to weakening of hydrogen bonds via first and third effects, and to its strengthening via the second one.

III. THE HYDROGEN BOND EFFECT ON ANHARMONICITY OF PROTONIC VIBRATIONS It is commonly known and accepted that the most widespread role in spectroscopic phenomena in hydrogen bonded systems is the anharmonicity of n (AH) vibrations. The basis for understanding these phenomena was established by Sandorfy.19 The main parameter introduced was named the anharmonicity constant and defined as: X12 ¼ n02 =2 2 n01

ð10:9Þ

where n01 and n02 are the frequencies of transitions between the fundamental and first and second excited levels of the stretching n (AH) vibrations. Sometimes, the opposite sign of X12 is assumed. Let us take as an example the data related to phenol and its complexes with amines. For phenol itself, the X12 constant equals to 2 85 cm21, while for a relatively weak complex with pyrazine it equals to 2 105 cm21.26 On increasing the interaction strength (changing the acid-base properties expressed by the DpKa value) the absolute value of X12 increases as shown in Table 10.1,27 where the data for pyridine complexes with phenol derivatives are collected. For relatively strong hydrogen bonds, the increase of the proton donor-acceptor ability has no visible influence on the X12 anharmonicity constant. Further, for very strong hydrogen bonds, when approaching the so called critical region of DpKa,28 i.e., when the second potential energy minimum appears, the 2 X12 value decreases, indicating that this parameter is not sufficient for the description of the proton dynamics and, as a consequence, for the description of the isotope effects. As can be

288

Isotope Effects in Chemistry and Biology

TABLE 10.1 The Frequencies and Anharmonicity Constants (cm21) for n(OH) Vibrations in Pyridine Complexes of Various Phenols According to Ref. 27

4-CH3-phenol Phenol 4-Cl-phenol 4-Br-phenol 3-Br-phenol 3,4-di-Cl-phenol 3,5-di-Cl-phenol 3-CF3-4-NO2-phenol

n01

n02

3130 3115 3070 3060 3020 2910 2860 2700

5850 5800 5700 5680 5570 5350 5280 5040

–X12 205 215 220 220 235 220 220 180

seen in Section VIII, after exceeding the critical region, for very short hydrogen bonds, the change of the sign of anharmonicity constant pffiffi may occur. This means a positive anharmonicity and, as a consequence, n ðAHÞ=n ðADÞ . 2: In the case of weak hydrogen bonds the situation is more complicated, as shown by Rossarie, Gallas, Binet and Romanet.23 They found that the anharmonicity constant starts to decrease at low values of relative frequency shifts, reaches a minimum, and then continuously increases as shown in Figure 10.3. The correlation embraces a large number of systems composed of various alcohols with either oxygen or nitrogen weak proton acceptors. These results demonstrate that a diatomic oscillator approximation does not well describe the proton dynamics, and confirm that, in order to explain the isotope effects in hydrogen-bonded systems, one should take into account a coupling between stretching and bending modes, as was emphasized in the previous section.

−X12

100

50

1

5

10 ∆nΓ ⋅102

FIGURE 10.3 Correlation between the anharmonicity constant X12 and relative frequency shift Dnr ¼ Dn(OH)/no(OH) (adapted from Ref. 23).

Vibrational Isotope Effects in Hydrogen Bonds

289

TABLE 10.2 The Frequencies and Anharmonicity Constants (cm21) for n(OH), n(OD) Vibrations in Deuterated and Non-Deuterated Alcohols and Phenols

Triphenylsilanol Trimethylcarbinol Adamantanol 2,6-Dimethylphenol 2,6-Dimethyl-4-chlorophenol

n01(OH)

n01(OD)

n02(OH)

n02(OD)

2X12(OH)

2X12(OD)

3683 3618 3607 3622 3617

2716 2669 2662 2674 2673

7199 7061 7041 7069 7067

5344 5248 5234 5258 5253

83.5 87.5 86.5 87.5 83.5

44 45 45 45 46.5

As can easily be expected, the absolute value of the anharmonicity constant for n (AD) vibrations is less than that for n (AH). We quote here, in Table 10.2, some data for various OH-group-containing compounds, which show that the anharmonicity constant 2 X12 for deuterated phenols and alcohols is half as large as for non-deuterated molecules (CzarnikMatusewicz, personal communication). The data in Table 10.2 show that the anharmonicity constant does not depend on the chemical species but is related to fundamental n (AH) frequency. In quantitative analysis, one should consider the relative anharmonicity constants, i.e., the X12 value divided by n (AH) or n (AD). Thus, for instance, in the case of trimethylcarbinol, the corresponding relative anharmonicity constants are equal to 0.024 and 0.017. There is little known about the anharmonicity of the d(AH) and g(AH) modes. The data cited by Sandorfy19 refer to the results based on combination bands. To our knowledge, there are two papers in which the frequencies of the librational modes overtones of proton-donor groups in hydrogenbonded complexes were reported.29,30 In Table 10.3 we compiled these data obtained for the HF complexes with weak proton acceptors in Ar matrices. The anharmonicity constants for librational modes cannot be compared directly with anharmonicity constants for stretching vibrations because vibration contributes to the libration, in addition to the H atom, as well as heavy atom motion. Nevertheless, in this case, one also observes half as large 2 X12 values for deuterated complexes than for non deuterated ones. The relative anharmonicity constants for librational modes are markedly higher as compared to the stretching modes. Thus, for instance, the relative value of 2 X12 for the librations in HCN complex with HF equals to 0.098 while that in HCN with DF is 0.067.

TABLE 10.3 The Frequencies (cm21) and the Isotopic Frequency Ratios (ISR) for the Stretching ðnÞ and Librational (nl ) HF, DF Modes and Anharmonicity Constants (cm21) for the Librational Modes in HF, DF Complexes with HCN, PH3, AsH329,30 HCN

n 2nl nl –X12

PH3

AsH3

HF

DF

ISR

HF

DF

ISR

HF

DF

ISR

3626 1062 589 58

2669 811 434 29

1.36

3627 843 477 55

2673 678 366 27

1.37

3694 710 409 54

2718 580 317 27

1.36

1.36

1.30

1.29

290

Isotope Effects in Chemistry and Biology

IV. POTENTIAL ENERGY FUNCTIONS FOR THE PROTON-STRETCHING VIBRATIONS The main subject of our interest is the stretching n (AH) vibrations and their behaviour in hydrogen bonding so a diatomic oscillator model can be applied with a good approximation of a description of the proton motion. The general expression for the one-dimensional potential energy function for the proton motion has a form:

›V VðrÞ ¼ Vðre Þ þ ›r

1 ›2 V ðr 2 re Þ þ 2! ›r 2 r¼re

!

1 ›3 V ðr 2 re Þ þ 3! ›r 3 r¼re

!

2

r¼re

ðr 2 re Þ3 þ · ··

ð10:10Þ

where r; vibration coordinate, is the distance between H and A, while re is the equilibrium A –H distance. Several empirical approaches to the description of the one-dimensional potential energy function for the proton motion can be found in literature. Most popular, (not only for the A –H proton donor), in cases where anharmonicity is not too high, is the Morse potential shown in Figure 10.4, in which two empirical parameters, Do and a; describe the depth and width of the potential well. Do is the bond energy and De is the dissociation energy. VðrÞ ¼ Do ½1 2 expð2aðr 2 re ÞÞ

2

ð10:11Þ

The Morse function describes quite well the potential for r . re but fails for r , re : A strong repulsion at short distances should be taken into account. An improved version of the Morse

V A

De

Do

Vo Γe

Γ

V B

Γ

FIGURE 10.4 One-dimensional Morse potential (A) and double-minimum symmetrical potential (B).

Vibrational Isotope Effects in Hydrogen Bonds

291

potential was proposed31 in the form: VðrÞ ¼ Do ½1 2 expð2nðr 2 re Þ2 =2rÞ

ð10:12Þ

where n replaces the Morse parameter a: n ¼ ke re =Do (ke is the force constant) In the cases of double-minimum potential occurring in strong hydrogen bonds, in a critical region,28 the double Morse function can be used as proposed in24 VðrÞ ¼ Do {exp½22aðr 2 ro Þ 2 2 exp½2aðr 2 ro Þ }

ð10:13Þ

The best description of a real two-dimensional potential for the proton motion in hydrogen-bonded systems is the Lippincott – Schroeder potential. The reasoning for that potential was formulated for the first time in Ref. 31. The function consists of three parts, attributed to A – H, Bþ – H (after proton transfer) and A· · ·B interactions: 0 V2 ðr 0 ; RÞ V3 ðr 0 ; RÞ Vðr 0 ; RÞ ¼ V1 ðr Þ þ # þ # # A–H Bþ – H A· · ·B

ð10:14Þ

The geometrical parametrization, according to Ref. 33 is shown in Scheme 10.2. The respective contributions to Equation 10.14 have a form: " !# 2nðr 0 2 ro cos aÞ2 0 V1 ðr Þ ¼ D 1 2 exp 2r 0 cos a " 0

V2 ðr ; RÞ ¼ 2D

p

np ½R 2 r 2 rop cosða þ QÞ 2ðR 2 r 0 Þcosða þ QÞ

V3 ðr 0 ; RÞ ¼ Aðr 0 Þexpð2bRo Þ exp½2bðR 2 Ro Þ 2

2

ð10:15Þ

# ð10:16Þ

1 ðR =RÞm 2 o

ð10:17Þ

where: "

0

p

p

Aðr Þ ¼ 2n D {1 2

½rop =ðRo

np ½Ro 2 r 0 2 rop cosða þ QÞ 2 r Þ }exp 2ðRo 2 r 0 Þcosða þ QÞ 0

2

½2 cosða þ QÞexpð2bRo Þðb 2 m=2Ro Þ

2

#

21

n ¼ ko ro =D where ko is the force constant of the free A –H vibrations; np ¼ kop r p =D where kop is H θ

Γ∗

Γ A

α

Γ'

SCHEME 10.2

B

I

Γ'* R

292

Isotope Effects in Chemistry and Biology

TABLE 10.4 Values of the Lippincott – Schroeder Potential Parameters for OH· · ·O, NH· · ·O, and N – H· · ·N Hydrogen Bonds

D(kcal.mol21) n(108 cm21) n p(108 cm21) ˚) ro(A p ˚ ro(A) b(108 cm21)

OH· · ·O

NH· · ·O

N –H· · ·N

118.6 9.18 13.32 0.97 0.97 4.8

112.4 8.60 13.15 0.99 0.97 4.8

112.5 9.01 13.49 1.038 1.038 4.8

the force constant of the Bþ – H vibrations; D and Dp are the AH and BþH bond energies; b is the repulsion parameter between A and B; the exponent m is close to unity. The values of the Lippincott –Schroeder potential parameters used in32 are given in Table 10.4. The Lippincott – Schroeder potential seems to be very useful in the description of the spectroscopic features of hydrogen bonds. Some attention should be paid to a simple empirical equation formulated for the first time by Samorjai and Hornig.33 It can be expressed in a general form Vðr; RÞ ¼ a2 ðRÞr 2 þ a3 ðRÞr 3 þ a4 ðRÞr 4

ð10:18Þ

where R is the bridge coordinate, while r is that of the proton. The coefficients a2 ; a3 ; and a4 are generally R dependent. The first member of the above equation expresses the harmonic single minimum potential, while the third one corresponds to a symmetric double-minimum curve. The second part of the equation expresses the asymmetry of the potential. As shown in Ref. 34 the coefficients a2 ; a3 ; and a4 in the above equation can be fit to experimental spectroscopic correlations for given types of hydrogen bonds. Equation 10.18 was successfully applied in semiquantitive analysis of infrared spectra,34 – 37 particularly to explain anomalous isotope effects (anomalous frequency isotopic ratio ISR). One should also mention that other empirical equations can be useful in the semiquantitative description of the anomalous dynamic behaviour of hydrogen bonds such as those applied by Laane38 for double-minimum potential with varying barrier height. In quantum mechanical studies of strong hydrogen bonds, the one-dimensional potential for the proton motion along the bridge may be approximated by an n-th order polynomial. Examples are the studies of the X H –NH3 system39; the one-dimensional potential for this system was obtained by fitting the ab initio energies for various proton positions along the bridge with a sixth-order polynomial.

V. THE SHAPE OF THE POTENTIAL AND EVOLUTION OF IR SPECTRA OF HYDROGEN-BONDED SYSTEMS From considerations in previous sections, it follows that, depending on the donor-acceptor properties of AH and B and the symmetry of charge distribution, the shape of the potential may vary according to the scheme presented in Figure 10.5. Limiting curves A and F are a mirror reflection when the proton transfer leads to a change of the proton donor-acceptor function of A and B. The dynamics, and thus IR spectra, should be similar for such limiting

Vibrational Isotope Effects in Hydrogen Bonds

293

FIGURE 10.5 Evolution of the potential for the proton motion with increasing proton-donor-acceptor ability.

cases, as has been shown for the series of pentachlorophenol complexes with N-bases of varying basicity.40 The evolution of IR spectra is shown schematically in Figure 10.6. The absorption related to n (AH) vibrations for nonperturbed AH group is characterized by a narrow band (above 3600 cm21 for OH and above 3500 cm21 for NH) as reflected in curve A. The formation of weak AH· · ·B bridges leads to a red shift, broadening and intensifying that band (curve B). In the case of mediumstrong bridges, further broadening, shifting, and intensifying of the band takes place. When it is shifted to the region of about 2800 – 3000 cm21, the structure of the band appears with a number of rather narrow submaxima, which are interpreted in terms of strong couplings and Fermi resonance theories.41 – 43 Here, one should emphasize that the fine structure starts to be visible, when the band first moment exceeds 3000 cm21. The bands for strong hydrogen bonds located at lower frequencies are deprived of such fine structure, but as can be seen in curve D, very broad

A

Absorbance

B

C

D

E

3400 3000 2600 2200 1800 1400 1000

n,cm−1

FIGURE 10.6 Various types of n (AH· · ·B) absorption bands in hydrogen-bonded systems.

294

Isotope Effects in Chemistry and Biology

submaxima (or subminima) may appear (Hadzi’s trio). Such structure of very broad bands can be interpreted in terms of two theories. Thus, one can ascribe broad subminima to the Fermi resonance interaction of the n (AH) with 2d(AH) and 2g(AH)43; but, as has been argued in several papers,44 such broad bands can arise from overlapping of the bands assigned to various transitions between levels created by the tunneling splitting. For potential energy curves C (low-barrier hydrogen bond) and D (single minimum), the n (AH) bands undergo narrowing, and are shifted to the region of ca 1000 cm21. However, in the case of NHN bridges, this band can be shifted to as low a frequency as 500 cm21 (Section VIII). Considering the isotope effects, one should always remember various phenomena accompanying the proton dynamics, described by a large variety of potential energy curves.

VI. FREQUENCY ISOTOPIC RATIO (ISR) AND ITS CORRELATION WITH OTHER PARAMETERS OF HYDROGEN BONDS The main parameter when discussing the isotope effects in infra-red spectra is the n (AH)/n (AD) value, where n (AH) and n (AD) are pthe ffiffi first band moments (centers pffiffi of gravity). For weak hydrogen bonds, this ratio is close to 2; the small deviation from 2 values is due to a small perturbation of harmonicity. As the perturbation increases (stronger hydrogen bond), the value of ISR decreases, approaching values close to unity for very strong hydrogen bonds, sometimes even less than 1. On the other pffiffi hand, for the strongest bridges, the ISR value quickly goes up, and in some cases exceeds 2: The first systematic data related to the correlation between ISR and OHO bridge length were reported by Novak.45 A minimum on the correlation curve ISR versus R(0· · ·0) takes place for ˚ exactly in the region where Matsushita and Matsubara predict the bridges of 2.50 –2.55 A maximum value of D based on the tunneling phenomenon. In Figure 10.7 are shown correlations of ISR versus nr for three types of hydrogen bonds, i.e., OH· · ·O, OH· · ·N, and N –H· · ·N; nr are defined as n (AH)/no (AH) where no (AH) is the frequency of the non-perturbed AH group. Data related to the O –H· · ·O bridges come mainly from Novak’s paper,45 while those related to O – H· · ·N and N – H· · ·N come from Refs. 46,47. The comparison of the correlations for the three types of hydrogen bonds shows that, in all cases, the minimum ISR value is around 1, but this minimum is located in different nr regions for considered types of hydrogen bonds. The essential difference consists in the extension of the three curves for the lowest nr values. pffiffi In the case of OH· · ·O bridges, there are only a few systems known with an ISR value above 2: In the case of N –H· · ·N systems (mainly protonated proton sponges), there are several ones known with very high pISR ffiffi values exceeding 2. We don’t know of any OH· · ·N hydrogen bonding with an ISR above 2: Very important results were accumulated for the complexes XH· · ·B (X ¼ F, Cl, Br, I) in argon matrices, where the environment effects are substantially reduced. Extended reviews on spectroscopy of XH· · ·B complexes in matrices were reported.48 – 50 In order to get a more generalized picture, the parameter describing the relative change of ISR defined as (ISRc –ISRo)/ ISRo was introduced where ISRc and ISRo are the isotopic ratios for complexes and free molecules.48 This relative change was correlated with the Pimentel normalized proton affinity, defined as:

DPAn ¼

PAðBÞ 2 PAðX2 Þ PAðBÞ þ PAðX2 Þ

where PA(B) and PA(X2) are proton affinities of the base B and anion X2.

ð10:19Þ

Vibrational Isotope Effects in Hydrogen Bonds

295

ISR 2.0 NHN 1.8

1.6

1.4 OHO 1.2 OHN 1.0

0.2

0.4

0.6

0.8

1.0

nΓ

FIGURE 10.7 Isotopic ratio ISR ¼ n (AH)/n (AD) plotted versus relative frequency n (AH)/no (AH) for three types of hydrogen bonding.

The FH· · ·FH· · ·B complexes were also included in the correlation. The estimated PA value for the HF2 2 anion is equal to 2 331 kcal/mol. The plot of D(ISR)/ISRo versus DPAn is presented in Figure 10.8. Although there is a considerable scattering of experimental values, undoubtedly a critical pffiffi region is visible, with positive D(ISR)/ISRo values that correspond to the region of ISR . 2 ∆(ISR) (ISRο) 0.1

0

−0.1

−0.2

−0.3

−0.2

−0.1 ∆PAn

FIGURE 10.8 Correlation between the relative isotopic ratio and the Pimentel generalized proton affinity for XH· · ·B complexes in the argon matrices (taken from Ref. 48).

296

Isotope Effects in Chemistry and Biology

shown in the previous correlation (Figure 10.7). After exceeding the critical region, D(ISR)/ISRo again drops. It seems justified to suspect that the second part of the correlation after exceeding the critical region is a mirror image of the first part. However, more experimental points are needed for the low absolute values of Pimentel proton affinity.

VII. THE ISOTOPE EFFECT UPON OTHER SPECTROSCOPIC PARAMETERS OF HYDROGEN-BONDED SYSTEMS There has been an enormous effort to study the shape (width, symmetry and substructure) of the band due to hydrogen stretching vibration in the spectra of hydrogen bonded systems. The basic theories concerning the weak and medium-strong hydrogen bonds are most recently reviewed by Henri-Rousseau, Blaise and Chamma.41,42 Here, we shall focus our attention mainly on those aspects which concern the isotope effect on the band shape and integrated intensity of the n(XH) band. Within the linear response theory, the infrared spectral density of the n (AH) mode is given by the Fourier transformation of the autocorrelation function G(t) of the dipole moment M of the AH· · ·B system: IðvÞ ¼

ð

GðtÞexpð2vtÞ

ð10:20Þ

where GðtÞ ¼ kMð0ÞMðtÞl The spectral density IðvÞ of the n (AH) band is influenced by four main band shaping mechanisms: The anharmonic coupling between the high frequency n (AH) mode (fast mode) and the low-frequency intracomplex (we use this term to emphasize the hydrogen-bonded complex as an independent entity) modes, in most cases it is the bridge stretching s mode † The Fermi resonances between states involving the n (AH) stretching and some other internal modes (usually overtones and combinations) † The resonance interaction between the vibrational levels of hydrogen bonds in neighbouring molecules † High polarizability of hydrogen bonds with double-minimum potential

†

The models that assume the strong anharmonic coupling theory well describe the behaviour of the n (AH) band with an isotopic substitution of the proton involved in the fast mode of the pffiffi hydrogen bond by deuterium: the band width and the band center are decreased by a factor 1= 2: The basic theory of strong couplings was formulated by Mare´chal and Witkowski51,52 and detailed discussion of the impact of this theory on the IR behaviour of hydrogen bonds, including isotope effects, was presented by Sokolov and Savelev.53 However, one should remember that the low frequency modes can be damped, as shown by Robertson and Yarwood,54,55 and in such cases a stochastic theory, that was formulated for the first time by Bratos,56 is justified. This theory can be applied not only for weak and medium-strong but also for very strong, double-minimum potential, hydrogen bonds.35 The role of electrical anharmonicity of the n (AH) mode in band shaping mechanism is not clear. Mare´chal and Bratos57 have shown that, within the limit of their theory, the spectral efficiency of the electrical anharmonicity is low in the case of liquids containing weak hydrogen bonds. Mare´chal and Ratajczak58 have studied the effect of electrical anharmonicity upon profiles of the

Vibrational Isotope Effects in Hydrogen Bonds

297

n (AH) band of liquids involving medium-strong hydrogen bonds, and showed that the mechanism generates the shift and some changes in the profile of bands. Recently Belhayara, Chamma, and Henri-Rousseau59 included the intrinsic mechanical anharmonicity of the hydrogen bond stretching mode (low frequency) in their linear response quantum theory. The model behaved well with an isotopic substitution of the proton involved in the pffiffi fast mode, i.e., the band-width decreased by a factor of 1= 2 after D substitution. The work also explicitly demonstrated that a particular Fermi resonance occurring in the spectrum of a H-bonded species may be inoperative with the deuterated species, and vice versa. The Fermi resonance, in our opinion, is the most important mechanism in the shaping of the n (AH) bands in hydrogen-bonded systems. The substructure of those bands is particularly well manifested when they are located in the region 2800– 3000 cm21, in which the overtones and combinations of the d (OH) or d (NH) appear. Of great importance, from this point of view, are the results reported by Wolff and Mu¨ller.60 Wo´jcik who developed the theory of strong coupling, also took into account the Fermi resonance as an important factor in the generation of substructure.61 The integrated intensities of the bands due to n (AH) hydrogen bond vibrations are sensitive quantities which have scarcely been measured, but which contain important information on the electronic effects of the studied system. Guissani and Ratajczak37 theoretically studied anomalous isotopic effects on the integrated intensity of the n (AH) band by applying standard methods of non-equilibrium statistical mechanics. They found that the anomalous isotopic effect on the IR integrated intensity is connected both with the mechanical and electrical anharmonicities, the latter one playing a role in the systems with moderately strong and strong hydrogen bonds. The most detailed experimental studies of the integrated band intensities of the n (AH) vibration were performed for the carboxylic acid dimers by Mare´chal and coworkers.62,63 It was found, in particular, that the ratio of the intensities of the n (OH) bands in the spectra of (RCOOH)2 and (RCOOD)2 cyclic dimers of formic and acetic acids in the gas phase, as well as in the spectra of adipic acid in crystals, show an anomalously large value equal to 2. The ratio of the intensities of the corresponding bands in the spectra of the monomers pffiffi was equal to ca. 1.35, which is close to the theoretical value for an harmonic oscillator ð 2Þ: In the first approach, this anomalous effect was explained by a strong electrical anharmonicity of the n (OH) mode and one of the intracomplex modes. However, the calculations performed afterwards63 indicated smaller electrical anharmonicity effect than proposed earlier. In acetic acid crystals, where the cyclic dimers are not formed, the intensity ratio of the n (OH), n (OD) bands assumed close to harmonic value, which suggested that the anomalous value of the intensity ratio observed for cyclic dimers might be connected with some peculiarities of cyclic rings (maybe due to cooperative effect). The renewed analysis of the collected experimental data64 and the results of quantum chemical calculations63 led to the suggestion that the anomalous isotopic effect originates in a nonadiabatic transfer of intensities between electronic and protonic transitions, which is favoured by the particular ring structure of the cyclic dimers. Recently Flakus65 proposed an explanation of the anomalous isotopic effect in cyclic dimers that is based on the contribution of the vibronicallyactivated, forbidden transition of the totally symmetric protonic motion to the overall band intensity. The fact that the Born-Oppenheimer approximation, i.e., separation of electronic and nuclear motion, might not work satisfactorily in the hydrogen-bonded systems was suggested by Witkowski.66,67 In his approach to the separation of electronic and nuclear motion, he went beyond approximation of infinitely quick electrons by including in the Hamiltonian oscillator the change of the nuclear coordinate as a function of the ratio of the nuclear to electronic velocity. In this way, the Hamiltonian of the simple quantum harmonic oscillator may be considered as corrected by a quadratic time-dependent term, the strength of which depends on the velocity ratio. The effect should appear in the vibrational motions of the lightest nucleus (hydrogen) performing large amplitudes, and in contact with electrons that can be easily displaced; these conditions are well

298

Isotope Effects in Chemistry and Biology

satisfied in hydrogen-bonded systems. The spectral density of the ns band obtained on the basis of this theoretical approach involves an asymmetric broadening. The effect is mass dependent, and should be diminished by substitution of hydrogen by deuterium. The coupling of electronic and nuclear motion, which is particularly important in the case of strong and moderately-strong hydrogen bonds, is a well-recognized fact now. It has even been reflected in new, comprehensive classifications of the strong and moderately-strong hydrogen bonds, as suggested by Gilli.68 Gilli distinguished three classes of strong hydrogen bonds: negativeor positive-charge assisted hydrogen bonds (CAHB) and resonance assisted or p-cooperative hydrogen bonds (RAHB). The coupling between the proton and electrons motion is particularly demonstrated in the behaviour of IR spectra of such intramolecular hydrogen bonds as those in Schiff bases. It was shown quantitatively69 that intensities of the n (OH· · ·N) bands are much lower as compared to analogous systems without p-electron conjugation. The hydrogen-bonded systems showing broad continuous infrared absorption deserve special treatment. This phenomenon is sometimes ascribed to unusually high polarizability of strong hydrogen bonds. The theoretical basis of such polarizability, named Zundel polarizability, of hydrogen bonds with double-minimum potential was formulated in Ref. 70, and its role in various processes, and in infrared spectra in particular, was discussed in Ref. 71. The existence of unusual polarizability was confirmed by direct experiments, such as Rayleigh light scattering,72 or measurements of dielectric dispersion in infrared.73 Direct measurements of vibrational (atomic) polarizability show that it can reach very high values, of the order of 15 cm3, that is close to the electron polarizability. This polarizability relates to the systems from the critical region, i.e., those characterized by a symmetrical or quasisymmetrical double-minimum potential. It is commonly known that such systems are distinguished by an intense infrared continuum, spread down to very low frequencies , 200 cm21. The role of Zundel polarizability in the creation of broad intense continua was discussed in many papers. Here we would like to mention the paper by Borgis, Tarjus and Azzouz,74 who analyzed the role of Zundel polarizability in proton dynamics of strong hydrogen bonds in solution, and the paper by Hayd and Zundel,75 who have shown the role of interactions of easily polarizable hydrogen bonds with phonons and polaritons in crystals. These interactions, that seem to be obvious, have a substantial influence on the broad absorption in far infrared. There is little known about the isotope effect on the Zundel polarizability. The only information comes from the indirect phenomenon, i.e., the intensity of continua. Particularly important seem to þ be the results of theoretical and experimental studies on H5Oþ 2 and D5O2 ions, the simplest 76 homoconjugated cations. They showed that deuteration reduces the overall intensity ca twice and the continuum becomes much narrower, both the high and low frequency wings are cut off. Particularly interesting seems to be the behaviour of the low frequency part of the broad absorption. Lower polarizability of deuteron bonds should cause weaker coupling of protonic vibrations with the lattice phonons.

VIII. LOW-BARRIER HYDROGEN BONDS The NH· · ·N hydrogen bonds present very good examples of systems where the anomalous isotope effects in IR spectra are strikingly expressed. In Figure 10.9, we have shown, as an example, the IR spectra of three systems containing NHN bridges of various strength. Three extreme cases were selected. The first one relates to a weak hydrogen bond with ISR ¼ 1.35, which deviates only a little from the harmonic oscillator. The second extreme case relates to the homoconjugated [NHN]þ cation, the behaviour of which is typical of strong, dynamically symmetric systems. In the IR spectrum, one observes a broad continuum, which can be explained in terms of several mechanisms listed at the beginning of Section VII. The isotopic ratio in such cases is close to unity.

Vibrational Isotope Effects in Hydrogen Bonds %T

299

ISR = 1.347

2000

(a) %T

ISR = 1

(b)

400

%T

ISR = 2.05

(c)

200

800

400

3000

1200

600

3500

1600

800

1000

cm−1

2000

cm−1

cm−1

FIGURE 10.9 Illustration of three extreme examples of the isotope effect in NHN hydrogen bonds. IR spectra of diphenylamine associates (a), quinuclidin-3-one-hemiperchlorate (b) and 1,8-bis(dimethylamino)naphthalene. HPF6 adduct (c), solid line NHN, dotted line NDN (according to Ref. 77).

The deuteration leads to a drop of intensity and width of the nas(NHN) band. In the case of very short NHN hydrogen bonds, such as in protonated proton sponges, one observes the absorption band ascribed to nas(NHN) vibrations at particularly low frequencies in the region , 500– 600 cm21. The isotopic ratio for such bridges is particularly high, exceeding 2. Unusual steric conditions in proton sponges cause NHN bridges to be markedly shortened as compared to free homoconjugated cations. This behavior leads to a substantial lowering of the potential barrier, in agreement with the Scheiner calculations78 for the model systems. The problem of low-barrier hydrogen bonds seems to be independently highly interesting, in light of the Perrin and Nielson79 as well as the Cleland and Kreevoy80 papers. In this section we would like to present the results of structural, IR spectroscopic and theoretical studies on protonated 2,7-dibromo-1,8-bis(dimethylamino)naphthalene (Br2DMAN),79 shown in Scheme 10.3.

300

Isotope Effects in Chemistry and Biology

H3C H3C

N

⊕ H H

CH3

N

CH3 Br

Br

SCHEME 10.3

14 12

E[kcal/mol]

9 8 6 4 2 0 0.8

0.9

1

1.1

1.2

1.3

1.4 r

1.5

1.6

1.7

1.8

1.9

2

FIGURE 10.10 Calculated potential and vibrational levels for the proton motion (H-solid lines, D-dotted lines) in protonated (deuterated) Br2DMAN (according to Ref. 81).

TABLE 10.5 Structural and IR Spectroscopic Characteristics of Protonated (Deuterated) Br2DMAN Experimental ˚ d(N· · ·N)/A ,NHN/o Barrier height/kcal/mol ˚ Distance between minima/A Distance between vibrational levels/cm21 HO!1

560

O!2 DO!1

340

O!2 ISR

1.65

2.547(2) 161.1(2) ,0.7

Calculated (MP2/6-31G(d,p)) 2.537 160.1 0.8 0.6 509 X12 ¼ þ 233 cm21 1484 284 X12 ¼ þ 141 cm21 850 1.76

Vibrational Isotope Effects in Hydrogen Bonds

301

The calculated potential energy curve with energy levels for both isotopologues for Br2DMAN is shown in Figure 10.10. The most striking result is that the distances between the vibrational energy levels increase going to higher levels. The anharmonicity in this case is highly positive, in contrast to the usual hydrogen-bonded systems. This is reflected in data presented in Table 10.5. The NHN hydrogen bonding in protonated Br2DMAN belongs to the shortest bonds known so far. The agreement between experimental and calculated data is relatively good. The experimental anharmonicity constant is not available, as the 0 ! 2 transition is not observed in IR spectra. The calculated relative X12 values (X12/n (0 ! 1) for both hydrogen and deuterium isotopologues are very close.

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19 Sandorfy, C., Anharmonicity and hydrogen bonding, In The Hydrogen Bond. Recent developments in theory and experiments, Vol. II, Schuster, P., Zundel, G., and Sandorfy, C., Eds., North Holland, Amsterdam, pp. 613–654, 1976. 20 Legon, A. C. and Millen, D. J., Systematic effect of D substitution on hydrogen-bond lengths in gas-phase dimers B· · ·HX and a model for its interpretation, Chem.Phys.Lett., 147, 484– 489, 1988. 21 Engdahl, A. and Nelander, B., On the relative stabilities of H- and D-bonded water dimers, J. Chem. Phys., 86, 1819–1823, 1987. 22 Sokolov, N. D. and Savel’ev, V. A., Isotope effect in weak hydrogen bonds. Allowance for two stretching and two bending modes of the A – H· · ·B fragment, Chem. Phys., 181, 305– 317, 1994. 23 Rossarie, J., Gallas, J.-P., Binet, C., and Romanet, R., Effet de la liaison hydrogene sur les vibrations du groupment OH (ou OD) de quelques alcools. I-vibration d’elongation n OH(D); anharmonicite mecanique, J. Chim. Phys., 74, 188– 198, 1977. 24 Matsushita, E. and Matsubara, T., Note on isotope effect in hydrogen bonded crystals, Progr. Theor. Phys., 67, 1– 19, 1982. 25 Ichikawa, M., Hydrogen-bond geometry and its isotope effect in crystals with OHO bonds-revisited, J. Mol. Struct., 552, 63– 70, 2000. 26 Rospenk, M., Leroux, N., and Zeegers-Huyskens, T., Assignment of the vibrations in the near-infrared spectra of phenol – OH(OD) derivatives and application to the phenol –pyrazine complex, J. Mol. Spectr., 183, 245– 249, 1997. 27 Rospenk, M. and Zeegers-Huyskens, T., FT-IR (7500 – 1800 cm21) study of hydrogen-bond complexes between phenols – OH(OD) and pyridine. Evidence of proton transfer in the second vibrational excited state, J.Phys.Chem. A, 101, 8428– 8434, 1997. 28 Zeegers-Huyskens, T. and Sobczyk, L., Critical behaviour of strong hydrogen bonds and the isotopic effects, J. Mol. Liq., 46, 263 –284, 1990. 29 Andrews, L., Fourier transform infrared spectra of HF complexes in solid argon, J. Phys. Chem., 88, 2940– 2949, 1984. 30 Arlinghaus, R. T. and Andrews, L., Infrared spectra of the PH3, AsH3 and SbH3· · ·HX hydrogen bonded complexes in solid argon, J. Chem. Phys., 81, 4341– 4351, 1984. 31 Lippincott, E. and Schroeder, R., One-dimensional model of the hydrogen bond, J. Chem. Phys., 23, 1099– 1106, 1955. 32 Saitoh, T., Mori, K., and Itoh, R., Two-dimensional vibrational analysis of the Lippincott – Schro¨der potential for OH· · ·O, NH· · ·O and NH· · ·N hydrogen bonds and the deuterium isotope effect, Chem. Phys., 60, 161–180, 1981. 33 Somorjai, R. L. and Hornig, D. F., Double-minimum potentials in hydrogen-bonded solids, J. Chem. Phys., 36, 1980–1987, 1962. 34 Romanowski, H. and Sobczyk, L., A stochastic approach to the IR spectra of the symmetrical OHO hydrogen bond, Chem. Phys., 19, 361– 370, 1977. 35 Romanowski, H. and Sobczyk, L., IR Stretching absorption bands in weak and medium strong hydrogen bonded complexes in solution, Acta Phys. Polon., A60, 545– 557, 1981. 36 Sobczyk, L., Quasi-symmetric O –H· · ·N hydrogen bonds in solid state, Mol. Phys. Rep., 14, 19 – 31, 1996. 37 Guissani, Y. and Ratajczak, H., Theoretical study on an anomalous isotopic effect on the position and integrated intensity of the IR stretching vibration band for medium-strong and strong hydrogen bonds, Chem. Phys., 62, 319– 331, 1981. 38 Laane, J., Isotopic shift for vibrations with potential energy barriers, J. Chem. Phys., 55, 2514– 2516, 1971. 39 Biczysko, M. and Latajka, Z., Accuracy of theoretical potential energy profiles along proton-transfer coordinate for XH – NH3 (X ¼ F, Cl, Br) hydrogen-bonded complexes, J. Phys. Chem. A, 106, 3197– 3201, 2002. 40 Malarski, Z., Rospenk, M., Sobczyk, L., and Grech, E., Dielectric and spectroscopic studies of pentachlorophenol-amine complexes, J. Phys. Chem., 86, 401– 406, 1982. 41 Henrie-Rousseau, O. and Blaise, P., Infrared spectra of hydrogen bond. Basic theories. Indirect and direct relaxation mechanisms in weak hydrogen-bonded systems, In Theoretical Treatments of Hydrogen Bonding, Hadzˇi, D., Ed., Wiley, Chichester, pp. 165– 186, 1997.

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42 Henrie-Rousseau, O., Blaise, P., and Chamma, D., Infrared lineshapes of weak hydrogen bonds: recent quantum developments, Adv. Chem. Phys., 121, 241– 309, 2002. 43 Hadzˇi, D. and Bratos, S., Vibrational spectroscopy of the hydrogen bond, In The Hydrogen Bond, Recent developments in theory and experiments, Vol. II, Schuster, P., Zundel, G., and Sandorfy, C., Eds., North Holland, Amsterdam, pp. 565– 611, 1976. 44 Zundel, G., Easily polarizable hydrogen bonds — their interactions with the environment IR continuum and anomalous large proton conductivity, In The Hydrogen Bond. Recent developments in theory and experiments, Vol. II, Schuster, P., Zundel, G., and Sandorfy, C., Eds., North Holland, Amsterdam, pp. 683– 766, 1976. 45 Novak, A., Hydrogen bonding in solid. Correlation of spectroscopic and crystallographic data, In Structure and Bonding, Vol. 18, Dunitz, J. D., Hemmerich, P., Holm, R. H., Ibers, J. A., Jorgensen, C. K., Neilands, J. B., Renen, D., and Williams, R. J. P., Eds., Springer-Verlag, Berlin, pp. 177– 216, 1974. 46 Odinokov, Ye.S., Spectroscopy of strong hydrogen bonds (in Russian), DSci Thesis, Lomonosov State University, Moscow, 1983. 47 Grech, E., Malarski, Z., and Sobczyk, L., Isotopic effects in NH· · ·N hydrogen bonds, Chem. Phys. Lett., 128, 259–263, 1986. 48 Zeegers-Huyskens, T. and Sobczyk, L., Isotopic ratio nHX/nDX of hydrogen halide complexes in solid argon, Spectrochim. Acta, 46A, 1693– 1698, 1990. 49 Barnes, A. J., Molecular complexes of the hydrogen halides studied by matrix isolation infrared spectroscopy, J. Mol. Struct., 100, 255– 280, 1983. 50 Barnes, A. J. and Legon, A. C., Proton transfer in amine-hydrogen halide complexes: comparison of low temperature matrices with the gas phase, J. Mol. Struct., 448, 101– 106, 1998. 51 Witkowski, A., Infrared spectra of the hydrogen bonded carboxylic acids, J. Chem. Phys., 47, 3645– 3648, 1967. 52 Mare´chal, Y. and Witkowski, A., Infrared spectra of H-bonded systems, J. Chem. Phys., 48, 3697– 3705, 1968. 53 Sokolov, N. D. and Savelev, V. A., Dynamics of the hydrogen bond: two-dimensional model and isotope effects, Chem. Phys., 22, 383– 399, 1977. 54 Robertson, G. N. and Yarwood, J., Vibrational relaxation of hydrogen bonded species in solution, I. Theory. Chem. Phys., 32, 267– 282, 1978. 55 Yarwood, J., Ackroyd, R., and Robertson, G. N., Vibrational relaxation of hydrogen bonded species in solution. II. Analysis of ns(XH) absorption bands, Chem. Phys., 32, 283– 299, 1978. 56 Bratos, S., Profiles of hydrogen stretching IR bands of molecules with hydrogen bonds: a stochastic theory. I. Weak and medium strength hydrogen bonds, J. Chem. Phys., 63, 3499– 3509, 1975. 57 Mare´chal, E. and Bratos, S., Infrared and Raman spectra of hydrogen bonded liquids. Effect of electrical anharmonicity on profiles of hydrogen-stretching bands, J. Chem. Phys., 68, 1825– 1828, 1978. 58 Mare´chal, E. and Ratajczak, H., Infrared and Raman spectra of moderatively strong hydrogen-bonded liquids. Effect of electrical anharmonicity upon profiles of hydrogen-stretching bands, Chem. Phys., 110, 103– 112, 1986. 59 Belhayara, K., Chamma, D., and Henri-Rousseau, O., Infrared spectra of weak H-bonds: Fermi resonances and intrinsic anharmonicity of the H-bond bridge, J. Mol. Struct., 648, 93– 106, 2003. 60 Wolff, H. and Muller, H., Substructure of the NH stretching vibrational band of imidazole, J. Chem. Phys., 60, 2938 –2939, 1974. 61 Wo´jcik, M. J., Theory of the infrared spectra of the hydrogen bond in molecular crystals, Int. J. Quant. Chem., 10, 747– 760, 1976. 62 Mare´chal, Y., Infrared spectra of cyclic dimers of carboxylic acids: the mechanism of H-bonds and related problems, In Vibrational Spectra and Structure, Vol. 16, Durig, J. R., Ed., Elsevier, Amsterdam, pp. 311– 356, 1987. 63 Bercmans, D., Figeys, H. P., Mare´chal, Y., and Geerlings, P., Ab Initio and LMO studies on the integrated intensities of infrared absorption bands of polyatomic molecules. 7. The formic acid dimer: influence of hydrogen bonding and isotopic substitution, J. Phys. Chem., 92, 66 – 73, 1988. 64 Mare´chal, Y., IR spectra of carboxylic acids in the gas phase: a quantitative reinvestigation, J. Chem. Phys., 87, 6344 –6353, 1987.

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65 Flakus, H. T. and Rogosz, K., On anomalous H/D isotopic effects for n(X– H) and n(X– D) band integral intensities in IR spectra of cyclic hydrogen bonded dimeric systems, J. Mol. Struct., 443, 265– 271, 1998. 66 Witkowski, A., On quantum molecular dynamics and electrodynamics, J. Chem. Phys., 79, 852– 860, 1983. 67 Witkowski, A., Separation of electronic and nuclear motions and the dynamical Schrodinger group, Phys. Rev. A, 41, 3511– 3517, 1990. 68 Gilli, P., Bertolasi, V., Ferretti, V., and Gilli, G., Covalent nature of the strong homonuclear hydrogen bond. Study of the O – H· · ·O system by crystal structure correlation methods, J. Am. Chem. Soc., 116, 909– 915, 1994. 69 Filarowski, A. and Koll, A., Specificity of the intramolecular hydrogen bond. The difference in spectroscopic characteristics of the intermolecular and intramolecular H-bonds, Vib. Spectr., 568, 123– 131, 1998. 70 Janoschek, R., Weideman, E. G., Pfeiffer, H., and Zundel, G., Extremely high polarizability of hydrogen bonds, J. Am. Chem. Soc., 94, 2387– 2396, 1972. 71 Zundel, G., Hydrogen bonds with large proton polarizability and proton transfer processes in electrochemistry and biology, Adv. Chem. Phys., 111, 1 – 217, 2000. 72 Danninger, W. and Zundel, G., Intense depolarized Rayleigh scattering in Raman spectra of acids caused by large proton polarizabilities of hydrogen bonds, J. Chem. Phys., 74, 2769– 2777, 1981. 73 Hawranek, J. P. and Muszyn´ski, A. S., Infrared dispersion of the pentachlorophenol-trioctylamine complex, J. Mol. Struct., 552, 205– 212, 2000. 74 Borgis, D., Tarjus, G., and Azzouz, H., An adiabatic dynamical simulation study of the Zundel polarization of strongly H-bonded complexes in solution, J. Chem. Phys., 97, 1390– 1400, 1992. 75 Hayd, A. and Zundel, G., The interaction of the easily polarizable hydrogen bonds with phonons and polaritons of the thermal bath - far infrared continua, J. Mol. Struct (Theochem.), 500, 421– 427, 2000. 76 Janoschek, R., Hayd, A., Weidemann, E. G., Leuchs, M., and Zundel, G., Calculated and observed isotope effects with easily polarizable hydrogen and deuterium bonds, J. Chem. Soc. Faraday Trans. II, 74, 1238– 1245, 1978. 77 Sobczyk, L., Presented at the Erasmus School Intermoleculaire Krachten, Leuven, Vol. II: pp. 436– 461, 1989. 78 Scheiner, S., Proton transfer in hydrogen-bonded systems. 4. Cationic dimers of NH3 and OH2, J. Phys. Chem., 86, 376– 382, 1982. 79 Perrin, Ch. and Nielson, J. B., Strong hydrogen bonds in chemistry and biology, Annu. Rev. Phys. Chem., 48, 511– 544, 1997. 80 Cleland, W. W. and Kreevoy, M. M., Low-barrier hydrogen bonds and enzymic catalysis, Science, 264, 1887– 1890, 1994. 81 Bien´ko, A. J., Latajka, Z., Sawka-Dobrowolska, W., Sobczyk, L., Ozeryanskii, V. A., Pozharskii, A. F., Grech, E., and Nowicka-Scheibe, J., Low barrier hydrogen bond in protonated proton sponge. X-ray diffraction, infrared, and theoretical ab initio and density functional theory studies, J. Chem. Phys., 119, 4313– 4319, 2003.

11

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics Michael Hippler and Martin Quack

CONTENTS I.

II.

III. IV.

V.

Introduction ...................................................................................................................... 306 A. Principles of Isotope Effects in Infrared Spectroscopy and Molecular Dynamics ......................................................................................... 306 B. Intramolecular Dynamics and Isotope Selective Spectroscopic Techniques: An Overview ............................................................................................................ 308 Intramolecular Redistribution Processes: From High-Resolution Spectroscopy to Ultrafast Intramolecular Dynamics ............................................................................. 311 A. Intramolecular Quantum Dynamics and Molecular Spectroscopy ......................... 311 B. Spectroscopic States and Intramolecular Dynamics: An Intuitive Perspective........................................................................................... 313 1. General Aspects................................................................................................. 313 2. An Example of Two-Level Dynamics.............................................................. 314 3. Coupling Many Levels in a Multistate Dynamics............................................ 316 The Experimental Approach to Infrared Spectroscopy with Mass and Isotope Selection (IRSIMS) ...................................................................................... 317 Mass Selective Overtone Spectroscopy by Vibrationally Assisted Dissociation and Photofragment Ionization: OSVADPI ...................................................................... 320 A. Mechanism of Vibrationally Assisted Dissociation and Photofragment Ionization......................................................................................... 320 B. Isotopomer Selective Overtone Spectroscopy of the Nj ¼ 42 CH Chromophore Absorption of CHCl3 ................................................................. 323 C. Isotopomer Selective Overtone Spectroscopy of the Nj ¼ 41 CH Chromophore Absorption of CHCl3: A Hierarchy of Time Scales and Isotope Effects in Intramolecular Vibrational Energy Redistribution (IVR) ................................................................................................ 325 Isotope Selective Overtone Spectroscopy by Resonantly Enhanced Two-Photon Ionization of Vibrationally Excited Molecules .......................................... 329 A. Overview .................................................................................................................. 330 B. Mechanism of Resonantly Enhanced Two-Photon Ionization of Vibrationally Excited Molecules......................................................................... 331 C. The N ¼ 2 NH Chromophore Absorption of Aniline Isotopomers Near 6750 cm21: Isotope Effects and Vibrational Mode Specificity in IVR and Tunneling Processes.......................................................................................... 334 D. 13C Isotope Effects in the IVR of Vibrationally Excited Benzene......................... 338 305

306

Isotope Effects in Chemistry and Biology

VI. Conclusions and Outlook ................................................................................................. 346 Acknowledgments ........................................................................................................................ 348 References..................................................................................................................................... 348

I. INTRODUCTION A. PRINCIPLES OF I SOTOPE E FFECTS IN I NFRARED S PECTROSCOPY AND M OLECULAR DYNAMICS Our chapter deals with isotope effects in intramolecular dynamics as observed and used in certain new spectroscopic techniques. Molecular isotope effects arise from four conceptually distinct origins connected to different properties of the isotopic nuclei: (i) (ii) (iii) (iv)

Mass differences Different spins Different “Pauli” identity Different electroweak charge of the isotopes.

While in practice the four qualities come as combinations when changing one isotope against another, one can nevertheless identify certain limiting situations where one quality becomes dominant, and we shall discuss these in turn in our general introduction to this chapter. (i) Mass differences of the isotopes are the most common source of molecular isotope effects and these are discussed in many chapters in this book; see also in particular the chapters by Jacob Bigeleisen (this volume, chapter “Theoretical Basis of Isotope Effects from an Autobiographical Perspective”) and Max Wolfsberg (this volume, chapter “Comments on Selected Topics in Isotope Theoretical Chemistry”). The difference in infrared spectra of isotopomers can be understood to lowest order by the change of harmonic wavenumber v~e of a harmonic oscillator sffiffiffiffi v 1 f ð11:1Þ v~e ¼ ¼ c 2pc m and to a change of rotational constant of a rigid rotor (as wavenumber). h Be ¼ 2 8p cmre2

ð11:2Þ

Here c is the speed of light in vacuum, f is the force constant of the oscillator as the second derivative of the potential energy V(r) at the equilibrium bond length re: ! ›2 VðrÞ f ¼ ð11:3Þ ›r 2 r¼re which is independent of isotopic substitution as is also the bond length re at equilibrium (minimum of V) in the Born –Oppenheimer approximation. On the other hand, the reduced mass m changes strongly for different isotopomers, say in diatomic molecules with

m¼

mA mB ¼ mA þ mB

1 1 þ mA mB

21

ð11:4Þ

and with the masses mA and mB of the two nuclei. Through the approximate energy level expressions of the harmonic oscillator – rigid rotor energies Ev 1 ¼ v~ v þ hc 2

ð11:5Þ

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

307

with vibrational quantum number v ¼ 0; 1; 2; … and EJ ¼ Be JðJ þ 1Þ hc

ð11:6Þ

with the rotational quantum number J ¼ 0; 1; 2; …; the change in the masses of isotopes of atom A or B through a change of m, Be lead v~e lead to obvious and, indeed, dramatic changes in the energy level structure and the corresponding transition wavenumbers observed in infrared spectra of diatomic molecules: H/D isotope substitution often leads to wavenumber shifts of hundreds of cm21, and even 13C/12C shifts are easily several tens of cm21, whereas 35Cl/37Cl isotope shifts are observable only at higher resolution, but nowadays in a standard way even in an undergraduate student spectroscopy laboratory. Extending these considerations from diatomic to polyatomic molecules,1 – 5 one frequently obtains comparable mass-dependent isotope shifts in the rigid rotor harmonic oscillator approximation. The situation can become more complex in reality, as some vibrations may show very little isotope shifts and spectral congestion may prevent their resolution, a point to which we shall return. However, the principles of the mass-dependent isotope effects are rather well understood even when including anharmonicity and nonrigidity as well as non-Born– Oppenheimer effects. The effect is generally large with large relative reduced mass differences and would vanish in the limit where lðm1 2 m2 Þ=m1 l approaches zero (or m1 =m2 approaches 1). Mass-dependent isotope effects are also important in reaction dynamics, particularly when tunneling or resonances become important.6 – 9 (ii) The second isotope effect in spectra and dynamics arises from different nuclear spins of the isotopes. In nuclear magnetic resonance spectra it is most obvious, determining even the existence (say for 13C isotopes) or absence (for 12C) of such spectra,10,11 but in combination with the Pauli Principle (iii) it also determines relative strength of spectral lines in microwave, infrared (IR), and optical spectra via nuclear spin statistical weights and strengths of spectral lines, including also the possibility of the presence or absence of such lines.1 – 5,10 – 15 It would in principle occur with isotopes of different spin even in the limit where ðm1 =m2 Þ ! 1: Although in practice there will always be a mass difference, for very heavy isotopes this ratio may approach 1 whereas the spin effects remain (see also the chapter “Hydrogen Bond Isotope Effects Studied by NMR”, by Limbach, H.H., Denisov, G.S. and Golubev, N.S., this volume). (iii) The third case of isotope effects in molecular dynamics is even more subtle as it would appear for isotopes with ðm1 =m2 Þ ! 1 and nuclear spins I1 ¼ I2 : It arises entirely through symmetry selection rules in spectra and dynamics of different isotopomers being different. Literally it might become dominant for nuclear isomers of almost the same mass and the same spin (say I1 ¼ I2 ¼ 0), a fairly exotic situation, but it may contribute also in other situations. It was apparently first postulated to be of potential importance in state-to-state chemical dynamics of heavy isotopes or nuclear isomers in 197714,15 and has since then been discussed to be at the origin of certain observations of isotope effects in chemical systems (Ref. 16 and this volume, chapter “Nonmass-Dependent Isotope Effects” by Ralph E. Weston). To our knowledge, no firm spectroscopic example of this type has really been studied in isolation from the other two effects (i) and (ii), but it is, of course, always present in combination with these in spectroscopy through the symmetry selection rules. (iv) While the three isotope effects mentioned so far can be understood based on quantum mechanics and physics of the first half of the 20th century, the last molecular isotope effect to be mentioned here is related to electroweak parity violation and the electroweak nuclear force influencing molecular dynamics. These phenomena have become the subject of theoretical investigations only during the last few decades, with some striking recent advances in the framework of new developments in electroweak quantum chemistry.17 – 21 While the electric charge qe ¼ ZA e for different isotopes of the same element is the same, the electroweak charge Qw ðAÞ ¼ ZA ð1 2 4 sin2 Qw Þ 2 NA

ð11:7Þ

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Isotope Effects in Chemistry and Biology

is different for different isotopes (with proton number ZA and neutron number NA), because of the different NA. Therefore the electroweak electron nucleus interaction mediated by the Z-Boson as a contact like interaction is different for different isotopes with the approximate parity violating Hamiltonian: ^ pv ¼ H

N n X pGF X pffiffi Qw ðAÞ ½~pi~si ; d3 ð~ri 2 ~rA Þ hme c 2 A¼1 i¼1

þ

ð11:8Þ

where [,]þ stands for the anticommutator (see Refs. 17 –23) and with the Fermi constant GF, electron mass me, its momentum p~ i and spin operators ~si ; the three-dimensional Dirac delta distribution d3 and positions ~ri of electrons and ~rA of the nuclei. This is perhaps the most exotic of the isotope effects in molecular spectroscopy and dynamics. It has been most recently quantitatively predicted to lead to a ground-state energy difference between chiral isotopomers such as PF35Cl37Cl (and similarly HCF35Cl37Cl, etc.).22 This difference is predicted to be very small (about 10213 cm21), but would be exactly zero for symmetry reasons if only (i) and (ii) are considered. While its spectroscopic observation is possible in principle,23,24 it has not yet been realized and should be seen somewhat in the future. After this introductory survey of the principles of molecular spectroscopic and dynamical isotope effects, we shall now turn to a discussion of isotope effects as observed and used in practice in some recently developed spectroscopic techniques where the mass effect (i) is of greatest relevance (combined with (ii) and (iii) to some extent).

B. INTRAMOLECULAR DYNAMICS AND I SOTOPE S ELECTIVE S PECTROSCOPIC T ECHNIQUES: A N OVERVIEW In homogeneous structures of high-resolution IR spectra, information on mechanisms and time scales of ultrafast intramolecular dynamic processes is encoded. By an analysis of homogeneous splittings and line widths, rovibrational Hamiltonian and time-evolution operators can be extracted which contain this relevant information.25 – 30 This allows the detailed study of dynamic processes on femtosecond to picosecond time scales, for example of Intermolecular Vibrational Redistribution (IVR), predissociation, or tunneling processes. In this context, it is particularly interesting to investigate the influence of vibrational excitation on intramolecular dynamics, since vibrational mode selectivity allows one to influence reactions by selective vibrational excitation, perhaps also after further electronic excitation.31,32 To understand isotope effects in chemical reactions, it is also important to investigate the influence of different isotope compositions of a molecule (isotopomers) on intramolecular dynamics. The ultimate aim of such studies is the better understanding of intramolecular primary processes and unimolecular reactions in polyatomic molecules at the fully quantum dynamical level, a level which goes far beyond simplifying statistical theories. This aim remains among the most challenging research questions in physics and chemistry, with applications also in biology and environmental sciences. In our approach to extract detailed molecular dynamics, a high-resolution analysis of complex and often weak band structures of rovibrational spectra is required.25 – 30 Unfortunately, IR spectra are often congested by hot-band transitions and due to the mixture of isotopomers, and thus reliable assignments were often not possible in the past: extensive hot-band congestion in room-temperature spectra of polyatomic molecules requires jet-cooling to obtain vibrationally resolved spectra. The resulting low number densities in conjunction with low absorption cross-sections will therefore necessitate very sensitive detection of the IR excitation. In addition, molecules often exist as a mixture of different isotopomers at natural abundance. The resulting congestion of rovibrational spectra then often prevents a detailed analysis; in many cases, isotopically pure samples are not available. It is thus desirable to use experimental techniques which allow the separation of spectral contributions from different isotopomers and which help to reduce the spectral congestion, for example by supersonic jet cooling.

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During the last ten years, we successfully developed new experimental techniques for the IR spectroscopy of gas-phase molecules, where the selective ultraviolet (UV) ionization of vibrationally excited molecules in IR þ UV double-resonance schemes allows indirect, but extremely sensitive detection of the IR excitation.30,33 – 38 Overtone transitions are intrinsically very weak. Instead of monitoring directly the IR absorption, one can couple IR excitation with further UV photon absorption steps leading finally to ionization. This allows indirect, but much more sensitive detection of IR excitation, since molecules are not ionized efficiently without vibrational excitation as resonance enhancement. The ionization yield thus mirrors the vibrational excitation. Ionization detection of IR excitation may increase the sensitivity and selectivity of IR spectroscopy, since electrostatic fields can extract all ions and even single ions can be detected in a mass spectrometer, in principle. The technique has been applied to overtone spectroscopy in supersonic jet expansions, where the cooling of vibrational and rotational degrees of freedom greatly reduces hot-band congestion and simplifies spectra. Ionization detection of the IR excitation can be coupled very efficiently with a time-of-flight (TOF) mass spectrometer, which allows the separation of spectral contributions of different components in a mixture. Mass spectrometry is thus added as a second dimension to IR spectroscopy, which greatly increases the selectivity. The IR excitation also increases the selectivity of mass spectrometry, since it allows the separation of species which have nearly the same masses (isobars), making use of the isotope effects (ii) – (iv) mentioned above in the case of isotopomers. Two different IR þ UV doubleresonance schemes are reviewed here; they are distinguished by the nature of the intermediate, electronically excited state: if it is dissociative, then overtone spectroscopy by vibrationally assisted dissociation and photofragment ionization (OSVADPI) allows the detection of the IR excitation in the parent molecule by observing fragment ions.30,33 – 36 With a bound intermediate state, resonantly enhanced two-photon ionization (RE2PI) of vibrationally excited molecules is very efficient via hot-band transitions of the excited state, which represents a quasi-continuum due to its high vibrational density of states.30,37,38 In both schemes, synchronous tuning of the UV laser during an IR spectral scan is not required to keep the resonance condition. Employing these techniques, we have studied isotope effects in IR spectroscopy and intramolecular dynamics of Cl isotopomers of some aliphatic chlorides, such as chloroform, CF2HCl, and tert-butyl chloride,30, 33 – 36 of H/D-isotopomers of aniline30,37 and of 12C/13C isotopomers of benzene,30,38 and we will review here some selected examples. We also review some recently developed alternative spectroscopic techniques, which are somewhat related to our own original developments. In infrared resonance enhanced multiphoton ionization (IR-REMPI), molecules are vibrationally excited and then ionized by multiphoton absorption in the intense IR laser field of one tightly focused laser beam. This vibrational preionization technique was demonstrated by us39 on C60 and later applied by others to the IR spectroscopy of fullerenes and other molecules.40,41 It is confined, however, to special cases where the ionization limit is low and comparable to the dissociation threshold (or lower); otherwise dissociation would predominate over ionization. In another technique, vibrationally excited molecules are ionized by one-photon ionization of a vacuum ultraviolet (VUV) laser.42,43 In general, however, it may be difficult to select a suitable VUV wavelength that allows sufficient discrimination between the ionization of ground state and excited state molecules:42 the additional IR energy is rather small compared to the VUV energy, and the ionization cross-section does often not change enough with energy, except at a sharp ionization threshold. In addition, the use of involved VUV laser systems may also not always be convenient. Alternatively, vibrationally excited molecules could be ionized by UV two-photon ionization, which is resonantly enhanced by an intermediate electronically excited state.42 Convenient frequency-doubled dye lasers can then be used. In previous implementations, distinct rovibrational levels of the resonance enhancing intermediate state were selected.42,44,45 Because of the increased selectivity by the additional selection rules imposed by the intermediate level, this double-resonance technique may aid the assignment of selected IR absorption features.44 As the UV ionization is via distinct, separated

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Isotope Effects in Chemistry and Biology

transitions, the UV laser has to be scanned simultaneously to keep the resonance during an IR scan. This is difficult to achieve, however, and no IR spectra obtained in this way have been reported. Recently, an IR þ UV absorption scheme was presented, where vibrationally excited phenol was ionized by nonresonant two-photon ionization.46,47 As discussed below, however, ionization presumably also occurred in these cases by RE2PI, as in the present scheme, which is based on the original development of OSVADPI.30,33 – 38 In another scheme based on photofragment spectroscopy, an IR laser promotes molecules to predissociating vibrational levels, and the fragments of this vibrationally mediated dissociation are probed by fluorescence detection, for example by laser-induced fluorescence (LIF) of a second UV laser.31,32,48,49 Since the yield of fragments mirrors the IR excitation, IR spectra are obtained indirectly, but very sensitively by scanning the IR laser while monitoring the photofragment yield. The scheme can also be applied to the IR spectroscopy of bound vibrational states. Vibrationally excited molecules are then dissociated by the further absorption of one IR or UV photon,31,48 – 50 or alternatively by multiphoton absorption of several CO2 laser photons — infrared laser-assisted photofragment spectroscopy (IRLAPS),51 – 55 which can also be used as a scheme for increasing selectivity in laser isotope separation.56,57 Besides allowing IR spectroscopy, vibrationally mediated dissociation also provides insight into the dynamics of the dissociation process.31,58 – 60 As a limitation, however, these schemes require fragments that can be probed by LIF. Dip spectroscopy is a double-resonance technique, where an UV laser is tuned to an electronic transition of ground-state molecules, which is monitored either by LIF or resonantly enhanced multiphoton ionization (REMPI) detection. If an IR laser depopulates ground state levels by rovibrational transitions first, a dip in the fluorescence or ionization signal is observed which mirrors the IR excitation.44,61 – 66 The main advantage of dip spectroscopy is the added selectivity provided by the UV excitation: UV transitions of specific isomers, isotopomers, clusters or species in a mixture can be selected, and the IR dip spectrum then corresponds to the selected species. Since dip signals are observed against a strong background, these methods suffer from poor signal-to-noise ratios. For one-photon IR absorption, the selection rules of electric dipole transitions within the electronic ground state allow only transitions changing several vibrational quanta due to anharmonicity, and these transitions are very weak. Many experimental studies in overtone spectroscopy are thus concerned with combination and overtone levels involving anharmonic hydride stretching vibrations, for example. In a different approach, vibrational levels of the electronic ground state are reached by two-photon excitation by stimulated transitions via excited electronic states (stimulated emission pumping, SEP)25,67 – 71 or virtual levels (stimulated Raman excitation, SRE).25,72 – 76 Since different electronic states are involved in the two-photon excitation, different selection rules apply; the transition strength is then mainly governed by the Franck –Condon principle, and thus different vibrational levels can be reached compared to direct one-photon IR absorption. SEP and SRE have been combined with photoacoustic spectroscopy,76 photofragment spectroscopy by vibrationally mediated dissociation,74,76 ionization gain72 and loss (dip) spectroscopy,70,73 LIF detection75 and fluorescence dip spectroscopy,71 for example. Double-resonance schemes, where two sequential IR absorption steps are involved to reach highly vibrationally excited levels, are also employed to increase sensitivity and selectivity.51 – 53,77 – 81 Since fundamental bands often show resolved rotational structure, a first IR photon may excite a selected rovibrational level via a fundamental transition, and a second IR photon of different wavelength then promote to a high overtone level, which allows a rotational state selected vibrational overtone spectroscopy.51,52,79 – 82 By the selection of the first absorption step, inhomogeneous congestion can be removed. To take advantage of the resolution obtainable with microwave spectroscopy, microwave transitions may also be included into the absorption chain.77, 83 – 85 For example, a microwave transition can provide the last increment of energy required for dissociation in photofragment spectroscopy.85 As a disadvantage, double-resonance schemes do not measure directly absolute absorbances. In addition, the more laser systems are involved in a set-up, the more complex and less robust the experiment will become in general.

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311

This short introduction of some recent developments of sensitive and selective IR spectroscopic techniques was meant to establish some concepts and to stress that each technique has its own merits and limitations. Ideally, an experimental technique would be as generally applicable as possible, provide high resolution and combine high sensitivity with selectivity, but yet would not be too involved. We think that our mass and isotope selective IR þ UV schemes which will be presented in this review have distinct advantages in this regard. This review is structured as follows: After this introduction, a short section is given on the theory of intramolecular redistribution processes, in particular to clarify how dynamic, timedependent processes can be deduced from high-resolution spectroscopy, which essentially probes time-independent states, without important time resolution. A section then follows on details of our IR þ UV double-resonance experiments for the IR spectroscopy with isotope and mass selection. In the remaining sections, the mechanisms of the IR þ UV schemes with dissociative and bound intermediate electronic states are discussed, and selected applications of these spectroscopic techniques to the study of isotope effects on intramolecular dynamics and of vibrational mode-specific IVR, tunneling, and stereomutation dynamics of chloroform, aniline, and benzene isotopomers are presented.

II. INTRAMOLECULAR REDISTRIBUTION PROCESSES: FROM HIGH-RESOLUTION SPECTROSCOPY TO ULTRAFAST INTRAMOLECULAR DYNAMICS A. INTRAMOLECULAR Q UANTUM DYNAMICS AND M OLECULAR S PECTROSCOPY The time-dependent quantum dynamics of molecules as of other quantum systems is governed by the time (t)-dependent Schro¨dinger equation i

h ›Cðq; tÞ ¼ H^ Cðq; tÞ 2p ›t

ð11:9Þ

where q represents the whole set of particle coordinates and spins and Cðq; tÞ is the corresponding time-dependent wavefunction. The molecular Hamiltonian is time independent in the absence of external fields, but may become time dependent in their presence. The general solution is given with ^ t0 Þ the time-evolution operator Uðt; ^ t0 ÞCðq; t0 Þ Cðq; tÞ ¼ Uðt;

ð11:10Þ

which satisfies an equation similar to Equation 11.9: i

^ h ›U ^ ¼ H^ U 2p ›t

ð11:11Þ

^ is solved by which with a time-independent Hamiltonian H ^ t0 Þ ¼ expð22piHðt ^ 2 t0 Þ=hÞ Uðt;

ð11:12Þ

In this latter case one can write the solution for Cðq; tÞ explicitly in terms of the time-independent molecular Schro¨dinger equation: H^ wk ðqÞ ¼ Ek wk ðqÞ

ð11:13Þ

312

Isotope Effects in Chemistry and Biology

with the stationary state eigenfunctions wk ðqÞ depending only on q (not on t) and the corresponding eigenenergies Ek. The general Cðq; tÞ takes then the form X Cðq; tÞ ¼ ð11:14Þ ck wk ðqÞexpð22piEk t=hÞ k

with time-independent complex coefficients ck determined by the initial condition. With time^ the ck in this solution would become time dependent as well.247 dependent H; In principle, high-resolution molecular spectroscopy provides relevant information on wk ðqÞ and Ek and thus on the time-dependent molecular quantum dynamics.25 In practice a first spectroscopic analysis relates to an effective Hamiltonian ZT Heff Z ¼ DiagðE1 ; …; En Þ

ð11:15Þ

where Z is the eigenvector matrix of Heff in some unknown basis x ¼ ðx1 · · ·x2 ÞT : In order to obtain the true variational molecular Hamiltonian in an eigenstate basis wk ; which can be described in ordinary coordinate space for the molecule, we have to find an appropriate transformation:

w ¼ ZT x

ð11:16Þ

MOLECULAR SPECTRA High-Resolution Molecular Spectroscopy Fourier Transform Spectroscopy Laser Spectroscopy (Symmetry Selection Rules)

Effective Hamilton Operators (Effective Symmetries)

Ab initio Hamilton Operators (Theoretical Summetries)

Rovibronic Schrödinger Equation

Molecular Hamilton Operators H (Observed Symmetries)

Electronic Schrödinger Equation

Ab initio Potential Energy Hypersurfaces and Hamilton Operators (Symmetries)

Time-Evolution Operator Û (Matrix)

Molecular Kinetics and Statistical Mechanics (Conservation Laws and Constants of Motion)

MOLECULAR DYNAMICS FIGURE 11.1 Scheme of the combined experimental – theoretical approach for the determination of molecular dynamics from molecular spectra (Source: From Quack, M., Chimia, 55, 753– 758, 2001. With permission).

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

313

Directly measurable by spectroscopy are the differences Ek 2 Ej : The scheme shown in Figure 11.1 illustrates the path from molecular spectra to molecular dynamics in the intermediate steps of the effective and true molecular Hamiltonians.25 – 29 The careful analysis of these steps was carried out by our group only in the years 1980– 1990, the difference between the “true” and effective Hamiltonian analysis being previously not taken into account by practicing spectroscopists. This point turns out to be particularly important for dynamical isotope effects which are properly accounted for only in the case of using true molecular Hamiltonians or else using the proper transformations.25,86 It may be noted that in general, if somewhat abstract terms, the time evolution operator equation (11.11) provides a general solution to the problem of time dependent molecular quantum dynamics including any type of time dependent observable in the Heisenberg equations of motion or the Liouville – von Neumann equation for the density matrix (operator) of an ensemble. Rather than pursuing these more formal aspects we shall turn now to a more elementary, intuitive perspective.

B. SPECTROSCOPIC S TATES AND I NTRAMOLECULAR DYNAMICS: A N I NTUITIVE P ERSPECTIVE 1. General Aspects Time-independent, molecular states are eigenstates of the time-independent Hamiltonian operator in the Schro¨dinger equation, Equation 11.13, with the time-dependent wavefunction for an eigenstate:

Ck ðtÞ ¼ wk e22piEk t=h

ð11:17Þ

The square of the wavefunction of such stationary states is time independent, and therefore nothing moves in the molecule, only the phase is time dependent. These eigenstates are the spectroscopic states, which are observable by high-resolution spectroscopy of isolated molecules (neglecting the natural lifetime and width due to spontaneous emission).25 – 30,87 – 90 A spectrum at highest resolution consists in principle of discrete lines having the natural line width corresponding to these eigenstates. The lines are in most cases overlapping to give rise to homogeneous and inhomogeneous structures.25,91,92 Homogeneous structures arise by absorption from one initial energy eigenstate. Inhomogeneous structures result from overlapping transitions from different initial quantum states, where the Doppler width of rovibrational lines, hot-band transitions or transitions from different species, isotopomers, or isomers are examples. Using special techniques to remove spectral congestion (double-resonance experiments, supersonic jet cooling), “eigenstate-resolved” spectra have been obtained in the past for some simple molecules.25,51,52,77,78,83,93 – 97 For vibrational states with small amplitudes, the harmonic approximation in the potential energy function is appropriate, and the normal modes resulting from a harmonic analysis correspond then to a good degree of approximation to the molecular eigenstates. This will in general be true for fundamental vibrations, except for large amplitude motions. For overtone and combination states at higher vibrational excitation, anharmonicity in the potential energy becomes more and more important, invalidating the harmonic approximation. The molecular eigenstates are then a mixture of the formal, “zeroth-order” normal mode overtone and combination states. The mixing of idealized normal mode vibrations, which are based on a harmonic approximation, is due to anharmonicity, hence the name “anharmonic resonances” between normal mode states of similar energy.1 – 5,25 The coupling or mixing coefficients can be calculated approximately by perturbation theory. They are frequently determined by the matrix elements of the perturbing, anharmonic potential terms in the basis of harmonic oscillator wavefunctions.5 Important for very fast relaxation processes are distinct anharmonic resonances, in particular Fermi resonances with an exchange of one quantum of excitation in one vibrational mode with two quanta in another mode, or Darling– Dennison

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Isotope Effects in Chemistry and Biology

resonances with an exchange of two quanta of vibrational excitation in one mode with two quanta in another mode.5,25,35,37,51 – 53,93 – 95,98 – 112 Fermi resonances were first observed and interpreted in the CO2 spectrum,112 and have been extensively studied since then in the IR spectra of the CH chromophore, for example Refs. 25,87,93,94,98– 108,113 – 115. Higher-order anharmonic resonances account for a further mixing and redistribution of vibrational excitation. All resonances become particularly effective if the zeroth-order energies arising from the involved idealized, normal mode vibrations are similar (resonance condition). In addition, rovibrational Coriolis resonances from the coupling of rotation with vibrational motions may also be present.1 – 5 The wavefunctions of highly excited eigenstates are in general spread over the entire molecule and not localized, and they are thus also named “global vibrational states.”90 Time-dependent states, where the atoms in the molecule are moving, are described as the superposition of time-independent eigenstates (Equation 11.14).25,27,28,77,86 – 89 The time dependence arises from the spreading of phases of the vibrational eigenstates from a concerted ordering of phases to give rise to an initial state Cðt ¼ 0Þ to essentially a random, unordered mixture. In this way, vibrational energy originally located in an initial vibrational state is distributed among other vibrational modes as a function of time: time-dependent IVR; it is the complement of the time-independent picture of IVR, visible in the nonseparability of the wk in a set of coordinates (normal modes).25,28 The time dependence in Equation 11.14 allows in principle recurrences of occupation probability of the initial state, but if the time-dependent state is composed of many eigenstates, Equation 11.14 describes basically a relaxation process. Highresolution spectroscopy identifies eigenstates, and from their analysis and assignment, an effective Hamiltonian and finally the true complete molecular Hamiltonian of the molecule can be constructed, perhaps supported by ab initio calculations.25 – 28,67 Zeroth-order states in perturbation theory are usually harmonic normal mode vibrations. Their energies provide the diagonal elements in a matrix representation; perturbation energies give the off-diagonal elements. Matrix diagonalization yields the energies and the vector composition of the true, perturbed states with respect to the basis functions used. Since the perturbation treatment is in general limited to a low-order approximation, the corresponding model Hamiltonian is not the true molecular Hamiltonian, but an effective Hamiltonian.25 – 28 The diagonal and off-diagonal elements in the matrix representation are usually considered as fit parameters to the observed spectrum. The effective Hamiltonian describes the observed spectrum in general very well, since mathematically it originates from a transformed molecular Hamiltonian with similar or the same eigenvalues as the true Hamiltonian. The basis functions, however, are also transformed and are then in general not known explicitly. Full knowledge of the true wavefunctions is only possible if the important step from an effective model Hamiltonian with unknown basis to the complete molecular Hamiltonian with known wavefunctions is made.25 – 28 This will finally allow the prediction of how an arbitrary initially created state will evolve in time in full detail, not only providing the time scale of decay of the initial state, but also the mechanisms of intramolecular redistribution processes. This approach of extracting dynamic information is summarized in Figure 11.1; it is much more than just “Fourier-transforming” spectra, as it invokes a crucial analysis step for the wavefunction.27 2. An Example of Two-Level Dynamics As the simplest case, one may consider two states w01 and w02 with energy E10 and E20 ; respectively, coupled by an off-diagonal matrix element (perturbation energy) 2 W: The matrix representation of the effective Hamiltonian is therefore E10

2W

2W

E20

!

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

315

The eigenfunctions w1 and w2 with energies given in Equation 11.18 are observable by highresolution spectroscopy in terms of the eigenenergies E1 and E2 : vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! u 0 u E 2 E0 2 E10 þ E20 1 2 t þW 2 ð11:18Þ 7 E1;2 ¼ 2 2 If the unperturbed states are degenerate (or nearly degenerate), E10 ¼ E20 ¼ E0 ; the solution is especially simple, E1;2 ¼ E0 7 W; and the corresponding wave functions can be chosen as w1;2 ¼ p1ffi ðw0 ^ w0 Þ: The perturbed wavefunctions are the symmetric and antisymmetric superposition of 1 2 2 the unperturbed states. By a coherent laser pulse, which, due to its time dependence, has a frequency distribution covering both transitions to the spectral eigenstates w1 and w2 ; a time-dependent coherent superposition state C can be created which corresponds to the unperturbed state w01 (Equation 11.19): 1 Cðt ¼ 0Þ ¼ pffiffi ðw1 þ w2 Þ ¼ w01 2

ð11:19Þ

As can be seen from Equation 11.14, CðtÞ ¼ p1ffi2 ðw1 2 w2 Þ ¼ w02 after t ¼ h=2DE; and after t ¼ h=DE; it recurs to w01 : C thus oscillates with a period

t¼

h DE

ð11:20Þ

between the two states w01 and w02 : In practice, further couplings not considered in the simple two-state model damp the oscillation and eventually transfer vibrational excitation out of these two states. A typical example where such a two-state model would be adequate is provided by a local mode resonance, that is, a relatively weak resonance, which becomes apparent due to an (accidental) frequency match of the two unperturbed vibrational states. Two equivalent, isolated oscillators provide an example for two-state coupling, for example the hydride stretchings in the NH2 or CH2 group. This “local mode” coupling can be the consequence of an entirely harmonic Hamiltonian and would not be apparent in a normal mode picture. At high vibrational excitation, one often observes two nearly degenerate pairs of spectral states, which correspond to the symmetric and antisymmetric combination of localized vibrations in one of the two bonds. By coherent excitation of these two eigenstates, a time-dependent, localized state can be created which will oscillate between the two equivalent localized vibrational modes with period t.25,26,90 In all these examples, timeindependent spectroscopy measures the energy difference DE, which allows the conclusion by quantum mechanics that a time-dependent state can in principle be created by coherent excitation by laser pulses, which will oscillate with period t ¼ h=DE: The nature of the migration of excitation between these two local modes or bond modes is of some interest. While it certainly corresponds to a migration of vibrational energy from one bond to the other bond, it is not necessarily linked to IVR.116 Indeed in a normal mode picture, the excitation energy would remain localized and conserved within one normal coordinate (or separately within each normal coordinate) and thus separability excludes true IVR.28 An apparently similar, but in fact quite different type of two-state coupling leads to true IVR: anharmonic Fermi resonances or Darling– Dennison resonances. If such resonances occur as two level resonances, which is the simplest case, then they are again described by Equation 11.18 to Equation 11.20 with a periodic oscillation. In a Fermi resonance the two coupled states correspond to an excitation of one normal mode with one quantum and another normal mode with two quanta, both levels of about the same energy, coupled by anharmonic potential terms. Here the separability of the dynamics in the normal modes is destroyed and the time dependence corresponds to a true flow of energy between different modes. In a Darling –Dennison resonance, the two coupled levels correspond both to an excitation of two normal vibrations, each with two quanta, the coupling being again anharmonic and nonseparable.

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3. Coupling Many Levels in a Multistate Dynamics In a more complex case, one state may be coupled to another state, which is coupled to a third state, and so on, leading to a chain-like coupling structure with a tridiagonal effective Hamiltonian matrix representation.25 – 28,98 A particularly well-studied example of this type is given by the Fermi resonance between stretching modes ns and bending modes nb in isolated hydride bonds, e.g., the CH chromophore. By this resonance, 3ns is coupled to the combination band 2ns þ 2nb ; which is coupled to ns þ 4nb ; which in turn is coupled to 6nb ; for example. All vibrational modes with polyad quantum number N ¼ vs þ 0:5vb are thus coupled together, where the v denote vibrational quantum numbers. Off-diagonal coupling matrix elements are given as function of a parameter ksbb, which would correspond at a low order of perturbation theory to the cubic anharmonic potential term Csbb. In practice, ksbb is just a fit parameter to the observed spectrum and may differ markedly from Csbb.25 – 28 Matrix diagonalization yields the energies of the molecular eigenstates. It is often assumed that only the zeroth-order state with the most stretching quanta carries noticeable transition strength (“bright state”), since hydride stretching modes are in general very anharmonic. The perturbed states then gain their transition strength through the admixture of bright state character in this simple model, which allows the prediction of transition energies and relative intensities within one polyad. With a very short laser pulse exciting all polyad members coherently, the pure hydride stretching mode can be prepared in principle, which will then redistribute vibrational energy to the stretching/bending manifold, since it is a time-dependent state due to the Fermi-resonance couplings. With increasing vibrational excitation, the density of vibrational states becomes very high, especially for molecules with low-frequency vibrational modes. In aniline, for example, the density of vibrational states is estimated to be about 1.7 £ 105 states per cm21 interval in the first NH-stretching overtone region. All these background states might be coupled at least weakly with the chromophore state giving rise to homogeneous spectral structures. The weak couplings typically govern the dynamics on time scales of psec to nsec. Although an exhaustive analysis of these couplings and thus of all details of the redistribution process appears very difficult, except perhaps for some simpler cases employing double-resonance experiments at very high resolution, a simplified interpretation on the basis of a statistical model is often still possible. Assuming a statistical coupling with some average coupling strength, one expects and often observes in experiment a Lorentzian envelope over the homogeneous structure, or even an apparent Lorentzian line width due to the overlapping of lines composing the homogeneous structure. In such a statistical model, which is similar to the Bixon – Jortner model of electronic relaxation processes,117 the full width at half maximum (FWHM) G of the Lorentzian distribution indicates an exponential decay of the initial vibrational chromophore state to the background states with relaxation time t ¼ h=2pG:90 An analogous interpretation applies to the special case of vibrational predissociation or preionization,39, 118 – 121 where the initial vibrational state decays exponentially to a true continuum of background states with relaxation time t; which can be again inferred from the Lorentzian line width G:90,122 In all cases, the accuracy of description of time-dependent and -independent states involved depends on the level of theory on which the effective Hamiltonian was derived. In general, several mechanisms may be active for intramolecular redistribution processes of an initially prepared timedependent state, and the interaction of different mechanisms can be studied.37 Sometimes a distinct ordering of coupling strengths and thus also of mechanisms can be observed, leading to a hierarchy of time scales in IVR.35,51,52,92,95,100 The understanding of spectra and the dynamics of highly excited vibrational states through overtone spectroscopy25 is among the most exciting current research subjects, since it is intimately related to primary processes in laser chemistry and reaction dynamics.26 It may appear at first glance that IVR is a molecule-specific process which cannot be controlled, and that therefore mode-specific laser chemistry is impeded by the very fast IVR processes. This is, however, not necessarily true. The knowledge of IVR time scales suggests first the use of very short (fsec to psec) laser pulses to “beat” IVR and thus to control the outcome

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of laser-driven chemistry. Furthermore, the molecular Hamiltonian derived from a detailed analysis of experimental spectra allows the prediction of very special initial time-dependent states with “unusual” properties, having for example slowed-down IVR processes or IVR processes, which proceed in a predefined way leading to the breaking of a selected chemical bond.77,89,123,124 Such “designed” initial states can be prepared in principle by the coherent superposition of eigenstates by laser pulses with specific amplitude and phase relationship between frequency components. Although at its very beginnings, this new approach to mode-specific laser chemistry is a “hot topic” of current research,125 – 127 but closely related to earlier suggestions on mode selective chemistry.6,27,29,31,104,123,238,244 – 247

III. THE EXPERIMENTAL APPROACH TO INFRARED SPECTROSCOPY WITH MASS AND ISOTOPE SELECTION (IRSIMS) Our experimental set-up for mass selective IR spectroscopy is shown schematically in Figure 11.2 and Figure 11.3. In short, IR radiation from a Ti:sapphire laser system or from an optical parametric amplifier (OPA) is focused into the core of a skimmed molecular beam within an ultrahigh vacuum (UHV) chamber. Counter propagating and after a suitable time delay, UV radiation is also focused to the same focal spot. Ions generated after IR þ UV absorption by vibrationally assisted dissociation and photo-fragment ionization (OSVADPI) or by RE2PI of vibrationally excited molecules are mass analyzed in a TOF mass spectrometer. Since the ionization yield mirrors the IR excitation, mass analysis of ions allows the separation of the spectral contributions of different isotopomers, or more generally, of different components in a mixture. For some experiments (e.g., to observe CHCl3 overtone spectra, see below), near IR light around 11,500 cm21 is created by a Ti:sapphire laser (STI optronics, HRL-100) pumped by the first harmonic output (532 nm, typically 500 mJ per pulse at 10 Hz) of a pulsed Nd:YAG laser (Continuum, NY82-10) (see Figure 11.2). The Ti:sapphire laser has single longitudinal mode

delay line PD 355 nm SHG TOF

Dye

Raman cell

L UV probe

− + UHV chamber

L

Nd:YAG

Ti:S

P PM

532 nm

IR pump

FIGURE 11.2 Experimental set-up with Ti:S laser and Raman shifter (Source: After Hippler, M. and Quack, M., Chem. Phys. Lett., 231, 75 –80, 1994. With permission). SHG: second harmonic generation of UV radiation, PD: photo diode to monitor the UV pulse energy; PM: pyroelectric monitor of the IR pulse energy; P: Pellin– Broca prism; L: lenses to focus UV and IR radiation, respectively; TOF: time-of-flight mass spectrometer.

318

Isotope Effects in Chemistry and Biology SHG PD

DM

OPA

Dye

Dye

Nd:YAG

TOF PM

L UV probe

− +

L

PA

IR pump

UHV chamber

FIGURE 11.3 Scheme of the experimental set-up with OPA as IR laser source (Source: From Hippler, M. and Quack, M., Ber. Bunsen-Ges. Phys. Chem., 101, 356– 362, 1997. With permission). SHG: second harmonic generation, PD: UV photodiode, L: lens, OPA: optical parametric amplifier, DM: dichroic mirror to separate the 355 nm pump radiation, PM: pyroelectric IR power monitor, PA: photoacoustic reference cell, TOF: timeof-flight mass spectrometer.

output with an essentially Fourier-transform limited Gaussian beam with specified bandwidth of less than 500 MHz (0.017 cm21) with ca. 20 to 40 mJ energy per 4.5 nsec pulse at a 10 Hz repetition rate. The IR radiation is focused with a 20 cm focal length lens into the TOF vacuum chamber with a focal spot diameter of 100 mm (full width at half-maximum, FWHM). At a pulse energy of 40 mJ, the power density in the focal spot is thus approximately 100 GW cm22. With the Ti:sapphire laser system, the 710 to 910 nm (11,000 to 14,100 cm21) red and near-IR spectral region is accessible. To extend this region further into the IR, a Raman shifting cell is inserted after the Ti:sapphire laser (Figure 11.2). The 70 cm long high-pressure cell is made of stainless steel and equipped with sapphire windows. It is filled with 20 to 30 bar of H2 as Raman active medium. A 40 cm focus lens focuses the pump radiation from the Ti:sapphire laser, and after the cell, the Raman-shifted radiation is made parallel again with another 40 cm focus length lens. Stimulated Raman scattering on the Q(1) line in v ¼ 1 of H2 with DE ¼ 4155 cm21 creates several Stokes and anti-Stokes lines from the pump radiation.128 – 132 With 60 mJ pulse energy at 790 nm as pump source, ca. 15 mJ of first Stokes radiation at 8500 cm21 is obtained. The second Stokes line is presumably also generated, although not observed since the glass of the lenses and the prism absorbs this radiation. The first Stokes line is separated from the fundamental pump light and other Stokes lines by a Pellin –Broca prism, and then focused into the TOF vacuum chamber with a measured spot diameter of 500 mm (FWHM). In this case a power density of roughly 1.5 GW cm22 is estimated. Time delayed and counter propagating to the IR, ca. 0.1 –0.5 mJ of UV laser light from a frequency doubled dye laser around 240 nm (Lumonics, HD-500) is also focused into the vacuum chamber with a 20 cm focal length quartz lens. The dye laser is pumped with the third harmonic output (ca. 50 mJ pulse energy at 355 nm) from the same Nd:YAG laser, which is used to pump the Ti:sapphire laser system. A fixed time delay of ca. 30 nsec between the IR and the following UV laser pulses in the vacuum chamber is achieved by optically delaying the pump light for the dye laser by several meters (Figure 11.2). In other experiments (e.g., to observe aniline and benzene overtone spectra, see below), IR idler radiation is generated in an OPA of a narrow-bandwidth dye laser operating at the signal wavelength (Figure 11.3). A frequency-tripled, injection-seeded Nd:YAG laser (Continuum, Powerlite 9010) pumps the OPA system (Lambda Physik, modified SCANMATE OPPO). In order to improve the conversion efficiency, two b-barium borate (BBO) crystals are employed for

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

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the optical parametric amplification. The laser system is tunable from 410 to 2500 nm, delivering typical IR pulse energies between 30 mJ at 800 nm to 10 mJ at 2000 nm and some mJ at 2500 nm after pumping the dye laser and the BBO crystals with 300 mJ/pulse of 355 nm radiation from the injection-seeded Nd:YAG laser. The temporal pulse profile is nearly Gaussian with 4 nsec FWHM. The spectra displayed here are recorded with an IR laser bandwidth of 0.15 cm21, since no additional structures are apparent with the improved bandwidth of 0.02 cm21 achieved using an intracavity e´talon in the dye laser in a series of control experiments at higher resolution. Remaining 355 nm pump light (typically 70 mJ/pulse), which is not converted in the OPA, is separated from the signal and idler radiation by a dichroic mirror (Figure 11.3). After a suitable, short optical delay, it is used to pump a dye laser, which is then frequency doubled generating typically 200 mJ/pulse of UV radiation around 240 to 300 nm (Lumonics, HD-500). The OPA IR idler radiation is separated from the visible signal radiation by a glass filter (Schott, RG715) and focused by a 25 cm lens into the UHV chamber. In order to avoid saturation of the aniline or benzene overtone transitions by IR excitation, the focus of the IR radiation is pulled slightly out of exact overlap with the jet region (by ca. 1 to 2 cm). Signals are then linearly dependent on the IR pulse energy. The UV laser is also focused into the vacuum chamber counter propagating to the IR by a 30 cm quartz lens. The time delay between IR pump and UV probe beams is 20 nsec. Both IR and UV laser radiation are linearly polarized within the same plane. In order to obtain IR þ UV double resonance signals, it is necessary to overlap both foci very carefully. The ionization signal is found to be proportional to the IR absorption, and by scanning the IR laser while monitoring the ionization yield, overtone spectra are obtained. Double-resonance ionization signals depend linearly on the IR pulse energy, and ionization signals are normalized accordingly to correct for laser pulse to pulse-energy fluctuations. Simultaneously with the IR þ UV ionization spectra, reference spectra are taken with a small photoacoustic cell filled with typically 100 mbar of the sample vapor or a reference gas (e.g., methane or water vapor). After a quadratic fit to reference lines known from literature133,134 or by comparing the photoacoustic spectrum with a corresponding room-temperature spectrum obtained separately with a FTIR spectrometer (BOMEM, DA-002),25,135 an absolute wavenumber accuracy of 0.1 cm21 is estimated for the IR spectra presented below. Sample substances are in general purified by distillation and degassed by several freeze – pump – thaw cycles, and the purity is checked with gas chromatography. Partially N-deuterated aniline is prepared by hydrogen exchange in a stoichiometric mixture with CH3OD (Cambridge Isotope Laboratories, 99% D), which is then distilled off. The sample vapor is typically diluted into 1 bar of Ar, e.g., for the chloroform experiments by preparing a mixture of saturated chloroform vapor (ca. 210 mbar, Fluka puriss. p.a.) in 1 bar argon or for the aniline experiments by bubbling 1 bar of the seed gas Ar or N2 through liquid aniline (Fluka), which has a vapor pressure of 0.74 mbar at room temperature.136 The gas mixture then expands through a pulsed solenoid nozzle with 1 mm circular orifice (General Valve, with Iota One pulse driver) into the first vacuum chamber. After passing a 0.5 mm skimmer (Beam Dynamics, model 2) at 3 cm distance to the nozzle orifice, the skimmed molecular beam enters the second UHV chamber. Ions created by IR þ UV absorption are guided by electric fields through a small aperture to the UHV chamber with the TOF mass spectrometer. All vacuum chambers are pumped differentially by turbo-molecular pumps. Room temperature spectra are obtained by flowing ca. 1024 mbar of sample vapor through the second chamber. Differential pumping keeps the pressure in the TOF chamber always below 1026 mbar. In a test run, NO seeded into 1 bar argon was probed by (1 þ 1) REMPI via the A 2S (v0 ¼ 0) ˆ X 2P1/2 (v00 ¼ 0) transition at 226 nm137,138 (without the IR laser operating). In the (R11 þ Q21) and R21 branch, only transitions from the lowest rotational state of NO with J ¼ 0.5 were observed; transitions from J ¼ 1.5 or higher J were not apparent. This indicates a rotational temperature of the skimmed molecular beam below 3 K, and a similar or perhaps somewhat higher rotational temperature is expected for expansions of chloroform or aniline seeded into Ar in the present experiment. Vibrational cooling in a supersonic expansion is frequently less effective than

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Isotope Effects in Chemistry and Biology

rotational cooling. Typical vibrational temperatures can be of the order of 100 to 200 K, depending on the molecule and type of vibration.139 – 143 The custom-built TOF mass spectrometer in our experiments is a Wiley – McLaren type reflectron (Kaesdorf, Munich)144 – 148 combining double-stage ion extraction with a double-stage ion reflector. With this TOF, the benzene ion parent peak with m ¼ 78 u is observed after a flight time Dt ¼ 27.89 msec with a FWHM of the ion peak of ca. DtFWHM ¼ 8 nsec, which is mainly caused by the UV pulse length. With the resolving power R of a TOF mass spectrometer defined as R ¼ m/Dm ¼ Dt/(DtFWHM £ 2),148 R ¼ 1743 at m ¼ 78 u is calculated for the present set-up. Ion signals are processed by boxcar integrators (Stanford Research, SR 250) and an oscilloscope (Tektronix, TDS 520A), where time windows for integrating ion signals are defined, corresponding to selected flight times of different masses. In this way, separated ion yields of different components are recorded simultaneously, which allows for mass- and isotopomer-selective overtone spectroscopy.

IV. MASS SELECTIVE OVERTONE SPECTROSCOPY BY VIBRATIONALLY ASSISTED DISSOCIATION AND PHOTOFRAGMENT IONIZATION: OSVADPI In this section, a new variant of IR photofragment spectroscopy is introduced and evaluated, which we developed about ten years ago: overtone spectroscopy by vibrationally assisted dissociation and photofragment ionization: OSVADPI.30,33 – 36 In this scheme, IR excitation is coupled with UV dissociation and ionization of photofragments which allows indirect, but extremely sensitive IR spectroscopy. Since ionization detection can be conveniently coupled with a mass spectrometer, mass- and isotopomer-selective IR spectroscopy is possible to study isotope effects in intramolecular dynamics. This new technique allows the separation of spectral contributions arising from naturally occurring isotopomers, or of different components in a mixture: isotopomerselective overtone spectroscopy (ISOS) or more generally, infrared spectroscopy with isotopomer and mass selection (IRSIMS). The combination of sensitive IR spectroscopy with mass spectrometry greatly increases the selectivity, which is important for both fundamental and analytical applications of spectroscopy. The decomposition of overlapping spectra from different components reduces inhomogeneous congestion; in many cases, it will allow for the first time a secure assignment, as in the examples discussed below. After a discussion of the excitation mechanism and scheme of absorption, we introduce as an exemplary application the overtone spectroscopy of Cl isotopomers of chloroform, revealing isotope effects and a hierarchy of time scales of IVR after excitation of the CH chromophore.35

A. MECHANISM OF V IBRATIONALLY A SSISTED D ISSOCIATION AND P HOTOFRAGMENT I ONIZATION Without the IR laser operating, a weak ion current is observed if UV radiation between 235 and 240 nm is focused into a cell or a molecular beam containing vapor of some aliphatic chlorides, such as chloroform, tert-butyl chloride or CF2HCl. The ion current increases, but is still very small, if the UV laser is tuned to one of the several (2 þ 1) REMPI transitions of Cl (2P3/2) or Cl (2P1/2) in this region.149 Aliphatic chloride compounds have a characteristic first UV absorption band below 200 nm.150 – 152 In this n ! sp transition, electron density is transferred from a nonbonding to an antibonding orbital around the C –Cl bond, which causes the bond to break releasing Cl (2P3/2) and Cl (2P1/2) fragments. Apparently UV radiation between 235 and 240 nm dissociates molecules on the very edge of this absorption band with low cross-section, or by a twophoton transition to a higher electronic dissociative state. Nascent Cl fragments are then ionized by the same UV laser beam in a (2 þ 1) REMPI process. Other heavier fragments might also be

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

321

ionized by REMPI via broad absorption bands. Since the UV laser wavelength is unfavorable for the first dissociation step in the absorption sequence, ionization signals are only weak. If the IR laser operates in addition and excites molecules first, a distinct increase of the resonant ionization yield is observed. Two possible schemes of excitation suggest themselves and need to be discussed: First, IR multiphoton absorption could lead to electronic ground-state dissociation. Nascent Cl fragments would then be ionized by the UV laser by (2 þ 1) REMPI, for example. IR multiphoton dissociation, however, appears not to apply in the present case. In CF2HCl or CHCl3, the energetically lowest dissociation channel via the electronic ground state is a,a-elimination of HCl.153 – 156 Scanning the UV laser through known HCl (2 þ 1) REMPI transitions did not reveal any HCl, however. Further, the ionization yield depends linearly on the IR pulse energy indicating that only one IR photon is absorbed. The presence of Cl fragments finally points to dissociation via an excited electronic state.156 This leads to the scheme of excitation summarized in Figure 11.4, OSVADPI: After the absorption of one IR photon, vibrationally excited molecules are UV photodissociated. Cl fragments are then ionized by (2 þ 1) REMPI, and possibly also some heavier fragments. The first IR absorption step effectively limits the ionization yield. Ionization detection in this scheme allows indirect, but very sensitive IR spectroscopy. The combined IR and UV radiation excites the dissociative electronic state much more efficiently than one-photon UV absorption alone. This is partly due to the higher total energy of IR þ UV photons: The absorption cross-section of the first UV absorption band increases by several orders of magnitude at the higher total energy of the IR þ UV photons compared to the UV photon alone. Another, essential enhancing mechanism is provided by the Franck –Condon principle: By vibrational excitation, the molecular wave function extends over a larger region, which allows a more efficient transition to the electronically excited state. This is indicated in the scheme of Figure 11.4, but the real multidimensional situation is, of course, much more complex than shown. Similar explanations are invoked to explain the enhanced absorption in vibrationally mediated dissociation of other molecules, where photofragments have been probed by LIF detection.31 In this respect the dissociation mechanism of OSVADPI is analogous to vibrationally mediated dissociation spectroscopy. The proposed mechanism of excitation is corroborated by studying the dependence of the ionization yield on the IR and UV pulse energies with CHCl3 as example.33 At an UV wavelength that corresponds to a (2 þ 1) REMPI transition of Cl, the ionization yield depends linearly on CI+ + e−

E

CI*

CHCI2 + CI

CI

u=4 u = 0 CHCI3 r

FIGURE 11.4 OSVADPI scheme of excitation with CHCl3 serving as example (Source: From Hippler, M. and Quack, M., Chem. Phys. Lett., 231, 75 – 80, 1994. With permission).

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Isotope Effects in Chemistry and Biology

the IR intensity and fluence, consistent with vibrational one-photon IR excitation as the first step in the absorption scheme. The dependence on the UV intensity and fluence is quartic at low intensity, cubic at intermediate levels, and finally quadratic at pulse energies over 40 mJ/pulse, corresponding to peak intensities of about 40 MW cm22 or fluences of about 0.4 J cm22. This is consistent with four UV photons being involved in the absorption sequence. The first photon induces vibrationally assisted photodissociation, the remaining photons ionize Cl fragments formed in their spin – orbit split ground state via (2 þ 1) REMPI. With increasing UV pulse energy, first the dissociation and the ionization step are saturated, leaving finally the quadratic dependence of the two-photon Cl excitation, which has a low absorption cross-section. At UV wavelengths not coinciding with a Cl atomic resonance, the ionization yield depends also linearly on the IR intensity. Again, the first step in the absorption sequence is the IR one-photon excitation. The dependence of the ionization yield on the UV intensity is found to be cubic at low intensities and quadratic at intensities over 50 MW cm22. The first absorbed UV photon causes vibrationally assisted photodissociation of CHCl3, followed presumably by two-photon ionization of fragments, possibly CHCl2. One important aspect in the OSVADPI scheme presented is that the electronically excited state is a dissociation continuum. This intermediate state in the ionization sequence does not show any apparent structure within the small energy range encompassed by scanning the IR laser to obtain an overtone spectrum. It was therefore in all cases entirely adequate to leave the UV laser at a fixed frequency while scanning the IR laser, which greatly simplifies the experiment. This is a very important advantage compared to other schemes with ionization detection of IR excitation via discrete electronically excited rovibrational states. The experimental set-up is also less involved than comparable schemes: The UV photons required for dissociation and REMPI detection of photofragments are all derived from one laser system operating in the UV range around 250 nm, which is conveniently accessible by standard frequency doubling of a dye laser. Both IR and UV laser systems are pumped by one single Nd:YAG pump laser, and an optical delay line achieves the synchronization of the laser systems (Figure 11.2 and Figure 11.3). Ion detection is extremely sensitive; OSVADPI is thus a very sensitive indirect technique for IR spectroscopy. Because low sample pressures are required and the absorption path is the focus region only, the OSVADPI technique is ideally suited to study overtone spectroscopy of samples having low vapor pressure or of jet-cooled samples in supersonic jet expansions. Compared to the much higher sample pressure required for FTIR or photoacoustic spectroscopy to obtain spectra of comparable quality, the low sample pressure greatly reduces the unwanted effect of pressure broadening. Coupling ionization detection with a TOF mass spectrometer finally allow mass- and isotope-selective IR spectroscopy of different components in a mixture, for example Cl isotopomers of chloroform at natural abundance. Although it would be in principle possible to chemically prepare isotope-pure Cl isotopomers of chloroform, this would be very expensive and not very practical; with the present scheme, however, this is not necessary, since the different Cl isotopes are separated in the mass spectrometer. Figure 11.5 shows the TOF mass spectrum after IR excitation at 11,384.6 cm21 and UV photodissociation and ionization. The UV wavelength at 235.32 nm corresponds to the (2 þ 1) REMPI transition of Cl, 3p44p (3P) 2Do3/2 ˆ ˆ 3p5 2Po3/2.149 The dominating ions are thus 35Cl and 37Cl. As shown below, the IR excitation corresponds to a transition of CH35Cl237Cl. The observed 35Cl/37Cl isotope ratio does therefore not reflect the natural abundance of Cl isotopes; it is rather about 2:1 as in the particular IR excited isotopomer. 12C, 12C35Cl, and 12C37Cl ions and a small fraction of the corresponding 13C containing ions are also observed. Possibly the second photofragment CHCl2 is ionized by (1 þ 1) REMPI, and further fragmentation leads to the appearance of CCl and C ions. Alternatively, secondary photodissociation of CHCl2 may yield CCl and HCl fragments (as for the 193 nm photodissociation of CHCl3156), and CCl is then ionized and further fragmented. Since the UV radiation cannot ionize C atoms efficiently, C ions result from the fragmentation of an ionic precursor. Mass gating the ion detection to 35Cl and

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

15

10 0

S / mV

−5 −10

20

(m/z) / u 25 30 35 40

60

12 37 12

C CI

C

12 35

C CI

−15

37

CI

−20 −25 10

50

323

35

CI

15

∆t / ms

20

25

FIGURE 11.5 Fragment mass spectrum after dissociation of vibrationally excited CHCl3 and ionization with UV radiation at 235.32 nm (Source: From Hippler, M., Habilitation thesis, ETH Zu¨rich, 2001. With permission). S denotes the ion signal, Dt the flight time in the time-of-flight mass spectrometer; m/z is the corresponding nonlinear mass scale.

37

Cl will not allow directly isotopomer selective spectroscopy of CHCl3, since there are four Cl isotopomers: CH35Cl3 (43.5% natural abundance), CH35Cl237Cl (41.7%), CH35Cl37Cl2 (13.3%), and the minor component CH37Cl3 (1.4%). 13CH and 12CH chromophore absorptions are well separated with 13CH transitions outside the range discussed below.102 Mass-selective detection of 35 Cl and 37Cl ions in OSVADPI allows a simple manipulation to assign spectral features to the three major Cl isotopomers: If twice the yield of 37Cl is subtracted from the yield of 35Cl fragments, the influence of CH35Cl237Cl on the total yield of 35Cl fragments is approximately subtracted since for every 37Cl two 35Cl appear for this isotopomer. Spectral features, which remain unaffected under this operation, are thus identified to belong to CH35Cl3, whereas features which disappear belong to CH35Cl237Cl. The influence of the third important isotopomer CH35Cl37Cl2, however, would be overestimated. “Negative” features after the subtraction are thus due to CH35Cl37Cl2. This subtraction scheme allows secure assignments, especially for resolved spectral features.30,35,36

B. ISOTOPOMER S ELECTIVE OVERTONE S PECTROSCOPY OF CH C HROMOPHORE A BSORPTION OF CHCl3

THE

N j 5 42

Because of the presence of four Cl isotopomers, CHCl3 overtone spectra have a very complex structure, which is also affected by vibrational hot-band transitions: at room temperature, the vibrational ground state of chloroform has only a population of 38%. n6 (the degenerate C – Cl3 bending vibration at 260 cm21,157) and 2n6 have a population of 22 and 10%, respectively, and n3 (the C –Cl3 “umbrella” vibration at 367 cm21,36,157,158) a population of 7%. This inhomogeneous congestion is especially apparent in the central part of parallel vibrational bands with sharp Q-branches of Cl isotopomers and of hot-band transitions. Without means to simplify spectra, secure assignment of features to genuine, anharmonic local resonances would be exceedingly difficult. The Nj ¼ 42 CH chromophore absorption (corresponding approximately to the 3n1 þ 2n4 combination band) is part of the N ¼ 4 polyad; in the order of descending wavenumbers, it is the second band after Nj ¼ 41 : The very weak band has only about 5% of the peak intensities compared to Nj ¼ 41 (see Figure 11.6). In a simple intensity model for the N ¼ 4 polyad, the v1 ¼ 4 CH-stretching overtone has all absorption intensity. All polyad states acquire absorption intensity according to the admixture of this CH-stretching overtone due to the Fermi resonance between

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Isotope Effects in Chemistry and Biology 600

σ / fm2

400 200 0 10980

11000

11020 11040 n / cm−1

11060

FIGURE 11.6 Nj ¼ 42 CH chromophore absorption of chloroform at room temperature, recorded by FTIR spectroscopy in a multipass absorption cell, l ¼ 49 m, 100 mbar CHCl3, 0.1 cm21 experimental bandwidth (recorded by H. Hollenstein, Zu¨rich group; From Hippler, M., Habilitation thesis, ETH Zu¨rich, 2001).

CH-stretching and -bending modes. The Nj ¼ 42 band has been observed before in an investigation of this strong Fermi resonance, but only an approximate band position at 11,019 cm21 was reported without any further details, for example discussing the position of different Cl isotopomers.159,160 Since a secure assignment of Nj ¼ 42 is essential to corroborate the identification of the local perturber in the Nj ¼ 41 overtone (see next section), this band was reinvestigated in detail.30 The room-temperature spectrum recorded by FTIR in a multipass absorption cell corresponds to the natural Cl isotopomer mixture of 100 mbar chloroform (Figure 11.6). The spectrum shows the parallel band with sharp Q-branch features and typical broad contours of the P- and R-branches located around the Q-branch at lower and higher wavenumbers, respectively. In addition, there are very weak features around 11,040 cm21 that cannot be assigned at present. Presumably, they are due to a different combination band. In Figure 11.7, the Q-branch region of Nj ¼ 42 is shown in more detail; in the lower panel the room-temperature FTIR spectrum (as in Figure 11.6) and in the upper panel the corresponding OSVADPI spectrum (without mass selection). OSVADPI spectra of ca. 1024 mbar chloroform vapor (leaking through the vacuum chamber) were obtained with the OPA system and with the UV laser at 235.32 nm corresponding to a (2 þ 1) REMPI transition of Cl. Both spectra are essentially identical, which indicates that OSVADPI is a faithful mirror of IR excitation. In the Qbranch, three features are apparent which are labeled A, B, and C for convenience. Whether they correspond to the Q-branches of the three major Cl isotopomers, to hot-band transitions, or perhaps to the homogeneous structure of a local anharmonic resonance, however, cannot be decided and assigned on the basis of the room-temperature spectrum without decomposition of the spectrum into Cl isotopomers. Mass gating the fragment ion detection in the OSVADPI scheme to the yield of 35Cl and 37Cl ions with the TOF mass spectrometer, isotopomer-selective overtone spectra (ISOS) have been obtained after application of the subtraction scheme introduced before35 (see Figure 11.8). It is clearly seen that feature C belongs to the CH35Cl3 isotopomer, since it does not contribute to the 37Cl fragment yield. Feature B has a 35Cl/37Cl isotope ratio of 2:1 and is therefore due to CH35Cl237Cl. Feature A corresponds to an isotope containing more 37Cl than 35 Cl, since the peak is negative after the subtraction; it is therefore assigned to CH35Cl37Cl2. The relative peak areas of features A, B, and C (normalized to the sum 100) are 15, 41, and 44, respectively, which in this assignment corresponds closely to the relative abundance of isotopomers. Q-branch maxima at 11,015.0, 11,017.2, and 11,018.7 cm21 are thus found by ISOS for the isotopomers CH35Cl37Cl2, CH35Cl237Cl, and CH35Cl3, respectively, with an estimated accuracy of 0.1 cm21. The maxima are good estimates for the band origins of Nj ¼ 42 : Relative to

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

325

5

Ssum / mV

4 3 2 1 0

σ / fm2

500

B C

400 200 0 11010

A

11015

~ / cm−1 n

11020

11025

FIGURE 11.7 Q-branch region of the chloroform Nj ¼ 42 CH chromophore absorption at room temperature (Source: From Hippler, M., Habilitation thesis, ETH Zu¨rich, 2001). The upper panel shows an OSVADPI spectrum of ca. 1024 mbar chloroform vapor (without isotopomer selection, Ssum corresponding to the sum of all ions), and the lower panel the FTIR reference spectrum (as in Figure 11.6). The three distinct spectral features have been labeled A, B, and C; they are discussed in the text.

the position of CH35Cl3, Cl isotopomer shifts are thus 1.5 cm21 for CH35Cl237Cl and 3.7 cm21 for CH35Cl37Cl2. These experimental shifts may serve as benchmark results for theoretical calculations.105,158 Since lighter isotopomers appear at higher wavenumbers, the order of band origins is as expected if no perturbations are effective. Vibrational hot-band transitions are probably hidden in the Q-branch contours, which have at room temperature a FWHM of about 0.5 to 1 cm21.

C. ISOTOPOMER S ELECTIVE OVERTONE S PECTROSCOPY OF THE N j 5 41 CH C HROMOPHORE A BSORPTION OF CHCl3 : A H IERARCHY OF T IME S CALES AND I SOTOPE E FFECTS IN I NTRAMOLECULAR V IBRATIONAL E NERGY R EDISTRIBUTION (IVR) The Nj ¼ 41 CH-overtone level of chloroform (approximately the third CH-stretching overtone) has a complex structure, which could not be assigned prior to our work.35 Figure 11.9 shows the central structure with the Q-branches of the different isotopomers of chloroform at room temperature. Outside the range displayed, the overtone has in addition broad P- and R-branches with a typical contour of a parallel transition. With the Ti:sapphire IR laser system and the UV laser operating at 235.32 nm (corresponding to a (2 þ 1) REMPI transition of Cl), OSVADPI spectra have been obtained as described before. The OSVADPI spectrum where all ions are collected without mass selection corresponds to the chloroform isotopomer mixture. It is essentially identical with photoacoustic and FTIR reference spectra, although they have been obtained using completely different detection schemes. Mass gating the detection in the TOF to 35Cl and 37Cl fragment ions and applying the subtraction scheme (upper panel in Figure 11.9), all major peaks in the complex

326

Isotope Effects in Chemistry and Biology 4

C

S diff / mV

3 2 1

CH35CI37CI2 A

0 −1

S 35

CI

/ mV

4

C B

3 2

CH35CI3

A

1 0

S 37

CI

/ mV

4

CH35CI2 37Cl

3 2

A

B

1 0 11010

11015 11020 ~ n / cm−1

11025

FIGURE 11.8 Isotopomer selective OSVADPI spectra (ISOS) of the Q-branch region of the Nj ¼ 42 CH chromophore component of chloroform at room temperature (Source: From Hippler, M., Habilitation thesis, ETH Zu¨rich, 2001). The peaks A, B, and C are assigned to the three major Cl isotopomers (see text).

room temperature spectrum have been assigned to Cl isotopomers as indicated in Figure 11.9 and summarized in Ref. 35. Hot-band transitions have been identified by comparison with jet cooled spectra (Figure 11.10) where the subtraction procedure has been applied to separate the Cl isotopomers. The P- and R-branches have a FWHM of only ca. 5 cm21, which indicates a very cold rotational temperature of the jet spectra. By comparison with simulated rotational band contours, a rotational temperature of about 5 K is estimated. No vibrational hot-band transitions are apparent in the expansion of chloroform seeded into Ar. Using He or N2 as seeding gas, hot-band transitions with n6 were just emerging corresponding to slightly higher vibrational temperatures of ca. 140 and 180 K, respectively. Under these different expansion conditions, however, no changes in band contours or positions are apparent, which indicates that the jet-cooled spectra are not affected noticeably by cluster formation. Compared to the room temperature spectrum, an interesting change of the rotational contour in the Q-branch of jet-cooled CH35Cl237Cl is noted: The peak maximum is shifted 0.5 cm21 to lower wavenumbers, and the FWHM of the contour increases to 1.3 cm21 compared to 0.4 cm21 at room temperature. This effect is genuine and not an experimental artifact due to insecure wavelength calibration or insufficient resolution, since jet-cooled and roomtemperature reference spectra were always recorded simultaneously using the same IR laser. Presumably a band head is formed at high J values at room temperature, which is removed at very low temperatures, thus changing the rotational contour. In the assignment, hot-band transitions

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

S diff / mV

80

/ mV CI

CH35CI3

40 0 −40

S 35

327

CH35CI 37CI2

80 60 40 20

S 37

CI

/ mV

0 40 30

CH35CI2 37CI

CH35CI 37CI2

20 10 0

S sum / mV

120 90 60 30 0 11370

11380 11390 ~ n / cm−1

11400

FIGURE 11.9 Central part of the Nj ¼ 41 CH chromophore absorption of CHCl3 at room temperature, smoothed graphical resolution 0.08 cm21 (actual resolution 0.02 cm21) (Source: After Hippler, M. and Quack, M., J. Chem. Phys., 104, 7426– 7430, 1996. With permission). The lower plot shows as reference the OSVADPI spectrum without mass selection, the other plots the corresponding isotope selective OSVADPI spectra (ISOS) with assignments to Cl isotopomers. Hot-band sequences are indicated by dashed lines, the splitting due to the local anharmonic resonance is indicated by solid lines (see also text).

arise from n6, 2n6, n3, or 3n6. The intensities in this assignment are consistent with the thermal population, and the shifts are compatible with the anharmonicity constants x~ 016 ¼ 0:32 cm21 and x~ 013 ¼ 0:98 cm21 :36,161,162 After removing all vibrational hot bands and isotopomer congestion in the isotopomer-selected jet spectra, the splitting of Q-branches of the isotopomers CH35Cl3 and CH35Cl237Cl is striking, which indicates a local anharmonic resonance (see Figure 11.10). For CH35Cl3, the Q-branch is split into two components of nearly equal intensity. In this case, the local resonance of the 41 state with a second vibrational state is almost perfect. In the isotopomer CH35Cl237Cl, the Q-branch is split into two components of different intensities, which indicates a detuning of the resonance due to a shift of vibrational frequencies in the different isotopomers. In the Q-branch of the third major Cl isotopomer, CH35Cl37Cl2, no splitting is apparent. Presumably, the local resonance is detuned even further and thus not as effective as for the other two isotopomers. In the vibrational hot-band

328

Isotope Effects in Chemistry and Biology 16

Sdiff / mV

12 8 4 0 −4

CI

/ mV

12

S35

CH35CI3

8

CH35CI 37CI2 CH35CI237CI CH35CI3

4 0 −4

S37

CI

/ mV

6

CH35CI237CI

4 2 0 −2 11370

11380 11390 ~ −1 n / cm

11400

FIGURE 11.10 Isotope selective OSVADPI spectra (ISOS) of the jet cooled Nj ¼ 41 CH chromophore absorption of CHCl3 with assignment of Q-branches to the three major Cl isotopomers (Source: After Hippler, M. and Quack, M., J. Chem. Phys., 104, 7426– 7430, 1996. With permission). The brackets indicate the splitting due to the local anharmonic resonance.

transitions of the room-temperature spectra, no such resonance splittings are apparent, which is presumably because the resonance is detuned in the combination band of 41 with the hot-band vibration. Assuming a two-level model for the local resonance, where the unperturbed state 41 carries all absorption intensity, a deperturbation for CH35Cl3 yields 11,384.1 cm21 as unperturbed position of 41 and 11,384.7 cm21 for the perturbing vibrational state. For the ~ ¼ 2:0 cm21 is found (the sign is undetermined). A similar deperturbation interaction energy, lWl 35 37 for CH Cl2 Cl gives the same interaction energy and 11,383.1 and 11,380.2 cm21 as unperturbed positions of 41 and of the perturber, respectively. Apparently, the same vibrational level is the perturber of 41 in both isotopomers. An assignment of the perturbing state may appear at first glance impossible: the density of vibrational states, which have A1 symmetry as has 41 and which can thus in principle interact via an anharmonic resonance, is about 340 states per cm21 interval at the energy of 41. The count of vibrational states and their assignment was performed by a computer program, which calculates all possible vibrational combination bands within a given energy interval using experimental anharmonic spectroscopic constants and experimental or estimated isotopomer shifts.101,157,162 To calculate positions of CH-stretching/bending modes within a Fermi resonance polyad, the program diagonalizes the corresponding effective Hamiltonian matrix with experimental parameters.101 Most of the vibrational background states involve many quanta of CCl3 frame modes, and thus have an isotopomer shift much larger than found in the experiment. At the 41 energy, only about 0.7 states per cm21 interval of A1 symmetry have an estimated isotopomer shift of less than 10 cm21 between CH35Cl3 and CH35Cl237Cl, estimating the shift by the sum of

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

329

the observed isotopomer shifts of fundamentals multiplied by the number of the corresponding vibrational quanta in the combination band. Among these few states, the likely perturber seems to be the combination of the second Fermi component Nj ¼ 42 and the C – Cl3 umbrella vibration n3. It is the only combination band of A1 symmetry within a 30 cm21 interval, which could interact via a quartic anharmonic resonance with 41; all other bands would involve the change of more than four vibrational quanta relative to 41. A coupling of CH-stretching and -bending vibrations with the C –Cl3 umbrella vibration also appeals to physical intuition. The n3 fundamental has been observed more recently at 367.296 cm21 for the isotopomer CH35Cl3, at 364.52 cm21 for CH35Cl237Cl, and at 361.68 cm21 for CH35Cl37Cl2,158 in good agreement with our own FTIR measurements.35 With large isotopomer shifts, the availability of rotationally resolved spectra and no apparent perturbations, an assignment to isotopomers was relatively unproblematic in this case. For the Nj ¼ 42 overtone absorption, an assignment would be less obvious, since the roomtemperature spectra are not rotationally resolved. With ISOS, however, the following peak positions of Q-branches were obtained: 11,018.7 cm21 for the isotopomer CH 35Cl3, 11,017.2 cm21 for CH35Cl237Cl and 11,015.0 cm21 for CH35Cl37Cl2; they may serve as good estimation for the band origins. Taking the sum of the n3 and 42 positions as estimation for the unperturbed position of the combination band 42 þ n3, almost perfect agreement with the values for the perturber in the local resonance in Nj ¼ 41 is found. Estimated positions are only 1.4 cm21 higher, and the estimated isotopomer shift equals the deperturbed value. The 4 cm21 almost perfect resonance splitting indicates a redistribution time of 4 psec to the CCl3 frame mode, independent of the particular assignment (half of the period in Equation 11.20). This is to be compared with the ultrafast ( ø 50 fsec) redistribution between CH-stretching and -bending modes in CHCl3 and generally in CHX3 compounds.25,26,93,98 – 102,104, 105,107,135,159,160,163 – 165 The present result for the local anharmonic resonance in CHCl3 thus indicates a clear separation of time scales in IVR by almost a factor of 100. Assuming further that the bandwidth of possible homogeneous structures, G~ (in wavenumber units), is caused by exponential decay to background states, further redistribution of vibrational excitation with decay time t ¼ h=ð2pG Þ ¼ 1=ð2pcG~ ) is calculated. Taking as upper bound of G~ the 0.4 cm21 FWHM of the sharpest feature in the Nj ¼ 41 overtone spectrum (the Q-branch of CH35Cl237Cl at 11,384.64 cm21), t $ 13 psec is thus obtained as lower bound. In summary, the following hierarchy of time scales in IVR emerges for the observed overtone Nj ¼ 41 : the v ¼ 4 CH-stretching overtone is the chromophore state coupled to the IR field. It is the “antenna” to receive vibrational energy. With a redistribution time of ca. 50 fsec, vibrational energy is redistributed to the CH-stretching/bending manifold by the strong Fermi resonance, resulting in the polyad state 41. This state is in local resonance with a CCl3 frame mode with redistribution time t ¼ 4 psec, which then further “decays” into the background of vibrational states with redistribution time t $ 13 psec. The resulting spectroscopic states are observable by high-resolution spectroscopy; they will eventually decay back to the ground state by spontaneous IR emission with decay times of the order of 1 msec.

V. ISOTOPE SELECTIVE OVERTONE SPECTROSCOPY BY RESONANTLY ENHANCED TWO-PHOTON IONIZATION OF VIBRATIONALLY EXCITED MOLECULES The OSVADPI technique introduced in the previous section is based on a photofragment ionization scheme which requires molecules to have a dissociative excited electronic state. Here, a new variant of ISOS will be introduced, which can also be applied to molecules with a bound excited electronic state, employing the same experimental equipment: RE2PI of vibrationally excited molecules. ISOS and IRSIMS are thus truly general experimental techniques for the gas-phase IR spectroscopy, which have a wide range of possible applications.

330

Isotope Effects in Chemistry and Biology

A. OVERVIEW After providing a discussion on the scheme of excitation and the mechanism of RE2PI of vibrationally excited molecules, we introduce as a selected application example the spectroscopy and intramolecular dynamics of aniline isotopomers after excitation of the first NH-stretching overtone. In aniline, the influence of different isotopes on IVR and on the internal tunneling motion of the NH2 group (inversion) can be studied by high-resolution spectroscopy. The study of tunneling or inversion motions is an important topic in intramolecular dynamics.6,26,27,37,122,136,166 – 170 The inversion at the N-atom with pyramidal threefold substitution represents one prototype system, where ammonia has been a textbook example for a long time.2,171,172 Because of its low barrier to inversion of the amino group,136,173 – 175 aniline is particularly well suited for studying inversion processes. In the past, the theoretical treatment and the analysis of experimental spectra have been mainly limited to effective one-dimensional models136,173,176 – 179 or to the two-dimensional subspace including the torsional motion,180,181 thus neglecting the influence of different excited vibrational modes on the inversion dynamics. Although the exact full-dimensional quantum mechanical treatment of such interactions is not feasible at present for molecules as large as aniline, the theoretical study of the vibrational mode-selective inversion motion has been possible within some approximations with the recent development of a reaction path Hamiltonian approach.6,136 The partially N-deuterated C6H5NHD is a particularly interesting isotopomer of aniline, since it is chiral in its equilibrium structure; the structure with defined handedness, however, is not an eigenfunction of the Hamiltonian and thus not stable.27,166,167 The inversion motion corresponds in this case to a stereomutation, i.e., to the isomerization between enantiomers of different handedness,27,166,167 and the analysis of stereomutation tunneling spectra can provide stereomutation times and their dependence on excitation of various vibrational modes for this prototypical chemical elementary reaction. Aniline has two high-frequency IR chromophores, the CH chromophore similar to benzene and the NH chromophore. Their absorptions are in general well separated in the IR spectrum. The study of overtone spectroscopy will give information on anharmonic resonances and IVR. Aniline C6H5NHD is particularly suited to study IVR processes, since it has an isolated NH chromophore, which facilitates a detailed analysis of its absorption spectrum. The nonplanar equilibrium structure of aniline is shown schematically in Figure 11.11.136 The dihedral angle between the benzene ring and the NH2 plane decreases from weq < 428 in the electronic ground state S0 to about 208 or less in the quasiplanar S1 state.173 – 175,182 Because of tunneling splitting with a relatively low effective barrier to NH2 inversion of about 450 cm21 in the electronic ground state, the vibrational ground state is split into two components with energy difference DE^ ¼ 40.9 cm21 for C6H5NH2 and DE^ ¼ 23.8 cm21 for C6H5NHD.136 These spectroscopic eigenstates have well defined parity, but no well-defined geometry and handedness.27,166,167 C6H5NH2 levels are classified by the Longuet –Higgins symmetry group MS4 for nonrigid molecules, which is isomorphic to the point group C2v ; and C6H5NHD by MS2, which x 180˚-j z

y

FIGURE 11.11 The equilibrium structure of aniline (Source: From Fehrensen, B., Luckhaus, D., and Quack, M., Z. Physik. Chem., 209, 1 – 19, 1999. With permission). The inversion angle w is defined as the angle between the NH2 plane and the CN bond. The a-axis is parallet to z, b-axis parallel to y, c-axis parallel to x (see also Refs. 136, 184).

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

331

is isomorphic to Cs :136,183 The torsional motion of the amino group is also a large amplitude motion, which could be in principle included in the symmetry classification, since it effects a permutation of atoms. Because of a high effective torsional barrier, however, the corresponding tunneling splittings are expected to be very small and not resolvable under the present experimental conditions.184 Therefore the torsional motion has been neglected in the symmetry classification as not feasible.183 In principle, the weak nuclear force will introduce a small asymmetry in the inversion potential of the chiral C6H5NHD.27,166,167 This effect, however, is extremely small compared to the much larger splittings induced by the low barrier to inversion and has also been neglected in the symmetry classification. Vibrational bands have a transition moment along the a, b, or c axis of the lowest, medium, or highest moment of inertia, respectively (see Figure 11.11), with a corresponding type A, B, or C rotational contour. In the IR spectrum of in-plane vibrations (e.g., NH chromophore), type B bands with a characteristic central minimum or type A bands with a strong central peak are prevailing. Because of the interaction of the inversion motion with vibrational excitation, energy differences DE^ of tunneling components in the ground and vibrationally excited state are in general different. At room temperature, both tunneling components of the ground state are almost equally populated. Vibrational transitions of in-plane vibrations have then satellites at a distance, which corresponds to the difference of the tunneling splittings in the ground and excited vibrational state. Vibrations of A1 symmetry (e.g., symmetric NH-stretching vibrations) have in addition to strong type A transitions much weaker c-polarized bands with satellites at a distance corresponding to the sum of tunneling splittings. With known energy difference DE^ in the vibrational ground state,136 the observation of “tunneling satellites” in the IR spectrum allows the calculation of the tunneling splitting in the vibrationally excited state.

B. MECHANISM OF R ESONANTLY E NHANCED T WO - PHOTON I ONIZATION OF V IBRATIONALLY E XCITED M OLECULES We shall discuss the absorption mechanism which allows us the observation of IR þ UV doubleresonance signals with aniline serving as example: aniline has a first UV absorption band with origin at 34,029 cm21 (293.87 nm) corresponding to the S1 ˆ S0 one-photon transition to the stable intermediate 1B2 (S1) electronic state.173,182,185 – 187 While the adiabatic ionization limit is at 7.720 eV or 62,265 cm21,188,189 the more relevant vertical ionization limit for efficient ionization is about 8.05 eV188 or 64,900 cm21. Since both absorption cross-sections for resonant electronic excitation S1 ˆ S0 and for the ionization of electronically excited aniline are large and comparable,190 strong (1 þ 1) REMPI (RE2PI) signals resonantly enhanced by the S1 state are obtained for UV absorption above 34,029 cm21.173,182,185 – 187 Applying UV radiation only, a very low background level of ions is observed at wavenumbers below 32,400 cm21 (about half the vertical ionization limit). Because of nonresonant two-photon ionization, the ionization yield increases above that threshold, and above 34,029 cm21, the origin of the S1 ˆ S0 one-photon transition, very strong ion signals are finally obtained by (1 þ 1) REMPI. If a vibrational level of the electronic ground state has been excited first by IR radiation, an enhancement of the ionization yield by several orders of magnitude is observed with UV radiation between about 30,000 and 32,400 cm21. Since ionization with UV radiation is very inefficient at these wavenumbers, the IR þ UV double-resonance ionization signals are virtually background free. For example, peak IR þ UV ionization signals for the N ¼ 2 NH chromophore absorption of C6H5NH2 at 6738 cm21 are ca. 25 mV compared with a nonresonant UV background of ca. 0.1 mV, only. The ion yield is easily saturated by the UV pulse energies employed, and no variation in the ion yield is then apparent when leaving the IR radiation resonant to a vibrational transition and scanning the UV laser over an extensive range. The ionization yield therefore mirrors the IR excitation, and by scanning the IR laser, IR spectra are obtained. Spectra obtained under room-temperature conditions, where aniline was leaking through a needle valve into the ionization

332

Isotope Effects in Chemistry and Biology

chamber, are essentially identical to FTIR reference spectra, without any apparent distortions. Compared to ionization depletion detection schemes, the spectra are almost background free and have an excellent signal to noise ratio, even for the reduced number densities in a skimmed supersonic jet expansion. Concerning the laser excitation mechanism, we propose a new absorption scheme to explain the enhancement in the UV ionization yield by IR excitation in our aniline or benzene experiments, as summarized in Figure 11.12:30,37,38 Without vibrational excitation, the UV radiation has not sufficient energy to promote the transition to the electronically excited state S1, which could resonantly enhance ionization ((1 þ 1) REMPI), and nonresonant two-photon ionization is also energetically not possible. After IR excitation, however, Franck– Condon factors for hot-band transitions are much more favorable for the one-photon transitions to the S1 state and the UV energy is sufficient to ionize vibrationally excited aniline or benzene molecules by RE2PI via vibrational hot bands on the S1 ˆ S0 transition. For aniline with IR excitation at 6750 cm21 and UV radiation at 31,220 or 32,100 cm21 in the double-resonance experiment, the excess energy in S1 is 3940 or 4820 cm21, respectively. At these excitation energies, the vibrational density of states in S1 is estimated to be roughly similar to the S0 ground-state density calculated184 as ca. 1.0 £ 103 states per cm21 interval at 4000 cm21, using the Beyer – Swinehart algorithm191 with a set of harmonic frequencies from ab initio calculations (B3LYP with 6-31Gpp basis set;192 the inversion mode is approximated by an effective harmonic oscillator with wavenumber 350 cm21; a more detailed calculation including anharmonicity and large amplitude torsional motion would be straightforward, but would not change the order of magnitude). For benzene with IR excitation at 6000 cm21 and UV radiation at 37,600 cm21, the excess energy in S1 is about 5500 cm21. At this energy, the vibrational density of states in the ground state S0 is about 700 states per cm21 interval, as calculated by the Beyer – Swinehart algorithm191 using a set of observed fundamental frequencies compiled in Refs. 193, 200. A similar density can be expected for the S1 state. In both cases, this corresponds to a dense set of states, which is accessible to one-photon excitation from the vibrationally excited ground state. Compared to previous schemes, such a quasicontinuum of resonance enhancing intermediate states has the distinct advantage that the UV laser does not have to be scanned simultaneously with the IR laser during an IR scan, since the resonance condition is always fulfilled. RE2PI of vibrationally

70000

C6H5NH2+ + e−

E / hc (cm−1)

60000 50000 50000

S1

S1

30000 20000 10000

(a)

0

S0

S0 (b)

C6H5NH2

FIGURE 11.12 Scheme of excitation, with aniline C6H5NH2 serving as example (Source: After Fehrensen, B., Hippler, M., and Quack, M., Chem. Phys. Lett., 298, 320– 328, 1998. With permission): (a) UV excitation only is not efficient for ionization; (b) resonantly enhanced two-photon ionization of vibrationally excited molecules.

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

333

excited molecules is similar to the OSVADPI scheme introduced before; it is its natural extension to molecules, which have bound intermediate electronically excited states. We have investigated the dependence of the ion yield on the UV and IR pulse energy (fluence) for aniline.37 The dependence on the UV radiation is consistent with the proposed resonant ionization mechanism, which involves two one-photon UV steps. The dependence is described satisfactorily assuming a kinetic two-step photoionization model.37,194 The relative absorption cross-sections obtained by this fit for the resonant excitation and ionization are in good agreement with literature values.190 At low fluence, the dependence is quadratic, at medium fluence, the S1 ˆ S0 transition is saturated, but not the ionization, and therefore a linear dependence is effective. At higher fluences, both one-photon steps are saturated. In a recent study of ionization-detected IR excitation of phenol and some other aromatic molecules under very similar experimental conditions, a nonresonant two-photon ionization mechanism of vibrationally excited molecules was suggested.46,47 However, in an UV scan with the IR kept at a vibrational transition, these authors observed distinct features in the ion yield, which they attributed to vibrational resonances in S1.47 This would imply that ionization occurs also in these experiments via RE2PI as suggested here for benzene and aniline. Aniline ionized by the IR þ UV double resonance scheme has a characteristic mass spectrum, which is dominated by the parent ion and some weaker fragment ion peaks ranging from Cx¼2 – 6Hy¼0 – 5 to C (see Figure 11.13). Obviously, fragmentation occurs after ionization, since the UV wavelength chosen is not efficient to ionize neutral C atoms or other fragments. By gating ionization-detected IR excitation to the parent ion mass peak, mass-selective IR spectra are obtained. With a natural abundance of 1.1% 13C, the aniline isotopomers with one 13C atom have 6.2% abundance in the natural isotopomer mixture. For the aniline isotopomer 13C12C5H5NH2, no shift of vibrational bands compared to 12C6H5NH2 is observed for the NH chromophore, and therefore the 13C isotopomer shifts must be less than 1 cm21. This is not unexpected, since the NHstretching and -bending vibrations are not affected much by the benzene ring. In a mixture of undeuterated, partially and fully N-deuterated aniline, gating the mass detection to m/z ¼ 93 and 94 u yields the separate vibrational spectra of 12C6H5NH2 and 12C6H5NHD in a single scan. 13 12 C C5H5NH2 is also present at m/z ¼ 94 u to a minor extent, but its transitions are easily discernible.

15 20

(m / z) / u 30 40 50 60 70

S / mV

−1 −2 −3 −4

C6Hx 12

C

C5Hx C4Hx

C2Hx C3Hx

−5 10 12 14 16 18 20 22 24 26 28 30 ∆t / ms

93

0 −20 −40 −60

S / mV

10 0

−80 −100 31

FIGURE 11.13 TOF mass spectrum after UV ionization (311.5 nm) of vibrationally excited jet-cooled aniline C6H5NH2 (N ¼ 2 NH chromophore at 6738 cm21) (Source: From Hippler, M., Habilitation thesis, ETH Zu¨rich, 2001). S denotes the ion signal, Dt the flight time in the time-of-flight mass spectrometer; m/z is the corresponding nonlinear mass scale. Note the different scales of the panel on the left (fragment spectrum) and on the right (parent peaks, 12C6H5NH2 and 13C12C5H5NH2).

334

Isotope Effects in Chemistry and Biology

C. THE N 5 2 NH C HROMOPHORE A BSORPTION OF A NILINE I SOTOPOMERS N EAR 6750 CM 21 : I SOTOPE E FFECTS AND V IBRATIONAL M ODE S PECIFICITY IN IVR AND T UNNELING P ROCESSES In Figure 11.14, the N ¼ 2 NH chromophore absorption between 6500 and 7100 cm21 of C6H5NH2 shows overtone and combination bands of symmetric (ns) and antisymmetric (nas) NH-stretching vibrations and anharmonic resonances with them. In the upper panel of Figure 11.14, the massgated jet spectrum in a very cold Ar expansion is shown. The rotational contours are characterized by a very low rotational temperature, and no vibrational hot bands are apparent. The middle panel displays the corresponding spectrum in an expansion with N2 as seeding gas. Compared to the Ar expansion, the rotational contours are broader due to a somewhat higher rotational temperature, and some bands clearly exhibit tunneling splittings. The lower panel shows for comparison an FTIR spectrum at room temperature. In Figure 11.15, the corresponding absorption spectrum of the N ¼ 2 NH chromophore of C6H5NHD is displayed, which is dominated by the NH-stretching overtone 2nNH. Mass-gated ionization detection of the IR excitation allowed isotopomer-selective spectroscopy of C6H5NHD in the isotopomer mixture with about 25% C6H5ND2 and C6H5NH2, respectively. Without isotopomer selection, no separated FTIR spectrum of C6H5NHD can be 40

S / mV

30

2bs+ns

2ns

2bs+nas

Ar

ns+nas

20 2nas

10 0

S / mV

∆ (inv)

20

N2

∆ (inv) ∆ (inv)

10 0

RT

s / pm2

0.5 0.4 0.3 0.2 0.1 0.0 6500

6600

6700 6800 ~ n / cm−1

6900

7000

FIGURE 11.14 N ¼ 2 NH chromophore absorption spectra of C6H5NH2 (Source: From Fehrensen, B., Hippler, M., and Quack, M., Chem. Phys. Lett., 298, 320– 328, 1998. With permission). The lower panel shows an FTIR reference spectrum of 0.34 mbar aniline at room temperature (l ¼ 28 m, res. 0.1 cm21). The middle and upper panels are ionization detected ISOS spectra of jet-cooled aniline, mass gated at m/z ¼ 93 u (12C6H5NH2), the middle trace in a jet expansion with 1 bar of N2, and the upper trace with 1 bar of Ar. Brackets indicate the Fermi resonance splitting (upper panel) and the tunneling splitting due to NH2 inversion (middle panel).

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics 80

2n NH

335

Ar

S / mV

60 40 20 0

S / mV

5

∆ (inv)

N2

4 3 2 1 0

In (Io / I )

0.08

RT

0.06 0.04 0.02 0.00 6500 6600 6700 6800 6900 ~ / cm−1 n

7000

FIGURE 11.15 N ¼ 2 NH chromophore absorption of C6H5NHD (Source: From Fehrensen, B., Hippler, M., and Quack, M., Chem. Phys. Lett., 298, 320– 328, 1998. With permission). The lower panel shows an FTIR reference spectrum of 0.4 mbar N-deuterated aniline (C6H5ND2) at room temperature (l ¼ 49 m, res. 0.1 cm21). The transitions observed in this region are expected to arise from small amounts of C6H5NHD due to incomplete deuteration. The middle and upper panels are ionization detected ISOS spectra of jet-cooled aniline, mass gated at m/z ¼ 94 u (12C6H5NHD), the middle panel in a jet expansion with 1 bar of N2, and the upper panel with 1 bar of Ar. The brackets in the middle panel indicate the tunneling splitting due to NHD inversion. The stars in the upper trace mark transitions, which are due to small amounts of 13C12C5H5NH2 present (compare Figure 11.14 and discussion in the text).

obtained. However, the absorptions (Fig. 11.15, lower panel) between 6500 and 7100 cm21 in a C6H5ND2 sample are expected to arise from small amounts of C6H5NHD due to incomplete deuteration; the ND chromophore of C6H5ND2 has strong absorptions in a different spectral region. A contour fit of 2nas with extrapolated rotational constants136 yields a rotational temperature Trot < 5 K for the Ar expansion and Trot < 60 K for the N2 expansion. From the absence of transitions from the tunneling split ground state level, which is higher in energy, Tinv , 15 K is estimated as a bound for the inversion vibrational temperature in the Ar expansion of both isotopomers. The tunneling splitting becomes apparent in the N2 expansion. From the intensity ratio of “tunneling satellites,” Tinv < 100 K is estimated. Three main transitions dominate the C6H5NH2 jet spectrum in the very cold Ar expansion. By extrapolation from the fundamentals136,184 and consistent with the band contours, they are tentatively assigned as the first overtones of the symmetric and antisymmetric NH-stretching vibrations 2ns and 2nas, respectively, and as the combination band ns þ nas. The NH-stretching modes act as an IR chromophore. Some of the weaker observed transitions presumably obtain their intensity by anharmonic resonance coupling with the NH-stretching modes. The two strongest transitions among them are assigned as combination bands ns þ 2bs and nas þ 2bs, which are

336

Isotope Effects in Chemistry and Biology

coupled to 2ns and ns þ nas, respectively, by a strong Fermi resonance exchanging two quanta of the symmetric NH-bending vibration bs (“scissor”) with one quantum of the symmetric NHstretching vibration ns. NH-stretching and bending vibrations are expected to be coupled by a Fermi resonance, as has previously been observed for other hydrides, whenever the 2:1 resonance condition is approximately fulfilled.25,26,93,94,98 – 102,104 – 107,135,159,160,163 – 165 With an observed position of the fundamental bs at 1610 cm21 and ns at 3421 cm21,136,184 the Fermi resonance coupling is quite effective. A harmonic approximation to estimate the unperturbed, zeroth-order positions of ns þ 2bs yields 6640 cm21 and for nas þ 2bs gives 6730 cm21. The true zeroth-order positions will have somewhat lower wavenumbers because of anharmonicity. The corresponding bands in the experimental jet spectrum at 6605 and 6670 cm21 appear at even lower wavenumbers because of the Fermi resonance. In the analysis of the anharmonic resonance system, the Fermi resonance exchanging one quantum of the symmetric NH-stretching vibration (ns) with two quanta of the symmetric NH-bending vibration (nb) and the Darling – Dennison resonance exchanging two quanta of the symmetric NH stretching with two quanta of the antisymmetric NH-stretching vibration (nas) have to be considered. Zeroth-order basis states lvs,vas,vbl of one symmetry group, which have the common polyad quantum number N ¼ vs þ vas þ 0.5vb are thus coupled and mixed. The interaction is described by an effective Hamiltonian with the usual matrix elements, Equation 11.21 to Equation 11.23: ~ Nv ;v ;v ;v ;v ;v ¼n~ 0s vs þ n~ 0as vas þ n~ 0b vb þ x~ 0s;s v2s þ x~ 0as;as v2as þ x~ 0bb v2b H s as b s as b þ x~ 0s;as vs vas þ x~ 0s;b vs vb þ x~ 0as;b vas vb

~ Nv ;v ;v ; H s as b

ðvs 21Þ;vas ;ðvb þ2Þ

¼

ð11:21Þ

1 1 k v ðv þ 1Þðvb þ 2Þ 2 sbb 2 s b

1=2

ð11:22Þ

~ N vs ; vas ; vb ; ðvs 2 2Þ; ðvas þ 2Þ; vb ¼ 1 gs;s;as;as {vs ðvs 2 1Þðvas þ 1Þðvas þ 2Þ}1=2 H 4

ð11:23Þ

Matrix diagonalization gives the perturbed positions, which are observed in the experiment. For convenience, the matrix for the N ¼ 2 polyad of A1 symmetry species is given explicitly in Equation 11.24, and the matrix of B2 symmetry in Equation 11.25, where G denotes the term values of unperturbed states (for axis conventions and symmetry operations in the B2 symmetry classification see Ref. 136): l2; 0; 0l l0; 2; 0l 0

B Gð2ns Þ B B B B N ¼ 2; A1 symmetry: B B B B @

gs;s;as;as Gð2nas Þ

l1; 0; 2l 1 pffiffi ksbb 2 0 Gðns þ 2nb Þ

l0; 0; 4l 0

1

C C C 0 C C C pffiffi C 6 C ksbb C A 2

ð11:24Þ

Gð4nb Þ l1; 1; 0l 0 B Gðns þ nas Þ N ¼ 2; B2 symmetry: @

l0; 1; 2l 1 1 k C 2 sbb A Gðnas þ 2nb Þ

ð11:25Þ

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

337

In the jet spectrum of C6H5NH2, both B2 states in the N ¼ 2 polyad have been observed. Assuming that the whole band strength arises from the chromophore state ns þ nas, while the combination band nas þ 2bs carries no oscillator strength, the unperturbed positions and the coupling matrix element can be determined from the observed perturbed positions and their relative intensity. The deperturbation yields zeroth-order positions at 6693 cm21 for nas þ 2bs and 6754 cm21 for ns þ nas. The A1 polyad with N ¼ 2 comprises four states, of which three have been observed. The effective Hamiltonian predicts in addition a very weak polyad component of A1 symmetry around 6400 cm21 (approximately 4bs), which we could not observe. At this stage, only a preliminary two-state analysis of the Fermi resonance between 2ns and ns þ 2bs can be attempted for the A1 polyad. In this model, 2ns as bright state is coupled to the dark state ns þ 2bs. The approximate deperturbation gives zeroth-order positions at 6626 cm21 for ns þ 2bs and 6718 cm21 for 2ns. In both symmetry groups, the off-diagonal coupling matrix element is about 47 cm21. ksbb is thus ca. 100 cm21 (Equation 11.25), corresponding to similar couplings for the CH chromophore and time scales of about 100 fsec for IVR in CHX3 or CHXY2 compounds, for example Refs. 25,26,93,94,98– 102,104 –107,135,159,160,163– 165. In the jet spectrum of C6H5NHD, no resonance splitting is evident in the 2nNH overtone region (Figure 11.15). The Fermi resonance is expected to be weak here, because the NH-bending fundamental frequency is much lower (1445 cm21,184) and the resonance will only become relevant at higher overtones (see also Ref. 184 for predictions of these resonances). Jet-cooled aniline bands observed in the Ar expansion have a FWHM between 3 and 5 cm21. Assuming a homogeneous rovibronic line width between 1 and 2 cm21, the rotational contour of 2nas is very satisfactorily reproduced for room temperature and jet conditions. The quasihomogeneous line width might be explained by the exponential decay of vibrational excitation into the quasicontinuum of background states near the statistical limit.91,117 In the 6740 cm21 region of C6H5NH2, a vibrational density of states of ca. 1.7 £ 105 per cm21 interval is estimated with the Beyer – Swinehart algorithm191 with harmonic frequencies from ab initio calculations184 (B3LYP, 6-31Gpp basis set;192 the inversion mode is approximated by an effective harmonic oscillator with wavenumber 350 cm21). The upper bound for the homogeneous bandwidth G~ , 2 cm21 provides td . 2.5 psec as a bound for the decay time of vibrational excitation into the dense background manifold. By these rough estimates, we do not imply that all background states are equally involved in the redistribution process. Some vibrational bands in the N2 expansion have a satellite at lower wavenumbers due to the hot-band transition from the upper component of the ground-state tunneling doublet, which is significantly populated compared to the very cold Ar expansion. The observed satellite shift corresponds to the difference of tunneling splittings in the ground and vibrationally excited NH-stretching states. A shift to lower wavenumbers indicates a smaller splitting DE^ of levels of different vibrational parity in the excited vibrational state. If no separate satellite is noticeable, DE^ is similar to the ground state. In the C6H5NH2 ground state, DE^ is 40.9 cm21,136 and a tunneling splitting of 20 ^ 2.5 cm21 in 2ns, 17 ^ 2.5 cm21 in ns þ nas and 11 ^ 1 cm21 in 2nas is thus found. No satellites are apparent for ns þ 2bs and nas þ 2bs. These bands have presumably a tunneling splitting close to the ground state value, approximately within 40 ^ 10 cm21. Since DE^ decreases in the NH-stretching overtone bands, the NH stretching is an inhibiting mode for the inversion, and the effective barrier height for inversion has therefore increased relative to the vibrational ground state. This result is in agreement with the previous analysis of the NH-stretching fundamentals.136 From an analysis of the NH-bending fundamental bs of C6H5NH2, the promoting nature of this mode has been established, since the tunneling splitting increases from DE^ ¼ 40.9 cm21 in the vibrational ground state to 63.3 cm21 in the bs fundamental.136,184 In the NH-bending/stretching combination bands ns þ 2bs and nas þ 2bs observed here, the inhibiting character of the NH stretching apparently balances the promoting character of the NH-bending vibration. With a ground-state value DE^ of 23.8 cm21 for C6H5NHD,136 DE^ ¼ 6.5 ^ 2.5 cm21 is calculated for 2nNH. The time of stereomutation in C6H5NHD is calculated as ts ¼ h/(2DE^).

338

Isotope Effects in Chemistry and Biology

NH-stretching excitation increases the effective barrier height for stereomutation relative to the vibrational ground state and therefore inhibits the stereomutation. Previously, values ts ¼ 700 fsec for the ground state and ts ¼ 1.33 psec for the NH-stretching fundamental were determined.136 With the present value ts ¼ 2.6 psec for 2nNH, an increase of approximately a factor of 2 for each quantum of NH stretching is thus found. At these energies and time scales, stereomutation is adiabatically separated from the vibrational motion, as theoretically analyzed in Ref. 136. A simple explanation for the inhibition of stereomutation by NH stretching is provided by the increase of the effective adiabatic barrier due to the increased NH-stretching frequency in the sp2 hybridized planar transition structure. Other, dynamical effects such as the increase of the mean N – H bond length by vibrational excitation are also important in a quantitative analysis.

D.

13

C I SOTOPE E FFECTS IN

THE

IVR

OF

V IBRATIONALLY E XCITED B ENZENE

Because of its high symmetry and as the archetypical aromatic compound, benzene occupies a special position in chemical physics. It has become a textbook example of an intermediate sized molecule to study structure,3,195 – 201 force fields,3,193,202,203 IVR,44,62,63,81,82,204 – 218 and normal mode/local mode behavior in the CH chromophore absorption.25,62,63,205,219 – 222 The benzene vibrational spectroscopy has thus a long tradition (see for a brief review Ref. 38). The study of dynamic IVR processes in benzene has started with the pioneering work of Berry and co-workers on the overtone spectroscopy of benzene in a room temperature cell.204,205 After an approximate deconvolution to correct for the rotational band contour, the remaining broadening of overtone bands was attributed to ultrafast IVR decay processes. This broadening was then explained by Sibert et al.206,207 assuming strong Fermi resonances between the CH-stretching oscillator and CH-bending modes as main mechanism of IVR (see, however, also Refs. 25,208), and more recent classical and quantum calculations have established the dominant role of these Fermi resonances for IVR on very fast time scales.206 – 209,212 – 218 In 12C6H6, an analysis of the IR active CH-stretching fundamental n20 (Wilson notation3) has confirmed the presence of strong Fermi resonance splittings.44,62,63,223 – 225 The calculations predict even finer details of IVR in 12C6H6, and a whole sequence of time-scales of redistribution of vibrational excitation among different modes has been suggested in recent publications.206 – 209,212 – 218 A comparison of calculated spectra with the few available jet overtone spectra of 12C6H6, however, shows at most qualitative, but no quantitative agreement. Recent jet measurements have also clearly shown that roomtemperature spectra are heavily affected by inhomogeneous rotational and vibrational hot-band congestion.38,44,62,63,81,83,210,211,224 Compared to 12C6H6, only few spectra of the isotopomer 13C12C5H6 are reported,200,226 – 228 and our spectra in Ref. 38 are the first overtone spectra published to our knowledge. Experimental data for 13C12C5H6 are relevant for structural studies, since the isotopic substitution in the carbon ring will provide structural information not available by deuterium labeling, and experimental 13C isotope shifts are also useful for force-field determinations. In a recent time-dependent quantum mechanical study on IVR from the second CH-stretching overtone in 12C6H6, nonstatistical energy redistribution has been attributed to the influence of the symmetry of the molecule.213 It is thus interesting to study the effect of the reduced symmetry in 13C12C5H6 on IVR. The shift of vibrational states in 13C12C5H6 compared to 12C6H6 will also affect Fermi and other anharmonic resonances and thus influence IVR processes. With the ISOS technique described, we have observed isotopomer resolved overtone spectra of 12C/13C isotopomers (at natural abundance) of benzene.30,38 In this case, two-photon (1 þ 1) ~ 10 transition and other vibronic levels of the S1 A~ state REMPI ionization occurs via the A~ ˆ X6 21 44,62,63,229,230 above 38,606 cm , (see Ref. 38 for a more detailed discussion of the absorption mechanism in the ISOS scheme of benzene). For the benzene spectra shown below, an UV wavelength of 266 nm was chosen for the IR þ UV double-resonance experiment. This particular UV wavelength is easily accessible and available by frequency tripling of a Nd:YAG laser,

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

339

for example by employing part of the output of the Nd:YAG laser, which is also used as pump laser for the generation of IR radiation. With a natural abundance of 1.1% 13C, the benzene or aniline isotopomers with one 13C atom have 6.2% abundance in the natural isotopomer mixture. Mass gating the ionization detection in the IR þ UV double-resonance experiment to the corresponding masses m/z ¼ 78 and 79 u allows obtaining the separate vibrational spectra of both isotopomers in a single scan of the natural isotopomer mixture. For the jet experiments, 500 mbar of Ar as seed gas is flowing over frozen benzene at 08C, which has a measured vapor pressure of 35 mbar. Certain experimental conditions promote the formation of clusters, for example high partial benzene sample pressure, high Ar stagnation pressure, or probing molecules in the middle or the end of the jet pulse train. Cluster formation makes itself conspicuous by additional mass peaks in the TOF spectra and specific peaks in the UV spectrum of the S1 ˆ S0 transition in benzene.63,231 – 233 IR spectra of benzene clusters differ greatly from the monomer spectra.63,66 Care was therefore taken that the spectra displayed here are not affected by cluster formation. Isotopically pure benzene 12 C6H6 (depleted benzene, Cambridge Isotope Laboratories, 12C6 99.95%) is also used for some FTIR measurements. Benzene 12C6H6 (D6h symmetry) has 20 normal vibrations, of which 10 are doubly degenerate.234 In D6h symmetry, only perpendicular IR transitions from the ground state to E1u vibrational states are allowed with an in-plane electric dipole moment, or parallel IR transitions to A2u vibrational states with an out-of-plane transition dipole moment.3 Seven fundamental vibrations are Raman active and four are IR active.235 Benzene has four fundamental normal modes (essentially CH-stretching modes) between 3048 and 3074 cm21, which are n2 (A1g symmetry), n7 (E2g), n13 (B1u) and the only IR active CH-stretching fundamental n20 (E1u) in the Wilson notation. In the region of the first CH-stretching overtone near 6000 cm21, only the three combination bands of CH-stretching modes (n20 þ n7, n7 þ n13, and n2 þ n20) have E1u symmetry components with allowed IR transitions from the ground state. All other combinations including the overtone 2n20 are IR inactive (because of the centre of symmetry, no overtone frequencies with v even can occur in the IR spectrum3). A normal mode description of vibrational overtone bands is, however, not entirely satisfactory, since all IR-active CH-stretching normal mode combination bands are strongly coupled by Darling –Dennison and related anharmonic resonances. The resultant mixed states probably resemble local mode excitations. In the local mode picture, each CH-stretching overtone is dominated by one dominant absorption feature corresponding to one local CH oscillator with all vibrational excitation. For the first overtone region, a recent calculation of the CH-stretching absorption spectrum by Iachello and Oss based on the algebraic “vibron” model has given three IR absorptions, at 6004, 6113, and 6128 cm21, with relative intensity 1, 0.0017, and 0.006, respectively.236 A pure local mode description of overtones, however, is also not satisfactory: theoretical studies206 – 209,212 – 218 and the analysis of the spectrum of the CH-stretching fundamental44,62,63,223 – 225 show that CH-stretching modes are in addition strongly coupled with CH-bending modes by Fermi resonances exchanging one quantum of CH-stretching excitation with two quanta of CH-bending excitation. All vibrational modes of a given symmetry with common polyad quantum number N are thus strongly coupled and mixed by Fermi, Darling – Dennison and related anharmonic resonances, where N ¼ vs þ 0.5vb; vs represents all CH-stretching and vb all CH-bending quantum numbers. We therefore refer to the IR absorption near 6000 cm21 as the N ¼ 2 CH chromophore absorption, instead of the more common “first CH-stretching overtone” (see also Ref. 25 for the IR-chromophore concept). The benzene isotopomer 13C12C5H6 is an asymmetric top of the point group C2v. The asymmetry parameter k ¼ 0:916 is close to 1;200,226 the replacement of one 12C by the 13C isotope introduces only a small perturbation. In C2v symmetry, many more normal modes are formally IR allowed: Among the 30 normal mode vibrations, 27 vibrations of A1, B1, and B2 symmetry species are IR active, and all modes are Raman allowed. Isotopic labeling will also cause in principle an alteration of normal mode characters (mode scrambling). The perturbation of the 13C, however, is so small that intensities in 13C12C5H6 cannot be expected to deviate much from the corresponding

340

Isotope Effects in Chemistry and Biology

transitions in 12C6H6, despite the relaxed IR selection rules, and the character of corresponding vibrational modes will also be similar. The 13C isotopic substitution will introduce a minor frequency shift to lower wavenumbers if no perturbations apply. If only the one carbon atom in question moves in a hypothetical vibration, a relative energy shift of 2 4.1% will occur due to the change in the reduced mass. The overtone vibrations of the CH chromophore will probably resemble local diatomic CH oscillators. The relative energy shift of 13CH harmonic oscillator vibrations compared to a 12CH harmonic oscillator is about 2 0.3%. The N ¼ 2 CH chromophore absorption spectrum of benzene obtained by FTIR spectroscopy at room temperature is shown in Figure 11.16. The spectrum of 12C6H6 is obtained in an isotopically pure sample. Depleted benzene 12C6H6 is readily available, since it is a solvent used 600000 13 12

C C5H6

σ / fm2

400000

200000 0

800000 12C

σ / fm2

600000

6H6

400000 200000 0 800000 C6H6

σ / fm2

600000 400000 200000

0 5800

5900

6000 n∼ / cm−1

6100

6200

FIGURE 11.16 FTIR spectra of the N ¼ 2 CH chromophore absorption of benzene at room temperature (Source: From Hippler, M., Pfab, R., and Quack, M., J. Phys. Chem. A, 107, 10743– 10752, 2003. With permission). The lower panel shows the spectrum of 2.20 mbar benzene (natural isotopomer mixture), the middle panel the corresponding spectrum of 2.14 mbar depleted benzene (12C6H6, 12C6 99.95%) at room temperature (l ¼ 28 m, res. 0.015 cm21). In the upper panel, the depleted spectrum has been subtracted from the normal benzene spectrum using appropriate weights, and rescaled. This difference spectrum is essentially due to the 13C12C5H6 isotopomer (note that absorption cross-sections for 13C12C5H6 are only approximate).

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

341

in NMR spectroscopy. For 13C12C5H6, no isotopically pure sample was available. To obtain its spectrum, the depleted spectrum is subtracted from the normal benzene spectrum using appropriate weights and rescaled. The difference spectrum is essentially due to the 13C12C5H6 isotopomer. This manipulation, however, is prone to noise and possibly also to systematic errors and distortions. The difference spectrum can therefore only be considered as an approximation to the proper 13C12C5H6 spectrum, in particular the absorption cross-sections indicated are only approximate. The room-temperature spectra improve upon previously published spectra of the natural benzene isotopomer mixture, which had been obtained at much lower instrumental bandwidth (6 to 12 cm21).205 The general appearance of both isotopomer spectra is similar. A main broad feature with FWHM of ca. 40 cm21 with a distinct central peak and some diffuse structure dominates the spectrum. The central peak is at 6005.3 ^ 0.4 cm21 for 12C6H6 and at 5991 ^ 1 cm21 for 13 12 C C5H6. Higher CH-stretching overtones of 12C6H6 are well described by a Morse oscillator with vm ¼ 3163 cm21 and xm ¼ 57.7 cm21, which has been interpreted as the local mode 12CH oscillator absorption.205,210 The 12CH Morse oscillator vibration for v ¼ 2 is then calculated at 5980 cm21, and at 5963 cm21 for the 13CH Morse oscillator taking into account the change of reduced masses,205 the shift of 2 17 cm21 being comparable to the 2 14 cm21 for band maxima mentioned above, although one must note that the spectrum should be dominated by the five 12CH local mode absorptions in the 13CH12C5H5 isotopomer (see below). A closer examination of the experimental spectra reveals some differences in relative intensities and rotational band contours of corresponding transitions, for example in the wing at higher wavenumbers of the main feature, where two additional features are apparent for 13C12C5H6. A further assignment or interpretation of the spectra is not possible due to inhomogeneous congestion. In addition to broad rotational contours, room-temperature spectra are heavily affected by vibrational hot-band transitions, since about 40% of benzene molecules are in vibrationally excited states. In Figure 11.17 and Figure 11.18, the ionization detected, jet-cooled ISOS spectra of benzene are displayed and compared with the corresponding FTIR room-temperature spectra. In the natural benzene isotopomer mixture, mass gating the ionization detection at m/z ¼ 78 and 79 u yield simultaneously the separated spectra of 12C6H6 and 13C12C5H6 isotopomers, respectively. In comparable jet experiments (for example on aniline, see previous section), rotational temperatures of about 5 K have been obtained, and vibrational hot-band transitions have been almost completely suppressed; similar conditions are expected to apply here. The spectral simplification afforded by jet cooling is striking. The FWHM of vibrational band contours decreases from about 40 cm21 at room temperature to about 4 to 5 cm21 for 12C6H6, and distinct resolved or partly resolved vibrational transitions become apparent. Line positions and intensities of observed transitions in the jet-cooled N ¼ 2 CH chromophore absorption spectra of the 12C6H6 and 13C12C5H6 isotopomers are summarized in Ref. 38. The 12C6H6 spectrum (Figure 11.17) is essentially in good agreement with the previously published jet spectrum of Page et al.,44,62,63 but it is not affected by saturation and has a better signal-to-noise ratio (see Ref. 38 for a detailed comparison). The spectrum of the 13 12 C C5H6 isotopomer in Figure 11.18 demonstrates that the 6.2% abundance in the natural isotopomer mixture of benzene is sufficient to obtain jet spectra with the ISOS technique. The strongest peak in the 12C6H6 jet spectrum is located at 6006.0 ^ 0.5 cm21, and the corresponding strongest peak in the 13C12C5H6 jet spectrum at 5993 ^ 1 cm21. These peak positions provide good estimates for the band origins giving a 13C isotopomer shift of 2 13 cm21 or a relative shift of 2 0.22%. This is close to the estimation from the room-temperature spectrum, but more reliable, since the value has been obtained from vibrational bands with much reduced inhomogeneous congestion. In 12C6H6, the main absorption feature and other features have a FWHM of about 4 to 5 cm21, whereas spectral contours in 13C12C5H6 are much broader, with a FWHM of about 15 cm21, but the features have sharp edges with a halfwidth of about 4 cm21 only. It is thus likely that part of the broadening is due to spectral congestion, for example by the different rotational contour of the asymmetric rotor or by the presence of additional vibrational bands that are IR active

342

Isotope Effects in Chemistry and Biology 35 30 S / mV

25 20 15 10 5 0 800000

σ / fm2

600000 400000 200000 0 5800

5900

6000 ∼ n / cm−1

6100

6200

FIGURE 11.17 N ¼ 2 CH chromophore absorption of 12C6H6 near 6000 cm21 (Source: After Hippler, M., Pfab, R., and Quack, M., J. Phys. Chem. A, 107, 10743– 10752, 2003. With permission). The lower panel shows an FTIR reference spectrum of 2.14 mbar depleted benzene (12C6H6, 12C6 99.95%) at room temperature (l ¼ 28 m, res. 0.015 cm21). The upper panel shows the ionization detected ISOS spectrum of jet-cooled benzene, mass gated at m/z ¼ 78 u (12C6H6), in a jet expansion with 500 mbar of Ar.

in the C2v symmetry. A degenerate E1u IR active vibrational band in D6h symmetry correlates to two IR active bands of A1 and B1 species in C2v symmetry,3 which may be affected by a strong Coriolis perturbation.226 This effect may also give rise to an anomalous intensity distribution. Previous theoretical studies provide some guidelines for a first tentative statistical interpretation of the 12C6H6 jet spectrum in terms of IVR processes: The calculations of Iachello and Oss have shown that one strong pure CH-stretching transition prevails in the IR spectrum of this region, which is calculated to occur at 6004 cm21.236 This CH-stretching mode is coupled to the IR field as a chromophore state. By anharmonic coupling, its vibrational excitation will be redistributed among many vibrational modes, giving rise to the observed vibrational bands in the IR jet spectrum. At 6000 cm21, the vibrational density of states is about 1560 states per cm21 interval, as counted in the harmonic approximation by the Beyer – Swinehart algorithm191 using an experimental set of fundamental vibrations (compiled in Refs. 193,200). E1u symmetry is required for an allowed anharmonic coupling with the E1u CH-stretching overtone level. In the regular limit,237 the density of E1u states is 2/24 ¼ 1/12 of the total density, about 130 states per cm21. Instead of the one-step coupling of the bright state to the bath, a sequential “tier” picture89,90,98,206,207,238 appears to provide a better description of IVR in benzene: In a first tier, the bright CH-stretching state is strongly coupled to a subset of vibrational states. A broad Lorentzian line shape obtained by a visual fit envelops all absorption features observed in the experimental jet spectrum (Figure 11.19, dotted curve) with FWHM G~ ¼ 45 cm21 corresponding to a decay time of vibrational excitation into a first tier of vibrational states of t < 120 fsec. Since CH-stretching and -bending modes are known to be coupled by strong Fermi resonances,44,62,63,206 – 209,212 – 218,223 – 225 the first tier presumably involves the CH-stretching/bending combination bands of E1u symmetry species in the absorption region. The inferred decay time is compatible with theoretical calculations206 – 209,212 – 218 and with typical redistribution times found, e.g., for CH-stretching/bending Fermi resonances in CHX3 or CHXY2

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

343

2.5

S / mV

2.0 1.5 1.0 0.5 0.0 600000

σ / fm2

400000 200000 0 5800

5900

6000 n∼ / cm−1

6100

6200

FIGURE 11.18 N ¼ 2 CH chromophore absorption of 13C12C5H6 near 6000 cm21 (Source: From Hippler, M., Pfab, R., and Quack, M., J. Phys. Chem. A, 107, 10743 – 10752, 2003. With permission). The lower panel shows a manipulated spectrum, where the FTIR spectrum of 2.14 mbar depleted benzene (12C6H6, 12C6 99.95%) at room temperature has been subtracted from the FTIR spectrum of 2.20 mbar benzene (natural isotopomer mixture) and rescaled (l ¼ 28 m, res. 0.015 cm21). The difference spectrum is essentially due to the 13C12C5H6 isotopomer (as in Figure 11.16). The upper panel shows the ionization detected ISOS spectrum of jet-cooled benzene, mass gated at m/z ¼ 79 u (13C12C5H6), in a jet expansion with 500 mbar of Ar.

compounds.25,26,93,94,98 – 102,104 – 107,135,159,160,163 – 165 The Lorentzian is centered at 5988 cm21, which corresponds to the position of the chromophore state. It is interesting to note that the Morse oscillator level v ¼ 2 is calculated at 5980 cm21, close to the position found here for the chromophore state in the N ¼ 2 polyad absorption. The main absorption feature at 6006 cm21 has accompanying weaker satellite lines. A second Lorentzian obtained by a visual fit envelops these absorption features (Figure 11.19, dashed curve) with G~ ¼ 15 cm21 corresponding to a decay time t < 0.35 psec of vibrational excitation into a second tier of vibrational states. This tier could be composed of vibrational states coupled by weaker higher order anharmonic resonances to the CH-stretching/bending manifold. The two-tier statistical model proposed here for IVR in the N ¼ 2 CH chromophore absorption is compatible with the experiment and with expectation. The analysis of the CH-stretching fundamental has provided a similar coupling scheme:44,62,63,223 – 225 the IR-active CH-stretching fundamental n20 is coupled to the IR field. By strong Fermi resonances with CH-stretching/bending combination bands, vibrational excitation is redistributed to an absorption triad, n20 ; n18 þ n19 ; and n1 þ n6 þ n19 : The latter band is in addition coupled by weaker anharmonic resonances with n3 þ n6 þ n15 : The 4 cm21 width (FWHM) of features in the jet spectrum of the N ¼ 2 absorption is an upper bound for the homogeneous bandwidth G~ giving a lower bound t . 1.3 psec for the decay time of further redistribution of vibrational excitation. Callegari et al. have obtained the “eigenstate resolved” benzene spectrum between 6004 and 6008 cm21 by IR– IR double resonance with cw-laser systems and bolometric detection.81,82 From the observed clustering of eigenstates, the authors deduced further time scales for IVR ranging from 100 psec to 2 nsec. The integral of a Lorentzian function with FWHM G~ centered around n~0 has its half height at n~0 ; and G~ extends from its 1/4 to the 3/4 heights. This suggests another useful and more quantitative

344

Isotope Effects in Chemistry and Biology

40

S / mV

30 20 10 0

(a)

5800

5900

6000 6100 ∼ n / cm−1

6200

800

G rel

900 400 200

(b)

0 5800

5900

6000 ∼ n / cm−1

6100

6200

FIGURE 11.19 (a) Jet-cooled, isotopomer-selective overtone spectrum of the N ¼ 2 CH chromophore absorption of 12C6H6 (as in Figure 11.17). The dotted curve is a Lorentzian function with 45 cm21 FWHM centered at 5988 cm21, the dashed curve a Lorentzian function with 15 cm21 FWHM centered at 6003 cm21; (b) relative Grel ðn~Þ (integral of the absorption cross-section) corresponding to the jet-cooled N ¼ 2 CH chromophore absorption of 12C6H6, in arbitrary units. The position of the half height (solid arrow) and of the 1/4 and 3/4 heights (dotted arrows) are indicated (see text for a discussion) (Source: After Hippler, M., Pfab, R., and Quack, M., J. Phys. Chem. A, 107, 10743– 10752, 2003. With permission).

analysis38,90 of the spectra considering the integral Sðn~Þ; Sðn~Þ ¼

ðn~

sðn~ 0 Þdn~ 0

n~0

ð11:26Þ

or more appropriately, but almost equivalently Gðn~Þ ¼

ðn~ n~0

0

sðn~ 0 Þn~ 21 dn~ 0 <

1 Sðn~Þ n~G

ð11:27Þ

sðn~Þ is the absorption cross-section. The lower integration limit n~0 is near 5850 cm21, outside the absorption range, and n~G is the center of gravity or effective band center. The integral Gðn~Þ corresponding to the jet-cooled N ¼ 2 CH chromophore absorption of 12C6H6 is shown in Figure 11.19. The effective band center obtained from the half height of this function is n~G ¼ 5995 cm21 ; whereas the effective width G~ ¼ 60 cm21 is obtained from the 1/4 and 3/4 heights, in reasonable agreement with the visual fit shown in Figure 11.19. We thus finally quote G~ ¼ ð50 ^ 10Þ cm21 ; and a resulting approximate decay time of vibrational excitation t < 100 fsec

Isotope Selective Infrared Spectroscopy and Intramolecular Dynamics

345

combining both approaches. The basis of this dynamic analysis is the assumption of one single chromophore absorption carrying all the oscillator strength in the integration range. This assumption has had some theoretical basis for a long time25,90,238,239 and has been justified again more recently, e.g., by the vibron model calculations of Iachello and Oss.236 For 12C6H6, this chromophore absorption probably resembles the excitation of local 12CH Morse oscillators. Because of the similarities in the general appearance of the jet spectra, the N ¼ 2 CH chromophore absorption of 13C12C5H6 can probably be interpreted on similar general lines as outlined here for 12C6H6. Since no theoretical calculations exist to our knowledge to guide the assignment, only a preliminary analysis of the IVR dynamics can be made at this stage using some simple models: On the basis of a local mode picture, one could expect two chromophore absorptions for 13C12C5H6 in this overtone region, a strong absorption due to local 12CH Morse oscillators, and a weaker absorption due to the local 13CH Morse oscillator. Assuming now that the absorption of the local 12CH Morse oscillators is dominating, one can attempt a similar quantitative analysis as before for the 12C6H6 isotopomer: In Figure 11.20, Gðn~Þ is shown for the jet-cooled N ¼ 2 CH chromophore

2.0

S / mV

1.5 1.0 0.5 0.0 5800

5900

6000 ~ n / cm−1

6100

6200

5900

6000 ~ n / cm−1

6100

6200

(a) 60

Grel

40

20 0 5800 (b)

FIGURE 11.20 (a) Jet-cooled, isotopomer-selective overtone spectrum of the N ¼ 2 CH chromophore absorption of 13C12C5H6 (as in Figure 11.18). The dotted curve is a Lorentzian function with 40 cm21 FWHM centered at 5993 cm21; (b) relative Grel ðn~Þ (integral of the absorption cross-section) corresponding to the jetcooled N ¼ 2 CH chromophore absorption of 13C12C5H6, in arbitrary units. The position of the half height (solid arrow) and of the 1/4 and 3/4 heights (dotted arrows) are indicated (see text for a discussion) (Source: After Hippler, M., Pfab, R., and Quack, M., J. Phys. Chem. A, 107, 10743– 10752, 2003. With permission).

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Isotope Effects in Chemistry and Biology

absorption of 13C12C5H6. The effective band center is obtained at n~G ¼ 5993 cm21 with width G~ ¼ 40 cm21 ; and a corresponding Lorentzian function envelops indeed all absorption features of the 13 12 C C5H6 jet spectrum. The center of gravity of the band is shifted by only 2 2 cm21 compared to the value for 12C6H6,38 presumably because the dominating chromophore absorption is in both cases due to 12CH Morse oscillators (as detailed before, the 13CH Morse oscillator has an expected shift of 2 17 cm21). Since the inherent assumption of one single chromophore absorption in this analysis is presumably not strictly valid for 13C12C5H6, the actual G~ due to IVR will be somewhat smaller than the 40 cm21 obtained by the integration method. We thus estimate G~ # 40 cm21 and a resulting approximate decay time of vibrational excitation t $ 130 fsec. This very fast decay of initial vibrational excitation is most likely due to strong anharmonic Fermi resonances between CHstretching and CH-bending modes, since similar resonances and redistribution times have been found for 12C6H6 (see above) and CHX3 or CHXY2 compounds.25,26,93,94,98 – 102,104 – 107,135,159,160,163 – 165 Zhang and Marcus212 have calculated the absorption spectrum of 12C6H6 in the 6000 cm21 region using an anharmonic force field together with dipole moment derivatives.202 More recently, Rashev et al.214 have also published a calculated spectrum based on anharmonic force fields202,203 and symmetrized complex coordinates. Relative intensities were calculated assuming local CH oscillators as zeroth-order “bright” states. A comparison with our experimental jet spectrum shows qualitative, but no quantitative agreement; also both calculations differ markedly (see Ref. 38 for more details). Further progress in the understanding of IVR in the important model molecule benzene could be achieved by alternative theoretical models to be developed in close comparison with experimental jet spectra. The isotopomer C6D5H might also be a promising candidate for further experimental and theoretical studies.

VI. CONCLUSIONS AND OUTLOOK In the present review we have stressed our recent developments of mass and isotope selective IR spectroscopy that have been carried out over about the last decade in the framework of our studies of intramolecular dynamics as derived from high-resolution spectroscopy. We have tried to relate this effort to alternative approaches by other research groups, wherever possible, and apologize for any possible omissions in the citation of work from our colleagues worldwide that may have occurred due to limited information on our side, due to the necessary restrictions being imposed on the length of our review, or that might have occurred inadvertently. We think that we have demonstrated in the foregoing sections the power of the methods developed in their application to various aspects of isotope effects in high-resolution spectroscopy. It should be clear that the methods described here open new avenues for highresolution spectroscopy. The selective ionization of vibrationally excited molecules in the IR þ UV double-resonance schemes allows indirect, but extremely sensitive detection of the IR excitation. The technique has been applied to overtone spectroscopy in supersonic jet expansions, where the cooling of vibrational and rotational degrees of freedom greatly reduces hot-band congestion and simplifies spectra, which is often essential for a reliable assignment. Ionization detection of the IR excitation can be coupled very efficiently with a TOF mass spectrometer, which allows the separation of spectral contributions of different components in a mixture. Mass spectrometry is thus added as a second dimension to IR spectroscopy, which greatly increases the selectivity. The IR excitation can also increase the selectivity of mass spectrometry, since it allows the separation of species which have nearly the same mass (isobars). Through their different IR spectra, isotope isomers can be distinguished in principle, although this aspect has not been further pursued here in detail. With these mass and isotopomer selective spectroscopic techniques it is possible to study isotope effects in intramolecular dynamics and energy

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redistribution240 with isotopomers in mixtures of natural abundance or enriched only to a limited extent. In considering the power of the presently discussed isotope selective techniques of OSVADPI, ISOS, and IRSIMS it may be useful to compare these with other powerful current techniques of high-resolution molecular spectroscopy as summarized in Table 11.1.241 As measures of spectroscopic power we use here first the obvious resolving power n=dn; and the effective resolution or instrumental bandwidth dn: Given the present limitations of our pulsed laser light sources, the current implementations of IRSIMS/ISOS perform only moderately well compared to FTIR spectroscopy, mid-IR diode laser spectroscopy or near-IR cw-diode laser spectroscopy. There is clearly room for improvement in this area and in the future powerful cw-optical parametric oscillators may be able to fill this gap and generate resolving powers larger than 108 with instrumental resolutions in the MHz or even sub-MHz range. The scanning range Dn~ and scanning power of ISOS/IRSIMS is excellent already today (Table 11.1). The sensitivity is not to be measured by effective absorption length l; as absorption signals are not measured directly, but rather an ionization signal. While this detection scheme has the usual disadvantage of lacking information about absolute absorption intensities, it is outstandingly sensitive, which more than compensates for the disadvantage. However, clearly the largest advantage in detection is the additional discrimination through the mass selectivity. An interesting future development might here be the combination of this advantage with the interferometric techniques thus providing hypothetical mass selective interferograms as proposed already some time ago.25 This would combine the great scanning power, scanning ranges, and high resolving powers of interferometric Fourier transform IR spectroscopy with the additional mass and isotope selectivity. However, the efficient implementation of such an experimental technique is highly nontrivial and not to be expected in the very near future. Besides an overview and outlook on possible extensions of the present techniques and their alternatives we should discuss also some further applications. Beyond the applications already discussed in some detail in this review, concerning molecular spectroscopy and dynamics, one can also conceive of mere analytical applications. One could think of ordinary analyses making use just of mass selected IR spectra of ordinary chemical mixtures (for instance in environmental or industrial applications). The isotopomer selected IR spectra could be used in special analyses studying isotope fractionation in natural processes or even in dating with isotopes. Here the IR molecular spectroscopy would provide the additional dimension to the isotope analysis. Another obvious application of ISOS/IRSIMS is in a preparatory phase of laser isotope separation. IR laser isotope separation has been discovered already in 1975/76242,243 and is now a well-established technique.194,242 – 250 Traditional spectroscopic techniques have been employed

TABLE 11.1 Powerful Spectroscopic Techniques to Apply to Atmospheric Analysis and Quantum Chemical Kinetics

FTIR Diode laser, direct absorption NIR-diode laser, cw-CRD ISOS/IRSIMS

RP

SP

#2 £ 106 * 2 £ 106 * 2 £ 108 * 6 £ 105

,107 ,5 £ 104 ,107 * 30,000 ($106)

dn (MHz) 20– 70 20– 30 ,1 ,500

Dn~ (cm21)

l (m)

20,000 20 (2500) 500 500 (20,000)

100 100 1000– 10,000 1023 (ion detection)

Resolving power RP ¼ n/dn, Scanning power SP ¼ Dn/dn, effective resolution or instrumental bandwidth dn, effective scanning range D~v; effective absorption length l; after Quack, M., Chimia, 57(4), 147 –160, 2003.

348

Isotope Effects in Chemistry and Biology

in optimizing the overall process, or else the laser isotope separation is optimized by simple trial of the laser frequencies used with respect to effective separation. ISOS/IRSIMS is obviously ideally suited to test for optimum frequency selection in appropriate laser isotope separation schemes. A particularly useful application of ISOS/IRSIMS would be in planning two-step laser isotope separation schemes.56,57 As there is now renewed interest in possible medical applications of stable isotopes245,246,250 – 253 one might think that this application might become more important in the future. In a more general context also laser isotope – isomer separation of organic molecules might become of interest as demonstrated in Ref. 254. A further future application might concern spectroscopic studies of molecules aiming at problems of fundamental physics. In this context, possible spectroscopic investigations of the new isotope effects based on differences in electroweak charges of isotopes in enantiomeric isotopomers22,23 discussed in Section I are obvious applications. ISOS/IRSIMS can be used here for preparatory spectroscopic studies of possible molecular candidates. If combined with appropriate additional experiments, it could also be used to study the effects of the weak nuclear interaction on energy differences of isotopic enantiomers, similar to the case of other enantiomers.24,27,166,255,256 There is clearly room for many exciting developments of isotope selective IR spectroscopy in the future.

ACKNOWLEDGMENTS We are particularly grateful to Benjamin Fehrensen, Hans Hollenstein, David Luckhaus, Robert Pfab, and Georg Seyfang for help and discussions, as well as other members of the Zu¨rich group as identified in the list of references. Our work is supported financially by the ETH Zu¨rich (including C4 and CSCS), the AGS project and the Schweizerischer Nationalfonds.

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196 Snels, M., Beil, A., Hollenstein, H., and Quack, M., Excited vibrational states of benzene: high resolution FTIR spectra and analysis of some out-of-plane vibrational fundamentals of C6H5D, Chem. Phys., 225, 107–130, 1997. 197 Angus, W. R., Bailey, C. R., Hale, J. B., Ingold, C. K., Leckie, A. H., Raisin, C. G., Thompson, J. W., and Wilson, C. L., Structure of benzene, J. Chem. Soc. London, 912– 987, 1936. 198 Tamagawa, K., Iijima, T., and Kimura, M., Molecular structure of benzene, J. Mol. Struct., 30, 243– 253, 1976. 199 Plı´va, J., Johns, J. W. C., and Goodman, L., Infrared bands of isotopic benzenes: n13 and n14 of 13 C6D6, J. Mol. Spectrosc., 148, 427– 435, 1991. 200 Hollenstein, H., Piccirillo, S., Quack, M., and Snels, M., High-resolution infrared spectrum and analysis of the n11, A2u (B2) fundamental band of 12C6H6 and 13C12C5H6, Mol. Phys., 71, 759– 768, 1990. 201 Gauss, J. and Stanton, J. F., The equilibrium structure of benzene, J. Phys. Chem. A, 104, 2865– 2868, 2000. 202 Pulay, P., Fogarasi, G., and Boggs, J. E., Force field, dipole moment derivatives, and vibronic constants of benzene from a combination of experimental and ab initio quantum chemical information, J. Chem. Phys., 74, 3999– 4014, 1981. 203 Maslen, P. E., Handy, N. C., Amos, R. D., and Jayatilaka, D., Higher analytic derivatives. IV. Anharmonic effects in the benzene spectrum, J. Chem. Phys., 97, 4233– 4254, 1992. 204 Bray, R. G. and Berry, M. J., Intramolecular rate processes in highly vibrationally excited benzene, J. Chem. Phys., 71, 4909– 4922, 1979. 205 Reddy, K. V., Heller, D. F., and Berry, M. J., Highly vibrationally excited benzene: overtone spectroscopy and intramolecular dynamics of C6H6, C6D6, and partially deuterated or substituted benzenes, J. Chem. Phys., 76, 2814– 2837, 1982. 206 Sibert, E. L. III, Reinhardt, W. P., and Hynes, J. T., Intramolecular vibrational relaxation and spectra of CH and CD overtones in benzene and perdeuterobenzene, J. Chem. Phys., 81, 1115– 1134, 1984. 207 Sibert, E. L. III, Hynes, J. T., and Reinhardt, W. P., Classical dynamics of highly excited CH and CD overtones in benzene and perdeuterobenzene, J. Chem. Phys., 81, 1135– 1144, 1984. 208 Shi, S. and Miller, W. H., A semiclassical model for intramolecular vibrational-relaxation of local mode overtones in polyatomic molecules, Theor. Chim. Acta, 68, 1 – 21, 1985. 209 Lu, D. H. and Hase, W. L., Classical trajectory calculation of the benzene overtone spectra, J. Phys. Chem., 92, 3217 –3225, 1988. 210 Scotoni, M., Boschetti, A., Oberhofer, N., and Bassi, D., The 3 ˆ 0 CH stretch overtone of benzene: an optothermal study, J. Chem. Phys., 94, 971– 977, 1991. 211 Scotoni, M., Leonardi, C., and Bassi, D., Opto-thermal spectroscopy of the benzene 4 ˆ 0 C– H stretch overtone, J. Chem. Phys., 95, 8655– 8657, 1991. 212 Zhang, Y. and Marcus, R. A., Intramolecular dynamics. III. Theoretical studies of the CH overtone spectra for benzene, J. Chem. Phys., 97, 5283– 5295, 1992. 213 Iung, C. and Wyatt, R. E., Time-dependent quantum mechanical study of intramolecular vibrational energy redistribution in benzene, J. Chem. Phys., 99, 2261– 2264, 1993. 214 Rashev, S., Stamova, M., and Djambova, S., A quantum mechanical description of vibrational motion in benzene in terms of a completely symmetrised set of complex vibrational coordinates and wave functions, J. Chem. Phys., 108, 4797– 4803, 1998. 215 Minehardt, T. J. and Wyatt, R. E., Quasi-classical dynamics of benzene overtone relaxation on an ab initio force field: 30-mode models of energy flow and survival probability for CH (v ¼ 2), Chem. Phys. Lett., 295, 373– 379, 1998. 216 Rashev, S., Stamova, M., and Kancheva, L., Quantum mechanical study of intramolecular vibrational energy redistribution in the second CH stretch overtone state in benzene, J. Chem. Phys., 109, 585– 591, 1998. 217 Minehardt, T. J., Adcock, J. D., and Wyatt, R. E., Quantum dynamics of overtone relaxation in 30-mode benzene: a time-dependent local mode analysis for CH (v ¼ 2), J. Chem. Phys., 110, 3326– 3334, 1999. 218 Minehardt, T. J., Adcock, J. D., and Wyatt, R. E., Energy partitioning and normal mode analysis of IVR in 30-mode benzene: overtone relaxation for CH (v ¼ 2), Chem. Phys. Lett., 303, 537– 546, 1999.

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¨ ber die Absorptionsspektren einiger Benzolderivate im nahen Ultraroten 219 Rumpf, K. and Mecke, R., U bei grober Schichtdicken, Z. Phys. Chem. B, 44, 299–312, 1939. 220 Henry, B. R. and Siebrand, W., Anharmonicity in polyatomic molecules. The CH-stretching overtone spectrum of benzene, J. Chem. Phys., 49, 5369– 5376, 1968. 221 Hayward, R. J., Henry, B. R., and Siebrand, W., Anharmonicity in polyatomic molecules: the problem of phase coincidence and applications to the CH-stretching overtone spectrum of benzene, J. Mol. Spectrosc., 46, 207– 213, 1973. 222 Halonen, L., Local mode vibrations in benzene, Chem. Phys. Lett., 87, 221– 225, 1982. 223 Plı´va, J. and Pine, A. S., The spectrum of benzene in the 3-mm region: the n12 fundamental band, J. Mol. Spectrosc., 93, 209– 236, 1982. 224 Snavely, D. L., Walters, V. A., Colson, S. D., and Wiberg, K. B., FTIR spectrum of benzene in a supersonic expansion, Chem. Phys. Lett., 103, 423– 429, 1984. 225 Plı´va, J. and Pine, A. S., Analysis of the 3-mm bands of benzene, J. Mol. Spectrosc., 126, 82 – 98, 1987. 226 Brodersen, S., Christoffersen, J., Bak, B., and Nielsen, J. T., The infrared spectrum of mono-13Csubstituted benzene, Spectrochim. Acta, 21, 2077– 2084, 1965. 227 Painter, P. C. and Koenig, J. L., Liquid phase vibrational spectra of 13C-isotopes of benzene, Spectrochim. Acta, 33A, 1003 –1018, 1977. 228 Thakur, S. N., Goodman, L., and Ozkabak, A. G., The benzene ground state potential surface. I. Fundamental frequencies for the planar vibrations, J. Chem. Phys., 84, 6642– 6656, 1986. 229 Boesl, U., Neusser, H. J., and Schlag, E. W., Visible and UV multiphoton ionization and fragmentation of polyatomic molecules, J. Chem. Phys., 72, 4327– 4333, 1980. 230 Boesl, U., Multiphoton excitation and mass-selective ion detection for neutral and ion spectroscopy, J. Phys. Chem., 95, 2949– 2962, 1991. 231 Weber, T., von Bargen, A., Riedle, E., and Neusser, H. J., Rotationally resolved ultraviolet spectrum of the benzene – Ar complex by mass-selected resonance-enhanced two-photon ionization, J. Chem. Phys., 92, 90 –96, 1990. 232 Weber, T. and Neusser, H. J., Structure of the benzene – Ar2 cluster from rotationally resolved ultraviolet spectroscopy, J. Chem. Phys., 94, 7689– 7699, 1991. 233 Henson, B. F., Hartland, G. V., Venturo, V. A., and Felker, P. M., Raman-vibronic double-resonance spectroscopy of benzene dimer isotopomers, J. Chem. Phys., 97, 2189– 2208, 1992. 234 Wilson, E. B., The normal modes and frequencies of vibration of the regular plane hexagon model of the benzene molecule, Phys. Rev., 45, 706– 714, 1934. 235 Wilson, E. B., A partial interpretation of the Raman and infrared spectra of benzene, Phys. Rev., 46, 146– 147, 1934. 236 Iachello, F. and Oss, S., Vibrational modes of polyatomic molecules in the vibron model, J. Mol. Spectrosc., 153, 225– 239, 1992. 237 Quack, M., On the densities and numbers of rovibronic states of a given symmetry species: rigid and nonrigid molecules, transition states and scattering channels, J. Chem. Phys., 82, 3277– 3283, 1985. 238 Quack, M., Statistical mechanics and dynamics of molecular fragmentation, Il Nuovo Cimento, 63B, 358– 377, 1981. 239 Mecke, R., Dipolmoment und chemische Bindung, Z. Elektrochem., 54, 38 – 42, 1955, and references therein. 240 Marquardt, R. and Quack, M., Energy redistribution in reacting systems, In Encyclopedia of Chemical Physics and Physical Chemistry, Moore, J. H. and Spencer, N., Eds., IOP Publishing, Bristol, pp. 897–936, 2001, chap. A.3.13. 241 Quack, M., Molecular spectra, reaction dynamics, symmetries and life, Chimia, 57(4), 147– 160, 2003. 242 Ambartsumyan, R. V., Gorokhov, Y. A., Letokhov, V. S., and Makarov, G. N., Separation of sulfur isotopes with enrichment coefficient 103 through action of CO2-laser radiation on SF6 molecules, JETP Lett., 21, 171–172, 1975. 243 Lyman, J. L., Jensen, R. J., Rink, J., Robinson, C. P., and Rockwood, S. D., Isotopic enrichment of SF6 in 34S by multiple absorption of CO2-laser radiation, Appl. Phys. Lett., 27, 87 – 89, 1975. 244 Lupo, D. W. and Quack, M., IR-laser photochemistry, Chem. Rev., 87, 181– 216, 1987.

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245 Quack, M., Infrared-laser chemistry and the dynamics of molecular multiphoton excitation, Infrared Phys., 29, 441–466, 1989. 246 Quack, M., Infrared laser chemistry, Infrared Phys. Technol., 36, 365– 380, 1995. 247 Quack, M., Multiphoton excitation, In Encyclopedia of Computational Chemistry, von Rague´ Schleyer, P., Allinger, N., Clark, T., Gasteiger, J., Kollman, P. A., Schaefer, H. F., and Schreiner, P. R., Eds., Wiley, Chichester, pp. 1775– 1791, 1998. 248 Outhouse, A., Lawrence, P., Gauthier, M., and Hackett, P. A., Laboratory scale-up of 2-stage laser chemistry separation of C-13 from CF2HCl, Appl. Phys. B-Photophys. Laser Chem., 36, 63– 75, 1985. 249 Chen, G. C., Wu, B., Liu, J. L., Jing, Y., Chu, M. X., Xue, L. L., and Ma, P. H., Scaling-up C-13 separation by infrared multiphoton dissociation of the CHClF2/Br2 system, Appl. Phys. B-Lasers Opt., 60, 583– 588, 1995. 250 Fub, W., Gothel, J., Ivanenko, M., Schmid, W. E., Hering, P., Kompa, K. L., and Witte, K., Macroscopic isotope-separation of C-13 by a CO2-laser, Isotopenpraxis, 30, 199– 203, 1994. 251 de Meer, K., Roef, M. J., Kulik, W., and Jakobs, C., In vivo research with stable isotopes in biochemistry, nutrition and clinical medicine: an overview, Isot. Environ. Health Stud., 35, 19 – 37, 1999. 252 Krumbiegel, P., Stable Isotope Pharmaceuticals for Clinical Research and Diagnosis, Gustav Fischer Verlag, New York, 1991, p. 149. 253 Widmer, R., Quantitative Untersuchung der 12C/13C Isotopentrennung mit Hilfe der Infrarotlaserchemie. Dissertation: ETH Zu¨rich, 1989. 254 Groß, H., Grassi, G., and Quack, M., The synthesis of [2-H-2(1)]thiirane-1-oxide and [2,2-H2(2)]thiirane-1-oxide and the diastereoselective infrared laser chemistry of [2-H-2(1)]thiirane-1oxide, Chem.-Eur. J., 4, 441– 448, 1998. 255 Quack, M., How important is parity violation for molecular and biomolecular chirality?, Angew. Chem. Int. Ed. Engl., 114, 4618– 4630, 2002, Angew. Chem., 114, 4812– 4825, 2002. 256 Gottselig, M., Quack, M., Stohner, J., and Willeke, M., Mode-selective stereomutation tunneling and parity violation in HOClH(þ) and H2Te2 isotopomers, Int. J. Mass Spectrom., 233, 373– 384, 2004.

12

Nonmass-Dependent Isotope Effects Ralph E. Weston, Jr.

CONTENTS I. II.

Introduction ...................................................................................................................... 361 Ozone Isotopologues ........................................................................................................ 364 A. Laboratory Experiments........................................................................................... 364 B. Atmospheric Ozone.................................................................................................. 367 C. Theoretical Explanations of the NMD Isotopic Fractionation in Ozone................ 368 III. Carbon Monoxide Isotopologues ..................................................................................... 372 A. Laboratory Experiments........................................................................................... 372 B. Atmospheric Carbon Monoxide............................................................................... 374 IV. Carbon Dioxide Isotopologues ........................................................................................ 374 A. Laboratory Experiments........................................................................................... 374 B. Atmospheric Carbon Dioxide .................................................................................. 375 C. Theoretical Models for NMD Isotopic Fractionation in Carbon Dioxide .............. 375 V. Nitrous Oxide Isotopologues ........................................................................................... 376 A. Laboratory Experiments........................................................................................... 376 B. Atmospheric Nitrous Oxide ..................................................................................... 376 C. Theoretical Explanations and Modeling Calculations ............................................ 376 VI. Oxygen and Sulfur Isotopic Fractionation in Terrestrial and Extraterrestrial Solids....................................................................................................... 377 A. Carbonates ................................................................................................................ 377 B. Sulfates ..................................................................................................................... 378 1. Laboratory Experiments .................................................................................... 378 2. Terrestrial Sulfates ............................................................................................ 379 3. Extraterrestrial Sulfur Compounds ................................................................... 379 C. Nitrate Aerosols ....................................................................................................... 380 VII. Other Molecules ............................................................................................................... 380 A. Hydrogen Peroxide Isotopologues........................................................................... 380 B. Atmospheric Oxygen Isotopologues........................................................................ 381 Acknowledgments ........................................................................................................................ 381 References..................................................................................................................................... 382

I. INTRODUCTION In 1999, I published a review article on this subject,1 in which I attempted to cover the literature up to that date as completely as possible. For this reason, in this chapter, I will emphasize results that have appeared more recently, referring the reader to the earlier review for a more complete coverage of the subject. Thiemens has also recently published review articles.2,3

361

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Isotope Effects in Chemistry and Biology

To begin with, some definitions are in order. I have always been somewhat uncomfortable with the term “mass-independent isotope effect”; it seems to be almost an oxymoron, since isotopes have different masses by definition. The term “mass-independent fractionation” was invented by Heidenreich and Thiemens4 to describe their observation of approximately equal enrichment of 18O and 17O relative to 16O in the formation of ozone in an electric discharge. However, the term has subsequently been applied to any deviation from the predictions of the conventional massdependent theory of isotope effects. In a recent comment by Kaiser et al.,5 it is suggested that a more appropriate term to replace “mass-independent isotope effects” would be “nonmass-dependent isotope fractionation”, which is certainly more appropriate, if a bit cumbersome. In this chapter it will be abbreviated as “NMD….” The best way for reporting relative isotope effects has been discussed by Kaiser et al.,5 Miller,6 and Young et al.7 Isotopic ratios are typically reported in the form shown here for oxygen isotopes:

d18 O ¼ 103 ½ðRsample =Rstandard Þ 2 1 ; where Rx ¼ ð18 O=16 OÞx

ð12:1Þ

For equilibria of the form X17 O þ Y16 O Y X16 O þ Y17 O; K17 18

16

16

and

18

X O þ Y O Y X O þ Y O; K18

ð12:2Þ

the isotopic fractionation of oxygen is given by:

a17 0 ¼ ð½Y17 O =½Y16 O Þ=ð½X17 O =½X16 O Þ ¼ K17 ¼ ½1 þ 1023 d17 OðYOÞ =½1 þ 1023 d17 OðXOÞ

ð12:3Þ

with a similar expression for the fractionation of 18O. Bigeleisen and Mayer8 showed that each of the above equilibrium constants can be expressed in terms of “reduced partition functions,” e.g., K17 ¼ QðY17 OÞQðX16 OÞ=QðY16 OÞQðX17 OÞ ¼ 17 fYO =17 fXO

ð12:4Þ

where the Qs and fs are partition functions and reduced partition functions, respectively. A similar expression can be written for K18. Each of the reduced partition functions can be expanded in a series in which the first term is, for example: 17

fYO ¼ 1 þ ð1=24Þð"=kb TÞ2 ð1621 2 1721 ÞCYO

ð12:5Þ

where CYO is a constant independent of isotopic species and temperature, but dependent on the individual molecule. Using the expansion lnð1 þ xÞ . x we have ln 17 fYO ¼ ð1=24Þð"=kb TÞ2 ð1621 2 1721 ÞCYO

ð12:6Þ

For each species, if we assume that the relative isotopic fractionation for 17O and 18O is given by 17

Rx ¼ ð18 Rx Þl ; ln17 Rx ¼ l ln 18 Rx

ln½1 þ 1023 d17 OðOXÞ ¼ l ln½1 þ 1023 d18 OðOXÞ

ð12:7Þ

If d is small, we can again use the approximation lnð1 þ xÞ . x; which leads to

d17 OðOXÞ ¼ ld18 OðOXÞ

ð12:8Þ

However from Equation 12.6 and Equation 12.7,

l ¼ ln 17 Rx =ln18 Rx ¼ 17 fOX =18 fOX ¼ ð1621 – 1721 Þ=ð1621 – 1821 Þ ¼ 0:529

ð12:9Þ

Nonmass-Dependent Isotope Effects

363

Equation 12.8 and 12.9 predict that a plot of d 17O against d 18O should be a straight line with slope l ¼ 0:529 if a mass-dependent isotope effect is involved. It should be pointed out that the masses in these approximations are the atomic masses of the exchanging isotope and not those of the molecular fragments at the site of the isotopic substitution. The case of isotope effects on reaction-rate coefficients involves a ratio similar to that of Equation 12.2, describing an isotope exchange equilibrium between the reactants and the transition state (see Chapter 10). The equilibrium constant is slightly more complicated, because one degree of freedom of the transition state corresponds to motion along the reaction coordinate with an imaginary frequency n‡ : Then Equation 12.4 will contain a term ðn‡16 =n‡17 Þ; and in the case of the three oxygen isotopes the final ratios derived from approximate expressions such as 12.6 will contain an additional temperature-independent term of the form ðln n‡18 2 ln n‡17 Þ: Stern and Wolfsberg9 discuss the effect of the imaginary frequency term in some detail, but neither in their work nor anywhere else that I am aware of has the effect of this on the expected “mass-dependent” kinetic isotope effect been discussed. Deviations from Equation 12.9, defined as 17 D ¼ d17 O 2 ld18 O; were taken to indicate that NMD isotopic fractionation was involved. However, as Kaiser et al.5 and Miller6 point out, the precision with which d values can now be determined leads to a deviation from this straight line at high values of d, even if the isotope effect is mass dependent. A better definition is obtained by using Equation 12.7:(see Ref. 6) 17

D ¼ 103 lnð1 þ 1023 d17 OÞ 2 103 l lnð1 þ 1023 d18 OÞ

ð12:10Þ

The expected difference between these two definitions of 17D is shown in Figure 12.1. Kaiser et al.5 propose an equivalent definition in their Equation 12.6. These differing definitions, together with the fact that different values of l are used by different authors, make it difficult to compare results from one group with those of another group, and the reader is cautioned accordingly. Swept under the rug in this discussion has been the fact that the expansion of the reduced partition function used in Equation 12.6 is already a source of deviation from an accurately calculated ratio of K17 =K18 : Matsuhisa et al.10 showed, in the very first report of a nonmassdependent isotopic fractionation, that in the CO2 –H2O isotopic exchange reaction the value of l varied from 0.5233 to 0.5251 as the temperature changed from 273 K to 1000 K. Skaron and

0.0

17

D O

−0.2

−0.4

−0.6

−0.8

0

5

10

15 d 18O

20

25

30

FIGURE 12.1 D 17O as a function of d 18O, calculated with d 17O ¼ d 18O/2. Solid line, linear approximation, Equation 12.8. Dotted line, Equation 12.9.

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Isotope Effects in Chemistry and Biology

Wolfsberg11 also discussed deviations in l that can result from the temperature dependence of isotopic exchange equilibrium constants. Considerations of this sort lead me to believe that the term “nonmass-dependent isotopic fractionation” is used somewhat indiscriminately. For example, it is found that l for a set of calcium and magnesium carbonates has a value of 0.5247 ^ 0.0018, statistically indistinguishable from the value of 0.5247 ^ 0.0007 obtained for terrestrial silicates, but the two parallel lines on a d 17O vs. d 18O plot have different intercepts on the d 17O axis (2 0.241 ^ 0.042‰ for carbonates and 0.008 ^ 0.009‰ for silicates).12 Is this difference really meaningful? In principle the intercept should be zero, because a finite value implies that one isotope is being fractionated and the other is not.

II. OZONE ISOTOPOLOGUES A. LABORATORY E XPERIMENTS It all began, when in 1983 Thiemens and Heidenreich13 reported that 17O and 18O were almost equally enriched in ozone produced by an electrical discharge in oxygen. Since that first report, a great deal of experimental work has been done on ozone formation and dissociation. The most complete set of experiments on isotope effects in the formation of ozone has been carried out by the Mauersberger group. Earlier experiments14 have been refined by the use of molecular beam mass spectrometry, analyzing ozone produced in the O þ O2 reaction directly, rather than molecular oxygen produced by decomposing the ozone product. By using molecular oxygen mixtures with different amounts of the three oxygen isotopes, it was possible to determine isotopic fractionation for ozone of masses 48– 54.15 The results shown in Table 12.1. are in good agreement with those obtained earlier. It is perhaps worth pointing out that deriving ratios of rate coefficients from observed ratios of isotopic fractionation is a nontrivial process. The exchange reaction between oxygen atoms and oxygen molecules has a significantly larger rate coefficient than does the recombination reaction, so that an experiment that begins with only 32O2 and 36O2 is soon dealing with 16O18O as well. The details of the kinetics involved are discussed by Anderson et al.16

TABLE 12.1 Fractionation Values (d%) for Ozone Isotopologues Obtained at a Pressure of 65 Torr and a Temperature of 300K dMean(%) Mass 48 49 50 50 51 51 52 52 53 54 a

a

Molecule

Expt.

Calc.b

16 2 16 2 16 16 2 16 2 17 16 2 17 2 17 16 2 16 2 18 16 2 17 2 18 17 2 17 2 17 16 2 18 2 18 17 2 17 2 18 17 2 18 2 18 18 2 18 2 18

0 10.6 ^ 0.4 11.0 ^ 0.3 13.0 ^ 0.3 19.8 ^ 0.4 21.5 ^ 0.4 16.1 ^ 0.5 9.4 ^ 0.4 8.9 ^ 0.5 23.9 ^ 0.5

0.0 12.3 12.2 12.7 17.4 22.1 12.7 10.4 9.2 24.7

From Jannsen, C., Gu¨nther, J., Krankowksy, D., and Mauersberger, K., J. Chem. Phys., 111, 7179–7182, 1999. With permission. b From Gao, Y. Q. and Marcus, R. A., J. Chem. Phys., 116, 137– 154, 2002. With permission.

Nonmass-Dependent Isotope Effects

365

Using tunable-diode infrared absorption spectroscopy, it has been possible to separate out the individual O þ O2 reactions in which both the atomic and molecular species contained 18O.17 Thus, all the factors contributing to masses 50 and 52 are determined. These experiments confirm earlier work in showing that only the end-on reaction is important.18 The results are shown in Table 12.2.

TABLE 12.2 Relative Rate Coefficients for the Formation of Ozone Isotopologues Relative Rate Reaction

Expt.

Calc.b

16 þ 16 2 16 ! 16 2 16 2 16 17 þ 17 2 17 ! 17 2 17 2 17 18 þ 18 2 18 ! 18 2 18 2 18 18 þ 16 2 16 ! 18 2 16 2 16 18 þ 16 2 16 ! 16 2 18 2 16 17 þ 16 2 16 18 þ 17 2 17 17 þ 18 2 18 16 þ 17 2 17 16 þ 18 2 18 16 þ 18 2 18 ! 18 2 16 2 18 16 þ 16 2 17 16 þ 16 2 18 ! 16 2 16 2 18 16 þ 18 2 16 ! 16 2 18 2 16 17 þ 16 2 17 17 þ 17 2 18 18 þ 16 2 18 ! 18 2 16 2 18 18 þ 17 2 18 16 þ 17 2 18 17 þ 16 2 18 18 þ 16 2 17 17 þ 16 2 16 18 þ 17 2 17 17 þ 18 2 18 16 þ 17 2 17 16 þ 18 2 18 16 þ 16 2 17 16 þ 16 2 18 17 þ 16 2 17 17 þ 17 2 18 18 þ 17 2 18 16 þ 17 2 18 17 þ 16 2 18 18 þ 16 2 17

1.00 1.02 1.03 0.92 ^ 0.04 0.006 ^ 0.005 1.03 1.03 1.31 1.23 1.53 0.029 ^ 0.006 1.17 1.45 ^ 0.04 1.08 ^ 0.01 1.11 1.21 1.04 ^ 0.02 1.09 — — — 1.03 1.03 1.31 1.23 1.53 1.17 1.27 1.11 1.21 1.09 — — —

1.00 1.02 1.03 0.93c — 1.03 1.07 1.39 1.38 1.53 — 1.19 1.48 1.04 1.04 1.20 1.04 1.05 1.43 1.21 1.01 1.03 1.07 1.39 1.38 1.53 1.19 1.25 1.04 1.20 1.05 1.43 1.21 1.01

a

a

Where error ranges are indicated, data from Jannsen, C., Gu¨nther, J., Krankowksy, D., and Mauersberger, K., J. Chem. Phys., 111, 7179–7182, 1999. With permission; otherwise from Mauersberger, K., Erbacher, B., Krankowksy, D., Gu¨nther, J., and Nickel, R., Science, 283, 370 –372, 1999. With permission. b Ref. 40; symmetry numbers for reactions with heteronuclear oxygen have not been included. c If products are not specified, the rate coefficient is for the sum of both possible reactions, e.g., 16 þ 16 2 17 ! 16 2 16 þ 17 and 16 þ 17 2 16.

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Two reports from the Mauersberger group provide information about pressure effects on relative ozone formation.19,20 Mixtures of 32O2 and 36O2 were illuminated at pressures of 37 to 3300 Torr, and the resulting concentrations of ozone of masses 48 – 54 were measured. The ratio of rate coefficients for 18 þ 16 2 16/18 þ 18 2 18 (this abbreviation for oxygen isotopologues is used hereafter) was constant over this pressure range, whereas that for the ratio 16 þ 18 2 18/ 16 þ 16 2 16 decreased monotonically, as had been found in earlier experiments.21 – 23 These same reactions have been studied with various bath gases (Ar, Kr, Xe, N2, O2, CO2, CH4), and absolute rate coefficients for ozone formation have been measured. With a total pressure of 200 Torr, 40 Torr of which was oxygen, the rate coefficient ratios were independent of the bath gas within the error limits. The rate coefficient ratio for 16 þ 18 2 18/16 þ 16 2 16 has an average value of 1.48 ^ 0.03 for all gases, and the ratio for 18 þ 16 2 16/18 þ 18 2 18 is 0.87 ^ 0.03. This is in spite of the fact that the absolute rate coefficient for O þ O2 is substantially larger with the polyatomic species as bath gases than in the rare gases. The temperature dependence between 230 and 350K of these same rate-coefficient ratios has been determined.24 Large rate-coefficient ratios, such as the value of , 1.5 at 300K for 16 þ 18 2 18/16 þ 16 2 16 show a very small effect (2 0.035% K21; this value is not given correctly in Table 12.1 of the paper). The ratio for 18 þ 16 2 16/18 þ 18 2 18 (0.9 at 300K) increases at a rate of 0.1% K21. The temperature dependence of isotopic fractionation in the former case is found to be in good agreement with that observed previously.21 The relevance of these measurements to fractionation in the stratosphere is discussed below. Chakraborty and Bhattacharya25 have presented new information about the photodissociation of ozone, and its reaction with O(1D), using both UV and visible light sources. Absorption in the Hartley band (, 300 nm) leads to a mass-dependent isotopic fractionation, with a slope of 0.54 ^ 0.01 in the usual three-isotope plot. On the other hand, radiation in the Chappuis band (520 and 630 nm) produces a NMD effect, with a slope of 0.63 ^ 0.01. Both processes are shown to follow the kinetics of a Rayleigh fractionation, i.e., lnð1 þ 1023 df Þ ¼ lnð1 þ 1023 d0 Þ þ ða 2 1Þln f

ð12:11Þ

where d0 and df refer to the initial ozone and the fraction f remaining and a is the fractionation factor for photolysis. An important feature of this paper is the determination of the importance of the reaction sequence with irradiation in the Hartley band: O3 þ hn ! Oð1 DÞ þ O2 Oð1 DÞ þ O3 ! 2O2 or O2 þ 2Oð3 PÞ Addition of nitrogen to the reaction mixture leads to electronic quenching of the O(1D), and to an increase in the slope of the three-isotope plot to a value of unity, constant at pressures of 40 –90 Torr. Thus, the value of 0.63 found in the absence of quenching is evidently the result of photolysis combined with the bimolecular reaction. The authors calculate shifts in the absorption spectra of the isotopically labeled ozone, using the method of Yung and Miller (see below) and calculated ZPE shifts. The results are not in agreement with experimental results, and the authors suggest that non-RRKM effects, as described by Marcus, are involved. Bhattacharya et al.26 report the results of photolysis experiments involving both formation and dissociation of ozone at pressures from 6.7 to 702 Torr. They find a gradual decrease of isotopic enrichment as the pressure increases, but a sharp peak in the region of 15 –50 Torr. They attribute this effect to secondary enrichment due to ozone dissociation combined with primary ozone formation, although they admit that estimates of the isotope effect on photolysis give values that are not large enough to be important. Bhattacharya et al. believe that the altitude dependence of isotopic enrichment measured by the Mauersberger group (see below) cannot be

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explained solely as a temperature effect. They suggest that the reaction O(1D) þ O3 is important, and point to an anticorrelation between enrichment and estimates of the turnover time in this reaction. However, as pointed out by Krankowsky et al.,27 under atmospheric conditions the electronic quenching of the oxygen atom is far more important than the O(1D) þ O3 reaction. They believe that artifacts are involved in the Bhattacharya et al. experiments at low pressures and with low yields of ozone. At higher yields, data from this work show a pressure dependence in good agreement with that obtained in other experiments19 and without the peak at , 15 Torr. This comment has been replied to by the authors of the original paper.28

B. ATMOSPHERIC O ZONE Early measurements of 18dO in the stratosphere gave values that were as high as 20– 40%,29 much larger than the values obtained in subsequent laboratory experiments. More recent measurements have given results that are in excellent agreement with laboratory experiments, especially when variations in temperature and pressure are taken into account. Figure 12.2 shows data from the Mauersberger group obtained by collecting samples during balloon flights from several locations.30 There are large deviations from the mass-dependent three-isotope plot, although the slope of a best fit line through these data has a slope that is smaller than would be predicted for mass-dependent fractionation. The dependence of the fractionation on altitude has been discussed by Krankowsky et al.,31 and both d 17O and d 18O increase with altitude over the range 22– 33 km. The results, plotted as d 49O3 vs. d 50O3 lie on a line that is identical with the same plot obtained from ozone formation and dissociation at different temperatures,21 although some of the higher enrichment values are considerably larger than the laboratory experiments predict. It is concluded that the change of temperature with altitude (, 217– 239K) is the major factor in determining the extent of fractionation. Johnson et al.32 report data obtained using the Smithsonian Astrophysical Observatory far-infrared spectrometer during balloon flights. Infrared spectroscopic measurements have one advantage over the more customary mass spectrometric measurements because they can distinguish

25

15

17

d O, %

20

10

5 0

0

5

10 15 18 d O, %

20

25

FIGURE 12.2 Oxygen isotopic fractionation in ozone collected in the stratosphere. Different symbols indicate different collection sites. Solid line, least-squares fit; dashed line, mass-dependent slope. (Data from Mauersberger, K., La¨mmerzahl, P., and Krankowsky, D., Geophys. Res. Lett., 16, 3155 –3158, 2001. With permission.)

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Isotope Effects in Chemistry and Biology

between pOOO and OpOO (pO ¼ 17O or 18O). They find the following values:

dð16 O18 O16 OÞ ¼ 6:08 ^ 1:83% dð16 O17 O16 OÞ ¼ 1:59 ^ 7:64% dð18 O16 O16 OÞ ¼ 12:17 ^ 0:99% dð17 O16 O16 OÞ ¼ 7:99 ^ 5:25% d18 OðtotalÞ ¼ 10:2 ^ 4:3% d17 OðtotalÞ ¼ 7:3 ^ 6:0% Within the rather wide error limits of these measurements, the total values agree with the measurements made by sample collection and mass spectrometric analysis. The 18O enrichment of the antisymmetric molecule is roughly double that of the symmetric molecule, as would be predicted statistically, and has been found in earlier laboratory experiments.33

C. THEORETICAL E XPLANATIONS OF

THE

NMD I SOTOPIC F RACTIONATION IN O ZONE

From the time of the first report on nonmass-dependent isotopic fractionation, theorists-and wouldbe theorists have found this a fertile field of research, or at least, speculation. Early attempts based on symmetry were shown to be incorrect, but symmetry does play a role in more sophisticated approaches to the problem. The main contenders at the present time for an explanation of the effects observed in ozone formation are discussed below. Gellene has observed very large and nonmass-dependent isotope effects in ion – molecule reactions of the general form Aþ þ A ! Aþ 2 . He has explained these effects as a result of symmetry, and has extended this model to the case of ozone formation.34 With the inclusion of an arbitrary “mixing” parameter b; this theory predicts that relative enrichments will be given by factors of ð1 þ 2bÞ=3; ð7 þ 2bÞ=9; and 1 for molecules containing one, two or three different isotopes, respectively. With a value of 0.78 for b; these factors are 0.853, 0.951, and 1.000. These values predict fairly well the enrichment for a number of cases (cf. Table 12.1), but fail to predict the observed difference between reactions 16 þ 18 2 18 (16.1%) and 17 þ 18 2 18 (8.9%), for example. In papers that have been generally overlooked, Robert et al.35 and more recently Robert and Camy-Peyret36 have derived relative rate constants for all the combinations of O þ O2 isotopologues. Their approach is based on the basic idea that in a “gedanken experiment” in which a beam of A and B atoms react with a mixture of MA and MB three types of reactions can occur: A þ MA ! A þ MAðatom exchange may or may not occurÞ B þ MB ! B þ MBðatom exchange may or may not occurÞ A þ MB ! B þ MAðatom exchange occursÞ B þ MA ! A þ MBðatom exchange occursÞ A þ MB ! A þ MBðatom exchange does not occurÞ B þ MA ! B þ MAðatom exchange does not occurÞ These reactions obviously parallel those involving oxygen isotopes, and this case is specifically treated in detail. In the A þ MA case, identical products will be observed for some scattering angle u and its supplement p 2 u: It is argued that this difference between A þ MA and B þ MA outcomes leads to a difference in differential scattering cross-sections (i.e., cross-sections as a function of scattering angle) and that this will lead to a difference in rate constant ratios. However,

Nonmass-Dependent Isotope Effects

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the authors admit that “if… integration is performed over all the scattering angles…, no isotopic fractionation is expected.” Since rate coefficients are derived from cross-sections by such integration, this would seem to argue against the validity of this model, except perhaps in (as yet unperformed) molecular beam experiments. Another strange feature of this model is that isotopic fractionation is predicted to depend on relative abundances of the isotopic species involved, and this is not confirmed experimentally. The actual calculation of rate-coefficient ratios is based on the assumption that the entire observed effect is due to differences in the ratio of rate constants for distinguishable and indistinguishable species (e.g., 16 þ 16 2 17 vs. 16 þ 16 2 16). This ratio depends on reduced mass ratios, together with four ad hoc parameters adjusted to fit some of the experimental values. The results of these calculations are in remarkably good agreement with many of the experimental values, but the assumptions required to carry out the calculation of relative rate coefficients and enrichment factors need a firmer theoretical basis. In a more recent paper,37 Robert extends this method to the calculation of isotope distributions for three-isotope elements in compounds found in meteorites. Marcus and coworkers have produced a very detailed treatment of the ozone formation isotope effects, based on the Rice –Ramsperger –Kassel– Marcus (RRKM) theory of unimolecular reactions. In the first of these papers, Hathorn and Marcus38 derive expressions for the dependence of product ratios on rate constants of elementary reactions, an exercise in careful bookkeeping that has not been carried out in detail previously. RRKM theory has previously been applied to many recombination reactions of the type X þ YZ ! XYZ, which has the more detailed mechanism X þ YZ ! XYZp ðformation of vibrationally excited adductÞ XYZp ! X þ YZ or Z þ XYðdissociationÞ XYZp þ Mðany molecule presentÞ ! XYZ þ Mðstabilization of adductÞ The important feature of these reactions is the competition between dissociation, either to the original reactants or to products that are different from reactants, and energy transfer by collision to stabilize the adduct. In RRKM theory, the rate coefficients of the two competing dissociation reactions depend on the density of states of both the excited molecule and the transition state for the reaction. Most of the experiments on these reactions has been done at relatively low pressures, in which case the rate coefficient is given by: ka ¼ v

X ð1 JK

0

Ya† r e2E=kb T dE=Qað1;2Þ

ð12:12Þ

In this equation, ka is the rate coefficient for recombination in channel a, v is the collision frequency for deactivation, J is the total angular momentum, K is any approximately conserved quantum number along the reaction coordinate, r is the density of states of the molecule, and Qð1;2Þ is the partition function for the atom-diatom pair. The factor Ya† is very important: Ya† ¼ a † Na =ðNa† þ Nb† Þ; where each N † is the number of quantum states of the transition state in each exit channel. If Z ¼ Y in the above scheme (unscrambled conditions), there is a single entrance channel corresponding to X þ YY ! XYY, but two exit channels corresponding to X þ YY and XY þ Y. The exit rates will determine the relative bimolecular rate constants for the addition reaction. These rates depend critically on small differences in the vibrational zero-point energies of the two exit channels which control the partitioning factors Ya† and Yb† for the a and b channels, where it will be assumed that a is the lower energy channel. At the lowest energies only this channel will be open, and Ya† ¼ 1 and Yb† ¼ 0: Once the zero-point energy of b has been reached, the number of states Nb† of the transition state grows approximately as the square of its excess energy above the zero-point energy, but meanwhile the number of states for the transition state

370

Isotope Effects in Chemistry and Biology

of the a channel has already been growing, so that Na† =ðNa† þ Nb† Þ may become quite large. This effect is found to be largely mass dependent. The partitioning factor was shown to disappear in the case of scrambled conditions where there are both two entrance channels and two exit channels. (For example, X þ XY ! XXY ! XX þ Y or X þ XY, X þ XY ! XYX ! X þ XY). In this case, it is shown that symmetry (long ago dismissed as a theoretical basis for nonmass-dependent isotopic fractionation) can be the determining factor. A fundamental assumption of RRKM theory is that the vibrationally excited molecule (XYZp in this case) lives long enough for vibrational energy to slosh around among many vibrational – rotational modes, limited only by the available energy and angular momentum restrictions. If for some reason the excited molecule does not live long enough for this to occur, or if some modes are not coupled to others, the molecule is said to show a non-RRKM effect. Suppose in the case of the symmetrical isotopologues of ozone some coupling terms are prohibited by symmetry, leading to non-RRKM conditions. In effect this reduces the density of states, and this in turn will affect the rate constant. Marcus and coworkers have used an adjustable parameter h that decreases the density of states of all symmetrical molecules equally relative to that of unsymmetrical isotopologues. The implementation of these principles differs slightly in the Hathorn and Marcus papers38,39 and the Gao and Marcus papers.40 – 42 Calculation of densities of states requires, among other things, a knowledge of molecular vibrational frequencies, and Hathorn and Marcus have developed a method for obtaining frequencies for the many unsymmetrical ozone isotopologues from known frequencies of the symmetrical ones.43 The accuracies of a few cm21 obtained by this method are adequate for the purpose of calculating state densities. In the second paper of this series,39 a loose transition state in which YZ rotates freely was assumed, and it was also assumed that every gas-kinetic collision was effective in decreasing the vibrational energy of the activated molecule below the dissociation limit. The Gao and Marcus paper40 modifies these assumptions slightly by using a transition state with a hindered rotor, and a model in which the average energy loss per collision ðDEÞ can be specified. The parameter h and DE were adjusted to provide the best fit to the measured relative reaction rates for the extreme cases: 1.53 for 16 þ 18 2 18/16 þ 16 2 16 and 0.92 for 18 þ 16 2 16/18 þ 18 2 18. With values of 1.18 for h and 210 cm21 for DE, the calculations of Gao and Marcus fit most of the experimental results very well. Calculated rate constant ratios and enrichments are shown in Table 12.1 and Table 12.2; in addition, the pressure and temperature dependence of some of the rate constant ratios is in good agreement with experiment. The major contribution to the state density for the transition state is rotational motion, and a further modification uses a potential energy surface for the hindered rotation based on ab initio calculations.42 The results differ very little from those obtained using the simpler models. Of course, the major shortcoming in this work is the use of the ad hoc factor h, which may in the future be derivable from first principles. A linear correlation between relative rate coefficients and the zero-point energy change (DZPE) of the exchange reaction that competes with ozone formation has been proposed.44,45 This seems to work in a number of cases: for example, when there is no DZPE there is little effect on rate coefficients, and the largest effect is found when DZPE is positive, i.e., the reaction is endothermic. Jannsen argues that this is the most important effect in the RRKM treatment by Marcus and coworkers. A very different approach has been used by Miklavc and Peyerimhoff,46 who assume that the important step in the ozone formation reaction is the transfer of kinetic energy of the incident O atom into vibrational energy of the O2 collision partner. This process is treated with a Landau – Teller formulation; thus, the probability of vibrational excitation depends on the vibrational frequency of the oxygen, the effective collisional mass, and the interaction potential. This dependence is similar to the results of the Marcus work, where a strong correlation between relative rate coefficients, differences in zero-point energy, and reduced mass was found. The resulting ratecoefficient ratios are in good agreement in cases where the ratio is greater than unity, but are consistently low by 20 –30% if it is less than unity.

Nonmass-Dependent Isotope Effects

371

Using an ab initio potential energy surface for O3, Charlo and Clary47 have calculated wave functions for scattering resonances arising from the O þ O2 collisions. These resonant states are then included together with bound vibrational states in considering the collisional energy transfer between excited ozone molecules and Ar. Cross-sections for individual vibrational states are obtained and show interesting differences among the different isotopic species. However, the rate constants obtained from these cross-sections do not reproduce the experimental values very well. Quantum scattering resonances again are an important part of the theoretical treatment of Babikov et al.48 Their work also required the development of a new PES49 based on that derived by Schinke and coworkers,50 but correcting two deficiencies of that surface: a dissociation energy for O3 that is about 10% low, and a barrier that is slightly above the dissociation limit (Figure 12.3). These features have a major influence on the reaction dynamics at low energies. Using their modified PES, the authors carry out full quantum mechanical scattering calculations including all six dimensions to calculate energies and lifetimes of resonances for 16O18O18O, 16O16O18O, and 16 16 16 O O O. They find a pronounced nonstatistical feature: within the narrow energy range below the DZPE there are many long-lived metastable states, and only a few (with shorter lifetimes) above this energy range, which of course is dependent upon the specific isotopologue in question. This presumably occurs because at energies above the DZPE all dissociation channels are open, while at lower energies some are closed. This is obviously closely related to the effects described by Marcus and coworkers. In a third paper from this group,51 lifetimes of metastable states are used to calculate state-specific rate constants for the recombination reaction. Comparing 16O þ 18O18O and 18 O þ 18O16O a ratio of 3.53 for the level J ¼ 0 is found compared with an experimental value of 1.56 (1.53/0.92). For the reactions 18O þ 16O16O/16O þ 16O18O, a ratio of 0.33 is found, compared with the experimental value of 0.73 (0.93/1.27). Additional theoretical investigation of the effect of including higher rotational levels is needed, but qualitative arguments indicate that this will improve agreement with experiment. It is interesting to note that no ad hoc symmetry-based factors are used in this theoretical work. Baker and Gellene have done trajectory calculations, classical and quasiclassical, on the O þ O2 system.52 They were able to obtain good agreement between calculated and experimental rate constants for ozone formation over the temperature range 130 –370K. They also discuss rate constants for the isotope exchange reaction, but do not tackle the problem of isotopic fractionation.

Rate: 0.98

Rate: 1.53

Rate: 1.27

Metastable O3°

Metastable O3° D ZPE

ZPE1618 1618+18

ZPE1818 Stable O3

161818 181618

Rate: 0.93

PES

16+1818

D ZPE

ZPE1616

ZPE1618 16+1618

Stable O3

1616+18 PES

161618 161816

FIGURE 12.3 Schematic potential energy surface for O þ O2 ! O3. From Babikov, D., Kendrick, B. K., Walker, R. B., and Pack, R. T., J. Chem. Phys., 118, 6298– 6308, 2003. With permission.

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Isotope Effects in Chemistry and Biology

III. CARBON MONOXIDE ISOTOPOLOGUES A. LABORATORY E XPERIMENTS Bhattacharya and Thiemens used the photodissociation of O2 to form oxygen atoms which then reacted with carbon monoxide to form carbon dioxide.53 Values of 17dO and 18dO as large as , 60‰ were observed, and a linear relation between the two with 17dO ¼ 0.82 18dO was found. No correlation of d values with the extent of reaction, initial CO/O2 ratio, or photolysis time is evident. Blank experiments showed that in the absence of oxygen no carbon dioxide was formed. The authors believe that the observed isotopic fractionation is the result of a combined exchange of oxygen atoms with CO, followed by the NMD recombination of O and CO. At 300K and the pressures of these experiments, the rates for these two processes are approximately equal. Given the known NMD oxygen isotopic distribution in atmospheric ozone, it is reasonable to look for a mechanism that will transfer this distribution to the oxygen atom in carbon monoxide. Ro¨ckmann et al. investigated various reactions in which this compound is a reactant or product.54 As many natural nonmethane hydrocarbons are oxidized by ozone rather than by OH, this is a potential source of such an effect. The hydrocarbons investigated were ethene (C2H4), isoprene (C5H8) and b-pinene (C10H16). Ozone was generated by the photolysis of oxygen with a lowpressure mercury lamp, and its isotopic composition was determined mass spectrometrically. After reaction with the hydrocarbon and removal of the unreacted ozone, the remaining CO was subjected to mass spectrometric analysis. A fairly wide range of isotopic enrichment relative to the oxygen used for ozone production was observed, from 18dO ¼ 0.46‰ for isoprene with water added to 0.82‰ for ethene (slightly greater than that for the ozone reactant). In all cases, the values of 17dO and 18dO were nearly equal. A very important reaction of carbon monoxide is the oxidation by hydroxyl: OH þ CO ! H þ CO2 This is the main atmospheric path for removal of carbon monoxide, and it is also important in combustion chemistry. Kinetic isotope effects of 13C and 18O were studied many years ago by Stevens et al., but there was no comparison of 18O and 17O.55 Such a comparison was made recently by Ro¨ckmann et al., who also repeated the earlier determinations of the carbon isotope effect.56 A mixture of He, CO, and hydrogen peroxide was circulated through a photochemical reactor in which OH was produced by photolysis of H2O2 by a Xe lamp (l . 190 nm). Ratios of rate coefficients were obtained from the isotopic composition of the unreacted CO. With the definition 1 mO (‰) ¼ 103 ½ð16 k=m kÞ 2 1 , values of 1 18O of ca. 2 10 were found; i.e., this is an inverse effect in which the heavier isotope reacts more rapidly. The 13C and 18O results are in good agreement with those from the earlier experiments of Stevens et al. A slight pressure dependence was found, as shown in Figure 12.4, and this was also evident in the earlier work. The effect of 17O is reported in terms of the quantity E 17O (‰) ; 1 17O 2 0.52 1 18O, and from this and the observed 18O effect I have calculated the values for 1 18O shown in Figure 12.4. Although the authors describe these results as evidence for NMD isotope fractionation, they differ from of other NMD effects that have been observed in ozone, in which 1 17 O . 1 18 O. If a massdependent isotope effect prevails, then 1 17O ¼ 0.52 1 18O. However, the observed values of 117O do not lie between these two limits, a result of the fact that 1 17O is even smaller than would be predicted by a normal mass-dependent effect. More recently, a group at the University of Copenhagen has determined ratios of rate coefficients for the reaction of OH with combinations of 12C16O/13C16O, 12C18O/12C16O, 12 16 C O/13C18O, and 12C17O/12C18O.57 These experiments differ from those done previously: instead of mass spectrometric isotopic analysis of the unreacted CO, FTIR spectroscopy was used. In addition, the source of hydroxyl was not the photodissociation of hydrogen peroxide, but the reaction of O(1D) (formed by photolysis of ozone) with water. As Figure 12.3 shows, where

Nonmass-Dependent Isotope Effects

373

0 −2

∋, 0/00

−4 −6 −8 −10 −12

0

200

400

600 800 1000 Pressure, mb

1200 1400

FIGURE 12.4 Kinetic isotope effects for the reaction CmO þ OH ! CmOO þ H. as 103 ð16 k=m k 2 1Þ; W, 181, Ref. 55; K, 181, Ref. 56; B, 171, Ref. 56; A, 181/10, Ref. 57; †, The solid line is a least-squares fit to the data of Refs. 55 and 56.

m

1 is defined 1/10, Ref. 57.

17

comparisons can be made the isotope effects are an order of magnitude larger than those previously reported; the error limits are also much larger than those reported in the experiments where mass spectrometric analysis is used. Although the authors did experiments which ruled out the possible exchange of O atoms with CO or CO2, there must be some additional complications due to the presence of O atoms in the reaction mixture and the more complicated reaction scheme involved when OH is generated less directly than by photolysis of hydrogen peroxide. This reaction, apparently a simple bimolecular reaction, actually has a more complicated mechanism similar to that of ozone formation: ka

kb

OH þ CO Y HOCOp ! H þ CO2 k2a

ð12:13Þ

M # kM HOCO

The formation of the energized radical HOCOp is followed by unimolecular dissociation to reactants or products, or alternatively by collisional deactivation. In the atmosphere, the stabilized HOCO radical will react rapidly with O2. In the absence of oxygen, the reaction OH þ HOCO ! H2O þ CO2 might be expected to occur, based on the fact that this reaction is , 40 kcal/mole exothermic. The stabilization of HOCO introduces a pressure dependence in the observed rate coefficient. Two recent papers have examined this in detail, and have derived expressions for the rate constants that apply to the above mechanism.58,59 Over the range of pressures at which the isotope effect was measured, the ratio of dissociation of HOCOp to stabilization changes by a factor of about two. Since the isotope effect 181 becomes smaller (in absolute value) with increasing pressure, the implication is that there is a smaller effect on the “complex” reaction (i.e., stabilization). However, it is difficult to reconcile this explanation for the pressure dependence of 18 1 with the much larger values observed when He is the bath gas.55 In fact, one would expect, on the basis of this model, that the 181 values for different gases would converge as the pressure is decreased and only the “direct” mechanism is operative. Stevens et al. also discussed the pressure dependence of the carbon and oxygen kinetic isotope effects.55 They point out that the high pressure limit of the observed rate constant is just ka ; so that (16k/18k)1 ¼ (16ka/18ka). Their data seem to indicate a leveling off at a value of about 0.990 for this ratio, and this “inverse” isotope

374

Isotope Effects in Chemistry and Biology

effect in a complex formation is not unreasonable. The fact that this reaction involves complex formation is reminiscent of the O þ O2 reaction, and as described earlier, a detailed understanding of the dissociation dynamics of the complex has provided the best explanation of the anomalies of the ozone isotope effects. One may need similar insight to provide the explanation here, although the important role of symmetry that is involved in the ozone example would not apply to this reaction.

B. ATMOSPHERIC C ARBON M ONOXIDE NMD isotopic fractionation anomalies were observed in atmospheric carbon monoxide by Hodder et al.60and by Huff and Thiemens.61 Ro¨ckmann et al.56 observed a seasonal variation in 17D values, with the maximum occurring during summer months and a minimum occurring during the winter, when carbon monoxide concentration is a maximum. Ro¨ckmann et al. 62 have determined the temporal variation in C18O and C17O at high latitudes (Spitsbergen, Norway, 798N and Alert, Canada, 818N). As in the earlier work, a pronounced anticorrelation between carbon monoxide concentration and 17D is observed. The major source of the nonmass-dependent fractionation is thought to be the reaction CO þ OH. In the spring and summer, CO is progressively enriched in 17O as a result of this process. Competing with this is the source of CO from fossil fuel combustion, which is known to be mass-dependent,61 and reaches its maximum effect in the winter months. This is in agreement with the measurements by Ro¨ckmann et al. of 18O, which in winter approaches the value of about 23‰ found for the burning of fossil fuels. Gros et al. have characterized air pollution events by examining the CO concentration and isotopic compositions of 13C16O, 14C16O, 12C17O, and 12C18O.63 The background range of D17O was 5.2 ^ 1.1‰, and this value was lowered to 2.8– 3.7‰ in air pollution events. Taking into account the combination of isotopic abundances, the authors were able to distinguish between local and distant air pollution events.

IV. CARBON DIOXIDE ISOTOPOLOGUES A. LABORATORY E XPERIMENTS Soon after the initial report of nonmass-dependent isotopic fractionation in ozone formation, Heidenreich and Thiemens observed similar effects when an electric discharge was used to dissociate carbon dioxide, forming carbon monoxide and oxygen.64 Analysis of the molecular oxygen indicated that the isotopic enrichment was NMD. Presumably the electric discharge produced oxygen atoms that could exchange with carbon dioxide through a symmetrical CO3 intermediate. This hypothesis was tested by Wen and Thiemens, who used UV photolysis to produce O(1D) atoms from ozone.65 A three-isotope plot of d 17O against d 18O for the oxygen product gives a slope of about 0.75. Bhattacharya et al. repeated these experiments, using either a mercury resonance lamp (185 and 254 nm) or a continuum Kr lamp (120 – 160 nm) to photodissociate carbon dioxide.66 The Hg lamp produced much more 17O enrichment in the residual carbon dioxide, and it is proposed that this wavelength dependence results from a transition from the ground 1Sg level of CO2 to a 1B2 level that is close to the point where the potential energy curve for this state crosses the curve for the 3B2 state. A spin forbidden intersystem crossing to the triplet state would lead to dissociation to CO þ O(3P), whereas the 185 nm photon does not have enough energy to lead to dissociation from the singlet state to CO þ O(1D). Direct evidence supporting the exchange mechanism with the CO3 intermediate has recently been reported by Perri et al., who studied the reaction using molecular beam techniques.67 The angular dependence of the product scattering provides evidence for a relatively

Nonmass-Dependent Isotope Effects

375

long-lived complex and also indicates that about one-third of the exchange takes place on the singlet surface without electronic quenching of O(1D). More recently, experiments similar to those of Wen and Thiemens and calculations similar to those of Yung et al. (see below) have been reported by Johnston et al.68 Mixtures of carbon dioxide and oxygen were irradiated with a Hg – Ar lamp that produces UV light peaking at 185 and 237 nm. Two sets of experiments were carried out: those with relatively large ratios of CO2/O2 (0.34 – 3.9) and those with ratios of about 1023, reproducing atmospheric concentrations. The high CO2 experiments showed that d 18O leveled off after a relatively short period of irradiation, whereas in the low CO2 experiments it increased approximately linearly with time of irradiation and reached much higher values. In these low CO2 experiments, it was also found that enrichment was inversely dependent on total pressure, but with no strong pressure dependence of the d 17O/d 18O ratio. Conversely, the high ratio experiments gave enrichment that was only weakly dependent on pressure. A three-isotope plot (d 17O vs. d 18O for both CO2 and O2) gave a straight line with a slope of 0.89 for the experiment with CO2/O2 ¼ 3.94 and a pressure of 435 mb. The authors believe that these results indicate that the isotopic composition of the carbon dioxide is controlled primarily by that of the O(1D). A numerical model was set up, including ten fundamental reactions, which expand to about 80 when isotopic variations are included. The model calculations corroborate the idea that the CO2 and O(1D) isotopic compositions are linked, but consistently overestimate the enrichment of heavy isotopes in the carbon dioxide. The d 17O vs. d 18O plot gives a slightly larger slope (1.05) than do the experimental data. The authors discuss possible improvements in the model, primarily dependent on a more extensive knowledge of the isotope effects in individual elementary reactions.

B. ATMOSPHERIC C ARBON D IOXIDE Several investigations of isotopic abundances in stratospheric carbon dioxide have been carried out, most recently by La¨mmerzahl et al., who determined 17O and 18O concentrations in both carbon dioxide and ozone collected simultaneously at altitudes from 19 to 33 km.69 References to earlier atmospheric measurements are given in this paper. Standard three-isotope plots (d 17O vs. d 18O) give slopes of 1.71 for CO2 and 0.62 for O3, indicating that the isotopic composition of these two species is not as closely linked as had previously been believed. The slope in the three-isotope plot is in agreement with that obtained in model calculations of Barth and Zahn,70 who assumed, however, 18O concentrations in stratospheric ozone that are at the upper limit of the observed values. Alexander et al. made isotopic measurements in CO2 contained in stratospheric samples obtained within the Arctic polar vortex.71 In these same samples, concentrations of SF6 and chlorofluorocarbons were measured. Contrary to the measurements of La¨mmerzahl et al., the three-isotope plot gave a line with a slope of , 1, in agreement with earlier work.72 Alexander et al. point out that the concentration of SF6 in these samples provides an indicator of the “age” of the atmospheric parcel containing it. Thus, an anticorrelation between the concentration of SF6 and the isotopic anomaly 17D is found, the increase in the latter quantity resulting from increased exchange with O(1D).

C. THEORETICAL M ODELS FOR NMD I SOTOPIC F RACTIONATION IN C ARBON D IOXIDE The Wen and Thiemens experiments65 were simulated in calculations by Yung et al.,73 who found that the final isotopic composition of the oxygen was nearly independent of that of the ozone because of the much larger reservoir of oxygen in the carbon dioxide under the experimental conditions used. Their ‘standard model’ produced oxygen with d 17O values of around 78‰, about 20% higher than experimental values. The problem seems to lie in the importance of exchange reactions such as O þ OpO ! Op þ OO, which have mass-dependent isotopic fractionation. Almost simultaneously with this paper, a report of similar calculations by Barth and Zahn

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Isotope Effects in Chemistry and Biology

appeared.70 In addition to modeling the laboratory experiments, they used a 1-D model to calculate the expected distribution of oxygen isotopes in the stratosphere and mesosphere. Their calculations differ from those of Yung et al. in that isotopic dependence of the rate coefficient for the exchange reaction: Op ð1 DÞ þ CO2 ! Op CO þ Oð3 PÞ was included. Adjustment of this and the appropriate scenario for the isotopic composition of ozone as a function of altitude led to results for the concentrations of 17O and 18O that were in good agreement with measurements of Zipf and Erdman74 and Thiemens et al.72

V. NITROUS OXIDE ISOTOPOLOGUES A. LABORATORY E XPERIMENTS Early laboratory experiments on the photolysis of nitrous oxide with a mercury resonance lamp (185 and 248 nm) indicated essentially no change in the 18O concentration in the N2O remaining after photolysis.75,76 In other experiments, the addition of ozone provided a source of O(1D) atoms, which rapidly react with N2O. This resulted in an enrichment of 18O in the residual nitrous oxide of about 6‰. In both sets of experiments, 17D values were less than 0.1‰, and the authors concluded that both processes were mass-dependent. More recent experiments indicate considerably larger enrichment of 18O in photolysis, consistent with analyses of stratospheric nitrous oxide (for further references, cf. Ref. 77). Ro¨ckmann et al. determined changes in both the nitrogen and the oxygen isotopic composition of nitrous oxide during photolysis in the UV, using both an ArF laser (193 nm) and a Sb lamp (185 –225 nm).78 Enrichments of up to 44.5‰ in 18O were observed, while simultaneous determinations of 17D again led to values # 0.1‰. The authors conclude that photolysis is a mass-dependent process.

B. ATMOSPHERIC N ITROUS O XIDE A few years ago, Cliff and Thiemens reported NMD isotopic fractionation of the oxygen isotopes in nitrous oxide samples obtained from four sampling locations on the Earth’s surface.76 A strong correlation between 17D and d 18O was found in these samples, as was a dependence on the distance from the primary source. However, the values of 17D were quite small, of the order of 1‰. Since that report, a number of both laboratory and atmospheric studies have focused on the abundances of the species 14N14N16O, 15N14N16O, 14N15N16O, 14N14N17O, and 14N14N18O. Ro¨ckmann et al.78 have also determined 17D values for samples from several terrestrial locations, extending the range of d 18O beyond that observed by Cliff and Thiemens. A reexamination of both sets of measurements by Kaiser et al.5 has led to D ¼ 0:9 ^ 0:1‰ and a value of l ¼ 0:516: using the definition of Equation 12.10. A few measurements have been made on samples obtained at 8 –11 km pressure altitude.79 The d 18O values for these were larger than for any tropospheric samples, and the differences from the mass-dependent prediction are also larger.

C. THEORETICAL E XPLANATIONS AND M ODELING C ALCULATIONS As the UV photolysis of nitrous oxide is the major atmospheric sink for this gas, several authors have attempted to calculate absorption cross-sections for the various isotopic species. First, Yung and Miller used a model (ZPE model) that approximates the spectrum of a heavy isotopologue by blue-shifting the spectrum of 14N14N16O by an amount corresponding to the zero-point energy difference between the two isotopic species.80 A more sophisticated approach (HP model) was used by Johnson et al., who used time-dependent Hermite propagation of the

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wave function on calculated potential energy surfaces.81 The absorption spectrum is obtained as the Fourier transform of the correlation function between the initial and the propagated wave functions. These calculations gave larger isotope effects than did the zero-point energy model, in better agreement with experimental values. 17D values are comparable to those calculated by the ZPE method. Whether or not photolysis is an NMD process is still being debated. McLinden et al. used a 3-D chemical transport model (CTM) to study stratospheric distributions and global budgets of the various nitrous oxide isotopologues.77 They calculated expected deviations from mass dependent fractionation of the oxygen isotopes, using the HP model for the absorption cross-sections, as well as a model in which the ZPE effect was doubled. Their CTM simulation produced ratios of 0.54 and 0.52 for 14N14N17O/14N14N18O with these two models for the absorption cross-sections. As pointed out by Kaiser et al.,5 over the important wavelength region of 185 –230 nm the calculated absorption cross section ratios lie between 0.518 to 0.529, which may be compared with the values of 0.507 and 0.518 obtained in the ArF laser photolysis and the Sb lamp photolysis, respectively. Since commercial N2O samples have a slope of 0.516 ^ 0.004 on a three-isotope plot, they conclude that the isotopic anomaly in atmospheric nitrous oxide does not result from photolysis; biological sources at the Earth’s surface may be partially responsible. If photolysis of nitrous oxide is mass dependent, what then is the source of the NMD isotopic fractionation observed in both tropospheric and terrestrial samples of nitrous oxide? Ro¨ckmann et al.78 suggest an indirect mechanism, of which the important steps are NO þ O3 ! NO2 þ O2 NO2 þ NH2 ! N2 O þ H2 O: The amidogen radical (NH2) is produced in the OH oxidation of ammonia in the atmosphere. These two reactions, in tandem, provide a mechanism for the transfer of the known nonmassdependent isotopic fractionation in stratospheric ozone to nitrous oxide. Only about 3% of the global N2O source would need to arise from this mechanism to account for the observed 17D in nitrous oxide, much smaller than that observed in ozone. This explanation depends on a number of atmospheric variables, such as the relative concentrations of ozone and HO2 as oxidants of nitric oxide, and of the NH2 radical as a reductant of NO2. Another mechanism for the transfer of O atoms from ozone into nitrous oxide has been proposed by Zipf and Prasad, who suggest that highly vibrationally excited ozone, formed in the O(1D) þ O2 reaction, can react with molecular nitrogen to form nitrous oxide. Their experiments confirm the existence of this possible reaction path.82 As N2O is not a symmetric molecule, the two N atoms are chemically different, and thus it is not surprising that 15N substitution in either the end or the central position leads to different isotope effects. This has led to a number of interesting applications to problems of atmospheric chemistry, the discussion of which is outside the scope of this chapter (cf. Chapter 13).

VI. OXYGEN AND SULFUR ISOTOPIC FRACTIONATION IN TERRESTRIAL AND EXTRATERRESTRIAL SOLIDS A. CARBONATES The very first observation of NMD isotopic fractionation was that of Clayton et al., who determined the oxygen isotopic composition of the high-temperature minerals in carbonaceous chondrite meteorites.83 Benedix et al.84 have recently examined carbonates in a number of chondrites, and find that the d 17O vs. d 18O values lie on a line that is significantly below the line for terrestrial carbonates and has a slope of 0.612. The authors find a correlation between the

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isotopic anomaly and a quantity defined as the mineralogic alteration index, which represents the Fe/Si ratio in phyllosilicates of the chondrite. References to earlier work on meteorites are contained in this paper. The Martian meteorite ALH 84001 was found to have 17D values of 0.8‰ in carbonates compared to values of 0.22 to 0.33‰ in associated silicates.85 This is believed to result from an atmospheric oxygen reservoir for carbonate formation, and NMD isotopic fractionation in the atmosphere that could result from a transfer of the isotopic signature from ozone to carbon dioxide. Thermal decomposition of calcium and magnesium carbonates from terrestrial minerals was shown to lead very small values of 17D (, 0.5‰) in the product oxides.12 The customary three-isotope plot gave a linear fit with a slope equal to that predicted for a mass dependent fractionation, but with a different intercept, and this was interpreted as an indicator of NMD isotopic fractionation.

B. SULFATES 1. Laboratory Experiments In addition to the oxygen isotopes in sulfur oxides and sulfates, sulfur compounds offer another possible indicator of NMD isotopic fractionation because there are four stable isotopes: 32S (95%), 33 S (0.75%), 34S (4.21%) and 36S (0.02%). With 32S taken as the reference isotope, deviations from mass-dependent isotopic fractionation are defined as 33

D ¼ 103 ½1 þ 1023 d33 S 2 ð1 þ 1023 d34 SÞ0:515 2 1

36

D ¼ 103 ½1 þ 1023 d36 S 2 ð1 þ 1023 d34 SÞ1:91 2 1

and

In the past few years a number of papers have described NMD fractionation of oxygen in terrestrial and extraterrestrial sulfates. A review by Thiemens et al.2 covers most of the work up to that date, which will only be discussed briefly here. Several investigations have been carried out in an attempt to determine the chemical source of nonmass-dependent isotopic fractionation in these compounds. In Chapter 13, Roth et al. discuss isotope effects in the atmospheric chemistry of sulfur. In the oxidation path from elemental sulfur or organic sulfur compounds to sulfate, sulfur dioxide (SO2) is a common intermediate. Savarino et al. have studied oxygen isotope effects in the gas-phase oxidation of sulfur dioxide by OH, and oxidation in solution by H2O2, O3 and O2 catalyzed by Fe (III) and Mn (II), as well as the hydration of sulfur trioxide.86 It was known previously that SO2 and by extension SO3 do not have a NMD isotopic label, and this is also the case for all the oxidants but aqueous hydrogen peroxide and ozone. If the oxidant has a mass-dependent isotopic label, NMD isotopic labeling in oxidation can only occur if these is an anomalous effect in the reaction itself. In the oxidation by H2O2 two of the O atoms in sulfate are derived from the hydrogen peroxide, and since atmospheric peroxide has been shown to be NMD labeled (see below), this provides a route for incorporation of the label into sulfate. A similar finding for ozone provides another possible route for transfer of the anomalous isotopic distribution. Sulfur isotope effects were not investigated in this work. A possible source of fractionation of sulfur isotopes, the UV photolysis of SO2, was investigated by Farquhar et al., using light sources covering the range 185 – 248 nm.87 At the longer wavelength end of this range, excited singlet and triplet states are formed, but the only chemical process is the reaction 3SO2 þ SO2 ! SO3 þ SO. At shorter wavelengths, the direct dissociation into SO þ O becomes possible, followed by photodissociation of SO itself. Isotopic distributions following photolysis in these two wavelength ranges are quite different. In the long wavelength region, NMD fractionation is observed, and sulfate is enriched in both 33S and 34S relative to residual sulfur dioxide. In the short wavelength region, the sulfate product is enriched in 34S with little change in 33S while product sulfur is enriched in 33S and depleted in 34S. The implication is that more than one fractionation process is operating simultaneously.

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2. Terrestrial Sulfates Sulfates formed in the combustion of various materials, as well as those from vehicular exhaust, were examined for both oxygen and sulfur isotopic composition.88 Only one sample (taken at some distance from the combustion source) showed a significant 17D value, in spite of d 18O values as high as , 22‰. As found here for combustion, most sulfate formation processes on the Earth have been shown to be mass dependent. Some interesting exceptions are found in sulfates from several arid regions, however: The Namib desert,89 Antarctic dry valleys,90 and Death Valley, California.91 Oxidation of sulfur dioxide, as described above, is proposed as the source of these effects. Aerosol sulfates from several regions were investigated for both sulfur and oxygen isotope effects by Romero and Thiemens.92 There appears to be little or no correlation between the two effects. Positive 33D values as large as 0.47 and negative 36D values as large as 2 1.6 were observed in aerosols from three locations in California, both near the Pacific and inland near the desert. A possible source of these effects is the UV photolysis of SO2, described above. Oxygen isotope fractionation in sulfates from rainwater and from aerosols has been used to determine the relative contribution of oxidation of sulfur dioxide by H2O2 and O3.93 Aerosol sulfates collected at a mountain site (elevation, 3801 m) were highly enriched in 17O. A similar observation was made by Johnson et al., who determined isotopic concentrations in sulfate from snowmelt and from mountain streams at elevations of 2300 to 3600 m.94 The sulfate that originated in snow had large values of 17 D compared to that in the mountain streams, and the size of the NMD fractionation can be used as an indication of the amount of sulfate coming from the atmosphere as compared with that produced from organic sulfides. Bao et al. have determined oxygen isotope concentrations in sulfates from Oligocene ash deposits in the northern High Plains, and find deviations from a mass-dependent fractionation.95 Since sulfates from volcanic ash falls typically do not show such deviations, it is suggested that they result from deposits of dry fogs (clouds of volcanic sulfate aerosols derived from SO2 oxidation in the troposphere and trapped in the planetary boundary layer). Oxygen isotope distributions in sulfates as old as 130,000 years have been determined by Alexander et al. and NMD isotopic fractionation is found to be greater during interglacial periods than during glacial intervals.96 This variation is attributed to mixing of mass-dependent oxidation of sulfur dioxide by OH and NMD oxidation by H2O2. The short wavelength photolysis results described above87 show a fractionation pattern that is similar to that found for Archean sulfate and sulfide rocks with ages of 2.45 to 3.9 billion years ago (Ga). This leads the authors to speculate that these rocks carry the isotopic signature of atmospheric processes, in particular, UV photolysis of sulfur dioxide. This would require some transparency of the atmosphere in spectral regions where O2 and O3 are the principal absorbers, and the authors present arguments that the Archaean atmosphere was much lower in oxygen than today’s atmosphere. Further evidence for this picture is presented by Farquhar et al.,97 who show that the isotopic signature in sulfates changes in the region of 2.09– 2.45 Ga. 33D values from more recent sulfides and sulfates are essentially null, whereas older rocks show a wide scatter between 2 1.29 and 2.04‰. These large values would rule out oxidative and microbial weathering of continental sulfides as a major source of sulfates. Criticism of this work98 has been replied to by the authors.99 Sulfide inclusions in diamonds from Botswana were found to have significant 33D values, relatively independent of d 34S fractionation.100 These results provide an insight into the Archean sulfur cycle, indicating that sulfur-containing gas emissions from volcanoes were photolyzed with NMD isotopic fractionation, and the resulting sulfur species were transferred to surface reservoirs. 3. Extraterrestrial Sulfur Compounds Thiemens and coworkers have investigated several classes of Martian meteorites.2,85,101 The sulfur isotope fractionation measured in sulfates and sulfides from these rocks provides information about the Martian atmosphere.

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C. NITRATE A EROSOLS Nitrate aerosols are formed largely by the oxidation of NO2 in the reactions NO2 þ OH þ M ! HNO3 þ M NO2 þ O3 ! NO3 þ O2 NO3 þ NO2 ! N2 O5 N2 O5 þ H2 OðsurfaceÞ ! 2HNO3 ðsurfaceÞ In addition, nitrogen dioxide enters a photostationary state that equilibrates the oxygen atoms with HO2 and O3: NO2 þ hn ! NO þ Oð3 PÞ NO þ O3 ! NO2 þ O2 NO þ HO2 ! NO2 þ OH Oð3 PÞ þ O2 þ M ! O3 þ M The known 17D data for ozone were combined with a photochemical box model to calculate the expected 17D values for nitrates over a time interval of 20 months.102 The results of the calculations are compared with measurements of the isotopic composition of nitrates contained in atmospheric aerosols collected over a one-year period. The agreement was very good for the winter and spring months, but the model values were 2 –4‰ high for fall and summer. The seasonal variation correlates with the relative importance of heterogeneous nitrogen pentoxide hydration compared with the gas-phase oxidation of NO2 by OH. Lyons has made model calculations to estimate the transfer of the NMD isotopic distribution found in stratospheric ozone to other atmospheric species.103 Large values of 17D are calculated for NO and NO2, with the result that HNO3 will also show this effect. The predicted values are in excellent agreement with the measurements on aerosol nitrates by Michalski and Thiemens.102 Transfer of the ozone isotopic anomaly to OH is possible, but only if oxygen exchange reactions between OH and O2 can be neglected. Exchange with oxygen also removes the isotopic signature from HO2, which otherwise is predicted to form H2O2 with nonzero values of 17D, as found in rainwater by Savarino and Thiemens.104

VII. OTHER MOLECULES A. HYDROGEN P EROXIDE I SOTOPOLOGUES As discussed above, oxidation of sulfur dioxide by hydrogen peroxide is postulated as one source of NMD isotopic fractionation in sulfates. d 17O and d 18O measurements were made on five samples of commercial hydrogen peroxide, and a plot of d 17O vs. d 18O gave a straight line with a slope of 0.511, which was taken to be the mass-dependent reference.104 Analysis of the H2O2 contained in a large number of rainwater samples gave values that lay on a line parallel to that observed for commercial H2O2, but with slightly higher values of d 17O. None of the major sinks for hydrogen peroxide in the aqueous phase exhibits NMD isotopic effects, so this behavior is assumed to result from the gas-phase formation of hydrogen peroxide, principally in the reaction HO2 þ HO2 ! H2O2 þ O2. In a related study, H atoms produced in an RF discharge were mixed with flowing oxygen, and the products (hydrogen peroxide and unstable hydrogen superoxides) were trapped at 2 1968C. Upon warming, the superoxides decomposed to liberate gaseous oxygen, and isotopic compositions of both this and the hydrogen peroxide were determined. A typical three-isotope plot gave a straight line through both sets of data points, and with a slope of

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0.84 compared with the slope of 0.513 previously obtained with commercial hydrogen peroxide samples. This system is chemically very complex. It is known, for example, that hydrogen peroxide is not formed in the absence of a cold trap.105 I have made calculations with a kinetic model of the system, using known rate constants for the gas-phase reactions, and these calculations indicate that essentially no hydrogen peroxide would be produced. More detailed information is needed before any sensible conclusions can be drawn. Model calculations of atmospheric peroxide formation and depletion rates indicate that the major radical source is photolysis of ozone, producing an O(1D) atom that reacts with water to form two hydroxyl radicals.106

B. ATMOSPHERIC O XYGEN I SOTOPOLOGUES It has long been known that there is an isotopic disequilibrium between molecular oxygen in the atmosphere and in the oceans: atmospheric oxygen is enriched in 18O with respect to ocean water. This has been labeled “the Dole effect”, referring to the scientist who discovered it.107 This difference is attributed largely to discrimination against the heavier isotope by respiration in plants, as photosynthesis does not show an isotope effect. Both of these processes lead to massdependent fractionation. Atmospheric oxygen, on the other hand, has the signature of a NMD isotope effect, with a value of 17D of , 2 0.34‰.6 In the stratosphere, the isotopic anomaly in ozone is transferred to carbon dioxide, which does not exchange with molecular oxygen. However, the ultimate source of the oxygen in ozone and carbon dioxide is molecular oxygen, which becomes depleted in a way that reflects the isotopic anomaly. Luz et al. have determined the isotopic fractionation and 17D values for atmospheric oxygen contained in a terrarium with growing plants.108 The isotopically anomalous atmospheric oxygen is removed by respiration and is replaced with isotopically normal oxygen by photosynthesis. Thus, the magnitude of 17D provides tracer for biological processes. Using estimates of the rate of production of the oxygen isotopic anomaly in the stratosphere, the authors calculate the rate of global biosphere production. Using isotopic analysis of air from an ice core, they predict that this quantity has changed by $ 10% over the past 80,000 years. This work has been extended by Luz and Barkan,109 who point out that the 17D value of oxygen dissolved in sea water depends on the competition between two rates: that of the air– water gas exchange, which tends to equalize the isotopic composition of atmospheric and dissolved oxygen, and the rate of in situ production of oxygen by biological activity which tends to decrease 17D because such processes are mass dependent. They use this concept to determine the gross production rate. Further studies of the isotopic composition of atmospheric oxygen as affected by photorespiration and dark respiration have been carried out by Angert et al.110 They find slightly different values of ln17 d=ln18 d for these pathways, and for diffusion in air. The current global rate of photorespiration can be estimated if the current global isotopic balance is determined. Note added in proof: Since the submission of this chapter, the following two important review articles have appeared: Brenninkmeijer, C. A. M., Jannsen, C., Kaiser, J., Ro¨ckmann, T., Thee, T. S., and Assonov, S. S., Isotope effects in the chemistry of atmospheric trace compounds. Chem. Rev., 103, 5125–5161, 2003. Mauersberger, K., Assessment of the ozone isotope effect, In Advances in Atomic, Molecular, and Optical Physics, vol. 50, Bederson, B. and Walther, H., Eds., Academic Press, New York, 2005.

ACKNOWLEDGMENTS This work was performed at Brookhaven National Laboratory under Contract DE-AC0298CH10886 with the U.S. Department of Energy and supported by its Division of Chemical Sciences, Office of Basic Energy Sciences.

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50 Siebert, R., Schinke, R., and Bitterova, M., Spectroscopy of ozone at the dissociation threshold: quantum calculations of bound and resonance states on a new global potential energy surface, Phys. Chem. Chem. Phys., 3, 1795– 1798, 2001. 51 Babikov, D., Kendrick, B. K., Walker, R. B., Pack, R. T., Fleurat-Lesard, P., and Schinke, R., Formation of ozone: metastable states and anomalous isotope effect, J. Chem. Phys., 119, 2577– 2589, 2003. 52 Baker, T. A. and Gellene, G. I., Classical and quasi-classical trajectory calculations of isotope exchange and ozone formation proceeding through O þ O2 complexes, J. Chem. Phys., 117, 7603– 7613, 2002. 53 Bhattacharya, S. K. and Thiemens, M. H., New evidence for symmetry dependent isotope effects: O þ CO Reaction, Z. Naturforsch., 44a, 435–444, 1989. 54 Ro¨ckmann, T., Brenninkmeijer, C. A. M., Neeb, P., and Crutzen, P. J., Ozonolysis of nonmethane hydrocarbons as a source of the observed mass independent oxygen isotope enrichment in tropospheric CO, J. Geophys. Res., 103, 1463– 1470, 1998. 55 Stevens, C. M., Kaplan, L., Gorse, R., Durkee, S., Compton, J., Cohen, S., and Bielling, K., The kinetic isotope effect for carbon and oxygen in the reaction CO þ OH, Int. J. Chem. Kinet., 12, 935– 948, 1980. 56 Ro¨ckmann, T., Brenninkmeijer, C. A. M., Saueressig, G., Bergamaschi, P., Crowley, J. N., Fischer, H., and Crutzen, P. J., Mass independent fractionation of oxygen isotopes in atmospheric CO due to the reaction CO þ OH, Science, 281, 544–546, 1998. 57 Feilberg, K. L., Selleva˚g, S. R., Nielsen, C. J., Griffith, D. W. T., and Johnson, M. S., CO þ OH ! CO2 þ H: the relative reaction rate of five CO isotopologues, Phys. Chem. Chem. Phys., 4, 4687– 4693, 2002. 58 Golden, D. M., Smith, G. P., McEwen, A. B., Yu, C-L., Eiteneer, B., Frenklach, M., Vaghjiani, G. L., Ravishankara, A. R., and Tully, F. P., OH (OD) þ CO: measurements and optimized RRKM fit, J. Phys. Chem. A, 102, 8598– 8606, 1998. 59 Fulle, D., Hamann, H. F., and Troe, J., High pressure range of addition reactions of HO. II. Temperature and pressure dependence of the reaction HO þ CO ! HOCO ! H þ CO2, J. Chem. Phys., 105, 983– 1000, 1996. 60 Hodder, P. S., Brenninkmeijer, C. A. M., and Thiemens, M. H., Mass independent fractionation in tropospheric carbon monoxide, ICOG Proceedings, US Geological Circular, 1107, 1994. 61 Huff, A. K. and Thiemens, M. H., 17O/16O and 18O/16O isotope measurements of atmospheric carbon monoxide and its sources, Geophys. Res. Lett., 25, 3509– 3512, 1998. 62 Ro¨ckmann, T., Jo¨ckel, P., Gros, V., Bra¨unlich, M., Possnert, G., and Brenninkmeijer, C. A. M., Using 14 C, 13C, 18O and 17O isotopic variations to provide insights into the high northern latitude surface CO inventory, Atmos. Chem. Phys., 2, 147– 159, 2002. 63 Gros, V., Jo¨ckel, P., Brenninkmeijer, C. A. M., Ro¨ckmann, T., Meinhardt, F., and Graul, R., Characterization of pollution events observed at Schauinsland, Germany, using CO and its stable isotopes, Atmos. Environ., 36, 2831– 2840, 2002. 64 Heidenreich, J. E. III and Thiemens, M. H., The non-mass-dependent oxygen isotope effect in the electrodissociation of carbon dioxide: a step toward understanding NoMaD chemistry, Geochim. Cosmochim. Acta, 49, 1303– 1306, 1985. 65 Wen, J. and Thiemens, M. H., Multi-isotope study of the O(1D) þ CO2 exchange and stratospheric consequences, J. Geophys. Res., 98, 12801– 12808, 1993. 66 Bhattacharya, S. K., Savarino, J., and Thiemens, M. H., A new class of oxygen isotopic fractionation in photodissociation of carbon dioxide: potential implications for atmospheres of Mars and Earth, Geophys. Res. Lett., 27, 1459– 1462, 2000. 67 Perri, M. J., Van Wyngarden, A. L., Boering, K. A., Lin, J. J., and Lee, Y. T., Dynamics of the O(1D) þ CO2 oxygen isotope exchange reaction, J. Chem. Phys., 119, 8213– 8216, 2003. 68 Johnston, J. C., Ro¨ckmann, T., and Brenninkmeijer, C. A. M., CO2 þ O(1D) exchange: laboratory and modeling studies, J. Geophys. Res., 105, 15213– 15229, 2000. 69 La¨mmerzahl, P., Ro¨ckmann, T., Brenninkmeijer, C. A. M., Krankowksy, D., and Mauersberger, K., Oxygen isotope composition of stratospheric carbon dioxide, Geophys. Res. Lett., 292002, doi: 10.1029/ 2001GL014343,012002. 70 Barth, V. and Zahn, A., Oxygen isotope composition of carbon dioxide in the middle atmosphere, J. Geophys. Res., 102, 12995– 13007, 1997. 71 Alexander, B., Vollmer, M. K., Jackson, T., Weiss, R. F., and Thiemens, M. H., Stratospheric CO2 isotopic anomalies and SF6 and CFC tracer concentrations in the Arctic polar vortex, Geophys. Res. Lett., 28, 4103– 4106, 2001.

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72 Thiemens, M., Jackson, T., Zipf, E. C., Erdman, P. W., and van Egmond, C., Carbon dioxide and oxygen isotope anomalies in the mesosphere and stratosphere, Science, 270, 969– 972, 1995. 73 Yung, Y. L., Carbon dioxide in the atmosphere: isotopic exchange with ozone and its use as a tracer in the middle atmosphere, J. Geophys. Res., 102, 10857 –10866, 1997. 74 Zipf, E. C. and Erdman, P. W., Studies of Trace Constituents in the Upper Stratosphere and Mesosphere Using Cryogenic Whole Air Sampling Techniques, Goverment Printing Office, Washington, DC, 1994. 75 Johnston, J. C., Cliff, S. S., and Thiemens, M. H., Measurement of multioxygen isotopic (d18O and d17O) fractionation factors in the stratospheric sink reactions of nitrous oxide, J. Geophys. Res., 100, 16801 –16804, 1995. 76 Cliff, S. S. and Thiemens, M. H., The 18O/16O and 17O/16O ratios in atmospheric nitrous oxide: a massindependent anomaly, Science, 278, 1774–1776, 1997. 77 McLinden, C. A., Prather, M. J., and Johnson, M. S., Global modeling of the isotopic analogues of N2O: stratospheric distributions, budgets, and the 17O– 18O mass-independent anomaly, J. Geophys. Res., 1082003, doi: 10.1029/2002JD002560. 78 Ro¨ckmann, T., Kaiser, J., Crowley, J. N., and Brenninkmeijer, C. A. M., The origin of the anomalous or “mass-independent” oxygen isotope fractionation in tropospheric N2O, Geophys. Res. Lett., 28, 503– 506, 2001. 79 Cliff, S. S., Brenninkmeijer, C. A. M., and Thiemens, M. H., First measurement of the 18O/16O and 17 O/16O ratios in stratospheric nitrous oxide: a mass-independent anomaly, J. Geophys. Res., 104, 16171 –16175, 1999. 80 Yung, Y. L. and Miller, C. E., Isotopic fractionation of stratospheric nitrous oxide, Science, 278, 1778– 1780, 1997. 81 Johnson, M. S., Billing, G. D., Gruodis, A., and Jannsen, M. H. M., Photolysis of nitrous oxide isotopomers studied by time-dependent hermite propagation, J. Phys. Chem. A, 105, 8672– 8680, 2001. 82 Zipf, E. C. and Prasad, S. S., Experimental evidence that excited ozone is a source of nitrous oxide, Geophys. Res. Lett., 25, 4333– 4336, 1998. 83 Clayton, R. N., Grossman, L., and Mayeda, T. K., A component of primitive nuclear composition in carbonaceous chondrites, Science, 182, 485– 488, 1973. 84 Benedix, G. K., Leshin, L. A., Farquhar, J., Jackson, T., and Thiemens, M. H., Carbonates in CM2 chondrites: constraints on alteration conditions from oxygen isotopic compositions and petrographic observations, Geochim. Cosmochim. Acta, 67, 1577– 1588, 2003. 85 Farquhar, J., Thiemens, M. H., and Jackson, T., Atmosphere-surface interactions on Mars: 17O measurements of carbonate from ALH 84001, Science, 280, 1580–1582, 1998. 86 Savarino, J., Lee, C. C. W., and Thiemens, M. H., Laboratory oxygen isotopic study of sulfur (IV) oxidation: origin of the mass-independent oxygen isotopic anomaly in atmospheric sulfates and sulfate mineral deposits on Earth, J. Geophys. Res., 105, 29079– 29088, 2000. 87 Farquhar, J., Savarino, J., Airieau, S., and Thiemens, M. H., Observation of wavelength-sensitive massindependent sulfur isotope effects during SO2 photolysis: implications for the early atmosphere, J. Geophys. Res., E12, 32829– 32839, 2001. 88 Lee, C. C-W., Savarino, J., Cachier, H., and Thiemens, M. H., Sulfur (32S, 33S, 34S, 36S) and oxygen (16O, 17O, 18O) isotopic ratios of primary sulfate produced from combustion processes, Tellus, 54B, 193– 200, 2002. 89 Bao, H., Thiemens, M. H., and Heine, K., Oxygen-17 excesses of the central namib gypcretes: spatial distribution, Earth Planet Sci. Lett., 192, 125–235, 2001. 90 Bao, H., Campbell, D. A., Bockheim, J. G., and Thiemens, M. H., Origins of sulphate in Antarctic dry valley soils as deduced from anomalous 17O compositions, Nature, 407, 499– 502, 2000. 91 Bao, H., Michalski, G. M., and Thiemens, M. H., Sulfate oxygen-17 anomalies in desert varnishes, Geochim. Cosmochim. Acta, 65, 2029– 2036, 2000. 92 Romero, A. B. and Thiemens, M. H., Mass-independent sulfur isotopic compositions in present-day sulfate aerosols, J. Geophys. Res., 108(D16), 000, 2003, doi: 10.1029/2003JD003660,002003. 93 Lee, C. C-W. and Thiemens, M. H., The d17O and d18O measurements of atmospheric sulfate from a coastal and high alpine region: a mass-independent isotopic anomaly, J. Geophys. Res., 106, 17359 –17373, 2001.

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94 Johnson, C. A., Mast, M. A., and Kester, C. L., Use of 17O/16O to trace atmospherically-deposited sulfate in surface waters: a case study in alpine watersheds in the Rocky mountains, Geophys. Res. Lett., 28, 4483 –4486, 2001. 95 Bao, H., Thiemens, M. H., Loope, D. B., and Yuan, X-L., Sulfate oxygen-17 anomaly in an oligocene ash bed in mid-North America: was it the dry fogs?, Geophys. Res. Lett., 30(16)2003, doi: 10.1029/ 2003GL016869. 96 Alexander, B., Savarino, J., Barkov, N. I., Delmas, R. J., and Thiemens, M. H., Climate driven changes in the oxidation pathways of atmospheric sulfur, Geophys. Res. Lett., 29(14)2002, doi: 10.1029/ 2002GL014879. 97 Farquhar, J., Bao, H., and Thiemens, M., Atmospheric influence of Earth’s earliest sun cycle, Science, 289, 756– 758, 2000. 98 Ohmoto, H., Yamaguchi, K. E., and Ono, S., Questions regarding precambrian sulfur isotope fractionation, Science, 292, 1959a, 2001. 99 Farquhar, J., Bao, H., Thiemens, M. H., Hu, G., and Rumble, D. III, Response to “Questions Regarding Precambrian Sulfur Isotope Fractionation” by Ohmoto, H., Yamaguchi, K. E., and Ono, S., Science, 292, 1959a, 2001. 100 Farquhar, J., Wing, B. A., McKeegan, K. D., Harris, J. W., Cartigny, P., and Thiemens, M. H., Massindependent sulfur of inclusions in diamond and sulfur recycling on early earth, Science, 298, 2369 –2372, 2002. 101 Farquhar, J., Jackson, T. L., and Thiemens, M. H., A 33S enrichment in ureilite meteorites: evidence for a nebular sulfur component, Geochim. Cosmochim. Acta, 64, 1819– 1825, 2000. 102 Michalski, G., Scott, Z., Kabiling, M., and Thiemens, M. H., First measurements and modeling of 17O in atmospheric nitrate, Geophys. Res. Lett., 30(16)2003, doi: 10.1029/2003GL017015,012003. 103 Lyons, J. R., Transfer of mass-independent fractionation in ozone to other oxygen-containing radicals in the atmosphere, Geophys. Res. Lett., 28, 3231– 3234, 2001. 104 Savarino, J. and Thiemens, M. H., Analytical procedure to determine both d 18O and d 17O of H2O2 in natural water and first measurements, Atmos. Environ., 33, 3683– 3690, 1999. 105 McKinley, J. D. Jr. and Garvin, D., The reactions of atomic hydrogen with ozone and with oxygen, J. Am. Chem. Soc., 77, 5802– 5805, 1955. 106 Kleinman, L. I., Photochemical formation of peroxides in the boundary layer, J. Geophys. Res., 91, 10889 –10904, 1986. 107 Dole, M., The relative atomic weight of oxygen in water and air, J. Am. Chem. Soc., 57, 27 – 31, 1935. 108 Luz, B., Barkan, E., Bender, M. L., Thiemens, M. H., and Boering, K. A., Triple-isotope composition of atmospheric oxygen as a tracer of biosphere productivity, Nature, 400, 547– 550, 1999. 109 Luz, B. and Barkan, E., Assessment of Oceanic productivity with the triple-isotope composition of dissolved oxygen, Science, 288, 2028– 2031, 2000. 110 Angert, A., Rachmilevitch, S., Barkan, E., and Luz, B., Effects of photorespiration, the cytochrome pathway, and the alternative pathway on the triple isotopic composition of atmospheric O2, Global Biogeochem. Cycles, 17(1), 1030, 2003, doi: 1010.1029/2002GB001933,002003.

13

Isotope Effects in the Atmosphere Etienne Roth, Rene´ Le´tolle, C. M. Stevens, and Franc¸ois Robert

CONTENTS I. Introduction ...................................................................................................................... 388 II. Isotopes in Geochemical Cycles ..................................................................................... 389 III. Isotope Effects in the Water Cycle ................................................................................. 390 A. The Reservoir Model ............................................................................................... 390 B. Exchange between Different Phases of Water ........................................................ 390 C. Vapor – Liquid Isotope Fractionation and the Study of Reservoirs ........................ 391 D. Water in Precipitation ............................................................................................. 392 1. The Isotope Composition of Rain .................................................................... 392 2. Migration Effects, Altitude Effects, Seasonal Effects, Reevaporation Effects ....................................................................................... 392 3. The Case of Hailstorms .................................................................................... 393 a. Early Tenets of the Method ........................................................................ 393 i. Experiments ......................................................................................... 393 ii. Results, Further Models, and Discussion ............................................ 393 4. The d2 H – d18 O Relation in Precipitations ....................................................... 395 IV. Archives of Atmospheric Isotopic Effects Retained by Ice Caps .................................. 396 V. Isotopic Effects on Atmospheric Carbon in the Carbon Cycle ...................................... 398 A. Isotopes of Atmospheric Methane .......................................................................... 398 1. Sources .............................................................................................................. 399 2. Discussion ......................................................................................................... 399 3. Removal Processes ........................................................................................... 399 4. Atmospheric 14CH4 ........................................................................................... 400 5. Atmospheric d D ............................................................................................... 400 B. Isotopes of Atmospheric Carbon Monoxide ........................................................... 400 1. Sources and Sinks ............................................................................................. 400 2. Atmospheric Concentration and Isotopic Composition ................................... 401 3. Summary ........................................................................................................... 401 C. Isotopes of Atmospheric Carbon Dioxide .............................................................. 402 VI. Isotope Effects of Atmospheric Nitrogen ....................................................................... 402 VII. Isotope Effects of Atmospheric Oxygen ......................................................................... 403 A. Air Oxygen .............................................................................................................. 403 B. Ozone ....................................................................................................................... 403 C. Nitrous Oxide .......................................................................................................... 403 D. Atmospheric Sulfates .............................................................................................. 403

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VIII. Isotope Effects of Atmospheric Sulfur ............................................................................ 403 A. Introduction ............................................................................................................. 404 B. Turnover and Inventory ........................................................................................... 405 C. Nature, Isotopic Composition, and Atmospheric Chemistry of Sulfur .................. 405 D. Effects during Removal of Sulfur from the Atmosphere ....................................... 407 1. Archean Isotope Atmospheric Chemistry of Sulphur and Nonmass-Dependent Isotope Effect ................................................................. 407 IX. Isotope Effects on Zinc and Lead in the Atmosphere .................................................... 407 X. Deuterium Enrichments in the Organic Molecules of the Interstellar Medium ............ 407 XI. Constraints in Using Deltas, Capital Deltas, and Reference Samples ........................... 410 A. Possible Evolution of Measurements of Isotope Effects ....................................... 411 Acknowledgments ........................................................................................................................ 411 References .................................................................................................................................... 411

I. INTRODUCTION This chapter deals with isotope effects observed in the troposphere, and with effects on hydrogen and nitrogen isotopes occurring in interstellar media. Nonmass-dependent effects are studied in Chapter 12 of this book, which as a consequence covers most isotope effects in the stratosphere. Observations of nonmass-dependent isotope effects in the troposphere, which are often signatures of effects transferred from the stratosphere, contribute to the understanding of past and present troposphere chemistry and lead us to recall in this chapter consequences of effects studied in Chapter 12 of this book. The discovery of 17O and 18O by Giauque and Johnston1,2 in 1929 inspired that of deuterium by Urey3 in 1931, which was followed by first efforts to find natural isotope abundance variations in water by delicate density methods. Later, with the increasing precision of mass spectrometers, variations in isotope composition were observed for almost all multi-isotopic elements. As was already known for Pb, but found also for He, Ar, Sr, all or part of the isotope variation is linked to the addition of a radiogenic isotope. Isotope effects take place in the hydrosphere and atmosphere, the external envelopes of the Earth. This is due to at least three factors: temperatures favoring isotope exchange fractionation (the mean temperature of Earth is 128C), large movements of matter by diffusion and mixing, and the importance of biochemical reactions. Natural abundance specificity of stable isotope sources enables the tracing of such movements. Almost all important isotope abundance variations in the atmosphere for which much information exists are O and H and elements forming anions. Most elements forming anions exist 2 in the form ExOy (e.g., SO2 3 , SO4 ). Most metal ions, even in the atmosphere, are in a hydrated form Ex(OH2)y with the consequence that the mass of the reacting ions is much higher than that of the isolated metal ion and may influence the magnitude of isotope effects. Isotopes of elements which have been explicitly considered in this chapter are those of hydrogen (H, D), oxygen (16O, 17O, 18O), carbon (12C, 13C), nitrogen (14N, 15N), and sulfur (32S, 34S, 33S, 36S). Isotope effects of other elements such as Pb that have not been as extensively studied in the atmosphere are more briefly discussed. Geochemical cycles along which isotope abundances are modified is described, and the isotope abundance of molecules entering the atmosphere is mentioned. In the water cycle, the most important one, large effects on hydrogen and oxygen isotopes, mainly due to interactions between water vapor and condensed phases, have long been studied. The chemistry and photochemistry of molecules of minor components of the troposphere is sometimes complex and accompanied by isotope effects. Some effects have been found to be nonmass-dependent, which is defined as isotope fractionations that are not related to a unique function of mass. These can only be observed in elements having at least three isotopes. Since the turn of the millennium they have attracted growing attention and have become a tool for evaluating transfers between the atmosphere and

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either the stratosphere or the Earth’s surface, but also for estimating the composition of paleoatmospheres. The intriguing mechanisms of these effects have led to a number of theoretical approaches and laboratory experiments. As late as November 2003, publications on isotope abundance in nature usually deemed it necessary to redefine deltas, and a few introduce capital deltas to quantify nonmass-dependent isotope effects. A capital delta D(x) is the difference between a measured d (x) not obeying the ðm0 2 mÞ=mm0 rule, and a calculated mass-dependent d (j ). These are expressed by some authors in megs, i.e., thousandths of ordinary deltas. Section XI lists constraints and uncertainties attached to their use. Caution to be observed in interpreting published data, especially when parts per million are quoted, is recalled in Section XI, which also discusses ways of reporting data. Studies of isotope effects, though only a small part of an immense literature reporting variations of isotope abundances, number in the thousands and multiply exponentially. To shorten the list of references reviews are included: Brenninkmeijer et al.4 study in depth both stratosphere and troposphere. They consider HCHO, H2O2, and OCS, and provide complementary approaches to this chapter for CH4, CO, N2O, and others. The Handbook of Environmental Isotope Geochemistry edited by Fritz and Fontes5 contains detailed descriptions of facts, fundamental interpretations and references to all important works up to 1980; International Union of Pure and Applied Chemistry (IUPAC)’s reports6 contain isotopic compositions and ranges of natural variations of every element; and Coplen et al.7 contains a detailed discussion of isotope abundance variations of the following elements: H, He, Li, B, C, N, O, Mg, Si, S, Cl, Ca, Cr, Fe, Cu, Tl. For each element variations found in a number of natural sources, or sometimes manufactured compounds, expressed in deltas, atom% and atomic weights, are given and represented graphically. Isotope mole fractions in this chapter are taken from Coplen et al.7 In Roth8 references are found to publications up to 1997, calculations of isotope effects are outlined, numerical values of isotope effects at equilibrium in exchange reactions or vaporization, and values of kinetic effects are reported.

II. ISOTOPES IN GEOCHEMICAL CYCLES Geochemical cycles of elements on Earth can be considered as transfers through a succession of reservoirs that are linked by matter fluxes. Reservoirs are characterized by their dimensions, the number of atoms, and residence time or turnover, which is a function of dimension, and exchange kinetics. Small reservoirs, such as water vapor of the atmosphere, have a much shorter residence time than big reservoirs, such as the ocean. “Mother” reservoirs A, B, C, and D are respectively: A. B. C. D.

For hydrogen and oxygen: the ocean water (150 £ 109 km3 or 150 £ 1018 tons) For nitrogen: the atmosphere For carbon: charcoal and carbonates For sulfur: gypsum, evaporite strata

There is a hierarchy of processes. Cycles considered as entities with a more or less rapid turnover time are embedded in bigger cycles such as the water cycles between surface water (ocean) and atmospheric water vapor, liquid water on continents and ice, or between biomass and sediments and deep rocks that are linked first to water involved in friction planes of global tectonics, and second to deep “hydrothermal water” coming from the Earth’s mantle. Knowledge of isotopic rates and time constants or of the isotope content of a given reservoir serves to establish some of the characteristics of the systems: e.g., temperature of atmospheric precipitation, where the mean isotope ratio D/H varies by 6‰ per 8C.

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Changes in isotope composition from a reservoir to another are linked to: The overall rate of reactions involved The different reaction rates of isotopic molecules The relative dimensions of successive reservoirs Isotope effects provide information on the intensity of fluxes from one reservoir to another and also on mean residence times (turnover times) or on intrinsic mechanisms of exchange between reservoirs, whatever their place in the “cycle hierarchy.” Nonmass-dependent effects such as those first discovered in 17O are an additional source of data. Relations between the hydrosphere and the atmosphere considered in this chapter are a part of the isotope cycle of elements that are very similar to the geochemical superficial element cycle.

III. ISOTOPE EFFECTS IN THE WATER CYCLE Isotope mole fractions of hydrogen and oxygen isotopes in the reference material VSMOW are respectively 1H ¼ 0.99984426 (Roth8), 2H (or D) ¼ 0.00015574 (Lorius et al.9), 16O ¼ 0.9976206 (Roth8), 17O ¼ 0.0003790 (9), 18O ¼ 0.0020004 (Roth8). The explanation of mechanisms of transitions between condensed phases of water and water vapor is one of the best results of studies of isotope fractionation effects, where multi-isotope studies, i.e., simultaneous measurements of D/H and 8O/16O ratios, give access to information not obtainable otherwise. Early researches tried to evaluate the abundance of deuterium in nature, but only after specialized mass spectrometers were built by Nier and Thode did an accurate perspective of the water isotope cycle begin to emerge. Several monographs have dealt with the isotope water cycle since the 1960s, especially in an IAEA report,10 in Gat,11 and in a short review of progress in hydrology by Le´tolle and Olive.12

A. THE R ESERVOIR M ODEL The water cycle is composed of several boxes, and the intensity of exchange fluxes between them is well known, except as concerns underground waters. Different boxes have different isotope compositions. The “ocean box” is rather homogeneous due to mixing and presents small local variations only. Some compositions are due to evaporation, precipitation, and mixing with continental waters in estuaries and with ice melting from ice caps. The atmospheric box is very small, with residence time 12 days, and has a complex structure due to atmospheric turbulence. Continental water is very heterogeneous due to climatic influences and geographic features. “Deep water” is very heterogeneous and well known only as concerns the first kilometer of depth. It exchanges more or less slowly with rock mineral oxygen and eventually hydrogen, as temperature increases with depth. At greater depths, water and solid may eventually come to isotopic equilibrium.

B. EXCHANGE BETWEEN D IFFERENT P HASES OF WATER Specific effects result from the mixture of equilibrium and out-of-equilibrium exchanges. The model for water evaporation has been applied to most natural isotope exchanges, for example in the study of C, N, and S isotopes in natural systems, with the existence of several “boxes” in series having various interfaces. The isotope fractionation between liquid and water vapor, when written as 1L=V ¼ ðaL=V 2 1Þ is always small and numerically equal to lna. For instance 1L=V ¼ 9:2 £ 1023 for 18O in the equilibrium exchange between liquid and vapor at 258C. When water evaporates in a closed system, the direct diffusion flux of water to the vapor phase equals the flux of inverse diffusion of vapor to the liquid phase. The partial pressure of water vapor

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is the saturation pressure qs. The equilibrium fractionation factor aL/V is linked to the differences 1 16 1 2 16 between saturation vapor pressures of 1H18 2 O, H2 O, and H H O. 1 16 1 18 1 16 1 2 18 As qs ð H2 OÞ . qs ð H2 OÞ and qs ð H2 OÞ . qs ð H H OÞ; there is an enrichment of the liquid reservoir in heavy isotopes due to the preferential departure of lighter molecules towards the vapor phase. The magnitude of this effect was linked by Majzoub13 to the temperature in the Earth’s atmosphere by relations Equation 13.1 and Equation 13.2, with u in K. Lnaeq ð18 OÞ ¼ 1137u22 2 0:4156u21 2 0:00207 2

Lnaeq ð HÞ ¼ 24844u

22

2 76:248u

21

þ 0:05261

ð13:1Þ ð13:2Þ

The kinetic isotope fractionation factor ak is a function of the ratio of the molecular diffusion coefficients of isotopic species in the vapor phase, the aerodynamic resistance of the medium by the way of the Schmidt number, and resistances to molecular and turbulent diffusion. When the air humidity h becomes zero, the difference between the molecular diffusion coefficients D favors the removal of light molecules towards the vapor phase and the enrichment of the liquid phase in heavy isotopes, because14 Dð1 H2 16 OÞ . Dð1 H2 18 OÞ and Dð1 H2 16 OÞ . Dð1 H2 H16 OÞ: In the case of intermediary conditions of humidity ð0 , h , 1Þ, a model presented by Craig and Gordon15 simulates the evolution of the isotope composition of water by a system of layers. Above these layers, “free” atmosphere is not influenced by evaporation. During evaporation, the water molecule crosses five layers from bottom to top: 1. The liquid zone where mass transfer operates through turbulent diffusion and the isotope composition is considered as uniform. 2. A liquid film where mass transfer is produced by molecular diffusion only. 3. An interface where humidity is considered as saturated. Liquid and vapor are considered to be at equilibrium for the water –vapor system, and isotope fractionation is at equilibrium. 4. A vapor film where transfer operates through Fick’s first diffusion law, and where the supplementary kinetic effect ak exists due to diffusion. 5. The vapor layer where turbulent diffusion occurs without fractionation effects as resistance to turbulent diffusion is the same for all isotope species. In the vapor, every layer has a relative humidity h, normalized to temperature and pressure at the liquid – vapor interface, a resistance to diffusion rx imposed by the transfer mode relative to every layer, and an isotopic composition dx. All models presented since this study finally derive from it. They are used when an isotopic balance of reservoirs is calculated where an inverse diffusion flux of vapor, which condenses water molecules coming from the atmosphere at the liquid surface, is opposed to this global evaporation flux. This leads to the limitation of the isotope enrichment of water in natural systems. Existence of a permanent hydric regime does not systematically imply that an isotopic stationary equilibrium will be reached in a free water reservoir, although the existence of an isotopic stationary state implies the existence of a permanent hydric regime in the evaporating reservoir.

C. VAPOR –L IQUID I SOTOPE F RACTIONATION AND THE S TUDY OF R ESERVOIRS Equilibrium being established, let Vr be a reservoir volume, vi the input flow of water, ve the evaporation rate, and vo other losses. The corresponding deuterium contents, in atoms per million, are Dr, Di, De, and Do. When a permanent regime is established, water flows and deuterium contents are related by vi ¼ ve þ vo

ð13:3Þ

vi Di ¼ ve De þ vo Do

ð13:4Þ

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Except for exceptional situations, Do is equal to Dr, and thus Equation 13.3 and Equation 13.4 lead to ve =vi ¼ ðDr 2 Di Þ=ðDr 2 De Þ

ð13:5Þ

At equilibrium when De is not measured, it is assumed to be Dr/aL/V from Equation 13.5 when losses occur exclusively by evaporation Dr ¼ aL=V Di . This treatment of data was applied in the real case of Mono Lake in the U.S.16 It showed that nearly 90% of Mono Lake water loss appears to occur through evaporation. Evaluations of evaporation from lakes in arid countries have been made by similar calculations.17 At equilibrium such water pools are enriched in deuterium. The time to reach equilibrium is roughly evaluated by dividing the excess deuterium content of the pool when equilibrium is obtained by the additional deuterium introduced with time by the incoming flow. When pools and input flows are initially at the same isotope content, if the reservoir of volume Vr is subject only to evaporation losses, the excess deuterium content at equilibrium will be Vr Di ðaL=V 2 1Þ: As the deuterium content of the vapor is initially Di/aL/V and is Di at the end of the process, the additional deuterium introduced with time could be roughly ! Di =aL=V þ Di vi Di 2 ð13:6Þ 2 The time to reach equilibrium with 18O is not the same as initial concentrations and aL/V are different. R. Gonfiantini, in Volume 2, Chapter 2 of Fritz and Fontes5 gives a detailed account of environmental isotopes in lake studies.

D. WATER IN P RECIPITATION Isotope effects in precipitation have first been interpreted in rain. Isotope abundance measurements were made of oxygen 18 or deuterium according to the technique mastered by each laboratory. Establishing the contributions of temperature, altitude, distance from an ocean, etc., to these effects enables one to build general circulation models, and is the subject of Subsection 1. Isotope studies of mechanisms of formation of droplets of rain, snowflakes, or hailstones are presented in Subsection 2, with hailstone formation as an example. Subsection 3 presents relations between deuterium and oxygen 18 concentrations that bring information on the conditions of precipitation, including formation of the water vapor and formation of droplets. A useful reference is J. Gat, Isotopes of hydrogen and oxygen in precipitations, in Fritz and Fontes,5 Volume 1, Chapter 1). 1. The Isotope Composition of Rain Dansgaard,18 in a pioneering work considering a Rayleigh condensation mechanism forming rain drops in isotope equilibrium with cloud vapor at a given instant, explained the mechanism which impoverishes the vapor in heavy isotopes. Gravity separates falling drops that are enriched in the heavier isotope, and with fractionation between vapor and liquid remaining in isotopic equilibrium as precipitation goes on, vapor and condensed drops become increasingly lighter. Shifts towards lower values with the duration of showers and the amount of precipitated water are actually observed. 2. Migration Effects, Altitude Effects, Seasonal Effects, Reevaporation Effects Migration effect: As a mass of vapor migrates, precipitation becomes progressively isotopically lighter. Pushed by oceanic winds across continents, the vapor is subject to an isotope gradient from the ocean to the interior of continents, called the “continentality” effect. This is reflected, for instance, in a dO18 gradient from the ocean to the interior in Europe of 2 0.3‰ per 100 km in winter, and of 2 1.3‰ per 100 km in summer.19 In absolute deuterium content, the D/H of an

Isotope Effects in the Atmosphere

393

Antarctic ice can be as low as 80 £ 1026 in comparison with ocean water at 155 £ 1026 (Coplen et al.7). Altitude effect: Gradients in dO18 of 2 0.16‰ to 2 0.4‰ per 100 m have been observed in regional studies and compared with a calculated value of 2 0.13‰, as reported by Hoffman et al.20 Gradients are due to rain starting above lower ground before clouds ascend along slopes, losing the greater part of their water content. On a continental scale there is almost no water vapor above an altitude of 3000 m in calm weather, and generally small quantities of water precipitate, but the lower temperature with altitude increases fractionation between vapor and liquid during condensation. Seasonal effects: Variations of temperature in a given place create the same effect as altitude: in winter precipitations are lower in heavy isotopes than in summer ones. Reevaporation effect: During their fall, drops may evaporate and produce vapor which remixes with ambient local vapor and may blur other effects. Models interpret isotope data from the whole world published by IAEA at regular intervals together with innumerable statistics of the repartition of isotopes in rain at the Earth’s surface.10 Hoffman et al.20 have published a synthesis of the many models of the global circulation of water in the atmosphere that takes into account all acquisitions relative to isotope fractionation mechanisms in the atmospheric cycle, including interpretations of the role of local vapor of continental origin that only a few authors have dealt with. The accuracy of today’s circulation models is checked by models of paleo-circulation to which isotopic studies on ice and sediments contribute. 3. The Case of Hailstorms a. Early Tenets of the Method In the 1960s the mechanism of hail formation was in need of new methods of investigation in order to establish thermodynamic conditions of its growth and to distinguish, for instance, between condensation of saturated vapor on initial nuclei in the cloud and capture of droplets. As hailstones were supposed to grow by successive layers, like belemnites in Urey and Epstein’s work on paleotemperatures,21 isotopic composition of these layers would provide information on the temperature of formation, and be related to thermodynamic and meteorological parameters in the cloud, provided fractionation conditions between vapor and solid phase could be established.22,23 i.

Experiments

ii.

Results, Further Models, and Discussion

Studies were first made by deuterium analysis, which the Saclay laboratory performed routinely. Hailstones came from many hailstorms. Hailstones 19 mm in diameter were first collected in France, and larger stones, 9– 11 cm in diameter, were collected later from North America. These were kept, sometimes for several years, in a dry ice box to avoid melting in case of failure of electrical power. To prevent sublimation they were placed in jars filled with a nonexchanging fluid. Finally cutting into small samples to obtain a three dimensional map of hailstones was done with a precision rotating saw. From the start isotope data were related to meteorological data from radiosonde probes, and to structural aspects obtained from examination of hailstone slabs under polarized light. Deuterium concentration of a large stone decreases first from the center to a relative minimum, and undergoes several subsequent fluctuations. Interpretation was made developing the MNR model.24 It follows the Chisholm claim that in hailstorm clouds the characteristic updraft core follows an adiabatic model. In addition it rests: (a) on conservation in an ascending air volume of the total mass of vapor plus condensed water, and of the amount of deuterium, and (b) on isotope equilibrium between vapor and condensed phases. This enables one to correlate deuterium concentration in the

394

Isotope Effects in Chemistry and Biology

condensed phase and temperature. Meteorological data convert temperatures into altitudes. At the lowest temperature, i.e., at the maximum altitude and the outside of the hailstone, all the vapor has been condensed and the solid-phase deuterium content is the same as that of the initial water vapor. The maximum temperature is at the base of the cloud and that is where the center of the stone is formed. The deuterium concentration of this center is just one alpha above that of this water vapor, because the amount of solid phase is small. Figure 13.1 where temperatures have been converted to altitudes illustrates the following results: hailstones start growing at the bottom of the cloud and, new information, are subject to downward and upward episodes. There is also a correlation between the direction of the vertical movement and structural observations. Interpretation of isotopic measurements gives the altitude of formation of each layer, which is combined with calculations of a maximum falling rate of stones to provide estimations of updraft velocities and growth time of stones. Validation of this first model was made by checking that oxygen 18 and deuterium fractionations were in the ratios of equilibrium factors for condensation. A specially built mass spectrometer source made it possible to analyze deuterium and oxygen 18 simultaneously on samples of a few cubic millimeters. Tritium analysis confirmed the general growth picture. Nevertheless in cases where the range of isotopic content in the hailstones is large, the MNR model leads to contradictions. Half a dozen conditions of validity of the MNR model were established, among which there was to be no mixing of the ascending parcel with environmental air, no depletion by drops and hailstones, etc. A new model was developed; the Isotope Cloud Model

11

200 500 600

Time (sec) 800 1000

1200

−30°

−25°

−20° 9 −15°

Dry growth Wet growth

8

Alberta B

7

5

10 15 Radius (mm)

20

Temperature (°C)

Z Altitude (km)

10

−10° −7°

−4°

FIGURE 13.1 Altitude of hailstone B from a storm in Alberta, Canada, versus its radius. Jouzel, Merlivat, and Roth,24 J. Geophys. Res. 80 (36) 5015– 5030 (1975). Reproduced by permission of American Geophysical Union.

Isotope Effects in the Atmosphere

395

(ICM), an improved Hirsch one-dimensional cloud model, was used.25 Calculations of the isotope content of five different forms of water were considered: vapor, cloud water, rainwater, cloud ice, and “graupel.” Hailstone collections were made in seven storms well documented by radar measurements. Growth rates of embryos and hailstones agreed generally with the storms’ radar structure. Many difficulties related to assumptions in the MNR model, such as that of constant isotope concentration at the base of clouds, were removed.25,26 To sum up, it was especially complex to identify and take into account every physical parameter influencing the observed isotope effects in the study of hail. Long and detailed calculations, required when trying to quantify the influence of each parameter, are found in the references, which contain also a number of general works on precipitation. 4. The d2 H – d18 O Relation in Precipitations Friedman27 had observed in 1953 a relation between the isotope composition of deuterium and 18O of various waters. In 1960, a precise linear relationship between 18O and deuterium distribution was found in ice of a Greenland iceberg.28 Craig29 gave a well-known relation for precipitation. It is largely empirical, although the coefficient 8 is close to the ratio of isotope effects at evaporation for deuterium and oxygen 18 around 258C:

dD ¼ 8d18 þ 10

ð13:7Þ

In 1964 Dansgaard described the fundamentals of the history of isotopes in precipitation. In particular, he found, from a series of meteorological stations in Greenland and the Atlantic, a relation between mean soil temperatures ta and mean isotope compositions dD and d18:

dD ¼ 5:6ta 2 100

and

d18 ¼ 0:695ta 2 13:6

ð13:8Þ

which, when combined, give the following:

dD ¼ 8:06d18 þ 9:6

ð13:9Þ

Moreover, based on the data for 34 continental stations in Western Europe, Near East, and North Africa stations, he found:

dD ¼ ð8:1 ^ 0:1Þd18 þ ð11 ^ 0:1Þ

ð13:10Þ

According to Dansgaard, mean isotopic weighted values for precipitation obey relation 13.10. This corresponds to a fractionation factor between vapor and liquid vapor essentially temperature dependent, and to isotope equilibrium established between vapor and liquid. Other precipitation conditions cause deviations from the slope. In dry atmospheres, where evaporation of rain drops occurs, the kinetic fractionation effect discussed above occurs, and this leads to a further enrichment in heavy isotopes of the remaining water, and to lines with a lower slope, which, in extreme cases, may shift to a value of two30 (and see Fritz and Fontes5 in Volume 2, Chapter 2). Factor 8 in Equation 13.7 or (8.1 ^ 0.1) in Equation 13.10 is related to condensation processes at ambient temperatures in temperate regions. The intercept d, or deuterium excess, at the origin of the line representing these relations, is related to evaporation conditions of the source of water vapor condensing in the cloud, usually the ocean. It has been interpreted by Merlivat and Jouzel,31 who consider evaporation from the oceans on a global scale to be under steady-state conditions. In addition to temperature, evaporation flux, precipitation, and flux fractionation coefficients, parameters of physical conditions are introduced, principally wind velocity at the water – air interface, water surface rugosity, and relative humidity. The isotopic composition of water vapor leaving the ocean surface is calculated. The isotope distribution in humid air, where condensed water is partially eliminated by precipitation, is evaluated and a table of deuterium excess at three relative humidity values, 0.85, 0.75, 0.60, is constructed. At around 258C, taken as the temperature at which the bulk of evaporation takes place, within natural ranges of variation, the dependence on temperature,

396

Isotope Effects in Chemistry and Biology

on evaporation, or on parameters such as wind velocity is small. When the source region warms up by 18C, d increases by approximately one delta unit. Thus deuterium excess d in Craig’s equation leads to a relative humidity of 81% and to a temperature of 268C, while Dansgaard’s relation gives 79.5% humidity and 25.48C. Lines drawn from analysis of Sahara ground waters older than 20,000 years have a deuterium excess of 5‰. The model indicates a paleo-humidity of the air over the ocean of 90% in comparison to around 80% presently. In summary, while Craig’s or Dansgaard’s relations are widely used to interpret precipitation conditions, d reflects meteorological source properties, mainly surface temperature and humidity.

IV. ARCHIVES OF ATMOSPHERIC ISOTOPIC EFFECTS RETAINED BY ICE CAPS Ice caps keep records of isotope effects on deuterium and oxygen 18 occurring in the atmosphere during precipitation and, to a lesser extent, of atmospheric effects affecting sulfur isotopes. Isotope abundances of 40Ar and 15N are used in temperature assessments. Isotope abundance variations of deuterium and oxygen 18 record, along a single ice core, seasonal, geographical, and altitude effects on the temperature of precipitation. Effects on sulfur isotopes are reflected in the isotope composition of sulfates of layers deposited on ice. Dating icecore levels makes a record of ice-cap climatic changes and of corresponding changes in air composition by analysis of imprisoned gas bubbles. Early deuterium or 18O measurements in Greenland and later in Antarctica on ice cores several tens of meters long recorded seasonal effects. Their amplitude, evaluated in deltas of deuterium, can be 20 to 40‰. They make it possible to evaluate accumulation rates. Ice may flow to the drilling point from a location with a different average isotope value. In Antarctica, a more than seasonal decrease of deuterium content at a depth of 106 m on an ice core extracted near the coast (Figure 13.2) signaled the origin of ice at some distance on the polar plateau.9,32 Later ice cores from deeper drillings gave evidence of isotopic abundance variations linked to climatic changes. Dating of temperature variations was made in several ways. Tritium at well defined depths or horizons was produced by fallout of the atmospheric nuclear explosions in the 1950s, and provided a check on years counted by seasonal oscillations of isotope signatures. Over longer periods, the count of seasonal variations becomes unreliable as their visibility decreases with compression of the ice and a certain amount of diffusion. Volcanic dust due to historically well-documented eruptions produces dated layers.31 In a more distant past temperature variations linked to cyclic modifications of the orbital course of the Earth, called the astronomic forcing, establish a time scale. In addition, stratigraphic observations, conductivity measurements, and other experiments are made. Temperature changes along isotope profiles are thoroughly discussed by Jouzel.34 Comparisons for geologically recent times with carbon 14 or Beryllium 10, or for past epochs with dating by 18O analysis in marine sediments, help to confirm the scale. The Vostok ice core reaches depths of more than 3300 m, where the ice is more than 420 000 years old. Dansgaard et al.35 gives many applications of dating methods and of climate assessment based on the record of preserved atmospheric isotope composition. Well-correlated temperature variations to ice ages provide information and climatic variations appear to be sometimes very brutal. A number of temperature changes of more than 108C in less than 100 years were observed; these are called Dansgaard – Oeschger events. These are part of measurements of deuterium at 10 cm intervals on the Vostok ice core36 corresponding to a time resolution of ca. 20 years. Extreme values of deltas of this ice core differ by about 80‰ in deuterium and 8‰ in 18O. Contemporary variations in polar ice have been correlated with the behavior of El Nin˜o current in Pacific equatorial ocean waters.35,37 Patris et al.38 have analyzed sulfur isotopes of ice cores in Antarctica39 and Greenland.38 Signatures of sulfur of various origins, preindustrial and anthropogenic, and in particular, fall-out from volcanic explosions, were identified. Volcanic plumes are known to reach the stratosphere

Isotope Effects in the Atmosphere

Profoundeur (m)

D/Hppm

397

132

134

136

140

20

40

60

80

1a 100 6D 200

− 180

− 170

− 160

− 150

− 140

− 160

FIGURE 13.2 Deuterium concentration versus depth of a106 m long firn core from Antarctica Y axis: depth in meters, top X axis deuterium in ppm of D/H ratios; lower X axis, deuterium In d versus SMOW. The first 98 m record seasonal variations of about ^2 ppm of D/H ratios. From 98 m down a sharp decrease in D concentration, of 1 ppm of D/H per meter, reflects climatic variations, or a different local origin of ice. Hagemann, R., Nief, G., and Roth, E., Teneur en deute´rium le long d’un profil de 106 m dans le ne´ve´ Antarctique. Application a` l’e´tude des variations climatiques. Earth Planet. Sci. Lett., 4, 237– 244, 1968. North Holland Publishing Comp., Amsterdam.

where nonmass-dependent fractionation of sulfur occurs during oxidation leading to differences between first and later fall-outs. Signatures of these nonmass-dependent fractionations were looked for in sulfur. Because sulfur concentration is of the order of 100 ng per gram in Antarctica, sulfur had to be collected from ice core sections several meters long, averaging compositions of early and late sulfur deposits. In Greenland, volcanic sulfur largely depleted in 34S may reflect heterogeneous oxidation of SO2 on large particles within the volcanic plume, but this requires further investigations. Savarino et al.40 have made a detailed study of mass-independent oxygen isotopic composition of volcanic sulfates collected in ice cores. They found evidence of dramatic changes in atmospheric oxidation processes following volcanic eruptions. Removal from the atmosphere by wet or dry deposition, either of rain or aerosol sulfates, are mass-dependent processes. Thus mass-independent sulfates result from atmospheric oxidation of SO2. This signature is due to the oxidizing agent that cannot be OH in the troposphere because it equilibrates rapidly with oxygen. A wet oxidation from H2O2 or O3 must be responsible. When volcanic dust is the fall out of emissions into the stratosphere where there is practically no water, OH becomes the oxidation agent. However, after very massive eruptions mass-independent stratospheric sulfates are not produced. This suggests that OH has been scavenged by volcanic dust and is no longer the oxidizing agent, as has also been proposed for other reasons. A different oxidation process must take place that has not yet been identified.

398

Isotope Effects in Chemistry and Biology

The NTP volume of air imprisoned in bubbles is about 10% of the ice volume. Greenhouse gases are analyzed in this air. Because isotope analysis of ice cores provides estimates of temperatures and rates of temperature changes in the past, the correlation of air bubble composition with isotope analysis is now central to studies of paleo-climates. It has thus been possible to investigate whether the increase in greenhouse gas concentration of the atmosphere trapped in bubbles was prior or consecutive to warming. A difficulty arises because bubbles are formed at the bottom of the firn, and may be younger than enclosing ice. At Vostok, because accumulation is small, this age difference could be as large as 7000 years with an uncertainty of 1000 years. Caillon et al.41,42 have linked Ar deltas in air bubbles to ice temperature, and showed that 15N deltas play a similar role. The fractionation mechanism is not clearly understood, but the method eliminates the problem of estimating age differences. At Vostok, the increase of CO2 concentration in bubbles lagged behind a rise in temperature by an estimated 800 ^ 200 years, confirming that the deglaciation that produces CO2 emissions is probably initiated by some insolation forcing. The CO2, by amplifying the initial orbital forcing, plays a key role in the greenhouse effect.42 In Greenland, an increase in CO2 precedes deglaciation, to whose initiation it probably has contributed. An increase of methane in air is nearly simultaneous with a well-dated fast rise in temperature, which also provides an evaluation of the time lag between the rise in temperature in Vostok and in Greenland. In summary, simultaneous analysis of deuterium and 18O in ice cores provides information on the intensity and the temperature of precipitations in the past. The intercept of the d18 O=dD line gives the temperature and humidity of that part of the ocean serving as a source of atmospheric water vapor. Analysis of ice cores and air bubbles in Greenland and Antarctica is used when establishing general circulation models for the past. Their importance stems from the confidence gained in applying models to the evolution of the present climate when data on paleo-climates are well represented by models.

V. ISOTOPIC EFFECTS ON ATMOSPHERIC CARBON IN THE CARBON CYCLE Carbon isotope mole fractions in the reference material V PDB are 12 C ¼ 0:988944, 13 C ¼ 0:011056, ^ 0.000028. The isotope carbon cycle is more complex than that of water isotopes, and almost dominated by kinetic effects, in photosynthesis, carbonate synthesis, etc. The regulating reservoir, atmospheric CO2, is 40,000 billion tons, and thus much less important than the ocean for water as the total mass of mobile water on Earth is 150 £ 1018 tons. Carbon exists on the Earth in three forms: native, as graphite and part of coal, oxidized as carbonates and carboxylic acids, and reduced as hydrocarbons and derivatives. Most of the chemical reactions between carbon compounds are redox reactions. In the oxidized range carbonates dominate. For reduced carbon, apart from the rare occurrence of carboxylic acids and alcohols and their polymers, polycyclic molecules such as lignin and derivatives are major components. In the atmosphere, CO2 is rather homogeneous, except under active vegetation covers where air circulation is poor.43 Minor components, CH4 at about 1 ppmv, and CO at 70 to 120 ppbv, undergo fractionation.

A. ISOTOPES OF ATMOSPHERIC M ETHANE An in-depth discussion of all topics related to isotopic studies of atmospheric methane was published by Quay et al.44 The information presented in the following review is largely based on this publication.

Isotope Effects in the Atmosphere

399

1. Sources The annual fluxes and isotopic compositions of the major sources of atmospheric methane are shown in Table 13.1. The flux values in this table were based on a scenario45 using several constraints including loss rates of the sinks of OH oxidation, uptake in soils, and loss in the stratosphere.46 The ratio of modern to fossil sources is based on 14CH4 data,47 and independent estimates of the emissions of 35 ^ 17 Tg yr21 from biomass burning.48 The strengths of these sources are categorized as biogenic 76 ^ 12%, fossil 18 ^ 9%, and biomass burning 6 ^ 3%.44,45,49,50 2. Discussion Between 1990 and 1995 the mean global d13C of atmospheric methane was 2 47.33 ^ 0.04‰ with an average difference of þ 0.23‰ between the northern and southern hemisphere,44 it was increasing at 0.024 ^ 0.10‰ yr21 globally as well as in both hemispheres. Because of the isotopic fractionation effect of the main removal mechanism, i.e., by OH oxidation, and taking this effect to be 2 5.4‰, experimentally measured,64 the average d13C of the sources was 53.4 ^ 2‰. 3. Removal Processes There are three processes for the loss of atmospheric methane: reaction with OH radicals, uptake in soils, and loss to the stratosphere. The rates of loss are 540 ^ 35, 30, and 6 Tg yr21, respectively. The main removal mechanism by OH has just been cited. Oxidation of methane to carbon monoxide by action of molecular oxygen on methyl radicals to form first methylperoxy has been investigated by Weston.65 A kinetic isotope effect enriches the methylperoxy on 18O; later stages do not lead to isotopic fractionation of oxygen.

TABLE 13.1 Annual Emissions and Average Isotopic Values of Sources of Atmospheric Methane Annual Emissionsa Source Biogenic Wetlands Ruminants Rice paddies Landfills Fossil Natural gas Coal mining Biomass burning Total flux (g)

TgCH4 yr

21

13

12

d C/ C (‰)b

14

CH4 (pM)c

dD (‰)d

232 ^ 14 90 ^ 10 69 ^ 12 40 ^ 8

260 ^ 5 260 ^ 5 263 ^ 5 250 ^ 2

116 ^ 5 120 ^ 5 112 ^ 5 120 ^ 5

2320 ^ 20 2300 ^ 10 2320 ^ 30 2310 ^ 10

70 ^ 14 33 ^ 5 41 ^ 6

243 ^ 7 236 ^ 7 224 ^ 3

0 0 130 ^ 5

2185 ^ 20 2140 ^ 20 2225 ^ 5

580 ^ 28

—

—

—

Reprinted from Ref. 44. Data from: a 45; b 47,51–58,61; c 54,56,59,60,62; d measured in the late 1980s and early 1990s in Refs. 47,56,57,62. PM means percent modern carbon.

400

Isotope Effects in Chemistry and Biology

4. Atmospheric

14

CH4

The 14CH4 of atmospheric CH4 is contributed from biogenic sources of wetlands (natural and rice paddies), at , 112 –120 ^ 5% of modern carbon content of 14C (pM), fossil fuel sources from which pM ¼ 0; biomass burning , 130 ^ 5 pM; and pressurized light water reactors 25– 30% of the annual atmospheric flux of 14CH4.44 The atmospheric mean global value from 1990 to 1995 was 124 pM,44 which is in accord with the dominant source from wetlands. 5. Atmospheric dD The mean dD of atmospheric CH4 between 1989 and 1995 was 2 86‰. Quay et al.44 estimated the fractionation of the overall loss rate at 2 218 ^ 50‰, which would lead to dD of the sources of 2 275% ^ 41‰ taking into account the annual trend of the CH4 concentration. B. Isotopes of Atmospheric Carbon Monoxide Two very comprehensive discussions of the isotopic composition of atmospheric carbon monoxide (CO) have been published by Brenninkmeijer et al.66 and Conny.67 The following review will present a brief summary of these. 1. Sources and Sinks The tropospheric budget of CO with the source strengths and respective isotopic composition and the sinks with its corresponding isotopic fractionation amounts are shown in Table 13.2.66 TABLE 13.2 The Tropospheric Budget of CO, with the Source Strengths and Respective Isotopic Compositions and the Sink Rates with the Corresponding Isotopic Fractionation Constants Source/Sink Source

21

Tg yr

13

d C (‰) V-PDB

14

CO (pMC)

d18O (‰) V-SMOW

d17O (‰)

23.5a,b ,16.3b ,18d

0.0 0.0 —

0b,c 14.9d ,0b,c

0? 0?

Fossil fuel combustion Biomass burning

300 –500 300 –700

227.5a 221.3e 224.5

0.0 ,115 —

CH4 oxidation NMHC oxidation

400 –1000 200 –600

252.6h 232.2d

,125 ,110

Ozonolysis Biogenic Oceans Total Sources Sinks Reaction with OH Soil uptake Loss to stratosphere

80 –100g 60 –160 20 –200

,13.5i

,110 ,110

25 –40 0.0

1800 –2700 1400 –2600 250 –640 ,100

5j

10j

10k

4

Reprinted from Ref. 66, Copyright 1999, with permission from Elsevier. Data from: a 69; b 72; c 73; d 74; e 75; f 76; g 77; h value based on the CH4 value of d13C at 247.2‰,77 and the amount of fractionation for CH4 þ OH 5.4‰, i79; jat atmospheric pressure; kinferred from the value for 13C, assuming mass-dependent behavior. V-PDB and V-SMOW are reference materials calibrated by IAEA versus PDB (an extinct sample of Peedee belemnite) and Craig’s SMOW. pMC, as well as PM, means percent modern carbon; NMHC are nonmethane hydrocarbons.

Isotope Effects in the Atmosphere

401

The major direct sources are incomplete combustion from engines and biomass burning; comparable amounts of CO are produced in the oxidation of CH4 and nonmethane hydrocarbons by OH. Reaction with OH is the principal loss rate; for 12C/13C the reaction rate ratio of k12C/k13C can vary from 0.997 to 1.008 depending on pressure; for 16O/18O it is almost constant at 0.990.68 2. Atmospheric Concentration and Isotopic Composition Because of the relatively short residence time of CO of 1 to 3 months, depending on the seasonal variation of oxidation by OH free radicals, the atmospheric concentration and isotopic composition are variable depending on season, hemisphere, and geographic proximity to the main continental combustion sources, from autos and burning of agricultural and forest biomass.66 In the southern hemisphere the concentration varies between 45 and 65 ppb and the 13CO isotopic composition from approximately 2 32 to 2 15‰. In the northern hemisphere the concentration is higher at , 150 to 300 ppb depending on the season, while the 13C/12C isotopic ratio is in the range of , 2 29 to 2 25‰.66 In the highest latitudes of the southern hemisphere, Stevens et al.69 and Brenningkmejier et al.66 found CO depleted in 18O with respect to atmospheric oxygen (whose d is ca. 2 30‰) with respect to VSMOW by 20‰ and more than 30‰, respectively. Weston65 calculated a kinetic fractionation effect in the oxidation of methane by OH leading to an enrichment in 18O. This disagreement calls for more studies. The concentration of CO measured at the west coast of Oregon has declined in the last decade.70,71 3. Summary Analysis of the atmospheric cycle for the fluxes of CO from the different source species and its seasonality and geographic variations are made complex by the short lifetime, the varied seasonal and geographic distributions, transport effects, and even the fractionation effect of the main sink of oxidation of CO by OH radicals, which has a large pressure dependence and so varies with altitude. Such an analysis requires 3D modeling with adequate resolution in time and space. There are an abundance of isotopic measurements, too numerous to describe here, that have been carried out on all the isotopic species of atmospheric CO: 13CO, C18O, 14CO, and C17O. Atmospheric C17O undergoes mass-independent enrichment as a result of the CO þ OH sink process in which the surviving CO gains excess 17O.79 The conclusions of these measurements, as enumerated by Brenninkmeijer et al.,66 are presented as follows: 13

CO

a. Local combustion sources generally do not give a clear signal. b. The annual cycle in the southern hemisphere shows the effect of enhanced methane oxidation in summer. As a rule d 13C has a clear annual cycle; with little scatter. c. The kinetic isotope fractionation effect (KIE) seems to generally have a small impact; it may be noticeable in the northern hemisphere where the effects of the CH4 play a lesser role. d. The reaction of CH4 þ Cl can be detected in the stratosphere and troposphere thanks to the large KIE combined with the 13C depletion of CH4. e. Generally d 13C values in the remote southern hemisphere are not as low as would be expected from the importance of the CH4 oxidation. There may be some fractionation in the CO formed relative to the parent CH4. Before drawing conclusions, laboratory measurements on CO from CH4 þ OH are necessary. C18O a. Combustion sources possess a distinctly enriched signal which can be used with success for calculating its contribution accurately.

402

Isotope Effects in Chemistry and Biology

b. The large kinetic fractionation offers the possibility of assessing the degree to which CO has been removed by OH. c. The impact on 18O from biomass burning-derived CO in the remote southern hemisphere is relatively small. d. The d(C18O) value of CO from CH4 and nonmethane hydrocarbon oxidation needs to be measured under controlled conditions. C17O a. CO in background air possesses mass-independent enrichment in the oxygen isotopes, which is witnessed by a small but clear excess in 17O. b. All previously published 13C data need a correction. c. Ozonolysis is a small source, producing comparatively strong mass-independent enriched (MIF) CO. d. The major source for MIF is CO þ OH, which makes this signal a direct measure for exposure to OH.80 14

CO

a. b.

14

CO measurements can be performed routinely on samples as small as 100 l. CO provides a very detailed picture of the effects due to transport combined with its removal by OH. c. It is probable that the biogenic 14CO can be adequately determined based on the concentration and stable isotope measurements. d. There is an as yet unexplained difference in the isotopic composition between the northern and southern hemispheres. 14

C. ISOTOPES OF ATMOSPHERIC C ARBON D IOXIDE The 13C composition of CO2 varies in air and may serve to identify its sources, but the main reaction accompanied by isotope effects in the atmosphere is an exchange reaction in high-temperature gases depleting methane in 13C by around 20 delta units with respect to the associated CO2.81 Oxygen in CO2 equilibrates rapidly with the oxygen in water.

VI. ISOTOPE EFFECTS OF ATMOSPHERIC NITROGEN Nitrogen has two stable isotopes of mole fractions 14N, 0.996337,7 15N, 0.003663.7 Atmospheric nitrogen is homogeneous within analytical uncertainties82 and is used as the origin of deltas. The inventory of nitrogen 15 abundance in the natural environment has been reviewed by Le´tolle,83 and isotope variations are found in Hu¨bner84 in precipitation among others. Variations are due to differences between sources of nitrogen entering the atmosphere. Redox processes monitor the nitrogen cycle just as for carbon, and the nitrogen cycle essentially exists through the intervention of plants and bacteria which possess the Nife gene. This permits direct fixation of atmospheric nitrogen in organic compounds, and partly by direct assimilation of NHþ 4 by plants. All known chemical steps logically favor the exit of 14N. The destruction of nitrogen organic compounds gives ammonia, which is either reincorporated directly in metabolic chains or oxidized to urea, nitrogen oxides, nitrite, and then nitrate ions. The last step to get back to molecular nitrogen is denitrification by bacteria such as Sulfobacter denitrificans. N2O, a 312 ppbv component emitted from marine and terrestrial ecosystems, shows a d15N of þ 15‰,85 but no isotope effect is reported taking place in it during the nitrogen cycle which parallels closely the carbon isotope cycle.

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VII. ISOTOPE EFFECTS OF ATMOSPHERIC OXYGEN As the isotope composition of atmospheric oxygen is homogeneous except for local fractionation effects in O2 production by photosynthesis or by respiration, many laboratories use air oxygen as the origin of deltas instead of VSMOW, but it is understandable that nonmass-dependent isotope fractionations were first observed on compounds of oxygen, the lightest element having three stable isotopes. In the troposphere effects are observed in O2, O3, CO, CO2, and N2O molecules.

A. AIR O XYGEN Luz et al.86 establish that ocean water has a þ 155 per meg D17O with respect to air (see notations in Section XI: one meg is 1026 delta unit). Ocean water is biologically equilibrated with CO2, as confirmed by laboratory experiment, but oxygen air has a nonmass-dependent isotope composition. It is explained by transfer of CO2 from the stratosphere having a þ 1280 D17O per meg with respect to air (Figure 13.3).

B. OZONE Krankowsky et al.87 have measured ozone isotope composition at ground level of an unspecified site over several months. The ozone concentration was in the range 10 to 100 ppvb. Its average delta of 18O, with respect to air, was þ 9.1 ^ 0.2% and that of 17O þ 7.1 ^ 0.3% showing an indisputable nonmass-dependent fractionation. Laboratory experiments from other authors are in agreement with these observations, but no known mechanism of ozone formation could explain them. Johnston and Thiemens88 did similar experiments at three different locations in California over periods of 1 to 3 months. At the three sites, enrichment in 18O varied, with maximum deltas with respect to air around 90‰. The isotope line in Pasadena had a slope of 0.54 ^ 0.1 implying mass-dependent fractionation. At White Sand Missile Range (WSMR) it was approximately 1.1 ^ 0.3, definitely nonmass-dependent. La Jolla results gave a 0.7 ^ 0.1 slope. The only marked differences between sites are the daily ozone concentrations that fluctuate around 40 ^ 10 ppbv at La Jolla or 36 ^ 10 at WSMR, but present a midday peak of 120 ppbv at Pasadena. No correlation to meteorological parameters other than these that reflect differences in the ozone budget have been found. No interpretation is yet established, but contrary to much higher enrichments of stratospheric ozone, these are comparable to enrichments produced in laboratory experiments. There is a useful review on ozone by Mauersberger et al.89

C. NITROUS O XIDE All samples of atmospheric N2O collected at La Jolla between January 1993 and February 1997 were independently fractionated by an excess D17O of 1 unit. It could be caused by the production of a small fraction of the nitrous oxide by oxidation by ozone having an enrichment of 100‰. No final explanation is proposed however.90

D. ATMOSPHERIC S ULFATES Some aerosol sulfate samples and rain water sulfates collected at La Jolla presented excess D17O of about 1.6 unit.91 The anomaly must be due to the only oxidants in the atmosphere that can transfer mass-independent signatures: O3 and H2O2. Others equilibrate too rapidly with water. Implications for past atmospheres are discussed.

VIII. ISOTOPE EFFECTS OF ATMOSPHERIC SULFUR From Coplen et al.,7 mole fractions of sulfur isotopes (uncertainties in parentheses) in the reference material V CDT, are 32 S ¼ 0:9503957 (90), 33 S ¼ 0:0074865 (12), 34 S ¼ 0:0419419 (87), 36 S ¼ 0:0001459.25

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Isotope Effects in Chemistry and Biology

Stratosphere Photochemical transfer of 17O and 18O from O2 via O3 to CO2 (∆17OCO2=1280 ∆17Oo =−1.5) 2

CO2 flux = 0.408(1280)

CO2 flux = 0.408(155)

O2 flux = 306 (−1.5)

O2 flux = 306 (0)

Troposphere (∆17OCO2=155 ∆17Oo =0) 2

CO2 fluxes Rapid hydration-dehydration H2O-CO2 exchange in ocean and leaf water eliminates the stratospheric anomaly of CO2

O2 flux = 2.97(0)

O2 flux = 2.97(155)

Respiration Photosynthesis

Biosphere

FIGURE 13.3 Simplified O2, CO2 and D17OCO2 cycle Ds per meg with respect to tropospheric oxygen, are specified or given in parenthesis without a D. Fluxes are in unit of 1016 mol yr21. The D value of tropospheric CO2 is taken as equal to D17O of the ocean, because the stratospheric anomaly is eliminated by exchange. The net positive anomaly of CO2 from the stratosphere has the same magnitude as the negative anomaly flux in O2, which is balanced by the positive anomaly from the biosphere. Luz et al.,86 reprinted by permission from Nature 400 August, 547– 550 (1999). Copyright 2004 Macmillan Publishers Ltd.

A. INTRODUCTION Sulfur abundance in the atmosphere is low but has almost doubled as a consequence of industrial activities, and this has important implications for the environment. Sulfur is introduced into the atmosphere under reduced, oxidized, or elemental form. Sources range from volcano emissions and products of industrial processes, to bacteria-induced reduction oxidations on land and in the oceans. Emissions, chemical reactions, or bacterial actions are generally accompanied by either kinetic or equilibrium isotope effects producing marked differences in the isotope abundance of 34S between different chemical species entering the atmosphere or, for a given source, according to location or season. S isotope effects are not easily observed in the atmosphere, either because of sampling difficulties or because end compounds of total transformations have, necessarily, the same isotope composition as starting molecules. Isotope analyses of oxygen usefully complement those of sulfur in the study of oxidations. Evidence of nonmass-dependent effects in the Precambrian atmosphere has recently been given.92 Publications on sulfur isotopic abundance and isotopic effects try partly to unravel the complex reactions that maintain the sulfur cycle, partly to find the origin of sulfur at given locations and to help fight atmospheric pollution. Two books provide fundamental data and nearly exhaustive compilations of results at the date these were published.5,6,93,94 While giving a schematic account of transformation reactions, we report only isotope effects measured on atmospheric constituents.

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B. TURNOVER AND I NVENTORY A global emission of 102 Tg yr21 of sulfur gases entering the atmosphere was estimated in 1980: 76% due to fuel combustion and industrial activities, 12% to marine biosphere, 9% to volcanoes, 2% to biomass burning, and 1% to terrestrial biosphere. Anthropogenic sources account for 84% of emissions in the Northern Hemisphere and for 50% in the Southern Hemisphere.95 Higher figures are found in Krouse and Grinenko94 with a comparable breakdown, adding that the average sulfur over the oceans has remained constant since the preindustrial era. According to Ryaboshapko96 and Coplen et al.7 total sulfur in the atmosphere is between 2.64 and 6.45 Tg. Concentrations vary according to the chemical nature of compounds, location, and season. Reduced forms over low productivity locations can be as low as 0.05 ^ 0.3 mg m23. SO2 concentrations over urban sites may rise to 20 ^ 10 mg m23. The concentration range at a sampling height from 10 to 25 km is 0.015 to 0.1 mg m23. Table 13.3 gives concentrations and residence times of different forms of sulfur in the troposphere.

C. NATURE, I SOTOPIC C OMPOSITION, AND ATMOSPHERIC C HEMISTRY OF S ULFUR Whereas thousands of atmospheric sulfur isotope abundance analyses are made to pinpoint sources of sulfur in the atmosphere or to study the transport of sulfur, taking isotope effects on sulfur into consideration is rarely necessary. The main sulfur gases entering the atmosphere are SO2, from combustion, industrial sources and volcanoes, dimethylsulfide (DMS) from biota, and H2S principally from biota. Concentrations of COS, CS2, mercaptans, and dimethyldisulfide are small. Figure 13.4 (from Newman et al.97) summarizes spreads of d values of different sources of atmospheric sulfur. The spread of d values on each line reflects seasonal and local variations and isotope effects that may occur before introduction in the atmosphere, such as large ones associated with biogenic emissions. The total spread of d34S on Earth is about 150.8,98 Isotope effects in atmospheric reactions are tied to oxidizing reactions from reduced forms to SO2, and SO2 is oxidized to SO3 and sulfates. The transformation of reduced to oxidized forms takes place under the influence of OH radicals, photons, or in droplets by O2 molecules in the presence of water and metals acting as catalysts or by oxidants such as ozone or hydrogen peroxide. Reactions are accompanied with isotope effects, as proved in the laboratory. Reaction rates vary, as do residence times of reduced compounds as seen in Table 13.3. Reaction 13.11 shows the initial

TABLE 13.3 Concentration and Residence Time of Different Forms of Sulphur in the Troposphere Species DMS H2S SO2 SO22 4 CS2 COS

Concentration (mg S m23)

Residence time (d)

0.04 0.05 0.2 0.5 0.7 0.2

,1 1 3 4 70 500

L Newman, H.R. Krouse and V.A.Grienko, Sulphur isotope variations in the atmosphere Scope 43 chapter V p.134 SCOPE 43 2 Stable Isotopes: Natural and Anthropogenic Sulphur in the Environment, edited by H. R. Krouse and V. A. Grinenko, copyright Scientific Committee on Problems of the Environment 1991, John Wiley & Sons Ltd, Chichester, UK.

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Isotope Effects in Chemistry and Biology

{ { Marine

Continental

SO2

Urban air Marine aerosols − 40

Sea spray Volcanic gases

{ {

H2S, DMS: Bacterial H2S, DMS: Plant decay H2S, DMS: Plant decay H2S, DMS: Bacterial

{

Coal Oil, Gas Ore

{

SO2

SO42 −

{ − 20

0 d 34S ( )

+ 20

+ 40

FIGURE 13.4 Variations of d34S values for different sources of atmospheric sulphur compounds. (Newman, L., Krouse, H. R., and Grienko, V. A., Sulphur isotope variations in the atmosphere Scope 43 Chapter V p. 149 SCOPE 43-Stable Isotopes: Natural and Anthropogenic Sulphur in the Environment, Krouse, H. R. and Grinenko, V. A., Eds., copyright Scientific Committee on Problems of the Environment 1991, Wiley, Chichester, UK.)

attack of H2S by an OH: OH þ H2 S 7 ! HS þ H2 O

ð13:11Þ

HS produced goes quantitatively to SO2 by one of several further paths. SO2 that is not removed from the atmosphere by dry deposition (about one half the content) is oxidized, hydrated, and condensed, though SO2 may not be oxidized before some droplets fall. Reactions 13.12 and 13.13 are possible examples only of first steps of homogeneous oxidation. OH þ SO2 7 ! HOSO2

ð13:12Þ

HOSO2 þ O2 7 ! HOSO2 OO

ð13:13Þ

Oxidation proceeds to sulfate further, for instance in droplets. The transformation of SO2 to sulfate could be accompanied, in laboratory experiments, by a 2% isotope effect on 34S. In stack plumes, analysis of both species has never shown more than a few per mil enrichment in 34S sulfates. Oxidation of biogenic H2S generates sulfates of isotopic composition similar to that of the starting molecule, whereas ambient SO2 is about 5‰ depleted.94 Sampling difficulties are an obstacle to better determinations. In contradistinction to sulfur isotopic composition of sulfate that does not change much in those processes, oxygen isotope composition may vary by more than 1% depending on relative amounts of oxygen coming from O2 or H2O molecules. Within plumes from hot stacks, it has been shown that in the generation of sulfates the proportion of oxygen from H2O with respect to oxygen from O2 increased with the distance of the stack. Seasonal 18O fluctuations in sulfates have been detected which parallel those of 18O in rain. Sulfur isotopic signatures of marine aerosols point to a possible long-range transport of sulfate particles through the troposphere.98 The last development in atmospheric isotope chemistry of

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407

sulfur is the discovery of nonmass-dependent effects in atmospheric sulfate aerosols99 following the discovery of such effects in terrestrial material a few years earlier.

D. EFFECTS DURING R EMOVAL OF S ULFUR FROM THE ATMOSPHERE One half of SO2 is removed by dry deposition, one half by oxidation, hydration, and/or condensation at rates varying with location, season, etc., at a rate of around 1 to 3% per hour. Sulfur dust falls rapidly before being oxidized, and no isotope effect is entailed in the process. Isotope composition of sulfur in dust is difficult to interpret because of mixing with large quantities of detritic sulfate from gypsum carried by wind. 1. Archean Isotope Atmospheric Chemistry of Sulfur and Nonmass-Dependent Isotope Effect Total isotopic sulfur analysis of Precambrian sulfates reported in Farquhar et al.92 has shown nonmass-dependent isotope effects. Suggestions attributing some effects to direct bacterial action were refuted100 and later interpreted.101 The basic explanation backed by precise laboratory experiments102 is to assign these effects to oxidation of SO2 by photochemical reactions. In archean epochs these were made possible by higher intensities of UV radiation in the lower atmosphere due to lower oxygen concentration entailing lower ozone concentration. In more recent times, layers of volcanic dust in ice cores whose ages cover the past 400,000 years may, when exhibiting nonmassdependent effects, identify gases from eruptions entering the stratosphere.42 Laboratory experiments establish the influence of different wavelengths of ultraviolet radiation producing nonmass-dependent effects on 33S and 36S, sometimes of different signs. Information on the composition of the atmosphere in oxygen and organic molecules, in altitude and at different epochs, is thus derived from these experiments and new measurements. It has also been shown that information carried with isotopic composition that precipitate on Earth from the atmosphere could be further transferred to other molecules, for instance to sulfates, possibly by bacterial action. Farquhar et al.103 has reviewed the collected evidence of mass independent sulfur isotope signatures in the geological record and have compared results with laboratory experiments. These conclude that once an isotopic signature is passed on to a given reservoir it is preserved unless there is addition of sulfur with a different composition. Implications, especially on studies of the evolution of the Earth’s atmosphere, are developed. Mechanisms of sulfur nonmass-dependent effects are discussed by R. Weston in Chapter 12.

IX. ISOTOPE EFFECTS ON ZINC AND LEAD IN THE ATMOSPHERE Zinc: Isotope 66: the abundance in volcanic gases of the Merapi Volcano in Indonesia varied in a 1‰ range. The reference used was JMC 3-0749 L. Further work on sampling techniques is planned (A. Nonel, Observatoire Midi-Pyre´ne´es, personal communication). Lead: the lead isotopic composition entering the atmosphere is characteristic of the source. During transport in the atmosphere, the interaction and coagulation of particles are evidenced by lead isotope analysis coupled with carbon isotope analysis that distinguishes organic from inorganic phases.104

X. DEUTERIUM ENRICHMENTS IN THE ORGANIC MOLECULES OF THE INTERSTELLAR MEDIUM The discovery of high deuterium enrichments in the organic molecules of the interstellar medium;105 D/H £ 1000 relative to the Universal D/H ratio, i.e., relative to D/H ¼ 1.5 £ 1024,106 and D/H £ 12

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in the macromolecular organic matter of the carbonaceous meteorites107,108 had brought to light the exceptional efficiency of ion –molecule reactions in isotopic fractionation processes. Such extreme isotopic enrichments must correspond to temperatures so low (, 100 K) for an isotope exchange that most chemical reactions can be excluded because of its endothermicity or, if exothermic, because these possess an activation energy. Ion – molecule reactions being barrierless constitute the major class of reactions able to overcome this constraint. In such reactions, a charged species encounters a neutral molecule under the influence of a long-range attractive potential. When exothermic, they are rapid at low temperatures down to 10 K.109 The rate constants of these ion – molecule reactions are typically in the range of 1 to 3 £ 1029 cm3 sec21.110 Such rates are at least 10 orders of magnitude higher than those for neutral reactions at room temperature and, contrary to the Arrhenius behavior of neutral reactions, are independent of the temperature. Let us take the example of the reaction: n

AB þmAC ¼ mAB þnAC

ð13:14Þ

where A, B, and C are chemical elements and m and n refer to the atomic masses of the various isotopes of A. The equilibrium coefficient is the isotopic fractionation factor designated by a(T):

aðTÞ ¼

kf ½m AB ½n AC ½m A=n A ¼ n ¼ kr ½ AB ½m AC ½m A=n A

B C

ð13:15Þ

with kf and kr the rate constants for the forward and reverse reactions, respectively. For ion – molecule reactions a(T) is expressed as

aðTÞ ¼ f ðB; mÞ expðDE=TÞ

ð13:16Þ

DE is the exothermicity expressed in K ðDE ¼ DE0 =kÞ; f(B,m) refers to the ratio of partition functions and, in most cases, is close to the usual molecular symmetry factor.111 Since this factor is implicitly expressed in isotope ratios, the symmetry of the molecules involved in the isotopic fractionation reactions can be ignored and a broad approximation of a(T) can be written as:

aðTÞ ¼ expðDE=TÞ

ð13:17Þ

In the literature, the physical interpretation of Equation 13.17 is often reported as the following: the reverse reaction of Equation 13.14 is negligibly small relative to the forward rate and thus the isotopic enrichment is entirely accounted for by the forward rate. In the case of ion – molecule reactions, this forward rate derives from the cross section for capture by an attractive potential which pertains to the long-range interaction between a charged ion and a neutral species; hence the high value of this forward rate at all temperatures. Let us take two classical examples of deuterium isotopic exchange reactions (e.g., Millar et al.112): þ Hþ 3 þ HD ¼ H2 D þ H2 þ CHþ 3 þ HD ¼ CH2 D þ H2

ðDEH Þ ðDEH Þ

ð13:18Þ ð13:19Þ

The products H2Dþ and CH2Dþ are then transferred to stable molecular species by a series of subsequent reactions which cause almost no additional isotopic fractionation. The rate constants of these reactions give a typical time to reach the equilibrium of 1014 sec at 10 K for an interstellar cloud gas density of 103 H cm23. For Reaction 13.18 calculated DEH values lie between 180 K105 and 230 K,113 230 K being the value adopted by Millar et al.112 For Reaction 13.19 calculated DEH values lie between 370 K113

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409

and 290 K (0.025 eV).114 Snell and Wooten115 determined an overall DEH value of 240 ^ 60 K based on DNC/HNC ratios measured in 13 interstellar clouds and for which the fractionation was assumed to be dominated by Reactions 13.18 and 13.19. In the case of the nitrogen isotopes, Terzieva and Herbst111 have calculated DEN values (the subscript N stands for nitrogen). These range from 2.25 to 35.9 K for: 15 þ N15 N þ HNþ 2 ¼ N2 þ H NN

ð13:20Þ

and 15

N þ HCNHþ ¼ N þ HC15 NHþ

ð13:21Þ

respectively, all other reactions lying between these two extreme values. A recent application of this theoretical approach has been proposed by Ale´on and Robert.116 To interpret the isotopic variations observed in the solar system organic matter embedded in carbonaceous meteorites, in comets, and in the interplanetary dust particles (IDP) collected in the Earth’s stratosphere. Results are reported in Figure 13.5 in which aN(T) and aH(T) were calculated using Equation 13.17 and the linear relations between log[aH(T)] and log[aN(T)] are reported for different values of DEN with DEH ¼ 240 K. The different equilibrium temperatures are indicated on these lines. Available isotopic determinations on natural samples are reported in Figure 13.5, so that theoretical predictions can be compared with solar system data. According to this figure, the isotopic variations in H and N may result from a common isotopic fractionating pathway taking place at different temperatures during the formation of the solar system. 0.7 0.6

∆E

Log (αN)

0.5

N

EN

∆

0.4

53

=

=

∆E N

K

43

=3

K

3K

0.3 50

0.2

70 190

0.1 0

150

90 110

PSN 250

0

0.5

1

1.5 Log (αH)

2

2.5

3

FIGURE 13.5 Isotopic fractionation of nitrogen versus hydrogen in organic molecules of meteorites (open symbol; Robert et al.117), Interplanetary dust particles (gray symbol; Ale´on et al.118; Messenger119) and Comets (black symbol; Meier et al.120; Arpigny et al.121). The fractionation is calculated relative to the value of the protosolar nebula (for N cf. Hashizume et al.122; for H cf. Geiss and Gloeckler106). Dotted line corresponds to theoretical values assuming a DEH ¼ 240 ^ 60 K. DEN that fit observational data are reported above each line. Graduations correspond to the temperature of isotopic equilibrium in K. Error bars are mostly due to the uncertainties on the initial protosolar nebula values (shown as bold lines in the bottom left corner).

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Isotope Effects in Chemistry and Biology

XI. CONSTRAINTS IN USING DELTAS, CAPITAL DELTAS, AND REFERENCE SAMPLES Isotope abundance variations in nature are usually made by comparing aRx, abundance ratio of isotope a to the more abundant isotope of the element studied in sample x, to aRr in reference material r. The ratio a Rx =a Rr is near unity of the form 1 þ 1 except for some hydrogen compounds. The use of ratios of two isotopes only to identify a sample increases precision by avoiding determinations of abundances of all isotopes, in particular of small ones. The fact that 17O was not easily measured was historically an argument for the adoption of deltas. The ratio to a reference ratio eliminates many systematic errors that might mar single ratio measurements. Delta (d), defined by Equation 13.22 (in ‰ unless specified), is easier to handle than a Rx =a Rr :

dðaÞ ‰ ¼

a a

Rx 2 1 £ 1000 Rr

ð13:22Þ

Writing the excess over one with three decimal digits, and naming it 1; Equation 13.23 follows:

dðaÞ ‰ ¼ 10001

ð13:23Þ

A first constraint upon the use of deltas is that they are not additive. Equation 13.3 expresses dðxÞs ; delta of sample x versus standard s, as a function of dðxÞr ; delta of sample x versus a secondary reference r; and of dðrÞs ; delta of r versus s:

dðxÞs ¼ dðxÞr þ dðrÞs þ dðxÞr £ dðrÞs =1000

ð13:24Þ

Nevertheless, when mixing a volume V1 of delta d(1) with a volume V2 of the same compound of delta d(2), the delta of the mixture, d(3), is the arithmetic mean

dð3Þ ¼

V1 dð1Þ þ V2 dð2Þ V1 þ V 2

ð13:25Þ

The multiplicative term in Equation 13.25 may contribute several delta units to large deltas of hydrogen or carbon. It also enters precision determinations of small deltas. For example, the PDB scale for ds of carbon being tied to an extinct sample of the Pee Dee belemnite, carbon abundance is usually measured against a reference prepared by the IAEA, with a delta value of 1.95‰ versus PDB. In conversion to the PDB scale, the multiplicative term affects even the second decimal place of measured deltas of 5‰. Other uncertainties arise when inhomogeneous or multiple references are used: Canon Diabolo Troilite (CDT) for sulfur presents deltas of 0.4‰ between different parts.7 For oxygen 18 air and V-SMOW are used as references, and d(18O air) is 22.960‰ versus V-SMOW. Constraints upon characterizing nonmass-dependent isotope effects with D. In an equilibrium reaction, if masses of exchanging isotopes are m and m0 , a mass-dependent 1 is closely proportional to ðm0 2 mÞ=mm0 .123 In a three isotope equilibrium reaction where masses of isotopes are m1 ; m2 ; m3 ; and isotope 1 is used as reference for Rð2Þ1 and Rð3Þ1 ; 1ð2Þ will be proportional to ðm2 2 m1Þ =m2 m1 ; and 1ð3Þ to the corresponding ratio. Thus 1ð3Þ m 2 m1 m ¼ 3 £ 2 1ð2Þ m2 2 m1 m3

ð13:26Þ

dð3Þ 1 ¼ 3 dð2Þ 12

ð13:27Þ

and from Equation 13.23

Thus, in an equilibrium reaction of oxygen, d(17O) versus

16

O would be 0.53 d(18O) versus

16

O.

Isotope Effects in the Atmosphere

411

A nonmass-dependent effect D(3) in a three-isotope reaction is Dð3Þ ¼ dð3Þ measured 2 dð2Þ £

m3 2 m1 m £ 2 m2 2 m1 m3

ð13:28Þ

Calculations from formulae proposed, e.g., in Romero and Theimens,99 differ at most by a factor 12 from the above. Ds, differences of ds per mil, are expressed permils of ds, by some authors in a D per meg notation.86 We use it by convenience. It must be remembered that initial measurements are ratios of isotope peaks and that deltas use ratios of ratios, and therefore 1 meg represents a millionth of such ratios of ratios. Thus, precision to one D per meg is not significant at the present state of the art, because of the uncertainties in measurements and of the difficulty of calculating exactly massdependent deltas. Either one uses Equation 13.26, and minor factors other than masses are omitted and mass numbers are used instead of exact masses, or one draws a d(3) versus d(2) line, called a three isotope line, of mass-dependent samples of different isotope composition. The slope of this line is used instead of mass ratios in Equation 13.21. In either procedure, an uncertainty of 5‰ induces, in D(3), an uncertainty of 5 D per meg multiplied by d(2). Such considerations lead, for instance in Luz et al.,86 authors to considers to a 155 D per meg not to be significantly different from a 184 D per meg.

A. POSSIBLE E VOLUTION OF M EASUREMENTS OF I SOTOPE E FFECTS More and more reference samples are getting exhausted, making reference to original scales increasingly unreliable. It does not affect the validity of series of data published by one laboratory at a given time, but it certainly prevents making inter-laboratory comparisons of data with precision, and comparing older and recent data, even perhaps from the same laboratory. The discovery of nonmass-dependent isotope effects leads to measurements of the abundance of every isotope of elements having three or more isotopes. Measured abundances should be accurate, or “absolute.” The availability of reference samples, certified in absolute isotope compositions, would enable laboratories to achieve accuracy by using them to calibrate their mass spectrometers. Scales based on absolute values would become permanent. This could be done because mass spectrometers can analyze with precision even rare isotopes, which was not possible when deltas were introduced. When they are available, it might help to start publishing isotope absolute ratios in addition to deltas, as is done in Coplen et al.5

ACKNOWLEDGMENTS We thank Dr. Ralph Weston Jr. for his help in reading our manuscript in detail, for correcting many errors and for suggesting useful additions.

REFERENCES More than 50,000 papers on stable isotope in earth sciences have been published; references are unable to cover exhaustively even the subject of isotope effects in the atmosphere. Review papers are quoted to palliate this difficulty. 1 Giauque, W. F. and Johnston, H. L., An isotope of oxygen of mass 18, J. Am. Chem. Soc., 51, 1436, 1929. 2 Giauque, W. F. and Johnston, H. L., An isotope of oxygen of mass 17 in the Earth’s atmosphere, Nature, 123, 831, 1929. 3 Urey, H. C., Brickwedde, F. G., and Murphy, G. M., A hydrogen isotope of mass 2, Phys. Rev., 39, 164– 165, 1932.

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Isotope Effects in Chemistry and Biology 4 Brenninkmeijer, C. A. M., Janssen, C., Kaiser, J., Rockmann, T., Rhee, T. S., and Assonov, S. S., Isotope effects in the chemistry of atmospheric trace compounds, Chem. Rev., 103, 5125– 5161, 2003. 5 Fritz, P. and Fontes, J. Ch., Eds., Handbook of Environmental Isotope Geochemistry, Vol. 3, Elsevier, Amsterdam, 1980. 6 Coplen, T. B., Isotopic composition of the elements, Pure Appl. Chem., 73, 667– 683, 2001. 7 Coplen, T. B. et al. Isotope abundance variations of selected elements (IUPAC Technical Report), Pure Appl. Chem., 74(10), 1987– 2017, 2002. 8 Roth, E., Critical evaluation of the use and analysis of stable isotopes (IUPAC Technical Report), Pure Appl. Chem., 69(8), 1753– 1828, 1997. 9 Lorius, C., Hagemann, R., Nief, G., and Roth, E., Teneur en deute´rium le long d’un profil de 106 m dans le ne´ve´ Antarctique. Application a` l’etude des variations climatiques, Earth Planet. Sci. Lett., 4, 237– 244, 1968. 10 Stable isotope hydrology deuterium and oxygen18 in the water cycle, STI/DOC/10/210, pp. 340, IAEA, Vienna, 1981. 11 Gat, J. R., Oxygen and hydrogen isotopes in the hydrologic cycle, Annu. Rev. Earth Planet. Sci., 24, 225– 262, 1996. 12 Le´tolle, R. and Olive, P., A brief story of isotopes in hydrology, In The Basics of Civilisation, Water Science?, Proceedings of Unesco/IAHS/WAS symposium, Rome, December 2003, IAHS pub. 286, 49 – 66, 2004. 13 Majzoub, M., Fractionnement en oxyge`ne 18 et en deuterium entre l’eau et sa vapeur, J. Chim. Phys., 68, 1423– 1436, 1971. 14 Craig, H., Gordon, L. I., and Horibe, G., Isotope exchange effects in the evaporation of water—low temperature experimental results, J. Geophys. Res., 68, 5079– 5087, 1963. 15 Craig, H. and Gordon, L. I., Deuterium and oxygen 18 variations in the ocean and the marine atmosphere. Stable isotopes in oceanic studies and paleotemperatures, Cons. Nat. del. Ric. Lab. Geol. Nucl. Pisa, 1965. 16 Bonner, F. T., Roth, E., Schaeffer, A. O., and Thomson, S. G., Chlorine 36 and deuterium study of Great Basin Lake waters, Geochim. Cosmochim. Acta, 25, 261– 266, 1962. 17 Rodriguez, M., Almecija, C., Cruz-San Julian, Benavente, J., Caballero, E., and Jimenez de Cisneros C., Stable isotopes and major ion chemistry of continental salt lakes from Southern Spain, IAEA-CN104; IAEA-CN-104/P-164, 2003. 18 Dansgaard, W., Stable isotopes in precipitation, Tellus, 16, 436– 468, 1964. 19 Rozanski, K., Sonntag C., and Muennich, K. O., Factors controlling stable isotope composition of European precipitation, Tellus, 34, 142– 150, 1982. 20 Hoffmann, G., Jouzel, J., and Masson, V., Stable water isotopes in atmospheric general circulation models, Hydrol. Process., 14, 1385– 1406, 2000. 21 Epstein, S., Buchsbaum, R., Lowenstam, H. A., and Urey, H. C., Revised carbonate-water isotopic temperature scale, Geol. Soc. Am. Bull., 64, 1315– 1326, 1953. 22 Merlivat, L., Nief, G., and Roth, E., E´tude de la formation de la greˆle par une me´thode isotopique, C.R. Acad. Sci. Paris, 258, 6500– 6502, 1964. 23 Merlivat, L., Nief, G., and Roth, E., Formation de la greˆle et fractionnement isotopique du deute´rium, Abh. Deut. Akad. Wiss., Berlin, 7, 839– 853, 1965. 24 Jouzel, J., Merlivat, L., and Roth, E., Isotopic study of hail, J. Geophys. Res., 80(36), 5015– 5030, 1975. 25 Federer, B., Brichet, N., and Jouzel, J., Stable isotopes in hailstones. Part I. The isotope cloud model, J. Atmos. Sci., 39(6), 1323– 1335, 1982. 26 Federer, B., Thalmann, B., and Jouzel, J., Stable isotopes in hailstones. Part II. embryo and hailstone growth in different storms, J. Atmos. Sci., 39, 1336– 1355, 1982. 27 Friedman, I., Deuterium content of natural waters and other substances, Geochim. Cosmochim. Acta, 15, 218– 228, 1953. 28 Dansgaard, W., Nief, G., and Roth, E., Isotopic distribution in a Greenland Iceberg, Nature, 135, 232, 1960. 29 Craig, H., Isotopic variations in meteoritic waters, Science, 133, 1702, 1961. 30 Fontes, J. C. and Gonfiantini, R., Comportement isotopique au cours de l’e´vaporation de deux bassins sahariens, Earth Planet. Sci. Lett., 3, 258– 266, 1967.

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75 Conny, J. M., Verkouteren, R. M., and Currie, L. A., Carbon 13 composition of tropospheric CO in Brazil: A model scenario during the biomass burn season, J. Geophys. Res., 102, 10693, 1997. 76 Conny, J. M., The isotopic characterization of carbon monoxide in the troposphere, Atmos. Environ., 32(14 – 15), 2609 –2683, 1998. 77 Rockmann, T., Measurement and interpretation of 13C, 14C, 17O, and 18O variations in atmospheric carbon monoxide, Ph.D. thesis, University of Heidelberg, Heidelberg, 1998. 78 Quay, P. D., King, S. L., Stutsman, J., Wilbur, D. O., Steele, L. P., Fung, I., Gammon, R. H., Brown, T. A., Farwell, G. W., Grootes, P. M., and Schmidt, P. H., Carbon isotopic composition of atmospheric CH4, Global Biogeochem. Cycles, 5(1), 25 – 47, 1991. 79 Manning, M. R., Brenninkmeijer, C. A. M., and Allan, W., Atmospheric carbon monoxide budget of the southern hemisphere: implications of 13C/12C measurements, J. Geophys. Res., 102(D9), 10673– 10682, 1997. 80 Rockmann, T., Brenninkmeijer, C. A. M., Saueressig, P., Bergamaschi, J., Crowley, U., Fischer, H., and Crutzen, P., Mass independent fractionation of oxygen isotopes in atmospheric CO due to the reaction CO þ OH, Science, 281, 544– 547, 1998. 81 Deines, P., The isotopic composition of reduced carbon, In Handbook of Environmental Isotope Geochemistry, Vol. 1, Fritz, P. and Fontes, J. Ch., Eds., Elsevier, Amsterdam, p. 384, 1980. 82 Mariotti, A., Germon, J., Hubert, P., Kaiser, P., Letolle, R., Tardieux, A., and Tardieux, P., Experimental determination of nitrogen kinetic isotope fractionation, Soil Sci. Soc. Am. J., 60(79), 1 – 800, 1981. 83 Le´ tolle, R., Nitrogen 15 in the natural environment, In Handbook of Environmental Isotope Geochemistry, Vol. 1, Fritz, P. and Fontes, J. Ch., Eds., Elsevier, Amsterdam, pp. 407– 409, 1980. 84 Hu¨bner, H., Isotope effects of nitrogen in the soil and biosphere, In Handbook of Environmental Isotope Geochemistry, Vol. 2, Fritz, P. and Fontes, J., Eds., Elsevier, Amsterdam, pp. 362– 363, 1986. 85 Sowers, T., N2O record spanning the penultimate deglaciation from the Vostock ice core, J. Geophys. Res., 106(D23), 31903– 31914, 2001. 86 Luz, B., Barkan, E., Bender, M. L., Thiemens, M. H., and Boering Kristie, A., Triple-isotope composition of atmospheric oxygen as a tracer of biosphere productivity, Nature, 400(5), 547– 550, August, 1999. 87 Krankowsky, D., Bartecki, F., Klees, G. G., Mauersberger K., Schellenbach, K., and Stehr, J., Measurement of heavy isotope enrichment in tropospheric ozone, Geophys. Res. Lett., 22(13), 1713– 1716, 1995. 88 Johnston, J. C. and Thiemens, M. H., The isotopic composition of tropospheric ozone in three environments, J. Geophys. Res., 102(D21), 25395– 25404, 1997. 89 Mauersberger, K., Krankowsky, D., and Jannsen, C., Oxygen isotope processes and transfer reactions, Space Sci. Rev., 106, 265– 279, 2003. 90 Cliff, S. S. and Thiemens, M. H., The 18O/16O and 17O/16O ratios in atmospheric nitrous oxide: A mass-independent anomaly, Science, 278, 1774– 1776, 1997. 91 Lee, Ch. C. W., Savarino, J. and Thiemens, M. H., Mass independent oxygen isotopic composition of atmospheric sulfate: origin and implications for the present and past atmosphere of earth and Mars, Geophys. Res. Lett., 28(9), 1783–1786, 2001. 92 Farquhar, J., Bao, H., and Thiemens, M., Atmospheric influence of earth’s earliest sulfur cycle, Science, 289, 756– 758, 2000. 93 Ivanov, M. V. and Freney, J. R., Eds., The Global Biogeochemical Sulphur Cycle, Scope Report 19, Wiley, New York, 1983. 94 Krouse, H. R. and Grinenko, V. A., Eds., Stable Isotopes, Natural and Anthropogenic Sulphur in the Environment, Scope Report 43, Wiley, New York, 1991. 95 Spiro, P. A., Jacob, D. J., and Logan, A., Global inventory of sulfur emissions with 18 p 18 resolution, J. Geophys. Res., 97(D5), 6023– 6036, 1992. 96 Ryaboshapko, A. G., The atmospheric sulphur cycle, In The Global Biogeochemical Sulphur Cycle, Scope Report 19, Ivanov, M. V. and Freney, J. R., Eds., Wiley, Chichester, UK, pp. 224– 230, 1983. 97 Newman, L., Krouse, H. R., and Grienko, V. A., Sulphur isotope variations in the atmosphere, In Stable Isotopes, Natural and Anthropogenic Sulphur in the Environment, Scope Report 43, Krouse, H. R. and Grinenko, V. A., Eds., Wiley, Chichester, UK, pp. 132– 176, 1991.

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14

Isotope Effects for Exotic Nuclei Olle Matsson

CONTENTS I. II.

Introduction ...................................................................................................................... 417 Isotope Effects with Short-Lived Radionuclides............................................................. 418 A. Fluorine Kinetic Isotope Effects .............................................................................. 418 B. Carbon Kinetic Isotope Effects................................................................................ 418 III. Synthesis of Compounds Labelled with Short-Lived Radionuclides ............................. 419 A. Labelling with 11C ................................................................................................... 419 B. Labelling with 18F.................................................................................................... 419 IV. Kinetic Methods — A Combination of Liquid Chromatography and Liquid Scintillation ................................................................................................... 420 V. Determination of Rate-Limiting Steps ............................................................................ 421 A. Using Leaving Group F KIEs — Nucleophilic Aromatic Substitution.................. 421 1. The Effect of Solvent on the Rate-Limiting Step............................................. 422 2. The Effect of Steric Hindrance on the Rate-Limiting Step.............................. 422 B. Concerted or Stepwise Reaction? The Use of F KIEs and Double Labelling for a Base-Promoted Elimination ............................................................................ 422 VI. Probing Transition-State Structure — Nucleophilic Aliphatic Substitution .................. 423 A. Relative Carbon KIEs .............................................................................................. 423 B. Labelled Central atom: Probing Steric Effects........................................................ 424 C. Labelled Nucleophile ............................................................................................... 425 1. The Effect of Substitution in the Substrate....................................................... 425 2. The Effect of Substitution in the Leaving Group............................................. 426 VII. The Determination of Secondary Deuterium KIEs by the Aid of Radioactive Carbon... 426 VIII. Secondary Carbon KIE in a Proton-Transfer Reaction................................................... 427 IX. Carbon Isotope Effects for Enzyme-Catalysed Reactions .............................................. 428 Acknowledgments ........................................................................................................................ 428 References..................................................................................................................................... 428

I. INTRODUCTION This chapter describes the use of the short-lived radionuclides 11C and 18F in the study of reaction mechanisms. The isotopes of the elements are utilized in different ways in the investigation of reaction mechanisms: (i) as tracers and (ii) by determining kinetic isotope effects (KIEs), i.e., effects on reaction rate caused by isotopic substitution. The study of KIEs1 yields answers to two fundamental types of mechanistic questions: (i) Which atoms undergo a rate-limiting bonding change (forming or breaking of bonds)? Do the bonding changes take place in a concerted fashion or proceed stepwise? Which step is rate limiting in a stepwise mechanism? 417

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Isotope Effects in Chemistry and Biology

(ii) What is the structure of the activated complex? Is the transition state (TS) constant or does it vary when the system is perturbed by, say, substitution? Is the TS reactantlike or product-like, or is it ‘symmetric’? Do bonding changes take place synchronously or not? There has been a tremendous development in the field ever since the pioneering work by Bigeleisen, Westheimer and Melander at the end of the 1940s. The applications today range from simple gas-phase reactions2 to complex enzyme reactions,3 and a large number of organic reactions have been studied with KIEs. Great progress has been made regarding experimental techniques as well as theoretical understanding and methodology; even the small but significant effects of heavy elements may be determined with high precision.4 A line of development which will be addressed in this chapter is the possibility of using accelerator-produced short-lived radionuclides — exotic nuclei — in the study of isotope effects on reaction rate. Certain short-lived radionuclides are today largely used in biomedical research and clinical diagnosis in connection with so-called positron emission tomography (PET).5 The nuclides which have been utilized are mainly 11C (t1/2 ¼ 20.4 min), 18F (t1/2 ¼ 109.7 min), 13N (t1/2 ¼ 9.96 min), 15 O (t1/2 ¼ 2.1 min) and 76Br (t1/2 ¼ 16 h). The discussion in this chapter focusse on the use of 11 C and 18F, which both decay almost exclusively by positron emission (98.1% bþ and 97% bþ, respectively), thus producing g-photons by annihilation.

II. ISOTOPE EFFECTS WITH SHORT-LIVED RADIONUCLIDES A. FLUORINE K INETIC I SOTOPE E FFECTS Organofluorine chemistry is a very active field; some recent monographs are listed in reference.6 In investigations of chemical reaction mechanisms fluoride ion is commonly employed, e.g., as a leaving group in elimination and substitution reactions. It was therefore important to try to include fluorine among the heavy atoms used for KIE measurements. Since natural fluorine consists of 100% of the isotope 19F and since no long-lived radioisotopes are available, the only way to accomplish determination of F KIEs is to use a short-lived radionuclide. Among these the accelerator-produced 18F has a convenient half-life of 109.7 min, and is routinely produced in many laboratories; the isotope 18F is used in the labelling of radiopharmaceuticals and other compounds used for biomedical research and clinical diagnosis utilizing the PET imaging technique.7 Nucleophilic as well as electrophilic labelling reagents are available and quite a number of compounds have been labelled with 18F; these include, e.g. carbohydrates, alkyl halides, fatty acids, and steroids (see Section III.B below).7,8 The maximal 18F/19F KIE for breaking of a carbon –fluorine bond has been estimated to be ca 3% based on the ratio of reduced masses for isotopic diatomic oscillators (12C – 18F and 12C – 19F, respectively) and a C – F stretching frequency of 1250 cm21.9

B. CARBON K INETIC I SOTOPE E FFECTS The advantage of using the short-lived carbon isotope 11C in combination with the long-lived radioisotope 14C is that the observed isotope effect is then maximised as compared to the ordinarily used 12C/13C or 12C/14C KIEs. Heavy-element isotope effects like those for carbon are small so it is very valuable to increase the mass ratio in order to determine the KIEs with the highest possible precision. This is particularly important when studying the often small changes of the KIEs caused by systematic variations such as change of substituent, different steric requirements, or choice of solvent that are generally performed in investigations of reaction mechanisms.

Isotope Effects for Exotic Nuclei

419

The main drawbacks with this approach are that special equipment is needed (e.g., cyclotron for the nuclide production, laboratory for radiochemical work, etc.), radiation protection is necessary, and the radionuclides used have short half-lives.

III. SYNTHESIS OF COMPOUNDS LABELLED WITH SHORT-LIVED RADIONUCLIDES Most traditional organic chemistry methods can be adapted to the synthesis of radiolabelled compounds.7,8,10 However, modifications of the original synthetic method are often needed for application to labelling chemistry and sometimes new synthetic strategies must be designed. Normally the synthetic route is chosen so that only a few rapid steps remain when the radionuclide has been incorporated. Synthesis of tracers labelled with short-lived radionuclides involves production of the radionuclide and the labelling precursor, synthesis of the tracer molecule, purification, and analysis. When synthesising compounds labelled with short-lived radionuclides, one of the most important parameters is time. While the build-up of the product is governed by kinetics of the chemical transformation, there is always a competing decay of the radionuclide. Consequently the optimal reaction time with regard to radiochemical yield is determined by both of these two parameters.11 Some examples of special technical approaches that have proven to be useful in radiolabeling syntheses are: remote- or processor-controlled devices; one-pot and on-line procedures; miniaturization; on-column preparations; sonication and microwave-assisted syntheses; and supercritical media.10 The labeling syntheses of the particular substrates used in the work described in this chapter are reported in the original papers cited.

A. LABELLING

WITH

11

C

The most commonly used starting material in 11C synthesis is [11C] carbon dioxide.10 It is usually produced by the bombardment of nitrogen gas with high-energy protons, the 14N(p,a)11C nuclear reaction. Trace amounts of oxygen gas in the target gas combine with the high-energy 11C atoms to produce [11C] carbon dioxide as a primary precursor.10 This primary precursor is then generally converted to a more reactive secondary precursor which can be further used to label a molecule of interest. Some of the secondary precursors available from [11C] carbon dioxide are summarised in Figure 14.1.

B. LABELLING

WITH

18

F

Introduction of 18F into a molecule can be performed by electrophilic or nucleophilic addition of the radionuclide.8 A common method for 18F preparation is the 18O(p,n)18F nuclear reaction. This can be adapted to production of either [18F] F2 or [18F] fluoride ion. 11CO 11CHO

RCH211COOM 11CO

H11CN

11CH 4

FIGURE 14.1 Some useful

R11CH2I

2

11CH

11

R11CH2OH

3OH

11CH

R11CH2COCl

3I

C labelled secondary precursors obtained from [11C] carbon dioxide.

420

Isotope Effects in Chemistry and Biology

Electrophilic fluorination with [18F] F2 provides a facile means of introducing 18F into electron-rich compounds such as alkenes or aromatic rings. Because of its rather unspecific reactivity the use of electrophilic fluorine is limited. Improvements have been achieved by deactivating the fluorine by dilution with an inert gas or formation of new fluorinating agents, [18F] CH3COOF being the most useful.8 Nucleophilic fluorinations with [18F] fluoride ion are usually accomplished via two main approaches: † †

Nucleophilic displacement of a halide or a sulphonate in unhindered aliphatic systems. Nucleophilic aromatic substitution of a suitable leaving group such as nitro or trialkylammonium in activated aromatic systems.

Since the nucleophilicity of the fluoride ion is low in aqueous and protic solvents due to strong solvation, reactions are normally performed in aprotic solvents. Phase transfer reagents such as quaternary ammonium salts or a potassium counterion complexed by macrocyclic ethers or the kryptand Kryptofix 2.2.2 are used to activate and increase the solubility of the fluoride.8

IV. KINETIC METHODS — A COMBINATION OF LIQUID CHROMATOGRAPHY AND LIQUID SCINTILLATION The methodology for determining KIEs using short-lived radionuclides is based on the separation of reactants and products by liquid chromatography followed by radioactivity measurements using liquid scintillation. The method for determination of carbon KIEs has certain advantages in addition to the already mentioned fact that the largest practical mass ratio of carbon isotopes is used: (i) the HPLC technique is usually easily applied to different chemical systems; (ii) no workup or degradation of the samples is required; (ii) the analyses are insensitive to unlabelled impurities as long as these does not cause any quenching of scintillation; (iii) both isotopic species can be quantitatively determined with high precision using the same instrument; (iv) the large difference in half-life for the two carbon isotopes used simplifies the measurement. Experiments with 11C and 18F require rapid execution and the experiments have to be carefully prepared before starting of the synthesis. The determination of a 11C/14C KIE typically involves the following steps:12 The kinetic reaction is started by mixing the 11C-labelled substrate, which has been prepared immediately prior to the kinetic experiment, and the previously synthesized 14C-substrate with any other necessary reactants. At suitable time intervals samples are withdrawn and analyzed by HPLC. The fractions containing the labelled reactant and product are collected in scintillation liquid. The HPLC instrument is equipped with a radioactivity detector as a complement to the UV detector. The total radioactivity (11C þ 14C) of each fraction is immediately measured by liquid scintillation counting, usually for 1 –2 min. After decay of the 11C, typically the next day, the 14 C-radioactivities of the samples are measured for long enough to keep the statistical error at a tolerable level. The 11C-radioactivity is obtained as the difference between the total and the 14 C-radioactivity and is then corrected for radioactive decay. Having these data at hand, the KIE for each sample is calculated as KIE ¼ lnð1 2 f11 Þ=lnð1 2 f14 Þ where f11 and f14 are the fractions of reaction for the isotopic reactions, calculated from the radioactivities for reactant and product fractions. For the fluorine KIEs9,13 the protocol is basically the same, except that, of course, 19F is not radioactive. The unlabelled substrate may, under favorable conditions, be quantitatively detected

Isotope Effects for Exotic Nuclei

421

by means of the HPLC UV detector provided that careful use of an internal standard and repeated injections are performed.9,13 Remote labeling might be a way to increase precision in certain cases. By labeling the 19 F-substrate with 14C in a position remote from the reacting center it is possible to determine both 18F- and 19F-labeled molecules by liquid scintillation counting. This strategy was used in a study of a HF elimination reaction (see below Section V.B).15

V. DETERMINATION OF RATE-LIMITING STEPS A. USING L EAVING G ROUP F KIEs — N UCLEOPHILIC A ROMATIC S UBSTITUTION Leaving group KIEs are fairly easy to interpret and have been utilized for a long time in mechanistic investigations of nucleophilic aliphatic substitution16 and elimination reactions.17 A leaving group KIE is expected to increase monotonously with increasing degree of bond breaking between the isotopic leaving group atom and the a-carbon in the transition state of the rate-limiting step. Thus sulfur, oxygen, and chlorine leaving group KIEs have been reported.16 Fluoride has been employed as the leaving group in many other mechanistic studies. Displacement reactions on activated aromatic molecules have been the subject of considerable mechanistic interest over the years.18 The generally accepted mechanism for nucleophilic aromatic substitution of activated substrates (the SNAr mechanism) is an addition– elimination and involves the formation of a Meisenheimer type of intermediate.18 Whether the rate-limiting step of this mechanism is the formation of the intermediate or expulsion of the leaving group has been found to depend on the character of the nucleophile and the leaving group as well as on the solvent. Decomposition of the intermediate may be base-catalysed, as indicated in Scheme 14.1 (k3[B]). The observation of base catalysis has been used as a mechanistic criterion of whether the formation or the decomposition of the intermediate is rate limiting. The nucleophilic substitution of 2,4-dinitrofluorobenzene (DNFB) with secondary amines appeared to be a good candidate as a model system for the demonstration of a fluorine KIE. The existence of a significant leaving group F KIE of 1.0262 ^ 0.0007 for the reaction of DNFB with piperidine in tetrahydrofuran (THF) at 308C (Scheme 14.2) unequivocally demonstrates that C –F bond cleavage is rate limiting in that system.9 The value is close to what is expected from F NO2 +

+

NHR1R2 NO2

F k1

NHR1R2

−

k−1

NO2

NR1R2

k2

NO2 +

k3[B] NO2

NO2

SCHEME 14.1

F

H NO2

N NO2

N +

NO2

SCHEME 14.2

+ NO2

HF

HF

422

CH3

Isotope Effects in Chemistry and Biology

NH2

+

NO2

F

DMSO

CH3

NH

+

NO2

+

HF

NO2

NO2

CH3

CH3 NH2

NO2

+

NO2

F

DMSO

NO2

NH

HF

NO2

SCHEME 14.3

an estimate of the maximal Section II.A).9

18

F/19F KIE for complete loss of zero-point energy (see above

1. The Effect of Solvent on the Rate-Limiting Step The leaving group F KIE probe offers an opportunity to learn whether the rate-limiting step is affected by change of solvent. On the basis of an investigation of base catalysis, Nudelman earlier concluded that a shift from rate-limiting elimination to rate-limiting addition takes place when the solvent is changed to one with slightly hydrogen-bond accepting properties, e.g., acetonitrile, from one lacking such properties, e.g., THF.19 The resulting isotope effect when changing the solvent is therefore interesting and permits conclusions regarding the rate-limiting step to be drawn; it also further proves that the observed KIE is real and not an artifact. The isotopic rate constant ratio obtained from the kinetic experiments in acetonitrile was 0.9982 ^ 0.0004.13 Thus, in acetonitrile no significant fluorine KIE is observed, although the small inverse value determined might be attributed to the very small secondary effect expected for ratelimiting formation of the intermediate involving sp2 to sp3 rehybridization. 2. The Effect of Steric Hindrance on the Rate-Limiting Step For reaction of methylanilines with DNFB it has been reported that the rate was reduced by a factor of 198 when the position of the methyl substituent was changed from para to ortho in the nucleophile (Scheme 14.3).20 Onyido and Hirst ascribed this to a steric effect and from studies of base-catalysis they concluded that a change in the rate-limiting step was induced by the steric effect of the o-methyl group.20 A change in the rate-limiting step induced by changing the steric properties of the nucleophile should be confirmed by determination of the F KIEs. The KIE determined in DMSO at 308C was 1.0005 ^ 0.0030 for 4-methylaniline and 1.0119 ^ 0.0037 for 2-methylaniline.21 The significant F KIE observed for the reaction between DNFB and the sterically more hindered nucleophile 2-methylaniline shows that expulsion of the nucleofuge is at least partially rate limiting. The F KIE for the reaction of 4-methylaniline is virtually nil and is thus consistent with rate-limiting addition of the nucleophile to the substrate.

B. CONCERTED OR S TEPWISE R EACTION? T HE U SE OF F KIEs AND D OUBLE L ABELLING FOR A B ASE- PROMOTED E LIMINATION The elimination of hydrogen fluoride from the fluoroketone shown in Scheme 14.4 is base promoted and, as demonstrated by Schultz and, coworkers, also antibody catalyzed.22 The observation of a significant primary deuterium KIE rules out the stepwise E1 mechanistic alternative (i.e., that with rate-limiting carbocation formation).14,22 The mechanism is thus either

Isotope Effects for Exotic Nuclei F

423 O

O base

+ F

O2N

O2N

SCHEME 14.4 F

O

F +B

k1 k−1

O2N

O

O + BH

O2N

k2

+F O2N

SCHEME 14.5

concerted (E2) or stepwise (E1cB). Using the leaving group KIE as a mechanistic probe was initially thought to discriminate between these alternatives. A significant but rather small F KIE of 1.0047 ^ 0.0012, which is consistent with the E2 as well as the E1cB mechanisms, was observed using acetate base (75% methanol(aq); 388C).14 However, by using a double isotopic fractionation methodology to determine the F KIE for a substrate where deuterium has been substituted for leaving protium in the 3-position, it was possible to show that the mechanism was stepwise (see Scheme 14.5). The deuterium substitution selectively slows down the reversal to reactant (k21), thus making fluoride detachment less rate limiting. The observed value of 1.0009 ^ 0.0010 for the F KIE for the deuterated substrate therefore provides very strong evidence in favour of the stepwise mechanism.15 For a concerted mechanism the F KIE would not be expected to be affected by deuterium substitution.

VI. PROBING TRANSITION-STATE STRUCTURE — NUCLEOPHILIC ALIPHATIC SUBSTITUTION A. RELATIVE C ARBON KIEs The first 11C/14C KIE reported was the primary carbon isotope effect for the methylation of N,Ndimethyl-p-toluidine with methyl iodide12,23 (Scheme 14.6). One of the reasons for choosing this reaction in that methodological study was that a fairly large 12C/14C KIE of 1.117 ^ 0.011 (in methanol at 48.58C) had been reported by Buist and Bender.24 At 308C the 11C/14C KIE was found to be 1.202 ^ 0.008.12 In another study hydroxide ion was used as nucleophile. In 50% dioxane/water at 258C the 11C/14C KIE was determined to 1.192 ^ 0.001.25 As exemplified by these two SN2 reactions of methyl iodide, where different nucleophiles have been used, the primary carbon KIE is large, which demonstrates that the method can undoubtedly be used as a tool for obtaining mechanistic information. Carbon isotope effects in the reaction of hydroxide ion with methyl iodide have earlier been studied by Bender and Hoeg,26 who determined the 12C/14C KIE to be 1.088 ^ 0.010 in 50% dioxane/water at 258C, and by Lynn and Yankwich,27 who determined the 12C/13C KIE to 1.035 ^ 0.006 in water at 318C. The latter value is smaller than expected from the 12C/14C KIE H3C

SCHEME 14.6

N(CH3)2 + ∗CH3I

MeOH 30°C

H3C

N+(CH3)2∗CH3 I−

424

Isotope Effects in Chemistry and Biology

value using a simple theoretical model presented by Bigeleisen,28 according to which the relative strengths of 12C/13C and 12C/14C KIEs are expressed by Equation 14.1. r ¼ lnðk12 =k14 Þ=lnðk12 =k13 Þ < 1:9

ð14:1Þ

It has been suggested that this deviation was caused by experimental errors or because the KIEs were determined in different solvents.29 A corresponding relation between 11C/14C and 12C/14C KIEs can be derived yielding the value of r < 1.6. From the experimentally determined 11C/14C KIE, the 12 14 C/ C KIE can thus be predicted to be 1.116. Again the experimental value (1.088) is significantly smaller than the one predicted from the theoretical model. Most experimental data reported regarding the relative strength of carbon KIEs have been shown to conform with Equation 14.1,30 but this is obviously not the case for the reaction between hydroxide ion and methyl iodide. Applying the relation above to the reaction of N,N-dimethyl-p-toluidine with methyl iodide yields a predicted 12C/14C KIE of 1.122 calculated from the experimentally determined 11C/14C value. This is, within the experimental errors, equal to the KIE observed by Buist and Bender (1.117).

B. LABELLED C ENTRAL ATOM: P ROBING S TERIC E FFECTS The quaternization reaction of tertiary amines by alkyl substrates (the Menshutkin reaction) has been extensively studied in solution31 with respect to effects of variations in substituents, leaving group, solvent, etc., and reactivity– selectivity relations have been considered as indicators of structural variations of the transition state. The rates of Menshutkin reactions using pyridines as nucleophiles are strongly affected by the size of substituents in the ortho positions. Hence, the Menshutkin reaction is one for which steric effects have been systematically studied, and for which steric and electronic effects can be separated.32 KIE techniques have earlier been applied to the Menshutkin reaction also with respect to steric effects. Thus primary 35Cl/37Cl leaving group KIEs were determined for the quaternizations of triethylamine and quinuclidine33 and of pyridine and 2,6-lutidine34 by methyl chloride. Such KIEs are in principle easily interpreted since they decrease monotonically with the magnitude of the stretching force constant for C– X within the TS. However, these KIEs are very small and difficult to measure accurately, and in the mentioned cases show opposite results with regard to the anticipated steric effect. One KIE probe which might yield information on TS variation due to steric perturbation is for the primary central carbon KIEs. These KIEs were determined for the quaternization reactions of pairs of tertiary amines, which exhibit essentially the same electronic contribution. Thus variations in reaction rate are sterically induced. The resulting KIEs determined in acetonitrile at 30.008C are 1.220 ^ 0.009 and 1.189 ^ 0.012 for 2,6-lutidine and 2,4-lutidine, respectively.35 These observed primary 11C/14C KIEs are large for both reactions, close to the maximal as estimated on the basis of loss of zero point energy in going from initial state to TS. As for the case of deuterium KIEs on hydron transfer, maximal primary carbon KIEs are expected for “symmetric ” TS where donor and acceptor are bound with equal strength to the isotopic atom in transfer.36 Furthermore, a small increase in primary carbon KIE is observed for 2,6-lutidine as compared to 2,4-lutidine. Thus steric hindrance seems to be reflected in a higher primary central carbon KIE for this reaction system. Steric hindrance has often been associated with large primary deuterium KIEs, a fact which has usually been explained by a larger amount of tunneling.37 More information regarding the structural variation of the TS may be obtained if the present results are viewed within the context of previously published KIE data for other labeled positions in similar reaction systems. The combined evidence from incoming and leaving group KIEs for the reactions of the substituted pyridines thus seems to demonstrate a structural variation of the TS as the steric demand of the nucleophile changes. Steric hindrance as modelled by dimethyl

Isotope Effects for Exotic Nuclei

425

substitution in the 2- and 6-positions to the nucleophilic nitrogen increases the C – X bond distance as reflected in the larger leaving group chlorine KIE. Thus the sterically more hindered nucleophile has a slightly looser TS which should be expected to yield the larger primary central carbon KIE that is actually observed experimentally.

C. LABELLED N UCLEOPHILE 1. The Effect of Substitution in the Substrate Isotope effects arising from labelling of the incoming group (nucleophile) of a substitution reaction are interesting since they would provide knowledge concerning the amount of bond forming of the new bond in the TS. However, such KIEs are very small. Therefore, this is an ideal case to utilize the fact that the carbon KIEs are maximized by using 11C/14C. The SN2 reactions between a series of p-substituted benzyl chlorides and carbon labelled cyanide ion (Scheme 14.7) were chosen as a model system to determine whether one could measure a significant (larger than the experimental error) incoming group KIE and to determine whether these isotope effects could be used to model the SN2 TS.38

∗CN

+ S

S

CH2Cl

CH2∗CN + Cl

SCHEME 14.7

The bonding to the labelled carbon atom of the nucleophile will be greater in the TS than in the reactants because the nucleophile – a-carbon bond is forming in the SN2 TS. As a result the primary incoming group KIE will decrease with increasing bond formation. The KIE has been estimated to be between 1.02 and 0.87 for TS ranging from reactant- to product-like.38 The actual value will, of course, be determined by the amount of nucleophile – a-carbon bonding in the SN2 TS. The 11C/14C KIE found in the benzyl chloride – cyanide ion SN2 reaction is large enough (1.0105 ^ 0.002, see Table 14.1)38 to suggest that these isotope effects can be used to learn how substituents on the benzene ring of the substrate affect the amount of bond formation to the nucleophile in the TS. The 11C/14C KIEs for the SN2 reactions between a series of p-substituted benzyl chlorides and cyanide ion in 20% aqueous DMSO at 308C, are displayed in Table 14.1.38 The very small change in the magnitude of these incoming group KIEs, suggests that there is little or no change in the nucleophile –a-carbon bond in these SN2 TS when the p-substituent on the

TABLE 14.1 Variation of the Reacting System by Substitution in the Substrate. The Carbon Incoming Group and Chlorine Leaving Group KIEs for the SN2 Reactions between Labelled Cyanide Ion and a Series of p-Substituted Benzyl Chlorides p-Substituent CH3 H Cl NO2 a b

k 11/k 14a

k 35/k 37b

1.0104 ^ 0.001 1.0105 ^ 0.002 1.0070 ^ 0.001 —

1.0079 ^ 0.0004 1.0072 ^ 0.0003 1.0060 ^ 0.0002 1.0057 ^ 0.0002

Measured at 308C in 20% (v/v) aqueous DMSO. Data from Hill and Fry.39 Measured at 208C in 20% (v/v) aqueous dioxane.

426

Isotope Effects in Chemistry and Biology

benzene ring is altered. Further light is shed on the structural variation of the TS for this reaction by considering the leaving group chlorine KIEs reported by Hill and Fry.39 2. The Effect of Substitution in the Leaving Group The incoming group KIEs have also been used to determine how a change in substituent in the leaving group affects the structure of the SN2 TS. The system chosen for this investigation is the SN2 reactions between a series of m-chlorobenzyl p-substituted benzenesulfonates with cyanide ion in 0.5% aqueous acetonitrile40 (Scheme 14.8). Cl

Cl ∗CN +

CH2OSO2

Z

CH2∗CN + OSO2

Z

SCHEME 14.8

The primary incoming group 11C/14C KIEs, found when isotopically labelled cyanide ion was reacted with m-chlorobenzyl p-substituted benzenesulfonates, are presented in Table 14.2. The KIE decreases slightly as a more electron-withdrawing substituent is added to the leaving group, i.e., they decrease from 1.0119 for the p-methylbenzene sulfonate leaving group to 1.0096 for the p-chloro. However, the change in the isotope effect with the substituent is small. These isotope effects suggest that the amount of cyanide – a-carbon bond formation increases slightly as a more electron-withdrawing group is added to the leaving group. The isotope effects found in the reaction between cyanide and the m-chlorobenzyl benzenesulfonates are slightly larger than those found for the benzyl chlorides, indicating that there is less cyanide – a-carbon bond formation in the arene sulfonate SN2 TS.

TABLE 14.2 Variation of the Reacting System by Substitution in the Leaving Group. The Nucleophile KIEs for the SN2 Reactions between a Series of m-Chlorobenzyl p-Substituted Benzenesulfonates and Labelled Cyanide Ion in 0.5% Aqueous Acetonitrile at 08C k 11/k 14

p-Substituent CH3 H Cl a

1.0119 ^ 0.0010a 1.0111 ^ 0.0020 1.0096 ^ 0.0005

The error is the standard deviation of the isotope effects found in at least four different experiments.

VII. THE DETERMINATION OF SECONDARY DEUTERIUM KIES BY THE AID OF RADIOACTIVE CARBON Secondary 1H/2H KIEs have proved to be informative on the TS structure of many reactions including nucleophilic aliphatic substitutions.41 Since a secondary deuterium isotope effect results from an isotopic substitution at a position where no bond formation or breaking occurs these KIEs are usually close to unity. As a consequence careful kinetic measurements are required to

Isotope Effects for Exotic Nuclei

427

k14D

+ OH

−

14CD I 3

k11D

+ OH

−

11CD I 3

k14

+ OH

−

14CH I 3

k11

+ OH

−

11CH I 3

3OH

+ I

−

3OH

+ I

−

11CH

14CH

11CD

3OH

+ I

−

14CD

3OH

+ I

−

SCHEME 14.9

determine these effects with an acceptable degree of accuracy. The use of 11C in combination with 14 C offers an unusual but quite interesting way to determine secondary deuterium KIEs with high precision. This has been demonstrated for the SN2 reaction between methyl iodide and hydroxide ion, which was studied by a double-labelling technique.25 The a-carbon KIE was first determined by the 11C/14C method. The experiment was then repeated, but now using a substrate mixture where either the 11C- or the 14C-labelled methyl iodide was doubly labelled with deuterium, i.e., the substrate was either 11CH3I and 14CD3I or 11CD3I and 14CH3I, where D is 2H. The notation for the different rate constants is given in Scheme 14.9. The different a-carbon KIEs obtained in this way for the reaction performed in 50% dioxane – water at 258C were k11/k14 ¼ 1.192 ^ 0.001, k11/k14D ¼ 1.051 ^ 0.006, and k11D/k14 ¼ 1.326 ^ 0.005. Assuming that the carbon and deuterium KIEs are multiplicative, i.e., that the rule of the geometric mean holds, the secondary deuterium KIEs could be calculated from these isotope effects. The deuterium content corrected secondary a-deuterium KIEs were thus found to be 0.881 ^ 0.012 and 0.896 ^ 0.011 when the double-labelled substrates were 11CD3I and 14CD3I, respectively.25 The origin of secondary isotope effects is due to changes in the force field upon going from reactants to the TS of the rate-determining step of the reaction. For the most part, secondary isotope effects depend on the change in zero-point energy. Any isotopically sensitive vibrational frequency that decreases on going from reactant to TS contributes to a normal isotope effect (KIE . 1). A corresponding increase in vibrational frequency decreases the KIE and may even yield an inverse effect (KIE , 1). Inverse secondary a-deuterium KIEs are often observed for reactions proceeding via the SN2 mechanism.41 Until recently, these small KIEs have been attributed to steric interference by the leaving group and/or the nucleophile with the C – H(D) out-of-plane bending vibrations of the trigonal bipyramidial SN2 TS. However, this view has had to be modified in the light of recent results obtained from several different theoretical calculations, which have shown that the C – H(D) stretching vibration contribution to the isotope effect is more important than previously thought.41

VIII. SECONDARY CARBON KIE IN A PROTON-TRANSFER REACTION Since secondary carbon KIEs are always very small, the potential value of their use to probe mechanistic details is, of course, dependent on the possibilities of making accurate measurements. An approach to the solution of this problem is to make use of the full mass range of carbon isotopes, i.e., 11C/14C. The amine catalyzed 1,3-proton transfer in indenes occurs via hydrogen-bonded ion-pair intermediates. A credible mechanism for the case of 1-methylindene is shown in Scheme 14.10.42 The proton-abstraction step is partially rate determining as demonstrated by the observation of a primary 1H/2H KIE of 5.03 (in toluene at 208C).43 The secondary 11C/14C KIE (with the carbon

428

Isotope Effects in Chemistry and Biology

H3∗C H

H

_

H3∗C

+ .. N

H N

H

H3∗C

_

H

H3∗C

H N

H + .. N

H

SCHEME 14.10

label in the methyl group), was found to be 1.010 ^ 0.005 (in benzene at 208C).44 This isotope effect might have its origin in the hybridization change. Other effects that contribute to the secondary isotope effect are hyperconjugative and/or inductive interactions between the negative charge on C-1 of the indene ring and the methyl group in the TS.

IX. CARBON ISOTOPE EFFECTS FOR ENZYME-CATALYSED REACTIONS The 11C/14C KIE method has proven to be useful also for the determination of enzyme isotope effects. The enzyme tyrosine phenol-lyase catalyses the a,b-elimination reaction of tyrosine to phenol, pyruvic acid, and ammonia and has earlier been subjected to mechanistic investigations including determination of deuterium KIEs.45 The tyrosine was labeled in the b-position via a multienzymatic synthetic route and the 11C/14C KIE was determined to be 1.067 ^ 0.009 for tyrosinase from Citrobacter freundii.46 Thus, carbon –carbon bond breaking is shown to be partially rate limiting.

ACKNOWLEDGMENTS Professor Bengt La˚ngstro¨m, Uppsala Imanet, is gratefully acknowledged for placing the excellent facilities of the Uppsala PET center at our disposal. ˚ ke Engdahl, Dr. Jonas I thank former and present coworkers Dr. Svante Axelsson, Dr. Kjell-A Persson, Dr. Per Ryberg, Dr. Nicholas Power, and MSc Susanna MacMillar for their invaluable contributions to the success of this project. I also thank my collaborators over the years, Professors Ken C. Westaway, Piotr Paneth, Ulf Berg, Michael L. Sinnott, and Dale E. Edmondson. Financial support from the Swedish Science Research Council (NFR/VR) and a grant from the Carl Trygger Foundation is gratefully acknowledged. Linguistic improvements by Professor Daniel M. Quinn are highly appreciated. Finally I wish to thank Professor em. Go¨ran Bergson for his continuous human and intellectual support. This chapter is dedicated to my family Anna, Jan, and Agnes.

REFERENCES 1 Melander, L. and Saunders, W. H. Jr., Reaction Rates of Isotopic Molecules, Wiley, New York, 1980. 2 Kaye, J. A., Ed., Isotope Effects in Gas-Phase Chemistry, ACS Symposium Series, Vol. 502, American Chemical Society, Washington, 1992. 3 Cook, P. F., Ed., Enzyme Mechanism from Isotope Effects, CRC-Press, Boca Raton, 1991. 4 (a) Paneth, P., How to measure heavy atom isotope effects: general principles, In Isotopes in Organic Chemistry, Heavy Atom Isotope Effects, Vol. 8, Buncel, E. and Saunders, W. H., Eds., Elsevier, Amsterdam, 1992, Chap. 2; (b) Singleton, D. A. and Thomas, A. A., High-precision simultaneous determination of multiple small kinetic isotope effects at natural abundance, J. Am. Chem. Soc., 117, 9357– 9358, 1995.

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5 (a) Greitz, T., Ingvar, D. H., and Wide´n, L., Eds., The Metabolism of the Human Brain Studied with Positron Emission Tomography, Raven Prees, New York, 1985; (b) Phelps, M., Mazziotta, I., and Schelbert, H., Eds., Positron Emission Tomography and Autoradiography: Principles and Applications for the Brain and Heart, Raven Press, New York, 1986; (c) Brennan, M. B., Positron Emission Tomography Merges Chemistry with Biological Imaging, pp. 26 – 33, C&EN; Feb. 19, 1996. 6 (a) German, L. and Zemskov, S., Eds., New Fluorinating Agents in Organic Synthesis, Springer Verlag, Berlin, 1989; (b) Hudlicky, M., Chemistry of Organic Fluorine Compounds, Ellis Horwood, New York, 1992; (c) Filler, R., Kobayashi, Y., and Yagupolskii, L. M., Eds., Organofluorine Compounds in Medicinal Chemistry and Biomedical Applications, Elsevier, Amsterdam, 1993; (d) Welch, J. T. and Eswarakrishnan, S., Fluorine in Bioorganic Chemistry, Wiley Interscience, New York, 1991; (e) Knunyants, I. L. and Yakobson, G. G., Synthesis of Fluoroorganic Compounds, Springer-Verlag, Berlin, 1985; (f) Liebman, J. F., Greenberg, A., and Dolbier, W. R. Jr., Eds., Fluorine-Containing Molecules, VCH, Weinheim, 1988; (g) Ojima, I., McCarthy, J. R., and Welch, J. T., Eds., In Biomedical Frontiers of Fluorine Chemistry, ACS Symposium Series, Vol. 639, American Chemical Society, Washington, D.C., 1996; (h) Taylor, N. F., Ed., Fluorinated Carbohydrates, ACS Symposium Series, Vol. 374, American Chemical Society, Washington, D.C., 1988; (i) Soloshonok, V. A., Ed., Enantiocontrolled synthesis of fluoro-organic Compounds, Wiley, Chichester, 1999; ( j) Hudlicky´, M., Fluorine Chemistry for Organic Chemists, Oxford University press, Oxford, 2000; (k) Ramachandran, P. V., Asymmetric Fluoroorganic Chemistry, American Chemical Society, Washington, D.C., 2000; (l) Hiyama, T., Ed., Organofluorine Compounds, Springer, Berlin, 2000. 7 (a) Ferrieri, R. A., Production and application of synthetic precursors labelled with carbon-11 and fluorine-18, In Handbook of Radiopharmaceuticals, Welch, M. J. and Redvanly, C. S., Eds., Wiley, New York, pp. 229– 282, 2003; (b) Fowler, J. S. and Wolf, A. P., Working against time: rapid radiotracer synthesis and imaging the human brain, Acc. Chem. Res., 30, 181– 188, 1997; (c) Feliu, A. L., Carbon-11: where familiar chemistry still holds new challenges, J. Chem. Educ., 67, 364– 367, 1990. 8 (a) Kilbourn, M. R., Fluorine-18 Labeling of Radiopharmaceuticals, Nuclear Science Series NASNS-3203, National Academy Press, Washington, D.C., 1990; (b) Snyder, S. E. and Kilbourn, M. R., Chemistry of fluorine-18 radiopharmaceuticals, In Handbook of Radiopharmaceuticals, Welch, M. J. and Redvanly, C. S., Eds., Wiley, New York, pp. 195– 228, 2003. 9 Matsson, O., Persson, J., Axelsson, B. S., and La˚ngstro¨m, B., Fluorine kinetic isotope effects, J. Am. Chem. Soc., 115, 5288– 5289, 1993. 10 Antoni, G., Kihlberg, T., and La˚ngstro¨m, B., Aspects of 11C-labelled compounds, In Handbook of Radiopharmaceuticals, Welch, M. J. and Redvanly, C. S., Eds., Wiley, New York, pp. 141–194, 2003. 11 La˚ngstro¨m, B. and Bergson, G., The determination of optimal yields and reaction times in syntheses with short-lived radionuclides of high specific activity, Radiochem Radioanal Lett., 43, 47 – 54, 1980. 12 Axelsson, B. S., Matsson, O., and La˚ngstro¨m, B., The 11C/14C kinetic isotope effects method. The 11 14 C/ C kinetic isotope effect in the SN2 Reaction of N,N-dimethyl-p-toluidine with labelled methyl iodide, J. Phys. Org. Chem., 77 –86, 1991. 13 Persson, J., Axelsson, S., and Matsson, O., Solvent dependent leaving group fluorine kinetic isotope effect in a nucleophilic aromatic substitution reaction, J. Am. Chem. Soc., 118, 20 – 23, 1996. 14 Ryberg, P. and Matsson, O., The mechanism of base-promoted HF elimination from 4-Fluoro-4(4’-nitrophenyl)butan-2-one: a multiple isotope effect study including the leaving group 18F/19F KIE, J. Am. Chem. Soc., 123, 2712– 2718, 2001. 15 Ryberg, P. and Matsson, O., The mechanism of base promoted HF elimination from 4-fluoro-4(4-nitrophenyl)-butan-2-one is E1cB. Evidence from double isotopic fractionation experiments, J. Org. Chem., 67, 811– 814, 2002. 16 Shiner, V. J. Jr. and Wilgis, F. P., Heavy atom isotope rate effects in solvolytic nucleophilic reactions at saturated carbon, Isotopes in Organic Chemistry, Vol. 8, Elsevier, Amsterdam, Chap. 6, 1992. 17 Melander, L. and Saunders, W. H. Jr, Reaction Rates of Isotopic Molecules, Wiley, New York, Chap. 92, 1980. 18 Terrier, F., Nucleophilic Aromatic Displacement. The Influence of the Nitro Group, VCH Publishers, Weiheim, 1991. 19 Nudelman, N. S., Mancini, P. M. E., Martinez, R. D., and Vottero, L. R., Solvent effects on aromatic nucleophilic substitutions. Part 5. Kinetics of the reactions of 1-fluoro-2,4-dinitrobenzene with piperidine in aprotic solvents, J. Chem. Soc. Perkin Trans., 2, 951– 954, 1987.

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20 Onyido, I. and Hirst, J., Influence of some steric and electronic effects on the mechanism of SNAr reactions in dimethyl sulphoxide, J. Phys. Org. Chem., 4, 367– 371, 1991. 21 Persson, J. and Matsson, O., Use of fluorine kinetic isotope effects in the study of steric effects in nucleophilic aromatic substitution reactions, J. Org. Chem., 63, 9348 –9350, 1998. 22 (a) Shokat, K., Uno, T., and Schultz, P. G., Mechanistic studies of an antibody-catalyzed elimination reaction, J. Am. Chem. Soc., 116, 2261–2270, 1994; (b) Romesberg, F. E., Flanagan, M. E., Uno, T., and Schultz, P. G., Mechanistic studies of an antibody-catalyzed elimination reaction, J. Am. Chem. Soc., 120, 5160–5167, 1998. 23 Axelsson, B. S., La˚ngstro¨m, B., and Matsson, O., 11C/14C Kinetic isotope effects, J. Am. Chem. Soc., 109, 7233– 7235, 1987. 24 Buist, G. J. and Bender, M. L., Carbon-14 kinetic isotope effects. IV. The effect of activation energy on some carbon-14 kinetic isotope effects, J. Am. Chem. Soc., 80, 4308– 4311, 1958. 25 Axelsson, B. S., Matsson, O., and La˚ngstro¨m, B., Primary 11C/14C and secondary 1H/2H kinetic isotope effects in the SN2 reaction of hydroxide ion with methyl iodide. The relationship between different carbon isotope effects, J. Am. Chem. Soc., 112, 6661 –6668, 1990. 26 Bender, M. L. and Hoeg, D. F., Carbon-14 kinetic isotope effects in nucleophilic substitution reactions, J. Am. Chem. Soc., 79, 5649– 5654, 1957. 27 Lynn, K. R. and Yankwich, P. E., Cyanide carbon isotope fractionation in the reaction of cyanide ion and methyl iodide. Carbon isotope effect in the hydrolysis of methyl iodide, J. Am. Chem. Soc., 83, 53 – 57, 1961. 28 Bigeleisen, J., The effects of isotopic substitution on the rates of chemical reactions, J. Phys. Chem., 56, 823– 828, 1952. 29 Melander, L. and Saunders, W. H. Jr, Reaction Rates of Isotopic Molecules, Wiley, New York, p. 242, 1980. 30 Stern, M. J. and Vogel, P. C., Relative 14C– 13C kinetic isotope effects, J. Chem. Phys., 55, 2007– 2013, 1971. 31 Abboud, J.-L., Notario, R., Bertran, J., and Sola, M., One century of physical organic chemistry: the menshutkin reaction, Prog. Phys. Org. Chem., 19, 1– 182, 1993. 32 (a) Berg, U., Gallo, R., Metzger, J., and Chanon, M., Experimental evidence for geometrical variations in the transition state of the SN2 reaction, J. Am. Chem. Soc., 98, 1260– 1262, 1976; (b) Berg, U., Gallo, R., Klatte, G., and Metzger, J., Demethylations of quaternary pyridinium salts by a soft nucleophile, triphenylphosphine. Electronic and steric accelerations, J. Org. Chem., 41, 2621 –2624, 1976; (c) Berg, U. and Gallo, R., Steric effects in SN2 reactions. Determination of transition state structures for the quaternization of 2-alkylpyridines and -thiazoles by a combined experimental and molecular mechanics procedure, Acta Chem. Scand., B37, 661– 673, 1983; (d) Gallo, R., Roussel, C., and Berg, U., The quantitative analysis of steric effects in heteroaromatics, Adv. Heterocycl. Chem., 43, 173– 299, 1988. 33 Swain, C. G. and Hershey, N. D., Effect of steric hindrance on the structure of transition states, J. Am. Chem. Soc., 94, 1901– 1905, 1971. 34 le Noble, W. J. and Miller, A. R., Kinetics of reactions in solutions under pressure. 49. Chlorine kinetic isotope effects in the methylation of pyridine and 2,6-lutidine, J. Org. Chem., 44, 889– 891, 1979. 35 Persson, J., Berg, U., and Matsson, O., Steric effects in SN2 reactions. Primary carbon kinetic isotope effects in menshutkin reactions, J. Org. Chem., 60, 5037– 5040, 1995. 36 (a) Yamataka, H. and Ando, T., Carbon-14 kinetic isotope effects in the Menschutkin-type reaction of benzyl benzenesulfonates with N,N-dimethylanilines. Variation of the effects with substituents, J. Am. Chem. Soc., 101, 266– 267, 1979; (b) Yamataka, H., and Ando, T., Variations of carbon-14 kinetic isotope effects in the Menschutkin type reaction of benzyl arenesulfonates, Tetrahedron. Lett., 16, 1059– 1062, 1975. 37 Lewis, E. S. and Funderburk, L. H., Rates and isotope effects in the proton transfers from 2-nitropropane to pyridine bases, J. Am. Chem. Soc., 89, 2322– 2327, 1967. 38 Matsson, O., Persson, J., Axelsson, B. S., La˚ngstro¨m, B., Fang, Y., and Westaway, K. C., Incoming group KIEs in the reaction of cyanide with some benzyl chlorides, J. Am. Chem. Soc., 118, 6350– 6354, 1996.

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39 Hill, J. W. and Fry, A., Chlorine isotope effects in the reactions of benzyl and substituted benzyl chlorides with various nucleophiles, J. Am. Chem. Soc., 84, 2763– 2769, 1962. 40 Westaway, K. C., Fang, Y., Persson, J., and Matsson, O., Using 11C/14C incoming group and secondary alpha deuterium kinetic isotope effects to determine how a change in leaving group alters the structure of the SN2 transition state, J. Am. Chem. Soc., 120, 3340– 3344, 1998. 41 Matsson, O. and Westaway, K., Secondary deuterium kinetic isotope effects and transition state structure, Adv. Phys. Org. Chem., 31, 143– 248, 1998. 42 Husse´nius, A., Matsson, O., and Bergson, G., Primary 2H kinetic isotope effects in the base-catalysed 1, 3-prototropic rearrangement of 1-methylindene and 1,3-dimethylindene, J. Chem. Soc. Perkin Trans., 2, 851–857, 1989. 43 Matsson, O., Primary and secondary b-deuterium kinetic isotope effects in the 1,3-prototropic rearrangement of 1-methylindene using tertiary amines as catalysts in the solvents toluene and dimethyl sulphoxide, J. Chem. Soc. Perkin Trans., 2, 221– 226, 1985. ˚ ., La˚ngstro¨m, B., and Matsson, O., A secondary 11C/14C kinetic isotope 44 Axelsson, B. S., Engdahl, K.-A effect in the base-catalyzed prototropic rearrangement of 1- to 3-methylindene, J. Am. Chem. Soc., 112, 6656– 6661, 1990. 45 Kiick, D. and Phillips, R. S., Mechanistic deductions from kinetic isotope effects and pH studies of pyridoxal phosphate dependent carbon – carbon lyases: erwinia herbicola and citrobacter freundii tyrosine phenol-lyase, Biochemistry, 27, 7333– 7338, 1988. 46 Axelsson, B. S., Bjurling, P., Matsson, O., and La˚ngstro¨m, B., 11C/14C kinetic isotope effects in enzyme mechanism studies. 11C/14C kinetic isotope effect of the tyrosine phenol-lyase catalyzed a,b-elimination of L -tyrosine, J. Am. Chem. Soc., 114, 1502– 1503, 1992.

15

Muonium — An Ultra-Light Isotope of Hydrogen Emil Roduner

CONTENTS I.

Physical Properties and the Chemical Nature of Mu in Comparison with H and D ............................................................................................................................ 434 II. Chemically Bound Mu States: Structural Isotope Effects of Vibrating Species ............ 435 A. Zero-Point Energy and Anharmonicity Effects ....................................................... 435 B. Isotope Effects in Vibrationally Averaged Bond Lengths and Bond Angles............................................................................................................. 436 C. Isotope Effects in Equilibrium Conformations........................................................ 437 D. Isotope Effects in Hyperfine Interactions of Free Radicals .................................... 439 E. The Validity of the Born –Oppenheimer Approximation ....................................... 439 III. Kinetic Isotope Effects: The Competing Effects of Zero-Point Energy and Tunneling...................................................................................................... 441 A. The Mu Reaction with Molecular Hydrogen: The Dominance of Zero-Point Energy ............................................................................................... 441 B. Mu Addition to Benzene: Evidence of Tunneling .................................................. 442 C. Mu Addition to Dioxygen: Cross-Over of Isotope Effects ..................................... 443 D. Mu Transfer: A World Record of a Kinetic Isotope Effect.................................... 444 E. A Reaction Proceeding over a Solvent-Induced Barrier: A Dynamic Solvent Effect ........................................................................................................... 445 IV. Mass Effect on Diffusion ................................................................................................. 446 A. Coherent and Incoherent Tunneling of Mu Diffusion in Crystals .......................... 446 B. Diffusion of Mu in Liquid Water ............................................................................ 447 V. Concluding Remarks........................................................................................................ 447 References..................................................................................................................................... 448

Most chemists are probably more familiar with H, D, and T as isotopes of hydrogen than with muonium (Mu), the youngest member of the family. The nucleus of this light atom is a positive muon (mþ ; Muþ), an elementary particle with a mass of approximately 1/9th the proton mass. Mu ; mþe2 thus possesses an unprecedented mass ratio of 1/9 compared with H, 1/18 with D, and 1/27 relative to T. As we shall see it is nevertheless a well-behaved hydrogen isotope with regard to all properties of interest in chemistry, except when the mass plays a role. Mass effects, of course, are greatly enhanced over those observed with H, D, and T. Mu is therefore most sensibly used to probe all kinds of mass dependencies, like zero-point energy effects and tunneling, with a very high sensitivity.

433

434

Isotope Effects in Chemistry and Biology

I. PHYSICAL PROPERTIES AND THE CHEMICAL NATURE OF Mu IN COMPARISON WITH H AND D Negative and positive muons form a particle –antiparticle pair. Together, they are the major constituents of “natural” radioactivity in the atmosphere. They are formed on the encounter of high-energy protons from interstellar space with nuclei of nitrogen and oxygen molecules in the upper atmosphere. In the laboratory they are generated in an analogous way at the targets of suitable accelerators.1 The negative muon has the behavior of a heavy electron. When captured by an atom, it cascades through its energy levels down to its ground state “orbit” which has a very small radius since the muon mass is large, 207 times that of an ordinary electron. During its cascade it loses its energy via emission of x-rays, and — as long as available — Auger electrons. In its ground state it is found with a high probability inside the nucleus. This causes the nucleus to break up so that the life time of the bound negative muon is shortened below that of the free state ðtm ¼ 2:19703ð4Þ msec), reaching in the order of 80 ns in the environment of heavy elements. In contrast, the positive muon keeps its natural life time independent of environment, even though it is part of antimatter, since it does not find its counterpart for annihilation. It is the positive muon which has found widespread applications as a probe in condensed matter physics and in chemistry. Like the proton it is a spin-1/2 particle. Its magnetic moment is a factor of 3 larger (see Table 15.1 for a compilation of its magnetic properties). It is available from the ports of accelerators in the form of mono-energetic beams with a spin polarization close to 100%. This makes it an ideal magnetic probe that can be implanted in any environment. The experimental method for its detection rests on a single particle counting technique that has been borrowed from particle physics. It takes advantage of the fact that the decay positron is emitted preferentially along the instantaneous direction of the muon spin at the moment of its decay. The information that is obtained from experiments in an applied external field is a free induction decay of spin polarization and thus akin to the information obtained from magnetic resonance-type experiments. The advantage of the muon is that it is available with full polarization at all temperatures, and even in zero magnetic field. Details of the experimental technique are given elsewhere.1 – 3 From the point of view of particle physics, the muon is classified as a heavy electron, but for a chemist it is more useful to regard the positive muon as a light proton which in a “chemical” environment does everything that a proton would do. Injected into matter it slows down, dissipating its energy in an ionization track. It thermalizes and comes to rest near the end of this track, where it may solvate by polarizing the nearby atoms and molecules, or by populating nonbonding electron pairs and forming Brønsted acid-type structures. These muons are in diamagnetic environment. They are thus characterized in experiments in transverse magnetic fields by a precession at the muon Larmor frequency (Table 15.1).

TABLE 15.1 Properties of Bare Isotopic Hydrogen Nuclei Isotope Mu H D T

Mass (a.u.) 0.113428917(3) 1.0072764669(2) 2.0135532127(4) 3.0155007

Refs. 4,5 for data on tritium.

Spin 1/2 1/2 1 1/2

Magnetic Moment (mp)

Larmor Frequency (MHz T21)

3.1833454(1) 1 0.307012208(5) 1.06664

135.53883(3) 42.577483(2) 6.5359035(2) 45.4148

Muonium — An Ultra-Light Isotope of Hydrogen

435

TABLE 15.2 Atomic Properties of Hydrogen Isotopes Isotope Mu H D T H1

Mass (a.u.)

Red. Mass (me)

IP (eV)

2D IP a (cm21)

Bohr Radius (pm)

Hyperfine Coupling (MHz)

0.1139775 1.0078250 2.0141018 3.0160493 1

0.9951869 0.9994557 0.9997276 0.9998181 1.0000000

13.540207 13.598433 13.602134 13.603365 13.605692(1)

528.17 58.54 28.70 18.77 0

53.1736509 52.9465396 52.9321395 52.9273483 52.9177208(2)

4463.3029(2) 1420.405751773(1) 218.2663 1516 —

Values derived from Ref. 4 and Table 15.1. Ionization potential relative to that of the atom with infinite nuclear mass.

a

In electrically insulating materials and in semiconductors a significant fraction of muons recombines with one of the last electrons from its thermalization track, leading to Mu formation. In this paramagnetic state, the muon spin is coupled by hyperfine interaction to the unpaired electron, which has a far larger magnetic moment. In the experiment this leads to characteristic precession frequencies which are well distinguishable from those of muons in diamagnetic environments. Chemically, Mu is in every sense a light hydrogen isotope,3 which is seen best by comparing its ionization potential and its Bohr radius with those of its heavier brothers, H, D, and T (Table 15.2). It is a reactive species and behaves chemically the same way as H, but the rate constants of its reactions are often considerably influenced by its low mass. One of the reactions that it undergoes is addition to double and triple bonds, as in ðH3 CÞ2 C ¼ CH2 þ Mu ! ðH3 CÞ2 C 2 CH2 Mu

ð15:1Þ

This places the muon in a free radical where it takes the position of a proton, in the same way as a deuteron could. If the experiment is done properly, the muon is substituted in the radical as a fully polarized spin label. It can be used to observe this radical, again by a magnetic resonance-type technique, and to obtain information about its structure, its reorientational dynamics, and the kinetics of its reactions.2 Of special interest in the present context are the isotope effects which are observed in these bound states.

II. CHEMICALLY BOUND Mu STATES: STRUCTURAL ISOTOPE EFFECTS OF VIBRATING SPECIES A. ZERO- POINT E NERGY AND A NHARMONICITY E FFECTS Within the Born – Oppenheimer (BO) approximation the anharmonic vibrational energies are given approximately by the expansion as a function of the quantum number v:6 GðvÞ ¼ ve ðv þ 1=2Þ 2 ve xe ðv þ 1=2Þ2 þ ve ye ðv þ 1=2Þ3

ð15:2Þ

The expansion coefficients are given for a set of isotopic hydrogen molecules and molecular ions in Table 15.3. ve scales approximately as m 21/2 (m is the molecular reduced mass), ve xe with m 21, and ve ye with m 23/2,7 but for the molecular ions the entries of the table are given from McKenna and Webster in a better approximation (see Section II.E).7,8 The energy level diagram for H2, HMu, and the hypothetical Mu2 is given in Figure 15.1. There are 15, 6, and 5 bound states for H2, HMu, and Mu2, respectively. Based on Equation 15.2 the zeropoint energies amount to 2170.4 cm21 (25.96 kJ mol21) for H2, 4785 cm21 (57.2 kJ mol21) for HMu, and 6499 cm21 (77.7 kJ mol21) for Mu2. Compared with thermal energy (2.5 kJ mol21 for

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Isotope Effects in Chemistry and Biology

TABLE 15.3 Morse Parameters for Isotopic Hydrogen Molecules and Molecular Ions Red. Mass (a.u.)

ve (cm21)

2 vexe (cm21)

veye (cm21)

Tþ 2 TDþ Dþ 2 HTþ HDþ Hþ 2 MuHþ Muþ 2

1.507 7504 1.207 3585 1.006 7766 0.755 0612 0.671 4065 0.503 6383 0.101 9485 0.056 7145

1342.92 1500.69 1643.36 1897.25 2012.05 2322.71 5238.30 7276.53

22.46 28.03 33.59 44.72 50.38 67.06 399.91 897.42

0.16 0.21 0.28 0.41 0.51 0.77 23.86 89.39

T2 TD D2 HT HD H2 MuH Mu2

1.507 7504 1.207 3585 1.006 7766 0.755 0612 0.671 4065 0.503 6383 0.101 9485 0.056 7145

2543.70 2842.58 3112.89 3594.51 3811.87 4401.20 9925.8 13788.0

Isotopic Species

40.5 50.6 60.7 80.9 91.0 121.3 723.4 1623.3

0.16 0.21 0.28 0.41 0.51 0.77 23.86 89.39

Values for H2 taken from Ref. 8 and scaled for the other species. For molecular ions: values obtained from a Dunham analysis of nonadiabatic energies, taken from Ref. 7.

FIGURE 15.1 Born – Oppenheimer potential energy curve9 and vibrational energies of H2, HMu, and Mu2.

RT at 298 K) these are of course highly significant, and they lead to dramatic isotope effects in chemical properties.

B. ISOTOPE E FFECTS IN V IBRATIONALLY AVERAGED B OND L ENGTHS AND B OND A NGLES Besides the energy, geometrical parameters are affected by isotopic substitution. Within the BO approximation the equilibrium bond length Re (given by the minimum in the BO potential energy curve) is independent of nuclear masses. A harmonic potential is symmetric with respect to Re, and the vibrationally averaged bond lengths are therefore the same, i.e., kRlv ¼ Re : However, one has to be more cautious with bond lengths which are taken from rotational spectra. These measure

Muonium — An Ultra-Light Isotope of Hydrogen

437

TABLE 15.4 Calculated Zero-Point Vibrational Corrections for the Internuclear Distance kDRl and Values for the Root-Mean-Square Displacements kDR2 l1=2 and kDu 2l1/2 of Isotopologues of Water Molecules 11 Isotopologue D2O DOH, O –H DOH, O –D H2O MuOH, O –H MuOH, O –Mu Mu2O

kDRl0,0,0 (pm)

kDR 2l1/2 (pm)

kDu 2l1/2 (degrees)

0.967 1.204 1.047 1.330 1.580 2.811 3.849

5.55 6.51 5.55 6.51 6.51 11.06 11.06

7.43 8.12 8.12 8.68 12.73 12.73 14.71

the root-mean-square (rms) average, kR2 l1=2 v ; which is larger than Re since positive elongations contribute more to kR2 lv : For anharmonic potentials, the wave function adapts to the broken symmetry, so that kRlv . Re : The effect was first estimated for a C –H (C– Mu) fragment using a Morse potential with realistic parameters. It was predicted that the vibrationally averaged bond length in the ground state is longer for C – Mu than for C –H by 4.9%.10 Accurate vibrationally averaged bond lengths and angles were calculated by Buttar and Webster 11 for a number of isotopologuesp of the water molecule. Deviations from the calculated equilibrium bond length of 94.3 pm and bond angle of 1068 are given in Table 15.4. The bond of hydrogen to oxygen is slightly stiffer than that to carbon, and the O – Mu bond is only 2.7% longer than O –H. It is instructive to see that in the mixed isotopologue MuOH the O – Mu bond does not gain its full elongation, but O – H is longer than in H2O. This is because all three atoms are involved in all normal modes. There is no pure O – Mu stretching mode, for example. The rms displacements are suitable measures for the vibrational amplitudes. This amounts to nearly 12% of Re in the bond direction of O – Mu, compared with 7% for O – H. The bond angle fluctuates by almost 14% in the case of Mu2O, and by 8% in H2O. The overall conclusion is that bound Mu in a molecule is considerably more space filling than H, because of its dynamics.

C. ISOTOPE E FFECTS IN E QUILIBRIUM C ONFORMATIONS In polyatomic species the situation becomes even more interesting. There is clear experimental evidence that the two isotopologues of the t-butyl radical, (CH3)2C˙CH2X (X ¼ D, Mu) adopt different equilibrium conformations.12,13 This is explained in Figure 15.2, which shows the reduced (i.e., scaled to the proton magnetic moment) magnetic hyperfine coupling between the unpaired electron and the nucleus of the atom X, A0X ¼ AX mp =mX : The experimental hyperfine coupling is a vibrational average over all internal degrees of freedom, and it is temperature dependent. All of these modes can be neglected to a good approximation, except for the internal rotation about the carbon – carbon bond. The temperature dependence is then given by a Boltzmann average over ground and excited internal rotational states. Empirically, the hyperfine coupling of a nucleus in bposition to the unpaired electron obeys the McConnell relation kA0X l ¼ A0 þ Bkcos2 ul p

ð15:3Þ

Isotopologue is the recommended nomenclature by IUPAC for what has conventionally often been called isotopomer.

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Isotope Effects in Chemistry and Biology

FIGURE 15.2 Reduced hyperfine coupling of t-butyl radicals, (CH3)2C˙CH2X, with X ¼ Mu, H, and D.12 Equilibrium conformations are shown as Newman projections.

where A0 is a small constant that reflects the effect of spin polarization. B represents the major term which is ascribed to hyperconjugation between the pz-orbital that hosts the unpaired electron at the radical center and the b-C –X bond. It is therefore modulated with the dihedral angle u between the pz-orbital and the C – X bond direction. In the high temperature limit, the system has an energy far above the potential that hinders internal rotation, and all angles u contribute equally to AX, so that kA0X l approaches A0 þ 0:5 B: For symmetry reasons, the protons in a CH3 group adopt the same value, kAH l ¼ A0 þ 0:5 B; independent of temperature (see Figure 15.2). Substitution breaks the symmetry of the group. If a substituent atom X has an equilibrium position at 458 , lu0 l , 1358 then its hyperfine coupling will decrease with decreasing temperature, otherwise it will increase. It is obvious from Figure 15.2 that the reduced muon hyperfine coupling is always above and for the deuteron coupling it is below that of the protons in a CH3 group, and the deviation increases with decreasing temperature. This is clear evidence that Mu avoids the nodal plane of the pz-orbital (u0 ¼ 0 for symmetry reasons), and D prefers to be in this plane (u0 ¼ 908). Within the BO approximation the potential energy surface is clearly the same for both isotopologues. So, what is the reason for the different equilibrium structures? The origin was traced to a zero-point energy effect due to a dependence of the force constant in particular of the C –X stretching mode on the dihedral angle.14,15 For a Morse potential the force constant is given approximately by f ¼ ð4xe De Þ2 m; and thus it scales with the square of the dissociation energy. It is clear that a substituent X can dissociate off more easily for u0 ¼ 08, where the p-bond can snap in, and thus f is smaller than for the orthogonal conformation with u0 ¼ 908. It is the light isotope which dominates zero-point energy. Thus, in order to minimize the total energy, the system chooses a conformation in which the light isotope vibrates with the lowest possible force constant. It leads to the equilibrium conformations displayed in Figure 15.2. The twofold rotational barrier which is induced by this zero-point energy effect amounts to 2.1 kJ mol21 for the Mu-substituted radical. A more detailed experimental and theoretical analysis was reported for several isotopologues of the ethyl radical.15 – 17

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D. ISOTOPE E FFECTS IN H YPERFINE I NTERACTIONS OF F REE R ADICALS In the temperature range displayed in Figure 15.2 the muon – proton hyperfine isotope effect, Am 0 =Ap ; varies in the range between 1.4 and 1.8, which is typical for Mu atoms substituted in methyl groups which are nearly freely rotating. In rigid systems, such as the cyclohexadienyl radical (the H/Mu adduct to benzene), the corresponding number is more typically a factor of ca. 1.2. A considerably smaller effect was reported recently for the muoniomethyl radical,18 C˙H2Mu, where the reduced muon coupling is only 3% larger than that of the proton.19 Since the isotopologues have the same BO energy surface the hyperfine isotope effect is only a consequence of vibrational averaging with isotope-dependent amplitudes and zero-point energies.

E. THE VALIDITY OF THE B ORN – O PPENHEIMER A PPROXIMATION Since the muon mass is only 207 times the electron mass one wonders how good the BO approximation is for muonated systems. To assess this question let us start with a simple set of charge exchange reactions: Dþ þ H Muþ þ H

KD

KMu

D þ Hþ

ð15:4Þ

Mu þ Hþ

ð15:5Þ

Within the BO approximation these reactions are energetically degenerate, but since in reality the electron is more strongly bound on D than on H by 29.84 cm21 (see Table 15.2), and on Mu it is less strongly bound by 469.63 cm21, this degeneracy is lifted purely as a consequence of the failure of the BO approximation. The entropy change associated with these reactions is zero, and thus we can directly convert the above energies to equilibrium constants, which gives KD ¼ 1.15 and KMu ¼ 0.105 at 300 K. The above reactions are actually somewhat hypothetical since there is a bound intermediate state, the hydrogen molecular ion, HDþ or MuHþ, respectively, which is much more stable. It is these molecular ions which have been investigated in nonadiabatic calculations beyond the BO approximation.7,8 Some of the results are given in Table 15.3 and Table 15.5. In addition to TABLE 15.5 Nonadiabatic Lowest Rovibrational Energies e(0,0) and D0 for the Isotopologues of the Hydrogen Molecular Ion, Together with Equilibrium and Vibrationally Averaged Bond Lengths 8 Isotopologue Tþ 2 TDþ Dþ 2 HTþ HDþ Hþ 2 MuTþ MuDþ MuHþ Muþ 2

e(0,0) (cm21) 2131 576.6 2131 494.1 2131 419.0 2131 284.5 2131 223.4 2131 056.9 2129 574.1 2129 546.0 2129 469.0 2128 420.3

Y00 ; D0 (a.u.)

e(0,0) 2 Y00 (kJ mol21)

Re (pm)

kRl00 (pm)

2132 242.5 2132 237.4 2132 232.4 2132 222.1 2132 217.0 2132 201.6 2132 001.9 2131 998.4 2131 991.2 2131 845.4

7.966 8.893 9.730 11.215 11.885 13.694 29.068 29.336 30.172 40.974

105.7289 105.6840 105.7245 105.7059 105.7305 105.7336 106.2885 106.3126 106.4291 107.2269

107.7077 107.9488 108.1675 108.5583 108.7355 109.2176 113.4637 113.5371 113.7549 116.6949

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TABLE 15.6 Reaction Energies between Rovibrational Ground States (DE00) and Purely Electronic Contribution beyond BO (DEBO,EL) and Approximate (DEBO,AT)

(1) (2) (3) (4) (5) (6) (7) (8)

Reaction

DE00 (cm21)

DEBO,EL (cm21)

DEBO,AT (cm21)a

þ D þ Hþ 2 ! DH þ H þ Mu þ Hþ ! MuH þH 2 þ þ H þ HD ! D þ H2 þ Hþ 2 þ HD ! HD þ H2 D þ H2 ! DH þ H Mu þ H2 ! MuH þ H 2 HD ! H2 þ D2 CHþ þ HD ! CDþ þ H2

2136.7b þ 1118.3b þ37.6c þ302.2b,c 2287.1c 22614.7c 254.7c —

þ14.5b 2259.2b 225.3d 210.9d þ 1.2c,d — 0.0d þ19.1d

þ15.0 2234.7 þ29.9 þ14.9 0.0 0.0 0.0 —

1 cm21 ¼ 11.96 J mol21. Sum of atomic contributions based on Table 15.2 (see text). b Nonadiabatic values, based on Table 15.2 and Table 15.5. c Nonadiabatic values, based on Table 15.2 and Ref. 9. d Adiabatic values, taken from Ref. 20. a

the atomic corrections as demonstrated for the equilibria, 4,5 there is now a correction term which scales with the derivative of the energy with respect to the internuclear distance, 2dE=dR:8 This derivative is zero, both at the minimum of the BO potential energy curve and for R ! 1. It has the consequence that the correction changes sign at Re(BO), and it leads to a shift of Re that increases þ from its BO value (105.26 pm) over 105.7 pm for Tþ 2 to as much as 107.2 pm for Mu2 . This correction adds to the bond length extension that was already attributed in Section II.B to vibrational averaging in the anharmonic potential. The energies associated with the isotope exchange reactions in the rovibrational ground states (DE00) and the purely electronic contributions beyond the BO approximation (DEBO,EL) are given in Table 15.6 for a series of reactions with an increasing number of electrons. The trend of the DE00 values is ascribed to zero-point energy corrections, and hence it is the same as within the BO approximation. It is therefore not further commented upon. However, DEBO,EL is often a significant fraction of DE00, although sometimes of opposite sign. In addition we give an estimate of DEBO,EL which is denoted DEBO,AT and based on the sum of atomic contributions as listed for DIP in Table 15.2. Only 50% of the atomic contributions is taken for the molecular ions, on the grounds that each nucleus is populated by approximately one half of the electron. It is interesting to see that the estimate based on atomic corrections, DEBO,AT, is within 10% of the calculated value for the first two entries in Table 15.6. For entries (3) and (4) the positive correction to the BO energy must be larger for H2 than for HD, and therefore the negative sign reported for DEBO,EL20 for these two reactions is likely to be wrong. Under this premise the estimate is again in qualitative agreement with the calculation. In reactions in which the charge on the isotopes does not change the estimate predicts no significant net electronic effect beyond BO. For entry (7) this is also the result of the exact calculation, so that we may assume that the estimate based on a sum of atomic terms is not so bad also for reactions (5) and (6). Unfortunately, calculations beyond the BO approximation are so far available for Mu-substituted species only for the isotopologues of the hydrogen molecular ions (Table 15.5). Thus, we have at present to rely on estimates. It seems that as long as the electron population on the muon nucleus does not change during a reaction we should expect little influence of the BO approximation, but when it changes by a full unit then the energy correction amounts to 5.6 kJ mol21, which is certainly significant on a chemical scale.

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The estimate for DEBO,AT assumes a symmetric charge distribution in molecules like HD and HMu. This is so only within the BO approximation, but it is well known that in reality the symmetry is broken and that these molecules have a small dipole moment. It amounts to 8.1(5) £ 1024 Debye for HD.21 The energy imbalance between the 1s orbitals of Mu and H is nearly 16 times larger than between H and D. The same factor may approximately also hold for the HMu/ DH dipole moment ratio.

III. KINETIC ISOTOPE EFFECTS: THE COMPETING EFFECTS OF ZERO-POINT ENERGY AND TUNNELING A. THE Mu REACTION WITH M OLECULAR H YDROGEN: T HE D OMINANCE OF Z ERO- P OINT E NERGY For quantum mechanical predictions of absolute rate constants, H þ H2 has been the standard prototype reaction for the past several decades. Since it involves only three atoms with a total of three electrons, its potential energy surface has been known at the highest level of accuracy that was available at any time. Moreover, comparison of theoretical and experimental results based on several isotopic variants of the reaction provide sensitive tests of approximations made in the treatment of the dynamics as well as on the potential energy surface. In view of this it is clear that the rate constants for the reactions of Mu with H2 and with D2, which were measured in the seminal work by Reid et al., 22 represent one of the most valuable sets of data ever obtained with Mu. Their results are given in Figure 15.3 and Table 15.7 along with those of the most recent combined experimental and theoretical quantum dynamical study of D þ H2 and H þ D2.23 Experimental values are now available between 167 and 2112 K for the former and between 274 and 2160 K for the latter reaction, covering more than eight orders of magnitude in the rate constant. Over this large temperature range the Arrhenius plots show clear curvature. Therefore, care has to be taken when activation energies and preexponential factors representing data in a limited temperature range are compared (in Table 15.7 this is done for the range where Mu data are available). According to classical transition-state theory the ratio of frequency factors is expected to correspond to the inverse square root of the reduced mass ratio. Within experimental error this is the case for the entries to these reactions in Table 15.7. This observation agrees with the expectation that tunneling should not be too important, in particular not for the Mu reactions, because of their

FIGURE 15.3 Arrhenius plot for the reaction of Mu with H2 and with D2 (see Ref. 22) in comparison with D with H2 and H with D2.23

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TABLE 15.7 Kinetic Isotope Effects: Rate Constants at 298 K and Arrhenius Parameters for the H Abstraction of Hydrogen from Molecular Hydrogen and Format, and for the Addition to Benzene Reaction

k298 (M 21 s21)

log10(A), M 21 s21

Ea (kJ mol21)

T range (K)

Ref.

450 –1000 473 –843 450 –1000 598 –843

23 22 23 22

640 –818 625 –820

25 26

D þ H2 Mu þ H2 H þ D2 Mu þ D2

9.84 £ 104 a 26.54a 5780a 2.005a

10.76(4) 11.17(8) 10.79(4) 11.1(1)

H þ CH4 Mu þ CH4

40a 3 £ 1025 a

11.3(3) 13.5(3)

D þ C6H6 H þ C6H6 Mu þ C6H6

2.9(2) £ 107 a 3.4(2) £ 107 65(2) £ 107

10.72(14) 10.65(6) 9.99(5)

19.4(1.2) 18.0(1.0) 6.7(3)

379 –788 298 –470 296 –500

28 28 29

D þ HCOO2 H þ HCOO2 Mu þ HCOO2

22.4(1.1) £ 107 19.1(8) £ 107 0.56(8) £ 107

11.40(5) 11.89(8) 11.92(15)

17.4(9) 20.6(4) 29.5(8)

280 –350 280 –350 276 –370

36 36 36

a

32.9(4) 55.6(9) 40.1(4) 61.6(1.6) 55(4) 103(4)

Extrapolated from Arrhenius parameters.

significant endothermicity. Much of the kinetic isotope effect is attributed to zero-point energy differences between reactant and transition state. This is reflected by the large differences in activation energies which amount to more than 20 kJ mol21 between the reactions of Mu and those of either H or D. The data for the Mu þ H2 reaction were in good agreement already with an early 3D quantum mechanical calculation by Schatz.24 It was also reported that corrections to the BO energy surface raise the barrier height by 578 J mol21 and 377 J mol21 for D þ H2 and H þ D2, respectively.23 For Mu þ H2 this correction was suggested to amount to as much as 2 kJ mol21.22 For D þ H2 this non-BO effect reduces the rate constant by 34% at 167 K. Since experiment and theory now agree perfectly within experimental error over the entire temperature range, it was concluded that the H þ H2 thermally averaged reaction rate constant can be added to the list of solved problems.23 A more up to date and also more involved example of H abstraction by Mu that has been investigated in detail both experimentally25,26 and theoretically on the basis of variational transition state theory with multidimensional tunneling 27 is the reaction with methane (Table 15.7). When H is the attacking species the agreement between experiment and theory is quite gratifying, but the kinetic isotope effect for Mu is overestimated by a factor of 3– 6. It is in particular the unusually high Arrhenius preexponential factor which is difficult to explain.

B. Mu ADDITION TO B ENZENE: E VIDENCE OF T UNNELING While the reaction of Mu with H2 is slower than that of H or D by more than three orders of magnitude at room temperature, which is typical for H abstraction reactions, addition of the same isotopes to benzene shows the opposite effect (see Figure 15.4 and Table 15.7). Here, Mu is faster than its heavier isotopes by about a factor of 20. The activation energy of the Mu reaction is only 37% of that of H, and the preexponential factor is also significantly lower. Both effects are clear indicators for a significant contribution of tunneling, which offsets any zero-point energy effects in the transition state.

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FIGURE 15.4 Arrhenius plots for the addition reaction of hydrogen isotopes to benzene in the gas phase.28,29

It is well known from calculations on model barriers that tunneling is promoted by a low mass, and a narrow and not too high barrier.30 Addition reactions are often considerably exothermic, so that their transition state is reactant-like and the barrier is narrow and relatively low. Therefore, tunneling plays a dominant role, rendering the reaction of the light isotope faster than that of the others. H abstraction reactions are less exothermic or even endothermic, so that their barrier is normally higher, and particularly at low energies often much wider. This impedes tunneling, so that the isotope effect in abstraction reactions is dominated by zero-point energy. In reactions where the isotope is the attacking species the difference arises only in the transition state. Thus, the barrier increases for the light isotope and makes its reaction slower.

C. Mu ADDITION TO D IOXYGEN: C ROSS- OVER OF I SOTOPE E FFECTS Recombination reactions of small species on a predominantly attractive surface need a third body (moderator gas, M ¼ N2 in the present case) to carry away the binding energy. They are therefore pressure dependent, and the capture dynamics is particularly sensitive to certain parts of the potential energy surface. The reaction H þ O2 (þ M) ! HO2 (þ Mp) plays a central role in combustion and in atmospheric chemistry. It has therefore been measured by many workers, including its isotopic variant with Mu (see Ref. 31). Again, the Mu reaction is crucial for a complete theoretical understanding, and it provides a sensitive test of the potential energy surface.32 – 34 The low-pressure limiting rate constant for Mu at room temperature is almost a factor of 7 below the corresponding value for H (Figure 15.5a). In contrast to H, the typical fall-off behavior was not observed for Mu. Owing to its low reduced mass the Mu –O2 collision complex has a higher vibrational frequency near the minimum of the reaction coordinate than H –O2, and thus a shorter lifetime. Therefore, a higher pressure is needed to stabilize all collision complexes before redissociation. At room temperature the effective rate constant of the Mu reaction crosses that of H 1 near a pressure of 300 bar, and the ratio of high-pressure limiting rate constants, k1 Mu/kH , was 33 predicted to amount to a factor of 2.8. Interestingly, there is also a cross-over of the high-pressure limiting values as a function of 1 32 temperature (Figure 15.5b). RRKM theory predicts k1 Mu , kH below about 230 K. Unfortunately, 1 1 there is not yet a reliable determination of either kMu or kH , and the temperature range over which kH was determined has little overlap with that of kMu. Remaining differences in the details of the

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Isotope Effects in Chemistry and Biology

FIGURE 15.5 (a) Cross-over of the pressure dependence of the effective rate constants of the addition reaction of H and Mu to O2 at room temperature.31 (b) Cross-over of the RRKM predicted temperature dependence for the high-pressure limiting rate constants of the addition reaction of H and Mu to O2.32

potential energy surfaces 33,34 and uncertainties in some of the approximations involved in theoretical approaches clearly call for further experiments.

D. Mu TRANSFER: A W ORLD R ECORD OF A K INETIC I SOTOPE E FFECT In the vast majority of reactions that have been investigated Mu is the attacking species. There is a single example where chemically bound Mu is transferred to another molecule. The Mu-substituted cyclohexadienyl radical clearly reacts with 2,3-dimethyl-1,3-butadiene (DMBD): C6 H6 Mu þ DMBD ! C6 H6 þ DMBD2Mu

ð15:6Þ

This reaction is exothermic by ca. 80 kJ mol21, driven by the high reactivity of DMBD and the recovery of the resonance energy in benzene.35 The reaction is nearly unactivated and has an unusually low preexponenital factor, log10(A/M21s21) ¼ 6.9(4), indicating that it proceeds by tunneling. In line with this hypothesis it was not possible to measure a significant rate for the transfer of H from C6H7 to DMBD under similar conditions, demonstrating that H transfer is

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suppressed over that of Mu by at least a factor of 75,000 at 298 K.35 We are not aware of any other reaction which shows a kinetic isotope effect of this magnitude and has been directly observed at room temperature.

E. A R EACTION P ROCEEDING OVER A S OLVENT- I NDUCED B ARRIER: A DYNAMIC S OLVENT E FFECT Hydrogen atoms react with the formate ion in aqueous solution formally by H-abstraction: MuðH; DÞ þ HCOO2 ! MuHðH2 ; DHÞ þ z COO2

ð15:7Þ

At a first glance the behavior looks like that of a normal H-abstraction: Mu is slower than H or D by a factor of 34 at room temperature (Figure 15.6). However, the activation energies for H and D are more comparable with those of the addition to benzene. For such barriers tunneling was expected to dominate the reaction, and it proved difficult to explain why Mu is so much slower.36 High quality QCISD calculations with an aug-cc-pvDZ basis set that includes diffuse functions to accommodate the negative charge then revealed that in the gas phase the reaction should proceed without a barrier. Addition of water in a polarizable continuum solvation model brought the calculated barrier back to the experimental value. The calculation also showed that the reaction is to a large extent an electron transfer followed immediately by proton transfer. This leads to a large change, literally an inversion, of the dipole moment of the reaction complex as it moves along the reaction coordinate, and it is this fact which gives raise to the solvation barrier. It was suggested that a dynamic (nonequilibrium) solvation effect is responsible for the kinetic isotope effects. Reorientation of the water molecules can probably just follow when the H and D reaction proceeds along the reaction path, while it cannot follow in case of the light Mu atoms.36 Thus, the transition state should be stabilized to a greater extent for the H and D than for the Mu reaction. In fact, a similar effect was shown to be responsible for the addition of hydrogen isotopes to benzene in aqueous solution.37,38

FIGURE 15.6 Arrhenius plots for the H abstraction reaction from aqueous formate ions by hydrogen isotopes. The D abstraction from deuterated formate shows a similar behavior, displaced to lower rate constants.36

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Isotope Effects in Chemistry and Biology

IV. MASS EFFECT ON DIFFUSION A. COHERENT AND I NCOHERENT T UNNELING OF Mu D IFFUSION IN C RYSTALS The diffusion of light interstitial impurities in solids at low temperature has attracted wide interest because of the expected intrinsic quantum mechanical nature of the transport mechanism. Because of their low mass the positive muon as well as the Mu atom are ideal for testing quantum theories of diffusion. A detailed review of experimental results and of theoretical aspects has been given by Storchak and Prokof’ev.39 Here, only some of the fundamental aspects will be introduced. At low temperature the interstitial particle is delocalized and forms a band in the ideal periodic lattice. The band width is proportional to the coherent tunneling rate of its motion in the absence of perturbations. Coherent propagation is observed in superconductors and insulators at low temperatures. Lattice defects, thermal excitations of lattice phonons, and thermal excitation of electrons act as perturbations which impede this coherent propagation. The motion slows down with increasing temperature and becomes that of a coherent hopping process of basically localized particles. Above a characteristic minimum in the hopping rate at a temperature T p the picture changes to that of an incoherent (stochastic) hopping process, which is thermally activated and often described well by the Arrhenius equation. As an instructive example which shows all three regimes Figure 15.7 displays the hopping rate of Mu in ultrapure single-crystal GaAs.40 Just as in chemical kinetics, the high temperature regime is often still dominated by quantum effects. For example, Mu diffusion in KCl has an activation energy of 3.2 kJ mol21, which is more than five times less than the corresponding value for H (19 kJ mol21). Along with the low preexponential factor (8.2 £ 109 sec21) this is clear evidence of tunneling.39 In the same way, for m þ diffusion in Cu in the temperature range 100 –250 K, Ea amounts to 4.6 kJ mol21, about an order of magnitude less than for proton diffusion, and the preexponential factor (3 £ 107 sec21) is also much less than the muon vibration frequency in the interstitial potential well (about 1013 sec21).39 If the transition state of diffusive motion represents a bottleneck this leads to a higher zero-point energy and thus a higher activation energy for the lighter particle.30 An example for this is D vs. H diffusion along the hexagonal channel in ice.41 D atoms diffuse faster than H below 200 K, and more slowly above this temperature. Thus, the high temperature regime of diffusion shows similar phenomena as activated chemical reactions, except that reaction barriers are nonperiodic and therefore there can be incoherent but no coherent tunneling.

40 FIGURE 15.7 Mu hop rate t21 c in a high resistivity GaAs single crystal. The three regimes relate to the bandlike delocalized state (I), coherent hopping (II), and incoherent hopping (III).

Muonium — An Ultra-Light Isotope of Hydrogen

B. DIFFUSION oF Mu

IN

447

L IQUID WATER

It is common knowledge that proton diffusion in liquid water is very fast, and there is no doubt that it can take advantage of a special effect, the Grotthuss mechanism. Less well known is the fact that the diffusion coefficient of the uncharged hydrogen atom, DH ¼ (7.0 ^ 1.5) £ 1029 m2 sec21 at 298 K,42 is as much as 75% of that of the proton, although the atom can clearly not benefit from a Grothuss mechanism. The question arises what the mechanism is that permits such a fast diffusion of the atom. Much of the diffusion data in liquids has been interpreted on the basis of the Stokes– Einstein equation, which describes the diffusion of macroscopic spheres with a hydrodynamic radius R0 in a liquid with viscosity h. In the slip limit (for solute and solvent of comparable size) it reads D¼

kB T 4phR0

ð15:8Þ

Based on this we derive from the above value for DH a hydrodynamic radius of the hydrogen atom of R0 ¼ 52 pm, which coincides with the atomic Bohr radius. This is certainly not physically meaningful, especially if one considers the relatively slow radial decay of the 1s wave function. On the other hand, it is known that hydrogen atoms in liquid water form hydrophobic bubbles with a radius of the order of 250 pm or more.43 For Stokes – Einstein diffusion of this bubble one would predict diffusion coefficients far smaller than found experimentally. It thus becomes clear that the Stokes– Einstein relation breaks down, which is not unexpected for such small particles. According to equation 15.8, diffusion depends only on the hydrodynamic radius but not on the mass of the diffusing solute. Of course, in the gas and solid phases it is normal to have a mass dependence of diffusion. The question has been addressed in a comparison of the (diffusion controlled) electron spin exchange reaction of Mu, H, and D with Ni2þ. A significant isotope effect was found, with DH/DD ¼ 1.074, and DMu/DH ¼ 1.34 at room temperature.44 This study also revealed a H2O/D2O solvent isotope effect on the diffusion coefficients. Both of these effects will be essential to clarify the mechanism which permits such an efficient diffusion of small hydrophobic species such as noble gases or hydrogen atoms in water.

V. CONCLUDING REMARKS Even though the Mu atom has an antimatter nucleus its chemical behavior is basically the same as that of H. Its isotope effects are explained mostly on the basis of conventional concepts, but due to the unprecedented Mu/H mass ratio of 1/9 they are strongly enhanced. The large zero-point energy leads to pronounced anharmonicity effects which are sensitively observed in radical hyperfine coupling constants. Most remarkably, it leads to different equilibrium conformations, even though the electronic potential energy surface is the same within the BO approximation. The extent of the breakdown of the BO approximation for Mu has so far been investigated only for the hydrogen molecular ions. Under normal circumstances the effect is expected to be small, although it may reach thermal energies and thus become chemically relevant in cases where the electron density at the muon changes strongly along the reaction coordinate. More dramatic effects are expected when there is an avoided crossing of electronic states that is passed along a Mu reaction coordinate so that nuclear motion is strongly coupled to a major rearrangement of electrons. Such a case has not yet been identified. In principle, H abstraction from formate (reaction 15.7) is a candidate, but it has not been investigated in this respect. Of prime interest in chemistry are kinetic isotope effects. It has been shown that they can span several orders of magnitude in both directions, depending on whether they are dominated by zeropoint energy in the transition state ðkMu =kH , 1Þ or by tunneling ðkMu =kH . 1Þ: Isotope effects can cross-over as a function of temperature and pressure, as demonstrated for the hydrogen reaction with dioxygen and for atomic hydrogen diffusion in ice.

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Isotope Effects in Chemistry and Biology

Owing to its low mass the Mu atom is particularly prone to quantum effects. Tunneling can occur coherently in Mu diffusion across the periodic potential of a crystal at low temperature, or incoherently at higher temperatures and in chemical reactions through a single barrier. Mu has provided excellent data for sensitive tests of theoretical models of quantum diffusion. There are often various contributions to observed effects which scale differently with mass. The enhancement of the Mu/H isotope effect over that of H/D is therefore not always explained by a simple mass scaling. This is seen most prominently in the H abstraction from the formate ion (Figure 15.6) where H and D are nearly indistinguishable but Mu a factor of 34 slower at room temperature, or by the addition of Mu to benzene (Figure 15.4) where tunneling enhances Mu far more than H. The availability of a further isotope can thus often serve to separate different contributions, and Mu has been very sensitive in this respect.

REFERENCES 1 Brewer, J. H. and Cywinski, R., mSR: an introduction, In Muon Science — Muons in Physics, Chemistry and Materials, Lee, S. L., Kilcoyne, S. H., and Cywinski, R., Eds., Institute of Physics, Bristol, pp. 1– 9, 1999. 2 Roduner, E., Polarized positive muons probing free radicals: a variant of magnetic resonance, Chem. Soc. Rev., 60, 337– 346, 1993. 3 Walker, D. C., Muonium: a light isotope of hydrogen, J. Phys. Chem., 85, 3960–3971, 1981. 4 Mohr, P. J. and Taylor, B. N., CODATA recommended values, Rev. Mod. Phys., 72, 351– 495, 2000. 5 Holden, N. E., Table of the isotopes, Handbook of Chemistry and Physics, 76th ed., Lide, D. R., Ed., CRC Press, Boca Raton, Section 11, pp. 38 – 43, 1995. 6 Hollas, J. M., Basic Atomic and Molecular Spectroscopy, Royal Society of Chemistry, Cambridge, 2002. 7 McKenna, D. and Webster, B., Non-adiabatic calculations upon the hydrogen molecular ion isotopically substituted by tritium, deuterium and muonium, J. Chem. Soc., Faraday Trans. 2, 80, 589– 600, 1984. 8 McKenna, D. and Webster, B., Muonic isotope effects and non-adiabatic natural orbitals for the isotopically substituted hydrogen molecular ion, J. Chem. Soc., Faraday Trans. 2, 81, 225– 234, 1985. 9 Wolniewicz, L., Relativistic energies of the ground state of the hydrogen molecule, J. Chem. Phys., 99, 1851– 1868, 1993. 10 Roduner, E. and Reid, I. D., Hyperfine and structural isotope effects in muonated cyclohexadienyl and cyclopentyl radicals, Isr. J. Chem., 29, 3 –11, 1989. 11 Webster, B. and Buttar, D., Zero-point vibrational corrections for the geometry, electric dipole moment and 17O nuclear quadrupole coupling constant calculated for the muonium isotopologues, MuOH and Mu2O, of the water molecule, J. Chem. Soc., Faraday Trans. 2, 88, 1087– 1092, 1992. 12 Roduner, E., Strub, W., Burkhard, P., Hochmann, J., Percival, P. W., Fischer, H., Ramos, M., and Webster, B. C., Muonium substituted organic free radicals in liquids. Muon– electron hyperfine coupling constants of alkyl and allyl radicals, Chem. Phys., 67, 275– 285, 1982. 13 Percival, P. W., Brodovitch, J.-C., Leung, S.-K., Yu, D., Kiefl, R. F., Luke, G. M., Venkateswaran, K., and Cox, S. F. J., Intramolecular motion in the tert-butyl radical as studied by muon spin rotation and level-crossing spectroscopy, Chem. Phys., 127, 137– 147, 1988. 14 Claxton, T. A. and Graham, A. M., Vibrationally-induced barriers to hindered rotation, J. Chem. Soc. Chem. Commun., 1167, 1987. 15 Claxton, T. A. and Graham, A. M., A Chemical interpretation of vibrationally induced barriers to hindered internal rotation, J. Chem. Soc., Faraday Trans. 2, 83, 2307– 3217, 1987. 16 Ramos, M. J., McKenna, D., Webster, B. C., and Roduner, E., Muon spin rotation spectra for muonium isotopically substituted ethyl radicals, J. Chem. Soc., Faraday Trans. 1, 80, 255– 265, 1984. 17 Ramos, M. J., McKenna, D., Webster, B. C., and Roduner, E., The barriers to internal rotation for muonium substituted ethyl radicals, J. Chem. Soc., Faraday Trans. 1, 80, 267– 274, 1984.

Muonium — An Ultra-Light Isotope of Hydrogen

449

18 Koppenol, W. H., Names for muonium and hydrogen atoms and their ions (IUPAC recommendations 2001), Pure Appl. Chem., 73, 377–380, 2001. 19 McKenzie, I., Addison-Jones, B., Brodovitch, J.-C., Ghandi, K., Kecman, S., and Percival, P. W., Detection of the muonated methyl radical, J. Phys. Chem., 106, 7083– 7085, 2002. 20 Kleinman, L. I. and Wolfsberg, M., Corrections to the Born – Oppenheimer approximation and electronic effects on isotopic exchange equilibria, J. Chem. Phys., 60, 4740– 4748, 1974, see also pages 4749– 4754. 21 Wishnow, E. H., Ozier, I., and Gush, H. P., Submillimeter spectrum of low temperature hydrogen: the P translational band of H2 and the R(0) line of HD, Astrophys. J., 392, L43 – L46, 1992. 22 Reid, I. D., Garner, D. M., Lee, L. Y., Senba, M., Arseneau, D. J., and Fleming, D. G., Experimental tests of reaction rate theory: Mu þ H2 and Mu þ D2, J. Chem. Phys., 86, 5578– 5583, 1987. 23 Mielke, S. L., Peterson, K. A., Schwenke, D. W., Garrett, B. C., Truhlar, D. G., Michael, J. V., Su, M. C., and Sutherland, J. W., H þ H2 thermal reaction: a convergence of theory and experiment, Phys. Rev. Lett., 91, 063201-1– 063201-4, 2003. 24 Schatz, G. C., A quantum reactive study of Mu þ H2 ! MuH þ H, J. Chem. Phys., 83, 3441– 3447, 1985. 25 Sepehrad, A., Marshall, R. M., and Purnell, H., Reaction between hydrogen atoms and methane, J. Chem. Soc., Faraday Trans. 1, 75, 835– 843, 1978. 26 Snooks, R., Arseneau, D. J., Fleming, D. G., Senba, M., Pan, J. J., Shelley, M., and Baer, S., The thermal reaction rate of muonium with methane (and ethane) in the gas phase, J. Chem. Phys., 102, 4860– 4869, 1995. 27 Pu, J. and Truhlar, D. G., Tests of potential energy surfaces for H þ CH4 $ CH3 þ H2: deuterium and muonium kinetic isotope effects for the forward and reverse reaction, J. Chem. Phys., 117, 10675– 10687, 2002. 28 Nicovich, J. M. and Ravishankara, A. R., Reaction of hydrogen atom with benzene: kinetics and mechanism, J. Phys. Chem., 88, 2534– 2541, 1984. 29 Roduner, E., Louwrier, P. W. F., Brinkman, G. A., Garner, D. M., Reid, I. D., Arseneau, D. J., Senba, M., and Fleming, D. G., Quantum phenomena and solvent effects on addition of hydrogen isotopes to benzene and to dimethylbutadiene, Ber. Bunsenges. Phys. Chem., 94, 1224– 1230, 1990. 30 Roduner, E., Aspects of muon chemistry, In Muon Science — Muons in Physics, Chemistry and Materials, Lee, S. L., Kilcoyne, S. H., and Cywinski, R., Eds., Institute of Physics, Bristol, pp. 173– 209, 1999. 31 Himmer, U., Dilger, H., Roduner, E., Pan, J. J., Arseneau, D. J., Fleming, D. G., and Senba, M., Kinetic isotope effect in the gas phase reaction of muonium with molecular oxygen, J. Phys. Chem. A, 103, 2076– 2087, 1999. 32 Himmer, U. and Roduner, E., The addition reaction of X to O2 (X ¼ Mu H, D): isotope effects in intraand intermolecular energy transfer, Phys. Chem. Chem. Phys., 2, 339– 347, 2000. 33 Harding, L. B., Troe, J., and Ushakov, V. G., Classical trajectory calculations of the high pressure limiting rate constant and of specific rate constants for the reaction H þ O2 ! HO2: dynamic isotope effects between tritium þ O2 and muonium þ O2, Phys. Chem. Chem. Phys., 2, 631– 642, 2000, and ibid. 3, 2630– 2631, 2001. 34 Marques, J. M. C., Llanio-Trujillo, J. L., and Varandas, A. J. C., Isotope effect on unimolecular dissociation of MuO2: a classical trajectory study, Phys. Chem. Chem. Phys., 2, 3583– 3589, 2000, and ibid. 3, 2632– 2633, 2001. 35 Roduner, E. and Mu¨nger, K., Reaction of the muonium substituted caclohexadienyl radical with 2,3dimethyl-1,3-butadiene, Hyperfine Interact., 17 – 19, 793– 796, 1984. 36 Lossack, A. M., Roduner, E., and Bartels, D. M., Solvation and kinetic isotope effects in H and D abstraction reactions from formate ions by D, H and Mu atoms in aqueous solution, Phys. Chem. Chem. Phys., 1, 2031– 2037, 2001. 37 Roduner, E. and Bartels, D. M., Solvent and isotope effects on addition of atomic hydrogen to benzene in aqueous solution, Ber. Bunsenges. Phys. Chem., 96, 1037– 1042, 1992. 38 Garrett, B. C. and Schenter, G. K., Nonequilibrium solvation for an aqueous-phase reaction: kinetic isotope effects for the addition of hydrogen to benzene, In Structure, Energetics and Reactivity in Aqueous Solution, Cramer, C. J. and Truhlar, D. G., Eds., American Chemical Society, Washington, 1994.

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39 Storchak, V. G. and Prokof’ev, N. V., Quantum diffusion of muons and muonium atoms in solids, Rev. Mod. Phys., 70, 929– 978, 1998. 40 Schneider, J. W., Kiefl, R. F., Ansaldo, E. J., Brewer, J. H., Chow, K., Cox, S. F. J., Dodds, S. A., Duvarney, R. C., Estle, T. L., Haller, E. E., Kadono, R., Kreitzman, S. R., Lichti, R. L., Niedermayer, C., Pfitz, T., Riseman, T. M., and Schwab, C., Quantum motion of muonium in GaAs and CuCl, Mater. Sci. Forum, 83 – 87, 569– 574, 1992. 41 Bartels, D. M., Han, P., and Percival, P. W., Diffusion and CIDEP of H and D atoms in solid H2O, D2O and isotopic mixtures, Chem. Phys., 164, 421– 437, 1992. 42 Benderskii, W. A., Krivenko, A. G., and Rukin, A. N., Anomalous mobility of hydrogen and deuterium atoms in aqueous solutions of electrolytes, Chim. Vysok. Energ., 14, 400– 405, 1980, English translation: High Energy Chem., 14, 303– 308, 1980. 43 Roduner, E., Percival, P. W., Han, P., and Bartels, D. M., Isotope and temperature effects on the hyperfine interaction of atomic hydrogen in liquid water and in ice, J. Chem. Phys., 102, 5989– 5997, 1995. 44 Roduner, E., Bartels, D.M., and Mezyk, S.P., to be published.

16

The Kinetic Isotope Effect in the Photo-Dissociation Reaction of Excited-State Acids in Aqueous Solutions Ehud Pines

CONTENTS I. II.

Introduction ...................................................................................................................... 451 General Kinetic Models for Acid – Base Reactions in Solutions .................................... 452 A. The Two-State Proton-Transfer Reaction Model (The Eigen – Weller Model) ..................................................................................... 452 B. Free-Energy Correlations of the Proton (Deuteron) Transfer Rates....................... 454 C. The Isotope Effect in a Series of Similar Reactions ............................................... 455 III. The Isotope Effect in the Equilibrium Constant of Photoacids ...................................... 456 IV. The KIE in Photoacid Dissociation ................................................................................. 459 V. Concluding Remarks on the KIE in Photoacid (Phenol-Type) Dissociation ................. 462 References..................................................................................................................................... 462

I. INTRODUCTION The effect of isotope substitution (H/D) on the kinetics of acid – base reactions has long been a primary mechanistic tool in proton-transfer research.1 – 7 The relative ease with which such experiments may be carried out in aqueous solutions of strong acids has made this mechanistic approach very appealing to experimentalists. From the standpoint of theory, the isotope effect on proton transfer dynamics gives access to the microscopic mechanistic details of the process. Laser-induced photoacidity has been recognized for decades as the major experimental tool in the research of bimolecular proton transfer reactions in aqueous solutions. In such experiments, which were pioneered by Fo¨rster and Weller,8,9 one uses suitable organic molecules which are weak acids in the electronic ground state. Upon optical excitation the acidity of the molecules increases considerably and rivals the acidity of strong mineral acids. Molecules which undergo such a transition in their acidity are usually termed photoacids.10 The most commonly used photoacids are simple hydroxyarenes (phenol-like photoacids) such as the 1- and 2-naphthols which increase their acidity by more than a factor of a million when excited to their first electronic singlet state.11 – 16 Combining isotope substitution with photoacid research is potentially a very promising avenue toward unveiling the important mechanistic details of bimolecular proton transfer reactions. Despite extensive experimental and theoretical efforts many principal questions about bimolecular proton-transfer reactions in solutions are still open: in particular, is the reaction coordinate in the transition-state region the hydrogen-bond coordinate or, as expected for sufficiently highly 451

452

Isotope Effects in Chemistry and Biology

exothermic and thereby almost barrierless reaction, is it the coordinate of solvent motion? Some related questions are: (a) what is the rate limiting step of proton transfer in solution; (b) how is proton motion coupled to the solvation process; (c) how important is proton tunneling; (d) are bimolecular proton-transfer reactions between acids and bases in aqueous solutions through-solvents or innersphere processes? Particularly valuable information about molecular mechanism can be obtained by determining the influence of the isotopic substitution on the rate constants and the activation energy of proton transfer. Below we consider the basic kinetic models for acid – base reactions in solutions and apply them to the proton-dissociation reaction of photoacids.

II. GENERAL KINETIC MODELS FOR ACID – BASE REACTIONS IN SOLUTIONS A. THE T WO- STATE P ROTON- TRANSFER R EACTION M ODEL (T HE E IGEN – WELLER M ODEL )9,17 To describe the photoacid dissociation rate in aqueous solution one may apply a general two-state reaction model first explored by Eigen17 and Weller9 in acid –base reactions in the ground state (Scheme 16.1). ROH

kd kr

RO− ··· H+

ks kdiff

RO− + H+

SCHEME 16.1

where ROH is the photoacid, kd ; kr is dissociation and recombination intrinsic rate constants, respectively, ks ; kdiff are the diffusion controlled rate constants for forming and separating the reactive (acid – base) complex. The overall dissociation and the recombination rate constants of an acid (kdis and krec ; respectively) may be found using the steady-state approximation and are given by: ks kd ð16:1Þ kdis ¼ ks þ kr and

kdiff kr ks þ kr

ð16:2Þ

4pDRD N 0 expðRD =aÞ 2 1

ð16:3Þ

krec ¼ kdiff ¼

is the diffusion-limited rate constant of the recombination reaction18,19 which is approached by krec when kr q ks : N 0 is Avogadro number per cm3. RD is the Debye radius, given by z1 z2 e 2 1kB T z1 and z2 are the charge numbers of the proton and the base, e is the electron charge, kB is the Boltzmann constant, T is the temperature, D is the mutual diffusion coefficient, a is the reaction radius (contact distance), and 3DRD ks ¼ 3 ð16:4Þ a ð1 2 expð2RD =aÞÞ is the diffusion limited rate constant of separation of the pair of reactants from their contact distance to infinite separation.

The Kinetic Isotope Effect

453

The overall equilibrium constant of the proton-transfer reaction is given by the ratio of the intrinsic (chemical) equilibrium constant, Ki and KF which is the stability (Fuoss) constant of the pair at contact separation:20 Ka ¼

kdis k k ¼ d s ¼ Ki =KF krec kr kdiff

ð16:5Þ

here, Ki ¼

kd ; kr

KF ¼

kdiff ¼ 4pN 0 a3 expð2RD =aÞ=3 ks

ð16:6Þ

so one can write for Ki and kr ; Ki ¼ Ka KF ¼ 102pKa KF kr ¼

ð16:7Þ

kd 10pKa KF

ð16:8Þ

For strong acids in aqueous solutions at room temperature kr p ks ; and by Equation 16.1 kdis < kd

ð16:9Þ

so for strong acids the apparent dissociation constant is practically equal to the intrinsic protondissociation rate. The situation is different for weak acids. In this case ks p kr and kdis ¼

ks kd kk ø s d ks þ kr kr

ð16:10Þ

so the apparent dissociation rate of weak acids depends on both the intrinsic equilibrium constant Ki and ks : Similarly, for deuterated water one can write, D kdis <

ksD kdD krD

ð16:10aÞ

and the apparent kinetic isotope effect (KIE) of the weak acid dissociation may be estimated using Equation 16.11 D H kdis =kdis ¼

ksH kdH ksD kdD ksH KiH ksH KaH DH DpKa ¼ ¼ 1:4 £ 10DpKa = ¼ H D D D D D ¼ D 10 kr kr ks Ki ks K a D

ð16:11Þ

here DpKa ¼ pKaD 2 pKaH and the ratio DH =DD is calculated for room temperature (< 1.4 in aqueous solutions) and is assumed the only reaction parameter in ksH =ksD which is isotope dependent. The important conclusion from this simple kinetic analysis is that there is a fundamental difference in the observed dissociation rates of strong and weak acids. While the actual proton dissociation stage may be observed directly in strong-acid dissociation, it is impossible to do so in weak-acid dissociation. In the latter case a multistage reaction rate is the directly accessed observable which is made of both the chemical and the diffusive stages of the dissociation reaction.

454

Isotope Effects in Chemistry and Biology

B. FREE- ENERGY C ORRELATIONS OF THE P ROTON (D EUTERON ) T RANSFER R ATES Brønsted relation21,22 represents the earliest example of a linear free-energy relation between reaction rates and equilibrium constants: DG# ¼ aðDG0 Þ #

ð16:12Þ

0

where DG is the free energy of activation and DG is the standard free-energy change in the proton-transfer reaction, DG0 ¼ RT lnð10ÞpKa : The curvature of the Brønsted plot, a, determines how the potential surface of the reactions varies with DG0 : Most modern theories treat proton transfer in aqueous solution as a dynamic process closely coupled with solvent motion and solvent relaxation.23 – 30 These theories assume that the proton-transfer rate between a proton donor and a proton acceptor has a general form akin to a transition-state rate constant which may be written as: ! w DG# kd ¼ exp 2 ð16:13Þ 2p RT where w is the frequency factor of the proton transfer reaction which may also depend on solvent relaxation frequencies. As early as 1924 Brønsted and Pedersen22 argued that for a highly exothermic proton transfer reaction DG# ¼ 0; so that at this limit the curvature of the Brønsted plot approaches zero and kd becomes a constant independent of DG0 : These ideas were refined by Marcus using a reaction model originally developed for outer-sphere (nonadiabatic) electron-transfer reactions. According to the Marcus theory,23 charge-transfer reactions proceed along the solvent coordinate with an intrinsic activation energy equal to 1/4 of the total solvent reorganization energy. The potential energy of such a reaction along the reaction coordinate involves a pair of intersecting parabolas which gives at their intersection point the activation energy needed for the transfer from the reactant to the product state: !2 DG0 # DG#0 ð16:14Þ DG ¼ 1 þ 4DG#0 where DG#0 is the intrinsic activation energy of a symmetric transfer where the total free-energy change ðDG0 Þ following the charge transfer is equal to zero. Marcus theory gives solvent reorganization central role in charge-transfer reactions in polar solvents where solvent reorganization energy is large. After being developed for weak-overlap electron transfers Marcus and Cohen,24 subsequently applied it in a semiempiric way to proton transfer reactions by using a Bond-Energy –Bond-Order (BEBO) model for the proton-transfer coordinate along a preexisting hydrogen bond. The semiempiric BEBO treatment represents the opposite extreme to nonadiabatic electron transfer. Here, bond rupture – bond formation is the principal contributor to the reaction coordinate. In this case the potential energy along the reaction coordinate is initially constant and then rises to a maximum at the transition state and than falls to another constant value. The activation energy is given by: ! DG0 DG#0 DG0 ln 2 # # DG ¼ þ DG0 þ ln cosh ð16:15Þ 2 ln 2 2DG#0 Remarkably, the two different approaches (i.e., Equation 16.14 and Equation 16.15) yield almost identical results in the endothermic branch of the reactions. However, at the exothermic branch the approaches differ considerably. While Marcus theory predicts an “inverted region” where, because of the quadratic nature of Equation 16.14, activation energy “reappears” when the driving force of the reaction becomes very large, it is not so with the BEBO model, Equation 16.15, where the reaction rate assumes a constant (maximal) value in this limit.

The Kinetic Isotope Effect

455

A nontraditional picture for adiabatic proton-transfer reaction in polar environment was recently developed by Kiefer and Hynes.29 Proton nuclear motion was treated quantum mechanically but the proton did not tunnel as the reaction barrier was found to fluctuate because of solvent rearrangements to below the zero-point energy of the proton. This description of nonadiabatic proton-transfer strongly differs from Marcus semiempiric treatment of proton transfer in quantization of the proton and consideration of solvent coordinate as the reaction coordinate. Remarkably, despite the different treatment, Kiefer and Hynes showed that the reaction path of the proton transfer characterized by the values of the quantum-averaged proton coordinate is very similar to the BEBO pathway. A quadratic free energy relationship resembling that of Marcus was found between the activation free energy DG# and the reaction asymmetry DG0 : DG# ¼ DG#0 þ aDG0 þ a0

ðDG0 Þ2 2

ð16:16Þ

With a0 ¼ 1=2; the Brønstead coefficient evaluated at DG0 ¼ 0 and a0 the Brønstead coefficient slope evaluated at DG0 ¼ 0:

C. THE I SOTOPE E FFECT IN A S ERIES OF S IMILAR R EACTIONS It was first suggested by Westheimer and Melander31,32 to correlate KIE with the free-energy change in reaction. They have argued that the KIE, kH =kD ; given by Equation 16.17, would have a maximum value for a symmetrical transition state ðDG0 ¼ 0Þ: kH ¼ expð2ðDG#H 2 DG#D Þ=RTÞ kD

ð16:17Þ

Marcus, assuming that DG #0 is the only isotope-dependent quantity in Equation 16.14 and Equation 16.15 derived a relationship between lnðkH =kD Þ and the free-energy change of the proton transfer. By Equation 16.14, the free-energy dependence of the KIE is given by a symmetric curve having a maximum at DG0 ¼ 0: kH ø exp kD

DG#0H 2 DG#0D RT

!

2

DG0 12 16DG#0H DG#0D

!! ð16:18Þ

or assuming ðDG#0 Þ2 ¼ DG#0H DG#0D in term of maximal KIE for the symmetric reaction k ln H kD

k ø ln H kD

" max

12

DG0 4DG#0

!2 # ð16:18aÞ

The BEBO equation24 (Equation 16.15) results with a more complex relationship: ! !1 0 DG0 ln 2 DG0 ln 2 DG0 ln 2 !B þ ln cosh tanh C 2DG#0 2DG#0 2DG#0 C kH DG#0H 2 DG#0D B B1 2 C ø ln B C RT kD ln 2 @ A ! !1 DG0 ln 2 DG0 ln 2 DG0 ln 2 tanh þ ln cosh C B 2DG#0 2DG#0 2DG#0 C B C B1 2 C B ln 2 A max @ 0

¼ ln

kH kD

ð16:19Þ

456

Isotope Effects in Chemistry and Biology

This yields a KIE curve with maximum at DG0 ¼ 0 which decreases in a bell-like fashion for large lDG0 l Kiefer and Hynes30 using the Equation 16.16 have obtained: kH ðDG0 Þ2 ¼ expð2ðDG#0H 2 DG#0D Þ=RTÞexp ða00H 2 a00D Þ kD 2RT

! ð16:20Þ

and in term of maximal KIE for the symmetric reaction kH ¼ kD

kH kD

max

exp

ða00H

2

a00D Þ

ðDG0 Þ2 2RT

! ð16:20aÞ

The decrease from the maximum KIE value is characterized by a symmetric Gaussian falloff (Equation 16.20). Although Equation 16.20 results from quantum mechanical considerations of the protontransfer reactions it has a similar functional form to the empiric Marcus treatment of the KIE in proton transfer. I thus review the available KIE data of photoacids proton-dissociation using the functional form of Marcus treatment Equation 16.18.

III. THE ISOTOPE EFFECT IN THE EQUILIBRIUM CONSTANT OF PHOTOACIDS The isotope effect on the equilibrium constant of acids, KaH =KaD ; is directly related to the KIE in the acid-dissociation rate constant, kH =kD ; (Equation 16.11). R. P. Bell summarized in 1959 the available data at that time on the isotope effect in acids equilibria and concluded in the first edition of his classic book The Proton in Chemistry1a that a linear correlation exists between the acid strength ðKa Þ and KaH =KaD : Bell retreated from this observation in the second edition of his book published some 14 years later.1b In Bell’s words “Early results for a number of weak acids suggested that the value of KaH =KaD decreased regularly with increasing acid strength, but when more extensive experimental data are considered there appears to be no real basis for this generalization, except perhaps as an ill-defined qualitative trend.” However, after being so decisive in his judgment Bell relaxes this statement and immediately continues referring indirectly to the correlation he had published in the first edition of his book: “There is some evidence that such a relation holds approximately for a closely related series such as the phenols and alcohols.” In 1991, G. Wilse Robinson33 rediscovered the apparent relation between the equilibrium constant of phenols and alcohols and the isotope effect in their equilibria and extended it to include excited-state photoacids of the phenol family. The correlation he found was over 11 pKa units in acid strength and looked impressively linear. I have reviewed for this contribution the data appearing in Bell’s book and in Robinson’s paper and added additional recent data on photoacids that were mainly gathered in the past 10 years (Tables 16.1– 16.3).33 – 40 Figure 16.1 shows that the general trend observed by Bell and Robinson of decreasing isotope effect with increasing acidity of the acid also appears to exist for very strong photoacids having negative pKa’s. However the spread in the data points is larger than in ground-state acids (Table 16.1) and larger than the one found by Robinson in the smaller set of moderately strong photoacids (Table 16.2). The correlation of the equilibrium constant of the full set of photoacids (including the data appearing in Table 16.3) with their isotope effect is shown in Figure 16.2. The observed spread in the data set of the excited-state equilibrium constant of deuterated photoacids in D2O may be the result of the considerable experimental error in their determination by several experimental methods of which none is of analytic accuracy.11,15 In comparison, the equilibrium

The Kinetic Isotope Effect

457

TABLE 16.1 D pK H a and pK a of Ground-State Acids Compound

pK H a

pK D a

Reference

2,6-Dinitrophenol Bromothymol blue 3,5-Dinitrophenol p-Nitrophenol Water Phenol 4-Phenolsulfonate 2,4-Dinitrophenol 2,5-Dinitrophenol 4-Nitrophenol 4-Bromophenol 3-Methoxyphenol 4-Methoxyphenol 4-Hydroxyphenyltrimethylammonium chloride 2-Naphthol

3.58 6.15 6.70 7.24 15.74 10 8.97 4.06 5.19 7.22 9.35 9.62 10.24 8.34 9.47

4.03 6.68 7.31 7.80 16.55 10.62 9.52 4.55 5.70 7.77 9.94 10.20 10.85 8.90 10.06

35 35 35 35 36 34 34 34 34 34 34 34 34 34 34

TABLE 16.2 D ¨ rster Cycle Calculations pK H a and pK a of Excited-State Acids Found by Fo Compound

pK pH

PK pD

Reference

Phenol 4-Phenolsulfonate 4-Bromophenol 3-Methoxyphenol 4-Methoxyphenol 4-Hydroxyphenyltrimethylammonium chloride 2-Naphthol

4.1 2.3 2.9 4.6 5.7 1.6 3.0

4.6 2.7 3.4 5.1 6.2 2.0 3.5

34 34 34 34 34 34 33

TABLE 16.3 D pK H a and pK a of Excited-State Acids Found by Kinetic Measurements Compound

pK H a

pK D a

Reference

5-Cyano-2-naphthol 6-Cyano-2-naphthol 7-Cyano-2-naphthol 8-Cyano-2-naphthol 1-Naphthol 1-Naphthol 2-Naphthol 1-Naphthol 2-sulfonate

20.75 20.37 20.21 20.76 0.4 20.16 2.72 1.58

20.44 20.11 0.10 20.57 0.81 0.30 3.12 2.02

37 37 37 37 38 39 38 38

458

Isotope Effects in Chemistry and Biology 0.9 0.8

Ground and excited-state acids

pK aD − pK aH

0.7 0.6 0.5 0.4

Förster cycle calculations kinetic measurements ground-state acids

0.3 0.2 0.1 0.0

−4 −2

0

2

4

6

8

10 12 14 16 18

pK aH

FIGURE 16.1 Isotope effects on equilibrium dissociation constants in acid dissociation. Open triangles: ground-state acids. Circles: excited-state acids, open symbols: Fo¨rster cycle calculations. Full symbols, kinetic calculations. Full line: linear regression DpK ¼ 0:34 þ 0:03pKH a . R ¼ 0:93:

constant of deuterated ground-state acids was determined directly by titration in D2O, a much more reliable method than the methods used for the estimation of excited-state equilibrium constants. These methods have included Fo¨rster cycle calculations and analysis of the reversible deuterontransfer kinetics of photoacids in their excited state.11,15 My conclusion is that within the set of phenol-type photoacids that has been investigated so far the linear correlation between the isotope effect in their equilibria and their acidity constant seems to be a valid observation. The observed correlation is similar (but not necessarily identical) to the one observed for ground-state phenols and alcohols. With the aid of Equation 16.11 this apparent correlation makes a useful experimental tool for predicting the KIE in the dissociation of weak photoacids which is discussed below.

0.60

Excited-state acids

0.55 pK aD − pK aH

0.50 0.45 0.40 0.35 0.30

direct kinetic measurements Förster cycle calculations

0.25 0.20 0.15

−2

−1

0

1

2

3

4

5

6

7

pK aH

FIGURE 16.2 Isotope effects on equilibrium dissociation constants in excited-state acid dissociation. Open symbols: Fo¨rster cycle calculations. Full symbols: kinetic calculations. Full line: linear regression DpK ¼ 0:32 þ 0:04pKH a . R ¼ 0:88:

The Kinetic Isotope Effect

459

IV. THE KIE IN PHOTOACID DISSOCIATION The proton-dissociation reaction of strong photoacids is readily measured directly in aqueous solutions. For typical time-resolved dissociation profiles of a strong photoacid (1-naphthol 5-sulfonate)41 in water and D2O see Figure 16.3. In this case, the dissociation rate of the strong photoacid is much faster than its fluorescence decay. The situation changes dramatically for photoacids having pKpa . 2, at which the dissociation rate of the phenol-type photoacids becomes smaller than the decay rate of the excited state which is usually 1– 10 nsec. This limits the observation range of photoacid dissociation to relatively strong photoacids. Until our own recent observation of the proton dissociation reaction of hydroxypyrene in water,42 there has been no report on the dissociation rate of a phenol-type photoacid having a pKpa higher than 3.5. The weakest photoacid for which the KIE has been reported was 2-naphthol, having a pKpa of 2.7 –2.8 in H2O. The available data on the KIE of photoacids were recently reviewed by Formosinho, Arnaut, and Barroso.43 The KIE was analyzed using the intersecting-state model (ISM) of proton transfer which is an empiric, BEBO-type, reaction model. The fit between the calculated and observed KIE was found to be only qualitative. Not intentionally, the review also gave an opportunity to examine critically the experimental KIE values reported by several research groups. The experimental KIE value given for HPTS (8-hydroxypyrene-1,3,6-trisulfonate), one of the most investigated photoacids, was 4.71 while the calculated value using ISM correlation was 2.68. However, closer examination of the literature reveals considerable variations in the reported value of the KIE of HPTS. Politi et al.44 reported a value of 5.6 while the Huppert group using much more sophisticated kinetic analysis reported in a variety of experiments extending over a 10-year period KIE values between 2.4 and 4.1,45 the best value being most probably just in the middle of this range, about 3.2 at 293 K extracted by us from revaluating the kinetic data reported in Ref. 45. The example of HPTS points to the fact that one needs to be critical when analyzing experimentally found KIE values even for strong, well-behaved photoacids such as HPTS. The dependence of the KIE on the pKa values of a family of similarly structured acids is predicted by Equation 16.18 to be bell-shaped and symmetric about the center given at pKa ¼ 0. Figure 16.4 shows a schematic drawing of the functional form of the KIE using Equation 16.18. Such a correlation, spanning over a range of 25 pKa units, was shown by Bell in his book (page 265 in Ref. 1b) for a family of carbon acids. The subject was further discussed in the 1975 symposia of

Normalized Intensity

1

1N5S

1 D2O H2O

1 1 0 0

260

280

300

320

340

Time (ps)

FIGURE 16.3 The isotope effect in the proton dissociation of 1-naphthol-5-sulfonate (1N5S), T ¼ 208C. Upper decay: 1N5S deuteron dissociation in D2O, t1 ¼ 31 psec (93%). Lower decay: 1N5S proton dissociation in water, t1 ¼ 15 psec (99%).

460

Isotope Effects in Chemistry and Biology 4.0 (k H /kD)max

Kinetic isotope effect

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 −10

−8

−6

−4

−2

0 H pK a

2

4

6

8

10

FIGURE 16.4 The functional form of the KIE predicted by the Marcus equation, Equation 16.18.

the Faraday Society by Bell and Kresge46 with references to Bell’s1b and Bordwell and Boyle’s data.47 Several less spectacular examples of the bell-shape dependence of the KIE were discussed more recently by Su¨hnel.48 Although the situation is far from being clear-cut, the usefulness of Equation 16.18 seems to be established for families of closely-related ground-state proton-transfer reactions. The situation is different for excited-state acids. The limited available data on the KIE of photoacids (Table 16.4) need to be analyzed with caution. One may start such a kinetic analysis by using Equation 16.1, assuming the isotope effect on the intrinsic proton-transfer rate, kd ; has the functional form given by Equation 16.15. Assuming that ks q kr up to pKpa ¼ 0 and that kr q ks from pKpa ¼ 4 onward (see Figure 16.5 for details) one may arrive by using Equation 16.10 and Equation 16.11 at a projected profile for the observed KIE in photoacid dissociation. Clearly, the measured KIE of photoacids having a pKpa below 1 is expected to drop sharply as a function of increased photoacidity (Figure 16.4), while the measured KIE of photoacids with a pKpa above 4 is predicted to increase monotonously with the isotope effect in their equilibria (Figure 16.1). The experimental data gathered so far are shown in Figure 16.6 and seem to confirm the first prediction: The isotope effect

TABLE 16.4 The KIE in the Observed Proton-Dissociation Rate of Some Phenol-Like Photoacids Compound

kH/kD

References

1-Naphthol 5-sulfonate 5-Cyano-1-naphthol 1-Naphthol 1-Hydroxypyrene 2-Naphthol HPTS (8-hydroxypyrene-1,3,6-tri-solfunate) 5-Cyano-2-naphthol 6-Cyano-2-naphthol 7-Cyano-2-naphthol 8-Cyano-2-naphthol 2-Naphthol-3,6-disulfonate 1-Naphthol-2-sulfonate

2 1.6 3.2 3.9 2.64 3.16 2.33 3.33 3.06 2.08 3.6 2.8

41 46 39 42 38 45 37 37 37 37 40 38

The Kinetic Isotope Effect

461

12

log (k[sec −1])

10 8

kd kr ks kdis

6 4 2

−8

−6

−4

0

−2

2

4

6

8

pK aH

FIGURE 16.5 The correlation between the excited-state dissociation rate constants of phenol-type photoacids ðkd Þ and their corresponding equilibrium constant values using Marcus (BEBO) correlation, Equation 16.15 with DG#0 ¼ 2:5 kcal/mol and ðkd Þmax < ð2 psecÞ21 (dashed line).16b The full line is the overall dissociation rate of the reactions, kdis ; (Equation 16.1), as correlated with the experimental data assuming kd is the intrinsic proton dissociation rate, Equation 16.15. The dashed-dotted line is the calculated proton recombination rate constants, kr using Equation 16.8. The dotted line is the diffusion controlled rate constant of separation of the products calculated from the contact distance to infinite separation, ks (Equation 16.4, z1 ; z2 ¼ 21).

of phenol-type photoacids having a pK pa value of about zero has been consistently found to be between 3 and 3.6, while stronger photoacids of this family exhibit much smaller KIE’s down to about 1.6.49 At the other limit of the correlation there is only one data point so far, the one for hydroxypyrene (HP, pK pa ¼ 3.8– 4.1). The apparent KIE of HP is the largest found so far for a phenol-type photoacid, KIE (HP) ¼ 3.8 ^ 0.1.42 This value agrees well with the KIE value calculated using Equation 16.11. 6

kH /kD

4

2

0 −10 −8

−6

−4

−2

0

2

4

6

8

10

pK aH

FIGURE 16.6 The KIE of the proton dissociation rate of phenol-type photoacids. Dotted-dashed line: Marcus equation, Equation 16.18. Dashed line: Marcus BEBO expression, Equation 16.19 with ðkdH =kdD Þmax ¼ 3:5: Full line, the predicted isotope effect combining the two-state model, Equation 16.1 with the experimental values of the isotope effect on the acid equilibria DpKa (Figure 16.1 and Equation 16.11).

462

Isotope Effects in Chemistry and Biology

These findings, although summarizing experimental studies which have been carried out over a period of roughly two decades, are still not fully decisive. However, these findings allow some general observations which are discussed below.

V. CONCLUDING REMARKS ON THE KIE IN PHOTOACID (PHENOL-TYPE) DISSOCIATION The following conclusions may be drawn from the kinetic data gathered so far on the KIE in photoacid dissociation: a. The KIE found in strong photoacid dissociation is relatively small and peaks around pKap ¼ 0: b. The KIE tends to decrease rapidly below pKap ¼ 0: c. Judging from the measured KIE values and the isotope effect in the photoacid equilibria, the KIE in the opposite proton-recombination reaction should be close to unity. This conclusion is supported by the kinetic analysis of reversible excited-state proton-transfer reactions.50 In all reported cases where such a kinetic analysis was undertaken the KIE in the proton-recombination rate was found to be between unity and 1.6.37 – 40,45 This important observation about the significant difference in the KIE of kd and kr still awaits detailed explanation. d. Judging from some early measurements on a different class of photoacids, the amine photoacids, the observed KIE in photoacid dissociation may well depend on the chemical nature of the photoacid. In the case of amine photoacids a large isotope effect of about 5 was observed and was found, in most cases studied so far, to be independent of the photoacid pKpa in the acidity range, 27 , pKap , 0:41,51 This behavior contrasts with the one found for phenol-type photoacid suggesting that proton dissociation from neutral oxygen acids proceed by a different mechanism than proton dissociation from protonated nitrogen acids which are positively charged prior to dissociation.16b Currently, there is no theoretical modeling of these “cationic” proton-transfer reactions. Transient solvation effects and large inner-sphere rearrangements may be the crucial factor in determining the pathway of cationic proton-dissociation reaction of amine acids. These observations on the KIE of amine photoacids draw further attention to Bell’s remarks given in Section III of this chapter. As Bell pointed out it has indeed become clear that one should be extremely cautious when trying to generalize processes as sensitive to environment (solvent) and intramolecular structure as the KIE in photoacids proton dissociation. A similar cautionary remark may be made about any theoretical modeling (at any level of theory) that suggests one general mechanistic treatment for the KIE in all proton-transfer reactions. A more realistic view is to assume that there are several possible pathways leading to proton transfer and to the observed KIE in proton-transfer. Generalizations should be limited to specific groups of acids and may become realistic only after extensive data gathering and after critically examining the various experimental findings. It was the aim of this contribution to point out that such a goal is probably at hand for the phenol-type family of excited-state acids.

REFERENCES 1 (a) Bell, R. P., The Proton in Chemistry, 1st ed., Cornell University Press, Ithaca, NY, Chap. 6, 1959. (b) Bell, R. P., The Proton in Chemistry, 2nd ed., Cornell University Press, Ithaca, NY, Chap. 6, 1973. 2 Caldin, E. and Gold, V., Proton-Transfer Reactions, Chapman & Hall, London, 1975. 3 Hibbert, F., Adv. Org. Chem., 26, 255, 1990.

The Kinetic Isotope Effect

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4 Kresge, A. J., Isotope Effects on Enzyme-Catalyzed Reactions, Cleland, W. W., O’Leary, M. H., and Northrop, D. B., Eds., University Park Press, Baltimore, MD, p. 37, 1977. 5 Kresge, A. J. and Silverman, D. N., Methods Enzymol., 308, 726, 1999. 6 Silverman, D. N., BBA, 1458, 88, 2000. 7 Kohen, A. and Klinman, J. P., Acc. Chem. Res., 31, 397, 1998. 8 Fo¨rster, Th. Z., Elecrtochem. Angew. Phys. Chem., 42, 531, 1952. 9 Weller, A. Z., Electrochemistry, 58, 849, 1954. Weller, A. Z., Phys. Chem. N.F., 17, 224, 1958. 10 Tolbert, L. M. and Haubrich, J. E., J. Am. Chem. Soc., 116, 10593, 1994. 11 Ireland, J. F. and Wyatt, P. A. H., Adv. Phys. Org. Chem., 12, 131, 1976. 12 Shisuka, H., Acc. Chem. Res., 18, 41, 1985. 13 Arnaut, L. G. and Formosinho, S. J., J. Photochem. Photobiol., A75, 21, 1993. 14 Wan, P. and Shukla, D., Chem. Rev., 93, 571, 1993. 15 Tolbert, L. M. and Solntsev, K. M., Acc. Chem. Res., 35, 19, 2002. 16 (a) Pines, E., UV-Visible spectra and photoacidity of phenols, naphthols and pyrenols, In The Chemistry of Phenols, Rappoport, Z., Ed., Wiley, New York, pp. 491– 529, 2003; (b) Pines, E. and Pines, D., Ultrafast Hydrogen Bonding and Proton Transfer Processes in the Condensed Phase, Elsaesser, T. and Bakker, H. J., Eds., Kluwer, Dordrecht, pp. 155–184, 2002. 17 (a) Eigen, M., Kruse, W., and De Maeyer, L., Prog. React. Kinet., 2, 285, 1964; (b) Eigen, M., Angew. Chem. Int. Ed. Engl., 80, 5059, 1958. 18 Smoluchowski, M. Z., Phys. Chem., 92, 129, 1917. 19 Debye, P., Trans. Electrochem. Soc., 82, 265, 1942. 20 Fuoss, R. M., J. Am. Chem. Soc. Electrochem., 54, 531, 1950. 21 (a) Brønsted, J. N., Rev. Trav. Chim., 42, 718, 1923; (b) Brønsted, J. N., Chem. Rev., 5, 231, 1928. 22 Brønsted, J. N. and Pedersen, K. Z., Phys. Chem., 108, 185, 1924. 23 (a) Marcus, R. A., J. Chem. Phys., 24, 966, 1954; (b) Marcus, R. A., J. Phys. Chem., 67, 853, 1963; (c) Marcus, R. A., J. Ann. Rev. Phys. Chem., 15, 155, 1964; (d) Marcus, R. A., J. Phys. Chem., 44, 679, 1965; (e) Marcus, R. A., J. Phys. Chem., 72, 891, 1968; (f) Marcus, R. A., J. Am. Chem. Soc., 91, 7244, 1969; (g) Marcus, R. A., Faraday Symp. Chem. Soc., 10, 60, 1975; (h) Marcus, R. A., Faraday Discuss. Chem. Soc., 74, 7, 1982. 24 Cohen, A. O. and Marcus, R. A., J. Phys. Chem., 72, 4249, 1982. 25 Levich, V. G., Dogonadze, R. D., German, E. D., Kuznetsov, A. M., and Kharkats, Yu. I., Electrochim. Acta, 15, 353, 1970. 26 Kuznetzov, A. M., Charge Transfer in Physics Chemistry and Biology: Physical mechanisms of Elementary Processes and an Introduction to the Theory, Gordon and Breach Pubs., Amsterdam, 1995. 27 (a) Kornyshev, A. A., Kuznetsov, A. M., Spohr, E., and Ultstrup, J., J. Phys. Chem., B107, 3351– 3366, 2003; (b) Kuznetsov, A. M., Spohr, E., and Ultstrup, J., Can. J. Chem., 77, 1085, 1999. 28 Krishtalik, L. I., BBA, 1458, 6 – 27, 2000. 29 (a) Kiefer, P. M. and Hynes, J. T., J. Phys. Chem., A106, 1834, 2002; (b) Kiefer, P. M., and Hynes, J. T., J. Phys. Chem., A106, 1850, 2002; (c) Kiefer, P. M. and Hynes, J. T., Sol. St. Ion., 168, 219, 2004. 30 (a) Kiefer, P. M. and Hynes, J. T., J. Phys. Chem., A107, 9022, 2003; (b) Kiefer, P. M. and Hynes, J. T., Isr. J. Chem., 44, 171, 2004. 31 Westheimer, F. H., Chem. Rev., 61, 265, 1961. 32 Melander, L., Isotope Effects on Reaction Rates, Ronald Press, New York, 1960. 33 Robinson, G. W., J. Phys. Chem., A95, 1038, 1991. 34 Wehry, E. L. and Rogers, L. B., J. Am. Chem. Soc., 88, 351, 2003. 35 Martin, D. C. and Butler, J., J. Chem. Soc., 1366, 1939. 36 Kingerly, R. W. and LaMer, V. K., J. Am. Chem. Soc., 63, 3256, 1941. 37 Huppert, D., Tolbert, L. M., and Linares-Samaniego, S., J. Phys. Chem., A101, 4602, 1997. 38 Krishnan, R., Lee, J., and Robinson, G. W., J. Phys. Chem., 94, 6365, 1990. 39 Pines, E., Tepper, D., Magnes, B.-Z., Pine, D., and Barak, T., Ber. Buns. fur Phys. Chem., 102, 504, 1998. 40 Masad, A. and Huppert, D., J. Phys. Chem., 96, 7324, 1992. 41 Pines, D. and Pines, E., Thesis, Ben-Gurion University, 2005. 42 (a) Pines, D., Barak, T., Magnes, B. Z., and Pines, E., The Photoacidity of 1-Hydroxypyrene from Fo¨rster cycle to Marcus Theory, Summer School, Molecular Basis of Fast and Ultrafast Processes,

464

43 44 45

46 47 48 49 50 51

Isotope Effects in Chemistry and Biology Algarve (Portugal), 12 – 15 June 2003; (b) Kombarova, S. V. and Il’ichev, Y. V., Langmuir, 20, 6158, 2004. Barroso, S. J., Arnaut, L. G., and Formosinho, S. J., J. Photchem. Photobiol., A154, 13, 2002. Politi, M. J., Brandt, O., and Fendler, J. H., J. Phys. Chem., 89, 2345, 1985. (a) Agmon, N., Pines, E., and Huppert, D. J., Chem. Phys., 88, 5631, 1988; (b) Pines, E., Huppert, D., and Agmon, N., Chem. Phys., 88, 5620, 1988; (c) Huppert, D., Pines, E., and Agmon, N., J. Opt. Soc. Am., B7, 1545, 1990; (d) Pines, E., Huppert, D., and Agmon, N., J. Phys. Chem., 95, 666– 674, 1991; (e) Poles, E., Cohen, B., and Huppert, D., Isr. J. Chem., 39, 347, 1999. Kresge, A. J. and Bell, R. P., in General Discussion, Proton Transfer. Simposia of the Faraday Society No. 10, Chemical Society, London, pp. 89 – 90, 91 – 92, 1975. Bordwell, F. G. and Boyle, W. J. Jr., JACS, 97, 3447, 1975. Su¨hnel, J., J. Phys. Org. Chem., 3, 62 – 68, 1990. Pines, E., Pines, D., Barak, T., Tolbert, L. M., and Haubrich, J. E., Ber. Bunsen-Ges. Phys. Chem., 102, 511, 1998. Pines, E. and Huppert, D., J. Phys. Chem., 84, 3576, 1986. Shiobara, S., Tajina, S., and Tobita, S., Chem. Phys. Lett., 380, 673, 2003.

17

The Role of an Internal-Return Mechanism on Measured Isotope Effects Heinz F. Koch

CONTENTS I. II.

The Internal-Return Mechanism for Hydron-Transfer Reactions................................... 466 The Internal-Return Mechanism for Alkoxide-Promoted E2 Dehydrohalogenations ................................................................................................ 468 III. Chlorine Isotope Effects vs. the Element Effect ............................................................. 470 IV. Hydron Transfer from Alcohols to Carbanions............................................................... 471 V. Conclusions ...................................................................................................................... 472 References..................................................................................................................................... 472 This chapter considers interpretation of the magnitude of experimental primary kinetic isotope effects (PKIEs) associated with alcholic alkoxide-promoted dehydrohalogenation reactions. The existence of a heavy isotope of hydrogen was first suggested around 1930 and by 1933 Lewis and Macdonald1 obtained a sample of water consisting of about 65% deuterium by electrolysis. Lewis, in a communication to the editor2 prior to reporting their experimental results, made the statement: The separation of any isotope in sufficient quantity to permit investigation not only of its spectroscopic but also of its other chemical and physical properties suggests a wide range of interesting experiments but the isotope of hydrogen is, beyond all others, interesting to chemists. I believe that it will be so different from common hydrogen that it will be regarded almost as a new element. If this is true the organic chemistry of compounds containing the heavy isotope of hydrogen will be a fascinating study. Physical organic chemists accepted the challenge, and in 1955 Wiberg3 published a review of deuterium isotope effects with 241 references. In 1940, the Ingold group4 postulated that the base-promoted dehydrohalogenations of alkyl halides to form alkenes are concerted (E2) with bonds to the b-hydrogen and the halide broken simultaneously in the transition structure. X b C a C

d c

H OR

465

466

Isotope Effects in Chemistry and Biology

In 1945, Skell and Hauser5 demonstrated that deuterium was not incorporated into b-phenethyl bromide during the ethoxide-promoted elimination of HBr when the reaction was carried out in C2H5OD. This was interpreted as experimental evidence that the reaction was synchronous and not stepwise. Cristol and Fix6 point out that a negative result for deuterium incorporation could come from two possible explanations. The process could be concerted; however, the carbanion intermediate could eliminate halide faster than hydron transfer and this would also be consistent with no deuterium incorporation. When the isomer of 1,2,3,4,5,6-hexachlorocyclohexane that has all the chlorines trans to each other [beta-benzene hexachloride], reacts with sodium ethoxide in deuterated ethanol for one half-life, a small amount of deuterium was found in the recovered starting chloride. This was interpreted as demonstrating the existence of a carbanion intermediate in that syn-elimination reaction. Earlier work by the Cristol group7 supplied the rates and activation entropies for the reaction of four of the isomers of benzene hexachloride with sodium hydroxide in 76% ethanol: alpha, 0.500 M21 s21 and 2 1.0 eu; beta, 2.11 £ 1025 M21 s21 and þ 20.2 eu; gamma, 0.151 M21 s21 and þ 3.6 eu; epsilon, 0.182 and 6.5 eu. The extremely slow reaction of the beta isomer is consistent with an E2 reaction occurring faster by a trans elimination of the hydrogen halide than for a syn elimination. Shiner8 used the hydrogen PKIE to investigate the mechanism of the alkoxide-promoted dehydrobromination of 2-bromopropane. A k H/k D ¼ 6.7 was obtained by comparing the rates of ethanolic sodium ethoxide-promoted dehydrobromination of CH3CHBrCH3 vs. CD3CHBrCD3. This confirmed that the carbon to hydrogen bond was broken in the rate limiting step of the mechanism. In the concerted elimination mechanism there should be both a hydrogen and leaving group isotope effect. Since the measurement of heavy atom isotope effects requires special mass spectroscopy equipment, Bunnett9 suggested an “element effect” as a substitute for the heavy atom isotope effect as a criterion of mechanism for aromatic nucleophilic substitution reactions. Bartsch and Bunnett10 applied this concept to the methoxide-promoted dehydrohalogenations of CH3CHBrCH2CH2CH2CH3 and CH3CHClCH2CH2CH2CH3. The element effect, k HBr/k HCl, for the E2 eliminations were 38 for the formation of 1-hexene, 51 for the formation of trans-2 hexene and 46 for the formation of cis-2-hexene. Since then it was accepted that a combination of a hydrogen PKIE and a significant k HBr/k HCl was experimental evidence that an alkoxide-promoted dehydrohalogenation occurred by the concerted E2 mechanism. The stereochemistry of the E2 concerted mechanism should also be anti to allow a proper alignment of the orbitals. Normal hydrogen isotope effects are largely due to zero-point energy differences of the stretching frequencies of C– H vs. C– D bonds. With allowance for bending vibrations this would result in a kH =kD ¼ 10 and kH =kT ¼ 27 at 258C.11 The effects for O – H vs. O – D bonds are slightly larger with k H/k D ¼ 13 and k H/k T ¼ 39. If only the stretching frequencies are considered the value for k H/k D would be 6.2 for C – H and 7.9 for O – H. Melander12 and Westheimer13 suggested that lower values of k H/k D could be due to residual zero-point energy in an asymmetric transition structure. For this reason the magnitude of the hydrogen PKIE was often used to assign early or late transition structures for hydron-transfer reactions. Smaller values that are due to asymmetric transition structures should still obey the Swain– Schaad relationship, k H/k D ¼ (k D/k T)2.26 (see Ref. 14). Experimental PKIE that are larger than the expected values were attributed to quantum mechanical tunneling through the barrier.11 To explain the near-unity experimental PKIE associated with some hydron-exchange reactions, Cram15 introduced the concept of internal return, a no-reaction process.

I. THE INTERNAL-RETURN MECHANISM FOR HYDRON-TRANSFER REACTIONS The internal-return mechanism for a methoxide-catalyzed hydron-exchange reaction of a carbon acid is shown in Scheme 17.1.

The Role of an Internal-Return Mechanism on Measured Isotope Effects DOMe C

H −O EC-h

CH3

DOMe

H

C− H

k −1

C

C− H O

DOMe

CH3

H k −2

C− + H

O CH3

CH3

DOMe

DOMe kExch

C− + D

O

CH3

HOMe

DOMe FC-d

FC-h DOMe D

O

CH3

HOMe FC-d

O

FC-h

DOMe H

DOMe

H

k2

DOMe HB-h

C−

CH3

HB-h DOMe

C− +

DOMe

H

k1

467

DOMe C− D

O CH3 HOMe

HB-d

SCHEME 17.1 The internal-return mechanism for methoxide catalyzed proton exchange.

After forming an encounter complex with a methoxide ion that is solvated by two methanols, EC-h, the proton can be transferred from the C – H to methoxide, kH 1 , forming a hydrogen-bonded carbanion intermediate, HB-h. The proton can return to carbon, kH 21, in the internal return step or break the hydrogen bond forming the intermediate FC-h, kH 2 . A slight solvent reorientation, kExch, leads to an O –D bond, FC-d, that is in the best position to form a new hydrogen-bonded species, HB-d. If kexc . kH 22, the kinetic expression for the internal-return mechanism is kObs ¼ ½k1 k2 =½k21 þ k2 ; which has two extremes. When k21 q k2 then kObs ¼ ½k1 =k21 k2 : Since the isotope effect associated with k1 is canceled by an isotope effect associated with k21 ; the result is a near-unity experimental PKIE. When k2 q k21 ; kObs ¼ k1 ; the experimental PKIE is the one associated with the breaking of a C – H bond. When the return and forward rates are competitive, all three rate constants contribute to kObs : The experimental PKIE can result in values similar to those predicted by an asymmetric transition structure with residual zero-point energy. The major difference would be that effects resulting from an internal-return mechanism would not obey the Swain – Schaad relationship. In 1971 the Streitwieser group16 made a significant contribution to analyzing experimental PKIE values resulting from reactions where both k21 and k2 contributed to kObs : Using single temperature rate constants for all three hydrogen isotopes and deviations from the Swain – Schaad relationship, they calculate an internal-return parameter, a ¼ k21 =k2 : This parameter can then be used to calculate the rate constant for the hydron transfer: k1 ¼ kObs ða þ 1Þ: The Streitwieser group applied this analysis to experimental data for the methanolic sodium methoxide-promoted hydron exchange of triphenylmethane to give aT ¼ 0:66: This resulted in increasing a ðkD =kT ÞObs of 1.34 at 97.78C to a k1D =k1T of 1.85 for the hydron-transfer step. The experimental PKIE data for the exchange reactions of 9-phenylfluorene at 258C gave a value for a T of 0.016. This suggests the amount of internal return is very small in the fluorene system, and the

468

Isotope Effects in Chemistry and Biology

value of the ðkD =kT ÞObs ¼ 2:50 would only increase slightly to a k1D =k1T of 2.54. A DpKa of 13 between triphenylmethane and 9-phenylfluorene should result in different transition structures for hydron transfer, and translate into a difference in the PKIE associated with their reactions. However, the extrapolation of k1D =k1T for 9-phenylfluorene to 1008C gives a value close to that calculated for triphenylmethane.16b

II. THE INTERNAL-RETURN MECHANISM FOR ALKOXIDE-PROMOTED E2 DEHYDROHALOGENATIONS The PKIE associated with the methanolic sodium methoxide-promoted dehydrochlorination of C6H5CiHClCF2Cl [iHyH or D] over a 508C range resulted in anomalous Arrhenius parameters with EaD 2 EaH ¼ 0:0 and AH =AD ¼ 2:4 for kH =kD ¼ 2:35:17 Isotope effects should result from the difference of experimental activation energies, EaH and EaD ; with AH =AD ¼ 1: Similar results were obtained for the methoxide-promoted dehydrobromination of C6H5CiHBrCF2Br with EaD 2 EaH ¼ 0:1 and AH =AD ¼ 3:0 for kH =kD ¼ 4:00: Both eliminations occur with an experimental hydrogen PKIE and an element effect, kHBr =kHCl ¼ 52:18 The single temperature data could have been interpreted as a concerted E2 mechanism with asymmetric transition structures. The smaller hydrogen PKIE for the dehydrochlorination than that for the dehydrobromination could be attributed to a later transition structure for the HCl elimination since the C – Cl bond is stronger than the corresponding C –Br bond. However, the reaction could proceed via an internal-return mechanism (Scheme 17.2), where the elimination of the leaving group, kElim, can occur from the hydrogen-bonded carbanion intermediate faster than the breaking of the hydrogen bond. What Arrhenius parameters are expected from a reaction that occurs via an internal-return mechanism? Using Equation 17.1, Dahlberg calculated the anticipated Arrhenius behavior for reactions proceeding by an internal-return mechanism.19 For dehydrohalogenations, k2 is replaced by kElim. " # " # H D H D kobsd ðE1D 2 E21 Þ 2 ðE1H 2 E21 Þ 1 þ ðA2 =A21 ÞexpððE21 2 E2 Þ=RTÞ ln D ¼ þ ln ð17:1Þ H RT kobsd 1 þ ðA2 =A21 ÞexpððE21 2 E2 Þ=RTÞ The first term in the equation is for the equilibrium isotope effect, DH1H 2 DH1D ¼ 46 cal/mol. The D H initial calculations then used a difference in E21 2 E21 of 1000 cal/mol and A2 ¼ A21 for both H D H isotopes. A maximum k =k of 6.02 was obtained when E21 2 E2 was 10,000 cal/mol (no internal H D H return) and a minimum k =k ¼ 1:08 for E21 2 E2 ¼ 210; 000 cal/mol (equilibrium condition). H With E21 2 E2 varying from 2 2000 to 1000 cal, the calculated Arrhenius parameters resulted in

a Cl C c

C

a

DOMe H −O

d EC-h

CH3

H k −1

DOMe

Cl C d

Cl

C c

b

C−

DOMe H O

d

kElim

CH3

a c

DOMe

C C

b d

HB-h a

c

H k1

b

DOMe

− C H O

CH3

DOMe HB-h

a H

k2

H k −2

Cl C c

b

C− + d

DOMe H O

CH3

DOMe

FC-h

SCHEME 17.2 The internal-return mechanism for methoxide-promoted dehydrohalogenation.

Cl−

The Role of an Internal-Return Mechanism on Measured Isotope Effects

469

AH =AD values between 1.31 and 0.75.20 There is no reason why A2 should equal A21 and varying this ratio can give isotope effects with mid-range values of kH =kD and normal Arrhenius behavior as well as kH =kD values of a normal magnitude with low AH =AD values that are normally considered to come from reactions that feature quantum mechanical tunneling. No model was found to explain the large AH =AD values obtained for the C6H5CHXCF2X systems; however, the model worked well for several other systems. Shiner and Smith21 obtained Arrhenius parameters for ethanolic sodium ethoxide-promoted dehydrobromination reactions of C6H5CH(CH3)CH2Br over a 508C range. The kH =kD of 7.51 at 258C was normal, but the AH =AD of 0.4 was anomalous and considered as experimental evidence for a reaction occurring with quantum mechanical tunneling. Shiner and Martin22 then reported tritium rates as well as correcting their earlier rates for small amounts of a substitution reaction that formed C6H5CH(CH3)CH2OC2H5. Corrected values at 258C were kH =kD ¼ 7:82 and kD =kT ¼ 2:65: To satisfy the Swain –Schaad relationship the ðkH =kT ÞObs of 20.6 should be 17 to 26% greater with a ðkH =kT ÞObs of 23 to 26.23 An equation similar to Equation 17.1 can be written to model the Arrhenius behavior of the experimental ðkD =kT ÞObs ; and their results were modeled using the following values: H A2 =A21 ¼ 0:1; DH1H 2 DH1D ¼ 46 cal/mol; DH1D 2 DH1T ¼ 20 cal/mol; E21 2 E2 ¼ 1950 cal/ 24 D T mol; E21 2 E2 ¼ 3290; E21 2 E2 ¼ 3862 cal/mol. Arrhenius parameters for methoxide-promoted eliminations of four YC6H4CiHXCH2X systems were modeled for the three hydrogen isotopes (Table 17.1).25 Results for the dehydrobromination of m-ClC6H4CiHBrCH2Br calculate such a small amount of internal return, aH ¼ 0:045 at 258C, that a concerted E2 mechanism can not be ruled out. On the other hand, the dehydrochlorination of m-ClC6H4CiHClCH2Cl gives a ten-fold increase in internal return, aH ¼ 0:59 at 258C, which is similar to that calculated for m-CF3C6H4CiHClCH2Cl, aH ¼ 0:59: The Swain– Schaad relationship requires that ðkH =kT Þ values should be 18 to 20% greater than the experiment ðkH =kT ÞObs : The data for p-CF3C6H4CiHClCH2F results in a calculated a H ¼ 2.1 at 258C and the experimental ðkD =kT ÞObs of 1.56 predicts a ðkH =kT Þ of 4.42 which is 25% higher than the experimental ðkH =kT ÞObs of 3.53. Single temperature rates for the dehydrochlorination of p-CF3C6H4CiHClCH2Cl results in aH ¼ 0:77 at 258C and the ðkD =kT ÞObs of 2.04 predicts a value of ðkH =kT Þ of 10.8, which is 40% greater than the experimental ðkH =kT ÞObs of 7.65. Eliminations occurring by the mechanism shown in Scheme 17.2 can therefore give many of the same experimental results that would be consistent with the concerted E2 mechanism. Without the tritium rates of elimination the difference of ðkHCl =kDCl Þ ¼ 3:75 vs. ðkHF =kDF Þ ¼ 2:21 for the reactions of p-CF3C6H4CHClCH2Cl and p-CF3C6H4CHClCH2F could be interpreted as reactions occurring by a concerted E2 elimination with asymmetric transition structures. The lower value of the kH =kD for the dehydrofluorination reaction would suggest a later transition structure due to the greater strength of the C – F bond compared to that of a C – Cl bond. However, the tritium data does not follow Swain– Schaad and allows the calculation of internal-return parameters. Neither of the deuterium compounds incorporate any hydrogen when reactions are carried out in CH3OH. TABLE 17.1 Internal-Return Parameters Associated with Methanolic Sodium Methoxide-Promoted Dehydrohalogenation Reactions at 258C Compound

k H M21s21

k D M21s21

k T M21s21

aH

aD

aT

m-ClC6H4CHClCH2Cl m-CF3C6H4CHClCH2Cl p-CF3C6H4CHClCH2Cl p-CF3C6H4CHClCH2F m-ClC6H4CHBrCH2Br

1.60 £ 1023 3.34 £ 1023 8.03 £ 1023 1.49 £ 1024 3.26 £ 1022

4.71 £ 1024 9.58 £ 1024 2.14 £ 1023 6.73 £ 1025 6.60 £ 1023

2.58 £ 1024 5.10 £ 1024 1.05 £ 1023 4.22 £ 1025 3.27 £ 1023

0.59 0.59 0.77 2.1 0.045

0.14 0.13 0.10 0.50 0.010

0.060 0.068 0.042 0.27 0.007

470

Isotope Effects in Chemistry and Biology

This means that both chloride and fluoride can leave faster from the hydrogen-bonded carbanion intermediate, kElim, than the breaking of the hydrogen bond to form FC-h, k2H : The hydrogen bond also allows the retention of any stereochemistry and results in an antielimination. This leaves the element effect as the only possible difference between a concerted reaction mechanism and one that occurs in two steps and features internal return as in Scheme 17.2.

III. CHLORINE ISOTOPE EFFECTS VS. THE ELEMENT EFFECT Comparing the rates for the dehydrohalogenation reactions of p-CF3C6H4CHClCH2X, XyF or Cl, results in an element effect of ðkHCl =kHF ÞObs ¼ 54: This is a normal value for the concerted E2 mechanism; however, when the experimental rate constants are corrected for internal return to obtain a value for the hydron-transfer step, k1 ; the calculated k1HCl =k1HF is 30. Therefore the major portion of the element effect is due to the hydron-transfer step, k1 ; and not to the elimination step, kElim. The element effect is used to replace the need to measure a heavy atom isotope effect like k35 =k37 for chlorine, which is more difficult.10 For the concerted E2 mechanism, the values of ðk35 =k37 ÞHCl should equal those for ðk35 =k37 ÞDCl ; however, for the two-step process, ðk35 =k37 ÞHCl H D . k21 the elimination step has a larger role in should be larger than that for (k35/k37)DCl. Since k21 the experimental rate constant, kObs ¼ ½k1 kElim =½k21 þ kElim ; for the elimination of HCl vs. DCl. Measuring k35 =k37 for both the protium and deuterium compounds should be able to distinguish between a concerted reaction or a two-step process that has internal return.26 There is a significant difference in the values obtained for ethoxide-promoted dehydrochlorination of C 6H 5C iHClCH 2Cl with ðk35 =k37 ÞHCl ¼ 1:00908 ^ 0:00008 vs. ðk35 =k37 ÞDCl ¼ 1:00734 ^ 0:00012: Similar differences are obtained for the reaction using methanolic sodium methoxide, and for the alkoxide-promoted dehydrochlorinations of C6H5CiHClCF2Cl, Table 17.2. The values of ðk35 =k37 Þ are very large. To explain this it was postulated that there is a considerable lengthening of the C –Cl bond during the formation of the hydrogen-bonded carbanion, k1 .26b In Table 17.2 the values of ðk35 =k37 ÞHCl are 22 to 26% larger than those for (k 35/k 37)DCl for both C6H5CiHClCH2Cl and C6H5CiHClCF2Cl. The element effects show much greater differences. For C6H5CiHClCH2X the (k HBr/k HCl) is 40 to 46% greater than (k DBr/k DCl) while C6H5CiHClCF2X has (k HBr/k HCl) 62 to 73% greater than ðkDBr =kDCl Þ: Table 17.1 has data to calculate the element effects for p-CF3C6H4CiHClCH2Cl vs. p-CF3C6H4CiHClCH2F. The ðkHCl =kHF ÞObs of 54 is 74% greater than the ðkDCl =kDF ÞObs of 31. Therefore measuring an element effect for both the hydrogen and deuterium compounds should also be able to distinguish between the concerted mechanism and one that is two-step and features internal return for the hydron transfer.

TABLE 17.2 Chlorine and Hydrogen Isotope Effects, and Element Effects Associated with Alcoholic Sodium Alkoxide-Promoted Dehydrochlorination Reactions Compound C6H5CHClCH2Cl C6H5CDClCH2Cl C6H5CHClCH2Cl C6H5CDClCH2Cl C6H5CHClCF2Cl C6H5CDClCF2Cl C6H5CHClCF2Cl C6H5CDClCF2Cl

Solvent EtOH MeOH EtOH MeOH

k 35/k 37 (8C)

k H/k D (8C)

k HBr/k HCl (8C)

1.00908 ^ 0.00008 [24] 1.00734 ^ 0.00012 [24] 1.00978 ^ 0.00020 [21] 1.00776 ^ 0.00020 [21] 1.01229 ^ 0.00047 [0] 1.01003 ^ 0.00024 [0] 1.01255 ^ 0.00048 [20] 1.01025 ^ 0.00043 [20]

4.24 [25]

35 [25] 24 [25] 35 [25] 25 [25] 38 [25] 22 [25] 47 [25] 29 [25]

3.83 [25] 2.73 [25] 2.28 [25]

The Role of an Internal-Return Mechanism on Measured Isotope Effects

471

IV. HYDRON TRANSFER FROM ALCOHOLS TO CARBANIONS We first became interested in the mechanism of hydron transfer by our studies on the nucleophilic reactions of fluoroalkenes with alkoxides. The reaction of ethanolic sodium ethoxide with C6H5C(CF3) ¼ CF2 proceeds rapidly at 2778C to form a carbanion, {C6H5C(CF3)CF2OC2H5}2, that is neutralized by proton transfer from ethanol forming C6H5CH(CF3)CF2OC2H5 (15%) or the loss of fluoride ion to give both vinyl ethers, 76% Z- and 9% E-C6H5C(CF3)yCFOC2H5.27 It was surprising that a poor leaving group like fluoride ion was ejected faster than the proton transfer from the solvent ethanol. When the reaction is carried out in C2H5OD, 13% of the saturated ether is still obtained.28 Therefore transfer of hydron from ethanol to the carbanion occurred with a near unity isotope effect at 2 778C! The carbanion, {C6H5C(CF3)CF2OC2H5}2, is also generated during the ethoxide-promoted dehydrofluorination of C6H5CH(CF3)CF2OC2H5 at 408C. That reaction occurs with a ðkH =kD ÞObs ¼ 1:4 which suggests a significant amount of internal return. When the reaction of C6H5CH(CF3)CF2OC2H5 is run at 408C in C2H5OD to 20% elimination there is 3 to 4% deuterium incorporation in the recovered starting compound. The ratio of fluoride lost to deuterium incorporated is similar for partitioning of {C6H5C(CF3)CF2OC2H5}2 regardless of it being generated during dehydrofluorination or the reaction of ethoxide and C6H5C(CF3)yCF2 in C2H5OD. Scheme 17.3 accounts for these results. The products are formed from {C6H5C(CF3)CF2OC2H5}2 generated directly during the reaction of C6H5C(CF3)yCF2, kN; however, it is not formed until after the second step, k2, during the reaction of C6H5CH(CF3)CF2OCH3. Hydron transfer does not occur in the rate-limiting step for either of these reactions. The Eaborn group reported excellent work on hydron transfer from methanol to benzylic anions generated by the reaction of methoxide and substituted benzyltrimethylsilanes.29 Reactions in known mixtures of CH3OH and CH3OD resulted in YC6H5CH3 with varying amounts of H and D that were used to calculate the kH =kD associated with the hydron-transfer step. The kH =kD for six of the substituted benzylic anions were 1.2 to 1.3; however, substituents like para-nitro that could form highly pi-delocalized anions had values of kH =kD ¼ 10: Fluorenyl and 9-methylfluorenyl anions also gave values of 10, while triphenylmethyl and diphenylmethyl had kH =kD ¼ 1:3: The result for triphenylmethyl caused them concern as they quoted a kH =kD of 4.2 at 1008C from the

CF3 C6H5 C2H5OCF2 k−1

C

H

−OC

C6H5

C

CF2

2H5

−

−

C HOC2H5

−OC

kNuc

k1

CF3 C6H5 C2H5OCF2

CF3

2H5

k2 k−2

CF3 C6H5 C C2H5OCF2

HOC2H5

kElim CF3 C6H5

C

CFOC2H5

SCHEME 17.3 The generation of {C6H5C(CF3)CF2OC2H5}2 from reaction of C6H5CH(CF3)CF2OC2H5 or C6H5C(CF3)yCF2 to regenerate C6H5CH(CF3)CF2OC2H5 or form E-C6H5C(CF3)yCFOC2H5.

472

Isotope Effects in Chemistry and Biology

exchange kinetics of triphenylmethane reported by the Streitwieser group.16b That value was for k1H =k1D and was calculated using internal-return parameters. The experimental kH =kD of 1.32 is consistent with their kH =kD of 1.3 at 508C for hydron transfer from methanol to the carbanion. Their results for fluorenyl anions agree with experimental values reported by the Streitwieser group that measure the isotope effect associated with C –H bond breaking. Differences between highly pi-delocalized carbanions and localized carbanions were found for carbanions generated from reactions of YC6H5CHyCF2 with methanolic sodium methoxide.30 The amount of saturated ether p-NO2C6H4CH2CF2OCH3 from the reaction of p-NO2C6H4CHyCF2 decreases from 96% at 2 778C to 42% at 288C while the reaction of m-NO2C6H4CHyCF2 to give m-NO2C6H4CH2CF2OCH3 shows a slight increase from 56% at 2 158C to 59% at 658C. The later trend is noted for many other YC6H5CHyCF2 reactions. There is also a significant difference in magnitude and temperature dependence of the isotope effects associated with hydron transfer from methanol to {YC6H5CHCF2OCH3}2.31 Reaction of p-NO2C6H5CHyCF2 results in a kH =kD ¼ 11:3 at 2 708C that decreases to kH =kD ¼ 6:44 at 258C. On the other hand, the reaction of m-NO2C6H5CHyCF2 has kH =kD ¼ 1:20 at 2 608C increase slightly to kH =kD ¼ 1:39 at 508C. The slight increase of the isotope effect with increasing temperature has also been observed for the reactions of several other YC6H5CHyCF2.

V. CONCLUSIONS The measurement of isotope effects associated with reactions is a powerful experimental tool to investigate the mechanisms of chemical reactions. Interpretation of the isotope effects has changed over the years as different systems are subjected to this analysis. The use of isotope effects is an extremely powerful tool in the study of detailed reaction mechanisms. The generality of concerted 1,2 eliminations reactions was questioned by Bordwell32 and supported by Saunders.33 The question really amounts to what is an intermediate during a chemical reaction. Jencks34 has given an excellent definition: “An intermediate is, therefore, defined as a species with a significant lifetime, longer than that of a molecular vibration of , 10213 s, that has barriers for its breakdown to both reactants and products.” When dealing with hydron transfer the use of isotope effects is the best method to determine if there is internal return, which is a no-reaction reaction. This requires using all three isotopes of hydrogen, and for elimination reactions the leaving group heavy atom isotope effects or element effects should be studied for both the protium and deuterium compounds. The effect of heavy atoms on the magnitude of experimental hydrogen isotope effects is still not fully defined and is a problem for the interpretation of the experimental results. The internal-return mechanism has complicated the picture and still needs more detailed studies to help in the interpretation of experimental results.

REFERENCES Lewis, G. N. and Macdonald, R. T., Concentration of H2 isotope, J. Chem. Phys., 1, 341– 344, 1933. Lewis, G. N., The isotope of hydrogen, J. Am. Chem. Soc., 55, 1297– 1298, 1933. Wiberg, K. B., The deuterium isotope effect, Chem. Rev., 55, 713– 743, 1955. Hughes, E. D., Ingold, C. K., Masterman, S., and McNulty, B. J., Mechanism of elimination reactions. Part V. Kinetics of olefin elimination from ethyl, isopropyl, tert-butyl and a- and b-phenylethyl bromides in acidic and in alkaline alcohol solution. Effects due to, and factors influencing the two mechanisms of elimination, J. Chem. Soc., 899–911, 1940. 5 Skell, P. S. and Hauser, C. R., The mechanism of b-elimination with alkyl halides, J. Am. Chem. Soc., 67, 1661, 1945. 6 Cristol, S. J. and Fix, D. D., Mechanisms of elimination reactions. X. Deuterium exchange in basepromoted dehydrochlorination of b-benzene hexachloride, J. Am. Chem. Soc., 75, 2647– 2648, 1953.

1 2 3 4

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473

7 Cristol, S. J., Hause, N. L., and Meek, J. S., Mechanisms of elimination Reactions. III. The kinetics of the alkaline dehydrochlorination of the benzene hexachloride isomers, J. Am. Chem. Soc., 73, 674– 679, 1951. 8 Shiner, V. J., Substitution and elimination rate studies on some deutero-isopropyl bromides, J. Am. Chem. Soc., 74, 5285– 5288, 1952. 9 Bunnett, J. F., Garbisch, E. W., and Pruitt, K. M., The “element effect” as a criterion of mechanism in aromatic nucleophilic substitution reactions, J. Am. Chem. Soc., 79, 385– 391, 1957. 10 Bartsch, R. A. and Bunnett, J. F., Kinetics of reactions of 2-hexyl halides and 2-hexyl p-bromobenzenesulfonate with sodium methoxide in methanol. Evidence that orientation of olefin-forming elimination is not determined by the steric requirements of the halogen leaving groups, J. Am. Chem. Soc., 90, 408– 417, 1968. 11 Bell, R. P., Recent advances in the study of kinetic hydrogen isotope effects, Chem. Soc. Rev., 3, 513– 544, 1974. 12 Melander, L., Isotope Effects on Reaction Rates, Ronald Press, New York, 24 – 32, 1960. 13 Westheimer, F. H., The magnitude of the primary kinetic isotope effect for compounds of hydrogen and deuterium, Chem. Rev., 61, 265– 273, 1961. 14 Swain, C. G., Stivers, E. C., Reuwer, J. F. Jr., and Schaad, L. J., Use of hydrogen isotope effects to identify the attacking nucleophile in the enolization of ketones catalyzed by acetic acid, J. Am. Chem. Soc., 80, 5885– 5893, 1958. 15 Cram, D. J., Fundamentals of Carbanion Chemistry, Academic Press, New York, 1965, pp. 28– 44. 16 (a) Streitwieser, A. Jr., Hollyhead, W. B., Pudjaatmaka, A. H., Owens, P. H., Kruger, T. L., Rubenstein, P. A., MacQuarrie, R. A., Brokaw, M. L., Chu, W. K. C., and Niemeyer, H. M., Acidity of hydrocarbons. XXXVII. The Bronsted correlation and hydrogen exchange kinetics of fluorenes, benzfluorenes, and indene with methanolic sodium methoxide, J. Am. Chem. Soc., 93, 5088– 5096, 1971; (b) Streitwieser, A. Jr., Hollyhead, W. B., Sonnichsen, G., Pudjaatmaka, A. H., Chang, C. J., and Kruger, T. L., Acidity of hydrocarbons. XXXVIII. Kinetic acidity and Bronsted correlation of di- and triarylmethanes with methanolic sodium methoxide, J. Am. Chem. Soc., 93, 5096– 5102, 1971. 17 Koch, H. F., Dahlberg, D. B., McEntee, M. F., and Klecha, C. J., Use of kinetic isotope effects in mechanism studies. Anomalous Arrhenius parameters in the study of elimination reactions, J. Am. Chem. Soc., 98, 1060– 1061, 1976. 18 The kinetic results for C6H5CHClCF2Cl and C6H5CHBrCF2Cl are similar [see note 14 in Ref. 17]. Since the synthesis of C6H5CHClCF2Cl is easier, the isotope effect studies were carried out with this compound to compare to those for C6H5CHBrCF2Br. 19 Koch, H. F. and Dahlberg, D. B., Use of kinetic isotope effects in mechanism studies. Effect of an internal return mechanism on the Arrhenius behavior of primary hydrogen isotope effects, J. Am. Chem. Soc., 102, 6102– 6107, 1980. 20 Koch, H. F. and Dahlberg, D. B., Use of kinetic isotope effects in mechanism studies. Effect of an internal return mechanism on the Arrhenius behavior of primary hydrogen isotope effects, J. Am. Chem. Soc., 102, 6103, 1980. See Figure 1. 21 Shiner, V. J. and Smith, M. L., The Arrhenius parameters of the deuterium isotope rate effect in a base-promoted elimination reaction: evidence for proton tunneling, J. Am. Chem. Soc., 83, 593– 598, 1961. 22 Shiner, V. J. and Martin, B., Tritium and deuterium Arrhenius parameter effects in a base-promoted elimination reaction: evidence for tunneling, Pure Appl. Chem., 8, 371– 378, 1964. 23 The original Swain – Schaad relationship was for ðkH =kD Þ1:442 ¼ ðkH =kT Þ which came from using only the mass of the hydrogen isotopes,14 and becomes ðkD =kT Þ3:26 ¼ ðkH =kT Þ: The treatment by the Streitwieser group16b has that exponent equal to 3.344 due to using the reduced mass. 24 Koch, H. F. and Dahlberg, D. B., Use of kinetic isotope effects in mechanism studies. Effect of an internal return mechanism on the Arrhenius behavior of primary hydrogen isotope effects, J. Am. Chem. Soc., 102, 6106, 1980. 25 Koch, H. F., Lodder, G., Koch, J. G., Bogdan, D. J., Brown, G. H., Carlson, C. A., Dean, A. B., Hage, R., Han, P., Hopman, J. C. P., James, L. A., Knape, P. M., Roos, E. C., Sardina, M. L., Sawyer, R. A., Scott, B. O., Testa, C. A. III, and Wickham, S. D., Use of kinetic isotope effects in mechanism studies. 5. Isotope effects and element effects associated with hydron-transfer steps during alkoxide-promoted dehydrohalogenations, J. Am. Chem. Soc., 119, 9965– 9974, 1997.

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26 (a) Koch, H. F., Koch, J. G., Tumas, W., McLennan, D. G., Dobson, B., and Lodder, G., Use of kinetic isotope effects in mechanism studies. 3. Measurement of hydrogen isotope effects on the primary chlorine isotope effect during elimination reactions, J. Am. Chem. Soc., 102, 7955– 7956, 1980; (b) Koch, H. F., McLennan, D. J., Koch, J. G., Tumas, W., Dobson, B., and Koch, N. H., Use of kinetic isotope effects in mechanism studies 4. Chlorine isotope effects associated with alkoxide-promoted dehydrochlorination reactions, J. Am. Chem. Soc., 105, 1930– 1937, 1983. 27 Koch, H. F. and Kielbania, A. J. Jr, Nucleophilic reactions of fluoroolefins. Evidence for a carbanion intermediate in vinyl and allyl displacement reactions, J. Am. Chem. Soc., 92, 729– 730, 1970. 28 Koch, H. F., Koch, J. G., Donovan, D. B., Toczko, A. G., and Kielbania, A. J. Jr., Proton-transfer Reactions. I. Partitioning of carbanion intermediates generated by reactions of alkenes with alkoxide ions in alcohol, J. Am. Chem. Soc., 103, 5417– 5423, 1981. 29 Macciantelle, D., Seconi, G., and Eaborn, C., Further studies of solvent isotope effects in the cleavage of substituted benzytrimethylsilanes by methanolic sodium methoxide. Intermediate kinetic isotope effects for reactions of carbanions with methanol, JCS Perkin II, 834– 838, 1978. 30 Koch, H. F., Koch, J. G., Koch, N. H., and Koch, A. S., Proton-transfer reactions. 3. Differences in the protonation of localized and delocalized carbanion intermediates, J. Am. Chem. Soc., 105, 2388– 2393, 1983. 31 Koch, H. F. and Koch, A. S., Proton-transfer reactions. 5. An observed primary kinetic isotope effect that increases with increasing temperature, J. Am. Chem. Soc., 106, 4536– 4539, 1984. 32 Bordwell, F. G., How common are base-initiated concerted 1,2 eliminations, Acc. Chem. Res., 5, 374– 381, 1972. 33 Saunders, W. H. Jr., Distinguishing between concerted and nonconcerted eliminations, Acc. Chem. Res., 9, 19 – 25, 1976. 34 Jencks, W. P., When is an intermediate not an intermediate? Enforced mechanisms of general acid – base catalyzed, carbocation, carbanion, and ligand exchange reactions, Acc. Chem. Res., 13, 161– 169, 1980.

18

Vibrationally Enhanced Tunneling and Kinetic Isotope Effects in Enzymatic Reactions Steven D. Schwartz

CONTENTS I. II. III.

Introduction ...................................................................................................................... 475 Theoretical Approaches to the Study of Chemical Dynamics in Complex Systems ..... 476 Promoting Vibrations and the Dynamics of Hydrogen Transfer .................................... 479 A. Promoting Vibrations and the Symmetry of Coupling ........................................... 479 B. Promoting Vibrations — Corner Cutting and the Masking of KIEs ...................... 480 IV. Enzymatic Hydrogen Transfer and KIEs ........................................................................ 482 A. Alcohol Dehydrogenase ........................................................................................... 482 B. Lactate Dehydrogenase ............................................................................................ 487 V. Hydrogen Transfer Coupled to Electron Transfer — Kinetic Trends in the Presence of a Promoting Vibration ....................................................................... 491 VI. Conclusions ...................................................................................................................... 495 Acknowledgments ........................................................................................................................ 495 References..................................................................................................................................... 495

I. INTRODUCTION Life of all forms depends on the acceleration of the rate of chemical reactions by enzymes. It is now known that many enzymes function by lowering the free-energy barrier to a reaction. This preferential binding of the enzyme to the transition state is a concept credited to Pauling.1 It is the origin of the extraordinary potency of transition-state inhibitors.2 It has recently been recognized that in some enzymes, the action of the enzyme includes coupling of the dynamic motions of the protein backbone to progress along the reaction coordinate.3 These effects have been deemed to be especially important in the case of tunneling reactions. In such a case, the geometry of the barrier may well be as important a determining feature of rate as the height of the barrier. This is because it is far easier to tunnel through a thin barrier than through a thick one. Thus, evolution has crafted enzymes to force reactants in close proximity to each other and a vibrational enhancement of the rate of light particle transfer. Understanding of light particle transfer in enzymatic systems had its origin in anomalous experimental signatures of tunneling behavior — kinetic isotope effects (KIEs). These experiments however, are complex to interpret rarely does evidence for tunneling come in the form expected — the primary KIE. Usually such indications must be found in more physically removed signatures such as KIEs in secondary and tertiary positions and temperature dependence of isotope effects. Recently, however, there has been a convergence in experimental and theoretical understanding of such systems. This has been possible only recently, as the theoretical methodology to study such 475

476

Isotope Effects in Chemistry and Biology

highly complex systems was not previously available. Reactions in these systems must be studied as in any chemical reaction with quantum mechanics — classical mechanics cannot make and break bonds. In addition, transition-state theory, one of the most popular approximate approaches to the study of reaction dynamics in complex systems, is not appropriate because at its heart it assumes a decoupling of reaction and all environmental modes when the reaction is poised at the transition state. It is exactly the interaction of these environmental effects that we seek to study.4 (We note there are ways to “correct” transition-state theory for such effects,5 but there is no rigorous way to derive these approaches, and to control for expected errors.) This chapter will review a set of such approaches developed in our group, and the application of these methodologies to a variety of enzymatic reactions. We will investigate the cause of some of the anomalous results, and the expected experimental signatures in enzymatic reactions that are still poorly understood. We stand at a unique point in the history of enzymology, where theory and experiments are needed to collaboratively elucidate the mechanism of action of some enzymes of interest. We will study protein coupling to reaction in alcohol dehydrogenase, in two different human isoforms of lactate dehydrogenase (LDH), and present a general investigation of expected experimental results in coupled hydrogen electron-transfer reactions such as amine dehydrogenase and lipoxygenase. The structure of this chapter is as follows: the next section will provide the theoretical underpinning for the approaches we will take. We will briefly derive the methodologies developed in our group for the quantum-mechanical study of chemical reactions in complex systems. Section III will describe how promoting vibrations may be incorporated into this picture, and how they change the basic chemical physics of the chemical reaction. In the next section we will present evidence for promoting vibrations in alcohol dehydrogenase, tying these results to the known experimental data, including evolutionary evidence that such approaches are highly conserved evolutionarily. We will then turn our attention to lactate dehydrogenase, in particular two human isoforms, and present a possible explanation of a previously unexplained mystery — how almost identical proteins exhibit radically different preferences for reactants or products in the two isoforms. We will then investigate coupled hydrogen electron transfer in an enzyme in the presence of a promoting vibration, and see how this more complex system varies. Those not wishing to delve into the theoretical details of the methodology can easily skip Section II. Section III is also fairly theoretical. It explains the important masking of KIEs in the presence of promoting vibrations; it can be skipped, with the reader proceeding directly to Section IV, application to enzymes.

II. THEORETICAL APPROACHES TO THE STUDY OF CHEMICAL DYNAMICS IN COMPLEX SYSTEMS There are two prerequisites for the theoretical study of the dynamics of a chemical reaction in a complex system. First, we need a model for the reaction; in other words a potential energy surface. Depending on the application we either use approximate potentials such as molecular mechanics potentials, or we can compute as accurately as possible quantum chemical surfaces. As we will describe, the former is appropriate when we are studying classical motion of atoms in the complex system, while the latter is needed if we wish to model the actual chemical event itself. This chapter will not be concerned with the methods we employ to compute potential energy surfaces. Rather we focus on the second need, which is for the development of theoretical methods to predict and analyze dynamics in complex systems. Our methods all start with an approach we term the Quantum Kramers methodology.6 Our ideas were motivated by the following approximations developed for study of the classical mechanics of large complex systems. It is known that for a purely classical system,7 an accurate approximation of the dynamics of a tagged degree of freedom (for example, a reaction coordinate) in a condensed phase can be obtained through the use of a Generalized Langevin Equation. The generalized Langevin equation is given by Newtonian

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dynamics plus the effects of the environment in the form of a memory friction and a random force:8 m€s ¼ 2

›VðsÞ ðt 0 þ dt g ðt 2 t0 Þ_s þ FðtÞ ›s 0

ð18:1Þ

Thus all the complex microscopic dynamics of all degrees of freedom other than the reaction coordinate are included only in a statistical treatment, and the reaction coordinate plus environment are treated as a modified one-dimensional system. What allows realistic simulation of complex systems is that the statistics of the environment can in fact be calculated from a formal prescription. This prescription is given by the Fluctuation-Dissipation Theorem, which yields the relation between the friction and the random force. In particular, this theory shows how to calculate the memory friction from a relatively short-time classical simulation of the reaction coordinate. The Quantum Kramers approach, in turn, is dependent on an observation of Zwanzig,9 that if an interaction potential for a condensed phase system satisfies a fairly broad set of mathematical criteria, the dynamics of the reaction coordinate as described by the Generalized Langevin Equation can be rigorously equated to a microscopic Hamiltonian in which the reaction coordinate is coupled to an infinite set of Harmonic Oscillators via simple bilinear coupling: X P2k P2 1 cs þ mk v2k qk 2 k 2 H ¼ s þ V0 þ 2mk 2 2ms mk vk k

!2 ð18:2Þ

The first two terms in this Hamiltonian represent the kinetic and potential energy of the reaction coordinate, and the last set of terms represent the kinetic and potential energy for an environmental bath. Here s is some coordinate that measures progress of the reaction (for example, in alcohol dehydrogenase where the chemical step is transfer of a hydride, s might be chosen to represent the relative position of the hydride from the alcohol to the NAD cofactor), ck is the strength of the coupling of the environmental mode to the reaction coordinate, and mk and vk give the effective mass and frequency respective of the environmental bath mode. A discrete spectral density gives the distribution of bath modes in the harmonic environment: JðvÞ ¼

p X c2k ½dðv 2 vk Þ 2 dðv þ vk Þ 2 k mk vk

ð18:3Þ

Here dðv 2 vk Þ is the Dirac Delta function, so the spectral density is simply a collection of spikes, located at the frequency positions of the environmental modes, convolved with the strength of the coupling of these modes to the reaction coordinate. Note that this infinite collection of oscillators is purely fictitious; they are chosen to reproduce the overall physical properties of the system, but do not necessarily represent specific physical motions of the atoms in the system. Now it would seem that we have not made a huge amount of progress: we began with many-dimensional systems (classical) and found out that it could be accurately approximated by a one-dimensional system in a frictional environment (the Generalized Langevin Equation). We have now recreated a manydimensional system (the Zwanzig Hamiltonian). The reason to do this is twofold. First, there is no true quantum mechanical analogue of friction, and so there really is no way to use the Generalized Langevin approach for a quantum system, as we would like to do for an enzyme. Second, the new quantum Hamiltonian given by Equation 18.2 is very much simpler than the Hamiltonian for the full enzymatic system. Harmonic oscillators are the one type of problem that can easily be solved in quantum mechanics. Thus, the prescription is, given a potential for a reaction, we model the exact problem using Zwanzig Hamiltonian, as in Equation 18.2, with distribution of harmonic modes given by the spectral density in Equation 18.3, and found through a simple classical computation of the frictional force on the reaction coordinate. Then using methods to compute quantum dynamics developed in our group,10 quantities such as rates or KIEs may be computed. These methods are an

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approximate but accurate way to compute the quantum mechanical evolution of any systems. The details are given in the literature,11 but in short, a general Hamiltonian is: _ _ _ _ H ¼ Ha þ Hb þ fða; bÞ

ð18:4Þ

where a and b are shorthand for any number of degrees of freedom. f ða; bÞ is a coupling, usually a function of coordinates, but this is not required. The operator resummation idea rests on the fact that because these three terms are operators, the exact evolution operator may not be expressed as a product: _

_

_

_

e2iHt – e2iHa t e2iHb tþfða;bÞ

ð18:5Þ

In fact equality may be achieved by application of an infinite order product of nth order commutators: _

_

_

_

e2iHt – e2iHa t e2iHb tþfða;bÞ ec1 ec2 ec3 · · ·

ð18:6Þ

This is usually referred to as the Zassenhaus expansion or the Baker Campbell Hausdorf theorem.12 As an aside a symmetrized version of this expansion terminated at the C1 term results in the Feit and Fleck13 approximate propagator. It is shown that an infinite order subset of these commutators may be resummed exactly as an interaction propagator: UðtÞresum ¼ UðtÞHa UðtÞHb þf ða;bÞ U 21 ðtÞHa þf ða;bÞ UðtÞHa

ð18:7Þ

The first two terms are the adiabatic approxmation, and the second two terms the correction. For example, if we have a fast subsystem labeled by the “coordinate” a, and a slow subsystem labeled by b, then the approximate evolution operator to first order in commutators with respect to the slow subsystem bð½f ða; bÞ; Hb Þ; and infinite order in the commutators of the “fast” Hamiltonian with the coupling ð½f ða; bÞ; Ha Þ is given by e2iðHa þHb þf ða;bÞÞt= h/ < e2iHa t= h/ e2iðHb þf ða;bÞÞt= h/ eþiðHa þf ða;bÞÞt= h/ e2iHa t= h/

ð18:8Þ

The advantage to this formulation is that higher dimensional evolution operators are replaced by a product of lower dimensional evolution operators. This is always a far easier computation. In addition, because products of evolution operators replace the full evolution operator, a variety of mathematical properties are retained, such as unitarity, and thus time-reversal symmetry. Combination of the Quantum Kramers idea with the resummed evolution operators results in a largely analytic formulation for the Flux autocorrelation function for a chemical reaction in a condensed phase. After a lengthy but not complex computation the quantum Kramers Flux autocorrelation function has been shown to be:14 Cf ¼ Cf0 B1 Zbath 2

ð1 0

dvk0f JðvÞB2 Zbath

ð18:9Þ

Here Cf0 is the gas-phase (uncoupled) Flux autocorrelation function, Zbath is the bath partition function, JðvÞ is the bath spectral density (computed as described above from a classical molecular dynamics computation), B1 and B2 are combinations of trigonometric functions of the frequency v and the inverse barrier frequency, and finally:

k0f ¼

1 lks ¼ 0le2iHs tc = h/ ls ¼ 0ll2 4m2s

ð18:10Þ

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As in other Flux correlation function computations, tc is the complex time (t 2 (ih/ b/2)). Thus, given the Quantum Kramers model for the reaction in the complex system, and the resummed operator expansion as a practical way to evaluate the necessary evolution operators needed for the Flux autocorrelation function, the quantum rate in the complex system is reduced to a simple combination of gas phase correlation functions with simple algebraic functions. This approach is able to model a variety of condensed phase chemical reaction with essentially experimental accuracy.14 There is one specific experimental system for which this methodology was not able to reproduce experimental results, and that is proton transfer in benzoic acid crystals. In developing a physical understanding of this system, we first identified the concept of the promoting vibration.

III. PROMOTING VIBRATIONS AND THE DYNAMICS OF HYDROGEN TRANSFER A. PROMOTING V IBRATIONS AND

THE

S YMMETRY OF C OUPLING

The Hamiltonian of Equation 18.2 contains only antisymmetric couplings. We have shown14a that this collection of bilinearly coupled oscillators is in fact a microscopic version of the popular Marcus theory for charged particle transfer.15 The bilinear coupling of the bath of oscillators is the simplest form of a class of couplings that may be termed antisymmetric because of the mathematical property of the functional form of the coupling on reflection about the origin. This property has deeper implications than the mathematical nature of the symmetry properties. Antisymmetric couplings, when coupled to a double-well-like potential energy profile, are able to instantaneously change the level of well depths, but do nothing to the position of well minima. This modulation in the position of minima is exactly what the environment is envisaged to do within the Marcus theory paradigm. As we have shown,16 the minima of the total potential in Equation 18.2 will occur, for a two-dimensional version of this potential, when the q degree of freedom is exactly equal and opposite in sign to cs=mv2 ; and the minimum of the potential energy profile along the reaction coordinate is unaffected by this coupling. Within Marcus’ theory, which is a deep tunneling theory, transfer of the charged particle occurs at the value of the bath coordinates that cause the total potential to become symmetrized. Thus, if the bare reaction coordinate potential is symmetric, then the total potential is symmetrized at the position of the “bath plus coupling” minimum. When this configuration is achieved, the particle tunnels, and in fact the activation energy for the reaction is the energy to bring the bath into this favorable tunneling configuration. The question is if such motions and their mathematical representations encompass all important motions in the coupling of dynamic motions to a reaction coordinate. We became aware of an example in which there is another significant contributor to the chemical dynamics, benzoic acid crystals. There is a long history of the study of proton transfer in crystalline benzoic acid.16 These experiments seemed to yield anomolous results when coupled with quantum-chemistry computations. That is, computations showed a reasonably high barrier while experiment showed a low activation energy. That is, of course, normally indicative of a significant contribution to the chemical reaction from quantum-mechanical tunneling. In this system, however, KIEs were quite modest — classical in behavior. It became clear to us that we could not model such behavior using the mathematical formalism we had developed. The reason for this is apparent in Figure 18.1. Motions of the carboxyl oxygens toward each other in each dimer that forms the crystal of benzoic acid modulate the potential for proton transfer through symmetric motions of the well bottoms toward each other. This environmental modulation both lowers and thins the barrier to proton transfer. This symmetric coupling of motion to the reaction coordinate requires modification of

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Isotope Effects in Chemistry and Biology

H

O

O

O

H

O

C

C O

H

O

FIGURE 18.1 A benzoic acid dimer showing how the symmetric motion of the oxygen atoms will affect the potential for hydrogen transfer.

the Hamiltonian in Equation 18.2: X P2k P2 1 cs þ mk v2k qk 2 k 2 H ¼ s þ V0 þ 2mk 2 2ms mk vk k

!2

P2Q 1 Cs2 þ þ M V2 Q 2 2 2M M V2

!2 ð18:11Þ

Note that in this case, the oscillator that is symmetrically coupled, represented by the last term in Equation 18.11, is in fact a physical oscillation of the environment.

B. PROMOTING V IBRATIONS — C ORNER C UTTING AND

THE

M ASKING OF KIE S

We were able to develop a theory17 of reactions mathematically represented by the Hamiltonian in Equation 18.11 and using this method and experimentally available parameters for the benzoic acid proton transfer potential, we were able to reproduce experimental kinetics as long as it included a symmetrically coupled vibration.18 The results are shown in Table 18.1. The twodimensional activation energies refer to a two-dimensional system comprised of the reaction coordinate and a symmetrically coupled vibration. The reaction coordinate is also coupled to an infinite environment as described above. KIEs in this system are modest, even though the vast majority of the proton transfer occurs via quantum tunneling. The end result of this study is that symmetrically coupled vibrations can significantly enhance rates of light-particle transfer, and also significantly mask kinetic isotope signatures of tunneling. A physical origin for this masking of the KIE may be understood from a comparison of the two-dimensional problem comprised of a reaction coordinate coupled symmetrically and antisymmetrically to a vibration. As Figure 18.2 shows antisymmetric coupling causes the minima (the reactants and products) to lie on a line, the minimum energy path, which passes through the transition state. In contradistinction, symmetric coupling causes the reactants and products to be moved from the reaction coordinate axis in such a fashion that a straight-line connection of reactant and products would pass nowhere near the

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TABLE 18.1 The 15 Active Site Amino Acids Most Strongly Correlated with Donor–Acceptor Motion Lactate Bound LDH: Donor 5 Lactate, Acceptor 5 NAD1 Heart, lactate 240D 94A 136A 194D 235D 31A 256A 236D 95A 35A 34A 32A 93A 253A 238D Muscle, lactate 255A 34A 29A 234D 236D 235D 160N 194D 94A 252A 33A 30A 193D 237D 31A

A.A.

n (cm21)

Amp. 3 105

E V V G V V V E T C A G V G A

271 173 196 271 173 283 271 179 173 173 185 266 173 208 179

9.1 9.3 9.4 9.4 9.7 10.4 10.4 10.6 10.8 12.8 13.2 13.2 14.8 15.8 16.3

A.A.

n (cm21)

Amp. £ 105

V C A V S E S D T G A V G A G

330 278 341 301 278 289 278 376 295 289 359 284 284 278 318

4.5 5.0 6.0 6.2 6.4 6.5 6.6 6.8 6.8 6.9 7.4 7.5 12.4 12.6 13.1

Pyruvate Bound LDH: Donor 5 NADH, Acceptor 5 Pyruvate Heart, pyruvate 237A 235A 104A 135D 94D 236A 136D 238A 95D 34D 239A 35D 256D 32D 93D Muscle, pyruvate 239D 34A 135A 252A 233D 94A 194D 234D 235D 236D 29A 193D 30A 237D 31A

A.A.

n (cm21)

Amp. 3 105

S V E V V E V A T A Y C V G V

98 93 93 104 133 110 75 139 93 127 93 145 75 139 104

11.3 11.5 11.7 12.0 13.0 13.4 13.5 14.1 14.1 14.1 14.4 14.7 15.0 16.8 17.3

A.A.

n (cm21)

Amp. £ 105

E C V G V T D V E S A G V A G

179 202 289 248 289 196 277 266 260 254 277 243 260 271 254

4.5 4.7 4.7 4.9 5.1 5.1 5.6 5.9 6.3 6.9 7.3 7.6 7.8 9.9 10.5

If lactate is bound, NADþ is the acceptor. If pyruvate is bound, NADH is the donor. Bold amino acids are among the top 15 for an isozyme, independent of substrate bound. Italic amino acids are among the top 15 for both isozymes, independent of substrate bound. The peak amplitude has been multiplied by 105 for convenience. Frequency screens: H2, lactate (175 to 315 cm21), H2, pyruvate (75 to 315 cm21), M2, lactate (175 to 375 cm21), M2, pyruvate (175 to 375 cm21). A.A. abbreviates amino acid.

transition state. This, in turn, results in the gas phase physical chemistry phenomenon known as corner cutting.19 Physically, the quantity to be minimized along any path from reactant to products is the action. This is an integral of the energy, and is a product of distance and depth under the barrier that must be minimized to find an approximation to the tunneling path. The action also includes the mass of the particle being transferred, and so in the symmetric coupling case, a proton will actually follow a very different physical path from reactants to products in a reaction than will a deuteron.

482

Isotope Effects in Chemistry and Biology q A

−s0

+s0 s

A, S

S

FIGURE 18.2 This diagram shows the location of stable minima in two-dimensional systems. The figure represents how antisymmetrically and symmetrically coupled vibrations effect position of stable minima, that is reactant and product. The x-axis, s, represents the reaction coordinate, and q the coupled vibration. The points on the figure labeled S and A are the positions of the well minimal in the two dimensional system with symmetric and antisymmetric coupling, respectively. An antisymmetrically coupled vibration displaces those minima along a straight line, so that the shortest distance between the reactant and product wells passes through the transition state. In contradistinction, a symmetrically coupled vibration, allows for the possibility of corner cutting under the barrier. For example, a proton and a deuteron will follow different paths under the barrier.

IV. ENZYMATIC HYDROGEN TRANSFER AND KIES A. ALCOHOL D EHYDROGENASE The low level of primary KIE in the benzoic acid crystal when tunneling is the dominant transfer mechanism suggesting a similarity between the proton-transfer mechanism in the organic acid crystal and that of hydrogen transfer in some enzymatic reactions. We note that there have been previous attempts to understand the anomalously low primary KIEs in alcohol dehydrogenases in the presence of a large body of experimental evidence that quantum tunneling is involved in the hydride transfer. Coupled motions of nearby atoms in enzymatic reactions have been shown to result in such anomalous KIEs in numerical experiments.20 These studies were classical kinetics with semiclassical tunneling (the Bell correction21) and they could not be used to account for enzymatic reactions in a deep tunneling regime. Klinman and coworkers have helped pioneer the study of tunneling in enzymatic reactions. The focus of their work has been the alcohol dehydrogenase family of enzymes. Alcohol dehydrogenases are NADþ-dependent enzymes that oxidize a wide variety of alcohols to the corresponding aldehydes. After successive binding of the alcohol and cofactor, the first step is generally accepted to be complexation of the alcohol to one of the two bound Zinc ions.22 This complexation lowers the pKa of the alcohol proton and causes the formation of the alcoholate. The chemical step is then the transfer of a hydride from the alkoxide to the NADþ cofactor. They23 have found a remarkable effect on the kinetics of yeast alcohol dehydrogenase (a mesophile) and a related enzyme from Bacillus stearothermophilus, a thermophile. A variety of kinetic studies from this group have found that the mesophile24 and many related dehydrogenases25 show signs of significant contributions of quantum tunneling in the rate-determining step of hydride transfer. Remarkably, their kinetic data seem to show that the thermophilic enzyme actually exhibits less signs of tunneling at lower temperatures. Recent data of Kohen and Klinman26 also show, via isotope exchange experiments, that the thermophile is significantly less flexible at mesophilic

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temperatures, as in the Petsko group’s results27 in studies of 3-isopropylmalate dehydrogenase from the thermophilic bacteria Thermus thermophilus. These data have been interpreted in terms of models similar to those described above, in which a specific type of protein motion strongly promotes quantum tunneling; thus, at lower temperatures, when the thermophile has this motion significantly reduced, the tunneling component of reaction is hypothesized to go down even though one would normally expect tunneling to go up as temperature goes down. Additionally, the Klinman group has investigated the catalytic properties of various mutants of Horse Liver Alcohol Dehydrogenase (HLADH). HLADH in the wild type is slightly less advantageous a system to study than yeast ADH because chemistry is not the rate-determining step in catalysis for this enzyme. The active site geometry of HLADH is shown in Figure 18.3. Two specific mutations have been identified, Val203 ! Ala and Phe93 ! Trp, which significantly affect enzyme kinetics. Both residues are located at the active site; the valine impinges directly on the face of the NADþ cofactor distal to the substrate alcohol. Modification of this residue to the smaller alanine significantly lowers both the catalytic efficiency of the enzyme, as compared to the wild type, and also significantly lowers indicators of hydrogen tunneling.28 Phe-93 is a residue in the alcohol-binding pocket. Replacement with the larger tryptophan makes it harder for the substrate to bind, but does not lower the indicators of tunneling.29 Bruice’s recent molecular dynamics calculations30 produce results consonant with the concept that mutation of the Valine changes protein dynamics and it is this alteration, missing in the mutation at position 93, which in turn changes tunneling dynamics. (We note that recent experimental results from Klinman’s group31 do not exhibit a decrease in tunneling as the temperature is raised.) With the suggestion of tunneling with a low KIE, we wish to investigate the dynamics of the enzyme to search for the possible presence of a promoting vibration. The quantity that naturally describes the way in which an environment interacts with a reaction coordinate in a complex condensed phase is the spectral density. In Equation 18.3, the spectral density could be seen to give a distribution of the frequencies of the bilinearly coupled modes, convolved with the strength of

FIGURE 18.3 A schematic of the active site of horse liver alcohol dehydrogenase. The bound substrate (in this case benzyl alcohol) and the cofactor NAD are shown along with several residues in the active site.

484

Isotope Effects in Chemistry and Biology

their coupling to the reaction coordinate. The concept of the spectral density is quite general and the spectral density may be measured or computed for realistic systems in which the coupling of the modes may well not be bilinear.32 We have also shown33 that the spectral density can be evaluated along a reaction coordinate. One only obtains a constant value for the spectral density when the coupling between the reaction coordinate and the environment is in fact bilinear. We have shown that a promoting vibration is created as a result of a symmetric coupling of a vibration mode to the reaction coordinate and this is quite a general feature of motions in complex systems. Analytic calculations demonstrated that such a mode should be manifest by a strong peak in the spectral density when it was evaluated at positions removed from the exact transition-state position, in particular in the reactant or product wells. In cases where there is no promoting vibration, while the spectral density may well change shape as a function of reaction coordinate position, there will be no formation of such strong peaks. Numerical experiments completed in our group have shown a delta function at the frequency position of the promoting vibration as the analytic theory predicted when we study a model problem in which a vibration is coupled symmetrically.34 ˚ crystal structures of Plapp and coworkers.35 This crystal Our analysis began with the 2.1 A þ structure contains both NAD and 2,3,4,5,6-pentafluorobenzyl alcohol complexed with the native horse liver enzyme (metal ions with both the substrate and cofactor). The fluorinated alcohol does not react and go onto products because of the strong electron-withdrawing tendencies of the flourines on the phenyl ring. So it is hypothesized that the crystal structure corresponds to a stable approximation of the Michaelis complex. We then replaced the fluorinated alcohol with the unfluorinated compound to obtain the reactive species as in Luo, Kahn, and Bruice.30 This structure was used as input for the CHARMM program.36 Both crystallographic waters29 (there are 12 buried waters in each subunit) and environmental waters were included via the TIP3P potential.37 The substrates were created from the MSI/CHARMM parameters. The NAD cofactor was modeled using the force field of Mackerell et al.38 The lengths of all bonds to hydrogen atoms were held fixed using the SHAKE algorithm. A time step of 1 fsec was employed. The initial structure was minimized using a steepest descent algorithm for 1000 steps followed by an adapted basis Newton– Raphson minimization of 8000 steps. The dynamics protocol was heating for 5 psec followed by equilibration for 8 psec and finally by data collection for the next 50 psec. Using CHARMM, we computed the force autocorrelation function on the reacting particle. The force is calculated in CHARMM as a derivative of the velocity. This is a numerical procedure, which can of course introduce error. We have recently found that spectral densities may also be calculated from the velocity autocorrelation function directly. These spectral densities exhibit exactly the same diagnostics for the presence of a promoting vibration as do those calculated from the force. In addition, the Fourier transform of the force autocorrelation function can be shown to be related to the Fourier transform of the velocity autocorrelation function times a square of the frequency. This square of the frequency tends to accentuate high frequencies. In a simple liquid this is not a problem because there are essentially no high-frequency modes. In a bonded system such as an enzyme, many high-frequency modes remain manifest in autocorrelation functions, and it is advantageous to employ spectral densities calculated from Fourier transforms of the velocity function. Application of this methodology to this model of HLAD yields the results shown in Figure 18.4. In fact we do see strong numerical evidence for the presence of a promoting vibration; intense peaks in the spectral density for the reaction coordinate are greatly reduced at a point between the reactant and product wells. Mutational experimental data can also be rationalized. Figure 18.5 shows the results of a mutation of Val203 to a smaller Ala. We note the intensity of the peak in the spectral density is reduced, indicative of a smaller force on the reaction coordinate. It is this mutant in which indicators of tunneling decrease. Figure 18.6 shows analogous results for a mutation of Phe93 to Trp. This mutation shows no experimental effect on tunneling (though it does affect rate by lowering binding of substrate) and in fact the two spectral densities are quite similar. These computational experiments were undertaken under the guidance of a large body of experimental literature on this enzyme. In some case all we have done is rationalize the known

Vibrationally Enhanced Tunneling and Kinetic Isotope Effects in Enzymatic Reactions

485

FIGURE 18.4 The spectral density of the hydride in the reactant well, product well, and at a point of MC.

experimental results. It is desirable to have a method to identify residues likely to be involved in the creation of a promoting vibration without prior experimental guidance. We have developed such an algorithm39 and we here sketch the approach and the results found for the HLADH system. The method depends on computing the projection of motions of the center of mass of individual residues along the reaction coordinate axis (by this we mean the donor acceptor axis; we do not mean to imply the actual sets of atomic motions needed for reaction are identified). A correlation function of this quantity with the donor acceptor motion is found. When Fourier transformed, strong peaks at the frequency location of an identified promoting vibration are indicative of the involvement of a residue in creation of the promoting vibration. The reader is referred to Ref. 39 for mathematical and implementation details. Eight residues are found to be strongly correlated in their motion to that of the donor and acceptor. They are shown in Figure 18.7. Some residues identified with this algorithm agree directly with experimental evidence. For example, Val203 has been identified by both Klinman40 and Plapp41 as being a residue that on mutagenesis changes kinetic parameters and signatures of tunneling. In addition, Val292,41,42 has been found by Plapp et al. to be similarly implicated in

FIGURE 18.5 The spectral density for the hydride in the reactant well for both wild-type HLADH, and one in which we mutate (in the computer) Val203 ! Ala. The smaller size of the alanine results in a much smaller effective force on the reaction coordinate.

486

Isotope Effects in Chemistry and Biology 300 Phe 93 → Trp mutant wild-type

J (w)

200

100

0 0

100

200 300 w (cm−1)

400

500

FIGURE 18.6 The spectral density for the hydride in the reactant well for both wild-type HLADH, and a one in which we mutate (in the computer) Phe93 ! Trp.

tunneling for the hydride reaction coordinate. Phe93 is found by Klinman to not change indicators of tunneling,43 and we find no evidence for coupling of the dynamics of this residue to the reaction coordinate. There is also agreement between the predictions of theory and experiment at location 292, where a Valine is found by both Plapp and predicted by our algorithm to impact enzyme kinetics.41 There are, however, some potential discrepancies — Plapp finds Thr178 to affect kinetics. Our algorithm found no evidence of dynamic coupling of this residue to the reaction coordinate. It is possible that there is no contradiction here — clearly static effect such as binding

FIGURE 18.7 Residues found computationally to be important in creation of a protein promoting vibration in HLADH.

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geometries can alter kinetics — they just do it in a different way than the dynamic coupling of residues to the reaction coordinate. It is important to point out that experimental evidence for the involvement of these residues in catalysis seems to indicate a moderate involvement of dynamics in the overall catalytic effect. For example, the mutation of Val203 ! Ala results only in about a factor of 4.5 decreases in V/Kb for HLADH.41 While it is expected that this mutation would only lower the dynamic contribution to catalysis, a factor of 4 or 5 is obviously not the entire catalytic effect. A final interesting question to address is the evolutionary importance of protein dynamics in the catalytic event. There has been a good deal of recent controversy regarding this issue. Finke44 has suggested that experimental evidence shows that dynamics coupled to a tunneling reaction coordinate has not been selected for because measurement of KIEs in enzymatic reactions and in related solution-phase reactions seem similar. This has been disputed45 on the basis that the reactions were not identical in the solution and the enzyme. Isotope effects are significantly repressed when there is a protein promoting vibration. We now present fairly direct evidence that evolution has at least conserved the architecture that creates the promoting vibration across a wide range of species and time. The eight residues shown in Figure 18.7 along with random residues in the intervening regions may be called the protein promoting generating sequence; that is: S-X(36)G-X(21)-V-G-X(2)-V-X(59)-E-X-I-X(22)-V. The PPV generating sequence, defined above, was used to search the PIR-NREF sequence database,46 which at the time of the search contained 1,042,859 protein sequences. Forty-four proteins were found to contain this pattern embedded within their amino acid sequences. They derived from the following organisms: Gallus gallus (chicken), Uromastyx hardwickii (Indian spiny-tailed lizard), Homo sapiens (human), Equus caballus (domestic horse), Oryctolagus cuniculus (European rabbit), Coturnix japonica (Japanese quail), Oryza sativa (rice), Zea mays (a plant), Octopus vulgaris (common octopus), Rana perezi (Perez’s frog), Struthio camelus (ostrich), Peromyscus maniculatus (deer mouse), Mus caroli (Ryukyu mouse), Mus musculus (house mouse), Rattus norvegicus (Norway rat), and Arabidopsis thaliana (mouse-ear cress). The interesting result is that despite their diverse origins, each protein shares the same function; each is an alcohol dehydrogenase. To what extent nature has conserved the PPV generating sequence is better appreciated by considering how much sequence conservation in this set of proteins is accounted for by it alone. By aligning the 44 sequences, using the CLUSTALW algorithm,16 we find that 68 residues are unchanged between these proteins. The average protein length in this set is 373 amino acids; thus, these 68 residues translate roughly to a total amino acid conservation of 18%. Clearly the eight residues that are perfectly conserved in all 44 proteins contribute a significant amount to the total sequence homology. Across such a wide range of sources, nature has applied strong selective pressure to maintain the PPV generating sequence. Further support for this theory is found when one makes a small change to the PPV sequence. If one changes S-X(36)-G-X(21)-V to S-X(36)-GX(20)-V or to S-X(36)-G-X(22)-V, in other words a single change to the length of the intervening residues in the PPV pattern, then no matches are found in the PIR-NREF database. Thus, this strong homology is not simply the result of these proteins being very ancient and conserved in all locations, but at the specific locations found to be of importance for HLAD.

B. LACTATE D EHYDROGENASE LDH catalyzes the interconversion of the hydroxy-acid lactate and the keto-acid pyruvate with the coenzyme nicotinamide adenine dinucleotide (see Figure 18.8).47 This enzyme plays a fundamental role in respiration, and multiple isozymes have evolved to enable efficient production of substrate appropriate for the microenvironment.48 Two main subunits, referred to as heart and muscle (skeletal), are combined in the functional enzyme as a tetramer to accommodate aerobic and anaerobic environments. Subunit combinations range from pure heart (H4) to pure muscle (M4).

488

Isotope Effects in Chemistry and Biology Arg-169

NH2

+

mobile loop

NH2

O

-

NH2 +

O C1

CH3

+ε C2

−ε

Arg-106 NH2

O

HN

His-193

+ NH

H 5 6

H 4

1N

3

O C-NH2

O

-

O C Asp-166

2

ADPR

FIGURE 18.8 Diagram of the binding site of LDH with bound NADH and pyruvate showing hydrogen bonds between the substrate and key catalytically important residues of the protein. The catalytic event involves the hydride transfer of the C4 hydrogen of NADH from the pro-R side of the reduced nicotinamide ring to the C2 carbon of pyruvate and protein transfer from the imidazole group of His-193 to pyruvate’s keto oxygen.

The reaction catalyzed involves the transfer of a proton between an active site histidine and the C2 bound substrate oxygen as well as hydride transfer between C4N of the cofactor, NAD(H), and C2 of the substrate. Remarkably, the domain structure, subunit association, and amino acid content of the human isozyme active sites are comparable. In fact the active sites have complete residue identity, with the overall subunits only differing by about 20%. What is astounding is that the kinetic properties of the two isozymes are quite different. The heart isoform favors the production of pyruvate from lactate in order to have the heart predominantly employ aerobic respiration. In contradistinction, muscles are quite comfortable under periods of stress undergoing anaerobic respiration. So the muscle isoform favors lactate production. The question that remains is how two proteins that are so strikingly similar in composition can possibly have such different kinetic behaviors. We will examine some very recent theoretical results that may shed light on this question.49 There are far fewer known experiments about this enzyme. For example, there are no reliable measurements of the rate of hydride transfer in both directions. In addition there are no reliable KIE experiments or mutagenesis experiments in the locations we will discuss, extant for the human enzyme to confirm our predictions, so the work described in this section is of a speculative nature. The first step in the theoretical study of this problem is a molecular dynamics computation on the human proteins. Our numerical methodology is described in detail in Ref. 49, but in brief: the starting point for computations were crystal structures solved by Read et al.50 for homo-tetrameric human heart, h-H4LDH, and muscle, h-M4LDH, isozymes in a ternary complex with NADH and ˚ resolution, respectively. Numerical analysis of molecular dynamics oxamate at 2.1 and 2.3 A computations followed our previously published approach.51 While the chemical step of lactate dehydrogenase and alcohol dehydrogenase is quite similar, transfer of a hydride from or to an NAD cofactor, there is no evidence for the involvement of protein dynamics in the mechanism of the enzyme. Thus, the first step in the analysis is the search for the presence of a protein promoting vibration. A Fourier transform of the correlation function of the donor – acceptor velocity in the two isoforms shows the relative motion that may be imposed on the reaction coordinate. The absence of strong peaks in a similar Fourier transform for the

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reaction coordinate at a point of minimal coupling (MC) — the putative transition state demonstrates the presence of a symmetrically coupled protein promoting vibration. Such computations are shown in Figure 18.9a and Figure 18.9b for the heart isoforms and in Figure 18.10a and Figure 18.10b for the muscle isoforms. These two sets of figures demonstrate convincing numerical evidence that there is in fact a protein promoting vibration present in both isoforms of this enzyme. What is strange is the relative intensity of the peaks in the reaction coordinate Figures (Figure 18.9b and Figure 18.10b). It seems that the strength of coupling of the promoting vibration in the heart isoform is larger when pyruvate is bound, and in the muscle isoform the signal is more intense when lactate is bound. This would seems to favor the production of the opposite chemical species to what is required for each tissue. The explanation is found in Figure 18.11a and Figure 18.9b. These figures show time series of the donor –acceptor distance in both the heart and muscle isoform, respectively. Note, for example, that in the heart ˚ isoform the distance between the donor and acceptor when lactate is bound is on average 0.6 A less than when pyruvate is bound. In contradistinction, in the muscle isoform, the donor –acceptor ˚ less when pyruvate is bound. Therefore the intensity of the promoting vibration distance is 0.6 A is in fact the product of the strength of the coupling times the distance from the point of MC, which is the putative transition state. The argument for these isoforms is that, for example in the heart isoform, when lactate is bound there is rapid conversion to pyruvate followed by relaxation ˚ . This could be a mechanism for ”locking in” the formed pyruvate. of the protein structure by 0.6 A For this mechanism to be viable, there either needs to be significant quantum tunneling in the hydride transfer step or significant dissipation to the protein medium as the hydride transfers. If there is tunneling, then clearly the longer distance for the “less preferred” substrate will significantly favor the other substrate. If there is no tunneling, but rather activated transfer across the barrier, the frictional dissipation could lower the probability of transfer across a longer distance. These concepts are all speculative because there are no KIE experiments to show a significant degree of tunneling; but given the chemical similarity of this enzyme to alcohol dehydrogenase, it is certainly plausible that there is a significant contribution to reaction from tunneling. What is also missing is a measurement of the equilibrium constant for the chemical step on the enzyme. This will show the relative rates of one direction versus the other; it may be, however, that there is never equilibrium, and microscopic rate measurements are needed. It is important to again point out that there are many factors other than chemistry that govern the production of one product or the other in these enzymes. For example, binding of reactants is strongly dependent on the pH of the medium which is strongly influenced by physiological conditions. It does, however, seem implausible that careful modulation of donor – acceptor distance has been developed by evolution to no purpose. A final question to answer by application of computational methods to this enzyme is the identification of residues involved in creation of the protein promoting vibration. Applying the methodology of Ref. [39] to this system shows slight variations in relative importance of residue depending on which substrate is bound. The results are given in Table 18.1. The results for the heart isoform with lactate bound are shown graphically in Figure 18.12. In many ways the results are similar to that generated for alcohol dehydrogenase, except for the clear involvement of two residues on the lactate side of the active site. In alcohol dehydrogenase, there is only one residue on the alcohol binding side, and the rest of the residues identified in creation of the promoting vibration are all on the cofactor side of the binding pocket. It is conceivable that greater involvement on the substrate side for lactate dehydrogenase is somehow involved in the relaxation of the entire pocket on production of the appropriate product (lactate or pyruvate). This is purely speculative, and in fact the cutoff for the strength of the signature in the spectral density is fairly arbitrary. Further investigations of lactate dehydrogenase are now under way. In these studies we employ an algorithm due to Chandler,52 which allows identification of all atomic motions necessary for movement across the transition state. This will more clearly identify the importance of these lactate side residues.

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FIGURE 18.9 (a) The spectral density GDA(v) for the donor – acceptor relative motion in the wild-type human heart lactate dehydrogenase isoform; it monitors the relative motion between the substrate C2 carbon and carbon C4N of the nicotinamide ring of the cofactor NADþ/NADH. The solid line represents the configuration where lactate and NADþ are bound. The dotted line is when pyruvate and NADH are bound. The dashed line is the MC simulation with lactate and NADþ bound and the (hydride – C2) and (hydride – C4N) distances restrained, and the dot-dash line is exemplary of the restrained hydride (RH) simulations to search for the point ˚ -hydride-A ˚ -C4N). The power spectrum is of MC. Distances are in Angstroms and defined in the form (C2-A reported in CHARMM units. (b) The spectral density Gs(v) for the reaction coordinate in the wild-type human heart lactate dehydrogenase isoform. The solid line represents the configuration where lactate and NADþ bound, the dotted line is when pyruvate and NADH are bound, the dashed line is the MC simulation with lactate and NADþ are bound and the (hydride– C2 and hydride – C4N) distances restrained, and the dot-dash line is exemplary of the RH simulations to search for the point of MC. Distances are in Angstroms and defined ˚ -hydride-A ˚ -C4N). The power spectrum is reported in CHARMM units. in the form (C2-A

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491

FIGURE 18.10 (a) Similar to Figure 18.9(a), but for the human muscle isoforms; (b) similar to Figure 18.9(b), but for the human muscle isoform.

V. HYDROGEN TRANSFER COUPLED TO ELECTRON TRANSFER — KINETIC TRENDS IN THE PRESENCE OF A PROMOTING VIBRATION Other important classes of enzymatically catalyzed reactions have recently been suggested to involve protein promoting vibrations. These reactions are characterized by the transfer of a hydrogen (that is, either a proton or neutral hydrogen radical) coupled to transfer of an electron. Two such enzymes are lipoxygenase, which has been studied by Klinman’s group,53 and amine dehydrogenase studied by Scrutton and coworkers.54 An interesting difference between these

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Isotope Effects in Chemistry and Biology

FIGURE 18.11 (a) Donor – acceptor distance for the wild-type human heart lactate dehydrogenase isoform; this is the distance between the C2 carbon of substrate and carbon C4N of the nicotinamide ring of the cofactor. The solid line represents the configuration where lactate and NADþ are bound, and the dashed line is when pyruvate and NADH are bound. (b) Donor – acceptor distance for the wild-type human muscle lactate dehydrogenase isoform; this is the distance between the C2 carbon of substrate and carbon C4N of the nicotinamide ring of the cofactor. The solid line represents the configuration where lactate and NADþ are bound, and the dashed line is when pyruvate and NADH are bound.

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FIGURE 18.12 Important residues in creation of a promoting vibration in lactate dehydrogenase in the heart isoform when lactate and NADþ is bound. The blue bonded molecules are lactate and NADþ. The purple bonded are behind lactate (donor region) and are strongly correlated with the donor – acceptor motion. The brown bonded are behind NADþ (acceptor region) and are strongly correlated with the donor– acceptor motion. The green bonded are weakly correlated residues.

reactions and those of alcohol dehydrogenase is the presence of rather large KIEs. The amine dehydrogenases exhibit KIEs in the range of 15 to 25 when soybean lipoxygenase has one of 100. The involvement of protein dynamics is suggested by unexpected temperature dependence in the KIEs. Fitting of the lipoxygenase to a simple few-parameter model of a promoting vibration, developed by Kuznetsov and Ulstrup55 and known as gating, yielded results. This seemed to imply an almost nonphysical forcing of acceptor and donor atoms toward each other.53 There have been previous model studies of these systems.56 These studies, while including the effects of environment, do not address the question of the effect of a promoting vibration. There has been recent work from Smedarchina and Siebrand’s group on this topic, addressing the issue of concerted versus sequential transfer of hydrogen and electrons.57 This is an extremely difficult problem to approach from a rigorous theoretical viewpoint because the electron transfer implies significant nonadiabatic electronic effects. In addition to any dynamics on a single surface (a hard enough problem for an enzyme) quantum-chemistry computations of the potential surfaces with the electron on the electron donor and acceptor sites is a minimal requirement. As a first step in trying to understand the nature of hydrogen transfer in such coupled hydrogen – electron-transfer systems, we have undertaken a study of a generic model system we feel will help explain the rather different kinetic behavior in these systems as compared to single particle hydrogen transfer enzymes. The starting point for the study is a simple model of the coupled process. This model is found from a generalization of the Hamiltonian in Equation 18.11 to include the modulation of hydrogen

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Isotope Effects in Chemistry and Biology

transfer potential as a result of electron transfer: X P2k P2 1 cs þ mk v2k qk 2 k 2 H ¼ s þ VD lDlkDl þ VA lAlkAl þ 2ms 2m 2 m k k vk k þ

!2

P2Q 1 D þ M V2 Q2 þ cQ ðs2 2 s20 ÞQ þ sz þ Vc ðsÞsx 2M 2 2

ð18:12Þ

The modifications to the Hamiltonian previously used are: 1. There are two different bare potentials for hydrogen transfer VD and VA : the potentials are chosen by the state of the electron given by the projection operators lDlkDl and lAlkAl: 2. These are an electron degree of freedom represented by a two-state spin system. The rate of electron transfer from one state to the other (from the electron donor to acceptor) is given by sx modulated by a term that can depend on the hydrogen position. We note that we still couple to an infinite bath of harmonic oscillators, which represent the bulk protein, and to a protein promoting vibration. For each of mathematical implementation, we here choose the zero of promoting vibration coupling to be in the well rather than at the barrier top, but this is arbitrary. We point out that we can tune this model to allow for both sequential and concerted hydrogen – electron transfer. Sequential transfer is found with a very high transfer rate, and concerted with a lower one. Our initial results are presented in Table 18.2 below. Introduction of the electronic degree of freedom significantly raises the KIE. In fact the results for 11 kcal mol21 are similar to those found by Scrutton et al. for amine dehydrogenase. The stronger the electronic coupling is, the less the enhancement of the KIE. This is understood by the result that very strong electron coupling yields results asymptotic to sequential transfer. In other words, hydrogen transfers in the presence of a promoting vibration in the electron acceptor state alone. This clearly rationalizes the high KIE found in the coupled electron – hydrogen systems with the possibility of the presence of a promoting vibration. In addition, the natural log of the KIEs versus 1=T over the biochemically accessible range is essentially temperature independent, which is in direct agreement with the amine dehydrogenase results. There is, however, a clear activation energy found from plots of k versus 1=T: the primary evidence suggested by the Scrutton group for significant extreme tunneling in amine dehydrogenase.

TABLE 18.2 Kinetic Isotope Effects from Exact Quantum Rate Computations on the Model of Equation 18.12 Kinetic Isotope Effect No Promoting Vibration

Moderate Promoting Vibration

Barrier height

BVc 5 0.45 eV

BVc 5 4.5 eV

BVc 5 0.45 eV

BVc 5 4.5 eV

11 kcal mol21 30 kcal mol21

56 1942

65 1256

38 1218

24 685

In one case there is no protein promoting vibration; in the second case there is a promoting vibration coupled with strength similar to that in our previous model studies. In each case there are two levels of electron coupling, essentially the rate of electron transfer between the two states. We have found that high coupling is asymptotic to sequential transfer.

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VI. CONCLUSIONS This chapter has focused on recent advances in the study of enzymes in which there is now both theoretical and experimental evidence that protein dynamics is in fact an integral part of a complex reaction coordinate. It is of course true in all enzymes that the protein must move; for example, hinge motions that lock substrates in active sites or breathing motions that allow substrates to bind through channels. The types of dynamics of which we speak are of a completely different nature. These motions are an integral part of the chemical/catalytic event. In fact the reaction coordinate, sometimes idealized to be a one-dimensional motion (for example, the motions of the hydride in alcohol dehydrogenase), is in fact far more complex. The hydride motion is only possible when coupled to motion of the protein backbone as the hydride is being transferred. This is analogous to the situation in the theory of liquid reaction dynamics. For example, in proton-transfer reactions, the barrier to reaction is caused not by the proton transfer itself, but by the reorientation of a polar solvent to allow solvation of the proton as it moves. The reaction coordinate is a complex agglomeration of the proton movement and the solvent movement. Having pointed out how important this motion is to catalysis, it is also important to remember that the dynamics effect is only part, and possibly a small part, of the overall catalytic effect of the enzyme. For example, in alcohol dehydrogenase, a large reason the chemical step can happen is the previous transfer of a proton from the alcohol to the coordinated zinc atoms to form the alcoxide. This alcoxide is a far more reactive species. In addition, local solvation of charge by active site residues is of significance. Thus, the action of the enzyme is a complex set of chemical capabilities. The importance of dynamics shows that set is even more complex than we previously knew.

ACKNOWLEDGMENTS The author gratefully acknowledges the support of the Office of Naval Research, The National Science Foundation, and the National Institutes of Health.

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26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

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V. I., and Makarov, D. E., Low-temperature chemical reactions. Effect of symmetrically coupled vibrations in collinear exchange reactions, Chem. Phys., 154, 407, 1991. Huskey, P. and Schowen, R., Reaction coordinate tunneling in hydride-transfer reactions, J. Am. Chem. Soc., 105, 5704 –5706, 1983. Bell, R. P., The Tunnel Effect in Chemistry, Chapman and Hall, New York, 1980. Agarwal, P. K., Webb, S. P., and Hammes-Schiffer, S., Computational studies of the mechanism for proton and hydride transfer in liver alcohol dehydrogenase, J. Am. Chem. Soc., 122, 4803– 4812, 2000. Kohen, A., Cannio, R., Bartolucci, S., and Klinman, J. P., Enzyme dynamics and hydrogen tunneling in a thermophilic alcohol dehydrogenase, Nature, 399, 496– 499, 1999. Cha, Y., Murray, C. J., and Klinman, J. P., Hydrogen tunneling in enzyme reactions, Science, 243, 1325, 1989. (a) Grant, K. L. and Klinman, J. P., Evidence that both protium and deuterium undergo significant tunneling in the reaction catalyzed by bovine serum amine oxidase, Biochemistry, 28, 6597, 1989; (b) Kohen, A. and Klinman, J. P., Enzyme catalysis: beyond classical paradigms, Acc. Chem. Res., 31, 397, 1998; (c) Bahnson, B. J. and Klinman, J. P., Hydrogen tunneling in enzyme catalysis, Methods Enzymol., 249, 373, 1995; (d) Rucker, J., Cha, Y., Jonsson, T., Grant, K. L., and Klinman, J. P., Role of internal thermodynamics in determining hydrogen tunneling in enzyme-catalyzed hydrogen transfer reactions, Biochemistry, 31, 11489, 1992. Kohen, A. and Klinman, J. P., Protein flexibility correlates with degree of hydrogen tunneling in thermophilic and mesophilic alcohol dehydrogenases, JACS, 122, 10738– 10739, 2000. Zavodsky, P., Kardos, J., Svingor, A., and Petsko, G. A., Adjustment of conformational flexibility is a key event in the thermal adaptation of proteins, Proc. Natl Acad. Sci. USA, 95, 7406– 7411, 1998. Bahnson, B. J., Colby, T. D., Chin, J. K., Goldstein, B. M., and Klinman, J. P., A link between protein structure and enzyme catalyzed hydrogen tunneling, Proc. Natl Acad. Sci. USA, 94, 12797– 12802, 1997. Bahnson, B. J., Park, D.-H., Kim, K., Plapp, B. V., and Klinman, J. P., Unmasking of hydrogen tunneling in the horse liver alcohol dehydrogenase reaction by site-directed mutagenesis, Biochemistry, 32, 5503– 5507, 1993. Luo, J., Kahn, K., and Bruice, T. C., The linear dependence of logðkcat =Km Þ for reduction of NAD þ by PhCH2OH on the distance between reactants when catalyzed by horse liver alcohol dehydrogenase and 203 single point mutants, Bioorg. Chem., 27, 289– 296, 1999. Tsai, S.-C. and Klinman, J. P., Probes of hydrogen tunneling with horse liver alcohol dehydrogenase at subzero temperatures, Biochemistry, 40, 2303– 2311, 2001. Passino, S. A., Nagasawa, Y., and Fleming, G. R., Three pulse stimulated photon echo experiments as a probe of polar solvation dynamics: utility of harmonic bath modes, J. Chem. Phys., 107, 6094, 1997. Antoniou, D. and Schwartz, S. D., Quantum proton transfer with spatially dependent friction: phenolamine in methyl chloride, J. Chem. Phys., 110, 7359– 7364, 1999. Caratzoulas, S. and Schwartz, S. D., A computational method to discover the existence of promoting vibrations for chemical reactions in condensed phases, J. Chem. Phys., 114, 2910– 2918, 2001. Ramaswamy, S., Elkund, H., and Plapp, B. V., Structures of horse liver alcohol dehydrogenase complexed with NAD þ and substituted benzyl alcohols, Biochemistry, 33, 5230 –5237, 1994. Brooks, B. R., Bruccoleri, R. E., Olafson, B. D., States, D. J., Swaminathan, S., and Karplus, M., CHARMM: a program for macromolecular energy, minimization, and dynamics calculations, J. Comp. Chem., 4, 187– 217, 1983. Jorgensen, W. L., Chandrasekher, J., Madura, J. D., Impey, R. W., and Klein, M. L., Comparison of 23 simple potential functions for simulating liquid water, J. Chem. Phys., 79, 926, 1983. Pavelites, J. J., Gao, J., Bash, P. A., Alexander, D., and Mackerell, J., A molecular mechanics force field for NAD þ NADH, and the pyrophosphate groups of nucleotides, J. Comput. Chem., 18, 221– 239, 1997. Mincer, J. S. and Schwartz, S. D., A computational method to identify residues important in creating a protein promoting-vibration in enzymes, J. Phys. Chem. B, 107, 366– 371, 2003. Bahnson, B. J., Colby, T. D., Chin, J. K., Goldstein, B. M., and Klinman, J. P., A link between protein structure and enzyme catalyzed hydrigen tunneling, Proc. Natl Acad. Sci. USA, 94, 12797– 12802, 1997.

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41 Rubach, J. K. and Plapp, B. V., Amino acid residues in the nicotinamide binding site contribute to catalysis by horse liver alcohol dehydrogenase, Biochemistry, 42, 2907– 2915, 2003. 42 Rubach, J. K., Ramaswamy, S., and Plapp, B. V., Contributions of valine-292 in the nicotinamide binding site of liver alcohol dehydrogenase and dynamics to catalysis, Biochemistry, 40, 12686– 12694, 2001. 43 (a) Bahnson, B. J., Park, D.-H., Kim, K., Plapp, B. V., and Klinman, J. P., Unmasking of hydrogen tunneling in the horse liver alcohol dehydrogenase reaction by site-directed mutagenesis, Biochemistry, 32, 5503– 5507, 1993; (b) Chin, J. K. and Klinman, J. P., Probes of a role for remote binding interactions on hydrogen tunneling in the horse liver alcohol dehydrogenase reaction, Biochemistry, 39, 1278– 1284, 2000. 44 Doll, K. M., Bender, B. R., and Finke, R. G., The first experimental test of the hypothesis that enzymes have evolved to enhance hydrogen tunneling, JACS, 125, 10877 –10884, 2003. 45 Kemsley, J., Enzyme tunneling idea questioned, C&EN, 81, 29 – 30, 2003. 46 Wu, C. H., Huang, H., Aminski, L., Castro-Alvear, J., Chen, Y., Hu, Z.-Z., Ledley, R. S., Lewis, K. C., Mewes, H.-W., Orcutt, B. C., Suzek, B. E., Tsugita, A., Vinayaka, C. R., Yeh, L.-S., Zhang, J., and Barker, W. C., Nucleic Acids Res., 30, 35 – 37, 2002, The URL for the PIR website is http://pir. georgetown.edu. 47 Gulotta, M., Deng, H., Deng, H., Dyer, R. B., and Callender, R. H., Toward an understanding of the role of dynamics on enzymatic catalysis in lactate dehydrogenase, Biochemistry, 41, 3353 –3363, 2002. 48 Meister, A., Advances in Enzymology and Related Areas of Molecular Biology, Vol. 37, Wiley, New York, 1973. 49 Bassner, J.E. and Schwartz, S.D., Donor acceptor distance and protein promoting vibration coupling as a mechanism for kinetic control in isozymes of human lactate dehydrogenase, J. Phys. Chem. B, 108, 444– 451, 2004. 50 Read, J. A., Winter, V. J., Eszes, C. M., Sessions, R. B., and Brady, R. L., Structural basis for altered activity of M- and H-isozyme forms of human lactate dehydrogenase, Proteins: Structure, Function, Genetics, 43, 175– 185, 2001. 51 Caratzoulas, S., Mincer, J. S., and Schwartz, S. D., Identification of a protein promoting vibration in the reaction catalyzed by horse liver alcohol dehydrogenase, JACS, 124(13), 3270– 3276, 2002. 52 (a) Csajka, F. S. and Chandler, D., Transition pathways in many body systems: application to hydrogen bond breaking in water, J. Chem. Phys., 109, 1125– 1133, 1998; (b) Bolhuis, P. G., Dellago, C., and Chandler, D., Sampling ensembles of deterministic pathways, Faraday Discuss., 110, 421– 436, 1998; (c) Dellago, C., Bolhuis, P. G., and Chandler, D., Efficient transition path sampling: application to Lennard-Jones cluster rearrangements, J. Chem. Phys., 108, 9236– 9245, 1998; (d) Bolhuis, P. G., Dellago, C., and Chandler, D., Reaction coordinates of biomolecular isomerization, Proc. Natl Acad. Sci. USA, 97, 5877– 5882, 2000. 53 (a) Knapp, M. J., Rickert, K., and Klinman, J. P., Temperature-dependent isotope effects in soybean lipoxygenase-1: correlating hydrogen tunneling with protein dynamics, J. Am. Chem. Soc., 124, 3865, 2002; (b) Knapp, M. J., and Klinman, J. P., Environmentally coupled hydrogen tunneling, Eur. J. Biochem., 269, 3113, 2002. 54 (a) Basran, J., Sutcliffe, M., and Scrutton, N., Enzymatic H-transfer requires vibration-driven extreme tunneling, Biochemistry, 38, 3218, 1999; (b) Sutcliffe, M. J. and Scrutton, N. S., A new conceptual framework for enzyme catalysis, Eur. J. Biochem., 269, 3096, 2002. 55 Kuznetsov, A. M. and Ulstrup, J., Can. J. Chem., 77, 1085– 1096, 1999. 56 (a) Cukier, R. I., Mechanism for proton-coupled electron-transfer reactions, J. Phys. Chem., 98, 2377, 1994; (b) Zhao, X., and Cukier, R. I., Molecular dynamics and quantum chemistry study of a protoncoupled electron transfer reaction, J. Phys. Chem., 99, 945, 1995; (c) Cukier, R. I., Proton-coupled electron transfer through an asymmetric hydrogen-bonded interface, J. Phys. Chem., 99, 16101, 1995; (d) Cukier, R. I., Proton-coupled electron transfer reactions: evaluation of rate constants, J. Phys. Chem., 100, 15428, 1996; (e) Shin, S. and Metiu, H., Nonadiabatic effects on the charge transfer rate constant: a numerical study of a simple model system, J. Chem. Phys., 102, 9285, 1995; (f) HammesSchiffer, S., Theoretical perspectives on proton-coupled electron transfer reactions, Acc. Chem. Res., 34, 273–281, 2001; (g) Iordanova, N. and Hammes-Schiffer, S., Theoretical investigation of large kinetic isotope effects for proton-coupled electron transfer in ruthenium polypyridyl complexes, J. Am. Chem. Soc., 124, 4848 –4856, 2002. 57 Smedarchina, Z. Siebrand, W., preprint.

19

Kinetic Isotope Effects for Proton-Coupled Electron Transfer Reactions Sharon Hammes-Schiffer

CONTENTS I. II.

Introduction ...................................................................................................................... 499 Theory and Methods ........................................................................................................ 500 A. Electron Transfer Theory......................................................................................... 500 B. Proton Transfer Theory............................................................................................ 501 C. Proton-Coupled Electron Transfer Theory .............................................................. 502 D. Methodological Developments ................................................................................ 505 III. Applications to Chemical and Biological Systems ......................................................... 506 A. PCET in Solution ..................................................................................................... 506 B. Enzyme Reactions.................................................................................................... 508 C. Role of Motion in Enzyme Reactions ..................................................................... 512 IV. Summary and Conclusions .............................................................................................. 514 Acknowledgments ........................................................................................................................ 515 References..................................................................................................................................... 515

I. INTRODUCTION The coupling between proton and electron transfer reactions is vital for a wide range of chemical and biological processes, including photosynthesis,1 – 7 respiration,8,9 and numerous enzyme reactions.10 In this chapter, reactions involving the simultaneous transfer of at least one proton and one electron are denoted proton-coupled electron transfer (PCET). Within the framework of this definition, hydrogen atom and hydride transfer reactions are viewed as specific types of PCET reactions in which the proton and electrons transfer in the same direction between the same centers. In general, however, the electron and proton could transfer between different centers and could transfer either in the same direction or in different directions. Nuclear quantum effects such as zeropoint motion and hydrogen tunneling play an important role in many types of PCET reactions. A powerful tool for probing the fundamental nature of nuclear quantum effects in PCET reactions is the investigation of the deuterium kinetic isotope effect (KIE), which is the ratio of the rate with hydrogen to the rate with deuterium. A wide range of magnitudes for the KIEs of PCET reactions in solution and enzymes has been measured experimentally. The KIE for PCET between iron bi-imidazoline complexes11 was measured to be 2.3, and the KIE for photo induced PCET in a tyrosine-substituted ruthenium – tris-bipyridine model system12 was measured to be 2.0 – 2.5. For a number of inorganic complexes in 499

500

Isotope Effects in Chemistry and Biology

solution, however, the KIEs are significantly larger in magnitude. The KIEs for PCET in oxoruthenium polypyridyl complexes13,14 were measured to be as high as 16, and the KIEs for PCET reactions between osmium complexes and benzoquinone15,16 were found to be as high as 400. Substantial KIEs have also been measured for enzymatic reactions. The KIEs for the hydride transfer reactions catalyzed by liver alcohol dehydrogenase (LADH)17,18 and dihydrofolate reductase (DHFR)19 were measured to be 3.8 and 3.0, respectively, and the KIE for the net hydrogen atom transfer catalyzed by soybean lipoxygenase-1 (SLO)20 – 24 was measured to be 81 at room temperature. We have developed two types of theoretical approaches to calculate and analyze the KIEs for PCET reactions. The first approach is based on a multistate continuum theory25 – 27 for multiple charge transfer reactions. In this theory, the solute is represented by a multistate valence bond model, the solvent is described by a dielectric continuum, and the transferring hydrogen nuclei are represented as quantum mechanical wavefunctions. We have applied the multistate continuum theory to a wide range of reactions, including iron bi-imidazoline complexes,28 tyrosine-substituted ruthenium bipyridyl complexes,29 ruthenium polypyridyl complexes,30 osmium –benzoquinone complexes,31 and the enzyme SLO.32 The second approach is based on hybrid quantum/classical molecular dynamics simulations33,34 that include the motion of the complete environment (i.e., solute, solvent, and protein). In this hybrid approach, the electronic quantum effects are included with an empirical valence bond potential,35 and the nuclear quantum effects are included by representing the transferring hydrogen nucleus as a three-dimensional vibrational wavefunction.36 We have applied this hybrid molecular dynamics approach to hydride transfer reactions in LADH33, 34 and DHFR.37 – 39 These theoretical studies of PCET reactions have provided physical explanations for the wide range of experimentally measured KIEs. This chapter provides an overview of our current understanding of KIEs for PCET reactions. The first part of Section II summarizes the theoretical descriptions of single electron transfer (ET), single proton transfer (PT), and PCET reactions. The latter part of Section II outlines the seminal aspects of the multistate continuum theory and the hybrid quantum/classical molecular dynamics approach. Section III summarizes the results from a variety of applications of these methods to PCET reactions. The general principles derived from these theoretical studies are discussed in Section IV.

II. THEORY AND METHODS A. ELECTRON TRANSFER T HEORY The most basic ET reaction may be described in terms of the following two diabatic states: ð1Þ D2 e ð2Þ

De

Ae 2

Ae

where De and Ae represent a general electron donor and acceptor. The free energy surface for a single ET reaction dominated by outer-sphere (solvent) reorganization can be calculated as a function of a single collective solvent coordinate ze :40 – 45 Rate expressions have been derived for ET in both the adiabatic and nonadiabatic limits.46 The adiabatic limit of ET corresponds to strong coupling between the diabatic electronic states, and the system remains in the lowest adiabatic electronic state (i.e., the solute electrons respond instantaneously to the solvent motion). The nonadiabatic limit of ET corresponds to weak coupling between the diabatic electronic states, and the excited adiabatic electronic states are involved. The conventional unimolecular rate expression for nonadiabatic ET is41,46 – 49 ! 2p 2DG† 2 ET 21=2 k ¼ exp ð19:1Þ lV l ð4plkB TÞ " 12 kB T

Kinetic Isotope Effects for Proton-Coupled Electron Transfer Reactions

501

where V12 is the coupling between the diabatic states, l is the outer-sphere (solvent) reorganization energy, and DG† is the barrier defined as DG† ¼

ðDGo þ lÞ2 4l

ð19:2Þ

An analogous rate expression has been derived for nonadiabatic electron transfer in the presence of a quantum mechanical inner-sphere (solute) mode that is not coupled to the solvent.47,50,51 In this case, the solute mode wavefunctions are calculated for each diabatic state, and the rate expression is:

k

ET

X X ð1Þ ð2Þ 2 2DG†1m;2n 2p ¼ P1 m lkxm lxn ll exp lV12 l2 ð4plkB TÞ21=2 kB T " m n

! ð19:3Þ

where the summations are over the vibrational states for diabatic states 1 and 2, P1m is the Boltzmann ð2Þ factor for state 1m, and DG†1m,2n is the barrier corresponding to states 1m and 2n: Here kxð1Þ m lxn l is the overlap between the reactant ð1mÞ and product ð2nÞ vibrational wavefunctions representing the innersphere solute mode and is often denoted the Franck –Condon overlap factor.

B. PROTON TRANSFER T HEORY A single PT reaction may also be expressed in terms of two diabatic states: ðaÞ Dp Hþ

2

ðbÞ

HAp

Dp

Ap

where Dp and Ap represent a general proton donor and acceptor. The free energy surfaces for a single PT reaction dominated by outer-sphere reorganization can be calculated as a function of a single collective solvent coordinate zp :33,35,52 – 54 A PT reaction may be electronically adiabatic or nonadiabatic. In the limit of strong coupling between the diabatic states, the PT reaction is electronically adiabatic, and the system remains in the electronic ground state. In the limit of weak coupling, the PT reaction is electronically nonadiabatic, and excited electronic states are involved. Typically single PT reactions are electronically adiabatic due to strong coupling between the diabatic states. Since the hydrogen has a small mass, nuclear quantum effects are often important for PT reactions. These effects may be included by representing the hydrogen nucleus as a quantum mechanical wavefunction and calculating the hydrogen vibrational states.36 A PT reaction may be vibrationally adiabatic or nonadiabatic, where vibrationally adiabatic reactions occur on the vibrational ground state and vibrationally nonadiabatic reactions involve excited vibrational states. The vibrational nonadiabaticity is determined by the coupling between vibronic states. In many cases, the donor –acceptor vibrational mode RDA plays an important role in PT reactions. This donor – acceptor mode strongly modulates the PT barrier and thereby influences the rates and KIEs. The impact of this mode may be included through the calculation of free energy surfaces as functions of both the solvent coordinate zp and the donor – acceptor vibrational mode RDA : The donor – acceptor mode can be treated classically or quantum mechanically, although typically a classical treatment is adequate.55 Rate expressions have been derived in the various limits of electronic and vibrational adiabaticity and nonadiabaticity, as well as the classical and quantum treatments of the donor –acceptor mode.35,52,54,56 – 60

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Isotope Effects in Chemistry and Biology

C. PROTON- COUPLED E LECTRON TRANSFER T HEORY PCET reactions have been studied with a variety of theoretical methods.25 – 27,61 – 63 As shown in Ref. 25, the most basic PCET reaction involving the transfer of one electron and one proton may be described in terms of the following four diabatic states: ð1aÞ D2 e

Dp H

···

ð1bÞ D2 e

D2 p

···

ð2aÞ

De

Dp H

···

ð2bÞ

De

D2 p

···

þ

þ

Ap

Ae

HAp

Ae

Ap

A2 e

HAp

A2 e

where 1 and 2 denote the ET state, and a and b denote the PT state. Within this notation, 1a ! 1b represents PT, 1a ! 2a represents ET, and 1a ! 2b represents EPT (where both the proton and the electron are transferred). Ref. 25 also shows that the free-energy surfaces for PCET reactions may be calculated as functions of two collective solvent coordinates zp and ze , corresponding to PT and ET, respectively. Typically the single PT reaction is electronically adiabatic, and often the single ET reaction is electronically nonadiabatic. Even for cases in which the single ET reaction is electronically adiabatic, the overall PCET reaction may be nonadiabatic because the coupling between the reactant and product vibronic states is small due to averaging over the reactant and product proton vibrational wavefunctions (i.e., due to the small overlap factor, analogous to the Franck –Condon factor in theories for single ET). In this limit, the ET diabatic free energy surfaces corresponding to ET states 1 and 2 are calculated as mixtures of the a and b PT states. The reactants (I) are mixtures of the 1a and 1b states, and the products (II) are mixtures of the 2a and 2b states. The proton vibrational states are calculated for both the reactant (I) and product (II) ET diabatic surfaces, resulting in two sets of two-dimensional vibronic free energy surfaces that may be approximated as paraboloids. In this limit, the PCET reaction is described in terms of nonadiabatic transitions from the reactant (I) to the product (II) ET diabatic surfaces. (Here the ET diabatic states I and II, respectively, may be viewed as the reactant and product PCET states.) Figure 19.1 provides

II

Free Energy

I

λIμ,IIυ

∆G°Iμ,IIυ

− Z pI

− Z IIp

− Z eI

− Z IIe

FIGURE 19.1 ET diabatic free energy surfaces as functions of two collective solvent coordinates, zp and ze , for a PCET reaction. The lowest energy reactant (I) and product (II) free-energy surfaces are shown. The minima for the reactant and product surfaces, respectively, are ðzIp , zIe Þ and ðzIIp , zIIe Þ: The free energy difference DGoIm,IIn and outer-sphere reorganization energy lIm,IIn are indicated.

Kinetic Isotope Effects for Proton-Coupled Electron Transfer Reactions

503

Free Energy (kcal/mol)

a schematic picture of the lowest energy reactant and product two-dimensional ET diabatic free energy surfaces. Figure 19.2 depicts a slice of a two-dimensional ET diabatic surface along the line connecting the two minima, as well as the proton potential energy curves and corresponding ground-state proton vibrational wavefunctions for specified solvent coordinates.

20

I

II

10 B 0

(a)

A

C

(Z− pI ,Z− eI )

(Z−IIp, Z−IIe )

Solvent Coordinate (kcal/mol) A

20 10

Free Energy (kcal/mol)

0 B

20 10 0

C

20 10 0 −0.3

(b)

0 rp (Å)

0.3

FIGURE 19.2 (a) Slice of a two-dimensional ET diabatic free energy surface along the line connecting the two minima. The lowest energy reactant (I) and product (II) free energy surfaces are shown. Points A, B, and C, represent the equilibrium reactant configuration, the intersection point, and the equilibrium product configuration, respectively. (b) Proton potential energy curves and corresponding ground-state proton vibrational wavefunctions as functions of the proton coordinate rp for the solvent coordinates associated with points A, B, and C indicated in (a). The proton potential energy curves (solid) and vibrational wavefunctions (dashed) are shown for the reactant and product ET diabatic free energy surfaces. Reproduced with minor modifications from Ref. 97.

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Isotope Effects in Chemistry and Biology

The unimolecular rate expression derived in Ref. 26 for nonadiabatic PCET is ! X 2DG†mn 2p X 2 21=2 PCET P lV l ð4plmn kB TÞ exp k ¼ " m Im n mn kB T

ð19:4Þ

P P where m and n indicate summations over vibrational states associated with ET states 1 and 2, respectively, PIm is the Boltzmann factor for state Im, and DG†mn ¼

o ðDGmn þ lmn Þ2 4lmn

ð19:5Þ

In this expression the free energy difference is o ¼ 1IIn ðzIIp n , zIIe n Þ 2 1Im ðzIpm , zIem Þ DGmn

ð19:6Þ

where ðzIpm , zeIm Þ and ðzpIIn , zeIIn Þ are the solvent coordinates for the minima of the ET diabatic free energy surfaces 1mI ðzp , ze Þ and 1IIn ðzp , ze Þ, respectively. Moreover, the outer-sphere (solvent) reorganization energy is

l ¼ 1Im ðzIIp n , zeIIn Þ 2 1Im ðzIpm , zIem Þ ¼ 1IIn ðzIpm , zIem Þ 2 1nII ðzIIp n , zIIe n Þ

ð19:7Þ

The free energy difference and outer-sphere reorganization energy are indicated in Figure 19.1. The coupling Vmn in the PCET rate expression is defined as Vmn ¼ kfIm lVðrp ; z†p ÞlfIIn lp

ð19:8Þ

where the subscript of the angular brackets indicates integration over rp , z†p is the value of zp in the intersection region, and fIm and fIIn are the proton vibrational wavefunctions for the reactant and product ET diabatic states, respectively. A schematic illustration of these proton vibrational wavefunctions is depicted in Figure 19.2. For many systems,28,30 the coupling is approximately proportional to the overlap between the reactant and product proton vibrational wavefunctions: Vmn / kfIm lfIIn lp

ð19:9Þ

The effects of inner-sphere solute modes have also been included in this theoretical formulation for several different regimes.26 In the high-temperature approximation for uncoupled solute modes, the inner-sphere reorganization energy is added to the outer-sphere reorganization energy in Equation 19.4.28,32 The KIEs for PCET reactions may be analyzed within the context of the rate expression given in Equation 19.4. Each term in this expression represents the rate of a nonadiabatic transition from a reactant to a product state. Based on Equation 19.9, the rate for each pair of states is approximately proportional to the square of the overlap between the reactant and product vibrational wavefunctions. Thus, the KIE for each pair of states is approximately proportional to the square of the ratio of the overlap for hydrogen to the overlap for deuterium. This ratio increases as the vibrational wavefunction overlap decreases. Moreover, since the vibrational overlap is smallest for the reactive channel involving the lowest energy reactant and product vibrational states, the overall KIE decreases as the contributions from channels involving higher energy vibrational states increase. An alternative formulation is possible for PCET reactions in which the electrons and protons transfer simultaneously between the same centers. Examples of such reactions include those that are conventionally denoted hydrogen atom transfer or hydride transfer reactions. (Note that this

Kinetic Isotope Effects for Proton-Coupled Electron Transfer Reactions

505

terminology depends on only the initial and final states and is not rigorous due to the delocalized nature of electrons.) In this case, the reactions may be described in terms of only two diabatic states (i.e., the reactant and product states), and the reactions are typically electronically adiabatic. The vibrationally adiabatic and nonadiabatic approaches developed for proton transfer reactions are also applicable to these types of PCET reactions. Conventional hydrogen atom transfer reactions, however, are expected to be dominated by inner-sphere (solute) reorganization more than outersphere (solvent) reorganization because they do not involve substantial solute charge redistribution.

D. METHODOLOGICAL D EVELOPMENTS We have developed a multistate continuum theory25 – 27 for the investigation of multiple charge transfer reactions. In this theory, the solute is represented by a multistate valence bond model, the solvent is described by a dielectric continuum, and the transferring hydrogen nuclei are represented by quantum mechanical wavefunctions. The free energy surfaces are obtained as functions of a set of collective solvent coordinates corresponding to the individual charge transfer reactions. This theory enabled the derivation of the PCET rate expression in Equation 19.4, as well as the analogous expressions including the effects of inner-sphere solute modes that are not coupled to the solvent.26 Recently, the effects of the donor –acceptor vibrational mode have been incorporated into this theoretical formulation, and rate expressions in various limits have been derived.64 Within the framework of this multistate continuum theory,25 – 27 the calculation of the rates and KIEs requires the gas phase valence bond matrix elements and the reorganization energy matrix elements. The two-dimensional free energy surfaces corresponding to the solvated reactant and product vibronic states are calculated from the gas phase valence bond matrix elements and the solvent reorganization energy matrix elements. The free energy differences DGomn and the solvent reorganization energies lmn are determined from these free energy surfaces, and the couplings Vmn are determined from the associated wavefunctions and the off-diagonal gas phase valence bond matrix elements. In practice, the gas phase valence bond matrix elements are represented by molecular mechanical terms fit to electronic structure calculations or experimental data.35 The inner-sphere (solute) reorganization energy may be calculated from the equilibrium force constants and bond lengths.65 The outer-sphere (solvent) reorganization energy matrix elements are calculated with an electrostatic dielectric continuum model. Our calculations utilize the frequencyresolved cavity model (FRCM),66,67 in which two effective solute cavities are formed from spheres centered on all of the atoms. We have also developed a hybrid quantum/classical molecular dynamics approach33,34 to include the effects of explicit solvent and protein, as well as the dynamical effects of both the solute and the environment. In this approach, the overall rate is expressed as the product of an equilibrium transition state theory rate, which is directly related to the activation free energy barrier, and a transmission coefficient, which accounts for dynamical recrossings of the barrier. The electronic quantum effects are included with a two-state empirical valence bond potential,35 and the nuclear quantum effects are included by representing the transferring hydrogen nucleus as a threedimensional vibrational wavefunction.36 The free energy profile is calculated as a function of a collective reaction coordinate comprised of motions from the entire solvated enzyme. The adiabatic nuclear quantum effects of the transferring hydrogen are included in this free energy profile. The transmission coefficient is calculated with a reactive flux scheme68 – 72 in which an ensemble of realtime dynamical trajectories is initiated at the top of the barrier and propagated backward and forward in time. The vibrationally nonadiabatic effects are incorporated into the transmission coefficient by combining the reactive flux scheme with the molecular dynamics with quantum transitions (MDQT) surface-hopping method.73,74 The motion of the complete environment (i.e., solute, solvent, and enzyme) is included during the generation of the free energy profiles and the propagation of the dynamical trajectories.

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Isotope Effects in Chemistry and Biology

This hybrid quantum/classical molecular dynamics approach enables the calculation of rates and KIEs and provides detailed mechanistic information. The calculations provide insight into the fundamental nature of nuclear quantum effects such as zero point motion and hydrogen tunneling. The simulations also elucidate the impact of specific motions on the activation free energy barrier and on the degree of dynamical barrier recrossing.

III. APPLICATIONS TO CHEMICAL AND BIOLOGICAL SYSTEMS A. PCET IN S OLUTION We have applied the multistate continuum theory to a variety of PCET reactions in solution. All of these applications were motivated by intriguing experimental data concerning rates and KIEs. The investigation of iron bi-imidazoline complexes was motivated by the similarity in the experimentally measured rates for single ET and PCET.11 The studies of ruthenium polypyridyl complexes were motivated by the experimental measurement of large KIEs that are strongly influenced by subtle modification of the ligands.13,14 Our investigation of osmium – benzoquinone complexes was motivated by the experimental measurement of unusually high KIEs that vary significantly with the identity of the proton donor.15,16 Finally, the application to tyrosine oxidation was motivated by the experimentally measured pH and temperature dependence of the rates for single ET and PCET.12 The rates and KIEs for the following single ET and PCET reactions between iron biimidazoline complexes in acetonitrile have been measured experimentally:11 ½FeII ðH2 bimÞ

2þ

½FeII ðH2 bimÞ

þ ½FeIII ðH2 bimÞ

2þ

þ ½FeIII ðHbimÞ

3þ

Y ½FeIII ðH2 bimÞ

2þ

Y ½FeIII ðHbimÞ

3þ

2þ

þ FeII ðH2 bimÞ

þ FeII ðH2 bimÞ

2þ

2þ

ð19:10Þ

ð19:11Þ

where H2bim denotes bi-imidazoline and Hbim denotes deprotonated bi-imidazoline. Equation 19.10 represents a single ET reaction, and Equation 19.11 represents a PCET reaction. Based on the experimental measurements,11 the rates for these ET and PCET reactions are similar, and the KIE for PCET is 2.3. Previously this result was explained in the context of adiabatic Marcus theory, and the PCET reaction was viewed as a hydrogen atom transfer involving negligible solute charge rearrangement, leading to zero solvent reorganization energy.11 The similarity of the ET and PCET rates was thought to be due to the compensation of the larger solvent reorganization energy for ET by a larger solute reorganization energy for PCET. Our calculations, which were based on the nonadiabatic rate expressions for ET and PCET given in Equation 19.1 and Equation 19.4, provide an alternative explanation for the experimental results.28 In the framework of the four-state valence bond model for PCET, the electron is transferred predominantly between the two iron centers, while the proton is transferred between the two nitrogen atoms on the ligands. The innersphere reorganization involving the Fe –N bonds is assumed to be the same for both ET and PCET. Our calculations illustrated that the solvent reorganization energies lmn for the dominant contributions to the PCET reaction were substantial and were < 1 –3 kcal/mol lower than the solvent reorganization energy for single ET. We found that the overall coupling for PCET is smaller than the coupling for ET due to averaging over the reactant and product hydrogen vibrational wavefunctions (i.e., multiplying by the vibrational overlap factor in Equation 19.9). The calculations indicated that the similarity of the rates for ET and PCET is due mainly to the compensation of the larger solvent reorganization energy for ET by the smaller coupling for PCET. The moderate KIE for PCET was determined to arise from the relatively large overlap factor and the significant contributions from excited vibronic states.

Kinetic Isotope Effects for Proton-Coupled Electron Transfer Reactions

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The rates and KIEs of the following two PCET reactions in oxoruthenium polypyridyl complexes in water have been measured experimentally:13,14 ½ðbpyÞ2 ðpyÞRuII OH2

2þ

Y ½ðbpyÞ2 ðpyÞRuIII OH ½ðtpyÞðpyÞRuII OH2

2þ

Y ½ðtpyÞðpyÞRuIII OH

þ ½ðbpyÞ2 ðpyÞRuIV O 2þ

þ ½ðbpyÞ2 ðpyÞRuIII OH

þ ½ðtpyÞðpyÞRuIV O 2þ

2þ 2þ

ð19:12Þ

2þ

þ ½ðtpyÞðpyÞRuIII OH

2þ

ð19:13Þ

where tpy ¼ 2,20 :60 200 -terpyridine, bpy ¼ 2,20 -bipyridine, and py ¼ pyridine. We refer to Equation 19.12 as CompA and Equation 19.13 as CompB. Note that these two reactions differ only by the replacement of two bipyridine ligands by a single terpyridine ligand. The experimental studies13,14 revealed that the CompB rate is nearly one order of magnitude larger than the CompA rate, and the CompA KIE of 16.1 is larger than the CompB KIE of 11.4. We performed density functional theory calculations on the ruthenium complexes and showed that the steric crowding near the oxygen proton acceptor is significantly greater for CompA than for CompB.30 Consistent with this observation, our multistate continuum theory calculations30 implied that the proton donor –acceptor distance is larger for CompA than for CompB, leading to a larger overlap between the reactant and product hydrogen vibrational wavefunctions for CompB than for CompA. The rate for CompB is larger than the rate for CompA because the rate increases as this overlap factor increases. The KIE for CompB is smaller than the KIE for CompA because the KIE decreases as this overlap factor increases. Both of these KIEs are larger than the KIE for the iron bi-imidazoline complexes described above because the vibrational overlap factor is smaller for the ruthenium systems. Experimental studies of PCET from a series of osmium complexes to benzoquinone identified unusually high KIE’s of up to , 400.15,16 Our theoretical calculations31 illustrated that these colossal KIEs arise from the relatively small overlap between the reactant and product hydrogen vibrational wavefunctions. The KIE increases as the vibrational overlap decreases and as the contribution of transitions between the lowest energy reactant and product vibronic states increases. The trends in the KIEs for a series of complexes were found to be determined by a balance among several factors, including the X –H frequencies and proton transfer distances for the different proton donors (X ¼ N, P, S), as well as the solvent reorganization energies and reaction free energies for the different complexes. These characteristics of the osmium systems influence the overlaps between the reactant and product hydrogen vibrational wavefunctions and the relative contributions of the excited vibronic states, which in turn impact the KIE. The compound depicted in Figure 19.3 was designed to model tyrosine oxidation in Photosystem II.3 – 7 In this model system,12 an electron is transferred to the ruthenium from the tyrosine, which is concurrently deprotonated (i.e., a proton is transferred from the tyrosine to a solvent water molecule). In contrast to the PCET reactions described above, the electron and proton transfer in different directions for this system. Experimental measurements12 indicated that the mechanism is PCET at pH , 10 when the tyrosine is initially protonated and single ET for pH . 10 when the tyrosine is initially deprotonated. The PCET rate was found to increase monotonically with pH, whereas the single ET rate was found to be independent of pH and two orders of magnitude faster than the PCET rate. For pH , 10, the KIE was measured to be 2.0– 2.5. As shown in Figure 19.4, our theoretical calculations reproduced the experimentally observed pH dependence of the rates.29 The calculations indicated that the larger rate for single ET arises from a combination of factors, including the greater exoergicity for ET, the smaller solvent reorganization energy for ET, and the averaging of the coupling for PCET over the reactant and product hydrogen vibrational wavefunctions (i.e., the vibrational overlap factor). The calculated temperature dependence of the rates and the deuterium KIEs were also consistent with the experimental

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Isotope Effects in Chemistry and Biology

PT

H O

3+

COOEt HN

HO

H ET

O

e− N

N

RuIII

N N

N N

FIGURE 19.3 PCET reaction in a model for tyrosine oxidation in photosystem II. In the first step of the experiment, the ruthenium-tris-bipyridine portion absorbs light, and the excited electron is transferred to an external methyl violegen acceptor. In the second step, which is shown here, the tyrosine portion transfers an electron to the ruthenium and is deprotonated. Reproduced from Ref. 29. 1.0E+08

Rate [1/s]

1.0E+07 1.0E+06 1.0E+05 1.0E+04 1.0E+03

5

6

7

8

9 pH

10

11

12

13

FIGURE 19.4 Experimental and theoretical data for the pH dependence of the rates for single ET and PCET in the tyrosine oxidation model shown in Figure 19.3. The experimental values are denoted with open circles. The theoretical PCET rates are denoted with filled circles, and the theoretical ET rate is represented by a solid line because it is independent of pH. Reproduced from Ref. 29.

results.29 An analysis of the hydrogen tunneling mechanism for this system is provided in Figure 19.5.

B. ENZYME R EACTIONS We have studied a number of enzyme-catalyzed PCET reactions. Recently we applied the multistate continuum theory to the PCET reaction catalyzed by SLO.32 This investigation was motivated by experimental measurements of unusually large KIEs at room temperature in conjunction with a weak temperature dependence of the rates.20 – 24 We calculated the temperature dependence of the rates and KIEs and investigated the role of the donor – acceptor vibrational mode in lipoxygenase. In addition to these multistate continuum calculations, we applied the hybrid quantum/classical molecular dynamics approach to hydride transfer in the enzymes LADH33,34 and DHFR.37 – 39 These applications were motivated by experimental measurements of rates and KIEs for wild-type and mutant enzymes.17 – 19,75,76 In our simulations, these hydride transfer reactions are described in terms of only two valence bond states, the reactions are assumed to be electronically

Kinetic Isotope Effects for Proton-Coupled Electron Transfer Reactions

509

Free Energy (kcal/mol)

40 30 20 10 0

−10

−0.3

0 rp [Å]

0.3

(Z− pI ,Z− eI )

(Z−IIp, Z−IIe )

Solvent Coordinate (kcal/mol)

−0.3

0

0.3

rp [Å]

FIGURE 19.5 Analysis of the free energy surfaces for the PCET reaction in the tyrosine oxidation model depicted in Figure 19.3. In the center frame are slices of the two-dimensional ET diabatic free energy surfaces as functions of the solvent coordinates. The slices were obtained along the line connecting the minima of the lowest energy reactant (I) and product (II) two-dimensional free energy surfaces. In the left frame is the reactant (I) proton potential energy curve and the corresponding proton vibrational wavefunctions as functions of the proton coordinate rp evaluated at the minimum of the ground-state reactant free energy surface. In the right frame is the product (II) proton potential energy curve and the corresponding proton vibrational wavefunctions as functions of the proton coordinate rp evaluated at the minimum of the ground-state product free energy surface. Reproduced from Ref. 29.

adiabatic, and vibrationally nonadiabatic effects are included in the transmission coefficient. The rates and KIEs were calculated, and the impact of enzyme structure and motion on activity was investigated. Lipoxygenases catalyze the oxidation of unsaturated fatty acids and have a wide range of biomedical applications.77,78 In mammals, lipoxygenases aid in the production of leukotrienes and lipoxins, which regulate responses in inflammation and immunity.77 Kinetic studies have shown that the hydrogen transfer from the carbon atom C11 of the linoleic acid substrate to the Fe(III)– OH cofactor is rate limiting above 32 8C for SLO.79 The deuterium KIE on the catalytic rate has been measured experimentally to be as high as 81 at room temperature.20 – 24,80 The rates for hydrogen and deuterium transfer were found to depend only weakly on temperature.24 Various tunneling models have been invoked to analyze the temperature dependence of the KIEs.20,21,24 Our multistate continuum theory calculations treat the net hydrogen transfer reaction catalyzed by lipoxygenase as a PCET mechanism, as illustrated in Figure 19.6.32 Quantum mechanical calculations indicate that the electron transfers from the p system of the linoleic acid to an orbital localized on the Fe(III) center, and the proton transfers from the donor carbon to the oxygen acceptor.81 Thus, the electron and proton are transferred between distinct donors and acceptors. Moreover, analysis of the thermodynamic properties of the single PT and ET reactions, as well as the concerted PCET mechanism, indicates that the single PT and ET reactions are significantly endothermic, whereas the PCET reaction is exothermic.24,32,81 These analyses imply that the mechanism is PCET, where the electron and proton transfer simultaneously between different donors and acceptors. As shown in Figure 19.7, the temperature dependence of the rates and KIEs predicted by the multistate continuum theory is in remarkable agreement with the experimental data.32 The calculations indicate that the weak temperature dependence of the rates is due to the relatively small free energy barrier arising from a balance between the reorganization energy and the reaction free energy. The unusually high KIE of 81 arises from the small overlap of the reactant and product proton vibrational wavefunctions and the dominance of the lowest energy reactant and product

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Isotope Effects in Chemistry and Biology

FIGURE 19.6 The PCET reaction catalyzed by SLO. This reaction entails a net hydrogen atom transfer from the linoleic acid substrate to the Fe(III)– OH cofactor. This conformation was obtained from docking calculations that included the entire SLO protein.32

vibronic states in the tunneling process. The hydrogen and deuterium vibrational wavefunctions are depicted in Figure 19.8. The proton donor – acceptor vibrational motion was included classically in these calculations. The dominant contribution to the overall rate was found to correspond to a proton donor –acceptor distance that is significantly smaller than the equilibrium donor –acceptor

Rate (s−1)

1000

Hydrogen

100 10

Deuterium

KIE

120 100 80 60 3.1

3.3 3.5 1000/T (K−1)

3.7

FIGURE 19.7 Temperature dependence of the rates and KIEs for the PCET reaction catalyzed by SLO. The theoretical results are denoted with open triangles, and the experimental data are denoted with closed circles. The theoretical results were generated with the multistate continuum theory including the proton donor – acceptor vibrational motion. Reproduced from Ref. 32.

Kinetic Isotope Effects for Proton-Coupled Electron Transfer Reactions

511

Energy (kcal/mol)

20

10

0 −0.5

0 rp (Å)

FIGURE 19.8 Reactant (left) and product (right) proton potential energy curves and the associated hydrogen (solid) and deuterium (dashed) vibrational wavefunctions for the lowest energy reactant and product states for the PCET reaction catalyzed by SLO. The wavefunctions were obtained at the lowest energy intersection point ˚ .32 of the two-dimensional free energy surfaces for R ¼ 2.67 A

distance. This dominant distance is determined by a balance between the larger coupling and the smaller Boltzmann factor as the distance decreases. Thus, the proton donor – acceptor vibrational motion plays an important role in decreasing the dominant donor – acceptor distance relative to its equilibrium value to facilitate the PCET reaction. The quantum mechanical and nonequilibrium dynamical aspects of the proton donor –acceptor vibrational motion are not essential for the description of the experimentally observed temperature dependence of the rates and KIES within the framework of this model. The enzyme LADH catalyzes the reversible oxidation of alcohols to the corresponding aldehydes and ketones. This enzyme is critical to many steps in metabolism and is relevant to the medical complications associated with alcoholism. The mechanism is thought to involve a hydride transfer from the benzyl alkoxide substrate to the cofactor NADþ, leading to the products benzaldehyde and NADH. The KIEs for this enzymatic reaction have been measured by Klinman and coworkers, who interpret the results as an indication of hydrogen tunneling.17,18 Klinman and coworkers also observed that the mutation of Val-203 to the smaller residue alanine decreases the rate.82 Numerous theoretical studies of this reaction have been performed on this system.83 – 88 We studied the hydride transfer reaction catalyzed by LADH with the hybrid quantum/classical molecular dynamics approach.33,34 The EVB potential was parameterized to reproduce the experimentally measured free energies of activation and reaction. Our simulations indicate that the nuclear quantum effects decrease the free energy barrier by , 2 kcal/mol. Analysis of the nuclear wavefunctions for configurations at the top of the free energy barrier suggests that hydrogen tunneling along the donor –acceptor axis is prevalent. The calculated deuterium and tritium KIEs obtained from the free energy profiles shown in Figure 19.9 are in agreement with the experimental values17 of 3.78 ^ 0.07 and 1.89 ^ 0.01. The transmission coefficient is calculated to be nearly unity, suggesting that dynamical barrier recrossings are not important for this reaction. Similar values for the KIEs and transmission coefficient were also obtained with a different computational approach.84 In addition, our simulations elucidate the role of enzyme motion in the LADH reaction and imply that the impact of the Val-203 mutation on the enzyme activity is due to the alteration of the equilibrium free energy barrier rather than the dynamical barrier recrossings. The enzyme DHFR catalyzes the conversion of dihydrofolate to tetrahydrofolate.89 This enzyme is essential for the maintenance of tetrahydrofolate levels needed to support the biosynthesis

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Isotope Effects in Chemistry and Biology

16.9

Free energy (kcal/mol)

15

16.4 15.4

T H

D

10

5

0 −200 −100 0 100 200 Collective reaction coordinate (kcal/mol)

FIGURE 19.9 Adiabatic quantum free energy profiles (including zero point motion of the hydride) as functions of the collective reaction coordinate for the hydride transfer reaction catalyzed by LADH. The curves are given for hydrogen (H), deuterium (D), and tritium (T) transfer.34

of purines, pyrimidines, and amino acids. The mechanism is thought to involve a hydride transfer from the NADPH cofactor to the protonated DHF substrate to form the products NADPþ and THF. This reaction has been studied extensively with experimental methods19,89 – 92 and more recently with theoretical methods.37 – 39,93 – 95 Crystallographic data imply substantial conformational changes along the reaction pathway,90 and NMR experiments have identified dynamic regions.91,92 Moreover, mutations in regions far from the active site have been observed experimentally to significantly impact the rate of hydride transfer.38,75,76 In particular, the mutation of Gly-121 to valine decreases the rate of hydride transfer by a factor of 163.38,75,76 Residue 121 is on the exterior of the ˚ from the transferring hydride. enzyme and is more than 12 A We investigated the hydride transfer reaction catalyzed by DHFR with the hybrid quantum/classical molecular dynamics method.37 – 39 As in the case of LADH, the EVB potential was parameterized to reproduce the experimentally measured free energies of activation and reaction. Representative three-dimensional vibrational wavefunctions obtained from these simulations are depicted in Figure 19.10, and a comparison between hydrogen and deuterium vibrational wavefunctions is provided in Figure 19.11. The experimentally measured19 deuterium KIE of 3.0 was reproduced by our calculations. As for LADH, the nuclear quantum effects were found to significantly decrease the free energy barrier, and hydrogen tunneling was observed. The transmission coefficient for DHFR was calculated to be 0.88, which is lower than that for LADH but still close to unity. Recently similar values for the KIE and transmission coefficient were obtained with a different computational approach.95 We also performed a detailed analysis of the impact of enzyme motion on the hydride transfer reaction catalyzed by DHFR. Furthermore, our simulations indicate that the decrease in the hydride transfer rate for the G121V mutant is due to an increase in the free energy barrier.39

C. ROLE OF M OTION IN E NZYME R EACTIONS Enzyme motion influences the activation free energy barrier and the dynamical barrier recrossings in enzymatic reactions.96 Motions influencing the activation free energy barrier are thermally averaged, equilibrium properties of the system, whereas motions influencing the barrier recrossings are dynamical properties of the system. Some motions may influence both the activation free energy

Kinetic Isotope Effects for Proton-Coupled Electron Transfer Reactions Ground state

513

Excited state

Reactant

TS

Product

FIGURE 19.10 Three-dimensional vibrational wavefunctions representing the transferring hydride for reactant, transition state, and product configurations obtained from simulations of the hydride transfer reaction catalyzed by DHFR. On the donor side, the donor carbon atom and its first neighbors are shown, whereas on the acceptor side the acceptor carbon atom and its first neighbors are shown.37

barrier and the barrier recrossings. Motions influencing the activation free energy barrier are expected to play a greater catalytic role than those influencing the dynamical barrier recrossings because the free energy barrier is in the exponential, whereas the transmission coefficient accounting for barrier recrossings is a prefactor in general rate expressions. For reactions involving quantum mechanical tunneling, enzyme motion can alter the probability of tunneling at configurations with degenerate quantum states. This type of motion directly modulates the tunneling barrier, typically increasing the rate by decreasing the width and height of the barrier along the tunneling coordinate. Several terms have been used to describe this phenomenon, including ‘vibrationally enhanced tunneling’,97 ‘rate-promoting vibrations’,98 and ‘gating’.24 These types of motions are subpicosecond vibrations and hence are much faster than the chemical turnover of the enzyme reaction. Although such subpicosecond vibrations are important for the modulation of the tunneling barrier, the equilibrium conformational changes Hydrogen

Deuterium

FIGURE 19.11 Comparison of the hydrogen and deuterium vibrational wavefunctions for a transition state configuration obtained from simulations of the hydride transfer reaction catalyzed by DHFR.37

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Isotope Effects in Chemistry and Biology

along the collective reaction coordinate leading to configurations with degenerate quantum states are thought to be rate limiting. Moreover, since the underlying principles of the quantum mechanical tunneling processes are the same in solution and enzymes, the quantification of the catalytic role of these subpicosecond vibrations is difficult. For hydrogen transfer reactions, the hydrogen donor – acceptor motion plays a particularly important role. The donor –acceptor motion influences both the activation free energy barrier and the dynamical barrier recrossings. The equilibrium, thermally averaged value of the donor – acceptor distance has been shown to change substantially along the collective reaction coordinate for hydrogen transfer reactions.34,37 This change in the average donor – acceptor distance impacts the free energy barrier and thereby the overall rate. For reasons discussed above, the thermally averaged motions influencing the activation free energy barrier are expected to have a greater impact on enzyme activity than do the dynamical motions influencing the barrier recrossings. In addition, the subpicosecond donor – acceptor vibrations significantly modulate the tunneling barrier, but the equilibrium conformational changes that are averaged over these fast vibrations are rate limiting in enzyme reactions. Recent studies indicate that thermally averaged, equilibrium motions representing conformational changes along the collective reaction coordinate play an important role in enzymatic reactions.34,37,38,96 These motions are averaged over the fast vibrations of the enzyme and occur on the time scale of the catalyzed chemical reaction. They reflect the conformational changes that generate configurations conducive to the chemical reaction and thereby influence the activation free energy barrier. For example, a network of coupled motions extending throughout the protein and ligands was identified in DHFR.37,38 These coupled motions represent equilibrium, thermally averaged conformational changes along the collective reaction coordinate leading to configurations conducive to the reaction. The equilibrium molecular motions in this network are not dynamically coupled to the chemical reaction, but rather give rise to conformations in which the hydride transfer reaction is facilitated because of short transfer distances, suitable orientation of substrate and cofactor, energetic matching of charge-transfer states, and a favorable electrostatic environment for charge transfer.99 This type of network has important implications for protein engineering and drug design.

IV. SUMMARY AND CONCLUSIONS This chapter describes theoretical approaches that have been developed to calculate KIEs for PCET reactions and summarizes the results of applications to a wide range of different types of PCET reactions in solution and enzymes. The experimentally measured deuterium KIEs for these PCET systems range from 2 to 400. The applications described in this chapter have reproduced the experimentally measured KIEs and have provided insight into the underlying physical principles leading to these KIEs. The unusually high KIEs are found to arise from nonadiabatic PCET reactions with a relatively small overlap between the reactant and product hydrogen vibrational wavefunctions and relatively low contributions from excited reactant and product vibronic states. The predominantly adiabatic hydride transfer reactions catalyzed by the enzymes DHFR and LADH exhibit moderate KIEs of 3– 4, whereas the nonadiabatic net hydrogen atom transfer reaction catalyzed by the enzyme SLO exhibits a higher KIE of 81. The KIEs for PCET reactions in condensed phase systems are determined by a complex balance among a variety of factors, including proton donor – acceptor distances, energetics, electronic couplings, and polarity of the environment. As a result, the prediction of KIEs for PCET reactions in solution and proteins is quite challenging. The theoretical framework described in this chapter enables the prediction of the KIE given sufficient information about the chemical and physical properties of the system. The ultimate objective is the ab initio prediction of KIEs for general PCET reactions in condensed phases.

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ACKNOWLEDGMENTS The work described in this chapter was supported by NSF grant CHE-0096357 and NIH grant GM56207.

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45 Calef, D. F. and Wolynes, P. G., Classical solvent dynamics and electron transfer: 1. Continuum theory, J. Phys. Chem., 87, 3387– 3400, 1983. 46 Newton, M. D. and Sutin, N., Electron transfer reactions in condensed phases, Annu. Rev. Phys. Chem., 35, 437– 480, 1984. 47 Bixon, M. and Jortner, J., Electron transfer from isolated molecules to biomolecules, Adv. Chem. Phys., 106, 35 – 202, 1999. 48 Barbara, P. F., Meyer, T. J., and Ratner, M. A., Contemporary issues in electron transfer research, J. Phys. Chem., 100, 13148– 13168, 1996. 49 Levich, V. G., Classical solvent dynamics and electron transfer: 1. Continuum theory, J. Phys. Chem., 87, 3387– 3400, 1983. 50 Kestner, N. R., Logan, J., and Jortner, J., J. Phys. Chem., 78, 2148, 1974. 51 Ulstrup, J. and Jortner, J., J. Chem. Phys., 63, 4358, 1975. 52 Borgis, D. and Hynes, J. T., Dynamical theory of proton tunneling transfer rates in solution: general formulation, Chem. Phys., 170, 315– 346, 1993. 53 Basilevsky, M. V. and Vener, M. V., Collective medium coordinates and their application in the theory of chemical reactions, J. Mol. Struct.: THEOCHEM, 398, 81, 1997. 54 Hammes-Schiffer, S., Proton-coupled electron transfer, In Electron Transfer in Chemistry, Principles, Theories, Methods, and Techniques, Vol. I, Balzani, V., Ed., Wiley-VCH, Weinheim, pp. 189– 214, 2001. 55 Kim, S.-Y. and Hammes-Schiffer, S., Molecular dynamics with quantum transitions for proton transfer: quantum treatment of hydrogen and donor – acceptor motions, J. Chem. Phys., 119, 4389 –4398, 2003. 56 Borgis, D. and Hynes, J. T., Curve crossing formulation for proton transfer reactions in solution, J. Phys. Chem., 100, 1118– 1128, 1996. 57 Kuznetsov, A. M. and Ulstrup, J., Can. J. Chem., 77, 1085 –1096, 1999. 58 Kiefer, P. M. and Hynes, J. T., J. Phys. Chem. A, 106, 1834, 2002. 59 Kiefer, P. M. and Hynes, J. T., J. Phys. Chem. A, 106, 1850, 2002. 60 Staib, A., Borgis, D., and Hynes, J. T., Proton transfer in hydrogen bonded acid – base complexes in polar solvents, J. Chem. Phys., 102, 2487– 2505, 1995. 61 Cukier, R. I., Proton-coupled electron transfer reactions: evaluation of rate constants, J. Phys. Chem., 100, 15428– 15443, 1996. 62 Cukier, R. I. and Nocera, D. G., Proton-coupled electron transfer, Annu. Rev. Phys. Chem., 49, 337– 369, 1998. 63 Mayer, J. M., Hrovat, D. A., Thomas, J. L., and Borden, W. T., Proton-coupled electron transfer versus hydrogen atom transfer in benzyl/toluene, methoxyl/methanol, and phenoxyl/phenol self-exchange reactions, J. Am. Chem. Soc., 124, 11142– 11147, 2002. 64 Soudackov, A.V., Hatcher, E., and Hammes-Schiffer, S., J. Chem. Phys., 122, 014505, 2005. 65 Zhou, Z. and Khan, S. U. M., Inner-sphere reorganization energy of ions in solution: a molecular orbital calculation, J. Phys. Chem., 93, 5292– 5295, 1989. 66 Basilevsky, M. V., Rostov, I. V., and Newton, M. D., A frequency-resolved cavity model (FRCM) for treating equilibrium and non-equilibrium solvation energies, Chem. Phys., 232, 189– 199, 1998. 67 Newton, M. D., Basilevsky, M. V., and Rostov, I. V., A frequency-resolved cavity model (FRCM) for treating equilibrium and non-equilibrium solvation energies 2: evaluation of solvent reorganization energies, Chem. Phys., 232, 201– 210, 1998. 68 Neria, E. and Karplus, M., A position dependent friction model for solution reactions in the high friction regime: proton transfer in triosephosphate isomerase (TIM), J. Chem. Phys., 105, 10812– 10818, 1996. 69 Bennett, C. H., Algorithms for Chemical Computation, Christofferson, R. E., Ed., American Chemical Society, Washington, DC, 1997. 70 Keck, J. C., Variational theory of chemical reaction rates applied to three-body recombinations, J. Chem. Phys., 32, 1035– 1050, 1960. 71 Anderson, J. B., Statistical theories of chemical reactions. Distributions in the transition region, J. Chem. Phys., 58, 4684– 4692, 1973. 72 Chandler, D., A story of rare events: from barriers, to electrons, to unknown pathways, In Classical and Quantum Dynamics in Condensed Phase Simulations, Berne, B. J., Ciccotti, G., and Coker, D. F., Eds., World Scientific, Singapore, pp. 3 – 23, 1998.

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73 Tully, J. C., Molecular dynamics with electronic transitions, J. Chem. Phys., 93, 1061– 1071, 1990. 74 Hammes-Schiffer, S., Mixed quantum/classical dynamics of hydrogen transfer reactions, J. Phys. Chem. A, 102, 10443– 10454, 1998. 75 Rajagopalan, P. T. R., Lutz, S., and Benkovic, S. J., Coupling interactions of distal residues enhance dihydrofolate reductase catalysis: mutational effects on hydride transfer rates, Biochemistry, 41, 12618– 12628, 2002. 76 Cameron, C. E. and Benkovic, S. J., Evidence for a functional role of the dynamics of glycine-121 of Escherichia coli dihydrofolate reductase obtained from kinetic analysis of a site-directed mutant, Biochemistry, 36, 15792– 15800, 1997. 77 Samuelsson, B., Dahlen, S.-E., Lindgren, J., Rouzer, C. A., and Serhan, C. N., Leukotrienes and lipoxins: structures, biosynthesis, and biological effects, Science, 237, 1171– 1176, 1987. 78 Holman, T. R., Zhou, J., and Solomon, E. L., Spectroscopic and functional characterization of a ligand coordination mutant of soybean lipoxygenase-1: first coordination sphere analogue of human 15lipoxygenase, J. Am. Chem. Soc., 120, 12564– 12572, 1998. 79 Glickman, M. H. and Klinman, J. P., Nature of rate-limiting steps in the soybean lipoxygenase-1 reaction, Biochemistry, 34, 14077– 14092, 1995. 80 Lewis, E. R., Johansen, E., and Holman, T. R., Large competitive kinetic isotope effects in human 5lipoxygenase catalysis measured by a novel HPLC method, J. Am. Chem. Soc., 121, 1395– 1396, 1999. 81 Lehnert, N. and Solomon, E. L., Density-functional investigation on the mechanism of H-atom abstraction by lipoxygenase, J. Biol. Inorg. Chem., 8, 294– 305, 2003. 82 Bahnson, B. J., Colby, T. D., Chin, J. K., Goldstein, B. M., and Klinman, J. P., A link between protein structure and enzyme catalyzed hydrogen tunneling, Proc. Natl Acad. Sci. USA, 94, 12797– 12802, 1997. 83 Rucker, J. and Klinman, J. P., Computational study of tunneling and coupled motion in alcohol dehydrogenase-catalyzed reactions: implication for measured hydrogen and carbon isotope effects, J. Am. Chem. Soc., 121, 1997– 2006, 1999. 84 Alhambra, C., Corchado, J. C., Sanchez, M. L., Gao, J. L., and Truhlar, D. G., Quantum dynamics of hydride transfer in enzyme catalysis, J. Am. Chem. Soc., 122, 8197– 8203, 2000. 85 Webb, S. P., Agarwal, P. K., and Hammes-Schiffer, S., Combining electronic structure methods with the calculation of hydrogen vibrational wavefunctions: application to hydride transfer in liver alcohol dehydrogenase, J. Phys. Chem. B, 104, 8884– 8894, 2000. 86 Luo, J. and Bruice, T. C., Dynamic structures of horse liver alcohol dehydrogenase (HLADH): results of molecular dynamics simulations of HLADH-NAD(þ)-PhCH2OH, HLADH-NAD(þ)-PHCH2O-, and HLADH-NADH-PhCHO, J. Am. Chem. Soc., 123, 11952– 11959, 2001. 87 Villa, J. and Warshel, A., Energetics and dynamics of enzymatic reactions, J. Phys. Chem. B, 105, 7887– 7907, 2001. 88 Caratzoulas, S., Mincer, J. S., and Schwartz, S. D., Identification of a protein-promoting vibration in the reaction catalyzed by horse liver alcohol dehydrogenase, J. Am. Chem. Soc., 124, 3270– 3276, 2002. 89 Miller, G. P. and Benkovic, S. J., Stretching exercises — flexibility in dihydrofolate reductase catalysis, Chem. Biol., 5, R105 – R113, 1998. 90 Sawaya, M. R. and Kraut, J., Loop and subdomain movements in the mechanism of Escherichia coli dihydrofolate reductase: crystallographic evidence, Biochemistry, 36, 586– 603, 1997. 91 Falzone, C. J., Wright, P. E., and Benkovic, S. J., Dynamics of a flexible loop in dihydrofolate reductase from Escherichia coli and its implication for catalysis, Biochemistry, 33, 439– 442, 1994. 92 Osborne, M. J., Schnell, J., Benkovic, S. J., Dyson, H. J., and Wright, P. E., Backbone dynamics in dihydrofolate reductase complexes: role of loop flexibility in the catalytic mechanism, Biochemistry, 40, 9846– 9859, 2001. 93 Radkiewicz, J. L. and Brooks, C. L., Protein dynamics in enzymatic catalysis: exploration of dihydrofolate reductase, J. Am. Chem. Soc., 122, 225– 231, 2000. 94 Rod, T. H., Radkiewicz, J. L., and Brooks, C. L. III, Correlated motion and the effect of distal mutations in dihydrofolate reductase, Proc. Natl Acad. USA, 100, 6980– 6985, 2003. 95 Garcia-Viloca, M., Truhlar, D. G., and Gao, J., Reaction-path energetics and kinetics of the hydride transfer reaction catalyzed by dihydrofolate reductase, Biochemistry, 42, 13558– 13575, 2003. 96 Hammes-Schiffer, S., Impact of enzyme motion on activity, Biochemistry, 41, 13335– 13343, 2002.

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97 Bruno, W. J. and Bialek, W., Vibrationally enhanced tunneling as a mechanism for enzymatic hydrogen transfer, Biophys. J., 63, 689– 699, 1992. 98 Antoniou, D., Caratzoulas, S., Kalyanaraman, C., Mincer, J. S., and Schwartz, S. D., Barrier passage and protein dynamics in enzymatically catalyzed reactions, Eur. J. Biochem., 269, 3103– 3112, 2002. 99 Benkovic, S. J. and Hammes-Schiffer, S., A perspective on enzyme catalysis, Science, 301, 1196, 2003.

20

Kinetic Isotope Effects in Multiple Proton Transfer Zorka Smedarchina, Willem Siebrand, and Antonio Ferna´ndez-Ramos

CONTENTS I. II.

Introduction....................................................................................................................... 521 Theoretical Methods ......................................................................................................... 523 A. Transition State Theory ........................................................................................... 523 B. Tunneling Preliminaries........................................................................................... 524 C. Approximate Instanton Method ............................................................................... 526 D. Isotope Effects.......................................................................................................... 528 E. Comparison of AIM with Other Methods ............................................................... 528 III. Stepwise Transfer ............................................................................................................. 529 A. Example: Porphine ................................................................................................... 529 B. Isotope Effects.......................................................................................................... 530 C. Temperature Effects ................................................................................................. 532 D. Applications ............................................................................................................. 533 IV. Concerted Transfer ........................................................................................................... 535 A. Example: Acetic Acid – Methanol Complex............................................................ 535 B. Hydrogen Bonded Dimers and Complexes ............................................................. 537 C. Water Wires ............................................................................................................. 537 D. The Proton Inventory Problem ................................................................................ 542 V. Conclusions....................................................................................................................... 543 Acknowledgments ........................................................................................................................ 544 References..................................................................................................................................... 544

I. INTRODUCTION Protons, like electrons, are light enough to tunnel through potential-energy barriers, an ability that ˚ or provides them with a high mobility. However, while electrons can jump over distances of 10 A ˚ more, proton jumps are typically limited to 1 to 1.5 A. Nevertheless, protons manage to travel over much larger distances, e.g., across cell membranes. For a long-time the mechanism of this longrange proton transport was controversial, but in recent years the idea that water molecules embedded in proteins can serve as proton conduits (water “wires”) has found wide acceptance. Such transport amounts to a relay process: While the first proton jumps from atom (or group) one to atom two, the second proton jumps from atom two to atom three, etc. This may be a cyclic process where the final proton upon reaching atom N jumps back to atom one, or a linear process in which atom one donates and atom N accepts a proton. Such processes can follow either a classical mechanism in which energy is conserved throughout or a tunneling path in which energy is conserved only at the start and finish of the jump. 521

522

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In this contribution, we analyze the dynamics of a number of reactions in which two or more protons transfer to a new site. These include systems of biological interest in which water molecules bridging the old and new site catalyze the transfer, a prominent example being the carbonic anhydrase II enzyme. We also examine double proton transfer in molecules such as porphine and complexes such as the formic acid dimer where proton exchange occurs between equivalent sites. Starting from the observed dynamic properties such as rate constants, tunneling splittings, and isotope effects, together with their dependence on external conditions such as temperature and environment, we will aim for a qualitative description of the mechanism of the transfer, augmented, where appropriate, by quantitative calculations. A new feature specific for multiple compared to single proton transfer is that the protons can jump one at a time (“stepwise”) or collectively (“concertedly”). Stepwise transfer implies the existence of a stable intermediate whose zero-point energy level lies below the tops of the barriers separating it from the initial and final equilibrium configurations. Concerted transfer implies the absence of such an intermediate, but it does not necessarily mean that the protons move with perfect synchronicity unless their paths are identical by symmetry. Thus, to the question whether the transfer occurs classically or by tunneling, we must add the question whether it occurs concertedly or stepwise. Where available, quantum-chemical calculations open an informative window on these processes. The form of the transfer potential indicates the prevailing mechanism. If the potential has a single maximum, i.e., if the transfer is characterized by a single transition state, a concerted mechanism is predicted. If that potential is symmetric, such that all the mobile protons are halfway through their transfer path in the transition state, the transfer will be synchronous. Two or more maxima predict a stepwise process with as many steps as there are maxima. If the barrier is high, tunneling will prevail and if it is low, classical transfer will be dominant; here, high and low are measured relative to the thermal energy kB T: As in all proton-transfer processes, an effective way to probe the transport mechanism is by replacing mobile protons by deuterons (or tritons) and measuring the kinetic isotope effect (KIE). Isotope effects have long been recognized as a major diagnostic indicator for hydrogen- and protontransfer mechanisms. In particular, KIEs have been used to find out whether the transfer occurs classically or by quantum-mechanical tunneling. Simple rules have been derived, either empirically or from more or less primitive models that proved useful as a framework for presenting experimental data. Typically, the rate constants measured in a limited temperature interval are expressed in the form of an Arrhenius equation: kðTÞ ¼ Ae2Ea =kB T

ð20:1Þ

where both the frequency factors A and the apparent activation energy Ea are temperature and isotope dependent. For tunneling reactions, neither A nor Ea has an obvious connection to the quantities calculated by quantum chemists. Rules derived from such parameters are necessarily empirical. To generalize our present understanding of single-proton transfer to multiple-proton reactions, we need a more solid basis. Therefore, we shall use theoretical methods based on parameters that represent physical quantities and can be calculated quantum chemically. Among the methods in common use are the Golden Rule approach,1 in which the transfer is treated as a nonadiabatic transition, transition-state theory (TST)2 to deal with classical transfer; and two semiclassical methods developed to deal with tunneling: TST with semiclassical tunneling corrections (TST/ST),3 and the Approximate Instanton Method (AIM),4 as implemented in the DOIT code,5,6 which will be our chosen method for quantitative tunneling calculations. For multiple proton transfer in which several chemical bonds are broken simultaneously, application of TST/ST, which we shall use occasionally for purposes of comparison, would usually require adoption of the large-curvature version (LCG4),7 which is computationally very demanding, although recently interpolation procedures have been developed to mitigate this demand.8 In many cases, tunneling transmission factors calculated in either the small-9 or the large-curvature

Kinetic Isotope Effects in Multiple Proton Transfer

523

approximation failed to produce satisfactory KIEs. For instance, for the formic acid dimer, the small-curvature approximation underestimated the transfer rate constant by three to six orders of magnitude in the range 300 to 100 K, while the large-curvature LCG3 version produced unreliable KIEs.10 – 12 The best TST/ST option for evaluating KIEs may be to use the microcanonical optimized transmission factor,13 which approximates the least-action path by chosing at each tunneling energy the larger of the small- and large-curvature tunneling probabilities. A more direct approach is to use AIM, since this method does not depend on calculated trajectories, but derives rate constants, tunneling splittings, and isotope effects directly from the calculated minimum action. Its efficiency allows application to large systems where other methods become unwieldy. Before discussing actual systems, we present a brief outline of this theory in the next section. We emphasize, however, that most of the arguments used in subsequent sections do not require more than a casual understanding of the method. The main purpose of this contribution is to investigate how isotope effects can help us to determine the mechanism of multiple proton-transfer reactions. After introducing the theoretical methods to be used, we first consider stepwise transfer and then concerted transfer. Our main focus will be on tunneling transitions, since this is the area to which most of the new results belong. The various mechanisms will be illustrated by examples taken from the literature.

II. THEORETICAL METHODS A. TRANSITION S TATE T HEORY Hydrogen atoms and protons can transfer by classical over-barrier processes as well as by quantummechanical through-barrier processes, depending on their thermal energy in relation to the obstructing barrier. First, we consider classical transfer. In standard TST the transfer rate constant is expressed as2 kTST ðTÞ ¼ ðkB T=hÞ

Q‡ 2U0 =kB T e QR

ð20:2Þ

where U0 is the barrier height, and QR and Q‡ are the vibrational partition functions for the reactant and the transition state, respectively. The corresponding KIE for deuterium substitution equals !H Q‡ QR ð20:3Þ hTST ðTÞ ; kH ðTÞ=kD ðTÞ ¼ !D Q‡ QR The partition functions are given by Q¼

Y j

1 2sinhð"vj =2kB TÞ

ð20:4Þ

the product being over all relevant degrees of freedom, which in the present case are the normal modes of the system with the exception of the reaction coordinate for the transition state. If all modes affected by the transfer have frequencies that are much higher than kB T; the hyperbolic sine can be replaced by an exponential, so that the rate constant assumes the familiar form ~

kTST ðTÞ ¼ ðkB T=hÞe2U0 =kB T

ð20:5Þ

524

Isotope Effects in Chemistry and Biology

~ 0 ¼ U0 2 DU0 is the barrier height corrected for zero-point energy changes DU0 between where U the reactant state and the transition state. The KIE is then given by the relation D

H

hTST . eðDU0 2DU0 Þ=kB T

ð20:6Þ

Hence, it is governed by the effect of deuterium substitution on the difference in zero-point energy between the initial state and the transition state. However, we note that this commonly used approximation is not valid if the system contains modes with frequencies of the order of kB T or lower, i.e., # 200 cm21 at room temperature. Thus in the absence of tunneling the KIE is defined in purely statistical terms and does not depend on any of the parameters that govern the dynamics of barrier crossing such as mass or barrier height and width.

B. TUNNELING P RELIMINARIES We now consider tunneling transitions. In our other chapter in this book,14 we introduced the dynamics of barrier crossing by protons via the Golden Rule, which describes proton transfer as a nonadiabatic transition between two crossing potentials. Here we use the alternative adiabatic approach, which has the advantage of being more easily turned into a quantitative method. To illustrate this approach, we consider a collinear reaction A –H· · ·A O A· · ·H –A, of proton transfer between two “heavy” atoms A taking place in a condensed phase or in a stable molecule or complex. The tunneling potential UðxÞ for fixed A…A separation R is the symmetric doubleminimum potential depicted in Figure 20.1. We choose x ¼ 0 for the center of the potential, i.e., the top of the barrier. The thermal rate constant of transfer between the two wells is the sum of Boltzmann-averaged partial rate constants of tunneling for each energy level. However, for illustrative purposes, we restrict ourselves in this subsection to temperatures where only the zero point level of this high frequency vibration is populated. The tunneling rate constant for this level will be proportional to the square of the overlap between the zero point wave functions in the two wells corresponding to AH stretching vibrations. Neglecting for simplicity the anharmonicity

U(x)

−2

−1 A−H

0 x

1 H−A

2

FIGURE 20.1 Two-oscillator model for hydrogen tunneling. The solid line represents the tunneling potential in the equilibrium configuration of the promoting mode and the dashed line the potential at an inner classical turning point of that mode.

Kinetic Isotope Effects in Multiple Proton Transfer

525

of these vibrations, we thus obtain ktun / S200 ¼ e2r

2

=2a20

¼ e2r

2

mv=2"

ð20:7Þ

where r ¼ R 2 2j is the transfer distance, j being the A –H bondffi length, and m and v are the mass pffiffiffiffiffiffiffi and frequency of the AH stretch oscillators, so that a0 ¼ "=mv is the zero point amplitude in the well. It leads to the well known result that the tunneling rate constant depends exponentially on the square root of the mass of the tunneling particle. The corresponding KIE is thus given by

h / e2ð

pffi 221Þr 2 =2a20

pffi 221Þr 2 mv=2"

¼ e2ð

ð20:8Þ

In reactions of practical interest, the atoms A serving as donor and acceptor of the transferred proton are often part of a complex reactive site and therefore subject to many vibrations, including (quasi)symmetric vibrations that can serve as promoting modes by modulating the proton donor – acceptor distance and thus the barrier height, as illustrated in Figure 20.1. In addition, there may be antisymmetric vibrations that are displaced by the transfer and thus cause a reorganization effect similar to that of a heat bath. This will tend to hinder the transfer, but since the effect will be either isotope independent or weakly isotope dependent, it is unlikely to affect the KIE substantially. Therefore, we will postpone treatment of these antisymmetric modes to the next subsection and focus here on the coupling of the tunneling mode to a single (symmetric) A· · ·A-stretching mode. As a result of this coupling, which is always present, R is not a fixed distance but subject to vibrations. Thus in order to calculate the thermal rate constant kðTÞ; we need to average the rate constant for fixed R; kðRÞ; over the distribution of R values. Assumingp the ffiffi A· · ·A vibration to be harmonic, we use the (quantum) distribution function PðR; TÞ ¼ pAðTÞ 21 exp{ 2 ½ðR 2 R0 Þ=AðTÞ 2 }; which leads to the thermal tunneling rate constant ktun ðTÞ ,

ð

S200 ðR 2 2jÞPðR; TÞdR ¼ exp

2a20

2r02 þ A2 ðTÞ

ð20:9Þ

where r0 ¼ R0 2 2j is the equilibrium transfer distance, R0 is the equilibrium value of R and A(T) is the amplitude of the A…A-stretch mode: A2 ðTÞ ¼ A20 coth

"V : 2kB T

ð20:10Þ

pffiffiffiffiffiffiffiffiffi Here A0 ¼ "=M V, M and V being the effective mass and frequency. The larger is the amplitude, i.e., the stronger is the coupling of the tunneling mode to this “promoting” mode, the larger will be the rate constant. However, this coupling reduces the KIE; replacement of the protons by deuterons gives rise to a KIE of the form

htun ðTÞ / exp

r02

pffiffi 221

2a20 þ A2 ðTÞ

ð20:11Þ

It follows from Equationp20.8 ffiffiffi and Equation 20.11 that the mass dependence of the rate constant and the KIE is weaker than m by a factor A2 ðTÞ=2a20 : This effect is universal, i.e., always present, and virtually independent of the theoretical model; the same result was obtained in the Golden Rule approach used in our other chapter.14 It is an essential part of every tunneling description.

526

Isotope Effects in Chemistry and Biology

C. APPROXIMATE I NSTANTON M ETHOD The simple two-dimensional model of the preceding subsection is convenient for dealing with general properties of tunneling reactions, but needs upgrading for quantitative calculations. Its principal shortcomings are that it replaces all vibrations that affect the separation of the proton donor and acceptor atoms by a single “effective” promoting mode and that it neglects vibrations that undergo permanent displacement as a result of the transfer. AIM is designed to deal with tunneling in multidimensional systems and includes coupling effects of both types for all relevant modes. It is an adaptation of the quasiclassical instanton theory of tunneling.15 The instanton concept is based on the recognition that, under specified conditions, among the trajectories that connect reactant and product, there is one, the Instanton path, which dominates the tunneling rate. While this trajectory follows the minimum energy path in the vicinity of the equilibrium configurations, it follows a shorter and higher-energy path in the region of the transition state, a behavior often referred to as corner cutting. This “tunneling” trajectory corresponds to a combination of path length and energy that minimizes the action; the corresponding Instanton Action SI ðTÞ (in units h) is the quantity of practical interest since the tunneling rate is proportional to e2SI ðTÞ : AIM is designed to allow direct application of this methodology to proton transfer in multidimensional systems for which the structure and vibrational force field of the stationary configurations along the reaction path can be evaluated quantum-chemically. AIM does not search for the Instanton trajectory explicitly, a feature that ensures its computational efficiency; instead, it approximates the multidimensional instanton action SI ðTÞ directly, using generalizations of exact instanton solutions for low-dimensional models.15 The method first generates a full dimensional potential energy surface in a form suitable for instanton dynamics, formulated in terms of the (mass-weighted) normal coordinates of the transition state. For simplicity, we formulate the approach for a symmetric potential; the generalization to asymmetric potentials, which is straightforward, can be found in the literature.16 For a symmetric potential the Hamiltonian is written in the form

H¼

1 2 1X 2 1X 2 2 x_ þ y_ s;a þ UC ðxÞ þ v ðy 2 Dy2s Þ 2 2 s;a 2 s;a s;a s X X 2 x2 Cs ðys 2 Dys Þ 2 x Ca ðya ^ Dya Þ s

a

ð20:12Þ

where the mode x with imaginary frequency iv * is chosen as the reaction coordinate and the modes ys;a represent symmetric and antisymmetric transverse modes, respectively, that are displaced during the tunneling; these are treated as independent harmonic oscillators. The antisymmetric modes have the same symmetry as the tunneling mode and undergo final reorganization during the process while the symmetric modes do not undergo such reorganization and may only be displaced between the equilibrium configurations and the transition state. The one-dimensional potential along the tunneling coordinate UC ðxÞ may be any smooth symmetric double-minimum potential with a maximum at x ¼ 0 and minima at x ¼ ^Dx: It is a “crude-adiabatic” potential in that it is evaluated with the heavy atoms fixed in the equilibrium configuration, i.e., with ya ¼ ^Dya ; ys ¼ Dys : The coupling coefficients Cs;a in Equation 20.12 are restricted to linear terms in the transverse mode displacements: Ca ¼ v2a Dya =Dx; Cs ¼ v2s Dys =Dx2 : The active modes are separated into high-frequency (HF) modes, treated adiabatically, and low-frequency (LF) modes, treated in the sudden-approximation limit, according to a criterion z @ 1 and z ! 1; respectively, where z (the so-called “zeta factor”) for each mode depends on the frequency and the coupling to the tunneling mode x.6,15 HF modes renormalize the one-dimensional motion leading to the effective one-dimensional potential UCeff ðxÞ and mass meff ðxÞ defined elsewhere.3 The Instanton

Kinetic Isotope Effects in Multiple Proton Transfer

527

action of this effective one-dimensional motion reduces to the classical action integral: SI0 ðTÞ ¼ SC ðEp Þ ¼

ðx2 x1

dx{2meff ½UCeff ðxÞ 2 Ep }1=2

ð20:13Þ

where x1;2 are the classical turning points for the energy E* of the one-dimensional Instanton, defined by the condition that the period of the classical motion t in the upside-down potential 2UCeff ðxÞ equals t ¼ h=kB T: LF modes introduce corrections to this action such that the multidimensional instanton action assumes the form SI ðTÞ ¼

ðLFÞ X S0I ðTÞ þ a da ðTÞ s ðLFÞ X a 1þ ds ðTÞ

ð20:14Þ

s

The da -term corresponds to a Franck –Condon factor arising from the reorganization of LF antisymmetric modes which act similarly to a heat bath. Symmetric LF modes, represented by the ds -term, effectively reduce the tunneling distance and thus enhance tunneling; the factor as , 1 in Equation 20.14 is the square of this reduced distance (in dimensionless units).3 The correction terms da;s ; as well as the renormalization corrections that turn the proton mass m into meff and the barrier UC into UCeff ; depend on the frequency of the corresponding transverse mode and are proportional to the square of the coupling coefficients Ca;s : The resulting tunneling rate constant is found from the relation ktun ðTÞ . kAIM ðTÞ ¼ ðvR0 =2pÞe2SI ðTÞ

ð20:15Þ

where vR0 is the effective frequency of the tunneling mode in the reactant state. Since SI ðTÞ enters the rate constant in the exponent, ds ðTÞ encompasses the promoting effect of the slow symmetric modes, while da ðTÞ contributes a Frank – Condon factor. To account for classical over-barrier trajectories, we must add the classical rate constant given by Equation 20.2, which in DOIT is evaluated from standard TST,2 to the tunneling rate constant. The thermal rate constant is thus given by the sum kðTÞ ¼ kAIM ðTÞ þ kTST ðTÞ

ð20:16Þ

The corresponding KIEs are then obtained as ratios of these rate constants. The thermal rate constant, expressed as the sum of its tunneling and classical components in Equation 20.16, is evaluated by the DOIT code5,6 based on standard electronic structure and force field data for the stationary configurations, which the code reads directly from conventional quantum-chemistry programs such as Gaussian. Rate constants corresponding to isotopic substitution (and thus KIEs) are also readily obtained by a simple change of atomic symbol (H to D or T) in the input file. The calculations are basically the same for single- and multiple-proton transfer as long as the transfer involves a single smooth barrier. For concerted multiple transfer the reaction coordinate is a collective coordinate corresponding to simultaneous displacement of all the mobile protons, but its properties are derived in the same way as those for single-proton transfer, namely as those of the mode with imaginary frequency in the transition state. Correspondingly, the couplings of the transverse modes, i.e., the modes with real frequencies in the transition state, are derived from the displacements of these modes between the equilibrium configurations and the transition state.

528

Isotope Effects in Chemistry and Biology

D. ISOTOPE E FFECTS The KIE for classical proton transfer, given by Equation 20.3, is typically , 3 for single-proton transfer, and thus also for stepwise multiple-proton transfer, but since it results from differences in zero-point energy between AH and AD oscillators, it will tend to increase linearly with the number of protons jumping simultaneously and may therefore be quite large for concerted multiple-proton transfer. The KIE for unassisted (one-dimensional) proton tunneling, given by Equation 20.8, is a dynamic effect proportional to the square root of the mass of the tunneling particle. This effect is not cumulative for multiple-proton transfer but will be basically independent of the number of protons involved. The contribution of promoting modes will reduce the tunneling KIE, as follows from Equation 20.11, the magnitude of the reduction being dependent on the strength of the coupling to this mode. To generalize this result to multidimensional systems, we first rewrite Equation 20.11 in terms of the AIM formalism. Using Equation 20.14 and Equation 20.15 we obtain pffiffi 0;D 0;H D H hH=D . 2eSI =ð1þds Þ2SI =ð1þds Þ

ð20:17Þ

H where normally dD s $ ds : If the coupling is weak or moderate, we can write 1=ð1 þ ds Þ . 1 2 ds ; so that H=D 2ds ðSI0;D 2SI0;H Þ

hH=D . h0

ð20:18Þ

e

H=D

where h0 represents the KIE in the absence of coupling. This confirms that coupling to promoting modes reduces the KIE, an effect that is always present and depends only on the frequency of the promoting mode and the strength of its coupling to the tunneling mode. For concerted double-proton transfer, Equation 20.17 can be generalized to lnhAB=CD <

S0;CD 2 S0;AB I I 1 þ ds

ð20:19Þ

where the capital superscripts are either H or D. Using the approximate but quite robust relation pffiffiffiffiffiffiffiffiffiffiffiffi SI0;AB . SI0 mA þ mB ; where SI0 ¼ SI0;HH with mH ¼ 1 and mD ¼ 2, one obtains pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi lnhAB=CD < ð mC þ mD 2 mA þ mB Þ

S0I 1 þ ds

ð20:20Þ

where the expression in parentheses equals 0.32 for HH/HD, 0.27 for HD/DD and thus 0.59 for HH/DD. It follows that in the Arrhenius plot the HD curve is roughly halfway between the HH and DD curves but slightly closer to the latter, a location not normally encountered for stepwise transfer. If more than one promoting mode is active, we can generalize these equations simply by summation of the correction factors ds : Since the reaction will become more delocalized the more protons are jumping simultaneously, the contribution of promoting modes is likely to increase accordingly. Thus, while for classical proton transfer the KIE tends to increase with the number of protons involved, for concerted tunneling transfer it tends to decrease. Hence, for multiple proton transfer, the magnitude of the KIE is not a reliable criterion to discriminate between classical transfer and tunneling.

E. COMPARISON OF AIM wITH O THER M ETHODS In earlier work and in our other chapter in this book,14 we have used the Golden Rule approach, in which the tunneling is described as a transition between two diabatic states. In practice, it is usually

Kinetic Isotope Effects in Multiple Proton Transfer

529

applied only to two-dimensional models consisting of a tunneling mode and a single promoting mode, whose properties are determined empirically. Despite its limitations, it constitutes an essential advance over one-dimensional tunneling models in that it can deal with realistic barrier heights and tunneling distances. It is particularly useful for deriving approximate relationships between tunneling parameters, but for quantitative calculations, it is less suitable than the Instanton approach. Exact Instanton solutions are known only for low-dimensional models.15 For simple twodimensional models, the AIM/DOIT approach reproduces these solutions, but for multidimensional systems, only approximate solutions have been obtained. For several such systems, the method was compared with TST/ST. For rate-constant calculations the methods were found to perform similarly if the potentials were calculated at the same level.17,18 This indicates that the two methods handle the multidimensional dynamics in equivalent ways. Because of its computational efficiency, the AIM/DOIT approach is well suited to deal with large systems. At the low temperatures relevant to spectroscopically observed tunneling splittings, AIM/DOIT was found to be superior to TST/ST in that it gave a more satisfactory account of KIEs and of the splitting of vibrationally excited levels.19,20 In a recent paper on tunneling splitting,21 we compared AIM/DOIT with a calculation performed with the least-action method of Liedl et al.,22 which is computationally much more demanding. In early AIM/DOIT calculations23 it was found that low levels of quantum chemistry leading to overestimated barrier heights still could produce good results with minimal scaling of the barrier. One reason may be a compensation of errors through the zeta factor for symmetric modes. Its underestimation for high barriers may affect the HF/LF assignment so as to yield more or stronger promoting modes. Other compensating effects such as high zero-point energies may also play a role. These problems indicate the need for a level of quantum chemistry that can deal realistically with hydrogen bonded systems.

III. STEPWISE TRANSFER A. EXAMPLE: P ORPHINE Among the early studies of double-proton transfer, that of the interchange of the two inner protons among the four nitrogens of free base porphines is one of the more informative. It was long regarded as a synchronous double-proton shift to which a vibrational frequency could be assigned. Limbach and his group24 showed that the rate of transfer near room temperature could be measured by nuclear magnetic resonance techniques. Initially they described the temperature- and isotopedependent rate constants to coherent transfer. However, Sarai25 presented strong arguments in favor of the stepwise process, illustrated in the insert of Figure 20.2, in the first step of which one of the protons moves from a trans to a cis configuration. This proposal is now widely accepted, especially after Butenhoff and Moore26 studied the transfer at much lower temperatures by optical methods. They observed a large KIE as expected for tunneling but also an apparent activation energy that was close to that measured at room temperature and showed no tendency to approach the zero activation limits typical for tunneling reactions. Direct dynamics calculations, first on the basis of a semiempirical two-oscillator model27 and later by ab initio methods that included coupling to all relevant vibrations,16,28 confirmed that in the range of temperatures for which measurements are available, stepwise transfer is always much faster than coherent transfer. In retrospect, this is not surprising. The protons in porphine are not hydrogen bonded and are shielded against solvent effects. Coherent transfer would require breaking two NH bonds simultaneously without compensation by hydrogen bonding. Breaking the bonds one at a time as in stepwise transfer therefore leads to a lower transfer barrier. The corresponding asymmetric cis intermediate has a higher energy than normal (trans) porphine, which means that the first step of the reaction is endothermic. The resulting activation energy will persist at low temperatures. The actual tunneling process contributes its own apparent activation energy, which is temperature dependent

530

Isotope Effects in Chemistry and Biology 4.0 2.0

log k

0.0 −2.0

TT

DD

HD

HH

−4.0 −6.0 −8.0 3.0

5.0

7.0 1000/T

9.0

11.0

FIGURE 20.2 Temperature dependence of the rate constant (in sec21) of double-proton transfer in porphine-d0, -d1, -d2 and -t2 evaluated for the stepwise mechanism. The symbols represent observed rate constants24,26 and the curves represent the results of a multidimensional AIM/DOIT calculation.16 The insert shows the stepwise and concerted transfer mechanisms.

and disappears at low temperatures. This temperature dependence is due to thermal activation of vibrations that support the tunneling, in particular vibrations that modulate the tunneling distance. An Arrhenius plot of the logarithm of the observed rate constant against the inverse temperature, illustrated in Figure 20.2, shows data points for HH and DD transfer fitting lines that are almost parallel and only weakly curved without converging to zero activation for T ! 0: The solid lines shown are the result of an AIM/DOIT calculation (Equation 20.16) that includes all vibrations linearly coupled to the tunneling; they confirm that the transfer is stepwise. Because stepwise transfer will generally include an endothermic step, a transition with an apparent activation energy that varies little with temperature is one of its revealing characteristics. Unfortunately, kinetic data covering a wide range of temperatures for a single species are rarely available. The question then arises whether the magnitude of the apparent activation energy by itself can provide sufficient evidence to assign the transfer to a stepwise process. This question will be considered in Section III.D. Another observation that may offer a clue to the actual mechanism is the effect of partial deuteration. For porphine, Figure 20.2 shows that log k HD is much closer to log k DD than to log k HH. Because of the symmetry of this molecule, the two steps of the stepwise process are equivalent in the HH and DD compounds, but not in the HD compound. In this mixed isotopomer, D transfer will be the rate-determining step. This explains why the transfer rate in the HD compound is close to that in the DD compound. Hence, the observed effect of partial deuteration supports the stepwise nature of the transfer. In the next subsection we will consider the question whether this argument can be generalized to less symmetric species.

B. ISOTOPE E FFECTS Stepwise transfer implies the existence of at least one intermediate structure that is “stable,” i.e., that can exist for a time longer than the vibrational period of the mode with the lowest frequency. A typical double-minimum potential of this type is illustrated in Figure 20.3. In the first step, a proton with an energy Em 2 Ei or higher in the initial state i tunnels through the barrier U1 with

Kinetic Isotope Effects in Multiple Proton Transfer

531

U2

U1

Energy

Em

Ei

Ef Reaction coordinate

FIGURE 20.3 A triple-minimum potential typical for stepwise double-proton transfer.

a rate constant k1 e2ðEm 2Ei Þ=kB T to the intermediate state m and relaxes. Then that proton can either return with a rate constant k21 ¼ k1 or the other proton can go forward to complete the second step by tunneling through the barrier U2 with a rate constant k2 to the final state f and relaxing. Elementary kinetics leads to an overall rate constant: kifHH ¼ 2

k1H k2H 2ðEm 2Ei Þ=kB T e k1H þ k2H

ð20:21Þ

For the reverse reaction, Ei must be replaced by Ef : For double-deuteron transfer, the same equation applies with the superscripts H replaced by D, but for mixed proton þ deuteron transfer the result is more complicated: ! k1H k2D k1D k2H HD e2ðEm 2Ei Þ=kB T ð20:22Þ þ D kif ¼ k1H þ k2D k1 þ k2H If U1 ¼ U2 as in the case of porphine, k1 ¼ k2 ; so that Equation 20.21 and Equation 20.22 reduce, respectively, to kifHH ¼ k1H e2ðEm 2Ei Þ=kB T

kifHD ¼ 2

k1H k1D 2ðEm 2Ei Þ=kB T e þ k1D

k1H

ð20:23Þ

ð20:24Þ

If hHH=DD ¼ kifHH =kifDD is large, then hHH=HD < ð1=2ÞhHH=DD and h HD/DD # 2. If the system is not symmetric, no such simple conclusions can be drawn. Not surprisingly, the key relation is that between the proton-transfer step across the higher barrier and the deuterontransfer step across the lower barrier. If these are about the same, one can readily show that h HH/HD < 4 and hHD=DD < ð1=4ÞhHH=DD : If the proton transfer remains faster despite the difference in barrier heights, not much can be said about the position of the HD Arrhenius curve

532

Isotope Effects in Chemistry and Biology 6 HH

log k, s−1

4

HD

2

0

DD

−2

−4 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 (U1 − U2)/(U1 + U2)

0.3

0.4

0.5

FIGURE 20.4 Relative magnitudes of the logarithms of HH, HD, and DD transfer rate constants as a function of the asymmetry of the potential illustrated in Figure 20.3.

except that it is likely to be somewhat closer to the DD than to the HH curve. If, on the other hand, the deuteron transfer is faster, we have h HH/HD # 2 and hHD=DD $ ð1=2ÞhHH=DD ; then the position of the HD curve relative to the HH and DD curves is the reverse of that observed for porphine. In Figure 20.4, we show a typical example of the dependence of the relative positions of the HH, HD, and DD rate constants as a function of the relative barrier heights of the two steps, calculated by the method discussed in the next subsection. The calculations show that for nearly equal barriers, U1 < U2 ; the HD rate constants tend to be close to the DD rate constants, while for very asymmetric barriers U1 ! or @ U2, these tend to be close to the HH rate constants.

C. TEMPERATURE E FFECTS For classical transfer, treated in Section II.A, the temperature dependence of the rate constant is dominated by the barrier height and that of the KIE by the difference in zero-point energy of the two isotopomers. The apparent activation energy Ea of Equation 20.1 for stepwise transfer has two contributions: Ea ¼ Em 2 Ei þ Eatun

ð20:25Þ

where Em 2 Ei is the energy required to reach the intermediate state and Eatun is the apparent activation energy of the actual tunneling process, which is always smaller than the barrier height. For a rough estimate of Eatun we use the simple approach of Section II.B. In Equation 20.9, we expressed the tunneling rate constant for a simple two-dimensional model potential in the form ln kðTÞ ,

2a20

2r02 þ A2 ðTÞ

ð20:26Þ

where r0 is the equilibrium tunneling distance and a0 and AðTÞ are zero-point amplitudes of the tunneling mode and the promoting mode, respectively. We assume that, due to the high frequency of the tunneling mode, a0 will be independent of temperature. The temperature dependence of AðTÞ is given by Equation 20.10.

Kinetic Isotope Effects in Multiple Proton Transfer

533

In the case of stepwise transfer the steps are generally far from thermoneutral, since the intermediate step tends to have a high energy. In that case, the potential of the final state at the intermediate state energy can be treated as an “absorbing wall.” Benderskii et al.29 have shown that Equation 20.26 then takes the alternative form ln kðTÞ ,

2rt2 a20 þ A2 ðTÞ

ð20:27Þ

where rt is the distance between the bottom of the intermediate well and the crossing point of the intermediate state and final state potentials: " # 2ðEm 2 Ef Þ 1 rt ¼ r0 1 2 < r0 =2 ð20:28Þ 2 mv2 r02 Since the only temperature-dependent parameter in Equation 20.27 is AðTÞ; we readily obtain Eatun ¼

d ln kðTÞ 1 A2 r 2 =sinh2 ð"V=2kB TÞ ¼ "V 2 0 t 2 21 2 dðkB TÞ ½a0 þ A0 cothð"V=2kB TÞ

2

ð20:29Þ

For high temperatures, i.e., kB T $ "V; in which case A2 ðTÞ , 2kB T=M V2 ; Equation 20.29 reduces to Eatun ¼

M V2 rt2 =2 ð1 þ M V2 a20 =2kB TÞ2

ð20:30Þ

For low temperatures, i.e., kB T # "V=4; in which case AðTÞ , A0 ; Equation 20.29 reduces to Eatun ¼

2rt2 A20 "V 2"V=kB T e ða20 þ A20 Þ2

ð20:31Þ

In practice, Eatun calculated from these equations near or below room temperature for reasonable values of the promoting-mode parameters is unlikely to exceed 10 kcal mol21. Since the equations neglect the anharmonicity of the potential deep in the wells, these will tend to overestimate of Eatun : Hence observation of an apparent activation energy of 10 kcal mol21 or higher will generally imply stepwise transfer.

D. APPLICATIONS We now apply the results of the preceding subsections to our porphine example. Because of the equivalence of the tunneling trajectories of the two protons, we have, according to Equation 20.24, for the KIE of the mixed isotope hHD=DD ; kHD =kDD # 2: It follows from Figure 20.2 that the observed rate constants agree with this prediction, a result consistent with the assumed stepwise transfer. From Equation 20.30 and Equation 20.31 we readily obtain that for T ¼ 100 K the tunneling contributes less than 1 kcal to the observed activation energies of 6.3 kcal mol21 for any reasonable set of parameter values, thus confirming the stepwise transfer mechanism. A detailed AIM/DOIT calculation,16 the results of which are represented by the solid line in Figure 20.2, confirms these conclusions. This calculation reveals that, in addition to coupled promoting vibrations that enhance the tunneling rate, there are coupled antisymmetric vibrations that reduce it. For the HH isotopomer, the reduction exceeds the enhancement, whereas for the other isotopomers the opposite holds. It follows that the overall effect of the couplings is a reduction of the KIE.

534

Isotope Effects in Chemistry and Biology

log k, s−1

−3

−4

−5 R 1

X

H

−6 2.4

X

R2

H

R1

SO2Ph SO2Ph

2.5

H

R2 H

H H PhO2S SO2Ph

1000/T

2.6

2.7

FIGURE 20.5 Temperature dependence of the rate constants of double-proton transfer in synsesquinorbornene disulfone-d0, -d1 and -d2, illustrated in the insert, evaluated for R1,2 ¼ CH2.31 The symbols represent observed rate constants and the solid lines the results of two-dimensional semiempirical Golden Rule calculations.

Next, we discuss several instances of double-proton transfer in bridged ring compounds, synthesized expressly to investigate the proton transfer mechanism. The idea behind this work was to construct stiff structures that would not allow the proton donor and acceptor groups to approach each other closely and thereby obscure the actual tunneling. A typical example, shown in the insert to Figure 20.5, is the series of syn-sesquinorbornene disulfones produced by Paquette and his coworkers.30 In a somewhat ambiguous analysis,31 the question of the mechanism of the transfer was not completely settled, although the authors leaned towards stepwise transfer. The compounds are not symmetric with respect to the transfer and show different forward and backward rate constants; an Arrhenius plot of the rate constants of the exothermic forward reaction is shown in Figure 20.5. The logarithms of the HD rate constants are closer to those of the HH than to those of the DD rate constants, consistent with moderately asymmetric barriers. The apparent activation energy is high (24 to 27 kcal mol21), much higher than in porphine and high enough to exclude coherent transfer. This interpretation was supported by an empirical Golden Rule calculation based on the two-oscillator model.31 The fitted parameters, apart from the standard values used for the frequency and anharmonicity of the CH-stretching vibrations, ˚ , slightly smaller than predicted on the basis of van der include: (i) a tunneling distance of 1.5 A Waals radii which seems appropriate for these strained molecules, (ii) an intermediate-state energy of about 20 kcal mol21, not far from the apparent activation energy, and (iii) a promotingmode frequency of about 100 cm21 and an effective mass of 40, which can be reasonably ascribed to a twisting mode. In this picture, the actual tunneling takes place near the top of the barriers because of the presence of the high-energy intermediate state. This accounts for the relatively small KIEs. The mixed isotope results were explained in terms of the observed30 exothermicities of the reactions. A more elaborate calculation aimed at interpreting the results in terms of either coherent or stepwise transfer used a model compound consisting of the carbon – hydrogen framework without the sulfon groups.31 Based on a potential and force field calculated semiempirically at the PM3 level, TST/ST calculations in the small curvature approximation were carried out, which again

Kinetic Isotope Effects in Multiple Proton Transfer

535

predicted stepwise transfer to be many orders of magnitude faster than coherent transfer. However, it was pointed out that PM3 is biased in favor of a stepwise pathway. An ab initio calculation of the potential at the CASSCF/3-21G level indicated that, after correction for dynamic correlation effects, transfer might be governed by a single transition state and thus coherent, as it is predicted to be for the ethylene –ethane system. However, the experimental evidence strongly favors a stepwise mechanism. The same applies to two bridged-ring compounds investigated by Mackenzie’s group.32 For these compounds, apparent activation energies of the order of 30 kcal mol21 were observed and KIEs somewhat smaller than 10 for double-proton transfer. These results are incompatible with coherent transfer. The small KIE indicates tunneling through a low barrier, whereas the high activation energy indicates a very high barrier. This paradox can only be resolved if it is assumed that tunneling occurs near the top of the barrier, i.e., at the energy imposed by an intermediate state, which energy contributes the bulk of the apparent activation energy, as shown by a Golden Rule calculation.33

IV. CONCERTED TRANSFER A. EXAMPLE: ACETIC ACID – METHANOL C OMPLEX Although many systems are known in which two or more protons transfer, direct evidence of concerted as opposed to stepwise transfer is not always easy to obtain. Failure to detect a stable intermediate experimentally does not provide such evidence, as the porphine case shows. Convincing experimental proof would be the observation of tunneling splitting, as observed recently for the formic and benzoic acid dimers in the vapor phase,34 since the presence of an intermediate state would destroy the phase coherence required for such splitting. More information, including the KIE of the mixed isotopomer, is available for the acetic acid –methanol complex illustrated in Figure 20.6; an Arrhenius plot of the rate constants observed in tetrahydrofuran solution35 is shown in Figure 20.7. These rate constants are products of the equilibrium constant of complex formation and the rate constant of double-proton tunneling. The acetic acid –methanol complex is a simple example of a system in which one of the partners, viz., methanol, bridges two active sites of the other partner, viz., acetic acid, and thereby facilitates transfer of a proton between these two sites through a relay mechanism, as discussed in the Introduction. Such processes, which may involve a bridge consisting of several molecules arranged in a chain, have been known for a long time, the oldest example being the Grotthus mechanism for

+

∆E‡ ∆EC O

O

HM H

O A

FIGURE 20.6 Synchronous double-proton transfer in the acetic acid – methanol (AA– Me) complex.

536

Isotope Effects in Chemistry and Biology 4

log k, s−1

2 0 2 4 6 2.5

3

3.5

4 1000/T

4.5

5

5.5

FIGURE 20.7 Temperature dependence of the rate constant of double-proton transfer in acetic acid – methanol – d0, –d1, and – d2 in tetrahydrofuran. The symbols represent observed rate constants35 and the solid and dashed lines represent calculated tunneling and classical rate constants, respectively, for concerted doubleproton transfer.36

proton conduction in ice. The relay mechanism is not limited to coherent transfer; in ice, it is a stepwise and almost purely classical process. Solvent-assisted proton transfer of this kind is particularly important in biological systems, where water “wires” may transport protons over considerable distances, as demonstrated for the enzymatic reaction catalyzed by carbonic anhydrase II, discussed below. The apparent activation energy of the tunneling rate constant of the acetic acid –methanol complex at room temperature is about 6.7 kcal mol21, i.e., much lower than that of the stepwise processes considered in Section III.D. This together with the KIE, which equals about 15 at room temperature, suggests coherent transfer. The activation energy is higher for the DD isotopomer, namely about 9.3 kcal mol21; Figure 20.7 clearly shows the divergence of the HH and DD Arrhenius curves towards lower temperatures, as expected for coherent tunneling. The corresponding HD curve is about halfway between the HH and DD curves, as illustrated by the room temperature KIE consistent with the values predicted by Equation 20.20 for coherent transfer. The complex is small enough to allow a high-level quantum-chemical evaluation of the transfer potential and the vibrational force field. The geometries of the stationary configurations were optimized at the QCISD level and the normal-mode frequencies were calculated at the MP2 level with a 6-31G(d,p) basis set.36 A single transition state was found, which confirms that the transfer is coherent. The effect of the solvent was included by single-point calculations over gas-phase geometries with the polarized continuum model. The proton-transfer rate constants were calculated with AIM as implemented in the DOIT 1.2 code;5 a minor adjustment of the calculated barrier height was applied to reach the values displayed as solid lines in Figure 20.7, where these are compared with the nuclear magnetic resonance results of Gerritzen and Limbach.35 The excellent agreement achieved both for the temperature dependence and the KIE, including that of the two mixed isotopes, lends strong support to the proposed coherent transfer mechanism. Although the barrier is symmetric, the two protons do not move synchronously, since the two hydrogen bonds are not equivalent. This is reflected in the (small) difference in the calculated rate constants for the two mixed isotopomers. In Figure 20.7, we also show Arrhenius curves calculated by TST for classical double-proton transfer.36 The results show that classical transfer is not competitive, even at room temperature.

Kinetic Isotope Effects in Multiple Proton Transfer

537

In general, coherent classical transfer is an inefficient process since the simultaneous breaking of several bonds will result in a relatively high barrier. Note that the KIE calculated classically for the mixed isotope does not follow the pattern shown by the tunneling process, the corresponding HD Arrhenius curve being closer to the HH than the DD curve.

B. HYDROGEN B ONDED D IMERS AND C OMPLEXES A closely related double-proton transfer process is observed in dimers of carboxylic acids.34 It is also expected to be coherent and thus should give rise to tunneling splitting in the spectra. Tunneling splittings of 0.0029 and 0.013 cm21 have been reported for (DCOOH)2 in the gas phase,34 one value applying to the zero-point level and the other to a CO-stretch fundamental. The authors could not assign their spectra with sufficient accuracy to decide which value applies to which level, but opted tentatively for assigning the smaller splitting to the lower level. Our estimates indicate that this assignment should be reversed.37 Proper adiabatic treatment of the HF CO-stretch vibration will lead to an increase in the effective mass of the tunneling proton pair and thus to a reduction of the splitting. The observation of tunneling splitting is of course direct proof of synchronous transfer. A tunneling splitting of 0.037 cm21 has been reported34 for the benzoic acid dimer. The dynamics of double-proton exchange in neat and doped crystals of benzoic acid has been studied by high-resolution spectroscopy and T1-NMR spin-lattice relaxometry. Double proton exchange is greatly enhanced in the crystal due to compression of the O· · ·O distances. The rate constant was found to vary from about 2 £ 108 sec21 at very low temperatures to about 8 £ 1010 sec21 at room temperatures with corresponding KIEs of order of 100 and 10, respectively.38 – 42 A rate of exchange lower by about an order of magnitude has been reported for nonaromatic carboxylic acid dimer crystals.43 For details, we refer to our recent theoretical study of these observations.44 The same methodology has been applied to calix[4]arenes,44 where four protons are exchanged between four hydroxyl groups in a stiff structure that is tightly hydrogen bonded. Comparison of the rates reported for these compounds and the crystalline benzoic acid dimers indicates unexplained discrepancies that may be related to the experimental method. For instance, the low-temperature rate constants and apparent activation energies reported for the two systems are almost identical despite the different numbers of OH bonds that are simultaneously broken. These conflicts with quantum-chemical calculations, which reasonably predict that the transfer barrier for the calixarene will be about double that for the benzoic acid dimer. It suggests that the origin of this discrepancy must probably be sought in the measurements. A recent AIM calculation of proton dynamics in calix[4]arene45 predicts synchronous quadruple proton transfer leading to a tunneling splitting of about 2 £ 1023 cm21, a value that is about an order of magnitude smaller than the splitting observed in the formic acid dimer discussed above. A tunneling splitting of 0.017 cm21, comparable to that in the formic acid dimer, has been reported for the 2-pyridone 2-hydroxypyridine complex,46 which, as illustrated in Figure 20.8, turns into itself upon double-proton transfer. Although the authors originally assigned this splitting to the excited state, our recent calculations indicate that it is a ground-state property.21 No deuterium effects have been measured thus far but the calculations predict a splitting in the DD compound of about 6.5 £ 1025 cm21,21 corresponding to a KIE of 260, a value that again reflects the high barrier associated with cleaving an OH bond and an NH bond simultaneously.

C. WATER W IRES Proton transfer via the relay mechanism, observed in the acetic acid – methanol complex treated earlier in this section, has been observed in many systems. A characteristic example is the heteroaromatic molecule 7-azaindole, which contains two nitrogen atoms, one of which carries a hydrogen atom. In the electronic ground state, this hydrogen atom is attached to the nitrogen in the fivemembered ring, a structure we refer to as canonical. The tautomeric structure with the hydrogen at

538

Isotope Effects in Chemistry and Biology

N

N

O

O

FIGURE 20.8 Transition state of synchronous double-proton transfer in 2-pyridone·2-hydroxypyridine in the electronic ground state.21

the nitrogen in the six-membered ring, which is readily produced via the excited state, has a higher energy but will not isomerize spontaneously at room temperature due to the high barrier separating the two structures. In aqueous solution, however, tautomerization is very rapid (, 109 sec21).47 Complexes of 7-azaindole with one to four water molecules have been studied in the gas phase.48,49 Specifically, the decay of the emission signal of the excited canonical structure produced by optical excitation has been measured for normal and deuterated complexes.49 On the basis of the observed KIEs, it was claimed that this structure tautomerizes during the lifetime of the excited state (about 6 nsec). However, closer inspection of the data shows that the observed decay is due to radiationless transition to a triplet state and not to proton transfer, which in these excited complexes is not fast enough to compete with the decay.50 To find out why the tautomerization is so much faster in aqueous solution than in these complexes, we carried out AIM/DOIT calculations on excited complexes with one, two, and five water molecules.51 The results indicate that with one and two water molecules proton transfer in the excited state is a concerted process, but that both the transfer rate and the degree of synchronicity are much smaller with two water molecules than with one. The asynchronicity leads to substantial charge separation, which in turn will give rise to a large solvent effect. With five water molecules in a bicyclic arrangement with the two nitrogens (see Figure 20.9), stabilization of intermediate structures is observed, which lowers the barrier to the point where classical transfer becomes a major contributor. This enhances the rate of tautomerization to a value approaching that observed in aqueous solution.51 In the ground state, these effects are muted.52 The rate constants for proton transfer in the 1:1 and 1:2 complexes are calculated to be very similar. The isotope effects, which are not previously reported, are small: 6.6 for the complex with one and 2.6 for the complex with two water molecules at room temperature. Such KIEs are typical for loose water wires and reflect the contributions of the motion of atoms heavier than hydrogen and deuterium to the transfer. Much larger KIEs appear if the hydrogen bonded framework is stiff, as found above for calix[4]arene. The deviation from perfect synchronicity is reflected in the effects of fractional deuteration. For the 1:1 complex replacing one of the two mobile protons by deuterium reduces the HD rate to a value close to the DD rate. For the 1:2 complexes, the HHD rate is close to the HHH rate and the HDD rate close to the DDD rate. The catalytic effect of water on tautomerization was studied also for complexes of guanine with one and two water molecules.52 This tautomerization is biologically important, since it provides a mechanism for the occurrence of point mutations during DNA replication.53 It also suggests a possible mechanism for enzymatic DNA repair of such mutations: passage of a hydroxyl group along the DNA chain may turn an erroneously embedded tautomeric guanine molecule into the correct canonical species. A recent study54 was reported of the mechanism of proton exchange between the amino and carboxyl groups of glycine in aqueous solution. The model included solvation by up to six discrete water molecules surrounded by a dielectric continuum. Both the discrete water molecules and the continuum were necessary to obtain a satisfactory value for the endothermicity of the

1.532 1.012

O O

N

0.956

2.153

1.044

1.455 1.096

0.954

1.976

TS2

1.623

0.982

1.779

1.724 1.029

TS1

2.091 0.998

O

O

1.741 0.974

1.394 1.123

1.270 1.201

O

1.646 1.040

N

N

539

1.604 1.051

Kinetic Isotope Effects in Multiple Proton Transfer

1.569 1.003

0.969 1.758

Int

T

FIGURE 20.9 Stepwise triple-proton transfer in 7-azaindole·5H2O.51

zwitterion-to-neutral interconversion. It was concluded that transfer through a water bridge is faster than direct transfer and is large enough to account for the rate derived from experiment. Interestingly, the calculated KIE for direct single-proton transfer was more than an order of magnitude larger than that for the indirect double-proton transfer, which suggests an experimental way to discriminate between the two mechanisms. The partial and total KIE for the indirect process turned out to be similar to those reported for the acetic acid – methanol complex discussed in the preceding subsection. However, a calculation of the rate of hydrogen abstraction from vitamin E (a-tocopherol) by a methyl radical by a direct mechanism and via a water bridge led to the conclusion that in that case the direct mechanism is much faster.55 Apparently, this is due to the free radical nature of the process. Water wires may be good conduits for protons but not necessarily for neutral hydrogen atoms. Although water wires have been proposed for many biological processes, few of these have been studied to the point where the results of isotopic labeling may be informative. An exception is 56 the catalytic conversion of CO2 to HCO2 3 by carbonic anhydrase II. The rate-determining step in this process is the transfer of a proton from the H2O ligand of a four-coordinated zinc ion to ˚ away. This transfer may occur via a water wire, a minimum of two a histidine residue some 8 A water molecules being required to make a proper connection; or by stepwise proton diffusion through solvent molecules in the enzyme pocket that act more or less like bulk water, a Marcus-type mechanism discussed by Silverman and Elder.56 The transfer exhibits a relatively weak isotope effect, the observed KIE at room temperature in heavy water being in the range three to four.57 Data are also available for H2O or D2O mixtures and show a rate constant whose logarithm depends linearly on the fraction of D2O in the mixture.58 Having chosen the water wire assumption as the most promising approach to this problem, we carried out a theoretical study of an active-site model of the enzyme with two water molecules

540

Isotope Effects in Chemistry and Biology

forming the proton conduit in a pocket whose size is constrained to fit the available x-ray data. The model is illustrated in Figure 20.10; the structure and vibrational force field were calculated by a density functional method after extensive testing; for details, we refer to the original publication.18 A single transition state was found, which indicates coherent triple-proton transfer. Our best estimate of the adiabatic barrier height amounts to about 6 kcal mol21. Tunneling rate constants were evaluated with AIM/DOIT and, on a more limited scale, with TST/ST. The resulting KIEs for partial and total deuteration, shown in Figure 20.11, agreed reasonably well with the observed

N

1.980 (1.994)

C

C

Zn N

C

O

1.977 (1.993)

C

N

2.115 (2.017) 1.738 (1.550)

C

N

1.983 (2.003)

C

C C

C

C C

C

C

N

1.761 (1.567)

O

N

1.976 (1.728)

O

C C N

N

C

R 1.995 (2.002)

C

N C

C

1.992 (2.021)

C

N

C

1.228 2.048 (1.190) (1.963) 1.292 (1.280)

1.195 (1.189)

O

C C

O

Zn

N

C

N

N

1.987 (2.006)

1.139 (1.154)

N

C C

C N

1.318 (1.321)

1.197 (1.239)

C

C

C

O

N

C

C

TS

2.025 (2.036) C

2.017 (2.039)

C N

C

N

N

C C

C

1.961 (1.893)

C

N

1.738 (1.640)

1.709 (1.587)

O

Zn

C N

O

C C

C N

O

C

1.814 (1.703)

N

2.012 (2.030)

C C C

C

N

P

FIGURE 20.10 Model of the reactive site of the enzyme carbonic anhydrase II, showing the reactant, the product, and the transition state for concerted triple-proton transfer.18 The insert depicts the mode with imaginary frequency.

Kinetic Isotope Effects in Multiple Proton Transfer

541

Kinetic isotope effect

17

13

9

5

1

0

0.2

0.4 0.6 Atomic fraction of D

0.8

1

FIGURE 20.11 Kinetic isotope effects as a function of the H2O or D2O ratio of the solvent in the reaction catalyzed by carbonic anhydrase II, as observed58 (solid circles) and calculated18 by tunneling dynamics (open circles) and classical dynamics (squares).

values, taking into account that the assumed water wire is probably not unique and that wires consisting of three and more water molecules may also contribute. Of special interest is the ability of the theoretical approach to account for the small KIEs, which contrast sharply with the enzymatic KIEs in the range 20 to 80 we discuss elsewhere in this volume.14 The calculations show that the KIEs are small because of strong participation of the motion of heavier atoms, especially the oxygen atoms of the water wire and the zinc ligand. This proton conduit is a loose structure that can be easily deformed without substantial expenditure of energy. Both symmetric and antisymmetric modes participate in the deformation, the former helping and the latter hindering tunneling. Because of the translational near-symmetry of the conduit, multiple transfer is highly synchronic. As a result, the KIE of partially deuterated wires does not depend strongly on the place of deuteration and the rate constant varies smoothly with the number of deuterium substituents. This is not necessarily true for longer and branched water wires, where trapping of the proton in the center or at a branching point may interfere with the synchronicity or even with the concertedness. Classical transfer, which prevails in ice, is not competitive in the enzyme, the classical rate constant calculated by TST being almost two orders of magnitude smaller than the tunneling rate constant.18 The absence of a sizeable classical contribution is confirmed by the calculated KIEs, which in the present system are much larger for classical than for tunneling transfer. Classically, KIEs are due to differences in zero-point energies and thus increase with the number of deuterium substituents. Quantum mechanically, KIEs are mostly due to the increase in the mass of the tunneling particle, but this effect is strongly mitigated by the participation of LF modes that are not subject to deuterium isotope effects. Hence, in the case of multiple proton transfer the “standard” rule that a large KIE implies tunneling and a small KIE implies classical transfer no longer applies. In the present system, the opposite holds: the observation of a small KIE indicates a concerted tunneling mechanism. The contribution of water wires to proton transport through cell membranes has been demonstrated for bacteriorhodopsin,59 where KIEs in the range two to seven have been measured for specific steps in the photocycle. From the dependence of the KIEs on the D2O mole fraction, it was tentatively inferred that some of the steps involve “cooperative” proton rearrangement, which presumably is the same mechanism as that labeled concerted proton tunneling in the present text.

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Isotope Effects in Chemistry and Biology

D. THE P ROTON I NVENTORY P ROBLEM To study the degree of concertedness of proton transfer through biological channels, the proton inventory technique has been a popular tool.60 It is based on partial deuteration of water molecules and others carriers of potentially mobile protons that form the channel, the idea being that the relation between the proton transfer rate and the fractional deuterium content may indicate whether multiple proton jumps are involved. Specifically, it has been proposed that a linear dependence of the logarithm of the transfer rate constant on the deuterium fraction will result if the observed KIE is made up of several small contributions and thus may indicate concertedness. For the time being, this proposal lacks a firm theoretical basis. In principle, the approach of Section II.C, specifically Equation 20.20, addresses the problem but this requires further development. In addition, the calculations of Section IV.A and Section IV.C provide some intriguing examples, which we will now discuss. In Figure 20.12, we have plotted the logarithm of the rate constant calculated for multiple proton transfer against the deuterium fraction for all the systems discussed in this chapter. Concerted transfer is represented by solid lines, stepwise transfer by dashed lines. Where available we have included experimental observations as solid circles. In addition to the systems treated above, we have included the triangular water trimer, in which the three mobile protons which transfer concertedly are strictly equivalent. This complex has been studied by several groups; for a discussion of these treatments, we refer to Ref. 5. In the present contribution, we extend our AIM/ DOIT calculations reported previously5 to fractional isotope effects by replacing one, two, and three mobile protons by deuterons, thus considering only primary KIEs; it turns out that secondary KIEs are very small in this case. This approach is somewhat different from that used for

6

CA II

4

7−azaindole.H2O AA−Me

porphine

log k

2 7−azaindole.2H2O 0

water trimer

−2

−4

−6 −0.25

bridged−ring compounds

0

0.25 0.5 0.75 Atomic fraction of D

1

1.25

FIGURE 20.12 Rate constants of multiple proton transfer plotted as a function of atomic fraction of D: The symbols depict observations, the lines calculations: solid lines refer to coherent transfer and dashed lines to stepwise transfer.

Kinetic Isotope Effects in Multiple Proton Transfer

543

carbonic-anhydrase II, discussed in the preceding subsection, where we replaced entire water molecules by heavy water, thus accounting for both primary and secondary KIEs, and averaged the results over all possible distributions. The error bars in the CA II curve in Figure 20.12 reflect this averaging. Inspection of Figure 20.12 shows that all the dashed curves are “broken,” indicating that stepwise transfer does not give rise to a linear dependence of ln kðTÞ on the D-fraction. Of the solid curves, some are linear or nearly so (CA II, water trimer, acetic acid – methanol complex), while the others are broken. This indicates that not all concerted transfer processes give rise to a linear plot. Those that do are characterized by a symmetric or nearly symmetric (CA II) tunneling potential, while those that do not are exothermic transfers characterized by strongly asymmetric potentials. Hence, these calculations lend support to the idea that a linear dependence of ln kðTÞ on the D-fraction indicates concerted transfer. These also suggest that this transfer is close to thermoneutral, or, perhaps more generally, that all transferring protons contribute roughly equally to the KIE. A nonlinear dependence, on the other hand, would indicate that either the transfer is stepwise or that it is concerted but strongly exothermic.

V. CONCLUSIONS Multiple proton transfer is intrinsically more difficult than single-proton transfer. If the transfer proceeds stepwise, at least one of the steps will be endothermic, so that the process requires thermal energy and will be blocked at 0 K. In order for the transfer to proceed concertedly, several chemical bonds must be broken simultaneously with the result that the transfer barrier will tend to increase linearly with the number of mobile protons. Thus one expects coherent multiple proton transfer to take place only if there is strong hydrogen bonding, so that the loss of valence bonding in the transition state can be compensated by an increase in hydrogen bonding. The high energetic demands of multiple proton transfer favor tunneling mechanisms in all but strong dielectric media. As a result, multiple proton transfer reactions usually show the characteristics of typical tunneling processes, such as curved Arrhenius plots and substantial isotope effects. However, if the proton conduit is a loose water wire, the isotope effect will be considerably reduced by coupling of the tunneling to vibrations of the wire and may be smaller than the cumulative effect found for classical transfer. To decide whether a given multiple proton transfer reaction proceeds stepwise or concertedly, one generally needs to measure the temperature and isotope dependence of the rate constant. In the previous sections, we have presented several criteria that can be used to discriminate between the mechanisms. Among these the temperature dependence at low temperatures is very easy to apply, since the apparent activation energy of the rate constant will converge to zero for concerted transfer and to a nonzero value for stepwise transfer. However, such data are not always available. The following, more qualitative criteria are useful at higher temperatures: 1. Coherent transfer reactions are unlikely to have apparent activation energies in excess of 10 kcal mol21. 2. Arrhenius plots of concerted reactions will generally show distinct curvature for temperature intervals of 50 K or more, while stepwise reactions tend to show quasilinear plots. 3. H· · ·H and D· · ·D Arrhenius plots will show divergence under these conditions, while the corresponding stepwise plots may appear to be parallel. 4. For double-proton transfer, specific equations that depend on the symmetry of the barrier are derived for the relationship of the HD Arrhenius plot to the HH and DD plots. 5. For hydrogen bonded water wires the logarithm of the rate constant of mixed isotopomers tends to be proportional to the fraction of H replaced by D if the transfer is coherent.

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Isotope Effects in Chemistry and Biology

The analysis will be greatly facilitated if the system is amenable to quantum-chemical calculation. In that case, the number of transition states along the transfer path determines the number of steps, so that a single transition state implies concerted transfer. If the energy of this state is not much higher than the thermal energy, the transfer will be classical; otherwise, the protons will tunnel. The validity of these conclusions depends of course on the level and reliability of the quantum-chemical methods used, a problem that is far from trivial for transition states of proton transfer reactions. For a more complete analysis, including isotope effects, quantum-dynamical calculations are essential. This involves evaluating the vibrational force field and using a dynamics method that includes all relevant vibrational modes in the calculation. Comparison of calculated and observed KIEs is probably the most reliable method of testing the assumed transfer mechanism, since these relative rate constants are less dependent on the potential and on the details of the calculation than absolute rate constants.

ACKNOWLEDGMENTS A. F-R. thanks the Ministerio de Ciencia y Tecnologia for a Ramon y Cajal Research Contract and Project No. BQU2003-01639.

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33 Smedarchina, Z. and Siebrand, W., Theoretical analysis of intramolecular double-hydrogen bond transfer in bridged-ring compounds, J. Mol. Struct., 297, 207– 213, 1993. 34 Madeja, F. and Havenith, M., High resolution spectroscopy of carboxylic acid in gas phase: observation of proton transfer in (DCOOH)2, J. Chem. Phys., 117, 7162– 7168, 2002. 35 Gerritzen, D. and Limbach, H-H., Kinetic isotope effects and tunneling in cyclic double and triple proton transfer between acetic acid and methanol in tetrahydrofuran studied by dynamic 1H and 2H NMR spectroscopy, J. Am. Chem. Soc., 106, 869– 879, 1984. 36 Ferna´ndez-Ramos, A., Smedarchina, Z., and Rodriguez-Otero, J., Double proton transfer in the complex of acetic acid with methanol: theory versus experiment, J. Chem. Phys., 114, 1567, 2001. 37 Smedarchina Z., Siebrand W., Ferna´ndez-Ramos A., Calculation of the tunneling splitting in the zero-point level and CO-stretch fundamental of the formic acid dimer, Chem. Phys. Lett., 395, 339– 345, 2004. 38 Oppenla¨nder, A., Rambaud, C., Trommsdorff, H. P., and Vial, J. C., Translational tunneling of protons in benzoic-acid crystals, Phys. Rev. Lett., 63, 1432– 1435, 1989. 39 Brougham, D. F., Horsewill, A. J., and Jenkinson, R. I., Proton transfer dynamics in the hydrogen bond: a direct measurement of the incoherent tunneling rate by NMR and the quantum-to-classical transition, Chem. Phys. Lett., 272, 69 –74, 1997. 40 Meier, B. H., Graf, F., and Ernst, R. R., Structure and dynamics of intramolecular hydrogen bonds in carboxylic acid dimers: a solid state NMR study, J. Chem. Phys., 76, 767– 774, 1982. 41 (a) Nagaoka, S., Terao, T., Imashiro, F., Saika, A., and Hirota, N., An NMR relaxation study on the proton transfer in the hydrogen bonded carboxilic acid dimers, J. Chem. Phys., 79, 4694– 4702, 1983; (b) Sto¨ckli, A., Meier, B. H., Kreis, R., Meyer, R., and Ernst, R. R., Hydrogen bond dynamics in isotopically substituted benzoic acid dimers, J. Chem. Phys., 93, 1502– 1520, 1990. 42 Horsewill, A. J., Brougham, D. F., Jenkinson, R. I., McGloin, C. J., Trommsdorff, H. P., and Johnson, M. R., The quantum dynamics of proton transfer in the hydrogen bond, Ber. Bunsen.-Ges. Phys. Chem., 102, 317– 325, 1998. 43 Smedarchina, Z., Ferna´ndez-Ramos, A., and Siebrand, W., Tunneling dynamics of double proton transfer in formic acid and benzoic acid dimers, J. Chem. Phys., 122, 134309– 134321, 2005. 44 Brougham, D. F., Caciuffo, R., and Horsewill, A. J., Coordinated proton tunneling in a cyclic network of four hydrogen bonds in the solid state, Nature, 397, 241– 243, 1999. 45 Ferna´ndez-Ramos, A., Smedarchina, Z., and Pichierri, F., Proton tunneling in calix[4]arenes: a theoretical investigation, Chem. Phys. Lett., 343, 627–632, 2001. 46 (a) Borst, D. R., Roscoli, G. R., Pratt, D. W., Florio, G. M., Zwier, T. S., Mu¨ller, A., and Leutwyler, S., Hydrogen bonding and tunneling in the 2-pyridone.2-hydroxypyridine dimer, effect of electronic excitation, Chem. Phys., 283, 341–355, 2002; (b) Roscioli, G. R., Pratt, D. W., Smedarchina, Z., Siebrand, W., and Ferna´ndez-Ramos, A., Proton transfer dynamics via high-resolution spectroscopy in the gas phase and instanton calculations, J. Chem. Phys., 120, 11351– 11354, 2005. 47 (a) McMorrow, D. and Aartsma, T. J., Solvent mediated proton transfer, the roles of solvent structure and dynamics on the excited-state tautomerization of 7-azaindole or alcohol complexes, Chem. Phys. Lett., 125, 581– 585, 1986; (b) Moog, R. S., and Maroncelli, M., 7-azaindole in alcohols: solvation dynamics and proton transfer, J. Phys. Chem., 95, 10359– 10369, 1991; (c) Chapman, C. F., and Maroncelli, M., Excited state tautomerization of 7-azaindole in water, J. Phys. Chem., 96, 8430– 8441, 1992; (d) Chen, Y., Rich, R. L., Gai, F., and Petrich, J. W., Fluorescent species of 7-azaindole and 7-azatryptofan in water, J. Phys. Chem., 97, 1770– 1781, 1993; (e) Mente, S. and Maroncelli, M., Solvation and the excited-state tautomerization of 7-azaindole and 1-azacarbazole: computer simulations in water and alcohol solvents, J. Phys. Chem. A, 102, 3860– 3877, 1998. 48 (a) Nakajima, A., Ono, F., Kihara, Y., Ogawa, A., Matsubara, K., Ishikawa, K., Baba, M., and Kaya, K., Spectroscopic study of hydrogen bonded 7-azaindole clusters, Laser Chem., 15, 167– 183, 1995; (b) Huang, Y., Arnold, S., and Sulkes, M., Spectroscopy and fluorescence lifetimes of jet-cooled 7-azaindole: electronic states and solvent complex geometry, J. Phys. Chem., 100, 4734– 4739, 1996; (c) Nakajima, A., Hirano, M., Hasumi, R., Kaya, K., Watanabe, H., Carter, C. C., Williamson, J. M., and Miller, T. A., High-resolution laser-induced fluorescence spectra of 7-azaindole-water complexes and 7-azaindole dimer, J. Phys. Chem. A, 101, 392– 399, 1997. 49 Folmer, D. E., Wisniewski, E. S., Stairs, J. R., and Castleman, A. W. Jr., Water-assisted proton transfer in the monomer of 7-azaindole, J. Phys. Chem. A, 104, 10545 –10550, 2000.

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50 Siebrand W., Ferna´ndez-Ramos A., Smedarchina Z., unpublished results. 51 Ferna´ndez-Ramos, A., Smedarchina, Z., Siebrand, W., and Zgierski, M. Z., Dynamics of the watercatalyzed phototautomerization of 7-azaindole, J. Chem. Phys., 114, 7518– 7526, 2001. 52 Smedarchina, Z., Siebrand, W., Ferna´ndez-Ramos, A., Gorb, L., and Leszczynski, J., A directdynamics study of proton transfer through water bridges in guanine and 7-azaindole, J. Chem. Phys., 112, 566– 574, 2000. 53 Gorb, L., Podolyan, Y., Leszczynski, J., Siebrand, W., Ferna´ndez-Ramos, A., and Smedarchina, Z., A quantum dynamics study of the prototropic tautomerism of guanine and its contribution to spontaneous point mutations in Escherichia coli, Biopolymers, 61, 77 – 83, 2002. 54 Ferna´ndez-Ramos, A., Smedarchina, Z., Siebrand, W., and Zgierski, M. Z., A direct-dynamics study of the zwitterion-to-neutral interconversion of glycine in aqueous solution, J. Chem. Phys., 113, 9714– 9721, 2000. 55 Zgierski M.Z., Siebrand W., Unpublished results. 56 Silverman D.N., Elder I., Solvent hydrogen isotope effects in catalysis by carbonic anhydrase: proton transfer through intervening water molecules, Chapter…of this volume. 57 (a) Steiner, H., Jonsson, B. H., and Lindskog, S., The catalytic mechanism of carbonic anhydrase, Hydrogen-isotope effects on the kinetic parameters of the human C isoenzyme, Eur. J. Biochem., 59, 253– 259, 1975; (b) Pocker, Y. and Bjorkquist, D. W., Comparative studies of bovine carbonic anhydrase in H2O and D2O, stopped-flow studies of the kinetics of interconversion of CO2 and HCO2 3, Biochemistry, 16, 5698– 5707, 1977. 58 Venkatasubban, K. S. and Silverman, D. N., Carbon dioxide hydration activity of carbonic anhydrase in mixtures of water and deuterium oxide, Biochemistry, 19, 4984– 4989, 1980. 59 Brown, L. S., Needleman, R., and Lanyi, J. K., Origins of deuterium kinetic isotope effects on the proton transfers of the bacteriorhodopsin photocycle, Biochemistry, 39, 938– 946, 2000. 60 Schowen, K. B., Limbach, H-H., Denisov, G. S., and Schowen, R. L., Hydrogen bonds and proton transfer in general-catalytic transition-state stabilization in enzyme catalysis, Biochim. Biophys. Acta., 1458, 43 – 62, 2000, and references therein.

21

Interpretation of Primary Kinetic Isotope Effects for Adiabatic and Nonadiabatic Proton-Transfer Reactions in a Polar Environment Philip M. Kiefer and James T. Hynes

CONTENTS I. II.

Introduction ...................................................................................................................... 549 Adiabatic Proton Transfer................................................................................................ 553 A. Adiabatic Proton-Transfer Free-Energy Relationship............................................. 553 1. General Adiabatic Proton-Transfer Picture ...................................................... 553 2. Adiabatic Proton-Transfer Free-Energy Relationship ...................................... 555 3. Further Analysis of the Intrinsic Barrier. Mass Scaling................................... 557 B. Adiabatic Proton-Transfer KIEs .............................................................................. 558 1. KIE Arrhenius Behavior ................................................................................... 559 2. KIE Magnitude and Variation with Reaction Asymmetry............................... 559 3. Swain –Schaad Relationship ............................................................................. 560 C. Further Discussion of Nontunneling KIEs .............................................................. 561 III. Nonadiabatic ‘Tunneling’ Proton Transfer...................................................................... 562 A. General Nonadiabatic Proton-Transfer Perspective and Rate Constant ................. 562 B. Nonadiabatic Proton-Transfer KIEs ........................................................................ 567 1. KIE Magnitude and Variation with Reaction Asymmetry............................... 567 2. Temperature Behavior....................................................................................... 568 3. Swain –Schaad Relationship ............................................................................. 571 IV. Concluding Remarks........................................................................................................ 572 Acknowledgments ........................................................................................................................ 573 References..................................................................................................................................... 573

I. INTRODUCTION In this contribution, we give an overview of some of the theoretical developments for primary kinetic isotope effects (KIEs) for proton-transfer (PT) reactions, primarily focusing on the efforts in the Hynes group.1 – 7 PT is of obvious importance in both chemistry and biology,8 and isotopic substitution — with comparison of the rates of transfer of a proton, deuteron, and triton — is a widely used and valuable experimental technique to probe PT reactions aspects. Information from KIEs has allowed identification of the rate-limiting step in complex reactions, and provided insight into the microscopic nature of the transition state (TS).8 – 12 In the present chapter, we present an 549

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Isotope Effects in Chemistry and Biology

account of a theory of KIEs, which differs fundamentally from ‘standard’ descriptions, for what we term adiabatic and nonadiabatic PT reactions, which as described within are characterized by a nontunneling and tunneling, respectively, quantum behavior of the proton motion.13 Widely used ‘standard’ descriptions for the origin of KIEs can be regarded to have both classical nontunneling and tunneling components.8,12,16 – 20 The standard nontunneling view for KIEs traces back to Westheimer and Melander (W –M),17 and is now briefly reviewed. In its simplest version, this picture uses a linear three-center molecular system for PT, illustrated here for an acid – base reaction within a hydrogen-bonded (H-bonded) complex, e.g., AH· · ·B ) A2 · · ·HBþ

ð21:1Þ

(Actually, most literature discussions of the standard description are more appropriate for hydrogen atom transfers; here we simply make the transcription to PT reactions of type Equation 21.1). The reaction potential surface is a function of two coordinates, the A – H proton distance and the distance between the two heavy moieties, A· · ·B. Along the minimum energy path, the reaction begins at large heavy atom separation with the A – H proton distance constant, proceeds through a TS A· · ·H· · ·B, and goes onto products with large A – B separation and constant H – B distance to produce A2· · ·HBþ. Using TS theory, the KIE arises from the exponentiated activation energies.8,12,16,17 For H versus D transfer, the KIE is given by kH =kD < exp½2ðDG‡H 2 DG‡D Þ=RT; DG‡H 2 DG‡D < ZPE‡H 2 ZPERH 2 ZPE‡D þ ZPERD ð21:2Þ As is of course well known, the KIEs originate, in this framework, from isotopic zero-point energy (ZPE) differences between the reactant and TS.8,12,16,17,21 The relevant reactant region ZPE is just the ZPE of the A – H vibration in the reactant A –H· · ·B complex, the motion transverse to the reaction coordinate in this region. The reactant ZPE is larger for H than for D, ZPERH . ZPERD, due to the lower reactant AH stretch vibration frequency for the more massive D. For a thermodynamically symmetric reaction (DGRXN ¼ 0, a very important reference situation), the reaction path through the TS consists solely of the proton’s classical motion over the barrier, so that there is no proton ZPE associated with this motion at the TS. Rather, the TS ZPE is associated with the transverse motion at the TS (just as for the reactant), which in the collinear model is a symmetric stretch, the heavy particle A –B vibration. Hence, at the TS, ZPE‡H ¼ ZPE‡D for such a symmetric reaction. The net result is the complete “loss” of the ZPE for the proton stretching mode on going from reactant to TS for a symmetric reaction. Typical reactant proton stretch frequencies (vCH , 3000 cm21) and a simple mass correlation between ZPEs pffiffi (i.e., ZPED < ZPEH = 2) give an H vs. D KIE of about 7 at room temperature.8,12,16,17 For an asymmetric reaction, the reaction coordinate at the TS in the traditional view includes both proton and heavy particle classical motions: consistent with the Hammond postulate,18 the TS becomes more geometrically similar to the product as the reaction becomes more endothermic, and more similar to the reactant as it becomes more exothermic. Thus the transverse vibration at the TS (whose ZPE is relevant for the rate) involves the proton motion more and more, and in either limit approaches the bound proton stretch vibration of the product BH or the reactant AH. This effect decreases the KIE, resulting in a kH =kD vs. DGRXN trend which is maximal at DGRXN ¼ 0 and drops off as the reaction becomes more endo- or exothermic. There are four common experimental observations which are consistent with this standard picture for nontunneling PT KIEs: (i) the Arrhenius temperature dependence of the KIE (as well as of the individual isotope rate constants); (ii) the KIE-DGRXN behavior described above; (iii) the KIE range is , 2– 10; and (iv) the wide applicability of the Swain – Schaad relationship16,22 connecting ratios of KIEs (e.g., kH =kT ¼ ðkD =kT Þ3:3 ), which follows directly from Equation 21.2 and its variants. These observations have done much to maintain the standard picture as a widespread perspective for KIEs.

Interpretation of Primary Kinetic Isotope Effects

551

The standard picture for classical proton motion over the barrier at the TS just described is sometimes supplemented with a quantum contribution via a tunneling correction.16,19,20,23,24 Addition of tunneling corrections to the standard PT rate will obviously affect the KIE, including its reaction asymmetry dependence.16,20 Indeed, it has been argued that variation of the tunneling contribution versus reaction asymmetry is primarily responsible for the broad range in magnitude of observed KIE versus reaction asymmetry plots, instead of the variation of ZPE at the TS described above.20 Various departures from the experimental observations listed in the previous paragraph, such as non-Arrhenius rate behavior or KIEs much in excess of 10, typically are taken as indicating tunneling.8,11,12,16,20,27 Despite the success of the standard picture described above, one can argue that a different picture would be more plausible. First, the standard description has a certain logical inconsistency in the TS description. In the symmetric case, proton motion is viewed as completely classical over the proton barrier. For any finite reaction asymmetry, however, the quantum character of the proton as a bound quantum vibration becomes extremely important, since it is that character that influences the frequency, and thus the ZPE of the transverse TS motion. It seems difficult to maintain that proton motion within an H-bond can be both classical and quantum. Second, the standard picture presented above makes no reference to the solvent (while we focus here on the case of a solvent environment, very similar ideas will apply to more general polar environments such as enzyme active sites). To the degree that the solvent is included in standard descriptions, it is imagined to alter the rate via a differential equilibrium solvation of the TS and the reactant, again all within the standard framework recounted above.24,28 However, the equilibrium solvation assumption — which requires, e.g., that the solvent motion is fast compared to the relevant motion of the reacting solutes in the TS region — is not at all plausible in the case of high-frequency quantum proton motion; indeed, the opposite situation is more appropriate: the solvent is generally slow compared to the proton motion.1,2,29 – 31 In the alternate, nontraditional picture of PT reactions1 – 7,29 – 31 employed within, the reaction is driven by configurational changes in the surrounding polar environment, a feature of much modern work on PT reactions,1 – 7,29 – 31 and the reaction activation free energy is largely determined by the environmental reorganization. In this picture, the rapidly vibrating proton follows the environment’s slower rearrangement, thereby producing a perspective in terms of the instantaneous proton potential for different environmental arrangements. (This statement needs to be expanded upon in the two different regimes of PT described below.) Figure 21.1 displays key features for a model overall symmetric PT reaction in a linear H-bonded complex. Proton potential curves versus the proton coordinate are shown for solvent configurations appropriate to that of the reactant pair, the TS, and the product pair. These different states of solvation — or more simply the solvent’s nuclear electrical polarization state — distort the ‡

R

P

G

A-H

A-H

A-H

FIGURE 21.1 Free-energy curves versus proton position at the reactant R, transition state ‡, and product P solvent configurations for a symmetric reaction. For both R and P, the ground state proton vibrational energy level (solid line) is indicated. For the transition state, the ground-state proton vibrational energy level is indicated for a small (solid line) and a large H-bond separation (dotted line), which correspond to adiabatic and nonadiabatic PT, respectively.

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Isotope Effects in Chemistry and Biology

potential from being initially asymmetric favoring the proton residing on the acid through an intermediate situation where a proton symmetric double well is established, and on to an asymmetric potential now favoring the proton residing on the base. The solvent motion is critical due to the strong coupling of the reacting pair’s evolving charge distribution to the polar solvent polarization field. Furthermore, the high-frequency quantum proton vibration adiabatically adjusts to the reorganizing solvent in the reactant and product regions; the reaction coordinate is a solvent coordinate, rather than the proton coordinate, and there is a free-energy change up to the TS activation free energy as the solvent rearranges. For each of the three proton potentials in Figure 21.1, the quantized ground proton vibrational energy, i.e., the ZPE, is indicated. For the TS solvent configuration, there are two possible cases for the zero-point level: if it is above the proton barrier, the system is in what we term the quantum adiabatic PT limit, while if instead the level is below the barrier top, the reaction would involve proton tunneling,32 which we term the quantum nonadiabatic limit. In either case, the proton motion is a bound quantum vibration; there is no classical barrier crossing of the proton, in contrast to a conventional TS theory for the standard description. The distinction between the two regimes, whether the ground proton vibrational level is below or above the barrier in the proton coordinate, is critically determined by another important coordinate Q, the acid– base (H-bond) A – B separation. Figure 21.2 displays proton potentials and proton vibrational levels for the environment’s TS configuration for three different proton donor – proton acceptor (H-bond) distances Q. Starting with small Q, Figure 21.2a has the proton ground and first excited vibrational levels above the proton barrier after solvent rearrangement; this is the case for adiabatic PT. As Q is increased, the proton barrier increases. In Figure 21.2b, the proton levels are now below the barrier. For further increases in Q, both levels are still below the barrier with a decreased level splitting (e.g., Figure 21.2c). In both Figure 21.2b and c, the levels are split by twice the resonance coupling C, which determines the tunneling probability. This coupling rapidly decreases as the H-bond coordinate Q increases, as the proton potential barrier for tunneling is higher and broader. In the latter two cases, the PT is in the nonadiabatic tunneling limit and the rate will have a prefactor involving the tunneling probability. The solvent generally does not adiabatically follow the proton in its actual tunneling motion.33 In succeeding sections, we recount the essential features of our arguments3 – 5 that all four of the experimental observations mentioned above can also follow directly from the quite different picture of PT reactions in a polar environment just outlined. A further emphasis is that generally the combination of all of these KIE probes is necessary to characterize whether the PT is adiabatic or nonadiabatic. Section II describes the adiabatic PT picture as well as the resulting KIE behaviors, while Section III presents the nonadiabatic PT picture and its KIE behaviors. Concluding remarks

G } 2C

(a)

AH

(b)

AH

} 2C

(c)

AH

FIGURE 21.2 Variation of proton potentials at the solvent reaction transition state configuration with increasing AB separation, going from (a) to (c). Both the ground and first excited proton adiabatic vibrational levels are indicated.

Interpretation of Primary Kinetic Isotope Effects

553

are offered in Section IV. The reader is referred to Refs. 1 – 5 for detailed discussions of the issues and results reviewed in the present contribution.

II. ADIABATIC PROTON TRANSFER We focus first on the adiabatic (quantum nontunneling) PT regime: the proton vibration adiabatically follows solvent fluctuations as the reaction proceeds from reactant to product, and at the TS the proton vibrational level is above the barrier in the proton coordinate. The reaction coordinate is a solvent coordinate, rather than the proton coordinate, and there is a free-energy change up to the TS activation free-energy as the solvent rearranges, as shown in Figure 21.3. The adiabatic PT regime picture has been supported in electronic structure calculations and simulation studies including acid ionizations in solution2,30,34 and elsewhere.2,7,30,31 (The abovethe-barrier proton zero point level at the reaction TS corresponds to what has been termed in the enzyme reaction literature a “low barrier H-bond”35 situation.) The general picture for adiabatic PT, including a description of the reaction free-energy barrier and its free-energy relationship (FER) is reviewed first, then the resulting KIE behaviors are presented. Comparison with the standard picture concludes this section.

A. ADIABATIC P ROTON-T RANSFER F REE- E NERGY R ELATIONSHIP 1. General Adiabatic Proton-Transfer Picture Our general path to the description of KIEs for adiabatic PT goes through the activation free energy – reaction free energy relation for this reaction class, now discussed. The total free energy of the reacting system versus the solvent reaction coordinate can be usefully decomposed into two basic contributions,3,7 as shown in Figure 21.3: G ¼ Gmin þ ZPE

ð21:3Þ

Reaction Free Energy

These are, respectively, a ‘bare’ free energy, Gmin, corresponding to the situation where the proton is located at its classical minimum position for any given solvent coordinate value, and the vibrational ZPE of the proton, measured from the latter potential energy minimum. The ZPE decreases as the TS in the solvent coordinate is approached, since the proton potential is becoming more symmetric and the proton is delocalized in a larger region. The decomposition Equation 21.3

TS R

P

ZPE Gmin

Solvent Reaction Coordinate

FIGURE 21.3 Free-energy curve for a symmetric adiabatic PT system4 with the proton quantized in its vibrational ground state versus solvent reaction coordinate (solid line). The free energy at the minimum of the proton potential along the solvent coordinate Gmin (dotted line) is also shown (see Figure 21.1).

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Isotope Effects in Chemistry and Biology

proves useful for analyzing KIEs because the isotope-dependent ZPE component of G is isolated from the isotope-independent classical free energy curve Gmin. This description (cf. Figure 21.1 and Figure 21.3) ignored for simplicity the influence of the H-bond coordinate Q as well as other nuclear vibrations, including any bending contribution. The H-bond coordinate has an important impact on the adiabatic PT regime.2,3,30,31,36 Namely, there is a significant H-bond coordinate –solvent coordinate coupling,3,34 i.e., Q compresses and the H-bond frequency increases as the system goes from the reactant complex to the TS in the solvent reaction coordinate. The additional effects from this and other modes can be added by including their zeropoint motion. We focus the discussion here solely on the key modes, the H-bond and proton stretch vibrational modes, quantizing them simultaneously for each solvent configuration, with a resulting one-dimensional free-energy curve similar to that in Figure 21.3. The PT rate constant expression including the quantized H-bond vibration is then the thermal average of the PT rate constant for each two-dimensional (2-D) state i(2c,3b), each having a TS theory form37 with the solvent coordinate as the reaction coordinate: ki ¼ ðvSi =2pÞexpð2DG‡i =RTÞ

ð21:4Þ

Here vSi is the reactant region solvent reaction coordinate frequency, and DG‡i is the free-energy barrier for PT for the ith 2-D vibrational state. The free-energy barrier is the barrier in the solvent coordinate, which also includes a difference in the ZPEs of the proton and the H-bond vibration between the reactant and the TS. However, since the proton stretch frequency (, 3000 cm21) is larger than the H-bond frequency (usually much less than 1000 cm21), the thermally occupied states primarily correspond to excitation of the H-bond vibration. Since the H-bond vibration frequency is larger at the TS than in the reactant and product states3,4 due to the H-bond – solvent coupling described above, the reaction barriers for excited H-bond vibrational states are larger than for the ground state i ¼ 0(3b). Consequently, the PT and DT rates are dominated by dynamics in the ground H-bond vibrational state,3 and hereafter we deal only with the i ¼ 0 case.3,4 Figure 21.4 displays the H and D free-energy curves for a symmetric and an asymmetric reaction for a model OH· · ·O system.3,4 Again, the total free energy G has a decomposition of the form in Equation 21.3. The free energy Gmin evaluated at the classical minima for both the proton and H-bond coordinates at a given solvent coordinate value is indicated as the lower curve in Figure 21.4a and b. The ZPE contributions for both H and D from Figure 21.4a and b are displayed as Figure 21.4c, and contain both the ZPE of the H or D vibration and that of the H-bond mode. The solvent coordinate DE in this picture is an alter ego of the solvent polarization, related to a certain energy gap defined such that for a thermodynamically symmetric PT reaction, it has a value of zero at the solvent TS where the proton potential is symmetric.38 The reaction barrier increases starting from an exothermic case (Figure 21.4b) to an endothermic case (reverse of Figure 21.4b). From Figure 21.4a and b, one can see that both the reactant well frequency vs and barrier height DG ‡ are isotope dependent. However, the contribution of vs to the KIE magnitude is minimal since it is largely governed by the solvent,4 and one can thus focus on the isotope dependence of the reaction activation free-energy barrier height. Figure 21.4a and b also indicate that the TS position DE ‡ along the solvent coordinate shifts with reaction asymmetry. The origin of this is that the addition of the ZPE to an asymmetric Gmin (cf. Figure 21.4b) shifts the maximum of G away from the maximum of Gmin at DE ¼ 0, in the direction consistent with the Hammond postulate,18 e.g., later for endothermic reactions DGRXN . 0. This indicates that the ZPE contribution at DE ‡ to the free-energy barrier DG ‡ in the solvent coordinate will increase with increasing reaction asymmetry, a crucial qualitative characteristic.3 Also visible in Figure 21.4b is the isotope dependence of the shift DE ‡ and the associated increase of ZPE at DE ‡ with increasing reaction asymmetry; the latter leads to a KIE reduction, since the ZPE contribution at DE ‡ will become more and more similar to that of the reactant. Here one can see that the variation in ZPE along the reaction coordinate and its isotopic

Interpretation of Primary Kinetic Isotope Effects

555

−116

G (kcal/mol)

H D

−124 −128

(a)

Gmin ∆E R

∆E ‡

∆E R

G (kcal/mol)

−116 H

−120

D −124 −128

(b)

Gmin ∆E R

∆E ‡

∆E R

5

ZPE (kcal/mol)

(c)

−120

H

4 3

D 2 1 −60 −40 −20 0 20 ∆E (kcal/mol)

40

60

FIGURE 21.4 Ground-state free-energy curves for adiabatic PT (solid lines H and dashed lines D) with both the proton/deuteron and H-bond vibrations quantized: (a) symmetric reaction and (b) exothermic reaction. Dotted lines show the free energy curves Gmin excluding the ZPE. (c) The ZPE for the proton (solid line) and deuteron (dotted line), including H-bond vibration, vs. DE. The dashed curves in (a) and (b) plus the ZPE in (c) give the full free energy G, the solid curves in (a) and (b). DE R, DE P, and DE ‡ denote the reactant, product, and TS solvent coordinate values, respectively.

difference plays a significant role in the reaction free-energy barrier variation, and hence the KIE as well. 2. Adiabatic Proton-Transfer Free-Energy Relationship Expressions for individual components of a DG ‡ vs. DGRXN free-energy relation (FER) were derived for adiabatic PT (cf. Equation 1.3 of Ref. 3b), DG‡ ¼ DG‡o þ

1 1 DGRXN þ a0o DG2RXN 2 2

ð21:5Þ

and proves useful for the KIE discussion. DG ‡o is the ‘intrinsic’ reaction barrier DG‡o ¼ DG ‡(DGRXN ¼ 0), and ao0 is the Brønsted coefficient slope evaluated at DGRXN ¼ 0.39 The linear term coefficient is the Brønsted coefficient for the symmetric reaction, ao ¼ 1/2. Various terms in this FER are described first before discussing the KIE.

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Isotope Effects in Chemistry and Biology

The intrinsic free-energy barrier DG‡o is composed of a contribution DG‡m,o due to a certain solvent reorganization and the difference Z‡o 2 ZRo ¼ DZPE‡o in the ZPEs of the protonic and H-bond coordinates between the reactant and the TS for the symmetric reaction (cf. Equation 1.4 of Ref. 3b) DG‡o ¼ DG‡m;o þ DZPE‡o

ð21:6Þ

The Brønsted coefficient

a¼

› DG‡ › DGRXN

ð21:7Þ

can be quantitatively described in the adiabatic PT picture by any of three differences between the TS and the reactant of the separation in the solvent coordinate, of the electronic structure, and of the quantum-averaged nuclear structure3,4

a¼

DE‡ 2 DER kc2P l‡ 2 kc2P lR kq 2 Q=2l‡ 2 kq 2 Q=2lR ¼ ¼ DEP 2 DER kc2P lP 2 kc2P lR kq 2 Q=2lP 2 kq 2 Q=2lR

ð21:8Þ

Here DE ‡ 2 DE R is the solvent reaction coordinate distance between the TS and reactant, and DE P 2 DE R is the corresponding distance between the product and reactant. kc2P l is the quantum average over the proton and H-bond vibrations of the limiting product contribution to the electronic structure c2P. (A two-state valence bond model is used3,6 to describe the PT system electronic structure, e.g., the limiting ionic contribution to the electronic structure in Equation 21.1.) The electronic structure for each critical point (c ¼ R, P, and ‡) is evaluated at the respective critical point position along the reaction coordinate DEc (DEc ¼ DER, DEP, or DE ‡). The structural element kq 2 Q/2l is the quantum-averaged proton distance (over both proton and H-bond coordinates) from the H-bond’s center. As noted above, the TS structure (and TS position along the reaction coordinate) for a symmetric reaction is halfway between that of the reactant and product, and ao ¼ 1/2, independent of isotope. The Brønsted coefficient a has often been used to describe TS structure via the Hammond postulate18 or the Evans – Polanyi relation,40 where a is the relative TS structure along the reaction coordinate, usually a bond order or bond length. The important point is that, although adiabatic PT has a quite different environmental coordinate as the reaction coordinate, Equation 21.8 is consistent with that general picture, with a proper recognition that quantum averages are involved. The variation of the TS structure with reaction asymmetry is described by the slope of the Brønsted coefficient, ao0 , the derivative of Equation 21.8 with respect to DGRXN evaluated for the symmetric reaction. In this manner, a for the FER in Equation 21.5 is linearly related to the reaction asymmetry 1 ð21:9Þ a ¼ þ a0o DGRXN 2 Expressions for ao0 have been explicitly derived in Ref. 3. One convenient expression is in terms of force constants of the free energy along the reaction coordinate, kR and k ‡, at the reactant and TS positions, and the reaction coordinate distance between the reactant and product DDE ¼ DE P 2 DE R (cf. Equation 1.5 of Ref. 3b)

a0o ¼

1 DDE2

1 1 þ kR k‡

ð21:10Þ

As seen from Figure 21.4, a key component of the TS structure variation is reflected in the variation of the ZPE along the reaction coordinate. This feature is incorporated in Equation 21.10 by the relation of the force constants to this ZPE variation.3,4

Interpretation of Primary Kinetic Isotope Effects

557

The components DG‡o and ao0 of the FER are isotope dependent. The isotope dependence of the intrinsic free-energy barrier DG‡o given by Equation 21.6 is, as is apparent in Figure 21.4a, due solely to the difference in the H and D ZPEs ‡ R ‡ R DG‡oD 2 DG‡oH ¼ ZoD 2 ZoD 2 ZoH þ ZoH ¼ DZPE‡oD 2 DZPE‡oH

ð21:11Þ

Recall that the ZPE contains both that of isotope L and that of the H-bond vibrational mode. The latter’s contribution to Equation 21.6 is actually a positive contribution (e.g., þ 0.4 kcal/mol for the model system used to generate Figure 21.4) to the ZPE difference DZPE‡o, because the H-bond mode in the TS has a higher frequency than in the reactant. This difference is, however, smaller in magnitude than the negative ZPE difference associated with the proton vibrational mode (2 2.5 kcal/mol in Figure 21.4) due to a larger proton vibrational frequency in the reactant compared with that at the TS. Thus, DZPE‡o is overall negative (e.g., 2 2.1 kcal/mol from Figure 21.4c). Furthermore, this ZPE difference decreases as the mass of the transferring particle L pffiffiffiffi increases, as one would expect from a ZPE / 1= mL mass dependence. The ZPE mass dependence will be very significant for the adiabatic PT KIEs. The isotope dependence of the Brønsted slope ao0 is most conveniently discussed in terms of the derivative of the expression involving force constants Equation 21.10. These force constants certainly depend on the variation of the ZPE along the solvent reaction coordinate via Equation 21.3. Accordingly, ao0 can be cast in terms of these slopes plus the variation in the ZPE value at the reactant and TS positions with reaction asymmetry4

a0o

1 ›ðDE‡ 2 DER Þ 1 1 › ZPE‡ ¼ ¼ › DGRXN DDE DDE a‡ ›DGRXN

1 › ZPER 2 a › DGRXN o

# ð21:12Þ

where a and a ‡ are the first derivatives of the ZPE in those two regions, respectively. The isotopedependent difference of ao0 is thus

a0oH 2 a0oD ¼

" 1 1 › ZPE‡ DDE a‡ ›DGRXN

› ZPE‡ 2 o;H › DGRXN

! o;D

1 › ZPERH 1 › ZPERD 2 H 2 D a › DGRXN a › DGRXN

!# ð21:13Þ

now explicitly described by the isotope-dependent ZPE variation at the reactant and TS with reaction asymmetry. (a ‡ is isotope independent, a property of the discontinuity at DE ¼ 0.3a) Since the ZPE variation is largest in the TS region, Equation 21.13’s first term is the most significant, and 0 0 thus, the essential point is that the isotope difference aoH 2 aoD is approximately proportional to ‡ the difference in the rate of increase of ZPE with increasing reaction asymmetry between H and D. 3. Further Analysis of the Intrinsic Barrier. Mass Scaling It is noteworthy that the ZPE‡ versus DGRXN behavior just described parallels the standard view in a general way: the proton ZPE‡ is minimal for a symmetric reaction and increases with reaction asymmetry. Again we stress that the cause of the ZPE‡ increase for the present and standard views is quite different: the reaction coordinates and physical picture for PT are completely different. While the positions of the reactant and product states, as seen in Figure 21.3, also shift upon addition of the ZPE, this is not a major effect: the change for the reactant ZPE going from a symmetric to an asymmetric reaction is smaller than the corresponding ZPE‡ change, because the ZPE variation is largest near DE ¼ 0. The important net result is that the difference ZPER 2 ZPE‡ decreases as the reaction becomes more asymmetric. Further analysis of the intrinsic barrier DG‡o’s isotope dependence is useful for the KIE discussion. The intrinsic barrier’s isotope dependence depends only on the difference in ZPEs

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Isotope Effects in Chemistry and Biology

(cf. Equation 21.11). DG‡oL2 2 DG‡oL1 ¼ DZPE‡oL2 2 DZPE‡oL1

ð21:14Þ

This illustrates a key common point of connection between the present and standard perspectives: in both cases, the difference in intrinsic barrier heights is related to the difference in a ZPE between the reactant and TS between both isotopes, resulting in a KIE which is maximal for DGRXN ¼ 0 and falls off with increasing asymmetry, as the intrinsic barrier plays a diminishing role in Equation 21.5. pffiffiffi In standard treatments, the isotope mass scaling for the L contribution is ZPE / 1= m; which assumes harmonic potentials. The Figure 21.1 proton potentials are not however harmonic, especially at the TS, where there is a double well, and thus one has no obvious reason to expect a priori that the simple ZPE mass scaling holds. However, it has been shown in Ref. 4 that this dependence in fact holds quite well numerically. In particular the relation which follows from pffiffiffiffi assuming that all ZPEs scale according to ZPEL / 1= mL ; i.e., via Equation 21.14 is DG‡oL2

2

DG‡oL1

¼

DZPE‡oL2

0sffiffiffiffiffiffi sffiffiffiffiffiffi1 sffiffiffiffiffiffi! mL2 mH A ‡ @ mH 2 12 ¼ DZPEoH mL2 m L1 mL1

ð21:15Þ

where all differences in DG‡o have been scaled to the ZPE difference DZPEo‡ for H in the last member, holds to better than 10%. This close agreement highlights an important point: the adiabatic PT picture also generates, from a numerical viewpoint, the mass scaling of standard KIE theory. This is especially important for the Swain – Schaad relations, which are entirely dependent on this mass scaling,16,22,27 discussed in Section II.B.3.

B. ADIABATIC P ROTON-T RANSFER KIEs The KIE for adiabatic PT is the ratio of individual rate constants, where each these is of the form in Equation 21.4, e.g., H versus D transfer kH < expð2ðDG‡H 2 DG‡D Þ=RTÞ kD

ð21:16Þ

Here, the reactant reaction coordinate frequencies vH,D in Equation 21.4 have been assumed equal S as their effect on KIE is minimal.4 From the FER analysis in Section II.A Equation 21.5, the explicit form for the KIE dependence on DGRXN is kH ¼ expð2ðDG‡oH 2 DG‡oD Þ=RTÞexpð2ða0oH 2 a0oD ÞDG2RXN =2RTÞ kD

ð21:17Þ

Further, an equivalent form re-expresses the first part of this in terms of the KIE for the symmetric reaction: kH k ¼ Ho expð2ða0oH 2 a0oD ÞDG2RXN =2RTÞ kD kDo

ð21:18Þ

The explicit form of these equations already suggests that the KIE has an Arrhenius temperature dependence and is maximal for the symmetric reaction DGRXN ¼ 0. These points are now briefly discussed as well as the KIE magnitude and Swain –Schaad behavior.

Interpretation of Primary Kinetic Isotope Effects

559

1. KIE Arrhenius Behavior The Arrhenius form for the adiabatic PT KIE in Equation 21.16 to Equation 21.18 is consistent with the first set of experimental results (i) stated in the Introduction, and the general form for the KIE is identical to that of the standard picture (i.e., the adiabatic PT Equation 21.16 is similar to the standard Equation 21.2), despite significant differences in ingredients between the present and standard pictures. The adiabatic PT rate constant has its temperature dependence governed by a temperature-independent DG ‡. While it is true that additional temperature dependence is in principle present in both the prefactor and DG ‡ of the above KIE expressions, these effects are negligible for highly polar solvents.41 2. KIE Magnitude and Variation with Reaction Asymmetry The KIE behavior versus reaction asymmetry in the adiabatic PT perspective follows directly4 from insertion of the isotopic difference between the FER curves described in Equation 21.16 to Equation 21.18. The general feature that the KIE is maximal for DGRXN ¼ 0 follows from a Brønsted coefficient for a symmetric reaction that is isotope independent, ao ¼ 1/2, which reflects the symmetric nature of the electronic structure of the reacting pair at the TS (cf., Equation 21.8).42 The decrease from the maximum, characterized by a gaussian fall-off with increasing reaction 0 0 . aoD , discussed in Section II.A. asymmetry, is due to the isotope dependence aoH These points can be made concrete via Figure 21.5, which displays the H versus D KIE (T ¼ 300 K) for the PT system in Figure 21.4. The calculated KIE is maximum at DGRXN ¼ 0 and drops off symmetrically as the reaction asymmetry is increased. The maximum KIE for the symmetric reaction and the KIE magnitude throughout the whole range for adiabatic PT are both consistent with experimental observations, (ii) and (iii), respectively, of the Introduction. The origin of this last aspect in the adiabatic PT picture is as follows. The intrinsic KIE magnitude in the adiabatic PT view is directly related to the isotopic difference TS-R ZPE difference DZPE‡o ¼ Z‡o 2 ZRo (see Equation 21.11 and Equation 21.14), whose special feature is the presence of the ZPE for the bound proton vibration at the solvent coordinate TS. Together with the mass dependence of DZPE‡o following from the mass-scaling of ZPEs discussed in Section II.A.3, the maximum KIE magnitude will automatically fall in the same general range as in the standard view. Further, the KIE will fall off due to the increase in TS ZPE with increasing reaction asymmetry, also similar to the standard view.

3

kH/ kD

2.5 2 1.5 1

−8 −6 −4 −2

0

2

4

6

8

DGRXN (kcal/mol)

FIGURE 21.5 KIE kH/kD versus DGRXN (T ¼ 300 K) for adiabatic PT system in Figure 21.3.

560

Isotope Effects in Chemistry and Biology

3. Swain – Schaad Relationship The Swain – Schaad relationship has been an important experimental probe for PT reaction KIEs.11,22,27 We have used4 one of its forms for discussion lnðkH =kT Þ ¼ 3:3 lnðkD =kT Þ

ð21:19Þ

pffiffiffi which assumes the ZPE mass correlation discussed in Section II.A.3 ZPE / 1= m to relate the H, D, and T ZPEs in Equation 21.2. Figure 21.6 displays the calculated adiabatic PT ln(kH/kT)/ ln(kD/kT) versus reaction asymmetry for the same PT systems as in Figure 21.5, and shows little variation from Equation 21.19. Thus, conventional Swain – Schaad behavior also follows from the adiabatic PT picture. We now recount the reasons for this.4 From the DGRXN-dependent form in Equation 21.17 for the KIE, the ratio of natural logarithms needed for the Swain – Schaad relation in Equation 21.19 can be written as ln

kH k =ln D kT kT

¼

DZPE‡oT 2 DZPE‡oH 2 DG2RXN ða0oH 2 a0oT Þ=2 DZPE‡oT 2 DZPE‡oD 2 DG2RXN ða0oD 2 a0oT Þ=2

ð21:20Þ

ln(kH/kT)/ln(kD /kT)

A first significant point is that the adiabatic PT form in Equation 21.20 has the same important feature as the standard picture, via Equation 21.2: the Swain – Schaad relation is independent of temperature. To examine more closely the magnitude of the Swain– Schaad slope, we examined4 first the symmetric case DGRXN ¼ 0, for which the adiabatic PT Equation 21.20 shows that the magnitude is related solely to the reactant and TS ZPE difference. In Section II.A.3, these ZPE differences were shown to obey the same mass scaling used to derive the Swain –Schaad relations (cf. Equation 21.15); hence the maximum of the Figure 21.6 plot is close to the traditionally expected value. However, Figure 21.6 also shows that there is a variation with reaction asymmetry, in the adiabatic PT perspective, of the Swain – Schaad slope. This can be understood by an examination4 of asymmetric reactions, for which the reactant, product, and TS positions differ between H, D, and T, because the variation of ZPE for each isotope along the reaction coordinate DE is different (see Figure 21.4). Consequently, the critical point ZPEs are not those of the same proton potentials for a given DGRXN. Thus, the ratio ln(kH/kT)/ln(kD/kT) should not be constant throughout the entire DGRXN range. But the net effect of these shifts is minimal, as illustrated in Figure 21.6, and thus, close adherence to the Swain –Schaad relationship occurs in the adiabatic PT picture, largely due to the mass scaling described above.

3.5 3.3 3 2.5 2 1.5 1

−8

−4 0 4 DGRXN (kcal/mol)

8

FIGURE 21.6 Swain – Schaad slope ln(kH/kT)/ln(kD/kT) versus reaction asymmetry calculated for PT system in Figure 21.5.

Interpretation of Primary Kinetic Isotope Effects

561

C. FURTHER D ISCUSSION OF N ONTUNNELING KIEs We have already repeatedly emphasized several important fundamental distinctions between the adiabatic PT and the standard view. Despite these distinct differences in physical perspective between adiabatic PT and the standard Westheimer – Melander (W – M) picture, we have emphasized4 that a remarkable general similarity exists between the two perspectives. For adiabatic PT, the symmetric reaction KIE depends on the difference in magnitude of the TSreactant difference DZPE‡o, and the KIE variation with reaction asymmetry is due to the variation of TS ZPE (and structure). These two points, in fact, are shared with the W – M picture (cf. Equation 21.2). In what follows, the numerical and physical differences between the two perspectives are further enumerated.4 The adiabatic PT maximum KIE in Figure 21.5 is in the range of KIEs commonly expected in the standard W – M picture, item (iii), but it is somewhat smaller than the higher KIEs 6 –10 that one would expect with the standard view. From Equation 21.15 and Equation 21.16, the maximum H/D KIE is that of the symmetric reaction qffiffiffi kHo ¼ exp½2ðDG‡oH 2 DG‡oD Þ=RT ¼ exp 2DZPE‡oH 1 2 12 =RT kDo

ð21:21Þ

where the second line follows from the mass scaling of the ZPEs. Equation 21.21 is also used8,12,16 as an estimate for the KIE in the standard W – M picture (cf. Equation 21.2). The different symmetric reaction KIE limits for the adiabatic PT and W –M pictures is entirely due to their different views of the TS reaction and transverse coordinates: for a symmetric reaction, there is always a finite proton TS ZPE contribution for adiabatic PT, whereas the proton TS ZPE is zero ‡ ðZoH ¼ 0Þ in the W –M description. The maximum KIE is thus always smaller in the adiabatic PT view. A further analysis can be given4 which requires attention to the magnitude of H stretch frequencies. In an H-bond with sufficient strength, the A –H stretch frequency is significantly red-shifted compared to a ‘free’ OH stretch frequency , 3600 cm21,43 e.g., a frequency vOH of ˚ .43 (This is a modest H-bond; stronger , 3200 cm21 for an O· · ·O H-bond distance of , 2.7A 21 H-bonds will have vOH , 3200 cm .) PT in a weaker O· · ·O H-bond with a larger equilibrium separation will most likely involve tunneling.3,36) With a reactant frequency vOH ¼ 3200 cm21, the W – M picture gives kH/kD < 10 at 300 K. Turning to other acids, a similar red shift is observed for N – H vibrations in H-bonds with vNH , 3000 cm21.43 Carbon acids will, of course, have smaller red shifts, but the C –H stretch frequency itself (, 3000 cm21) is typically less than that for O –H. Hence, for all common acids kH/kD < 10 is a good estimate for the maximum KIE for the standard picture, excluding tunneling. In adiabatic PT (with an appropriately small proton barrier height such that neither the proton nor deuteron tunnel), the TS ZPE for the proton is , 1 kcal/mol for H and , 0.7 kcal/mol for D. Using vR , 3200 cm21 as the maximum reactant frequency and , 1 kcal/mol ZPE for H at the TS, the maximum KIE without tunneling is , 6 at 300 K, a lower value than for the W – M picture. An experimental prescription to distinguish these perspectives would be to find the maximum observed KIE for which the reaction is known not to involve tunneling. Unfortunately, while the possible distinction between tunneling and nontunneling PT reactions has generated intensive experimental effort,8,11,12,16,20,27 this is not at all straightforward: the difficulty is that the magnitude of tunneling KIEs overlaps with that of nontunneling, especially in the 5 –10 range.44 A more promising experimental probe is the other, lower limit of the KIE, where tunneling is less likely to be present. In the discussion of this lower limit,4 a first key point is that the complete loss of the ZPE in the reactant going to the TS in a symmetric reaction in the standard picture also limits the minimum symmetric reaction KIE value. From Equation 21.21, the minimum value results from the smallest

562

Isotope Effects in Chemistry and Biology

reactant proton stretch frequency vR one could have and still have PT. A reactant proton stretch frequency of 2300 cm21 gives kH/kD < 5 for the standard picture. But such a frequency value would obviously correspond to a very strong H-bond,43,45 and it is questionable that smaller reactant proton frequencies would be plausible while still permitting activated PT: the reaction barrier for interconversion of proton states in the standard picture would be expected to be quite small. Consequently, the expected kH/kD range for a symmetric reaction in the standard picture is , 5 – 10. In the adiabatic PT approach, however, a finite symmetric reaction TS ZPE reduces the minimum value to less than 3 (cf., Figure 21.5). Experimentally kH/kD versus reaction asymmetry plots have been determined for a variety of PT systems, and the kH/kD maximum ranges from 3 to 10,9,12,46 the lower limit being the key focus here. Included in the lower part of this KIE range are an enzymatic PT,9c a nonenzymatic PT,9e and an excited state PT from a photoacid to a water molecule.46 All three have a kH/kD maximum , 3 – 4, a value which is not obtainable with the standard picture, at least within a linear H-bond model,47,48 but obtainable in the adiabatic PT picture.

III. NONADIABATIC ‘TUNNELING’ PROTON TRANSFER We now turn to the proton nonadiabatic, or tunneling, regime. We first briefly review the PT tunneling rate constant formalism,1 including the role of the H-bond mode, followed by a summary of the resulting KIE behaviors. We should first place our basic treatment1 in perspective with other descriptions, of which there are several, although we make no attempt to be exhaustive. First, there are efforts within the framework of corrections to a W – M approach.16,20 However, these picture tunneling as occurring through the potential energy along a miminum energy path — a perspective which has long since been replaced by ‘corner-cutting’ paths23 — and the role of the solvent is not taken into account. More recent approaches24,25 are based on a reference classical path, sometimes with the solvent equilibrated, with quantum features subsequently added. In our view, these suffer from some difficulties (for discussion, see Refs. 3 –5). Finally, there is the Russian school,29 who early on pioneered a perspective where the solvent played a key role, and related developments.49,50 While these latter approaches share some features with the perspective of Ref. 1, they assume that PT is electronically nonadiabatic, in very strong contrast to the present assumption of electronic adiabaticity. Recently, we have examined in detail the differences in predictions between the two perspectives for PT tunneling reactions, and they are very significant.5b Our view is that most PT reactions in the tunneling regime are electronically adiabatic, with rather strong electronic coupling — an aspect related to the fact that chemical bonds are broken and made — and that an electronically diabatic description is inappropriate.5b (So-called proton-coupled electron transfer is a quite different reaction class, involving transfer of an electron over larger distances, where different considerations apply.14,15) The focus of our attention in this section is on temperatures close to room temperature and above where the H-bond mode with frequency "vQ is significantly populated, i.e., "vQ , RT and "vQ p RT: Other regimes exist and the reader is referred to Ref. 1 for details.

A. GENERAL N ONADIABATIC P ROTON-T RANSFER P ERSPECTIVE AND R ATE C ONSTANT The physical picture for nonadiabatic PT is displayed in Figure 21.7, with a fixed H-bond separation, a constraint later relaxed. The system free energy as a function of the proton coordinate — involving the electronically adiabatic proton potential — is displayed with the reactant and product diabatic proton vibrational states indicated, for three values of the solvent coordinate characterizing different environmental configurations: reactant state R (Figure 21.7a), transition state (TS) ‡ (Figure 21.7b), and product state P (Figure 21.7c). The R and P proton diabatic levels in Figure 21.7a to c are found by solving the nuclear Schro¨dinger equation for the proton in each of the reactant and product wells, respectively. (Proton adiabatic proton levels are found by solving

Interpretation of Primary Kinetic Isotope Effects

(a)

AH

563

AH

(b)

R

(c)

‡

AH

P

ES

G

} 2C DG } DGRXN

(d)

Solvent Reaction Coordinate

FIGURE 21.7 Free-energy curves versus proton position at (a) the reactant R, (b) transition state ‡, and (c) product state P solvent configurations for nonadiabatic PT. In each case, the ground diabatic proton vibrational energy levels are indicated for both the reactant and product proton wells. (d) Free-energy curves versus the solvent reaction coordinate for both diabatic proton levels displayed in (a– c).

the Schro¨dinger equation for the entire proton potential.) The evolving diabatic ground proton vibrational states define free energies as a function of the environment rearrangement, shown in Figure 21.7d. At the thermally activated TS position, the proton reactant diabatic vibrational state is in resonance with the corresponding ground proton product state, and the proton can thus tunnel. For the picture in Figure 21.7, the rate constant for nonadiabatic PT between reactant and product proton ground vibrational states with the H-bond separation Q fixed is1,29 # rffiffiffiffiffiffiffiffiffi " C2 p DG‡ exp 2 k¼ ð21:22Þ " ES RT RT where the free-energy barrier DG ‡ is DG‡ ¼

ðDGRXN þ ES Þ2 4ES

ð21:23Þ

and ES is the solvent reorganization (free) energy.51 The tunneling probability is governed by the square of the proton coupling C.51 As described in Figure 21.2, the resonance splitting C between the proton diabatic levels increases as the H-bond separation decreases, e.g., going from Figure 21.2b to c, due to the decreased tunneling probability for a larger proton barrier. The Q dependence of C is predominantly linear exponential, with the form1 CL ðQÞ ¼ CeqL exp½2aL ðQ 2 Qeq Þ ; CeqL ¼ CL ðQeq Þ

ð21:24Þ

where Qeq is the equilibrium H-bond separation in the reactant state and aL is the exponent characterizing the exponential dependence(L ¼ H, D, and T). The mass dependence in

564

Isotope Effects in Chemistry and Biology

Equation 21.24 is contained the form pffiffi within aL and pffiffiCeqL. In particular, aL is expected to be of pffiffiffiffi ˚ 21,1 values aL / mL (e.g., aD < 2aH ; and aT < 3aH ), with typical values aH , 25– 35 A much larger than the corresponding quantity for electron transfer; the proton, while extremely quantum in character, is more localized than the electron.1 As shown in Figure 21.2, proton donor – proton acceptor modes that modulate the proton barrier are critical in nonadiabatic PT.1,5 We limit our discussion here to the single most important mode, the H-bond mode, but other modes that regulate the barrier through which the proton must tunnel, e.g., H-bond bending modes4,31,53 can be dealt with in a similar manner. For simplicity, a harmonic H-bond vibration UQ ðQÞ ¼ UQ;eq þ 12 mQ v2Q ðQ 2 Qeq Þ2 has been assumed, with an effective mass mQ and vibrational frequency vQ, and for the moment, we retain the restriction to PT between ground proton vibrational levels in the reactant and product. For extremely low temperatures "vQ q RT; the Q vibrational mode resides primarily in its ground state, and the PT rate expression is1 C2 kL ¼ 00 "

# rffiffiffiffiffiffiffiffiffi " p ðDGRXN þ ES Þ2 exp 2 ES RT RT4ES

ð21:25Þ

which is similar to Equation 21.22 except that the proton coupling C is replaced by its quantum average in the ground Q-vibrational state " 2 C00

2

¼ lk0lCðQÞl0ll ¼

2 CeqL exp

ðEaL 2 EQ Þ aL DQ þ "vQ

# ð21:26Þ

Here DQ ¼ QP;eq 2 QR;eq is the difference in product and reactant equilibrium Q positions, and EQ ¼ 12 mQ v2Q DQ2 is the associated reorganization energy. EaL is a quantum energy term associated with the tunneling probability’s variation with the Q vibration EaL ¼ "2 a2L =2mQ

ð21:27Þ

Even with DQ ¼ 0 (EQ ¼ 0), C is increased from its fixed value C(Qeq) by exp(Ea L/"vQ): there is a finite probability of smaller H-bond separations even at low T due to the zero-point motion of Q. The ratio EaL/"vQ describes Ea L as a quantum energy scale for the localization of the Q wavefunction.1,5 If EaL ="vQ p 1; the coupling C is essentially that for fixed Q ¼ Qeq. As EaL/"vQ increases, C increases, corresponding to increased quantum accessibility of smaller Q values. As T is increased, contributions from excited Q vibrational states become more significant. For moderate to high temperatures "vQ , RT and "vQ p RT; the focus here, many Q vibrational states are thermally excited. In this regime, the PT rate expression1 for DQ ¼ 0 is kC 2 l kL ¼ "

# rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " p DG‡L exp 2 RT ðES þ E~ aL ÞRT

ð21:28Þ

where the reaction activation free energy is given by ðDGRXN þ ES þ EaL Þ2 4ðES þ E~ aL Þ

ð21:29Þ

1 1 E~ aL ¼ EaL b"vQ coth b"vQ 2 2

ð21:30Þ

DG‡L ¼

Interpretation of Primary Kinetic Isotope Effects

565

The square proton coupling factor in Equation 21.28 is the thermal average over the Q vibrational states1 ! Ea L 1 2 2 kC l ¼ CL ðQeq Þexp 2 coth b"vQ ð21:31Þ "vQ 2

aL (via Ea L, Equation 21.27) contributes significantly to the average in Equation 21.31. In particular, the sensitivity of the coupling of C to Q dynamics is displayed in the ratio Ea L/"vQ so that kC 2l increases as this ratio increases. Ea L also appears in the reaction barrier in Equation 21.29; Ea L appears as an energetic contribution to that barrier due to thermal activation of the H-bond mode. The isotopic dependence Ea L / mL in the barrier plays a key role in isotope and temperature effects, but before recounting those, we describe the inclusion of excited proton vibrational levels.1,5 In the discussion above, PT has been assumed to occur from the reactant ground proton diabatic vibrational state to the corresponding state in the product. However, for very exothermic or endothermic reactions (lDGRXNl $ ES þ "vQ), excited proton vibrational states can become important: the proton can be transferred into an excited proton product vibrational state for an exothermic case and from a thermally excited reactant proton vibration for an endothermic case. Here, free-energy curves corresponding to excited diabatic proton vibrational states are added to the ground diabatic proton vibrational states in Figure 21.7d, displayed as Figure 21.8. Each freeenergy curve corresponds to a diabatic proton vibrational level, and now there are several transitions possible. A specific resonance situation is associated with each TS or intersection of the proton diabatic free-energy curves. Figure 21.9 displays the TS proton potentials for four such transitions.54 Figure 21.9b shows the symmetric proton potential for the ground state to ground state (0 – 0) transition and the corresponding first excited state transition (1 –1). The 1– 1 transition will have a higher transition probability (larger C) because the excited proton level is closer to the proton barrier top. But this increased 1– 1 transition tunneling probability comes at a cost of 1 quantum of proton vibration excitation, which is added to the activation energy. Figure 21.9a and c display the proton potentials with 1 –0 and 0 –1 transitions, respectively. Both will have an increased tunneling probability compared with the 0 –0 transition due to a smaller proton barrier through which to tunnel. Reactant proton vibrational mode thermal excitation leads to the 1– 0 transition, assisting endothermic reactions, while extra solvent activation leads to the 0– 1 transition, assisting exothermic reactions. The interplay between cost of thermal excitation and gain from increased tunneling probability, and their isotope dependence, plays a significant role in KIEs. Excited proton vibrational states are included in the PT rate as a sum over all state-to-state PT rates knR !nP from a proton reactant state nR to a product state nP XX kL ¼ PnR knR !nP ð21:32Þ nR

nR=1

nP

nP=1

nR=0 DGRXN

nP=0 Solvent Reaction Coordinate

FIGURE 21.8 Proton diabatic free-energy curves versus the solvent reaction coordinate for individual reactant (nR) and product (nP) proton vibrational states.

566

Isotope Effects in Chemistry and Biology

1−0

(a)

1−1 0−0

(b)

0−1

AH

(c)

FIGURE 21.9 Proton potentials for the solvent coordinate TS for four proton vibrational transitions (nR 2 nP): (a) 1 – 0, (b) 0 – 0 and 1– 1, and (c) 0 – 1. The lines indicate diabatic proton vibrational levels.

where each state-to-state P rate is weighted by the reactant state thermal occupation PnR(PnR ¼ expð2bEnR Þ= nR expð2bEnR Þ; and EnR ¼ "vR ðnR þ 1=2Þ). knR !nP is now a modified version of Equation 21.28

knR !nP

kCn2R ;nP l ¼ "

# rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " DG‡nR ;nP p exp 2 RT ðES þ E~ aL ÞRT

ð21:33Þ

where the reaction free-energy barrier DG‡nR ;nP is transition dependent DG‡nR ;nP ¼

ðDGRXN þ nP "vP 2 nR "vR þ ES þ EaL Þ2 4ðES þ E~ aL Þ

ð21:34Þ

The proton coupling CeqL(nR ! nP) is also transition dependent, and increases as the quantum numbers nR and nP increase (more properly, the difference), because the proton coordinate barrier’s width and height are smaller as the proton level sits higher in either well.5 The Q dependence of the coupling can still be approximated by the same form in Equation 21.245 CnR ;nP ðQÞ ¼ CnR ;nP ðQeq Þexp½2aL ðQ 2 Qeq Þ

ð21:35Þ

Interpretation of Primary Kinetic Isotope Effects

567

such that the thermal average of C 2 for Equation 21.33 is accordingly kCn2R ;nP l

¼

Cn2R ;nP ðQeq Þexp

E 1 2 aL coth b"vQ "vQ 2

! ð21:36Þ

B. NONADIABATIC P ROTON- TRANSFER KIES We now review the KIE behaviors that follow from the above nonadiabatic PT formalism, focusing on the four KIE observables (i –iv) listed in the Introduction.5 The KIE magnitude and its variation with reaction asymmetry are first summarized, which serves to demonstrate the importance of excited proton and H-bond vibrational states, and the temperature dependence is reviewed. The Swain –Schaad behavior concludes the KIE behavior discussion. 1. KIE Magnitude and Variation with Reaction Asymmetry Traditional treatments of KIEs, including those invoking tunneling along a minimum energy path, predict that the KIE is maximal for a symmetric reaction DGRXN ¼ 0.20 As now discussed, a similar behavior results in the present perspective. During our presentation, the magnitude of tunneling PT KIEs will also be discussed (focusing on tunneling PT systems that give smaller KIE magnitudes which might be confused with nontunneling PT). The KIE behavior can be illustrated via a PT system with T ¼ 300 K, "vQ ¼ 300 cm21, ES ¼ 8 kcal/mol,5 and using Equation 21.32 with Equation 21.33, the KIE is displayed as Figure 21.10. The behavior is maximal for DGRXN ¼ 0 and falls off with increasing reaction asymmetry. The maximal KIE behavior for tunneling PT is due to increased excitation in both the proton and H-bond modes, excitations that become more facile with increased reaction asymmetry.5 Proton excitation increases the tunneling probability, via the proton coupling C, and because the deuteron mode is easier to excite than H, "vH . "vD, this benefits D more than H. Similarly, H-bond excitation also benefits D more than H because the D tunneling probability is more sensitive to changes in Q: see the EaL/"vQ ratio in Equation 21.36 with E aH , E aD . The KIE magnitude in Figure 21.10 is actually fairly small compared to expectations for a PT tunneling reaction. In fact, the KIE magnitude for fairly asymmetric reactions might be considered consistent with nontunneling PT. To emphasize this important point, the KIE with a slightly lower H-bond vibrational frequency "vQ ¼ 275 cm21 is also included, where the KIE magnitude decreases by a factor of 3, emphasizing the sensitivity of the KIE to the donor – acceptor frequency. Again, this KIE behavior cannot be distinguished from that for nontunneling PT.

kH /kD

12 w Q =300 cm−1

8 4 0

w Q= 275 cm−1 −12

−8

−4

0

4

8

12

DGRXN (kcal/mol)

FIGURE 21.10 kH/kD for a nonadiabatic PT system with "vQ ¼ 300 cm21. The dotted line is the same PT system, except that "vQ ¼ 275 cm21.

568

Isotope Effects in Chemistry and Biology

2. Temperature Behavior We now review the T dependence of the rate in Equation 21.32,5 which in general is certainly not Arrhenius. However, one must bear in mind that most experiments are conducted over a reasonably restricted temperature range and the issue must be carefully examined. Two contributions to the T dependence of individual transition rates dominate in Equation 21.33. The first is contained within the exponential containing the reaction free-energy barrier, which gives Arrhenius behavior if the components of the reaction barrier, i.e., E~ aL (see Equation 21.30) and ES,5,41 have only a minor T dependence. An important point is that the impact of any such is suppressed if the reorganization energy is significant (ES . EaL). The second contribution comes from the thermally averaged square proton coupling Equation 21.36, and in principle is not Arrhenius. In addition to these T dependencies for the individual transition rate constants, the thermal sum over excited proton transitions for the full rate in Equation 21.32 is clearly also not Arrhenius in principle. Altogether, these contributions give rise to a nonlinear T dependence in an Arrhenius plot, as expected for tunneling PT.8,11,16,17,55 (We stress however that this is not a non-Arrhenius behavior associated with a transition from high temperature, classical “over the barrier” PT to tunneling PT at lower temperatures; the entire description here is in the tunneling regime.) Nonetheless, this T dependence was shown5 to be effectively linear in an Arrhenius plot for a limited but nonnegligible temperature range.56 In this analysis,5 the PT rate in proximity to a specific temperature To is written in an Arrhenius form kL ¼ kL ðTo Þexp½2ðb 2 bo ÞEAL

ð21:37Þ

where the Arrhenius intercept is just the extrapolation from the rate at T ¼ To to infinite temperature: AL ¼ kL ðTo Þexp½bo EAL ; and EAL is determined by the slope in an Arrhenius plot. For analysis purposes, the same system as in Figure 21.10 was taken, and the temperature was varied (T ¼ 300 –350 K), while keeping the reaction asymmetry constant, DGRXN ¼ 0. The apparent Arrhenius rate and KIE behavior obtained in this limited T range are displayed in Figure 21.11. The apparent activation energies for H and D differ considerably, with EAD almost twice EAH: EAH ¼ 5.7 kcal/mol and EAD ¼ 10.6 kcal/mol; this results in a significant effective

lnk L

20

H

18 D

16 3

(a)

lnkH /kD

2.5 2

1.5 1

(b)

1.4

1.5 1.6 1/RT (kcal/mol)

1.7

FIGURE 21.11 (a) ln kH (o) and ln kD (þ ) versus 1/RT (T ¼ 300– 350 K) for the PT system in Figure 21.10 with DGRXN ¼ 0. (b) ln(kH/kD) ( £ ) for rate constants in (a). Lines are linear fits to points. Slopes of lines give the activation energies (a) EAH ¼ 5.7 kcal/mol; EAD ¼ 10.6 kcal/mol and (b) KIE EA ¼ 5.0 kcal/mol.

Interpretation of Primary Kinetic Isotope Effects

569

activation energy for the KIE EAD 2 EAH ¼ 5.0 kcal/mol, displayed in Figure 21.11b. These slopes can be quantitatively analyzed5 to determine contributions from H-bond and proton vibration excitations. This analysis begins with the rate expression in Equation 21.32 including excited proton vibrational states, which can be written5 X

FnR ;nP ¼ kL00 rL

ð21:38Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " DG‡ # p L0;0 exp 2 ~ RT ðES þ EaL ÞRT

ð21:39Þ

kL ¼ kL00

nR ;nP

in terms of the 0 –0 transition rate constant kL00

2 kC0;0 l ¼ "

and the rate enhancement rL due to excited proton vibrational states. The coefficient of each transition FnR ;nP is5 FnR ;nP

" # DDG‡nR ;nP pð"vR nR þ "vP nP Þ ¼ PnR exp exp 2 RT "v‡

ð21:40Þ

DDG‡n R,n P is the difference between the general reaction barrier DG‡n R,n P Equation 21.34 and that, DG‡L0,0, for the 0– 0 case DDG‡nR ;nP ¼ DG‡nR ;nP 2 DG‡L0;0 ¼

ðnP "vP 2 nR "vR Þð2ðDGRXN þ ES þ EaL Þ þ nP "vP 2 nR "vR Þ 4ðES þ E~ aL Þ

ð21:41Þ

Here, properties from the TS proton potential (cf. Figure 21.2) are included, i.e., the curvatures in the wells, vR and vP, and at the top of the barrier v ‡, as a means to relate CnR,nP to C0,0.5 For the case under discussion, the 0 –0 rate kL00 and rL is evaluated at the mid-range temperature To koL ¼ kL00(To) (e.g. To ¼ 325 K in Figure 21.11), such that kL(To) ¼ rLkoL. Staring with Equation 21.38, an Arrhenius form for the rate constant Equation 21.37 has been derived in this limited T region,5 with AL ¼ koL rL ðTo Þexpðbo EAL Þ; EAL¼ EaL ½coth2 ðbo "vQ =2Þ 2 1 þ DG‡L0;0 þ kDDG‡nR ;nP lL

ð21:42Þ

DG‡L0,0 is the 0 –0 reaction free-energy barrier Equation 21.29, and kDDG‡nR ;nP lL is the activation free-energy barrier contribution from excited proton states XX kDDG‡nR ;np lL ¼

nR

nP

FoR ;P ð"vR nR þ DDG‡nR ;np Þ XX nR

nP

Fo R ; P

ð21:43Þ

where the symmetric reaction transition coefficient is FoR,P ¼ FnR,nP(T ¼ To) (cf. Equation 21.40). For the behavior in Figure 21.11, Equation 21.42 gives reasonable estimates for EAH and EAD, EAH ¼ 6.1 kcal/mol and EAD ¼ 11.2 kcal/mol, which differ by less than 10% from the obtained numerical values. The decomposition of these apparent activation energies via Equation 21.42

570

Isotope Effects in Chemistry and Biology

is useful5 in determining which contributions are more important and how these contributions change with T, "vQ, reaction asymmetry, and solvent reorganization energy ES, as now reviewed. The first term in Equation 21.42 is the activation energy contribution from the thermally averaged square coupling kC2l (see Equation 21.36), and as such is extremely sensitive to parameters affecting the H-bond mode-tunneling coupling, namely T, "vQ, and Ea L. For the present system, this term dominates the activation energy for both H (60%) and D (66%). Furthermore, since Ea L / mL is mass sensitive, the predominant contribution to the activation energy difference determining the EA for the KIE will be dominated by this first term. The coefficient {coth2(bo"vQ/2) 2 1} in this term is extremely sensitive to To and "vQ, increasing drastically as To is increased or "vQ decreases, and the ratio "vQ/RTo determines the relative contribution for this first term. The second term in Equation 21.42, the activation free-energy barrier DG‡L0,0, is for the present system also significant for both H (39%) and D (25%). Of course, the magnitude of this term changes with reaction asymmetry, decreasing as the reaction goes from endo- to exo thermic (cf. Equation 21.29). The last term in Equation 21.42 for EAL is the least important for the present system, , 1% for H and 9% for D. Its lack of importance correlates with the significance of the 0– 0 transition in the overall rate, described here by rL , 1, rH ¼ 1.004 and rD ¼ 1.25. rL, however, will obviously change as the reaction becomes more asymmetric as well as with increasing T. With the above individual isotope Arrhenius parameter results in Equation 21.42, those for the KIE can be examined. The KIE Arrhenius slope is thus determined by the isotopic difference of the apparent activation energies EAL EAD 2 EAH ¼ EaH ½coth2 ðb"vQ =2Þ 2 1 þ ½DG‡Do 2 DG‡Ho þ ½kDDGnR ;nP lD 2 kDDGnR ;nP lH

ð21:44Þ

For the chosen system, the first term contributes the most, 72% as predicted above, the final difference is next in significance at 20%, and the middle difference contributes only 8%. The minimal significance of the (middle term) difference in 0– 0 reaction barriers reflects the disparity ES . EaL. The increased contribution (third term) of the excited proton state contribution is due to the differential contribution of the 0– 0 transition to the total rate, rH , rD. Excited state contributions described by rL have a key characteristic in that they increase with increased reaction asymmetry DGRXN and decreased reorganization energy ES.5 Furthermore, the isotopic disparity rH , rD also increases with these trends, resulting in an increase in significance of the third difference in Equation 21.44 with increased DGRXN and decreased ES.5 The Arrhenius intercept AL in Equation 21.42 is the extrapolation from the rate at T ¼ To kL(To) ¼ koLrL to infinite temperature, and thus the ratio of intercepts (H vs. D) is the extrapolation of the KIE (H vs. D) at To to infinite temperature AH ¼ AD

kH kD

o

exp½2bo ðEAD 2 EAH Þ

ð21:45Þ

where kH kD

o

¼

koH rH koD rD

ð21:46Þ

The significant isotopic difference of Arrhenius intercepts, i.e., AH – AD, is a signature for a tunneling process.11 For the Figure 21.11a system, the Arrhenius prefactors have AH , AD, which is the case where Equation 21.45 is less than 1. If, however, the system had EAD 2 EAH , 1 (not , 5 kcal/mol as in Figure 21.11), then one would instead have AH . AD; tunneling itself imposes no relative size of this ratio. Clearly, the interplay between the (kH/kD)o magnitude and the

Interpretation of Primary Kinetic Isotope Effects

571

difference EAD 2 EAH determines whether AH . AD or AH . AD. Thus, an alternate yet equivalent isotope analysis of Arrhenius plots would be5 to analyze (kH/kD)o and EAD 2 EAH rather than AH/AD and ED 2 EH. The advantage of this alternative analysis is the direct connection between the KIE magnitude and Arrhenius slope with H-mode characteristics. Specifically, larger KIE magnitudes result from longer (large acid –base separations) and more rigid H-bond (large "vQ/ RT) complexes, and small Arrhenius slopes arise with less probability of H-bond and proton mode excitation (i.e., low T and especially higher "vQ/RT ratios). 3. Swain – Schaad Relationship

(a)

ln(kH/kT)/ln(kD/kT)

This final category of KIE behavior describes the relative KIEs between the three isotopes H, D, and T via the Swain– Schaad relationship (cf. Equation 21.19). Deviation from this relationship is regarded as a clear indication of tunneling.11,57 Traditionally, the relationship has been experimentally assessed by varying system parameters and plotting ln(kH/kT) versus ln(kD/kT) and determining whether this produces a line which goes through the origin and has a slope , 3.3.8,16,17,22 However, Equation 21.19 has also been assessed by plotting the ratio ln(kH/kT)/ln(kD/kT) versus a system parameter, such as temperature.11 If the ratio deviates significantly from the value of , 3.3, the PT system is said to be tunneling. Here we recount our examination5 of whether or not a nonadiabatic tunneling PT system can exhibit the behavior in Equation 21.19 as well as have a KIE magnitude that is normally consistent with adiabatic nontunneling PT, i.e., kH/kD # 6 at T ¼ 300 K.4,8,16,17 Figure 21.12a displays the ratio ln(kH/kT)/ln(kD/kT) for the same system in Figure 21.11 except that the H-mode frequency has been increased to "vQ ¼ 375 cm21 (T ¼ 300 K).5 4 3 2 1

kH/kD

100

0 −10 ln(kH/kT)/ln(kD/kT)

(b)

(c)

50

−5 0 5 DGRXN (kcal/mol)

10

5 4 3 1.5

1.7 1.9 1/RT (mol/kcal)

FIGURE 21.12 (a) Swain – Schaad ratio ln(kH/kT)/ln(kD/kT) versus reaction asymmetry for a nonadiabatic PT system with "vQ ¼ 375 cm21; (b) kH/kD for system in (a); (c) ln(kH/kT)/ln(kD/kT) versus 1/RT (T ¼ 250– 350 K) for symmetric reaction in (a).

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Isotope Effects in Chemistry and Biology

The Swain – Schaad ratio is at the expected traditional value, but the H versus D KIE displayed in Figure 21.12b is clearly large enough to indicate tunneling PT. Beyond this, the T-variation of the Swain –Schaad ratio for this system shown in Figure 21.12c displays a distinct deviation from Equation 21.19 for part of the temperature range, and thus also allows confirmation of tunneling PT.58 This example demonstrates the lesson that while the traditionally appropriate Swain– Schaad ratio can be obtained for a tunneling PT system, it is difficult to find a tunneling PT reaction with both the Swain –Schaad ratio and KIE magnitude of the traditionally appropriate values. This analysis indicates that the Schwain –Schaad ratio can be a clear indicator of tunneling,11 but examination over a parameter (e.g., T ) range and simultaneous examination of the KIE magnitude can be necessary to establish this.

IV. CONCLUDING REMARKS A theoretical description of primary KIEs for adiabatic PT reactions — where the proton motion is not tunneling but yet is quantum — has been reviewed utilizing a nontraditional picture in which the proton nuclear motion is treated quantum mechanically. This PT description markedly differs from the standard treatment of PT KIEs, based on the Westheimer – Melander description, in both the proton quantization (rather than, e.g., classical proton barrier crossing) and the identification of a solvent coordinate as the reaction coordinate (rather than the proton). This change in perspective focuses attention on the local environment for the PT event, and should have important consequences for both solution and enzymatic PT rates in this adiabatic PT limit, where the proton motion is quantum but is not tunneling. Four different KIE features experimentally observed that are believed to support the standard nontunneling interpretation of KIEs also follow from the adiabatic (nontunneling) PT picture. This occurs despite the pronounced differences between the adiabatic PT and standard pictures. A primary difference between the two perspectives for KIEs concerns the proton ZPE magnitude at the transition state (TS) of a symmetric reaction. In the standard picture, the TS contains no such contribution, while the adiabatic PT picture — reflecting the quantum nature of the proton motion — has a finite one (, 1 kcal/mol). This distinct difference defines a lower range for the KIE of a symmetric reaction for adiabatic PT than that for the standard picture, consistent with experimental results. It is important, however, to stress certain general features that the two perspectives share. Namely, a TS theory formalism applies for both pictures in which the KIE exhibits Arrhenius behavior and is primarily determined by the ZPE difference between the reactant and TS for both isotopes. Of course, the definitions of the TS differ considerably in the two pictures: the respective pffiffiffi reaction coordinates are entirely different. Further, the mass scaling of ZPEs ðZPE / 1= mÞ allows for similar magnitudes in KIEs and leads to approximate adherence to the well-known Swain –Schaad relationships. Also included in the similarity is the long-held physical organic perspective that the extent of PT in the TS correlates with reaction asymmetry, consistent with the Hammond postulate. For the adiabatic PT treatment (but not the standard one), the extent of PT at the TS in the solvent coordinate is described in terms of the electronic and geometric structure, averaged over the quantum motion of the proton and H-bond. KIE trends including KIE magnitude, temperature dependence, variation with reaction asymmetry, and adherence to the Swain – Schaad relationships have also been analyzed for tunneling PT reactions, here termed the nonadiabatic PT regime. While each of these four KIE behaviors can individually be consistent with nontunneling PT, the combination of these behaviors, with special emphasis on variation with reaction asymmetry or temperature, clearly allows for identification of tunneling PT systems. The sensitivity to excitation of the H-bond and proton vibrational modes is a key factor for these KIE behaviors.

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573

An additional criterion, which we have not discussed, does exist for identifying a tunneling PT system, namely the existence of an ‘inverted’ regime, where the rate constant decreases with increasing reaction asymmetry. An ‘inverted’ regime has been experimentally observed for some PT systems;59 but in our view, further elucidation is required for the conditions for its existence, especially considering the contribution of excited state reaction asymmetry dependencies to the total rate.5 The ‘inverted’ regime is subject of current work. The tunneling PT rate constant and KIE behaviors are significantly influenced by excitation of the H-bond and proton vibrational modes. These modes modes are more accessible for asymmetric reactions, and because these mode excitations increase the tunneling probability, asymmetric reactions are dominated by proton and H-bond mode excitation. The fact that the deuteron is more sensitive to the H-bond dynamics as well as being easier to excite than the proton implies that Hbond and deuteron vibrational excitation increase the D transfer rate more than that for H. The result is a kH/kD KIE decreasing with increased reaction asymmetry as well as temperature. Finally, a prescription for interpreting slopes in an Arrhenius plot for tunneling rate constants and KIEs was also reviewed for nonadiabatic PT,5 i.e., Equation 21.42. Thermal excitation of the proton and H-bond modes explicitly contribute to the effective activation energy, in addition to the actual reaction free energy barrier, and in fact these excitations can dominate the effective activation energy in some cases.

ACKNOWLEDGMENTS This work was supported in part by NSF grants CHE-9700419, CHE-0108314, and CHE-0417570.

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6 Timoneda, J. J. and Hynes, J. T., Nonequilibrium free energy surfaces for hydrogen-bonded proton transfer complexes in solution, J. Phys. Chem., 95, 10431– 10442, 1991. 7 Gertner, B. J. and Hynes, J. T., Model molecular dynamics simulation of hydrochloric acid ionization at the surface of stratospheric ice, Faraday Soc. Discuss., 110, 301– 322, 1998. 8 (a) Bell, R. P., The proton in chemistry, 2nd ed., Cornell University Press, Ithaca, NY, 1973; (b) Caldin, E. and Gold, V., Proton Transfer Reactions, Chapman and Hall, London, 1975; (c) Kresge, A., What makes proton transfer fast? J. Acc. Chem. Res., 8, 354– 360, 1975; (d) Hibbert, F., Mechanisms of proton transfer between oxygen and nitrogen acids and bases in aqueous solution, Adv Phys. Org. Chem., 22, 113– 212, 1986; (e) Hibbert, F., Hydrogen bonding and chemical reactivity, Adv. Phys. Org. Chem., 26, 255– 379, 1990. 9 (a) Kreevoy, M. M. and Konasewich, D. E., The Brønsted a and the primary hydrogen isotope effect Adv. chem. phys., 21, 243– 252, 1972; (b) Kreevoy, M. M. and Oh, S.-w., Relations between rate and equilibrium constants for proton-transfer reactions, J. Am. Chem. Soc., 95, 4805– 4810, 1973; (c) Kresge, A. J. and Silverman, D. N., Application of marcus rate theory to proton transfer in enzymecatalyzed reactions, Meth. Enz., 308, 276– 297, 1999; (d) Kresge, A. J., Chen, H. J., and Chiang, Y., Vinyl ether hydrolysis. 7. Isotope effects on catalysis by aqueous hydrofluoric acid, J. Am. Chem. Soc., 99, 802– 805, 1977; (e) Kresge, A. J., Sagatys, D. S., and Chen, H. L., Vinyl ether hydrolysis. 9. Isotope effects on proton transfer from the hydronium ion, J. Am. Chem. Soc., 99, 7228– 7233, 1977; (f) McLennan, D. J. and Wong, R. J., The carbanion mechanism of olefin-forming elimination. VII theoretical analysis of substituent and isotope effects in proton transfer from 2,2-diaryl-1,1,1trichloroethanes, Aust. J. Chem., 29, 787– 798, 1976; (g) Lee, I.-S. H., Jeoung, E. H., and Kreevoy, M. M., Primary kinetic isotope effects on hydride transfer from 1,3-dimethyl-2-phenylbenzimidazole to NAD þ analogues, J. Am. Chem. Soc., 123, 7492–7496, 2001. 10 (a) Bell, R. P. and Goodall, D. M., Kinetic hydrogen isotope effects, Proc. Roy. Soc. London Series A, 294, 273– 297, 1966; (b) Dixon, J. E. and Bruice, T. C., Dependence of the primary isotope effect (k H/ k D) on base strength for the primary amine catalyzed ionization of nitroethane, J. Am. Chem. Soc., 92, 905– 909, 1970; (c) Pryor, W. A. and Kneipp, K. G., Primary kinetic isotope effects and the nature of hydrogen-transfer transition states. The reaction of a series of free radicals with thiols, J. Am. Chem. Soc., 93, 5584– 5586, 1971; (d) Bell, R. P. and Cox, B. G., Primary kinetic isotope effects and the rate of ionization of nitroethane in mixtures of water and dimethyl sulphoxide, J. Chem. Soc. B, 783– 785, 1971; (e) Bordwell, F. G. and Boyle, W. J., Kinetic isotope effects for nitroalkanes and their relationship in proton transfer reactions, J. Am. Chem. Soc., 97, 3447– 3452, 1975. 11 (a) Cha, Y., Murray, C. J., and Klinman, J. P., Hydrogen tunneling in enzyme reactions, Science, 243, 1325– 1330, 1989; (b) Kohen, A. and Klinman, J. P., Enzyme catalysis: beyond classical paradigms, Acc. Chem. Res., 31, 397–404, 1998; (c) Kohen, A. and Klinman, J. P., Hydrogen tunneling in biology, Chem. Biol., 6, R191 –R198, 1999; (d) Kohen, A., Cannio, R., Bartolucci, S., and Klinman, J. P., Enzyme dynamics and hydrogen tunneling in thermophilic alcohol dehydrogenase, Nature, 399, 496– 499, 1999. 12 Kresge, A. J., Magnitude of primary hydrogen isotope effects, In Isotope Effects on Enzyme-Catalyzed Reactions, Cleland, W. W., O’Leary, M. H., and Northrop, D. B., Eds., University Park Press, Baltimore, MD, pp. 37 – 63, 1977. 13 We refer to proton transfer throughout, but the theory applies to H atom and hydride transfer as well with special considerations.1 – 5 For the solvent coordinate picture presented here, the formalism assumes that the electronic adiabatic proton potentials can be described by two electronic VB states, one for reactant and one for product. The nonadiabatic PT picture has previously been applied to H atom transfer,1e but for some H atom or hydride transfer reactions more than two VB states may be required. In particular, proton coupled-electron transfer reactions in which electron transfer over a large distance is coupled with proton transfer is thought to require multiple solvent coordinates, in addition to several VB states.14,15 14 (a) Cukier, R. I., Proton-coupled electron transfer reactions: evaluation of rate constants, J. Phys. Chem., 100, 15428– 15443, 1996; (b) Cukier, R. I. and Nocera, D., Proton-coupled electron transfer, Ann. Rev. Phys. Chem., 49, 337–369, 1998. 15 (a) Soudackov, A. and Hammes-Schiffer, S., Derivation of rate expressions for nonadiabatic protoncoupled electron transfer reactions in solution, J. Chem. Phys., 113, 2385– 2396, 2000; (b) Iordanova, N., Decornez, H., and Hammes-Schiffer, S., Theoretical study of electron, proton, and proton-coupled

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electron transfer in iron bi-imidazole complexes, J. Am. Chem. Soc., 123, 3723– 3733, 2001; (c) Iordanova, N. and Hammes-Schiffer, S., Theoretical investigation of large kinetic isotope effects for proton-coupled electron transfer in ruthenium polypuridyl complexes, J. Am. Chem. Soc., 124, 4848– 4856, 2002; (d) Hammes-Schiffer, S., Theoretical perspectives on proton-coupled electron transfer reactions, Acc. Chem. Res., 34, 273–281, 2001. Melander, L. and Saunders, W. H., Reaction Rates of Isotopic Molecules, Wiley, New York, 1980. (a) Westheimer, F. H., The magnitude of the primary kinetic isotope effect for compounds of hydrogen and deuterium, Chem. Rev., 61, 265– 273, 1961; (b) Melander, L., Isotope Effects on Reaction Rates, The Ronald Press Co., New York, 1960. (a) Hammond, G. S., A correlation of reaction rates, J. Am. Chem. Soc., 77, 33 – 338, 1955; (b) Lowry, T. H. and Richardson, K. S., Mechanism and Theory in Organic Chemistry, 3rd ed., Harper Collins Pubs., New York, 1987. More O’Ferral, R. A., Substrate isotope effects, In Proton Transfer Reactions, Caldin, E. and Gold, V., Eds., Chapman and Hall, London, pp. 201– 261, 1975. (a) Bell, R. P., The Tunnel Effect in Chemistry, Chapman and Hall, New York, 1980; (b) Bell, R. P., Sachs, W. H., and Tranter, R. L., Model Calculations of Isotope Effects in Proton Transfer Reactions, Trans Far Soc., 67, 1995– 2003, 1971. A KIE expression of the form in Equation 21.2 neglects any mass dependence in the prefactor, the contribution of which is usually negligible, except in the case of tunneling,8,11,20 as well as assumes that the proton primarily resides in it ground (stretching and bending) vibrational state. Swain, G. G., Stivers, E. C., Reuwer, J. F., and Schaad, L. J., Use of hydrogen isotope effects to identify the attacking nucleophile in the enolization of ketones catalyzed by acetic acid, J. Am. Chem. Soc., 80, 5885– 5893, 1958. (a) Babamov, V. K. and Marcus, R. A., Dynamics of hydrogen atom and proton transfer reactions. Symmetric case, J. Chem. Phys., 74, 1790– 1798, 1981; (b) Hiller, C., Manz, J., Miller, W. H., and Ro¨melt, J., Oscillating reactivity of collinear symmetric heavy þ light 2 heavy atom reactions, J. Chem. Phys., 78, 3850– 3856, 1983; (c) Miller, W. H., Tunneling corrections to unimolecular rate constants, with application to formaldehyde, J. Am. Chem. Soc., 101, 6810– 6814, 1979; (d) Skodje, R. T., Truhlar, D. G., and Garrett, B. C., Vibrationally adiabatic models for reactive tunneling, J. Chem. Phys., 77, 5955– 5976, 1982; (e) Marcus, R. A. and Coltrin, M. E., A new tunneling path for reactions such as H þ H2 ! H2 þ H, J. Chem. Phys., 67, 2609– 2613, 1977; (f) Garrett, B. C. and Truhlar, D. G., A least action variational method for calculating multidimensional tunneling probabilities for chemical reactions, J. Chem. Phys., 79, 4931– 4938, 1983; (g) Kim, Y. and Kreevoy, M. M., The experimental manifestations of corner-cutting tunneling, J. Am. Chem. Soc., 114, 7116– 7123, 1992. Calculations of KIEs derived from a classical reaction path (e.g., the MEP) in the presence of a solvent or polar environment typically add quantum corrections to that path.25 Such a reaction path, however, includes classical motion of the proton, especially near the TS, and thus this technique exhibits no difference in quantum corrections between H and D at the TS for a symmetric reaction (DGRXN ¼ 0),25b in contrast to the present picture. In variational TS theory for gas phase H atom transfer, the TS significantly deviates from the MEP TS and is isotope-dependent.26 This feature has been calculated for PT in an enzyme, where the KIE has been diminished because the TS position significantly differs between H and D even in a symmetric case.25e For numerical calculations consistent with the stanadard view, see (a) Alhambra, C., Corchado, J. C., Sa´nchez, M. L., Gao, J., and Truhlar, D. G., Quantum dynamics of hydride transfer in enzyme catalysis, J. Am. Chem. Soc., 122, 8197– 8203, 2000; (b) Hwang, J.-K. and Warshel, A., A quantized classical path approach for calculations of quantum mechanical rate constants, J. Phys. Chem., 97, 10053– 10058, 1993; (c) Hwang, J.-K., Chu, Z. T., Yadav, A., and Warshel, A., Simulations of quantum mechanical corrections for rate constants of hydride- transfer reactions in enzymes and solutions, J. Phys. Chem., 95, 8445– 8448, 1991; (d) Hwang, J.-K. and Warshel, A., How important are quantum mechanical nuclear motions in enzyme catalysis? J. Am. Chem. Soc., 118, 11745– 11751, 1996; (e) Alhambra, C., Gao, J., Corchado, J. C., Villa, J., and Truhlar, D. G., Quantum mechanical dynamical effects in an enzyme catalyzed proton transfer reaction, J. Am. Chem. Soc., 121, 2253– 2258, 1999; (f) Cui, Q. and Karplus, M., Quantum mechanics/molecular mechanics studies of triosephosphate isomerase-catalyzed reactions: Effect of geometry and tunneling on proton-transfer rate constants, J. Am. Chem. Soc., 124, 3093– 3124, 2002.

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26 Garrett, B. C. and Truhlar, D. G., Generalized transition state theory. Bond energy-bond order method for canonical variational calculations with application to hydrogen atom transfer reactions, J. Am. Chem. Soc., 101, 4534 –4547, 1979. 27 Saunders, W. H., Calculations of isotope effects in elimination reactions. New experimental criteria for tunneling in slow proton transfers, J. Am. Chem. Soc., 107, 164– 169, 1985. 28 For examples of the equilibrium solvation picture for PT in a complex system, see (a) Bash, P. A., Field, M. J., Davenport, R. C., Petsko, G. A., Ringe, D., and Karplus, M., Computer simulation and analysis of the reaction pathway of triosephosphate isomerase, Biochemistry, 30, 5826– 5832, 1991; (b) Cui, Q. and Karplus, M., Triosephosphate isomerase: a theoretical comparison of alternative pathways, J. Am. Chem. Soc., 123, 2284– 2290, 2001; (c) Alhambra, C., Corchado, J., Sa´nchez, M. L., Garcia-Viloca, M., Gao, J., and Truhlar, D. G., Canonical variational transition state theory for enzyme kinetics with the protein mean force and multidimensional quantum mechanical tunneling dynamics. Theory and application to liver alcohol dehydrogenase, J. Phys. Chem. B, 105, 11326– 11340, 2001; (d) Cui, Q., Elstner, M., and Karplus, M., A theoretical analysis of the proton and hydride transfer in liver alcohol dehydrogenase (LADH), J. Phys. Chem. B, 106, 2721– 2740, 2002. 29 (a) Dogonadze, R. R., Kuznetzov, A. M., and Levich, V. G., Theory of hydrogen-ion discharge on metals: case of high overvoltages, Electrochim. Acta, 13, 1025– 1044, 1968; (b) German, E. D., Kuznetzov, A. M., and Dogonadze, R. R., Theory of kinetic isotope effect in proton transfer reactions in a polar medium, J. Chem. Soc. Farad. Trans II, 76, 1128– 1146, 1980; (c) Kuznetzov, A. M., Charge Transfer in Physics, Chemistry and Biology: Physical Mechanisms of Elementary Processes and an Introduction to the Theory, Gordon and Breach Publishers, Amsterdam, 1995; (d) Kuznetzov, A. M. and Ulstrup, J., Proton and hydrogen atom tunneling in hydrolytic and redox enzyme catalysis, Can. J. Chem., 77, 1085– 1096, 1999; (e) Su¨hnel, J. and Gustav, K., Theory of proton-transfer reactions. The influence of adiabaticity on free energy relations, Chem. Phys., 87, 179– 187, 1984. 30 (a) Basilevsky, M. V., Soudackov, A., and Vener, M. V., Electron proton free energy surfaces for proton transfer reaction in polar solvents: test calculations for carbon –carbon reaction centres, Chem. Phys., 200, 87 – 106, 1995; (b) Basilevsky, M. V., Vener, M. V., Davidovich, G. V., and Soudackov, A., Dynamics of proton transfer reactions in polar solvent in the nonadiabtic two-state approximation: test calculations for carbon– carbon centre, Chem. Phys., 208, 267– 282, 1996; (c) Vener, M. V., Rostov, I. V., Soudackov, A., and Basilevsky, M. V., Semiemperical modeling free energy surfaces for proton transfer in polar aprotic solvents, Chem. Phys., 254, 249–265, 2000. 31 (a) Agarwal, P. K., Billeter, S. R., and Hammes-Schiffer, S., Nuclear quantum effects and enzyme dynamics in dihydrofolate reductase catalysis, J. Phys. Chem. B, 106, 3283– 3293, 2002; (b) Agarwal, P. K., Billeter, S. R., Rajagopalan, P. T., Benkovic, S. J., and Hammes-Schiffer, S., Network of coupled promoting motions in enzyme catalysis, PNAS, 99, 2794– 2799, 2002; (c) Hammes-Schiffer, S., Comparison of hydride, hydrogen atom, and proton-coupled electron transfer reactions, Chem. Phys. Chem., 3, 33 –42, 2002; (d) Hammes-Schiffer, S. and Billeter, S. R., Hybrid approach for the dynamical simulation of proton and hydride transfer in solution and proteins, Int. Rev. Phys. Chem., 20, 591– 616, 2001; (e) Billeter, S. R., Webb, S. P., Agarwal, P. K., Iordanov, T., and Hammes-Schiffer, S., Hydride transfer in liver alcohol dehydrogenase: quantum dynamics, kinetic isotope effects, and the role of enzyme motion, J. Am. Chem. Soc., 123, 11262– 11272, 2001; (f) Billeter, S. R., Webb, S. P., Iordanov, T., Agarwal, P. K., and Hammes-Schiffer, S., Hybrid approach for including electronic and nuclear quantum effects in molecular dynamics simulations of hydrogen transfer reactions in enzyme, J. Chem. Phys., 114, 6925– 6936, 2001. 32 While these two regimes are distinctly separated, there may exist real systems where different isotopes will be in different regimes. For example, the proton potential barrier at the solvent TS configuration may be small enough such that the proton ground vibration state may be above the barrier, but still large enough such that the deuteron ground vibration state, with a smaller ZPE, may be below the barrier. The present discussion assumes that all isotopes transfer in the same regime. 33 This statement refers to the solvent nuclear degrees of freedom; the electronic solvent polarization generally is an exception.1b 34 Thompson, W. H. and Hynes, J. T., A model study of the acid-base proton transfer reaction of the ClH· · ·OH2 pair in low polarity solvents, J. Phys. Chem. A, 105, 2582– 2590, 2001. 35 (a) Cleland, W. W. and Kreevoy, M. M., Low-barrier hydrogen bonds and enzymic catalysis, Science, 264, 1887– 1890, 1994; (b) Cleland, W. W., Frey, P. A., and Gerlt, J. A., The low barrier

Interpretation of Primary Kinetic Isotope Effects

36 37

38

39 40

41 42 43

44 45 46

47

48

49

50

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hydrogen bond in enzymatic catalysis, J. Biol. Chem., 273, 25529– 25532, 1998, For adiabatic PT, the “low barrier H-bond” situation occurs at an activated TS configuration in the solvent coordinate. Kiefer, P. M., Leite, V. P. B., and Whitnell, R. M., A simple model for proton transfer, Chem. Phys., 194, 33 – 44, 1995. (a) Hynes, J. T., The role of the environment in chemical reactions, In The Enzyme Catalysis Process, Cooper, A., Houben, J. L., and Chien, L. C., Eds., Plenum NATO ASI Series, New York, pp. 283– 292, 1989; (b) Hynes, J. T., The theory of reactions in solution, In The Theory of Chemical Reaction Dynamics, Vol. IV, Baer, M., Ed., CRC Press, Boca Raton, FL, pp. 171– 234, 1985, Remarks about recrossing corrections to TST can be found in Ref. 41 of Ref. 4. The literature concerning this solvent coordinate for both PT and ET is quite extensive. For examples, see references in Ref. 3. Although not pursued within, motion in the solvent coordinate has been explicitly connected to specific molecular rearrangements for HCl2a and HF2b to water, as well as H3Oþ to water.2a DGRXN is defined throughout as the free energy difference between reactant and product H-bond complexes, a free energy difference that is rarely experimentally determined. The full connection to experimentally determined quantities is discussed in Ref. 43 of Ref. 4. (a) Evans, M. G. and Polanyi, M., Further considerations on the thermodynamics of chemical equilibria and reaction rates, Trans. Faraday Soc., 32, 1333– 1360, 1936; (b) Evans, M. G. and Polanyi, M., Inertia and driving force of chemical reactions, Trans. Faraday Soc., 34, 11– 24, 1938, The EvansPolanyi relations were developed mainly for gas phase reactions but are also useful in solution in the context of the standard picture. Kiefer, P. M. and Hynes, J. T., Temperature dependent solvent polarity effects on adiabatic proton transfer rate constants and kinetic isotope effects, Israel J. Chem., 44, 171– 184, 2004. For real systems, ao is not expected to be exactly 0.5 due to ‘intrinsic’ asymmetry, but the deviation from 0.5 for either H or D is not expected to be significant. For further discussion, see Ref. 45 of Ref. 4. (a) Novak, A., Hydrogen bonding in solids. Correlation of spectroscopic and crystallographic data, Struct. Bond., 18, 177– 216, 1974; (b) Zeegers-Huyskens, T. and Huyskens, P., Proton transfer and ion transfer complexes, In Molecular Interactions, Vol. 2, Ratajcak, H. and Orville-Thomas, W. J., Eds., Wiley, New York, pp. 1 –106, 1980. In particular, a similar KIE versus DGRXN behavior can be observed for the tunneling regime of this new perspective.5 Pimentel, G. C. and McClellan, A. L., The Hydrogen Bond, W.H. Freeman, New York, 1960. Personal communication from E. Pines. For preliminary KIEs of photo-acids, see. (a) Krishnan, R., Lee, J., and Robinson, G. W., Isotope effect on weak acid dissociation, J. Phys. Chem., 94, 6365– 6367, 1990; (b) Pines, E., Pines, D., Barak, T., Magnes, B.-Z., Tolbert, L. M., and Haubrich, J. E., Isotope and temperature effects in ultrafast proton-transfer from a strong excited-state acid, Ber. Bunsenges. Phys. Chem., 102, 511– 517, 1998. The observed KIEs9c,9e,46 were measured by changing the solvent from H2O to D2O, and while this change in solvent introduces other possible solvent isotope effects (i.e., viscosity), the rate limiting step in each case has been shown to be a PT step (or a series of PT steps), and thus the measured KIE corresponds to PT. We need to stress that the above discussion has ignored vibrational contributions other than the AH stretch. We have already discussed in Section II the H-bond stretch contribution to DZPE‡, where it was shown that a H-bond frequency is larger for the TS compared than for the reactant. However, since this H-bond frequency difference does not significantly change between H and D transfer, the KIE magnitude is unaffected. A vibration which is isotope-dependent is the AH bending vibration, but it is not expected to significantly contribute to the KIE.4 (a) Siebrand, W., Wildman, T. A., and Zgierski, M. Z., Golden-rule treatment of hydrogen-transfer reactions. 1. Principles, J. Am. Chem. Soc., 106, 4083– 4089, 1984; (b) Siebrand, W., Wildman, T. A., and Zgierski, M. Z., Golden-rule treatment of hydrogen-transfer reactions. 2. Applications, J. Am. Chem. Soc., 106, 4089– 4096, 1984. (a) Cukier, R. I., A theory that connects proton-coupled electron-transfer and hydrogen-atom transfer reactions, J. Phys. Chem. B, 106, 1746– 1757, 2002; (b) Cukier, R. I. and Zhu, J., Simulation of

578

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52

53 54 55 56

57

58 59

Isotope Effects in Chemistry and Biology proton transfer reaction rates: the role of solvent electronic polarization, J. Phys. Chem. B, 101, 7180– 7190, 1997. The expression in Equation 21.22 is similar in form to that for weakly electronically coupled (electronically nonadiabatic) electron transfer,52 in that the proton coupling C is analogous to the electronic resonance coupling and the reorganization energy ES is analogous to the electronic diabatic solvent reorganization energy. Even though the reorganization energies and couplings are analogous, the physical picture behind the two reaction types is quite different. The reorganization energy for proton tunneling is the free energy difference associated with a Franck– Condon-like excitation (all nuclear and solvent modes other than the proton mode are held fixed) of the ground diabatic proton vibrational state at the equilibrium reactant solvent position to the ground product diabatic proton vibrational state, followed by relaxation along the solvent coordinate to the equilibrium solvent product position (see Figure 21.7d). (a) Marcus, R. A., On the theory of oxidation-reduction reactions involving electron transfer. I, J. Chem. Phys., 24, 966– 978, 1956; (b) Marcus, R. A., Electrostatic free energy and other properties of states having nonequilibrium polarization. I, J. Chem. Phys., 24, 979– 988, 1956; (c) Marcus, R. A. and Sutin, N., Electron transfer in chemistry and biology, Biochim. Biophys. Acta, 811, 265– 322, 1985; (d) Sutin, N., Theory of electron transfer reactions: insights and hindsights, Prog. Inorg. Chem., 30, 441– 498, 1983. Discussion of the influence of bending on these results can be found in Refs. 4 and 5a. The H-bond vibrational mode is assumed to remain significantly unchanged while the reaction coordinate fluctuates from the 0 – 0 TS to either the 0 – 1 or 1– 0 TS. This is particularly true for H atom transfer reactions because they are weakly coupled to a polar environment, i.e., small reorganization energies (cf. H atom transfer reaction in Ref. 1e). We note in passing that even for data over a broad temperature range where nonlinear behavior is observed, the analysis is useful to analyze different subregions in the nonlinear plot where the behavior is effectively linear, i.e., rate and KIE expressions for a given To and the local slope in an Arrhenius plot at To are obtained. (a) Antoniou, D. and Schwartz, S. D., Large kinetic isotope effects in enzymatic proton transfer and the role of substrate oscillations, PNAS, 94, 12360– 12365, 1997; (b) Antoniou, D. and Schwartz, S. D., A molecular dynamics quantum Kramers study of proton transfer in solution, J. Chem. Phys., 110, 465– 472, 1999; (c) Karmacharya, R. and Schwartz, S. D., Quantum proton transfer coupled to a quantum anharmonic mode, J. Chem. Phys., 110, 7376 –7381, 1999. Further remarks on the T dependence of the Swain – Schaad ratio can be found in Section 3c of Ref. 5a. (a) Peters, K. S., Cashin, A., and Timbers, P., Picosecond dynamics of nonadiabatic proton transfer: A kinetic study of proton transfer within the contact radical ion pair of substituted benzophenones/n,ndimethylaniline, J. Am. Chem. Soc., 122, 107– 113, 2000; (b) Peters, K. S. and Cashin, A., A picosecond kinetic study of nonadiabatic proton transfer within the contact radical ion pair of substituted benzophenones/n,n-diethylaniline, J. Phys. Chem. A, 104, 4833 –4838, 2000; (c) Peters, K. S. and Kim, G., Solvent effects for nonadiabatic proton transfer in the benzophenone/n,n-dimethylaniline, J. Phys. Chem. A, 105, 4177– 4181, 2001; (d) Andrieux, C. P., Gamby, J., Hapiot, P., and Saveant, J.-M., Evidence for inverted region behavior in proton transfer to carbanions, J. Am. Chem. Soc., 125, 10119– 10124, 2003.

22

Variational Transition-State Theory and Multidimensional Tunneling for Simple and Complex Reactions in the Gas Phase, Solids, Liquids, and Enzymes Donald G. Truhlar

CONTENTS I. II. III. IV.

Introduction ...................................................................................................................... 580 Previous Reviews ............................................................................................................. 583 Validation Against Accurate Quantum Mechanical Dynamics ...................................... 583 Theory .............................................................................................................................. 584 A. Gas Phase ................................................................................................................. 584 B. Reactions in the Solid State and at Solid Surfaces ................................................. 590 C. Reaction in Liquids .................................................................................................. 591 1. Solute –Solvent Separation................................................................................ 591 2. Reaction Coordinates and Nonequilibrium Solvation ...................................... 592 3. VTST/MT Methods for Condensed-Phase Reactions ...................................... 594 a. Implicit Bath................................................................................................ 594 b. Reduced-Dimensionality Bath .................................................................... 595 c. Explicit Bath................................................................................................ 596 D. Reactions in Enzymes .............................................................................................. 599 V. Applications to KIEs........................................................................................................ 600 A. Gas Phase ................................................................................................................. 600 B. KIEs in Liquid Phase ............................................................................................... 603 C. Enzymes ................................................................................................................... 603 VI. Software............................................................................................................................ 605 VII. Concluding Remarks........................................................................................................ 605 Acknowledgments ........................................................................................................................ 606 Glossary ........................................................................................................................................ 606 References..................................................................................................................................... 607

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I. INTRODUCTION The theory of absolute reaction rates, or transition-state-theory, forms the basis of almost all of our discussions of isotope effects in chemical reactions.1 This is as true now as it was 25 years ago, except that now we have a much better appreciation of the high accuracy afforded by transition-state theory in its modern form. This chapter is concerned with modern transition-state theory, which differs from that in use 25 years ago in two key ways: † †

Consistent incorporation of variational effects Inclusion of multidimensional corner-cutting tunneling contributions

By variational effects we mean the use of variational transition-state theory2 – 6 to optimize the transition state dividing surface. By multidimensional tunneling we mean, at a minimum, accounting for the change in vibrational frequencies of modes perpendicular to the reaction coordinate.7 For quantative accuracy though, we know that tunneling calculations must allow for the possibility of corner cutting, that is the tendency of the dominant tunneling paths to lie on the concave side of the minimum energy path.8 – 12 The incorporation of tunneling is usually accomplished by a transmission coefficient, which can also be used for other purposes, as discussed in Section IV.A. The consistent inclusion of quantization effects, especially zero-point energy, on perpendicular vibrations is implicit in the above description, and in fact for quantitative calculations of reaction rates and even for qualitative discussions of kinetic isotope effects (KIEs), purely classical variational transition state theory2 – 4 is only of historical and heuristic importance. The above discussion of quantum effects and classical transition-state theory prompts a few comments on definitions, notation, and some conceptual issues. First of all we use transition-state theory as a generic name for both conventional transition-state theory and generalized transitionstate theories, such as variational transition-state theory; however, in the present chapter, I restrict the acronym TST to mean conventional transition-state theory. Transition-state theory assumes that one can identify a hypersurface in phase space, separating reactants from products and called the transition-state dividing surface, that serves as a dynamical bottleneck (as discussed in more detail in Section IV.A). Conventional transition-state theory13,14 is the special case where the transitionstate dividing surface passes through the saddle point, and the transmission coefficient is taken as unity (thus tunneling is neglected). The transition-state dividing surface is often just called the transition state, although that term is used by some workers, especially electronic structure theorists who are not dynamicists, to denote a saddle point on the potential energy surface (PES). Conventional transition-state theory and conventional transition states are also denoted by the widely used double-dagger symbol (‡, also called a double cross), which dates back to Eyring’s secretary, Miss Lucy D’Arcy.15 We often use p or TS to denote a variational transition state. Transition states that are not located at saddle points are sometimes called generalized transition states (abbreviated GT or TS) to emphasize their nonconventional character. Variational transitionstate theory (VTST) in which the transition state is optimized for a canonical ensemble is called canonical VTST or, for short, canonical variational theory (CVT).16 Usually, in both TST and CVT, vibrations are quantized on the whole reaction path (including reactants and transition states), but rotation and reaction-coordinate motion are not. We call this quasiclassical (QC), by analogy to quasiclassical trajectories in which vibrations are initially quantized, but everything else is classical. (In our older papers, we sometimes said hybrid rather than quasiclassical.) Here is the first big disconnect: the isotope effect community tends to call this semiclassical. However, we (and a large number of chemical physicists) use the word semiclassical for WKB-like treatments of quantum phenomena such as tunneling9 – 12,17 – 19 [or, in other contexts, for methods where classical and quantal equations of motion are integrated in tandem]; thus when we say semiclassical we mean including tunneling, but when the isotope effect community says semiclassical they mean excluding tunneling. For that reason, we will not use the overworked and confusing word semiclassical in the

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rest of this chapter, except in references and at the end of Section III, where it has the chemical physics meaning, i.e., approximate quantal dynamics (not neglect of tunneling). There is another language issue which it is good to clarify at the outset, namely differences between barrier height, activation energy, activation free energy, and related terms. The potential energy surface (PES, really a hypersurface, and also called potential energy function) is the sum of the Born– Oppenheimer electronic energy and the nuclear repulsion. Chemical reactions, except barrierless ones (which are not considered in this chapter) have one or more saddle point on the PES between reactants and products. The energy difference between the lowest-energy saddle point and the energy of the reactant is called the classical barrier height or zero-point exclusive barrier height, denoted V ‡. Technically, if there could possibly be any confusion, this would be called the forward V ‡ to distinguish it from the reverse V ‡, which is the potential energy of the saddle point minus the potential energy of products. The smaller of the forward and reverse V ‡ is called the intrinsic V ‡. The energy of the products minus the energy of reactants is called the energy of reaction, denoted DE. Thus the intrinsic V ‡ is V ‡ in the direction in which DE is negative. Zero-point-inclusive barrier heights are obtained by adding in zero-point energy, recalling that the reaction coordinate has no zero-point energy at the saddle point (this coordinate is missing in the transition state) or at reactants for a bimolecular reaction (because it is a relative translational mode there), but the zero-point energy of the reaction coordinate is not zero at reactants for a unimolecular reaction. Adding zero-point energy converts V ‡ to the vibrationally adiabatic ground-state potential at the saddle point, which is called Va‡G : Subtracting the zero-point energy of reactants then yields DH0‡;o where H denotes enthalpy, the subscript denotes a temperature of 0 K, and the o in the superscript denotes standard state. Note that some workers write D‡ H0o instead of DH0‡;o : The addition of product zero-point energy to DE and the subtraction of reactant zero-point energy converts it to DH0o : Evaluating V ‡ and DH0‡;o at the canonical variational transition state instead of the conventional one converts them to V CVT and DH0CVT;o ; which can also be written as V p and DHop : Both DH0‡;o and DH0CVT;o are enthalpies of activation; technically though we should call these quasithermodynamic enthalpies of activation for three reasons. First of all, the use of thermodynamic language to discuss transition states is an analogy, not real thermodynamics. This is worthy of elaboration: Transition states are dividing surfaces in phase space; they are assumed to be in statistical mechanical equilibrium with forward-moving trajectories originating at reactants, but since the dividing surface has the wrong number of dimensions to correspond to a real molecule (the reaction coordinate z, is missing), this is not thermodynamic equilibrium. Following others,14 we call it quasiequilibrium, and the associated freeenergy difference is the (quasithermodynamic) free energy of activation. Although the quasiequilibrium language is well established, many workers omit the quasi.20 Second, the quasiequilibrium does not necessary apply to transition state phase points originating at products (which are not present in equilibrium amounts when forward rates are measured). There is a third reason why it is important to use the word quasithermodynamic. This is the use of transition-state language to express experimental results. The transition-state theory rate constant at temperature T may be written in the form k ¼ ðC o Þ12n gðTÞ

kB T exp½2DGTTS;o =RT h

ð22:1Þ

where C o is the standard-state concentration, n is the molecularity (2 for bimolecular, 1 for unimolecular), g is the transmission coefficient, kB is Boltzmann’s constant, h is Planck’s constant, DGTTS;o is the quasithermodynamic standard-state molar free energy of activation at temperature T, and R is the gas constant. However, when experimentalists express rate data in thermodynamic language (which is very common in the enzyme kinetics community, but also seen often in physical organic chemistry papers), they use k ¼ ðCo Þ12n

kB T exp½2DGoact ðTÞ=RT h

ð22:2Þ

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o Clearly DGTS;o (and its special cases DGTCVT;o and DG‡;o T T ) and DGact are different quantities. We call the former the quasithermodynamic standard-state free energy of activation and the latter the phenomenological free energy of energy. Of course one often shortens the notation; for example, “standard-state free energy of activation” becomes “activation free energy.” (When n ¼ 1, the dependence on choice of standard state is often smaller than experimental error.) The transmission coefficient is often far from unity. Even when g is included in a calculation, theory can predict the phenomenological free energy of activation. Comparing Equation 22.1 and Equation 22.2 yields

DGoact ¼ DGTS;o 2 RT ‘n gðTÞ T

ð22:3Þ

o DGoact ¼ DHact ðTÞ 2 TDSoact ðTÞ

ð22:4Þ

DGTS;o ¼ DHTTS;o 2 TDSTS;o T T

ð22:5Þ

Furthermore we have

The Gibb– Helmholtz equation also has analogies for the quasithermodynamic and phenomenological quantities. Thus o DHact ¼ 2T 2

DHTTS;o ¼ 2T 2

dðDGoact =TÞ dT

ð22:6Þ

dðDGTS;o T =TÞ dT

ð22:7Þ

Clearly, just as in Equation 22.3, all the phenomenological activation quantities have two contributions — one from the quasithermodynamic part of Equation 22.1 and one from the transmission coefficient. Another language that has been used for these contributions is substantial for the part of the former (quasithermodynamic) that can be calculated with a single temperatureindependent definition of the TS and nonsubstantial for the rest.21 This is a useful computational distinction, but it depends on the definition used for the TS (and the choice of temperature); see Section IV.C.2. Finally we come to activation energy, by which we mean the Arrhenius activation energy, denoted Ea. This is defined by replacing Equation 22.2 with the much simpler equation: k ¼ AðTÞexpð2Ea ðTÞ=RTÞ

ð22:8Þ

where Ea ; 2R

d ‘n k dð1=TÞ

ð22:9Þ

Thus this too is a phenomenological quantity. However, perhaps surprisingly, Tolman showed that Ea has a very simple interpretation. It is the average energy of molecules that react minus the average energy of all possible reactants in the system.22,23 This provides insight into the temperature dependence of Ea,24 which usually leads to a concave Arrhenius plot, that is, Ea increases as T increases. o Much confusion has been engendered by confusing V ‡, DH0TS;o ; DHTTS;o ; DGTS;o T ; DGact ; and Ea in various combinations or by neglecting the temperature dependences of the last four of these quantities. A good illustration of the danger is provided by the gas-phase reaction of O(3P) with H2 to form OH and H. Computing Ea from Equation 22.9 by using average slopes over various T ranges,

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one obtains 14 kcal/mol at 1400 – 1900 K, 8 kcal/mol at 318 – 471 K, and 6 kcal/mol at 250 –300 K,25 – 27 but V ‡ is estimated to be 13 kcal/mol.25,27,28

II. PREVIOUS REVIEWS Variational transition state theory with multidimensional tunneling has been reviewed several times, and it might be useful to provide a guide to some of these reviews that could be most useful to the present readers. A historical and conceptual overview6 has already been referenced; additional overviews of the current status of transition state theory were given in the Henry Eyring Memorial Issue of JPC29 and the Centennial Issue of JPC.30 An introductory and conceptual overview is available in Accounts of Chemical Research5 followed by an even shorter introduction;31 a more complete attempt at a self-contained derivation and explanation of the foundations is given in a NATO Advanced Study Institute volume,32 which is recommended reading for all new students of VTST. Other pedagogical chapters are available in the Techniques of Chemistry series33 and the Encyclopedia of Computational Chemistry.34 A handbook-style article that gives full details for gas-phase calculations (except for the final details of the corner-cutting tunneling methods, which were improved later) is also available.35 The above reviews are focused on fundamental theory and gas-phase applications. Reviews of condensed-phase applications are also available: vacuum –solid state interfaces,30,36 reactions in liquids,29,30,37 and reactions catalyzed by enzymes.38,39 A review specifically addressed to KIEs is also available.40

III. VALIDATION AGAINST ACCURATE QUANTUM MECHANICAL DYNAMICS A key attribute of VTST with multidimensional tunneling (VTST/MT) is that it has been well validated against quantum mechanical dynamics calculations. This is important because the uncertainties in the potential energy surface cancel out when one compares converged quantum dynamics to approximate dynamics (VTST/MT) for a given PES, whereas attempts to validate the theory by comparison to experiment always raise questions about the PES. An early review of the validations against accurate quantal dynamics was provided,41 and a very complete review of validations for 74 three-body reactions (collinear and three-dimensional) for which accurate quantal rate coefficients are available has also been written.42 The latter article includes a number of isotopically substituted reactions. A key issue in making the tunneling calculations reliable is to allow for sufficient corner cutting. We have developed two approximations, to be discussed in more detail below. One, called the small-curvature tunneling (SCT) approximation43 is valid when the curvature of the minimumenergy path (MEP) is small; in such a case the tunneling paths are close enough to the MEP that the potential energy along the optimum tunneling paths may be represented semiquantitatively (i.e., as well as for calculating transition state-zero-point energies) by harmonic expansions centered on the MEP. The other, called the large-curvature tunneling (LCT) approximation,44 which is required when the curvature of the MEP is large, involves tunneling in a broad region of coordinate space called the reaction swath, and in particular it requires information about the potential energy in regions too far from the MEP to be well represented in MEP-based coordinates. By microcanonically (i.e., at each energy) optimizing the tunneling path from among these two approximations, we obtain a method called optimized multidimensional tunneling (OMT) or microcanonical OMT (mOMT).44 To illustrate the kind of accuracy that may be obtained with this method, we may consider, as an example from the review42 mentioned above, the (real three-dimensional) reactions of O(3P) with

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H2, with HD to produce OH, with HD to produce OD, and with D2 at 300 K. Conventional transition state theory is in error by factors of 0.05, 0.08, 0.15, and 0.14, respectively, which would give errors of up to a factor of 3 in kinetic isotope effects. VTST with multidimensional tunneling that neglects corner cutting reduces the errors, and the factors become 0.16, 0.20, 0.38, and 0.36, which would reduce the maximum error in a KIE to a factor of 2.4. Including corner-cutting tunneling by the same microcanically optimized MT method, mOMT, that is practical for (and has been applied to) reactions in large systems such as polyatomic molecules and enzymes, reduces the error further, resulting in factors of 1.16, 1.16, 1.47, and 1.29 (the errors are reduced because the factors are closer to unity), which would give a maximum error in KIEs of 1.27. This is a typical example of the improvement afforded by VTST with optimized multidimensional tunneling. Validation against accurate quantum dynamics for larger systems is precluded in most cases by the lack of accurate quantal results for systems with more than three atoms. However such comparisons are available for H þ CH4 and O(3P) þ CH4 and they show similar agreement to what we found for atom – diatom reactions.45 – 47 For condensed-phase systems, we do not have comparisons to accurate quantal results, but we do have comparisons to other completely different semiclassical methods that show good agreement for hydrogen site hopping on rigid and non-rigid Cu surfaces, as reviewed elsewhere.30 Comparison has been made to accurate quantal dynamics for an Eckart barrier linearly coupled to a harmonic bath (which is a model of reaction in liquid solution); the calculations agree well with each other.48,49

IV. THEORY A. GAS P HASE The theory of VTST with multidimensional tunneling is most straightforward for gas-phase bimolecular reactions, and an understanding of this case provides a good foundation for understanding the condensed phase. The key historical references for the treatment we now use are as follows: † † † † † †

VTST16,50,51 more general reaction coordinates52 – 54 consistent incorporation of multidimensional tunneling7,55,56 small-curvature tunneling43,57 large-curvature tunneling44,58,59 optimized multidimensional tunneling44,59

The fundamental assumption of transition-state theory is the existence of a dynamical bottleneck. Consider a reaction A þ B ! products

ð22:10Þ

d½A ¼ k½A ½B dt

ð22:11Þ

with a rate coefficient k defined by 2

where t denotes time, and [X] is the concentration of X. Technically, transition-state theory replaces k by a one-way flux coefficient corresponding to the rate of passage of phase points (i.e., trajectories) through a hypersurface in phase space that separates reactants from products. The hypersurface is the transition state, and often it is just called the dividing surface. If reactants are in local equilibrium, and all trajectories passing through the dividing surface in the direction

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of products originated at reactants and will proceed fully to products without ever returning to the dividing surface, then TST is exact in a classical mechanical world. Otherwise it overestimates the rate. Therefore, in VTST2 – 6,16,29 – 35,41,42,60 one optimizes the location of the dividing surface to minimize the rate. The optimized dividing surface is called the variational transition state or the dynamical bottleneck. In practice, one quantizes the vibrational motions, which involves replacing classical partition functions, which are phase space integrals, by quantal ones, which are sums over states; because of quantum effects such as this the upper bound no longer strictly applies. Nevertheless the variational optimization of the dividing surface at this stage is still justified on physical grounds. As a final step, one may add a transmission coefficient ðgÞ to account for tunneling ðg . 1Þ or for trajectories that recross the transition state ðg , 1Þ; or both. We have found that the accuracy of VTST is typically very good42,45,46 even if we do not correct for recrossing; therefore in most of our work we include a transmission coefficient only to correct for tunneling, except that we also correct for nonclassical reflection whenever we correct for tunneling. (Nonclassical reflection is the wave phenomenon of diffraction by the barrier top, which means that particles with more than enough energy to surmount the barrier are nevertheless partially reflected by it. This is the obverse of tunneling, and it partly cancels the enhancement due to tunneling, but because tunneling occurs at energies with larger Boltzmann factors than nonclassical reflection, tunneling almost always dominates — except for very small intrinsic barriers.) In general we can write g approximately as a product:

g ¼ kðTÞGðTÞgðTÞ

ð22:12Þ

where kðTÞ accounts for tunneling (technically, in light of nonclassical reflection, for the net effect of treating the reaction coordinate quantum mechanically), G for classical or quasiclassical reflection, and g for the breakdown of the equilibrium assumption. In general for gas-phase reactions we have set G ¼ g ¼ 1; then g ¼ k: We shall discuss the transmission coefficient further below. We place the trial transition states at trial positions z0 along the reaction coordinate, and we specify the geometry (shape) and orientation of the dividing surface by V: Without tunneling, the VTST rate constant (see Equation 22.11) for a bimolecular reaction at temperature T may then be written as kVTST ¼ ðC o Þ21 ¼

kB T exp{ 2 ½GGT;o ðT; zp ; Vp Þ 2 GR;o ðTÞ =RT} h

kB T min QGT ðT; z0 ; VÞe2VRP =ðz0 Þ=RT hFR ðTÞ z0 ;V

ð22:13Þ ð22:14Þ

where GT (as already stated) is a synonym for TS, R denotes reactants (including their relative translation), FR ðTÞ is the reactants’ partition function per unit volume, z0 is the value of the reaction coordinate at which the dividing surface crosses the reaction path, V denotes the shape and orientation of the TS, QGT is the partition function of the generalized transition state (with its zero of energy on the reaction path), VRP is the molar potential energy on the reaction path, and zp and Vp are the values of z0 and V that minimize the calculated rate constant. (The overall translation of the entire system has no effect on the rate and is omitted in all quantities.) Here we include symmetry numbers in partition coefficients, although in most of our previous papers, we collected the symmetry numbers as a separate factor. If the nuclear motions were governed by classical mechanics, Equation 22.13 and Equation 22.14 would provide an upper bound to the accurate rate constant.4,16,32,61 If one completely optimized the dividing surface not just in coordinate space but as a function of all coordinates and momenta (that is, in phase space), one would obtain the exact local-equilibrium rate constant in a classical world. This is a rigorous result, but impossible to achieve. Anyway the real world is quantum mechanical, and we assume that all vibrational partition functions in Equation 22.14 are quantized. This makes the theory less rigorous but more relevant to real molecules.

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By comparing Equation 22.13 and Equation 22.14 to each other we see that minimizing the calculated rate constant is equivalent to maximizing the free energy of activation. The VTST free energy of activation may be written DGCVT;o ðTÞ ¼ max GGT;o ðT; z0 ; VÞ 2 GR;o ðTÞ z0 ;V

ð22:15Þ

We could also label DGCVT;o as DGp : The concept of reaction coordinate plays an important role in the theory. In fact, there is more than one reaction coordinate. Globally the reaction coordinate is defined as the distance s along the reaction path. This coordinate plays a critical role in tunneling calculations. Locally the reaction coordinate is the degree of freedom (called z) that is missing in the generalized transition state. Even locally, this may differ from s. A good general prescription for the reaction path is the steepest descent path in isoinertial coordinates or, more technically, the union of the steepest descent path from the saddle point to the reactants with the one from the saddle point to products.7,8,62 Isoinertial coordinates are coordinates where every direction of motion is associated with the same reduced mass. There are two essentially equivalent ways to obtain isoinertial coordinates, mass scaling and mass weighting. Spectroscopists prefer mass-weighted coordinates defined by63 xab ¼ m1=2 a Rab

ð22:16Þ

where Rab is the Cartesian coordinate (b ¼ x, y, or z) of atom a; and ma is the mass of atom a: ˚ . In these coordinates all directions of These coordinates have units of mass1/2 length, e.g., amu1/2 A motion have a unitless reduced mass of unity (which I find confusing). I prefer mass-scaled coordinates defined by xab ¼ ðma =mÞ1=2 Rab

ð22:17Þ

where m is an arbitrary constant with units of mass. In this coordinate system, coordinates have units of length, and all directions of motion have a reduced mass of m: If one sets m ¼ 1 amu, then xab ˚ has the same numerical value as qab in units of amu1/2 A ˚ . If one sets m equal to the mass in units of A of whatever atom dominates the polyatomic motion of interest, then xab has a numerical value of approximately the same magnitude as the actual distance moved by that atom, which is convenient for thinking about lengths of tunneling paths. Further discussion of these coordinates is provided in a recent comment.64 It can easily be shown that the steepest descent path is the same in all isoinertial coordinate systems;50 one special case is the “intrinsic” reaction path (usually abbreviated IRC) used by Fukui.65 We denote the steepest descents path in isoinertial coordinates7,8,62 by minimumenergy path (MEP). Note that at the saddle point the MEP is parallel to the imaginary-frequency normal mode. Elsewhere it is curved, and the curvature is physically meaningful because of the mass scaling; this is useful for understanding corner-cutting tunneling.10,66 The transition-state dividing surface is defined by the MEP only on the reaction path itself. The full definition of the transition-state dividing surface requires a global definition (a definition that is valid off the MEP as well as on it) of the reaction coordinate s or z,67 after which the oneparameter sequence of dividing surfaces (with parameter s0 or z0 ) is defined by s ¼ s0 or z ¼ z0. In our original work16,51 we took the dividing surface to be locally a plane orthogonal to the MEP in isoinertial coordinates, but we approximated the partition functions in a way50,51,61 that provided physical results even when this surface is unphysical beyond a certain distance from the MEP. This procedure leads to a reaction coordinate z that is rectilinear, e.g., it is the distance along a straight line in both Cartesian and isoinertial coordinates. In later work we defined the dividing surface at locations displaced from the MEP in terms of curvilinear coordinates,52 in particular the valence coordinates used in spectroscopic force fields.

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Such valence coordinates63 are the collection of bond stretches, valence angle bends, and bond torsions, and they are often redundant (which complicates the treatment68). For this kind of dividing surface, the reaction coordinate z is curvilinear, i.e., not the distance along a straight line. A further advance involves the optimization of the orientation of the dividing surface so that it need not be locally perpendicular to the reaction path. Our algorithms for doing this are called reorientation of the dividing surface (RODS)53,69,70 and variational reaction path (VRP).54 Since the reaction coordinate z is locally defined as the distance along a normal to the dividing surface, this corresponds to optimizing the definition of the reaction coordinate. In the context of association reactions, an algorithm in which the reaction coordinate definition is optimized along with the position of the dividing surface along a one-parameter sequence of paths is called “variable reaction coordinate” (VRC) variational transition-state theory.71,72 Thus the RODS, VRP, and VRC algorithms are instances of a more general variational prescription. One key difference, however, is that the RODS and VRP algorithms are used with quantized partition functions (a plus compared to VRC) but (a minus) have no simple way to include vibration – rotation coupling (which is handled by classical mechanics in the VRC algorithm). There are also differences in the treatment of anharmonicity and vibrational mode – mode coupling as well as the kinds of reactions to which the methods are well suited. Because quantization of the vibrational partition functions on the dividing surface is very important for simple barrier reactions with tight transition states, the formalism we have developed is well suited to such reactions. In such cases we have found that the variational transition state can be found by optimization of a oneparameter sequence of dividing surfaces orthogonal to the reaction path, where the reaction path is defined as the MEP. By using the RODS algorithm one can calculate rate constants for several isotopic versions of a reaction from a single reaction path,70 which is more efficient than using the MEP, which changes upon isotopic substitution or isotopic scrambling. In the last few years there has also been tremendous progress in the treatment of barrierless association reactions with strictly loose transition states. A strictly loose transition state is defined as one in which the conserved vibrational modes are uncoupled to the transition modes and have the same frequencies in the variational transition state as in the associating reagents.73,74 (Conserved vibrational modes are modes that occur in both the associating fragments and the association complex, whereas transition modes include overall rotation of the complex and vibrations of the complex that transform into fragment rotations and relative translations upon dissociation of the complex.) Progress has included successively refined treatments of the definition of the dividing surface and of the definition of the reaction coordinate that is missing in the transition state71 – 77 and elegant derivations of rate expression for these successive improvements.75 – 78 The recent variational implementation of the multifaceted dividing surface (MDS) variable reaction coordinate (VRC) version of VTST seems to have brought the theory to a flexible enough state that it is suitable for application to a wide variety of practical applications. Although some refinements (e.g., the flexibility of pivot point placement for cylindrical molecules like O277) would still be welcome, the dynamical formalism is now very well developed. However this formalism is probably less interesting for KIEs because the most interesting KIEs seem to be associated with reactions that have tight transition states. There is some work that suggests that VTST rate constants can be calculated accurately for low-barrier79 – 81 and even no-barrier associations82 – 84 with reaction-path methods such as those that we have employed for tight transition states. This requires further study. We now return to the transmission coefficient. Let the transition state (the dividing surface that yields the minimum in Equation 22.14) value of any quantity be denoted p . Then, multiplying Equation 22.14 by the transmission coefficient to account for multidimensional tunneling (MT) yields kVTST=MT ¼ kðTÞ

p kB T Qp ðTÞ 2VRP =RT e R h F ðTÞ

ð22:18Þ

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Isotope Effects in Chemistry and Biology

In a tunneling region, at least one component of the nuclear momentum is imaginary and is associated with a negative contribution to the kinetic energy. As mentioned in Section III, in multidimensional systems this leads to a negative centrifugal effect — also called the negative bobsled effect or the quantum bobsled effect.8,10 We usually call it corner cutting; the physical picture is that the tunneling path is not on the MEP but rather is on its concave side. In practice, as already mentioned in Section III, we have developed two very useful approaches to the multidimensional tunneling problem. In both of these we estimate k by a multidimensionnal WKB-like approximation as a function of tunneling energy, and the temperature-dependent transmission coefficient in Equation 22.18 is obtained by averaging the tunneling probabilities calculated for a set of tunneling energies and tunneling paths. In a more complete theory (for example, Ref. 12 or 17), one would optimize the tunneling paths; the optimum tunneling paths minimize WKB imaginary action integrals, which in turn maximizes the tunneling probabilities. We have found42 that sufficiently accurate results can be obtained by a simpler criterion44 in which, for each energy, we choose the maximum tunneling probability from two approximate results: one, called small-curvature tunneling (SCT),43,57 calculated by assuming that the curvature of the reaction path is small, and the other, called large-curvature tunneling (LCT),6,31 – 33,35,44,57 – 59,85 calculated by assuming that it is large. The result is called microcanonically optimized multidimensional tunneling (mOMT) or, for short, optimized multidimensional tunneling (OMT). The resulting VTST/OMT rate constants have been carefully tested against accurate quantum dynamics,42,45,46 and the accuracy has been found to be very good. The SCT, LCT, and OMT tunneling calculations differ from one-dimensional models of tunneling in two key respects: (i) These approximations include the quantized energy requirements of all the vibrational modes along the tunneling path; since the vibrational frequencies are functions of the reaction coordinate, this changes the shape of the effective potential for tunneling. (ii) These approximations include corner-cutting tunneling. Corner cutting has the effect that the tunneling path is shorter than the minimum-energy path. Therefore the optimum tunneling paths involves a compromise between path length and effective potential along the path. As a consequence, the optimum tunneling paths occur on the concave side of the minimum energy path, i.e., they cut the corner.8 – 12,31,32,66,85 – 91 The considerations summarized above have important implications. In order to calculate the VTST rate constant we must not only know the geometry, energy, and frequencies at the saddle point, but also we must know the reaction path, reaction-path potential, and vibrational frequencies over a wide enough range of s0 to find the minimum in Equation 22.3. Calculating tunneling contributions requires even more information about the potential energy surface because tunneling occurs over a wider region than the region where the variational transition state occurs. This is easy to understand if one recognizes that the variational transition state approximately corresponds to the maximum of the effective potential for tunneling in the smallcurvature limit. To calculate the whole barrier we need potential energy surface information over a wider range than is required to find the maximum of the barrier. When reaction-path curvature is large, corner cutting becomes extensive and one needs to know potential energy surface information not just on the reaction path but also over a broad swath on its concave side; this is called the tunneling swath.87,91 In recent work,92 we made the LCT tunneling calculation more efficient by developing an interpolation scheme that yields the required information about the PES in the large-curvature reaction swath with a much smaller number of electronic structure calculations than were required by the previous algorithm. The algorithm can be further improved93 by interpolating not only along each tunneling path but also from path to path. The improved algorithms mean that LCT calculations now involve much less work, compared to SCT calculations, than they used to, and these savings are important for applying high levels of electronic structure theory to complex reactions.

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A function that plays a key role in the tunneling calculations is the ground-state (G) vibrationally adiabatic potential curve VaG ðsÞ; which is defined by VaG ðsÞ ¼ VRP ðsÞ þ 1G ’ ðsÞ

ð22:19Þ

where s is distance along the reaction path, VRP is the potential energy along the reaction path, and 1G ’ is the ground-state energy (zero-point energy) of the vibrational modes perpendicular to the reaction path. This is called vibrationally adiabatic because the vibrational quantum numbers are conserved (they all remain zero) as one proceeds along the reaction path. A critical aspect of including a transmission coefficient is that it must be consistent with VTST. We have solved this problem by using the ground-state approximation5,55,56 for the transmission coefficient. In this approximation, the transmission coefficient is the ratio of an approximate quantum mechanical thermally averaged transmission probability for reaction in the ground state to the thermally averaged transmission probability implied for ground-state reaction in VTST. The transmission probability implied for ground-state reaction by VTST is a Heaviside step function at a total energy of VaG ðsp Þ where VaG is defined in Equation 22.19. One can treat the threshold region even more accurately by a method we call improved canonical variational theory (ICVT).35,56,94 For most reactions, the results are almost identical to CVT. Hence we usually just report the simpler CVT results, which is easier for readers to understand. Note that “probability of reaction in the ground state” in the previous paragraph refers to being in the ground state as one passes through the transition-state region. In general, vibrational quantum numbers are not conserved all the way from reactants to products. However we find that for smallcurvature of the reaction path one can calculate accurate transmission coefficients by considering VaG ðsÞ as a global effective potential for tunneling because, at the low energies where tunneling is important, the reactive flux in the transition-state region is primarily associated with the vibrational ground state although many low-energy vibrational and rotational states of the reactant and product may contribute to this flux.95 – 100 In comparison, the LCT case is more complicated in two respects: (i) We need nonadiabatic effective potentials for a variety of tunneling paths exhibiting extreme corner corning. (ii) On the low-energy side of the transition state we must connect the tunneling path to vibrationally excited states (of the product for an exoergic or symmetric reaction and of the reactant for an endoergic one).58,86,88 – 90 Pictures of tunneling paths for tunneling into vibrationally excited states are given elsewhere.44,86,89,90 Even when the tunneling proceeds formally into the ground state, the large-curvature algorithm involves a vibrationally nonadiabatic effective potential in the tunneling region. Note that we use the ground-state approximation for the transmission coefficient at all T. At low T (where tunneling is important) it is justified by the fact that the system passes through the transition region either in the ground state (at least on the high side of the transition state) or in an energetically similar state. At high T, k tends to unity for any reasonable choice of effective tunneling potential so the ground-state approximation remains acceptable. Another point to be emphasized here is the separability of the reaction coordinate. An intrinsic aspect of transition state theory without a transmission coefficient is the separability of the reaction coordinate. (A coordinate is called separable if there are no cross terms in the Hamiltonian that couple it or its conjugate momentum to other coordinates or their conjugate momenta.) A truly separable reaction coordinate would not exhibit recrossing, and so a transmission coefficient that corrects for recrossing is also a correction for nonseparability. Corner-cutting tunneling is also a manifestation of nonseparability. Thus VTST/MT can be much more accurate than a theory with a separable reaction coordinate. At temperatures where k . 2; the reaction is dominated by tunneling, and the nonseparable tunneling dynamics is a larger contributor to the calculated rate than the overbarrier part calculated with a separable reaction coordinate. Now we return to the consideration of the other factors, G and g, in Equation 22.12. First consider G: Recrossing of the TS dividing surface can in principle invalidate the use of

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transition-state theory, but in our experience this effect is usually small for simple barrier reactions in the gas phase if the TS is variationally optimized. The effect is expected to become more important as temperature increases,41,42,61 and our extensive tests against accurate quantum mechanics show that VTST tends to overestimate rate constants by about 25– 30% at 1500– 2400 K.42 It is hard to estimate the effect for polyatomic reactions because accurate quantal results are not generally available at high T. Furthermore it is not clear whether we can believe estimates based on classical or quasiclassical trajectory calculations because they tend to overestimate recrossing due to not enforcing quantization conditions at the dynamical bottleneck. Unimolecular reactions in the gas phase require an additional consideration, namely the competition between preferential reaction of the more energetic molecules in the reactant ensemble and replenishment of the equilibrium distribution by energy transfer. Gas-phase unimolecular reactions are well known to suffer a fall-off from the high-pressure limit, and there is general agreement that this is due to vibrational nonequilibrium.101,102 In fact, more often than not, the reaction cannot be studied at high enough pressure to actually observe the full high-pressure plateau, which corresponds to equilibrium conditions. For this reason we will not consider gasphase unimolecular reactions in this chapter. In contrast, nonequilibrium effects are generally assumed to be negligible for gas-phase bimolecular reactions, although detailed evidence for this is not as complete as one would like. There have been many studies of translational nonequilibrium, and they agree that that effect is small, although not totally missing for the most quantitative work.103 Vibrational nonequilibrium could be more of a problem given that vibrational spacings are large,104 but one careful study found that the effect is small, even for fast reactions,105 although further work with more general assumptions about the state-to-state transition probabilities is required for a definitive statement. Nevertheless, for reactions with free energies of activation of several RT or higher, the most credible evidence from studies of gas-phase reactions is that it is a good approximation to assume reactant equilibrium — with one exception: vibrational nonequilibrium is well known to be large for one very special class of bimolecular reactions: diatomic dissociation. This has been studied by many workers,106 – 111 and it is explained by the fact that vibrational relaxation is slow in a molecule with only one vibrational degree of freedom.

B. REACTIONS IN

THE

S OLID S TATE AND

AT

S OLID S URFACES

The gas-phase methods discussed above have been extended to reactions at crystalline surfaces36 by the embedded-atom method.112 – 115 The same formalism applies to solid-state reactions. The system is separated into a primary subsystem (cluster) and a secondary one (lattice), with the former embedded in the lattice. The lattice is frozen, while the atoms of the cluster are allowed to vibrate. For an N-atom cluster, the system has 3N degrees of freedom and the generalized transition states have 3N 2 1 degrees of freedom. This differs from a nonlinear gas-phase system where the corresponding numbers are 3N and 3N 2 7, respectively; the reason for this difference is that the fixed lattice removes translations and rotations from the problem. Consider the case of a reactive solute (i.e., a reactive guest molecule, e.g., an isomerizing complex) in a solid host. The cluster can be just the solute or even a portion of the solute, but the spirit of the method is that the cluster also includes a reasonable number (20 –100) of host atoms, with the rest of the host serving as the lattice. As one carves a larger and larger cluster out of the fixed lattice, the method (in principle) converges. A simpler way to treat reactions in solids is just to ignore the host.116,117 This replaces three low-frequency lattice vibrations by substrate rotations, and the effect of this on rate predictions has not been systematically tested. Other approximations that one makes in this approach may be more important; namely, one neglects the effect of the lattice on the reaction path and on the potential energy profile and solute vibrations along the reaction path. If one includes the effect of the host as a static field, one recovers the special case of the embedded cluster method where the solute is the cluster and the host is the lattice.

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Usually, in calculating the tunneling contributions, one averages over a continuous distribution of tunneling energies in the reaction coordinate. This makes sense because at the saddle point, the reaction coordinate is an unbound vibration with a continuous spectrum. At reactants, the reaction coordinate becomes a relative translational mode for a bimolecular reaction or a collision of a reactive species with a solid surface, but it becomes a bound vibration for a unimolecular reaction. Unimolecular reactions in solids and at solid surfaces are sometimes studied at very low temperatures where the discreteness of the reactant’s reaction-coordinate energy distribution must be considered. In such cases the integral over tunneling probabilities as a function of tunneling energy is replaced by a sum over tunneling probabilities at discrete energies.113,117,118 At low enough temperature, the reaction rate constants and KIEs become independent of temperature because all reaction occurs out of a single state (the ground state). In practice, this T-independent limit may be approached as follows. First, as T is lowered, the rate constant becomes almost independent of T as the guest molecule is cooled to its ground state, but a small temperature dependence will remain due to thermal excitation of low-frequency lattice vibrations (lowfrequency phonon modes).117 Then, as T is lowered further, even the lattice-state distribution tends to a single state. There has been some discussion in the literature of coherent quantum processes at low temperature, but when phenomenological rate constants have been observed they seem to be well described by the same kind of formalism that we use at higher T. One does need to reexamine the equilibrium assumption though, just as discussed for liquids in the next section.

C. REACTION IN L IQUIDS 1. Solute– Solvent Separation Reactions in liquids29,30,33,37,119 – 123 or amorphous solids are more complex than those in the gas phase or in well-ordered crystalline environments. Two types of transition-state theory treatments may be distinguished. In explicit solvent models, all solute and solvent molecules are treated on an equal footing, at least from the point of view of dynamics (one might still use hybrid methods to obtain the potential energy surface). In implicit solvent treatments33,37 the solute is treated explicitly and the solvent is treated by a continuum-solvation model; a special case would be to include only the solvent reaction field. However, there are many treatments that fall on the boundary between explicit solvent and implicit solvent. For example, one might freeze certain degrees of freedom of either the solute or the solvent in one or more steps of the calculation. For another example, in implicit solvent treatments, one sometimes singles out a few solvent molecules in the first solvation shell to be treated on the same footing as the solute; this yields the so called mixed discrete – continuum models. Methods for reactions in liquids can also be applied to amorphous solids and even crystalline solids, although I shall just say liquids in this section. Section IV.B discussed methods that take advantage of the ordered structure of a crystalline host. In the gas phase, in crystalline solids, and in approximations that take advantage of a solute – solvent separation with implicit solvent, the reactive molecule or solute often has only one saddle point or a small number of saddle points whose contributions can be treated individually and added. For example, the gas-phase reaction of OH with propane to make n-propyl radical and H2O proceeds through two transition states, gauche and trans.124 Similarly the aqueous Menshutkin reaction of NH3 with CH3Br passes through a single saddle point if only the solute is explicit.125 However, if we add explicit solvent there are innumerable saddle points differing from one another in the arrangement of solvent hydrogen bonds and in torsions around hydrogen bonds, not just in the first solvation shell but throughout the liquid. The key new issue that occurs in liquid-phase transition-state theory that does not occur in gases or in crystalline environments is the intrinsically statistical character of the medium. Unless one uses statistical methods (Monte Carlo or molecular dynamics) to average over the myriad of solvent configurations, one may miss the essence of the problem (at worst) or at least

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miss important quantitative contributions to the enthalpy and entropy of activation. In principle, continuum models, because they directly approximate the free energy of solvation (not just the solute –solvent interaction energy) include even the entropic contribution of the changing number of low-energy solvent configurations as the system proceeds along the reaction path (the free energy of solvation includes the change in internal free energy of the solvent126). However, in practice one cannot always be confident that the first solvation shell is well represented by a continuum model. That is the motivation for mixed discrete –continuum models. However, adding even a few (three or more) solvent molecules to the explicit subsystem greatly increases the number of floppy degrees of freedom so that statistical averaging becomes difficult again. In other words, even a small cluster consisting of a solute molecule and a small number of solvent molecules may require, for a proper treatment, the same kinds of statistical mechanical averaging that one requires to treat the whole liquid statistically. 2. Reaction Coordinates and Nonequilibrium Solvation For transition-state theory calculations on reactions in solution and solids, we must revisit the question of nonequilibrium effects, i.e., the magnitude of the g factor in Equation 22.12. There is one class of liquid-phase reactions where nonequilibrium effects are well known to be important. That is outer-sphere electron transfer (also called weak overlap electron transfer) where nonequilibrium bath contributions are a critical part of Marcus’ theory.127,128 For other reactions in liquids, the conventional assumption is that liquid reactions are at equilibrium, i.e., like the highpressure limit in gases. Like many assumptions that work quite well in practice, this assumption is often accepted without examining its basis, but there is work available that is relevant to the question of reactant disequilibrium in reactions with nuclear motion in the reaction coordinate. In seminal work,129 Kramers identified three regimes for liquid-phase reactions: weak solute– solvent coupling, intermediate solute –solvent coupling, and strong solute –solvent coupling. An extreme limit of weak coupling of a molecule to its surroundings would be a gas, but for liquids the weakcoupling regime is associated with a low-viscosity solvent. Strong coupling would be a highviscosity solvent, where the “frictional” effect of solvent is sometimes associated with small-scale spatial diffusion across the transition state region — since diffusion involves a lot of back and forth motion to accomplish small net motion, it leads to recrossing of the transition state, and this is what is meant when one says that friction decreases a rate. First we consider the limit of weak solute – solvent coupling, sometimes thought of as energydiffusion.130,131 In the energy-diffusion regime, energy flow within the solute (or into the solute) cannot keep up with reaction, and the rate slows down. This is the analog of the nonequilibrium effect that is well established for gas-phase unimolecular reactions. Can one observe this in liquids? Perhaps the only example where the interpretation is detailed enough to merit serious concern is the cyclohexane isomerization. This interpretation is based on the pressure dependence of the rate and it is not definitive, although it does provide caution about accepting the local equilibrium assumption uncritically.132 – 134 At the same time, one must also be cautious about invoking disequilibrium on the basis of an inadequate appreciation of transition state theory. For example, it is important to realize that even though some modes may be more effective at causing reaction than other modes are, TST fully includes this kind of effect. Thus, for example, an argument that tunneling or classical barrier recrossing is promoted by some particular vibration of the reactants is not a disequilibrium effect. Next we consider the nonequilibrium effect associated with strong solute –solvent coupling. This effect is sometimes called nonequilibrium solvation, but this term is often misunderstood. To sort out the issues it is important to recall the quasiequilibrium discussion in Section I. One might say that the fundamental assumption of transition state theory is that one can calculate the reaction rate by calculating the one-way flux toward products of an ensemble of trajectories in a transition state that is in quasiequilibrium with reactants. Now if the reactant states are in local

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equilibrium104 with each other, these states, by Liouville’s theorem135 of statistical mechanics, will evolve into an equilibrium distribution in the transition-state region. Label a sequence of trial dividing surfaces by z ¼ zi with reactants at large negative z; products at large positive z; and zi , ziþ1 : Suppose that z ¼ z2 is a perfect dynamical bottleneck, that is, all species with z ¼ z2 that are moving toward products came directly from reactants without ever having had z ¼ z2 before, and they will proceed to products without ever having z ¼ z2 again. Then transition-state theory will be perfect if we put the TS at z ¼ z2 : However, if we put the TS at z ¼ z1 ; the TS ensemble will include some nonreactive trajectories moving toward z2 that turn around before they get there and recross the z ¼ z1 dividing surface toward reactants. I would say that transition-state theory overestimates the rate constant because it counts those nonreactive recrossing trajectories. However, you could also identify those phase points (a phase point is a point in phase space, that is, it is a point on a trajectory) at z ¼ z1 ; that are moving toward large z but will recross the z ¼ z1 dividing surface and remove them from the TS ensemble. The resulting ensemble is missing some phase points that are present in the equilibrium ensemble, and so you might say that the error in the transition-state calculation is due to the fact that it uses the equilibrium ensemble — you would call the error a nonequilibrium effect. This would be correct if you define your terms, but it is not the language used here. It is more informative to recognize that the ensemble at z ¼ z1 really is equilibrated to a good approximation ðg < 1Þ but some members of the ensemble are points on recrossing trajectories ðG , 1Þ: Another way to correct the calculation with z ¼ z1 is to calculate how much the actual rate is reduced relative to the transition-state theory one (calculated at z ¼ z1 ) because of the recrossing trajectories and to multiply the transition-state theory rate constant by a transmission coefficient G; which is less than unity. Then the calculation at z1 ; with G , 1; and the calculation at z ¼ z2 ; with G ¼ 1; give the same answer. However, the division of the free energy of activation (see Equation 22.2) into quasiequilibrium and nonsubstantial contributions (as defined in Section I) will change. Thus this division is actually a description of a particular calculation that was done with a particular definition of the transition state, and one should be careful not to attribute a deeper significance to it. When one hears the statement that G < 1; the true meaning is that somebody has found a transition state definition that is good enough to set G < 1: However, one finds that if G ! 1; it does not prove that transition-state theory breaks down for this reaction; rather it means that transition-state theory breaks down with that particular definition of the TS. Defining the TS is equivalent to defining the reaction coordinate z since the TS dividing surface is the mathematical surface with z ¼ zi : Suppose that all surfaces defined by z ¼ zi include some recrossing trajectories. Then variational transition-state theory cannot be perfect with z chosen as the reaction coordinate. Suppose, however, that one can define a new coordinate s by s¼zþu

ð22:20Þ

where u is some other coordinate, such that s ¼ sp does define a perfect transition state. Then I would say that s is the best choice for reaction coordinate and that coordinate u participates in the reaction coordinate. The discussion above has particular relevance to reactions in solution. In particular suppose that z is a solute coordinate, and u is a solvent coordinate. Then I would say that the solvent participates in the reaction coordinate. The equilibrium ensemble that I would calculate without considering the solvent (u) cannot describe the rate process in a transition state sense. This is called nonequilibrium solvation. In other words nonequilibrium solvation is the participation of solvent coordinates in the reaction coordinate. If the solvent coordinates that participate in the reaction coordinate are manageable, for example the atomic coordinates that are necessary to describe the length of some solute – solvent hydrogen bond, then the most direct way to proceed is to allow these coordinates to explicitly

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participate in the reaction coordinate. Our experience has been that if those solvent molecules that participate in the reaction coordinate are included in the explicit degrees of freedom, and the reaction coordinate is identified with the path of steepest descents through the explicit degrees of freedom in isoinertial coordinates, i.e., the MEP, this procedure will “automatically” generate a reaction coordinate that includes the relevant degrees of freedom in the reaction coordinate. In other words it seems that one can take G < 1 if one uses the MEP as the reaction coordinate. Sometimes, however, it is not practical to do this. For example, for weak-overlap electron transfer, the best reaction coordinate seems to be a solvent electric polarization coordinate.127,128 Because of the long-range nature of electrostatic forces, the calculation of the electric field exerted on the reactive molecule by the solvent requires knowing the Cartesian coordinates of a large number of solvent molecules. Suppose this number is a hundred. Including all these solvent molecules in the explicit system would greatly increase the number of saddle points and hence the number of steepest descent paths. Furthermore the steepest descent paths might not all be independent, that is, there might be only small barriers separating the valleys surrounding each of these paths so that one cannot just sum and average over their independent contributions calculated with the harmonic approximation for vibrations perpendicular to the path. When we are unable to find a good enough reaction coordinate to make G < 1 because of difficulty in identifying the collective solvent motions that participate in the reaction coordinate, we say that solvent friction is responsible for slowing down the rate process. The treatment of this kind of effect in terms of friction and collective solvent coordinates is discussed elsewhere.29,30,37,119,121,122,126,136 – 146 In light of these complexities, there is more than one practical way to proceed, as discussed next. We will divide the methods into two categories: (i) those based on an MEP33,37,121,140,141,143 – 145,147 – 149 or a small number of MEPs whose fluxes are additive and (ii) a method based on a potential of mean force and an ensemble of MEPs.38,39,150 – 152 The first category includes summing over a small number of transition states with differing conformations, if necessary (see the hydroxyl plus propane example mentioned above), and it is especially suitable for implicit-solvent methods; whereas the second category is designed to handle the case of an uncountable number of saddle points as arise in explicit-solvent treatments of liquid-phase reactions. Furthermore the methods in the first category may be subclassified into implicit bath models and reduced-dimensional bath models. We consider two implicit bath methods, in particular separable equilibrium solvation (SES)148 and equilibrium solvation path (ESP);33,121,147,148 and we consider one reduced-dimensional-bath model, which is called the nonequilibrium solvation (NES) method.141,143 Finally we consider one ensemble-based method, called ensemble-averaged variational transition-state theory with multidimensional tunneling (EA-VTST/MT).38,151 Note that the EA-VTST/MT method was originally developed for enzyme reactions, but it is general enough to apply to any kind of reaction. 3. VTST/MT Methods for Condensed-Phase Reactions a. Implicit Bath In these methods the system consists of a primary system and a bath, and the reaction coordinate depends only on the coordinates of the primary system. The primary system includes the reactive solute for a unimolecular reaction and the reacting solutes for a bimolecular reaction; in addition it can include one or more solvent molecules. The solvent may be treated by a continuum solvent model or a discrete one, although if one makes the solvent discrete, the explicit solvent model of Section IV.C.3.b can also be used and may in fact be required. In the implicit solvent models we let x denote the 3N coordinates of the N-atom explicit subsystem (primary system). Then the multidimensional potential of mean force (PMF) is written as WðxÞ ¼ VðxÞ þ DGoS ðx; TÞ

ð22:21Þ

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where V is the gas-phase potential energy surface of the primary system, and DGoS is the standardstate free-energy37 of solvation of the rigid solute at temperature T: Note that W(x) is obtained by averaging over a thermal distribution of the implicit molecules of the solvent. At thermal equilibrium, the mean force of the implicit molecules on the 3N coordinates of the explicit ones is the negative of the gradient of W.153 In the separable equilibrium solvation (SES) method, we find the MEP and vibrational frequencies on the surface V, then add the second term of Equation 22.21 for geometries along the MEP. This gives VTST without tunneling. Tunneling can no longer be approximated with the ground-state approximation of Section IV.B since information about the liquid solvent is only available at finite T, and the system does not remain a liquid all the way to temperatures where only the ground state is populated. Furthermore the available information ðDGoS Þ about the solvent is thermally averaged, but the strictly correct procedure would be to tunnel through an unaveraged effective potential, then average the tunneling probabilities, not tunnel through a free-energy profile (it is physically incorrect to treat the entropic contribution to the free-energy as a potential energy that can be tunneled through). However, if the temperature dependence of DGoS is not too large, one can derive an approximation,147,148 called the zero-order canonical mean shape approximation (CMS-0), to the correct tunneling average that involves W(x) as the effective potential. In the SES approximation we use the CMS-0 approximation for both SCT and LCT tunneling calculations. As in the gas phase, care is taken147 to insure that the transmission coefficient is consistent with VTST so that it becomes unity if quantum effects on the tunneling coordinate motion are neglected. The equilibrium solvation path (ESP) approximation is the same as the SES one except that the MEP and vibrational frequencies are computed from W rather then from V. This can be very important because the low-energy region of the best liquid-phase variational transition state might be quite different from any geometry in the sequence of geometries along the gas-phase MEP.154 Another example of when it would be very important to use a reaction path optimized in liquid solution would be when the very nature of the transition state depends strongly on solvent.155 Extreme examples are when the reaction does not occur in the gas phase or occurs in a different way (or by a different mechanism), when the reactants or products are unstable in the gas phase, or when the reaction is barrierless in the gas phase but not in the liquid. b. Reduced-Dimensionality Bath Rather than treat all solvent degrees of freedom implicitly one may single out one or a few solvent coordinates to be included in the explicit coordinate set. If these coordinates consist of the atomic Cartesians of one or a few solvent molecules, then this yields a mixed discrete –continuum model156 – 159 that can be treated just as in Section IV.C.3.a and requires no further discussion. Another possibility, though is to use one or more collective solvent coordinates, which represent collective motion of the solvent. Collective solvent coordinates were already mentioned in Section IV.A in conjunction with electron transfer reactions. An especially broad class of collective solvent coordinates is based on energy gaps; we have recently provided a critical analysis of this kind of solvent coordinate and compared it to coordinates based on valence coordinates.145 Sometimes very useful collective solvent coordinates can be defined by using a specific model for the reaction, such as charge transfer.137,140,160 The formulation141,143 of NES theory that we have used is more general, and it does not assume a specific form for the solvent coordinate. For the case of a single solvent coordinate, one adds the following solvent Hamiltonian to the Hamiltonian of the 3N atoms of the primary system for a generalized transition state at a location s along the reaction path: Hsolvent ¼

p2y 1 þ DGoS ðx; TÞ þ F½ y 2 Cðx 2 x‡ Þ 2m 2

2

ð22:22Þ

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where py is the momentum conjugate to collective solvent y, m is the scaling mass of Equation 22.17, F is the force constant, C is the solute – solvent coupling vector, x (as before) is a vector of coordinates of the primary system, and x ‡ is the geometry of the saddle point of W. All vectors in Equation 22.22 are of order 3N. When C is a null vector, the solvent is equilibrated to the primary system; alternatively, if y were constrained to have the value yeq ¼ C·ðx 2 x‡ Þ

ð22:23Þ

the theory would also reduce to ESP theory. In the NES model, though, y is allowed to vibrate around its equilibrium position. One can obtain values for the new parameters F and C from specific solvent models or from general considerations involving general quantities like solvent viscosity, diffusion coefficients, and solvent relaxation times. After the parameters are fixed, the calculations proceed just as in the ESP case except that there are 3N þ 1 coordinates instead of 3N. In transition-state theory, the partition functions and quasithermodynamic free energy of the transition state are actually surrogates for a dynamical quantity, the one-way flux through the transition-state dividing surface. The reactant partition functions and free energy are, however, true equilibrium thermodynamic quantities. Thus nonequilibrium solvation enters the theory for the transition state but not the reactant, and C is nonzero for generalized transition states but C ¼ 0 for reactants. (Reactant disequilibrium is something different and, if present, enters through g, not G and not Equation 22.22.) c. Explicit Bath The theory for an explicit bath is called ensemble-averaged variational transition state with multidimensional tunneling (EA-VTST/MT). It was originally developed38,151 for reactions catalyzed by enzymes, and implementation details151,152,161 were also worked out in that context, but it is also applicable to other condensed-phase reactions. We review it here in the liquid-phase language of Section IV.C.3.a. The EA-VTST/MT model has two stages. In stage one, all atoms of the system are treated on an equal footing. In practice one would treat from 102 to 105 atoms with periodic or stochastic boundary conditions. The first step is to calculate a one-dimensional potential of mean force W(z) where z is the stage-one reaction coordinate. A one-dimensional PMF is very similar to the multidimensional PMF considered above except that only one coordinate is fixed (and all the rest averaged over) instead of fixing 3N coordinates. The coordinate z is fixed in the sense that W(z) corresponds to a particular value of z. It can also be fixed at a sequence of values along the reaction path in simulations used to compute W(z);162 – 171 simulations with fixed z are called constrained-z simulations. An alternative to constrained dynamics is restrained dynamics or restrained Monte Carlo in which one calculates W(z) from simulations in which z not fixed but varies freely.163,172 – 179 To improve the statistics one applies a bias potential (restraining potential) that ideally would be the negative of the true PMF. Furthermore it is convenient to break the simulation into a series of windows, in each of which one applies a harmonic restraining force whose purpose is to force the system to spend more time in a particular region near the minimum of the associated harmonic potential. After the simulation the probability distribution is corrected to remove the artificial bias(es), and the PMF is calculated from the unbiased distribution. This procedure is called umbrella sampling or multistage importance sampling. In the description in stage 2 below, we will assume that molecular dynamics calculations with umbrella sampling were used in stage 1. During the umbrella sampling run, configurations are saved at well-spaced time intervals for possible later analysis. A subset of these configurations will also be used in stage 2.

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Standard methods for calculating WðzÞ are purely classical mechanical. Our first goal, however, is to calculate a quasiclassical WðzÞ in which the vibrations that are strongly coupled to z are quantized. We have developed a procedure to do this,150 based on projected instantaneous normalmode analysis. From W(z) we can calculate a free energy of activation profile DGGT;o ðzÞ: If z is a rectilinear T coordinate, one simply needs to subtract the free energy of the reaction coordinate of the reactant and adjust the zero of energy. If z is not rectilinear, there is also a Jacobian term to be added;180 for z coordinates that are simple functions of valence coordinates, the Jacobian term is usually less than RT (and can be neglected in all but the most precise work) although if z is a collective solvent coordinate, the Jacobian term could be large. Maximizing DGGT;o ðzÞ with T respect to z yields DG CVT(T). Putting this into Equation 22.1 with g ¼ 1 yields a stage-1 approximation k (1) to the rate constant. This approximation is quasiclassical. In stages 2 and 3 we continue to use the stage-1 quasiclassical value of GCVT ðTÞ as part of the calculation, but we no longer assume that the transmission coefficient is unity. These two stages differ in the way that the field exerted by the secondary zone is included in the primary-zone dynamics calculations.151 In either stage, the transmission coefficient g is the average over a large number of transmission coefficients gi calculated for various possible reaction paths of the primary system corresponding to transition-state configurations i selected from the variational transitionstate window of stage 1 (or other nearby windows — see next paragraph). Each of these gi consists of two factors, gi ¼ Gi ki ; with one factor Gi accounting for dynamical recrossing4,29,33,35,136,181 and the other factor ki accounting for the increase in the rate due to quantum mechanical tunneling contributions. The latter factor is defined by extending the definitions of a consistent CVT transmission coefficient that were used earlier in the gas phase,7,56 in embedded clusters in solids,113 and in liquid solution.147 The reaction coordinate used in stage 1 may be inaccurate, and the transmission coefficient is designed to make up for that inaccuracy as well as to include tunneling. In stage 2, one samples the transition-state ensemble determined in stage 1. In particular, one selects the L saved configurations from stage 1 that have z values closest to the value zpð1Þ at which the quasiclassical W(z) of stage 1 has its maximum at the temperature of the simulation. In practice L might, for example, be in the range from 5 to 20. This selection process is sometimes called rare-event sampling; a difference of what is done here from the standard method181 – 185 is that in our case the PMF is quasiclassical with many of the vibrations perpendicular to the reaction coordinate being quantized.150 This is particularly important for quantitative work and for the calculation of KIEs. For each of the L selected members of the transition-state ensemble, one calculates a transmission coefficient gð2Þ i ; where i ¼ 1; …; L: This is called the stage-2 transmission coefficient or the static-secondary-zone (SSZ) approximation to the transmission coefficient. During stage 2, the system is divided into a primary zone and a secondary zone.151,179,186 The partition of atoms into one zone or another can be done in various ways, similar to the primary system/bath partition in Section IV.C.3.a. Then, for each configuration i of the entire system, selected from the transition-state ensemble as specified in the previous paragraph, one performs the following steps: (a) Freeze the secondary zone with its coordinates for this ensemble member. (b) Optimize the primary zone coordinates to the nearest saddle point. (c) Find the MEP of the primary zone in the field of the fixed secondary zone. (d) Carry out a VTST calculation without tunneling. This yields a new quasiclassical rate constant, and one defines a quasiclassical transmission coefficient by

Gi ;

kCVT ðT; iÞ kð1Þ ðTÞ

ð22:24Þ

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(e) Add multidimensional tunneling, still with the secondary zone frozen. This yields another transmission coefficient:

ki ;

kCVT=MT ðT; iÞ kCVT ðT; iÞ

ð22:25Þ

One then repeats this L times, after which one can calculate the stage-2 quasiclassical rate constant by ð2Þ kQC ¼ Gð2Þ ðTÞkð1Þ ðTÞ

ð22:26Þ

where

Gð2Þ ¼

L 1 X G L i¼1 i

ð22:27Þ

and the final stage-2 rate constant, including tunneling, by kð2Þ ¼ gð2Þ ðTÞkð1Þ ðTÞ

ð22:28Þ

where

gð2Þ ¼

L 1 X Gk L i¼1 i i

ð22:29Þ

Notice that each of the L stage-2 calculations is essentially an embedded cluster calculation of the type described in Section IV.C.2. However, although the secondary zone is static for each of these calculations, it varies from calculation to calculation. For a given i the reaction coordinate is the optimum one for that fixed secondary zone. Furthermore the reaction coordinate dependence on the secondary zone coordinates is not neglected, as in Section IV.C.3.a, but is built in by the dependence of the MEP on i. Thus the averaging of the transmission coefficient over the TS ensemble is a practical way to allow the secondary zone coordinates to participate in the reaction coordinate. Just as in Section IV.A and Section IV.C.3.a, care is taken151 to insure that the transmission coefficients of Equation 22.24 and 22.25 are consistent so that Gi becomes unity if there is no variational effect, and ki becomes unity if quantum effects on the reaction coordinate are neglected. In many (maybe even most) cases, it is probably sufficient to use the SSZ approximation. In fact, Hynes and coworkers concluded that what they call the nonadiabatic solvation limit, in which G is governed by frozen-solvent configurations at the transition state, often provides an excellent description of the barrier passage.138,142 Nevertheless, we have developed a stage-3 procedure151,161 that can be used to include bath relaxation along each MEP. In this stage, the primary-zone atoms are held fixed at a sequence of geometries on a stage-2 MEP, and, for each structure in the sequence, the secondary zone is equilibrated to the primary zone to obtain the relative change in free energy of the bath (recall that the secondary zone is already equilibrated to the primary zone at the stage-1 variational transition state). Adding the change in secondary-zone free energy to the effective potential along the path gives new values of Gi and ki : The former is averaged to obtain G ð3Þ ; and the product Gi ki is averaged to obtain gð3Þ : Then, a stage-3 quasiclassical rate constant and final rate constant are obtained by ð3Þ kQC ¼ G ð3Þ ðTÞkð1Þ ðTÞ

ð22:30Þ

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and kð3Þ ¼ gð3Þ ðTÞkð1Þ ðTÞ

ð22:31Þ

respectively. In stage 3, the effective potential includes the change in free energy of the secondary zone along the primary-zone reaction coordinate in an average way, by assuming that the bath is equilibrated to the primary system. Consequently, the transmission coefficient in stage 3 is called the equilibrium secondary zone (ESZ) approximation.151 One could imagine an even more sophisticated treatment in which the secondary zone motion is included but without assuming it is equilibrated. Such a treatment is available,145 but so far it has not been widely applied nor presented in completely general way. Such a treatment would be required to allow the secondary-zone atoms or the polarization of distant parts of the bath to participate explicitly in the reaction coordinate, rather than effectively through the ensemble average. However the evidence to date is that the EAVTST/MT method is adequate even for at least some systems in which secondary-zone motions play an important role in promoting reaction.187 It is worth emphasizing that in the SSZ approximation, the secondary zone is only static for a given i. Since it varies from i to i it is effectively not static. Similarly in the ESZ approximation, the secondary zone is equilibrated for a given i, but differently equilibrated for each i. Thus this approach includes nonequilibrium solvation, despite its name, which refers only to a given member of the TS ensemble.

D. REACTIONS IN E NZYMES Reactions in enzymes are intermediate between those in the liquid phase and those in a crystalline solid phase. In the implicit bath methods of Section IV.C.3.a, liquid-phase reactions are treated by the separable equilibrium solvation (SES) and equilibrium solvation path (ESP) approximations. Both of these approximations take advantage of the clear separation between solute and solvent. However there is no such clear-cut separation in enzyme kinetics unless we treat the whole enzyme, coenzyme (if any), and substrate as a solute, which is impractical. Therefore we use a somewhat arbitrary separation into primary subsystem and secondary subsystem,151,179,186 which is similar to the separation described in Section IV.B for processes at gas –crystal interfaces.114 When we do this, the nature of typical enzyme activation sites leads to a more structured environment for the primary system (substrate or part of substrate plus nearby of the protein and perhaps all or part of a prosthetic group) than we have when we carve a primary system consisting of a solute and a few solvent molecules out of a liquid. Aside from that the EA-VTST/MT method is applied to enzymes in the same way as already explained. In fact, as already stated, this method was originally developed in the context of enzymes.38,151,152,161 Enzyme reactions also have unique aspects that require special considerations not involved in modeling simple reactions. Foremost among these is the enzyme itself. As compared to a small solute or substrate, the enzyme, despite having well defined secondary, tertiary, and quaternary structure, can be floppy enough to make significant configurational entropic contributions to the free energy of activation, and the individual physical contributions to each stage of the calculation may be quite different than for liquids. One further comment specific to enzyme kinetics is that we must distinguish the kind of dynamical disequilibrium discussed in Section IV.C.2 (disequilibrium in the state space of the reactants of an elementary reaction step) from mechanistic disequilibrium. One might say that an enzyme reaction that has two fast steps that are not cleanly separated kinetically has some of the latter, which is often called kinetic complexity. However, we see no strong evidence in any enzyme mechanism we have studied of nonequilibrium distributions in the dynamical sense, and this does

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not surprise us based on the experience with nonbiological reactions summarized above. Thus we expect that such effects are small.

V. APPLICATIONS TO KIES Variational tradition state theory with multidimensional tunneling (VTST/MT) has been widely applied, and in several cases KIEs have been computed. This section gives a brief summary of some of these applications. Our notation is that X/Y denotes the ratio of the rate constants for X and Y. In addition to discussing TST (conventional transition-state theory), CVT (canonical variational transition-state theory), CVT/SCT (CVT with small-curvature tunneling), and CVT/OMT (CVT with optimized multidimensional tunneling), which are all explained above, we sometimes compare to CVT/ZCT where ZCT denotes zero-curvature tunneling. Zero-curvature tunneling calculations are the same as SCT except that reaction-path curvature is neglected. As a consequence this level of theory is the same as CVT/SCT and CVT/OMT with the one exception that it does not allow corner cutting in the tunneling.

A. GAS P HASE Comparison to accurate quantal results for KIEs in O þ H2 and isotopomers was already mentioned in Section III. For polyatomic reactions though we do not have accurate quantum mechanical KIEs for comparison. The [1,5] sigmatropic hydrogen shift in cis-1,3-pentadiene has an experimental KIE of 5.2 at 470 K.188 Dewar et al.189 proposed that this occurs by tunneling but calculated a KIE of 146 at 498 K. Although an accurate PES is not known, Austin Model 1 (AM1) gives a reasonable PES, and we used it to study this reaction. For 1H, k is 4.2 in the ZCT approximation but 6.5 with OMT. For D, these values are 2.5 and 3.0, respectively. Our calculated KIEs43 are 2.25 without tunneling, 3.7 with ZCT, and 4.9 with SCT (which is the same as OMT in this case). Note that the PES in the above calculation (and in most of our work since then) is implicit, not an analytic function. In particular the calculation was carried out by direct dynamics,43,44,87,190 – 193 which means that whenever we require the potential energy or its gradient or Hessian, we perform an electronic structure calculation. Thus the PES is defined implicitly by the level of electronic structure theory that is chosen. The experimental KIE for CF3 þ CD3H ! CF3H/CF3D is 3.2 at 627 K.194 We studied44 this reaction by using specific reaction parameters44,192,193,195 in AM1; the resulting implicit PES44 is called AM1-SRP-2. The KIE is 2.1 without tunneling, about 2.6 with ZCT tunneling, and 3.0 with OMT tunneling.44 The effects are much larger at 300 K where the KIE is 5.5 at the CVT level, 7.7 with CVT/ZCT, and 16 with CVT/OMT. For the reaction of H with CH4 and the SN2 reactions of Cl2, Cl(H2O)2, and ClðH2 OÞ2 2 ; with CH3Cl, we carried out a factor analysis of the vibrational partition functions to show which modes contribute to the secondary KIEs. We found non-negligible contributions to the KIEs from high-frequency modes, mid-frequency modes, and low-frequency modes. This work is reviewed elsewhere.40,196 We studied the reaction of OH with CH4, CD4, and 13CH4.197 We found that even without tunneling, VTST predicts quite different KIEs than TST. Thus these reactions provide a good illustration of a key difference between TST and VTST. In TST the geometry of the lowestenergy point in the TS in the same for all isotopic variations because that point is the saddle point of the PES, which is independent of isotopic substitution. Not only the geometry of the saddle point but also its force constant matrix (Hessian) is invariant to isotopic substitution. Much of the beauty of the classic TST formulation of KIEs1 follows from this fact. However, in real reactions, the dynamical bottleneck can depend on the isotopic constitution of the reagents. A measure of the difference in geometries is the difference in VMEP ðsp Þ; i.e., the potential energy on the reaction

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path at the variational transition state. For OH þ CH4 the PES used in our study has V ‡ ¼ 7.37 kcal/mol. The value of VMEP ðsp Þ for OH þ CH4 varies from 7.15 to 7.03 kcal/mol as T increases from 250 to 2400 K, whereas for OH þ CD4 it varies from 7.28 to 7.17 kcal/mol, and for OH þ 13CH4 it varies from 7.19 to 7.10 kcal/mol. The Boltzmann factor of this energy difference contributes as much as a factor of 1.31 to the CH4/CD4 KIE and a factor of 1.08 to the CH4/13CH4 KIE. For CH4/CD4 the contributions from high-, mid-, and low-frequency vibrations differ from unity by as much as 0.88, 0.57, and 6.40, respectively; the corresponding numbers for CH4/13CH4 are 1.002, 0.970, and 0.889. The calculation of heavy-atom KIEs requires high precision. For CH4/13CH4 our calculated overall KIE of 1.005 at 273– 353 K agrees with the experimental value198 at those temperatures. For CH4/CD4, the only experimental value199 was 11 at 416 K, and we calculated 4.5 at that temperature. However, after our paper was submitted, a new experimental value of 4.0 was reported.200 Further analysis of these KIEs was reported by other workers.201 – 203 A quasithermodynamic-type21 analysis of the perprotio case by Masgrau et al.204 is particularly informative in understanding the temperature dependence and magnitude of the OH þ CH4 rate constant. We also calculated CH4/CD4 and CH4/13CH4 KIEs, as well as CH4/CH3D KIEs, for Cl þ CH4. A direct dynamics study205 gave good agreement with experiment206 – 210 whereas a later study211 based on an analytic PES was less accurate. The CH4/CD4 KIEs were also calculated for O(3P) þ CH4,212 but no experiment is available. At room T, the CVT/OMT KIE is 4.0 times larger than the TST value because of tunneling, whereas at 2400 K it is 7% lower. Further analysis of the KIE is available.213,214 For the SN2 reaction of F(H2O)2 with CH3Cl, three experimental KIEs were reported215: H2O/ D2O ¼ 0.65, CH3Cl/CD3Cl ¼ 0.85, and [F(H2O)2 þ CH3Cl]/[F(D2O)2 þ CD3Cl] ¼ 0.56. We216 calculated 0.65, 0.83, and 0.54, respectively. We showed that these KIEs are dominated by the highfrequency stretching mode of the microsolvating water molecule, which is consistent with some,217 – 219 but not all, previous interpretations of bulk solvent isotope effects. The key to the interpretation is that the hydrogen bond to water is weaker at the transition state than at the reactant due to charge delocalization away from the ion. A review220 is available. In later work, the solute and solvent KIEs for methyl chloride solvolysis were studied with upto 13 explicit water molecules.221 We predicted the OD and ND3 KIEs for the OH þ NH3 reaction,222 but so far they have not been measured. The experimental CH3X/CD3X KIEs of the C12 þ CH3Br, Cl2 þ CH3I, and Br2 þ CH3I SN2 reactions at 300 K are 0.79 ^ 0.05, 0.84 ^ 0.02, and 0.76 ^ 0.03, respectively.223 – 225 We calculated 0.94 ^ 0.03, 0.91, and 0.94 ^ 0.02.226 It is encouraging that we correctly predicted the inverse nature of these KIEs, but the errors are disconcertingly large. The inverse character of the KIEs was attributed to high- and low-frequency modes, offsetting a normal contribution from midfrequency modes. Kato et al.225 agreed that our neglect of third-body collisions is reasonable under the experimental conditions and concluded that further theoretical work is required to fully understand the KIEs. We calculated C2H5Cl/C2D5Cl KIEs for the competitive E2 and SN2 gas-phase reactions of ClO2 with C2H5Cl as functions of T.227 The treatment required additional assumptions beyond VTST to predict the branching ratio. The KIE for the SN2 reaction is predicted to decrease with T, whereas the KIE for the E2 reaction is predicted to be very small at low T, then to increase with T. Experimental228 – 230 and calculated231 KIEs for various isotopic substitutions in Cl þ H2 ! HCl þ H are listed in Table 22.1 (when there is more than one experiment we compare to the average). The trends are all well reproduced by theory. As a follow up we calculated the accurate quantum mechanical rate constant for the 1H2 case for this potential energy surface and found that it agreed with our transition-state theory rate constants within 16% over a factor of 7.5 in T.232

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TABLE 22.1 Kinetic Isotope Effects for Cl 1 H2 KIE H2/D2

H2/T2 H2/HDa H2/HTb H2/DTb HD ! HCl/DCl

a b

T (K) 245 300 345 275 345 245 345 245 345 275 345 300 445

Experiment 13 ^ 2 8^1 7^1 34 18 3.4 2.5 6.5 4.1 21 12 1.8 1.4

Theory 15 10.5 8.3 51 26 4.4 3.1 7.5 6.0 26 15 2.1 1.8

sum of production of HCl and DCl. sum of production of HCl and TCl.

Experiments are available for the reaction of OH with C3H8, C3H6D2, C3H5D3, C3H3D5, and C3H2D6, and C3D8.233 Because of multiple sites, and conformations, simulating these experiments requires 22 rate calculations. The results124 exhibit some striking features. For example, the CVT rates are factors of 2 – 3 lower than the TST ones for C3H8 and 10 –30% lower than TST for C3D8; the errors of conventional TST for different isotopologs do not largely cancel, and the KIEs decrease by a factor of 2 when the variational effects are included. Abstraction of H from a secondary carbon has a classical barrier of only 2.2 kcal/mol, and k is actually greater for D than for H. This effect (that tunneling decreases the KIE) cannot be predicted by one-dimensional tunneling methods, but it is easy to see how it arises in multidimensional models. In particular, the zero-point-inclusive barrier is much smaller for H, and this eliminates most of the tunneling for H, but not for D. Thus at 295 K, the H/D KIE for a secondary site reaction is 5.6 with TST, 2.7 with CVT, and 2.6 with CVT/OMT. For the trans primary-site reaction these values are 6.6, 3.1, and 6.6, whereas for the gauche primary-site reaction they are 6.6, 4.4, and 5.0. In all three cases the CVT/mOMT KIE is less than the TST one. The deuterium KIEs for the association reaction of H with C2H4 presented a great challenge to theory. The very poor agreement of theory with experiment raised questions about the assumptions of TST and even about the validity of the Born –Oppenheimer approximation.234 – 236 An even greater challenge to theory was provided by the muonium þ C2H4 experiments of Garner et al.237 We calculated a more accurate PES and applied VTST/MT to the high-pressure rate constants79 – 81 and obtained good agreement with experiment for a large number of KIEs. Although the barrier is low, , 1.7 kcal/mol, tunneling is very important for calculating reliable KIEs for this reaction. The location of the variational transition state depends significantly on both isotopic composition and temperature. We also studied Mu KIEs with H2 and D2,238,239 HBr,240,241 and CH4.242 Muonium KIEs are extremely challenging because of the very large zero-point energy associated with bonds involving Mu and the concomitantly large anharmonicity and because the very light mass of Mu makes the tunneling very quantal, but we successfully predicted238 the reaction rate for Mu þ D2 before it was measured.

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In addition to Mu þ CH4, we studied ten other isotopic variants of the H þ CH4 and CH3 þ H2 reactions.242 Since one of the implicit PESs we used should be quite accurate the calculations provided a good test of the dynamical methods or of experiment, and indeed some of the experiments do seem to merit reexamination. An earlier paper on H þ CD3H243 employed an older, less accurate version of the multidimensional tunneling approximation that somewhat overestimates the effect of reaction-path curvature when it couples more than one perpendicular mode to the reaction coordinate (the current version of the SCT43,57 and OMT44,58,59 approximations were introduced later). Nevertheless the reader is referred to the earlier paper for a historical discussion of coupled-motion tunneling and its effects on secondary KIEs. A key point made in that discussion is that it is important to consider coupling due to reaction-path curvature along the whole portion of the MEP in the tunneling region, not just the mixing of various coordinates in the imaginaryfrequency normal mode of the saddle point. Furthermore, because tunneling is often delocalized over a large region, one should almost always avoid the parabolic approximation, unless one uses an effective frequency that takes account of the nonquadratic character of the barrier244 and the curvature of the reaction path.245

B. KIES

IN

LIQUID P HASE

The H2/D2 KIEs for liquid-phase hydride transfer reactions, in particular bimolecular hydride transfer between NADþ analogs88 and unimolecular hydride migration in a polycyclic hydroxy ketone246 were calculated in the SES approximation, the former using reduced-dimensionality analytic surfaces and the latter by dual-level direct dynamics with 81 vibrational degrees of freedom explicitly participating in the tunneling calculation. The bimolecular case is dominated by largecurvature tunneling and the unimolecular case by small-curvature tunneling. The reaction of H with methanol to produce H2 and CH2OH in water was studied in the SES,148,247 ESP,121,148,247 and NES141,143 approximations. The (H þ CH3OH)/(D þ CH3OD) secondary KIE is calculated to be 0.48, 0.51, and 0.37 in these three approximations,143,247 respectively, in comparison to an experimental value 248 of 0.7. The (H þ CH3OH)/(H þ CD3H) primary KIE was calculated to be 21.3, 20.2, and 19.5, respectively, in comparison to an experimental value249 of 20. In both cases we assumed a solvent relaxation time of 10 fsec. The solute – solvent coupling strengths were predicted by a model based on diffusion coefficients. The primary KIE is relatively insensitive to this solute – solvent coupling strength, but the secondary KIE is predicted to be larger, in particularly to be 0.44 (and hence in slightly better agreement with experiment) if we arbitrarily decrease the solute –solvent coupling strength by a factor of two. It is not clear though if the treatment of the nonequilibrium solvation effect is the largest error in the theory (the gas-phase potential energy surface and the free energy of solvation surface are also uncertain). This aqueous free radical reaction is a very interesting test case for theory because the solvation effects are smaller and apparently more subtle than those in ion reactions. Clearly more work is needed.

C. ENZYMES Many enzyme reactions involve proton or hydride transfer in the chemical step, and we know from experience with simpler reactions that multidimensional treatments of the tunneling process are essential for quantitative accuracy and sometimes even for qualitative understanding. Our experience with gas-phase reactions showed that we can obtain good accuracy for reactions dominated by tunneling including corner-cutting tunneling paths, but enzyme reactions bring in new issues such as the possible need for more complicated reaction coordinates. We have now used the EA-VTST/OMT method reviewed above to calculate KIEs for several enzyme-catalyzed reactions38,39,151,152,161,179,186,250 – 252 and the comparison of these results to experiment253 – 265 provides a test of whether the gas-phase theory can be successfully

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TABLE 22.2 Enzyme Systems for which KIEs have been Calculated by EA-VTST/OMT Enzyme

Coenzyme

H6

D

A

QPES

N1

Ntot

L

Enolase LADH MADH XyI SCAD DHFR

Mg2þ NADH, Zn2þ TTQ Mg2þ FAD NADPH

Hþ H2 Hþ H2 H2 H2

C C C C C C

N N O C N C

AM1 AM1 þ SEVB PM3 2 SRP PM3 þ SVB AM1 þ SVB AM1 þ SVB

25 21 25 32 43 40

8863 5506 11025 25317 23277 21468

1 18 6 5 15 13

Notation: H^, transferred ion; D, donor; A, acceptor; QPES, method used for quantum mechanical part of potential energy surface; N1, number of atoms in primary zone for the tunneling calculations; Ntot, total number of atoms; L, number of systems averaged in transition state ensemble. Abbreviations: LADH, liver alcohol dehydrogenase; MADH, methylamine dehydrogenase; XyI, xylose isomerase; SCAD, short-chain acyl CoA dehydrogenase; DHFR, dihydrofolate reductase; NADH, reduced nicotinamide adenine dinucleotide; TTQ, tryptophan tryptophylquinone; FAD, flavin adenine dinucleotide; NADPH, reduced nicotinamide adenine dinucleotide phosphate; AM1, Austin model 1; SEVB, semiempirical valence bond; SRP, specific reaction parameters; SVB, simple valence bond.

extended to complex reactions in solution. (The enolase calculations179 use a simpler version of the theory, but we expect that a calculation with the later EA-VTST/OMT formalism would give similar results.) The enzyme systems studied are summarized in Table 22.2, and the theoretical results are compared to the experimental ones in Table 22.3. The reader is referred to the original papers for discussion of such critical issues as pH, temperature of the experiments, mutations, and kinetic complexity. We simply note here that all the theoretical calculations are at 300 K and that these comparisons of theory to experiment have all the usual complications one finds in enzyme kinetics (for example, the intrinsic KIE may be higher than the experimental value due to kinetic complexity), but I hope that these details are not needed in an overview of the results.

TABLE 22.3 Experimental and Calculated KIEs for Enzyme Systems Enzyme

KIE

Experiment

Stage

TST

EA-VTST

EA-VTST/OMT

Enolase

Primary H/D Secondary H/D Primary H/T

3.3 n.a.a 7–8

4.7 0.89 6.6

Secondary H/T

1.36

MADH XyI SCAD

Primary H/D Primary H/D Primary H/D

17 3–4 7–53

DHFR

Primary H/D Secondary H/D

3 1.13

SSZ SSZ SSZ ESZ SSZ ESZ SSZ SSZ SSZ ESZ SSZ SSZ

3.7 0.96 6.7 6.6 1.09 1.09 5.9 1.8 3.2 3.7 2.5 1.03

3.5 0.96 6.9 7.5 1.27 1.36 18 3.8 4.1 70 2.8 1.13b

LADH

a b

n.a. denotes not available Published in Ref. 252 before the experiment was reported in Ref. 265.

1.08 5.9 1.8 3.5 2.7 1.00

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Next I give a few comments on the various cases. The enolase case179 is very interesting because it shows a significant variational effect on the ˚ longer at the variational transition for proton KIE. The breaking C – H bond distance is 0.18 A transfer than at that for deuteron transfer. This lowers the predicted primary KIE by 21%, which is essential for obtaining good agreement with experiment. The LADH case151,186 is a very significant one because previous models266 were unable to simultaneously fit the primary and secondary KIEs except by making assumptions about the force constants that eventually turn out to be incorrect. The problem is dissected in detail in our first paper on this enzyme186 and in a review;39 the conclusion that one draws is that multidimensional corner-cutting tunneling is essential to understanding the origin of the KIEs. The LADH case shows only a small effect of including secondary-zone entropy effects on the transmission coefficient, i.e., the SSZ and ESZ KIEs are similar, although the more complete ESZ theory does agree better with experiment. The MADH case251 is interesting because of the large amount of tunneling and the very large KIE. Tunneling increases the rate of proton transfer by a factor of 75, which lowers the phenomenological free energy of activation by 2.5 kcal/mol. The good agreement of the EA-VTST/ OMT KIE with experiment is very encouraging. The xylose isomerase case152,250 is important because the hydride transfer is strongly coupled to Mg2þ motion. Nevertheless our treatment with the simple difference-of-bond-lengths reaction coordinate gives a picture of the reaction that is very consistent with experiment. The tunneling in this reaction has also been studied by Nicoll et al.267 The SCAD reaction has the complication that the hydride transfer is strongly coupled to a proton transfer. Our calculations161 predict a nonconcerted mechanism, but the experimental situation is unclear. The extremely large effect of relaxing the secondary zone along the reaction path in the calculation of the transmission coefficient is probably an indication that we need a larger primary zone for this case. The DHFR case252 is interesting because, as for xylose isomerase, the hydride motion is strongly coupled to the structural changes in the protein. It is very interesting that for DHFR our results are in good agreement with those of Hammes –Schiffer and coworkers,268,269 who used a collective reaction coordinate depending on the atomic coordinates of all the atoms of the substrate and enzyme. The secondary deuterium KIE was a prediction published before the experiment was available.

VI. SOFTWARE Software for carrying out VTST/OMT and EA-VTST/OMT calculations is available at http://comp. chem.umn.edu/truhlar.

VII. CONCLUDING REMARKS This article has focused on the assumptions underlying the formalism that is available for practical calculations of the rate constants and KIEs of complex systems, with a special emphasis on conceptual and operational issues, such as the quasiequilibrium approximation, the reaction coordinate, quantization of vibration, the treatment of tunneling, the coupling of a primary system to a bath, and the different ways of putting the components of the calculations together for reactions in the gas phase, in liquids, and in enzymes. We also provided a brief review of some calculations of KIEs. One subject that is not covered is the use of quantum mechanical scattering theory to study the quantized energy levels of the transition state. This subject, reviewed elsewhere,97,99 has provided a look at the transition states, including their lifetimes and state-specific decay probabilities, with the finest level of detail allowed by the uncertainty limitations of quantum mechanics. The picture provided by this computational transition-state spectroscopy is fully

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consistent270 with the way that we use variational transition states and tunneling probabilities to calculate reaction rates, and in fact it provides deep support for the theory.

ACKNOWLEDGMENTS I am grateful to many collaborators (see references) for essential contributions to the work reported here and to Ralph Weston for helpful comments that enabled me to improve the manuscript. My work on gas-phase VTST is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, and the work on condensed-phase dynamics is supported by the National Science Foundation.

GLOSSARY The following is a glossary of acronyms used in this paper, except for those explained in Table 22.2, which are not repeated here: CVT DS EA ESP GT ICVT IRC JPC KIE LCT MDS MEP MT NATO NES OMT PES PMF RODS QC SCT SES SN2 SSZ TS TST VRC VRP VTST WKB

canonical variational theory (same as canonical VTST) dividing surface ensemble-averaged equilibrium solvation path generalized TS improved CVT intrinsic reaction coordinate (same as MEP) Journal of Physical Chemistry kinetic isotope effect large-curvature tunneling multifaceted DS minimum energy path (assumed to be in isoinertial coordinates) multidimensional tunneling North Atlantic Treaty Organization nonequilibrium solvation optimized MT potential energy surface (really a hypersurface) — same as potential energy function potential of mean force reorientation of the DS quasiclassical small-curvature tunneling separable equilibrium solvation bimolecular nucleophilic substitution static secondary zone (refers to the configuration of the secondary zone during the calculation of the transmission coefficient for a single member of the TS ensemble) transition state TS theory (in this chapter the acronym TST always means conventional TST, although the words “transition-state theory” are often used to include generalized transition-state theory including variational transition-state theory, with or without tunneling) variable reaction coordinate variational reaction path variational TS theory Wentzel –Kramers – Brillouin

Variational Transition-State Theory and Multidimensional Tunneling

607

ZCT zero-curvature tunneling mOMT microcanonically optimized MT (often just called OMT)

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205 Roberto-Neto, O., Coitin˜o, E. L., and Truhlar, D. G., Dual-level direct dynamics calculations of deuterium and carbon-13 kinetic isotope effects for the reaction Cl þ CH4, J. Phys. Chem. A, 102, 4568– 4578, 1998. 206 Chiltz, G., Eckling, R., Goldfinger, P., Huybrechts, G., Johnston, H. S., Meyers, L., and Verbeke, G., Kinetic isotope effect in photochlorination of H2, CH4, CHCl3, and C2H6, J. Chem. Phys., 5, 1053– 1061, 1963. 207 Wallington, T. J. and Hurley, M. D., A kinetic study of the reaction of chlorine atoms with CF3CHCl2, CF3CH2F, CF2ClCH3, CH3D, CH2D2, CHD3, CD4, and CD3Cl at 295 ^ 2 K, Chem. Phys. Lett., 189, 437– 442, 1992. 208 Saueressig, G., Bergamaschi, P., Crowley, J. N., Fisher, H., and Harris, G. W., Carbon kinetic isotope effect in the reaction of CH4 with Cl atoms, Geophys. Lett., 22, 1225– 1228, 1995. 209 Saueressig, G., Bergamaschi, P., Crowley, J. N., Fischer, H., and Harris, G. W., D/H kinetic isotope effect in the reaction CH4 þ Cl, Geophys. Res. Lett., 23, 3619– 3622, 1996. 210 Matsumi, Y., Izumi, K., Skorokhodov, V., Kawasaki, M., and Tanaka, N., Reaction and quenching of Cl(2PJ) atoms in collisions with methane and deuterated methanes, J. Phys. Chem. A, 101, 1216– 1221, 1997. 211 Corchado, J. C., Truhlar, D. G., and Espinosa-Garcia, J., Potential energy surface, thermal and stateselected rate constants, and kinetic isotope effects for Cl þ CH4 ! HCl þ CH3, J. Chem. Phys., 112, 9375– 9389, 2000. 212 Corchado, J. C., Espinosa-Garcia, J., Roberto-Neto, O., Chuang, Y. Y., and Truhlar, D. G., Dual-level direct dynamics calculations of the reaction rates for a Jahn – Teller reaction: hydrogen abstraction from CH4 or CD4 by O(3P), J. Phys. Chem. A, 102, 4899– 4910, 1998. 213 Espinosa-Garcia, J. and Garcia-Bernoldez, J. C., Analytic potential energy surface for the CH4 þ O(3P) ! CH3 þ OH reaction. Thermal rate constant and kinetic isotope effects, Phys. Chem. Chem. Phys., 2, 2345–2351, 2000. 214 Espinosa-Garcia, J., Capability of LEPS surfaces to describe the kinetics and dynamics of noncollinear reactions, J. Phys. Chem. A, 105, 134– 139, 2001. 215 O’Hair, R. A. J., Davico, G. E., Hacaloglu, J., Dang, T. T., DePuy, C. H., and Bierbaum, V. M., Measurements of solvent and secondary kinetic isotope effects for the gas-phase SN2 reactions of fluoride with methyl halides, J. Am. Chem. Soc., 116, 3609– 3610, 1994. 216 Hu, W. P. and Truhlar, D. G., Modeling transition state solvation at the single-molecule level: test of correlated ab initio predictions against experiment for the gas-phase SN2 reaction of microhydrated fluoride with methyl chloride, J. Am. Chem. Soc., 116, 7797– 7800, 1994. 217 Bigeleisen, J., Spectral and reactivity differences of transition metal ions in H2O and D2O, J. Chem. Phys., 32, 1583– 1584, 1960. 218 Bunton, C. A. and Shiner, V. J. Jr, Isotope effects in deuterium oxide solution. II. Reaction rates in acid, alkaline, and neutral solvations, involving only secondary solvent effects, J. Am. Chem. Soc., 83, 3207– 3214, 1961. 219 Newton, M. D. and Friedman, H. L., A proposed neutron diffraction experiment to measure hydrogen isotope fractionation in solvation, J. Chem. Phys., 83, 5210– 5218, 1985. 220 Sunko, D. E., Secondary deuterium isotope effects and neighboring group participation revisited, Croat. Chem. Acta, 69, 1275 –1304, 1996. 221 Aida, M. and Yamataka, H., An ab initio MO study on the hydrolysis of methyl chloride, Theochem, 461/462, 417– 427, 1999. 222 Corchado, J., Espinosa-Garcia, J., Hu, W. P., Rossi, I., and Truhlar, D. G., Dual-level reaction-path dynamics (the /// approach to VTST with semiclassical tunneling). Application to OH þ NH3 ! H2O þ NH2, J. Phys. Chem., 99, 687– 694, 1995. 223 Viggiano, A. A., Morris, R. A., Paschkewitz, J. S., and Paulson, J. F., Kinetics of the gas-phase reactions of chloride anion, Cl2, with CH3Br and CD3Br: experimental evidence for nonstatistical behavior?, J. Am. Chem. Soc., 114, 10477– 10482, 1992. 224 Boyd, R. J., Kim, C. K., Shi, Z., Weinberg, N., and Wolfe, S., Secondary H/D isotope effects and transition state looseness in nonidentity methyl transfer reactions. Implications for the concept of enzymatic catalysis via transition state compression, J. Am. Chem. Soc., 115, 10147– 10152, 1993. 225 Kato, S., Davico, G. E., Lee, H. S., DePuy, C. H., and Bierbaum, V. M., Deuterium kinetic isotope effects in gas phase SN2 reactions, Int. J. Mass Spectrom., 210/211, 223– 229, 2001.

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226 Hu, W.-P. and Truhlar, D. G., Deuterium kinetic isotope effects and their temperature dependence in the gas-phase SN2 reactions X2 þ CH3Y ! CH3X þ Y2 (X, Y ¼ Cl, Br, I), J. Am. Chem. Soc., 117, 10726– 10734, 1995. 227 Hu, W.-P. and Truhlar, D. G., Factors affecting competitive ion-molecule reactions: ClO2 þ C2H5Cl and C2D5Cl via E2 and SN2 channels, J. Am. Chem. Soc., 118, 860– 869, 1996. 228 Stern, M. J., Persky, A., and Klein, F. S., Force field and tunneling effects in the H – H– Cl reaction system. Determination from kinetic-isotope-effect measurements, J. Chem. Phys., 58, 5697– 5706, 1973. 229 Miller, J. C. and Gordon, R. J., Kinetics of the Cl – H2 system. III. The deuterium isotope effect in Cl þ H2, J. Chem. Phys., 79, 1252– 1258, 1983. 230 Kumaran, S. S., Lim, K. P., and Michael, J. V., Thermal rate constants for the Cl þ H2 and Cl þ D2 reactions between 296 and 300 K, J. Chem. Phys., 101, 9487–9498, 1994. 231 Allison, T., Lynch, G. C., Truhlar, D. G., and Gordon, M. S., An improved potential energy surface for the H2Cl system and its use for calculations of the rate constants and kinetic isotope effects, J. Phys. Chem., 100, 13575– 13587, 1996. 232 Mielke, S. L., Allison, T. C., Truhlar, D. G., and Schwenke, D. W., Quantum mechanical rate constants for the Cl þ H2 reaction, J. Phys. Chem., 100, 13588– 13593, 1996. 233 Droege, A. T. and Tully, F. P., Hydrogen-atom abstraction from alkanes by hydroxyl. 3. Propane, J. Phys. Chem., 90, 1949– 1954, 1986. 234 Nagase, S., Fueno, T., and Morokuma, K., An ab initio approach to organic reaction rates. Kinetic isotope effects in the reaction H þ C2H4 ! C2H5, J. Am. Chem. Soc., 101, 5849– 5851, 1979. 235 Sugawara, K., Okazaki, K., and Sato, S., Kinetic isotope effects in the reaction H þ C2H4 ! C2H5, Chem. Phys. Lett., 78, 259–263, 1981. 236 Sugawara, K., Okazaki, K., and Sato, S., Temperature dependence of the rate constants of hydrogen and deuterium atom additions to ethylene, ethylene-d1, ethylene-d4, ethyne, and ethyne-d2, Bull. Chem. Soc. Jpn, 54, 2872– 2877, 1981. 237 Garner, D. M., Fleming, D. G., Arseneau, D. J., Senba, M., Reid, I. D., and Mikula, R. J., Muonium addition reactions in the gas phase: quantum tunneling in Mu þ C2H4 and Mu þ C2D4, J. Chem. Phys., 93, 1732–1740, 1990. 238 Bondi, D. K., Clary, D. C., Connor, J. N. L., Garrett, B. C., and Truhlar, D. G., Kinetic isotope effects in the Mu þ H2 and Mu þ D2 reactions: accurate quantum calculations for the collinear reactions and variational transition state theory predictions for one and three dimensions, J. Chem. Phys., 76, 4986– 4995, 1982. 239 Blais, N. C., Truhlar, D. G., and Garrett, B. C., Quasiclassical trajectory (and variational transition state theory) study of the rates and temperature-dependent activation energies of the reactions Mu þ H2 (completely thermal) and H, D, and Mu þ H2 (v ¼ 0, j ¼ 2), J. Chem. Phys., 78, 2363– 2367, 1983. 240 Lynch, G. C., Truhlar, D. G., Brown, F. B., and Zhao, J.-g., Kinetic isotope studies of the gas-phase reaction (H,D,Mu) þ HBr((H,D,Mu)H þ Br, Hyperfine Interact., 87, 885– 898, 1994. 241 Lynch, G. C., Truhlar, D. G., Brown, F. B., and Zhao, J.-g., A new potential energy surface for H2Br and its use to calculate branching ratios and kinetic isotope effects for the H þ HBr reaction, J. Phys. Chem., 99, 207– 225, 1995. 242 Pu, J. and Truhlar, D. G., Tests of potential energy surfaces for H þ CH4 $ CH3 þ H2: deuterium and muonium kinetic isotope effects for the forward and reverse reaction, J. Chem. Phys., 117, 10675– 10687, 2002. 243 Lu, D. H., Maurice, D., and Truhlar, D. G., What is the effect of variational optimization of the transition state on a-deuterium secondary kinetic isotope effects? A prototype: CD3H þ H ! CD3 þ H2, J. Am. Chem. Soc., 112, 6206– 6214, 1990. 244 Skodje, R. T. and Truhlar, D. G., Parabolic tunneling calculations, J. Phys. Chem., 85, 624– 628, 1981. 245 Skodje, R. T., Truhlar, D. G., and Garrett, B. C., A general small-curvature approximation for transition-state-theory transmission coefficients, J. Phys. Chem., 85, 3019 –3023, 1981. 246 Nicoll, R. M., Hillier, I. H., and Truhlar, D. G., Quantum mechanical dynamics of hydride transfer in polycyclic hydoxy ketones in the condensed phase, J. Am. Chem. Soc., 123, 1459– 1463, 2001.

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247 Chuang, Y. Y., Radhakrishnan, M. L., Fast, P. L., Cramer, C. J., and Truhlar, D. G., Direct dynamics for free radical kinetics in solution: solvent effect on the rate constant for the reaction of methanol with atomic hydrogen, J. Phys. Chem. A, 103, 4893– 4909, 1999. 248 Lossack, A. M., Roduner, E., and Bartels, D. M., Kinetic isotope effects in H and D abstraction reactions from alcohols by D atoms in aqueous solution, J. Phys. Chem. A, 102, 7462– 7469, 1998. 249 Anbar, M. and Meyerstein, D., Isotope effects in the hydrogen abstraction from aliphatic compounds by radiolytically produced hydrogen atoms in aqueous solutions, J. Phys. Chem., 68, 3184– 3187, 1964. 250 Garcia-Viloca, M., Alhambra, C., Truhlar, D. G., and Gao, J., Quantum dynamics of hydride transfer catalyzed by bimetallic electrophilic catalysis: synchronous motion of Mg2þ and H2 in xylose isomerase, J. Am. Chem. Soc., 124, 7268– 7269, 2002. 251 Alhambra, C., Sa´nchez, M. L., Corchado, J. C., Gao, J., and Truhlar, D. G., Quantum mechanical tunneling in methylamine dehydrogenase, Chem. Phys. Lett., 347, 512– 518, 2001, reprinted correctly: 355, 388– 394, 2001. 252 Garcia-Viloca, M., Truhlar, D. G., and Gao, J., Reaction-path energetics and kinetics of the hydride transfer reaction catalyzed by dihydrofolate reductase, Biochem, 42, 13558– 13575, 2003. 253 Anderson, S. R., Anderson, V. E., and Knowles, J. r., Primary and secondary kinetic isotope effects as probes of the mechanism of yeast enolase, Biochemistry, 33, 10545– 10555, 1994. 254 Bahnson, B. J., Park, D. H., Kim, K., Plapp, B. V., and Klinman, J. P., Unmasking of hydrogen tunneling in the horse liver alcohol dehydrogenase reaction by site-directed mutagenesis, Biochemistry, 32, 5503– 5507, 1993. 255 Bahnson, B. J. and Klinman, J. P., Hydrogen tunneling in enzyme catalysis, Methods Enzymol., 249, 373– 397, 1995. 256 Brooks, H. B., Jones, L. H., and Davidson, V. L., Deuterium kinetic isotope effect and stopped-flow kinetic studies of the quinoprotein methylamine dehydrogenase, Biochemistry, 32, 2725 –2729, 1993. 257 Basran, J., Sutcliffe, M. J., and Scrutton, N. S., Enzymatic H-transfer requires vibration-driven extreme tunneling, Biochemistry, 38, 3218– 3222, 1999. 258 Lee, C., Bagdasarian, M., Meng, M., and Zeikus, J. G., Catalytic mechanism of xylose (glucose) isomerase from Clostridium thermosulfurogenes, J. Biol. Chem., 265, 19082– 19090, 1990. 259 van Tilbeurgh, H., Jenkins, J., Chiadmi, M., Janin, J., Wodak, S. J., Mrabet, N. T., and Lambeir, A. M., Protein engineering of xylose (glucose) isomerase from Actinoplanes missouriensis. 3. Changing metal specificity and the pH profile by site-directed mutagenesis, Biochemistry, 31, 5467– 5471, 1991. 260 van Bastelaere, P. B. M., Kersters-Hilderson, H. L. M., and Lambeir, A. M., Wild-type and mutant D-xylose isomerase from Actinoplanes missouriensis: metal-ion dissociation constants, kinetic parameters of deuterated and non-deuterated substrates and solvent-isotope effect, Biochem. J., 307, 135– 142, 1995. 261 Murfin, W.W., Mechanism of the Flavin Reduction Step in Acyl CoA Dehydrogenases, Ph.D. dissertation, Washington University, St. Louis, MO, 1974. 262 Schopfer, L. M., Massey, V., Ghisla, S., and Thorpe, C., Oxidation-reduction of general acyl-CoA dehydrogenase by the butyryl-CoA/crotonyl-CoA couple. A new investigation of the rapid reaction kinetics, Biochemistry, 27, 6599– 6611, 1988. 263 Reinsch, J., Katz, A., Wean, J., Aprahamian, G., and McFarland, J. T., The deuterium isotope effect upon the reaction of fatty acyl-CoA dehydrogenase and butyryl-CoA, J. Biol. Chem., 255, 9093– 9097, 1980. 264 Fierke, C. A., Johnson, K. A., and Benkovic, S. J., Construction and evaluation of the kinetic scheme associated with dihydrofolate reductase from Escherichia coli, Biochemistry, 26, 4085– 4092, 1987. 265 Sikorski, R.S., Wang, L., Markham, K.A., Rajagopalan, P.T.R., Benkovic, S.J., Kohen, A., Tunneling and coupled motion in the E. coli dihydrofolate reductase catalysis, J. Am. Chem. Soc., 126, 4778– 4779, 2004. 266 Rucker, J. and Klinman, J. P., Computational study of tunneling and coupled motion in alcohol dehydrogenase-catalyzed reactions: implication for measured hydrogen and carbon isotope effects, J. Am. Chem. Soc., 121, 1997– 2006, 1999.

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23

Computer Simulations of Isotope Effects in Enzyme Catalysis Arieh Warshel, Mats H. M. Olsson, and Jordi Villa`-Freixa

CONTENTS I. II.

Introduction ...................................................................................................................... 621 Methods for Simulations of Chemical Processes in Enzymes........................................ 623 A. QM/MM Molecular Orbital Methods ...................................................................... 624 B. EVB as a Reliable QM/MM Method ...................................................................... 625 III. Simulating Nuclear Quantum Mechanical Effects in Condensed Phase ........................ 626 A. The Dispersed Polaron (Spin Boson) Model........................................................... 626 B. Quantized Classical Path Simulations ..................................................................... 627 IV. Simulations of the KIE and Nuclear Quantum Mechanical Effects in Enzymatic Reactions ................................................................................................... 630 A. Systematic Studies of Hydride Transfer in Solutions ............................................. 630 B. Simulating NQM Effects in LDH by a Microscopically Based Quasiharmonic Model and a QCP Treatment .............................................. 630 C. Nuclear Quantum Mechanical Effects in Carbonic Anhydrase .............................. 631 D. Nuclear Quantum Mechanical Effects in Alcohol Dehydrogenase ........................ 632 E. Lipoxygenase and the Large Tunneling Limit ........................................................ 634 V. What is the Catalytic Contribution from Nuclear Quantum Mechanical Effects?......... 635 VI. What Can and What Cannot be Learned from Simulations of Isotope Effects?............ 636 A. The Use of Vibronic Models in Studies of Isotope Effects .................................... 636 B. Using Calculated and Observed Isotope Effects as a Tool for Validating Single Only Simulations of NQM and Determining the Catalytic Contributions of NQM Effects........................................................... 637 C. Determining the Concertedness of Enzymatic Reactions by the KIE.................... 638 D. Dynamical Effects and Promoting Modes............................................................... 638 VII. Concluding Remarks........................................................................................................ 639 Acknowledgments ........................................................................................................................ 640 References..................................................................................................................................... 640

I. INTRODUCTION Understanding enzymatic reactions and realizing what makes them so efficient is one of the challenges of modern biochemistry. Although important elements of this puzzle were thoroughly investigated by biochemical and structural studies, the source of the catalytic power of enzymes is not entirely understood. General statements that suggest that the enzyme binds the transition state better than the ground state do not really increase the knowledge about enzyme catalysis since the real question is how this differential binding is accomplished and which catalytic groups are 621

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Isotope Effects in Chemistry and Biology

responsible. To discuss this rate enhancement we will consider a generic enzymatic reaction using the equation k1

kcat

E þ S O ES ! ES‡ ! EP ! E þ P k21

ð23:1Þ

Here, E, S, and P are the enzyme, substrate and product, respectively, and ES, EP, and ES‡ are the enzyme –substrate complex, enzyme product complex and enzyme transition state, respectively. As was eloquently shown by Wolfenden and coworkers, e.g., Ref. 1 many enzymes have evolved by optimizing kcat =KM ; where KM ¼ ðk21 þ kcat Þ=k1 and can be approximated as KM ¼ k21 =k1 : However, this and related findings have not identified the factors responsible for the catalytic effect. As will be shown in the next section, the key question is related to the reduction of the activation barrier in the chemical step. Unfortunately, even mutation experiments, which were found to be extremely useful in identifying catalytic factors,2 cannot tell us in a unique way the origin of the catalytic effect (see e.g., Ref. 3). Instead, we need a quantitative tool that is capable to determine the relation between the structure and the function and that also gives us the individual contributions to the overall catalytic effect. It is becoming clear that computer simulation approaches are a powerful tool to address these questions. This chapter will review some of the advances in simulations of enzymatic reactions while emphasizing the use of these approaches in the analysis of kinetic isotope effects (KIE) and the corresponding nuclear quantum mechanical (NQM) effects. To address enzyme catalysis, it is first necessary to define the relevant questions. The first question to address is: “catalysis relative to what?” In other words, before proceeding it is essential to define a suitable uncatalyzed reference reaction. The most obvious reference is the reaction in water, where the effect of the enzyme is taken away whereas the aqueous environment is retained. Since the mechanism in water can be different than that in the enzyme, different mechanisms should be considered along with the effect of having different environments. Fortunately, a difference in mechanism can be classified as a “chemical effect” (e.g., having a histidine resudue instead of a water as a base), and such effects are well understood. Thus, we can focus on the effect of the environment and our task is to compare the rate constant of a reaction with the same mechanism and binds the same chemical groups but is conducted in water. Obviously, the water reaction is not the only conceivable reference reaction, e.g., often the gas-phase reaction is used instead. However, a proper thermodynamic analysis of the catalyzed reaction should in that case include the difference gas-phase ! enzyme as well as gas-phase ! solution and will also give the difference between the reaction in enzyme and in solution. We find the water reaction to be the best-suited and well-defined reaction. Regardless of what cycle we choose, understanding the catalytic power of enzymes boils down to determining the origin of the difference between the activation barrier in water ðDg‡w Þ; and the activation barrier in the protein ðDg‡enz Þ: At this point, it seems conceivable that the enzyme can reduce Dg‡enz both by binding the substrate with an equal strength in the reactant state (RS) and the transition state (TS) (in which case Dg‡enz 2 Dg‡w ¼ DGbind ) and reducing the activation barrier Dg‡cat for the chemical step (see Figure 23.1). Since the factors that control the binding step are well understood, the real puzzle is to determine the factors that govern the reduction of the activation barrier of the chemical step ðDg‡cat Þ: Although it was recognized by Polanyi4 and Pauling5 that Dg‡cat is reduced by the enzyme, they did not determine the origin of this reduction. This can in principle be achieved by either destabilizing the RS or stabilizing the TS (which is not easily resolved experimentally). In summary, the main focus of simulation studies should be on two questions: (i) which contributions are responsible for the difference between ðDg‡cat Þ and ðDg‡w Þ; and (ii) how do these contributions operate (i.e., do they destabilize the RS or stabilize the TS)? Recently, it was proposed that proteins utilized NQM effects to enhance its catalytic power (see e.g., Refs. 6,7). To assist in catalysis, NQM effects should reduce the reaction barrier in the protein more than in the uncatalyzed reference reaction. In general, it is difficult to quantify, and thereby deduce the NQM contributions to enzyme catalysis from current experimental approaches,

Computer Simulations of Isotope Effects in Enzyme Catalysis

+

∆g +enz

+

+ ∆gcage

+

∆g w+

623

+

+ ∆g cat

∆G bind

[A]w

(a)

+

[B]w

[AB] + [AB]cage

[A]w

(b)

[E]

[EA]

+ +

[EA]

FIGURE 23.1 The enzyme (b) and solution (a) reaction profiles and the relevant activation barrier.

since it is required to compare the NQM contributions to the given enzymatic reaction to those in the water reference reaction.8 In some cases, such as enzymes utilizing metal centers, it might be very hard to construct the relevant model system to study this reference reaction, making such a comparison impossible. In other cases the enzyme and solution reactions can be different. In these, the catalysis includes both the effect of changing the reaction mechanism from the one in water to that in the enzyme and the effect of changing the active-site environment from water solution to that of the protein. Finally, a complete analysis should also address the actual contributions of the protein that are responsible for enhancing the tunneling effect. This information is usually not available experimentally, whereas it is comparatively easy to estimate contributions and to determine their origin in simulation approaches. It is important to augment the experimental information about KIEs by simulation approaches that are capable of reproducing the observed effects and evaluate the role of the protein or the solution environment in modulating these effects. In doing so we must focus on the quantum mechanical correction to the activation free energy since this is the primary factor that determines the reaction rate (see below). In this chapter, we will address the insight on enzymatic reactions from computer simulation focusing on the role of NQM contributions in changing the difference between ðDg‡cat Þ and ðDg‡w Þ: We will also examine in a critical way the general issue of the information content of KIE studies in elucidating reaction mechanisms and establishing the role of dynamical effects in enzyme catalysis.

II. METHODS FOR SIMULATIONS OF CHEMICAL PROCESSES IN ENZYMES Although the main subject of this work is the role of NQM effects in enzyme catalysis, it is crucial to have a method that can correctly treat classical effects, such as electrostatics and configurational sampling, in the protein (this may turn out to be the most important issue as far as catalysis is concerned). In this section we will first consider general approaches for modeling chemical processes in enzymes and in solutions. In doing so, we will focus on strategies capable of providing activation barriers for enzymatic reactions. The study of chemical reaction processes is very challenging since the potential energy surfaces cannot generally be described by simple force fields when chemical bonds are being broken and formed during reactions. The most common way of obtaining viable surfaces for chemical reactions is the use of quantum mechanical computational approaches, which have become effective in treating small molecules in the gas phase (e.g., Ref. 9).

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Isotope Effects in Chemistry and Biology

However, here we are interested in chemical reactions in very large systems. In addition, since these energies are relatively small only a few approaches are practical and give reasonable results. We will consider below the main options and discuss their effectiveness.

A. QM/MM MOLECULAR O RBITAL M ETHODS The development of the hybrid Quantum Mechanics/Molecular Mechanics (QM/MM) approach10 originated from the realization that the effect of the environment on the reacting fragments must be included in studies of enzymatic reactions. QM/MM approaches divide the simulation system (e.g., the enzyme/substrate complex) into two regions. The inner region, region I, contains the reacting fragments which are represented quantum mechanically, whereas the surrounding protein/solvent region, region II, is represented by a molecular mechanics force field. The Hamiltonian of the complete system is then written as ^ ¼H ^ QM þ H^ QM=MM þ H ^ MM H

ð23:2Þ

^ QM is the QM Hamiltonian, H^ QM=MM is the Hamiltonian that couples regions I and II, and where H ^ MM is the Hamiltonian of region II. H ^ QM can be evaluated by any standard QM approach, either H ab initio or semiempirical. The total potential is then expressed as ^ QM þ H^ QM=MM þ H ^ MM lCl ¼ EQM þ kClH^ QM=MM lCl þ EMM Vtotal ¼ kClH

ð23:3Þ

The nature of the first two terms in the above equation can best be realized by thinking of a simple molecular orbital representation of a solute/solvent system. If we consider this system as a super molecule we can express its molecular orbital (MO) wave function as X S S X s s fi ¼ vm;i xm þ vl;i xl ð23:4Þ m

l

where S and s designate the solute and solvent, respectively, njX;k are the MO coefficients and xjX are atomic orbital (AO) wave functions. We also assume that the atomic orbitals x S and x s are orthogonal. The coefficients nmS;i can be obtained by solving the SCF equation for the super system: Fn i ¼ 1 i n

ð23:5Þ

where the matrix F can be separated into blocks, describing the solute – solute, solvent –solvent, and the solute –solvent interactions FS

F Ss

F Ss

Fs

ð23:6Þ

The matrix elements of F are given in Refs. 11,12. The assumption that the orbitals are orthogonal to each other implies that F Ss ¼ 0 (see Ref. 11). With some simple manipulations and with the assumption that the electron – electron repulsion integral between atoms A and B in the solute and solvent regions, respectively, is given by e2 =rAB we obtain for any solvent configuration: S S Fmm ; ðFmm Þ0 2

X e 2 qB S ¼ ðFmm Þ0 2 U A r AB B

ð23:7Þ

Here, m [ A and UA designates the total electrostatic potential from the solvent atoms at the site of atom A. This equation can be used in the more general case where the solvent charge distribution is

Computer Simulations of Isotope Effects in Enzyme Catalysis

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not fixed and can be polarized by the field of the solute.11 The leading term in the solute – solvent coupling Hamiltonian is obtained by adding the potential from the solvent atoms to the solute Hamiltonian. Now, the total potential energy is given by Vtotal ¼ ES ðFS Þ þ E0Ss þ Ess

ð23:8Þ

where the energy ES ðFS Þ is obtained quantum mechanically with the F matrix that includes the given electrostatic potential from the solvent (the vector of all the U 0A s). E0Ss is the nonelectrostatic solute – solvent interaction term and Ess is the solvent –solvent classical force field. At this level of approximation the nonelectrostatic term is evaluated by the standard classical van der Waals potential function. In studies of very large solute molecules, we sometimes divide the solute region into quantum and classical parts. The connection between the quantum and the classical regions is treated by a classical force field (which is included in E0Ss ) where the quantum atoms at the boundaries are connected to dummy hydrogen-like atoms in order to balance the electrons in the quantum system. The QM/MM methods are now widely used in studies of complex systems in general and enzymatic reactions in particular and we can only mention several works (e.g., Refs. 13– 24). Despite these advances, we are not yet at the stage where these approaches can readily be used in fully quantitative studies of enzyme catalysis. One of the major problems is that a quantitative evaluation of the potential energy surfaces for the reacting fragment should be performed by ab initio electronic structure calculations and such calculations are too expensive to allow for a configurational averaging needed for free-energy calculations. Specialized approaches can help in progressing toward ab initio QM/MM free-energy calculations (e.g., see Refs. 25,26), but even these approaches are still in a development stage. Fortunately, one can instead use approaches that are calibrated on experimental rates or key energies of the reference solution reaction to obtain reliable results with semiempirical QM/MM studies. One of the most effective methods of doing this is EVB described below.

B. EVB AS A R ELIABLE QM/MM M ETHOD Reliable studies of enzyme catalysis require accurate results for the difference between the activation barriers in enzyme and in solution. The early realization of this point led to a search for a method that could be calibrated using experimental and theoretical information of reactions in solution. It also became apparent that in studies of chemical reactions it is more physical to calibrate surfaces that reflect bond properties (i.e., valence bond-based, VB, surfaces) than to calibrate surfaces that reflect atomic properties (e.g., MO-based surfaces). Furthermore, it appears to be advantageous to force the potential surfaces to reproduce the experimental results of the broken fragments at infinite separation in solution. This can be easily accomplished with the VB picture. The resulting empirical valence bond (EVB) method has been discussed extensively elsewhere,27,28 but its main features will be outlined below. The EVB method is a QM/MM approach that describes reactions by mixing resonance states (or more precisely diabatic states) that correspond to classical valence-bond structures, which describe the reactant intermediate (or intermediates) and product states. The potential energies of these diabatic states are represented by classical MM force fields of the form: i i 1i ¼ aigas þ Uintra ðR; QÞ þ USs ðR; Q; r; qÞ þ Uss ðr; qÞ

ð23:9Þ

Here, R and Q represent the atomic coordinates and charges of the diabatic states, and r and q are those of the surrounding protein and solvent. aigas is the gas-phase energy of the ith diabatic state i (where all the fragments are taken to be at infinity); Uintra ðR; QÞ is the intramolecular potential of

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Isotope Effects in Chemistry and Biology

i the solute system at that given diabatic state (relative to its minimum); USs ðR; Q; r; qÞ represents the interaction between the solute (S) atoms and the surrounding (s) solvent and protein atoms. Uss ðr; qÞ represents the potential energy of the protein/solvent system (“ss” designates surrounding– surrounding). The 1i ; the diagonal elements of the EVB Hamiltonian ðHii Þ; are given by Equation 23.9. The off-diagonal elements of the Hamiltonian, Hij ; are usually represented by a simple exponential function of the distances between the reacting atoms. The Hij elements are assumed to be the same in the gas phase, in solutions and proteins. The ground state energy Eg is obtained by solving

^ EVB Cg ¼ Eg Cg H

ð23:10Þ

Here, Cg is the ground state eigenvector and Eg provides the EVB potential energy surface. The EVB treatment provides a natural picture of intersecting electronic states, which is useful for exploring environmental effects on chemical reactions in condensed phases.27,28 The ground state charge distribution of the reacting species (“solute”, S) polarizes the surroundings (“solvent”, s) and the charges of each resonance structure of the solute then interact with the polarized solvent.28 This coupling enables the EVB model to capture the effect of the solvent on the quantum mechanical mixing of the different states of the solute. For example, in cases where ionic and covalent states are describing the solute, the resulting ground state has more ionic character and more solvation energy when the solvent stabilizes the ionic state to a greater extent. MD trajectories on the EVB surface of the reactant state can provide the free energy function Dg that is needed to calculate the activation energy Dg‡ : However, since trajectories on the reactant surface will reach the transition state only rarely, it is usually necessary to run a series of trajectories on potential surfaces that gradually drive the system from the reactant to the product state.29 The corresponding statistics are used in the framework of a free energy perturbation (FEP) umbrella sampling (US) method and provide the actual free energy surface for the reaction. The EVB method satisfies some of the main requirements for reliable studies of enzymatic reactions. Among the obvious advantages of the EVB approach is the facilitation of proper configurational sampling and converging free-energy calculations. This includes the inherent ability to evaluate nonequilibrium solvation effects.30 Calibrating EVB surfaces using ab initio calculations was found to provide quite reliable potential energy surfaces.31,32 For example, it has been found that the EVB approach accurately reproduces the structure and energetics of water clusters around H3Oþ.33,34 The EVB approach has been used extensively in studies of different enzymatic reactions (for a partial list, see Ref. 30) and some of the studies concerning KIE and NQM contributions will be considered in subsequent sections.

III. SIMULATING NUCLEAR QUANTUM MECHANICAL EFFECTS IN CONDENSED PHASE A. THE D ISPERSED P OLARON (S PIN B OSON ) M ODEL Simulation studies of NQM effects of chemical reactions in condensed phases date back to studies of electron transfer (ET) reactions in solutions35 and proteins.36,37 In the case of outer sphere ET reactions the coupling between the reactant (a) and product (b) states is small and the rate constant can be expressed as:35 ka!b ¼ lHab ="l2

ð1 21

exp½i4ba t þ g ðtÞ dt

ð23:11Þ

Computer Simulations of Isotope Effects in Enzyme Catalysis

627

where 4ba ¼ kVba la =" and k· · ·la designates an average over trajectories on Va : The quantity gðtÞ is the QM correlation function,

gðtÞ ¼

X j

h i X D2j ðnj þ 1=2Þðcos vj t 2 1Þ þ i ðD2j =2Þsin vj t

ð23:12Þ

i

where Dj and vj are, respectively, the dimensionless coordinate displacement (origin shift) and frequency of the jth normal vibrational mode (vj is assumed to the same in state a and state b); nj is given by nj ¼ ½expð"vj =kB TÞ 2 1

21

ð23:13Þ

where kB is the Boltzmann constant and T is the temperature. The parameter Dj determines the overlap between the QM wave functions for the nuclear vibrations in state a and b. Equation 23.11 provides a direct prescription for calculating the temperature dependence of the rate constant through the temperature dependent nj values. Rather than evaluating the integral in Equation 23.11 directly, it is necessary to use an analytical continuation of this equation, which can be obtained by expanding the exponential term in Equation 23.13. Any realistic molecular calculation requires the origin shift Dj associated with the protein vibrations and these parameters are not available from direct experimental information. Fortunately, the time dependence of the energy gap DVba ¼ 1b 2 1a (where b and a are the VB states of the product and reactant state, respectively) contains the information needed for evaluating the Dj and vj values. When Hab is small, the NQM treatment is rigorous within the quasiharmonic approximation and there is no problem to consider the KIE in the simulated system. It is also straightforward to consider temperature effects in the simulated system (e.g., see Ref. 36). However, the problem is to extend the NQM treatment to the adiabatic limit when Equation 23.11 is invalid. The general issue of the transition from the diabatic to the adiabatic limit in molecular simulations has been considered by Warshel and Chu.38 Here, we will focus on one of the most effective ways of obtaining NQM corrections in adiabatic reactions in condense phases, namely the quantized classical path (QCP) approach that will be described below.

B. QUANTIZED C LASSICAL PATH S IMULATIONS With the analytical EVB surface of the reacting system and its surrounding protein þ water system, the challenge is to obtain the quantum correction to the classical activation free energy. Since most enzymatic reactions involve large coupling between the diabatic states that describe the bondbreaking/bond-making processes we have to treat these systems in the adiabatic limit. To do this, we modified the centroid-path integral-approach39 – 41 in a way that allows us to use classical trajectories as a convenient and effective reference for the corresponding centroid calculations. This QCP approach8,42 will be described briefly below. In the QCP approach, the NQM rate constant is expressed as kqm ¼ Fqm

kB T expð2Dg‡qm =kB TÞ h

ð23:14Þ

where Fqm ; kB ; T; and h are, respectively, the transmission factor, Boltzmann’s constant, the temperature, Planck’s constant, and b ¼ 1=kB T: The quantum mechanical activation barrier, Dg‡qm ; includes almost all the NQM effects, whereas only small effects come from the preexponential transmission factor in the case of systems with a significant activation barrier.43,44 The quantum mechanical free energy barrier, Dg‡qm ; can be evaluated by Feynman’s path integral formulation,45 where each classical coordinate is replaced by a ring of quasiparticles that

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Isotope Effects in Chemistry and Biology

are subjected to the effective quantum mechanical potential Uqm ¼

p X 1 1 M V2 Dx2k þ Uðxk Þ 2p p k¼1

ð23:15Þ

Here, Dxk ¼ xkþ1 2 xk (where xpþ1 ¼ x1 ), V ¼ p="b; M is the mass of the particle, and U is the actual potential used in the classical simulation. The total quantum mechanical partition function can then be obtained by running classical trajectories of the quasiparticles with the potential Uqm : The probability of being at the transition state is approximated by a probability distribution of the center of mass of the quasiparticles (the centroid) rather than the classical single point. The use of quasiparticles is a computational device whose relationship to quantum effects is not so straightforward. Nevertheless, in Figure 23.2 we provide a qualitative rationalization that the path integral approach is able to evaluate quantum mechanical effects. The figure compares the classical and quantized description, solid and dashed lines, respectively, of a particle for a simple schematic one-dimensional potential. As illustrated by the figure, a classical particle (dark gray balls) with a total energy E , U ‡ cannot pass from the left to the right side of the potential since its energy is lower than the value of the potential at the transition state, x‡ : On the other hand, a quantum mechanical particle (represented by the ring of light gray balls) can penetrate or “tunnel” through the barrier since each of the quasiparticles only experiences the potential Uðxk Þ=p rather than Uðx0 Þ: (Note that when p increases we have on average less energy per particle.) This results in a nonzero probability at the transition state in the quantum description even though the total energy is lower than the transition state energy. The only reason the tunneling does not occur so readily is the restoring force of the M V2 Dx2k =2p term that connects the quasiparticles close to each other at high temperature (small b), and when M is large, the system behaves classically (see Figure 23.3). However, at low temperature and when M is small, the quasiparticles can spread, and some of them can penetrate the barrier. The quantum mechanical probability that the system will reach x‡ is given by the chance that the center of mass of the quasiparticle ring will be at this point. Similarly the centroid path integral approach reproduces the quantum mechanical effect of the zero point energy. That is, in the classical limit at low temperature the particle will relax to x0 : On the other hand, at the quantum limit the systems will always have nonzero potential energy when the centroid position is x0 ; since some of the quasiparticles will be at xk – x0 : At low temperature the quasiparticles can be

x0

+

x+

FIGURE 23.2 A schematic behavior of the classical and quantum mechanical description of a proton transfer reaction. In the QCP quantum mechanical description, the nuclear wave function, and thereby the nuclear distribution, is given by an ensemble of quasiparticles (here depicted as a ring of light gray particles), whereas in the classical picture it is given by a point particle (here depicted as a dark gray particle). Similarly, the overall probability distribution is given for the classical and quantum mechanical description by the solid and dashed lines, respectively. Close to the reactant geometry, x0, the quasiparticles can be at positions higher in energy than the point particle, which reflects the zero point energy. In contrast, at an energy E , Uðx‡ Þ the classical point particle cannot reach the transition state region, whereas some of the quasiparticles in the quantum mechanical description can. This gives a nonzero probability to penetrate the reaction barrier and results in nuclear tunneling.

Computer Simulations of Isotope Effects in Enzyme Catalysis

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FIGURE 23.3 Showing a snapshot of the quasiparticles (depicted as a cluster of gray particles around the atom position of the five atoms involved in the hydrogen atom transfer) during the simulation. The quasiparticles are used to represent the probability density of the reacting atoms in lipoxygenase. This representation enables the atoms to be in classically forbidden configurations and tunnel through the barrier. It can further be seen that the distribution of quasiparticles are more spread out for lighter particles (i.e., the hydrogen atoms) than for the heavier (e.g., the carbon and oxygen atoms) and results in increased tunneling.

at points whose potential energy is larger than Uðx0 Þ and will have larger average potential energy than the corresponding classical particle. This effect is reflected by the zero-point energy. Actual calculations of centroid probabilities in the condensed-phase reactions are very challenging and may involve major convergence problems. The QCP approach offers an effective and rather simple way for evaluating this probability without significantly changing the simulation program. This is achieved by propagating classical trajectories on the classical potential surface of the reacting system and using the positions of the atom of the system to generate the centroid position for the quantum-mechanical partition-function. This treatment is based on the finding that the quantum-mechanical partition-function can be expressed as42,46 ** ( )+ + X Zqm ðxÞ ¼ Zcl ðxÞ exp 2ðb=pÞ Uðxk Þ 2 UðxÞ ð23:16Þ k

fp U

where x is the centroid position, k· · ·lfp designates an average over the free particle quantum mechanical distribution obtained with the implicit constraint that x coincides with the current position of the corresponding classical particle, and k· · ·lU designates an average over the classical potential U. Using Equation 23.16 we can obtain the quantum mechanical free energy surface by evaluating the corresponding probability by the same combined free-energy perturbation umbrella sampling approach that has been repeatedly applied in our classical simulations as well as in our quantum mechanical simulations, but now we use the double average of Equation 23.16 rather than an average over a regular classical potential. The actual equations used in our FEP umbrella sampling calculations are given elsewhere, but the main point of the QCP is that the quantum mechanical free-energy function can be evaluated by a centroid approach, which is constrained to move on the classical potential. This provides stable and relatively fast-converging results that have been shown to be quite accurate in studies of well-defined test potentials (where the exact quantum mechanical results are known). To further improve the stability, e.g., for calculating the temperature dependence in Section IV.E, it can be useful to formulate the reaction free energy as a regular classical

630

Isotope Effects in Chemistry and Biology

temperature expansion, in terms of enthalpy and entropy, and a quantum mechanical correction, DDg‡cl!qm ðTÞ; according to Dg‡qm ðTÞ ¼ DHcl‡ 2 TDS‡cl þ DDg‡cl!qm ðTÞ

ð23:17Þ

This approach uses the FEP/US procedure to calculate the classical reaction profile and adds the NQM correction. In the case of temperature dependence, it is important to realize that this correction term is temperature dependent and can also be expanded into a temperature-independent and temperature-dependent term ‡ DDg‡cl!qm ðTÞ ¼ DDHcl!qm 2 TDDS‡cl!qm

ð23:18Þ

This formulation is more appropriate for studying Arrhenius plots since the classical terms DHcl‡ and TDS‡cl depend on the temperature implicitly and usually need more extensive sampling than the quantum correction term, which depends explicitly on the temperature (V in Equation 23.15).

IV. SIMULATIONS OF THE KIE AND NUCLEAR QUANTUM MECHANICAL EFFECTS IN ENZYMATIC REACTIONS Although the experimental evaluations of KIEs and related measurements can lead to significant insight about NQM, it is crucial to use computer simulation approaches to interpret the meaning of the experimental findings. Our effort in this direction starts quite early with the first reported simulation of an adiabatic PT in solution38 followed by studies that to focused on enzyme catalysis. Some of these studies are reviewed below.

A. SYSTEMATIC S TUDIES OF H YDRIDE T RANSFER IN S OLUTIONS Perhaps the most systematic theoretical study of hydride transfer reactions in solutions involved the analysis of the linear free energy relationships (LFERs) for hydride transfer between NADþ analogues in solution.47 This study involved the evaluation of free-energy barriers, reorganization energies and the corresponding NQM effects in the above series. It was found that the reorganization energies are much larger that the values estimated from Marcus’ equation. This was established using well defined microscopic calculations and the full NQM corrected EVB freeenergy barrier. This finding is highly relevant to our conclusions about the reorganization free energy in carbonic anhydrase (see below).

B. SIMULATING NQM EFFECTS IN LDH BY A M ICROSCOPICALLY B ASED Q UASIHARMONIC M ODEL AND A QCP T REATMENT Our first study of NQM effects in enzymes (and probably the first simulation study of NQM effects in enzymes) involved simulations of the reaction of lactate dehydrogenase (LDH).46 In this study we used the dispersed polaron (DP) model to describe the reaction surface as a function of ten effective protein modes and five solute modes. This was done by using the quasiharmonic approximation: Hii ¼ Vi ø "=2ðq 2 lÞT vðq 2 lÞ þ DVi0 Hij ø Hij0 expð2Dqij mij Dqij Þ Vg ¼

ð23:19Þ

CTg HCg

where the components of the vectors q and l are, respectively, the normal modes and origin shifts of the system and v is a diagonal matrix whose elements are the frequencies of the system.

Computer Simulations of Isotope Effects in Enzyme Catalysis

631

Here, Dqij ¼ qij 2 lij and lij ¼ ðli þ lj Þ=2: The trick is to obtain the v’s and l’s for the entire enzyme þ substrate complex. This is done here by the DP method. With this simple and explicit model we used the QCP model (with classical trajectories on the quasiharmonic surface) and obtained the KIE for the protein and solution reaction. It was found that the quantum correction to the free energy is similar in the enzyme and solution reaction (2.6 and 2.9 kcal mol21, respectively).

C. NUCLEAR Q UANTUM M ECHANICAL E FFECTS IN C ARBONIC A NHYDRASE One of our most systematic studies involved the simulation of the rate determining proton transfer step in the catalytic reaction of carbonic anhydrase (CA). The NQM effects in this system have been studied by the QCP method. Although we obtained significant NQM effects, it was found again that the same effects exist in the reference solution reaction. In this case we tried to separate the effect of the reorganization energy and reaction free energy from those of the NQM effects. The corresponding results are outlined in Figure 23.4, indicating that the catalysis is due to the reduction in the reorganization energy and DG rather than to the NQM effects. This finding is of general importance since frequently we can get larger NQM effects when the classical activation barrier is reduced and one may argue that the enzyme catalyzes the reaction by reducing the barrier and the corresponding increase in NQM effects. However, we were able to show that the NQM effects are the same for the water reaction as for the corresponding enzyme with the same reduced classical barrier. It is important to point out that the reorganization energy obtained from the simulation is much larger than that estimated by phenomenological use of the Marcus’ formula.48,49 The problem with the phenomenological approach are outlined elsewhere50 and they reflect the fact that the Marcus formula misses the effect of the mixing term which is quite large in PT reactions. In conclusion, the tunneling contribution becomes more important when the barrier is lower, and in the enzyme the barrier is lower because of preferential binding of the transition state relative to what happens in solution. This would lead us to think erroneously that tunneling contributions should be more important in enzymes than in solution, but this overlooks the fact that first there exists a barrier decrease (the actual reason for catalysis) and only then a minor side effect will be the slightly higher tunneling effect that would be obtained in the enzyme relative to solution. 20 in water

in protein 10

+

∆g + (kcal/mol)

15

5 0 −20

−15

−10 −5 0 ∆G (kcal/mol)

5

10

FIGURE 23.4 The dependence of the quantum (squares) and classical (diamonds) Dg‡ on DG0 for PT and DT in the active site of carbonic anhydrase and in a solvent cage. The activation barriers are much larger in aqueous solution than in the enzyme site. The main reason for the difference in Dg‡ is the reorganization energies.

632

Isotope Effects in Chemistry and Biology

D. NUCLEAR Q UANTUM M ECHANICAL E FFECTS IN A LCOHOL D EHYDROGENASE The catalytic reaction of alcohol dehydrogenase (ADH) has been the subject of extensive experimental investigations (e.g., Ref. 6). These studies were used to imply that NQM and dynamical effects play a major role in catalysis (e.g., Refs. 6,7,51 –58). To explore this issue, we conducted systematic studies of this reaction in water and the protein active site. It was found that the catalytic effect of the enzyme is due to the electrostatic preorganization of the polar active site as is the case with other enzymes.3 As far as the dynamical proposal is concerned, it was found that the same dynamical effects operate in the enzyme and solution reaction, and the dynamical effects do not contribute to catalysis. Nevertheless, we investigated significant effort in exploring the role of NQM contributions to the catalytic process and our findings are discussed below. Klinman and coworkers have made extensive use of the Swain –Schaad relationship,59 and they have been successful in demonstrating the presence of tunneling effects in some enzymatic reactions. In particular, they have found an unusually high secondary kinetic isotope effect of the hydride-transfer step in the reaction catalyzed by ADH.6 This finding has partially motivated the number of studies devoted to the explanation of NQM effects in ADH which have appeared in recent years. Kohen and Jensen,60 for example, used the ADH model to propose an extension of the Swain –Schaad rule in experimental studies of KIEs, although there does not seem to be a necessary correlation between the breakdown of this rule and the tunneling contribution.61,62 It is interesting to note that the catalytic enhancement of ADH is relatively low. Thus, the hydride transfer in the direction depicted in Scheme 23.1 needs to surmount a free energy barrier, Dg‡ ; of approximately 20 kcal mol21 in water and approximately 15 kcal mol21 in protein, where the exothermicity of the process of 4 and 1.4 kcal mol21 in water and protein, respectively.63,64 Note also that there is no rate enhancement for the reaction in the opposite direction and the effect of the enzyme in that direction would be the trivial one of binding both ground state and transition state of the reverse process with the same affinity. From the convex curvature of the Arrhenius plots for a thermophillic ADH, Klinman and coworkers54 concluded that not only is tunneling an important factor in the enzymatic reactivity in ADH, but also dynamical effects are significant. Interestingly, they found an opposite trend in mesophilic ADH.65 They also concluded that the rigidity of a thermophilic ADH is increased at room temperature, not only as compared with its own rigidity at its physiological temperature, but also as compared with a mesophilic ADH at room temperature.53 These dynamical effects were associated to the possible existence of promoting vibrations, which could be correlated with the reaction coordinate in a coherent way.51 Antoniou and Schwartz55 performed MD RS 1

RS 2 L1

H11

C1

O1

H5

H6

C5 C6

C4

N1

L2

C3 C2

R'

H11

O1 L1 C1

SCHEME 23.1

NADH

R'

H2 R

R benzaldehyde (BA)

L2

benzyl alcoholate (BA−)

NAD+

Computer Simulations of Isotope Effects in Enzyme Catalysis

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simulations of the hydride transfer from an alcohol to NAD þ in an acetonitrile solution (reverse reaction in Scheme 34.1). They argued that the explanation for the convex Arrhenius plots found by Kohen and Klinman54 should be the existence of promoting vibrations that are crucial to catalysis (presumably by some coherent coupling). Caratzoulas et al.56 applied a methodology similar to the dispersed polaron model of Section II.A, but restricted to selective distances rather than the actual reaction coordinate, ADH. They found that the relative motions between the alcohol and NAD and between NAD and Val203 are in resonance, and this relative motion acts as a promoting vibration, since it modulates the distance between the donor and acceptor C atoms. The fact that there are vibrations that are coupled to the reaction coordinate of enzymatic reactions has been recognized and quantified long ago (e.g., Refs. 30,43,66). This includes the development of the dispersed polaron model to determine the modes with significant projection along the reaction coordinate. However, the same type of promoting modes exist in chemical reactions in solution and reflect the fact that the reaction coordinate is not simple in Cartesian representation. The issue is whether there is any special coherent excitation of the modes that contribute to the reaction coordinate. At present, there is no experimental evidence that some particular modes of the protein may participate in the reaction in a coherent way. Other interpretations of the experimental results are possible, and the temperature effects that have been offered as evidence for dynamical effects in alcohol dehydrogenase can be rationalized in a consistent way as entropic effects.30,43 Fluctuations similar to those that are supposed to facilitate tunneling in the protein also appear in the reference reaction in water. We found that in LADH, the spectral densities obtained from fluctuations of the reactive systems in protein active sites and water are similar, and thus the modes that are coupled to the reaction coordinate are similar. The effects of mutations (e.g., of Val203), which also were brought as evidence for the importance of tunneling contributions to catalysis,6 may not be relevant to studies of dynamical effects in the chemical step. That is, although the recent studies of Rubach and Plapp69 have finally provided an estimate of the effect of the V203A mutant on the chemical step (rather than on kcat/kM) they found only a 16-fold reduction in the rate which is rather small and does not provide a good benchmark for testing theories about tunneling contribution to catalysis. Furthermore, the primary isotope effect appears to stay constant upon mutation, see ref 69, thus making it hard to see how NQM contributions can change the barrier height. Moreover, the proposal that the compression of the donor-accept distance should increase the tunneling contribution is inconsistent with current findings of the opposite trend (see Section VI.A). It is also useful to point out that the MD studies of the effect of Val203 on the ground-state geometry may not have much bearing on the catalytic effect. In general, experiments67,68 and theoretical calculations58 based on mutations of residues in the active site that have been put forward as probes of the existence of promoting vibrations in the chemical step of ADH, do not provide any direct evidence of a dynamical effect (see discussion in Ref. 30). They indicate essentially that the reaction coordinate may be represented by different normal modes in the native and mutant protein (and as stated above, this is not a dynamical effect unless we have a coherent excitation of these modes). A recent study by Hammes-Schiffer and coworkers44 found that Val203 has no significant dynamical effect and that its effect on the rate constant is associated with an increase of the activation free energy. Recently, Alhambra et al. found small classical dynamical effects in LADH (the classical variational correction for recrossing was small) and the tunneling contribution to Dg‡ was calculated to be around 2 kcal mol21.70 When the protein fluctuations are taken into account by the QCP approach, the contribution of NQM effects to Dg‡ are also around 2 kcal mol21.30 Thus, it seems that the NQM effects is found to be very similar by using a method that includes explicitly the fluctuations of the protein (QCP) and others that do not. In a related theoretical study of the hydride-transfer reaction catalyzed by formate dehydrogenase, Torres et al.71 used MD simulations and showed a large dispersion of donor – acceptor distances in the ground state. Unfortunately, they

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Isotope Effects in Chemistry and Biology

have not studied the same reaction in water, which would tell us if the different dispersion of distances in the two environments have something to do with enzyme catalysis. The relationship between the activation barrier and the fluctuations is a well-known fact, which has also been pointed out repeatedly in earlier papers.

E. LIPOXYGENASE AND

THE

L ARGE T UNNELING L IMIT

In a recent study,72 we explored the performance of the QCP approach in the range of large NQM corrections by calculating the large kinetic isotope effect in soybean lipoxygenase (SLO-1) and its temperature dependence using the approach of Equation 23.17 and Equation 23.18. The basic reaction catalyzed by lipoxygenase enzymes is the peroxidation of 1,4-pentadiene containing fatty acids. This reaction is believed to proceed via a radical mechanism (though other suggestions have been proposed, it seems that the radical mechanism is presently the most widely supported73 – 75), where the initial and rate-limiting step consists of a hydrogen atom abstraction from the substrate to a nonheme iron-active site. This transfer is associated with an exceptionally large kinetic isotope effect, kH/kD , 80, 73 which is significantly larger than that which is commonly seen in enzymes, kH/kD , 3– 8. As was found, and can be seen from the solid black line with diamonds in Figure 23.5 (DHcl‡ ¼ 16 kcal mol21 and 2TDS‡cl ¼ 4 kcal mol21), our analysis reproduces the trend of a very large isotope effect and the large difference between the quantum mechanical and classical temperature dependence. The three values of DHcl‡ (15, 16, and 17 kcal mol21, circles, diamonds and triangles, respectively) reflect the error range in our EVB refinement procedure. All these are almost parallel to the experimental (black with no symbols) line. For comparison, we have also included classical (dashed) lines in the Arrhenius plot that correspond to the same DHcl‡ and DS‡cl ; but without the NQM correction term, DDg‡cl!qm in Equation 23.17. In general it is extremely challenging to obtain the detailed compensation between entropic and enthalpic contributions by computer simulations of protein76,77 and it is simpler to get stable activation free energy due in part to the well known enthalpy entropy compensation effect.76 It is reasonable to find that we can reproduce the overall NQM effects on the activation free energy of Equation 23.17 but may have difficulties providing the detailed decomposition to temperature

4

log kHcat

2

0

−2

−4 2.9

3.1

3.3 3

−1

3.5

3.7

10 /T (K )

FIGURE 23.5 Arrhenius plot of the free energy barrier for lipoxygenase; experimental (solid black line without symbols) and theoretical values (solid black line; DHcl‡ ¼ 15 (circles), 16 (diamonds) and 17 kcal mol21 (triangles), and 2TDS‡cl ¼ 4 kcal mol21). The dashed lines show the reaction with the same classical free energy contributions, DHcl‡ and DS‡cl ; but without the nuclear quantum mechanical correction, DDg‡cl!qm in Equation 23.17.

Computer Simulations of Isotope Effects in Enzyme Catalysis

635

dependence and temperature-independent terms. We would also like to mention at this point that we are not aware of any other simulation approach that can actually evaluate the temperature dependent of the quantum activation free energy in proteins. Here, we do not consider gas-phase studies of hydrogen transfer reaction that may reproduce the small observed temperature dependence of the isotope effect but cannot simulate the effect of the protein and thus cannot help in elucidating the possible role of the protein in enhancing NQM effects. It is also interesting to comment that our study reproduced the observed temperature dependence and activation enthalpy of the H-transfer reaction but did worse in reproducing the trend in the D-transfer reaction. It is well-known that in the case of significant tunneling effects the rate constant is temperature independent in a range and then start to be temperature dependent.43 The transition between tunneling to the activated region reflects the properties of the medium e.g., the reorganization energy and other factors that may be hard to simulate exactly, especially when there is a competition between the classical entropic effect (e.g., the DS‡cl of Equation 23.17) and the quantum corrections. The corresponding difficulty may be appreciated by realizing that it is extremely hard to obtain the freezing point of proteins by MD simulations. The difficulties of reproducing the temperature dependence of the isotope effect reflect a wellknown difficulty in reproducing the temperature dependence of free energies by microscopic simulation. It also seems to reflect an actual physical instability demonstrated by the effect of the Ile553Ala mutation that changes AH/AD by a factor of , 150 while leaving kcat (and therefore DDg‡ ) unchanged.73 Despite the difficulties of obtaining the temperature dependence of the KIE, we have demonstrated that the QCP approach provides a robust way of evaluating NQM contributions to activation free energies. Overall we feel that the study of lipoxygenase has accomplished our main aim of obtaining a reliable tool for studying the NQM contributions to catalysis. With this tool we are in a position to examine the effect of NQM contributions to kcat and on the rate constant for reactions in solution, and thus to explore the role of NQM effects in catalysis. The present work compared the NQM contribution to the activation free energy in both the enzyme and the gas-phase reaction and found the corresponding contributions to be similar. The same calculations were obtained with the solution reaction, which indicates that the enzyme does not use NQM effects to catalyze its reaction in the present case.

V. WHAT IS THE CATALYTIC CONTRIBUTION FROM NUCLEAR QUANTUM MECHANICAL EFFECTS? As clarified in Section I, a contribution to enzyme catalysis must change the ratio between the rate constant for the enzyme and reference solution reaction. This requires that the difference between Dg‡cat and Dg‡w will be affected by the given contribution, excluding the unlikely event that the catalysis is due to the preexponential factor (see below). With this in mind, we must conclude that NQM contributions to catalysis must involve different quantum contributions to the activation free energies of the enzyme and water reactions. In all the cases examined by us so far we could not find a case where the NQM effects provide a major contribution to catalysis. We are also not aware of any realistic theoretical study that leads to the opposite conclusions. We believe that, although NQM effects can be quite large, they are similar in the enzyme and in the reference solution reaction. In general, it is hard for enzymes, which are quite flexible, to change the width of the solute potential energy surface in a drastic way and thereby change the NQM effects. The solvent contribution to the potential energy surface and the reorganization energy can be changed significantly in cases of charge transfer reactions,79 but this does not seem to lead to large changes in the NQM effects.8 Thus, although NQM effects can be quite large, they are usually similar in the enzyme and solution reaction.

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Isotope Effects in Chemistry and Biology

VI. WHAT CAN AND WHAT CANNOT BE LEARNED FROM SIMULATIONS OF ISOTOPE EFFECTS? Isotope effects provide a powerful diagnostic tool that has helped tremendously in elucidating reaction mechanisms and solving key biological problems. However, in some cases the interpretation (which is always of a theoretical nature) of observed KIE has been confused with the reliability of the experimental observations. At this point, we believe that it is important to consider the validity of some frequently assumed interpretations and this issue is examined below.

A. THE USE

OF

V IBRONIC M ODELS IN S TUDIES OF I SOTOPE E FFECTS

One of the most popular ways to model NQM effects in solution and enzymes has been the use of a (phenomenological) vibronic formulation, where the rate constant is obtained as a function of the Franck Condon (FC) factors for vibrational transitions between the quantum vibration of the reactant and product states (see e.g., Refs. 78 and 80). Here, the rate constant for transitions from the ith to the jth vibronic state is scaled by (HabSij)2 rate where Sij is the FC factor for the given transition. As clarified repeatedly (see e.g., Ref. 38) the main problem with this expression is associated with the fact that it is only valid in the diabatic limit (where HabSij is small) and is therefore not applicable to most hydrogen-transfer and proton-transfer reactions (where HabSij is large) and one must use a method that is appropriate for the adiabatic limit. Another problem is associated with the fact that the proper FC factors cannot be obtained from phenomenological considerations. Nevertheless, we provide in Figure 23.6 the microscopically based FC factors (obtained by the DP treatment) for SLO to illustrate the relevant issues. The simulated FC factors for the solvent modes are not as relevant since, as mentioned elsewhere,72 the solvent reorganization energy is very small in SLO (, 2.5 ^ 1 kcal mol21) and does not contribute to the rate constant in a significant way. The main contribution to the total reorganization is associated with the solute modes, whose harmonic analysis is not simple. As seen from Figure 23.6, the DP results cannot be represented by a single set of origin shifts since the energy gap cannot be 15 160 Solid spectrum

Dashed spectrum

∆ (w ) (dimensionless)

140 120

10

100 80

0

2

1

5

0

0

1000

2000 w (cm−1)

3000

4000

FIGURE 23.6 The dispersed polaron analysis (Equation 23.12) for the catalytic reaction of lipoxygenase. As usual the origin shifts (the D:s) and the frequencies (the v:s) are obtained from the power spectrum of the time dependent energy gap DVba ¼ 1b 2 1a : However, in our case the energy gap gives different results for different time segments (this corresponds to fluctuations of the donor – acceptor distance and other variables). We provide DP results for the time segments of the 0 –1 and 1 – 2 ps.

Computer Simulations of Isotope Effects in Enzyme Catalysis

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decomposed in a unique way to a single set of shifted harmonic surfaces. The reason for this is associated with the fact that the hydrogen transfer barrier changes drastically when the donor acceptor distance and/or the transfer angel change. The best result is obtained by performing a separate DP analysis for different time segments while using the different FC factors in evaluating several vibronic rate constants (for different donor acceptor distances). Although the vibronic treatment is not expected to give valid results in cases where HabSij is large, since the system cannot be described by the diabatic limit, it is possible to get a useful insight fot the zero-zero transitions since S00 can be sufficiently small to move us close to the diatatic limit. Using such a treatment, we found recently (Olsson et al., to be published) that the temperature dependence of the isotope effect of lipoxygenase can be reproduced in a resonable way considering the potential for the donor – acceptor distance in a phenomenological way. More significantly, we found that the tunneling and isotope effect decrease (rather than increase) when the donor and acceptor are compressed together. This finding brings into question the idea that proteins can increase tunneling by compressing the reacting system.

B. USING C ALCULATED AND O BSERVED I SOTOPE E FFECTS AS A T OOL FOR VALIDATING S INGLE O NLY S IMULATIONS OF NQM AND D ETERMINING THE C ATALYTIC C ONTRIBUTIONS OF NQM E FFECTS The ability to correctly simulate the observed isotope effects is a prerequisite from any model that is aimed at elucidating the catalytic contribution of NQM effects. Obviously before judging the validity of the conclusions obtained from the QCP approach it is essential to examine its ability to reproduce the isotope effects observed in different enzymatic reactions and such an examination is presented in Table 23.1. As can be seen from this table, overall we obtain good agreement between the calculated and observed isotope effects in a vide class of enzymes. Similar agreement has been obtained by other approaches.44,81 However, the QCP method provides probably the most effective way of examining the effect of the protein fluctuations on the isotope and other NQM effects. More importantly, the QCP studies presently provide the most extensive comparison of isotope effects in enzymes and the corresponding solution reactions. All of these studies (i.e., Table 23.1) indicate that the NQM effects are similar in enzymes and solutions. This finding is particularly important because, in many cases the solution reaction is different from the corresponding enzymatic reaction. Note, however, that the proper reference reaction is defined82 by considering the same mechanism as the one that occurs in the enzyme in a reference solvent cage. Fortunately, this reference reaction

TABLE 23.1 Experimental and Calculated Isotope Effects Enzyme Enzyme

Calc

Obs, Ref.

Water Calc

Method

Year, Ref.

LDH Carbonic anhydrase Carbonic anhydrase Glyoxylase ADHa Lipoxygenase

5.0 2.3 3.9 5.0 — 86

2–392 3.893 3.895 3.096 3.898 8173

5.6 — — 3.6 — 100

QCP QCP QCP QCP QCP QCP

199146 199294 19968 200097 200130 200472

a

In the case of ADH, we only calculated the total NQM contribution for the hydrogen transfer and found it to be similar in the enzyme and water reactions. We assume that the KIE is similar for the enzyme and water reactions.

638

Isotope Effects in Chemistry and Biology

and the corresponding isotope effect are rigorously defined and easily examined by computer simulation approaches.

C. DETERMINING THE C ONCERTEDNESS OF E NZYMATIC R EACTIONS BY

THE

KIE

It has been assumed repeatedly that KIE can be used to detect concerted paths in enzymatic reactions.83 – 85 While the experimental origin of this assertion has been reasonable in some respects, the theoretical support has been problematic at best. Unfortunately, one can find theoretical studies that quote the experimental interpretation as a fact without any serious attempt to validate these interpretations. This includes replacing the proper potential surface in the enzyme by gas-phase calculations that tend to give unrealistic concerted surfaces.83 – 85 Proper calculations seem to give surfaces that are much less concerted than has been thought, based on gas-phase calculations. Furthermore, entropic effects tend to destabilize concerted pathways. The common argument that the observation of KIE can be used to prove the existence of concerted paths is problematic. One such case is that of carbonic anhydrase, where proper calculations that take the enzyme environment into account reproduced the observed KIE with a stepwise path. Thus, the argument of Cui and coworkers,83 who studied the concerted path in the gas phase, that their calculations support the concerted mechanism is problematic. More specifically, the energetic of the concerted and stepwise paths are usually very similar and the potential energy surface is shallow. The best way to examine the meaning of the observed KIE is to calculate it for both paths. Unfortunately, no such study has been reported by those who promoted the concerted mechanism as an established fact.

D. DYNAMICAL E FFECTS AND P ROMOTING M ODES The temperature dependence of the KIE has been used as an evidence for the importance of dynamical effects in enzyme catalysis. It is clear that the experimental information about KIE and its temperature dependence might provide interesting clues about the coupling between the protein modes and the reacting substrates. However, such clues cannot be extracted without the use of microscopic simulations of the type described here. With this in mind, we note that qualitative arguments that relate the temperature dependence of the KIE to dynamical contributions to catalysis are not based on a well-defined analysis and that such observations are more consistent with a nondynamical picture (see discussion in Ref. 30). More importantly, the same simulations that reproduced the observed KIE and NQM contributions to activation energies of enzymatic reactions demonstrated that dynamical effects do not contribute significantly to enzyme catalysis.43 Another study that has been considered as evidence that enzyme catalysis is significantly enhanced by dynamical contributions and involved a ground state simulation of dihydrofolate reductase (DHFR).86 It seems to us that investigations that do not consider the relevant reaction or activation barrier should not be used to support the dynamical proposal. However, a more recent simulation study by Brooks and coworkers that does consider the actual reaction of DHFR concluded that dynamical effects are not important in this system.87 It has also been suggested that there are special promoting modes that are associated with special catalytic effects (see e.g., Refs. 7,55). The existence of promoting modes has been established previously by observing peaks in the spectral distribution of, for instance, the dispersed polaron analysis,46 or even by examining peaks in the distribution of some distances. This observation does not provide by itself any evidence that dynamical effects contribute to catalysis. Clearly, all reactions can be described by a reaction coordinate that involves several modes (the reaction coordinate is not a straight line in the Cartesian space). However, the same types of promoting modes exist in both solution and proteins. Thus, we can conclude that promoting modes in enzymatic reactions is a well-known fact that should not be used to imply a new catalytic

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mechanism, unless it is demonstrated that these modes are excited in a coherent way in the enzyme. So far this has not been demonstrated. Finally, it is important to clarify some recent misunderstandings about the assertion that the reorganization energy and gate opening fluctuations are dynamical effects,88 as well as the assumption that the description of PT processes by curve-crossing formulations (e.g., Ref. 89) provide a new dynamical insight. The view of PT in solutions and proteins as a curve crossing process has been formulated in early realistic simulations studies38,66,90 with and without quantum corrections. The phenomenological formulation of such models has already been introduced quite early by Kuznetsov and others.80 Furthermore, that the fluctuations of the environment in enzymes and solution determined the rate constants of PT reactions has been demonstrated in realistic microscopic simulations of Warshel and coworkers.66,90 However, as clarified in these works, the time dependence of these fluctuations does not provide a useful way to determine the rate constant. That is, the electrostatic fluctuations of the environment are determined by the corresponding Boltzmann probability and do not represent a dynamical effect. In other words, the rate constant is determined by the time it takes the system to produce reactive trajectory multiplied by the time it takes such trajectories to move to the TS. The time needed for a generation of a reactive trajectory is determined by the corresponding Boltzmann probability and the actual time it takes the reactive trajectory to reach the transition sate (in the order of picoseconds) is more or less constant in different systems. Concerning our statement that neither the reorganization energy nor the gate opening fluctuations are dynamical effects; first, the solvent reorganization energy, which determines the amplitude of the solvent fluctuations, is not a statical dynamical effect (as proposed in Ref. 88), but a unique measure of the free energy associated with the reorganization of the solvent from its reactant to its product configuration (see Ref. 91 for a more rigorous definition). Second, describing the chance of reaching a given solvent configuration as a dynamical gate opening is not useful or predictive. This probability is entirely determined by the corresponding Boltzmann probability,91 which is determined by DG and l.

VII. CONCLUDING REMARKS This work has described microscopic simulations of NQM effects in enzymatic reactions. The simulations considered have demonstrated the ability to reproduce, in a quantitative way, the NQM contributions to the activation free energies of reactions in enzyme active sites and in-solution reference reactions. This unique ability provides a powerful tool to assess these contributions to catalysis. More specifically, our simulations that are capable of reproducing the observed KIE can go one crucial step further by quantifying the NQM contributions in both the enzyme and the corresponding solution reaction, even if they cannot be studied experimentally. Our studies have demonstrated that, although NQM effects are frequently significant, they do not lead to substantial catalytic effects since similar effects occur in both the enzyme and solution reactions. Of course, this cannot be considered as a general proof since we have not studied all possible enzymatic reactions. However, in the many cases studied by us to date, we have repeatedly found that the major contribution to the overall catalytic effect is due to electrostatic preorganization. The finding of our studies should not be taken as a suggestion that one should stop exploring the nature of enzymatic reactions by studying KIE. First, the contributions of KIE studies of enzyme mechanisms are of tremendous importance as is demonstrated in many chapters in this book. Second, the use of KIE experiments is essential for establishing the existence of NQM effects in enzymes. Furthermore, the finding of interesting temperature dependence of the KIE provides a major challenge for various theoretical models. With this in mind, however, we wish to clarify that the unique interpretation of KIE experiments requires powerful microscopic models and simplified interpretations of the meaning of KIE experiment should not be confused with experimental findings. We believe that one of the most important roles of experimental KIE measurements is to

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Isotope Effects in Chemistry and Biology

provide a challenging validation bench-mark for theoretical models. Also we believe that the role of modern simulation approaches is not so much in explaining the experimental observation, but rather in explaining how enzymes work while gaining credibility by being able to reproduce diverse experiments that may or may not be related to enzyme catalysis.

ACKNOWLEDGMENTS This work was supported by NIH grant GM24492. We gratefully acknowledge the University of Southern California’s High Performance Computing and Communications Center for computer time. J.V. was supported by the Spanish MCYT grant BQU 2003-04448.

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44 Billeter, S. R., Webb, S. P., Agarwal, P. K., Iordanov, T., and Hammes-Schiffer, S., Hydride transfer in liver alcohol dehydrogenase: quantum dynamics, kinetic isotope effects, and role of enzyme motion, J. Am. Chem. Soc., 123, 11262– 11272, 2001. 45 Feynman, R., Statistical Mechanics, Benjamin, New York, 1972. 46 Hwang, J. K., Chu, Z. T., Yadav, A., and Warshel, A., Simulations of quantum-mechanical corrections for rate constants of hydride-transfer reactions in enzymes and solutions, J. Phys. Chem., 95, 8445– 8448, 1991. 47 Kong, Y. S. and Warshel, A., Linear free-energy relationships with quantum-mechanical corrections — classical and quantum-mechanical rate constants for hydride transfer between Nad(þ) analogs in solutions, J. Am. Chem. Soc., 117, 6234– 6242, 1995. 48 Silverman, D. N., Marcus rate theory applied to enzymatic proton transfer, Biochimi. Biophys. Acta Bioenerg., 1458, 88 – 103, 2000. 49 Silverman, D. N., Tu, C. K., Chen, X., Tanhauser, S. M., Kresge, A. J., and Laipis, P. J., Rate equilibria relationships in intramolecular proton-transfer in human carbonic anhydrase-Iii, Biochemistry, 32, 10757– 10762, 1993. 50 Schutz, C. N. and Warshel, A., Analyzing free energy relationships for proton translocations in enzymes: carbonic anhydrase revisited, J. Phys. Chem. B, 108, 2066– 2075, 2004. 51 Mincer, J. S. and Schwartz, S. D., A computational method to identify residues important in creating a protein promoting vibration in enzymes, J. Phys. Chem. B, 107, 366– 371, 2003. 52 Maglia, G. and Alleman, R. K., Evidence for environmentally coupled hydrogen tunneling during dihydrofolate reductase catalysis, J. Am. Chem. Soc., 125, 13372– 13373, 2003. 53 Kohen, A. and Klinman, J. P., Protein flexibility correlates with degree of hydrogen tunneling in thermophilic and mesophilic alcohol dehydrogenases, J. Am. Chem. Soc., 122, 10738– 10739, 2000. 54 Kohen, A., Cannio, R., Bartolucci, S., and Klinman, J. P., Enzyme dynamics and hydrogen tunneling in a thermophilic alcohol dehydrogenase, Nature, 399, 496– 499, 1999. 55 Antoniou, D. and Schwartz, S. D., Internal enzyme motions as a source of catalytic activity: ratepromoting vibrations and hydrogen tunneling, J. Phys. Chem. B, 105, 5553– 5558, 2001. 56 Caratzoulas, S., Mincer, J. S., and Schwartz, S. D., Identification of a protein-promoting vibration in the reaction catalyzed by horse liver alcohol dehydrogenase, J. Am. Chem. Soc., 124, 3270– 3276, 2001. 57 Mincer, J. S. and Schwartz, S. D., Protein promoting vibrations in enzyme catalysis — a conserved evolutionary motif, J. Proteome Res., 2, 437–439, 2002. 58 Kalyanaraman, C. and Schwartz, S. D., Effect of active site mutation Phe93- . Trp in the horse liver alcohol dehydrogenase enzyme on catalysis: a molecular dynamics study, J. Phys. Chem. B, 106, 2002. 59 Swain, C. G., Stivers, E. C., Reuwer, J. F., and Schaad, L. J., J. Am. Chem. Soc., 80, 5885 –5893, 1958. 60 Kohen, A. and Jensen, J. H., Boundary conditions for the Swain – Schaad relationship as a criterion for hydrogen tunneling, J. Am. Chem. Soc., 124, 3858– 3864, 2002. 61 Villa`, J., Gonza´lez-Lafont, A., and Lluch, J. M., On kinetic isotope effects as tools to reveal solvation changes accompanying a proton transfer. A canonical unified statistical theory calculation, J. Phys. Chem., 100, 19389– 19397, 1996. 62 Cui, Q., Elstner, M., and Karplus, M., J. Phys. Chem. B, 106, 2721, 2002. 63 Va´rnai, P. and Warshel, A., Computer simulation studies of the catalytic mechanism of human aldose reductase, J. Am. Chem. Soc., 122, 9641– 9651, 2000. 64 Sekhar, V. C. and Plapp, B. V., Rate constants for a mechanism including intermediates in the interconversion of ternary complexes by horse liver alcohol dehydrogenase, Biochemistry, 29, 4289– 4295, 1990. 65 Tsai, S-C. and Klinman, J. P., Probes of hydrogen tunneling with horse liver alcohol dehydrogenase at subzero temperature, J. Am. Chem. Soc., 40, 2303– 2311, 2001. 66 Warshel, A., Dynamics of enzymatic-reactions, Proc. Natl. Acad. Sci. USA Biol. Sci., 81, 444– 448, 1984. 67 Rubach, J. K., Ramaswamy, S., and Plapp, B. V., Contributions of valine-292 in the nicotinamide binding site of liver alcohol dehydrogenase and dynamics to catalysis, Biochemistry, 40, 12686– 12694, 2001. 68 Rubach, J. K. and Plapp, B. V., Mobility of fluorobenzyl alcohols bound to liver alcohol dehydrogenases as determined by NMR and x-ray crystallographic studies, Biochemistry, 41, 15770– 15779, 2002.

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69 Rubach, J. K. and Plapp, B. V., Amino acid residues in the nicotinamide binding site contribute to catalysis by horse liver alcohol dehydrogenase, Biochemistry, 42, 2907– 2915, 2003. 70 Alhambra, C., Corchado, J. C., Sa´nchez, M. L., Gao, J. L., and Truhlar, D. G., Quantum dynamics of hydride transfer in enzyme catalysis, J. Am. Chem. Soc., 122, 8197– 8203, 2000. 71 Torres, R. A., Schiott, B., and Bruice, T. C., J. Am. Chem. Soc., 121, 8164– 8173, 1999. 72 Olsson, M. H. M., Siegbahn, P. E. M., and Warshel, A., Simulations of the large kinetic isotope effect and the temperature dependence of the hydrogen atom transfer in lipoxygenase, J. Am. Chem. Soc., 2004. 73 Knapp, M. J., Rickert, K., and Klinman, J. P., Temperature-dependent isotope effects in soybean lipoxygenase-1: correlating hydrogen tunneling with protein dynamics, J. Am. Chem. Soc., 124, 3865– 3874, 2002. 74 Borowski, T. and Broclawik, E., Catalytic reaction mechanism of lipoxygenase. A density functional theory study, J. Phys. Chem. B, 107, 4639– 4646, 2003. 75 Lehnert, N. and Solomon, E. I., Density-functional investigation on the mechanism of H-atom abstraction by lipoxygenase, J. Biol. Inorg. Chem., 8, 294– 305, 2003. 76 Levy, R. M. and Gallicchio, E., Computer simulations with explicit solvent: recent progress in the thermodynamic decomposition of free energies and in modeling electrostatic effects, Ann. Rev. Phys. Chem., 49, 531– 567, 1998. 77 Villa`, J., Strajbl, M., Glennon, T. M., Sham, Y. Y., Chu, Z. T., and Warshel, A., How important are entropic contributions to enzyme catalysis?, Proc. Natl. Acad. Sci. USA, 97, 11899– 11904, 2000. 78 DeVault, D., Quantum mechanical tunneling in biological systems, Q. Rev. Biophys., 13, 387– 564, 1980. 79 Warshel, A., Energetics of enzyme catalysis, Proc. Natl. Acad. Sci. USA, 75, 5250– 5254, 1978. 80 Kuznetsov, A. M. and Ulstrup, J., Proton and hydrogen atom tunneling in hydrolytic and redox enzyme catalysis, Can. J. Chem. — Revue Canadienne De Chimie, 77, 1085– 1096, 1999. 81 Kim, Y., Corchado, J. C., Villa`, J., Xing, J., and Truhlar, D. G., Multiconfiguration molecular mechanics algorithm for potential energy surfaces of chemical reactions, J. Chem. Phys., 112, 2718– 2735, 2000. 82 Shurki, A. and Warshel, A., Structure/function correlations of proteins using MM, QM/MM, and related approaches: current progress, Adv. protein Chem., 66, 249– 313, 2003. 83 Cui, Q. and Karplus, M., Is a “proton wire” concerted or stepwise? A model study of proton transfer in carbonic anhydrase, J. Phys. Chem. B, 107, 1071–1078, 2003. 84 Daggett, V., Schroder, S., and Kollman, P., Catalytic pathway of serine proteases — classical and quantum-mechanical calculations, J. Am. Chem. Soc., 113, 8926– 8935, 1991. 85 Schowen, R. L., Molecular Structure and Energetics. Principles of Enzyme Activities, VCH publishers, Veinheim, 1988. 86 Radkiewicz, J. L. and Brooks, C. L., Protein dynamics in enzymatic catalysis: exploration of dihydrofolate reductase, J. Am. Chem. Soc., 122, 225– 231, 2000. 87 Thorpe, I. F. and Brooks, C. L. III, Barriers to hydride transfer in wild type and mutant dihydrofolate reductase from E. coli, J. Phys. Chem. B, 2003, ASAP. 88 Knapp, M. J. and Klinman, J. P., Environmentally coupled hydrogen tunneling — linking catalysis to dynamics, Eur. J. Biochem., 269, 3113– 3121, 2002. 89 Borgis, D. and Hynes, J. T., Curve crossing formulation for proton transfer reactions in solution, J. Phys. Chem., 100, 1118– 1128, 1996. 90 Warshel, A., Dynamics of reactions in polar-solvents — semi-classical trajectory studies of electrontransfer and proton-transfer reactions, J. Phys. Chem., 86, 2218– 2224, 1982. 91 King, G. and Warshel, A., Investigation of the free-energy functions for electron-transfer reactions, J. Chem. Phys., 93, 8682– 8692, 1990. 92 Clarke, A. R., Wilks, H. M., Barstow, D. A., Atkinson, T., Chia, W. N., and Holbrook, J. J., An investigation of the contribution made by the carboxylate group of an active-site histidine aspartate couple to binding and catalysis in lactate-dehydrogenase, Biochemistry, 27, 1617– 1622, 1988. 93 Silverman, D. N. and Lindskog, S., The catalytic mechanism of carbonic-anhydrase — implications of a rate-limiting protolysis of water, Acc. Chem. Res., 21, 30 – 36, 1988. 94 Warshel, A., Hwang, J. K., and Aqvist, J., Computer-simulations of enzymatic-reactions — examination of linear free-energy relationships and quantum-mechanical corrections in the initial proton-transfer step of carbonic-anhydrase, Faraday Discuss., 225– 238, 1992.

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95 Steiner, H., Jonsson, B. H., and Lindskog, S., Catalytic mechanism of carbonic-anhydrase — hydrogen-isotope effects on kinetic-parameters of human C isoenzyme, Eur. J. Biochem., 59, 253– 259, 1975. 96 Ridderstrom, M., Cameron, A. D., Jones, T. A., and Mannervik, B., Mutagenesis of residue 157 in the active site of human glyoxalase I, Biochem. J., 328, 231–235, 1997. 97 Feierberg, I., Luzhkov, V., and Aqvist, J., Computer simulation of primary kinetic isotope effects in the proposed rate-limiting step of the glyoxalase I catalyzed reaction, J. Biol. Chem., 275, 22657– 22662, 2000. 98 Bahnson, B. J., Park, D. H., Kim, K., Plapp, B. V., and Klinman, J. P., Unmasking of hydrogen tunneling in the horse liver alcohol-dehydrogenase reaction by site-directed mutagenesis, Biochemistry, 32, 5503– 5507, 1993.

24

Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen Justine P. Roth and Judith P. Klinman

CONTENTS I. II. III. IV.

Introduction ...................................................................................................................... 645 Instrumentation................................................................................................................. 646 Equilibrium Isotope Effects ............................................................................................. 648 Applications...................................................................................................................... 650 A. Glucose Oxidase....................................................................................................... 650 B. Tyrosine Hydroxylase .............................................................................................. 653 C. Soybean Lipoxygenase ............................................................................................ 655 D. Methane Monooxygenase ........................................................................................ 657 E. Cytochrome P-450 ................................................................................................... 658 F. Dopamine b-Monooxygenase and Peptidylglycine a-Hydroxylating Monooxygenase ........................................................................... 660 G. Copper Amine Oxidases .......................................................................................... 662 V. Overview and Perspectives for the Future ...................................................................... 665 References..................................................................................................................................... 666

I. INTRODUCTION With the exception of specialized ecological niches, life on planet Earth is inextricably linked to the chemistry of molecular oxygen. The range of enzymes that uses O2 as cosubstrate is exhaustive and includes those which activate and transform organic substrates through the direct insertion of one or both oxygen atoms from O2,1 those which bind recyclable cofactors that use O2 as a two-electron sink and generate hydrogen peroxide2 and those which interconvert O2 among its variously reduced forms3: e2

2Hþ ; e2

H þ ; e2

H þ ; e2

† O2 O O2 O H2 O2 O OH† þ H2 O O H2 O 2

ð24:1Þ

Regarding cellular physiology, the most central of the enzymes in the latter class is cytochrome C oxidase, the terminal electron acceptor in aerobic cells that couples the four-electron reduction of O2 to water for the generation of the cellular fuel, ATP.4 Efficient utilization of O2 is achieved through diverse strategies and exquisite control over products formed. It is widely held that the reactivity of O2 is correlated to its spin state, which can be either a singlet or triplet. However, reactivity patterns may be better explained on thermodynamic grounds. The high energy of singlet O2 leads to rapid depletion of this species from the atmosphere, with the resulting accumulation of the less oxidizing triplet O2 as the dominant species.5 This form is relatively unreactive because of high kinetic barriers and the modest thermodynamic 645

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Isotope Effects in Chemistry and Biology

driving force associated with its fundamental redox reactions.6 The reduction of each mole of triplet O2 to HO†2 releases ca. 3 kcal/mol while reduction of the singlet releases ca. 25 kcal/mol of free energy.1a From an evolutionary perspective, the existence of life as we know it, has been critically dependent on the capacity of enzymes to increase the reactivity of O2. The enzymatic strategies that have emerged accommodate the step-wise transfer of reducing equivalents, and production of partially reduced free radicals and charged species as intermediates (Equation 24.1 above). The underlying chemistry of the various O2 utilizing enzymes may vary, involving the participation of organic cofactors such as flavins, pterins, quinones, or redox active metal ions, but in all cases radical species are produced. How, then, do enzymes achieve their remarkable chemistry while escaping damage due to the presence of active-site, reactive radical intermediates? As we discuss in this chapter, oxygen-18 isotope effects have been developed as a generic tool that can lead to the dissection of reaction mechanism and catalytic strategy among the large class of O2-activating enzymes. In particular, the O-18 kinetic isotope effects (KIEs) can indicate whether electron transfer to O2 is kinetically irreversible and how other steps, such as protein conformational changes and the coupled transfer of protons may control electron transfer rates. The chapter begins with a description of the instrumentation that has allowed a streamlining of O-18 isotope effect measurements and a summary of the equilibrium measurements that are currently used as benchmarks for interpretation of kinetic effects. This is followed by a summary of progress regarding the nature of O2 activation among numerous classes of enzymes. Future challenges are discussed in the context of the considerable progress, as well as current limitations, of this approach.

II. INSTRUMENTATION The first measurements of O-18 isotope effects in an enzyme reaction were conducted with ribulose bisphosphate carboxylase (or rubisco).7 This enzyme catalyzes the initial step in the dark reaction of photosynthesis, inserting CO2 into a carbanionic intermediate derived from the five-carbon sugar ribulose bisphosphate. It was recognized many years ago that rubisco is unable to prevent a competing reaction in which O2 traps the carbanionic intermediate to form a peroxy adduct. Subsequent decomposition yields only 1 mole of the desired product, 3-phoshoglycerate, as opposed to the 2 moles formed from CO2 insertion. The reduction in crop yields that follows from the O2 side reaction had led many researchers to attempt to understand the O2 process.8 Dr. Joe Berry at the Carnegie Institute (Palo Alto) was the first to undertake a systematic measurement of the O-18 isotope effect in the rubisco O2 reaction, or in any enzyme reaction for that matter, designing an apparatus which has been adapted for use in a number of academic laboratories including those of the authors.6,9,10 Adjoined vacuum manifolds are used for the isolation of O2 from a reaction mixture and the preparation of samples for manometric and isotopic measurements. The central elements of the apparatus, a reaction chamber and combustion loop, are shown in Figure 24.1. Additionally, a gas bubbler and a series of traps are used to separate O2 from water vapor and CO2. A key feature of the methodology is that it uses natural abundance O2 to monitor the relative rates of reactions involving 16O – 16O and 18O – 16O. Aside from the obvious advantage of not requiring isotope enrichment, this approach avoids any error that would arise from dilution of the enriched O2 by ambient O2 in the air. However, since the natural abundance of 18O is low (, 0.2%), isotope ratio mass spectrometry analysis requires 5 to 20 mmol of O2. In general, the 18O/16O content of a reaction solution is analyzed before and after introduction of the enzyme of interest, at which point the sample chamber is refilled for subsequent measurements. In a typical experiment, the O2 in a He carrier gas, is collected in the cold trap containing molecular sieves. Following removal of He, warming of the sieves liberates O2. Although in principle, the isotopic composition of O2 could be analyzed directly, CO2 is easier to

Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen

647

Helium in 4 9 CO2, to pressure measurement

O2 isolated

1

2 8

2 3 4

5 7

(a)

6

To vacuum line for isolation and conversion of O2 to CO2

3 1 (b)

FIGURE 24.1 Instrumentation for measurement of O-18 kinetic isotope effects. (a) The reaction chamber used to introduce solution samples into the vacuum line for isolation of O2 and conversion to CO2. The plunger of a conventional syringe, consisting of a plastic stem (1) and rubber piston (2), is used to seal the jacketed reaction chamber (3) leaving no headspace above the solution. A rubber septum (4) covering a small side-arm on the reaction chamber is used for injections. Mixing is achieved using (5) a magnetic stir bar and stirring plate (6). The sample is introduced from the chamber through a 908 cross-over stopcock (7) in the 9 o’clock configuration to a calibrated volume tube (8). Although the barrel of the stopcock has four openings, the bore is only open in two directions at any given time. Once filling is complete, the stopcock is turned to 3 o’clock, dispensing solution to a bubbler for quenching and gas removal. A needle valve (9) is opened at this point to introduce He carrier gas into the system at a fixed flow rate and pressure. (b) While under dynamic vacuum, the He gas helps displace O2 from the solution. The gas mixture passes through a series of traps, for CO2 and water vapor removal, prior to entering the molecular sieve trap (1). Here O2 collects in the sieves cooled to 173 K with liquid N2. After a period of time collection is terminated and the sieve trap is bypassed. The He that accumulates along with the O2 is removed under reduced pressure. Heating of the sieves with boiling water releases O2 which is passed into a combustion loop for conversion to CO2. The furnace contains a graphite rod wrapped in platinum wire (2). A liquid nitrogen-cooled finger (3) is used to condense CO2 away as the gas recirculates through the heated furnace and the conversion is monitored using a standard thermocouple gauge (4). The CO2 is transferred to a second tube (not shown) for pressure determination within a fixed volume of the vacuum manifold. The sample is then transferred to a glass tube which is flame sealed and subsequently analyzed by isotope ratio mass spectrometry.

manipulate and analyze. For this reason the O2 is converted to CO2 over platinum wire and graphite at 8008C. Under these conditions the by-products are minimized. Pure CO2 is condensed away from other gases and its total pressure determined. The CO2 is placed in a sealed tube for subsequent isotopic analysis. Isotope ratio mass spectrometry gives isotopic ratios designated as d 18O per mil (part per thousand) with precisions of ^ 0.2.9 Essentially two types of measurements have been performed, those which give rise to equilibrium isotope effects (EIEs)11 and those which provide KIE measurements.10 These differ in their approaches, with the EIEs being obtained in an open system that uses sufficient protein such that ca. 50% of the dissolved O2 will be bound. Since the isotope composition will be different for dissolved O2 than for O2 bound to protein, analysis of the total O2, relative to a buffer-only solution, provides a measurement of 18O fractionation. In kinetic measurements, the system is closed to allow a continuous consumption of O2. The increased 18O in the unreacted O2 as the reaction proceeds is then related back to the initial 18O content of O2 in buffer prior to addition of enzyme.

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Isotope Effects in Chemistry and Biology

III. EQUILIBRIUM ISOTOPE EFFECTS Interpretation of KIEs among the various O2 utilizing enzymes requires a context of boundary values that can be approximated from a knowledge of EIEs. In an early study, Tian and Klinman chose to examine proteins that both bound O2 reversibly, and had been characterized in their oxygenated forms via a range of spectroscopic methods.11 The proteins that fit these criteria were those copper and iron containing carriers of dioxygen, hemoglobin (Hb), myoglobin, (Mb), hemerythrin (Hr), and hemocyanin (Hc). The postulated structures for the oxygenated forms of three of these proteins are shown in Table 24.1, in which the two-electrons reduced peroxo form of O2 in Hc and Hr is bound either “side on” (Hc) or “end on” (Hr). In the case of Hb (and Mb) the oxygenated species is represented as the one-electron reduced, “end on” superoxo species in which a single iron has undergone oxidation to its þ 3 valence state. The EIEs for binding of O2 to each of these systems were measured and found to vary from 1.0039 (Hb) to 1.0184 for Hc. In the absence of protonation or interaction with metal ions, the introduction of each electron into O2 can be formalized using the concept of bond order: e− O2

0.5

e− O2_•

0.5

ð24:2Þ

O22−

However, other factors are clearly expected to influence the net change in bond order and the measured EIEs. In order to provide a more quantitative basis for the experimental data, isotope effects were also calculated for the conversion of O2 to the six species shown in Table 24.2. These computations relied on the ability to express an EIE in terms of the isotope dependent partition functions corresponding to zero point (ZPE), excited state (EXC) vibrational energies, and the mass, and moments of inertia (MMI)12: EIE ¼ ZPE £ EXC £ MMI

ð24:3Þ

It has been shown that all three terms in Equation 24.3 can be written as a function of vibrational frequencies, n16;16 and n18;16 : These were either obtained from the literature, calculated from known

TABLE 24.1 Postulated Structures for Oxygenated Proteins and Measured Equilibrium O-18 Isotope Effects Protein Hc

Proposed Structures O

Cu(II)

Cu(II) O Fe(III)

Hr

O

Fe(III) O H

Hb

− O2• Fe(III)

a

O

Bondinga H2O2 , Hc , HO2 2; 1.0089 , 1.0184 , 1.0343

H2O2 ø Hr; 1.0089 ø 1.0113

HB Hb , HO†2 ; 1.0039 , 1.011

Comparison of the O-18 isotope effect for each protein to computed values for various reduced forms of O2.

Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen

649

TABLE 24.2 Computed EIEs for the Reactions Leading to Reduced Dioxygen Species Reaction 2

2e

O2 O O22 2 1e2

O2 O O†22 2e2 ; Hþ

O2 O HO2 2 16 18 2

DBO/O atom

1e2 ; Hþ

O2 O2 a

2e2 ; 2Hþ

O

4e2 ; 4Hþ

O

(K )calc

ZPE

EXC

MMI

1

1.04962

1.05147

0.99761

1.00063

0.5

1.03309

1.03386

0.99921

1.00004

1.05164 1.01696 Net: 1.03401

1.05401 1.01165

0.99725 0.99718

1.00050 1.00809

1.02474 0.99590 Net: 1.01032a

1.02270 0.98967

0.99930 0.99935

1.00269 1.00695

0.99763

1.00837

1.00009

0.99226

0.5

H O– O H18O – 16O2

O2 O HO†2 H16O – 18O† H18O – 16O†

18

0

H2 O2

0

1.00890

1.00290

2H2 O

0

0.96822a

0.97568 2

1e ; H

þ

All numbers have been recalculated, and a small discrepancy observed for O2 O HO†2 to to give 1.010 vs. 1.011 in 4e2 ; 4Hþ Ref. 11. In the case of O2 O 2H2 O; the original calculation in Ref. 11 failed to take into account the production of two H2O molecules; after correction the final 18(K)calc is reduced to 0.968.

force constants for the 16O and 18O containing molecules or, when force constants were lacking for the 18O compound, from the value for n16;16 : The results in Table 24.2 show the dominance of the ZPE term in determining the size of the calculated EIE and the regular trend of the final EIE after taking into account compensating protonation as electrons are transferred to O2. With the exception of complete cleavage of the O –O bond (to form two water molecules), a net change of bond order of zero predicts an EIE of ca. 1%, a net change of one half predicts a value of ca. 3%, and a change of bond order of one gives an upper limit to the EIE of 5%. The actual size of the isotope effects reflects the change in vibrational frequency that occurs as O2 is converted to its variously reduced species. The measured values in Table 24.1 can now be rationalized in terms of the values in Table 24.2 and the structures for the oxygenated species at the active site of each O2 carrier. Starting with oxyhemoglobin, it can be seen that the measured value lies below that predicted for HO†2 . This can be attributed, at least in part, to the presence of a hydrogen bond from the distal histidine to the iron bound superoxide ion. In the case of oxygenated hemerythrin, its EIE is very close to H2O2. From all studies of Hr, it can be inferred that the two electron reduced O2 undergoes interaction with an active site iron (as Fe3þ) and the bridging oxygen that links the two irons; hydrogen bonding of a terminal hydroperoxide with the bridging oxygen is expected to elevate the size of the experimental EIE in relation to the value predicted for H2O2, precisely as observed. The unusual structure for oxygenated Hc makes it difficult to rationalize its EIE from the computed values in Table 24.2. The one feature that is very clear is that the change in bond order is far more extensive than that seen for either Hb or Hr, indicating a loss of between zero and one half bond order according to the systematization of Table 24.2. A number of important comments need to be made regarding the experimental and computed limits for the EIEs. First, the experimental values provided by Tian and Klinman pertain to metal complexes, whereas the computed values refer to protonated reduced oxygen species. There has been much discussion whether complexation of dioxygen to a metal can be approximated by the addition of a hydrogen atom. Burger and Tian have calculated the EIE for a FeIII(2O – OH)

650

Isotope Effects in Chemistry and Biology

intermediate, obtaining a value that is very close to H2O2, in support of the view that the values in Table 24.2 offer a reasonable approximation of well-established metal –oxygen complexes.13 There is no assurance, however, that this will always be the case and additional studies of experimental EIEs, using model metal complexes with well-established structures, are much needed at the current time. A second issue, when comparing the established EIEs as limits for measured KIEs (see below), concerns the nature of the electron transfer process. When O2 or a preformed oxygen complex undergoes a simple electron uptake process, the kinetics of the reductive step is best treated in the context of Marcus theory.14 Classical Marcus theory does not explicitly consider the differences in ZPE vibrational levels among isotopically labeled reactants, which are the major origin of semiclassical isotope effects in transition state theory.15 A modification of Marcus theory is therefore needed to formulate and predict the nature of the KIE. This is discussed in greater detail in Section IV.A. If the electron transfer occurs concomitantly with bond formation (in a single step), theory exists to treat the origin of the KIE, although in most instances the necessary vibrational data are unavailable. For heavy nuclei such as carbon or oxygen, the KIE expression can be simply expressed in terms of reactant and transition-state vibrational frequencies, but this also leads to a modifier term that contains the ratio of reaction coordinate frequencies, n‡L =n‡H ; where L is the light isotope and H is the heavy one. While we expect the magnitude of n‡16;16 =n‡18;16 to be close to unity when forming a metal oxygen complex, the value for this ratio is not known. This potential complication in the interpretation of O-18 KIEs is discussed in the context of tyrosine hydroxylase and P-450 chemistry (Section IV.B and Section IV.E).

IV. APPLICATIONS A. GLUCOSE O XIDASE Glucose oxidase uses O2 to activate a flavin adenine dinucleotide cofactor noncovalently bound ˚ below the protein surface. The oxidized flavin (FAD) effects the dehydrogenation of , 15 A glucose to gluconolactone by accepting the equivalent of a hydride ion and forming FADH2. The additional proton equivalent is presumably removed from the sugar by an active-site base. During catalytic turnover, the reaction of O2 with FADH2 produces one equivalent of H2O2 and regenerates FAD. Analysis of the enzyme structure,16 thermodynamics,17 and kinetics18 has indicated that the mechanism of O2 activation involves an irreversible electron transfer as shown in Scheme 24.1. Support for the outer-sphere electron transfer mechanism of Scheme 24.1 was originally based on thermodynamic arguments and the measured isotope effects. The absence of solvent KIEs on the reactions studied at pH extremes, where limiting rate constants are observed, ruled out contribution of proton transfer to the rate-limiting steps, while the large oxygen-18 KIEs at high and low pH † indicated rate-limiting formation of O2 2 . Two forms of glucose oxidase reduce O2 with kcat =KM (O2) values that range from 5.7 ( ^ 1.8) £ 102 M21 s21 at pH 12.5 to 1.5 ( ^ 0.3) £ 106 M21 s21 at pH 5. Site-directed

R N N H

N − O NH O

HisH+ + O2

R N

N

N • H O

HisH+ O + O2• NH

R N

N

N

His O + H 2 O2 NH

O

SCHEME 24.1 The mechanism whereby the reduced flavin cofactor in glucose oxidase is oxidized by O2.

Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen

651

mutagenesis studies have implicated an active-site His516 as a Lewis acid catalyst, which accounts ˚ away from the for the three order of magnitude difference in rate constants. His516 is located 4 A anionic N(1) of the FADH and appears to have a role in stabilizing the negative charge and especially in lowering the kinetic barrier to O2 reduction. The difference in rates observed with high and low pH enzyme forms has been ascribed to the influence of pH-dependent electrostatics. The electron transfer reactions in glucose oxidase occur with similar driving-forces ðDG80 ¼ 0:92 kcal mol21 at pH 12.5 and DG80 ¼ 21:3 kcal mol21 at pH 5.0) requiring the estimated outer sphere reorganization energies to differ by upwards of 14 kcal mol21 to account for the observed rates.17 The temperature dependence of kcat =KM ðO2 Þ; which indicates ca. 6 kcal mol21 difference in activation energies, supports this observation. In addition, the activation energy of the His516Ala mutant at low pH resembles the activation energy of the high pH enzyme form. Within the context of electron transfer theory these observations are explained by lowering of the outer-sphere reorganization energy, the energy needed to distort the surroundings such that the reactant state adopts the equilibrium nuclear configuration of the product state. Despite the large variation in activation barriers and consequently rates, the oxygen KIEs, 18 ½kcat =KM ðO2 Þ ¼ 1:028ð^0:003Þ at high pH and 18 ½kcat =KM ðO2 Þ ¼ 1:0279ð^0:0006Þ at low pH, are indistinguishable within the error limits. These data suggest that changes in reorganization energy are fairly independent of oxygen isotope, consistent with a dominant contribution from the outer-sphere component. Reconstitution of apo-glucose oxidase with chemically modified cofactors has further confirmed the electron transfer mechanism and allowed the first investigation of oxygen-18 KIEs on O2 activation as a function of reaction driving-force.19 Shown in Figure 24.2 is a comparison of rates and kinetic oxygen KIEs for the reaction between O2 and reduced glucose oxidase at low pH. The enzyme-bound flavins with electron withdrawing substituents react with O2 at significantly lower rates than the native flavin. Yet the oxygen-18 KIEs are barely distinguishable; the error-weighted data indicate a slight trend of decreasing isotope effect as the reactions become thermodynamically less favorable. Like the comparisons at high and low pH, these data may be explained by a minor contribution to the reaction barrier from the O –O mode and a dominant contribution from the reorganization of the surroundings.

6.5

1.03 1

2

1.02

cat

3

5.5

/KM

1.025

18 k

log kcat /KM(O2)

6

5

1.015

4

Y

R N

X

N H

O

N

NH O

1: X= CH3, Y= CH3 2: X= CH3, Y= CI 3: X= H, Y= CI 4: X= CI, Y= CI

4.5

2

1 0 ∆G°′ (kcal/mol)

−1

1.01 −2

FIGURE 24.2 Rate constants (W) and oxygen 18 kinetic isotope effects (A) as a function of reaction driving force from studies of chemically modified flavins bound to glucose oxidase. DG o0 refers to the driving force for single electron transfer from flavin to O2.

652

Isotope Effects in Chemistry and Biology

The results obtained with glucose oxidase underscore the need for a new paradigm for understanding oxygen-18 KIEs in electron transfer reactions. The current framework is based on semiclassical transition state theory where the KIE originates from the difference in ZPE and the maximum is given by the EIE. However, this view is inappropriate when the reaction coordinate does not derive from stretching of the O – O bond. This may be the case for outer-sphere electron transfer to O2 due to the high frequency of the O –O (1556.3 cm21) in relation to the energy available at room temperature (200 cm21). When "v .. 4kB T; vibrationally excited states are not populated to an extent that the nuclear rearrangement can readily occur by thermal activation. Standard explanations for heavy atom isotope effects on electron transfer reactions have invoked nuclear tunneling.20 The most clear-cut cases involve reactants and products that are characterized by vibrational frequencies significantly greater than the available thermal energy and large differences in bond lengths.21 KIEs during electron transfer to O2 can be explained by treatment of the O – O either semiclassically or quantum mechanically as depicted in Figure 24.3.19 In the semiclassical picture the difference in ZPE can be taken into consideration by using the force constants for O2 and O2z 2 for the reactant and product states, respectively, and offsetting the parabolas by the isotopic difference computed from Bigeleisen’s formula (cf., Equation 24.3). When DG8 ¼ 0 for reduction of 16,16O2, the calculated isotope effect is 1.0195 and increases as the reaction becomes less thermodynamically favorable. The calculated isotope effect is half that observed experimentally, exposing the inadequacy of the model. In quantum mechanical electron transfer theory, the high frequency O –O stretch does not contribute to the classical reaction coordinate. Rather the isotope effect results from the density weighted Franck –Condon factors which reflect a combination of (i) the differences in the overlap integral for the O – O vibrational wavefunctions and (ii) the density of states having the appropriate configuration for oxygen atom tunneling. In this model, the KIE is very sensitive to internuclear distance and can actually be larger than the EIE. Inverse KIEs may also be explained within this model by the increased density of states overriding the poorer vibrational overlap factor associated with the heavier isotopologue. It is interesting to speculate whether O2 activation can be understood in terms of a continuum with reactivity ranging from outer-sphere to inner-sphere electron transfer reactions. Glucose oxidase provides an example of the former case, where the electron moves between reactants that are coupled only through weak electronic interactions. When open-shell species such as transition metals and free radicals are involved, electron transfer may be more likely to occur with concomitant bond formation to oxygen. Ideally, electron transfer could be formulated for innersphere electron transfer using the same basic formalism as for glucose oxidase, but within semi-classical

quantum mechanical

P R

P R

16 , 16O ,O − 2 2 18 , 16O ,O − 2 2

FIGURE 24.3 Semiclassical and quantum mechanical descriptions of the origin of the kinetic isotope effect on electron transfer to O2.

Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen

653

the strong-coupling limit.22 However, an additional feature of inner-sphere electron transfer is the conversion of a translational degree of freedom within the reactant to a vibrational mode; in semiclassical treatments of KIEs, this leads to an isotope effect expression that contains the isotope effect on the reaction coordinate frequency term, n‡16;16 =n‡18;16 multiplied by a term that describes changes in vibrational frequency at the reacting bond(s). The extent to which the isotope effect on the reaction coordinate frequency contributes to the magnitude of the measured KIE for inner sphere reactions of redox metals with O2 is unclear. One of our goals is to develop a unified model that will incorporate a reaction coordinate relevant to electron transfer reactions and explain values of KIEs that can, in principle, vary from normal to inverse.

B. TYROSINE H YDROXYLASE Tyrosine hydroxylase catalyzes the oxidation of tyrosine to dihydroxyphenylalanine (DOPA) in the ˚ below first step of neurotransmitter biosynthesis. The x-ray structure reveals an iron active site 10 A the surface. The square pyramidal iron has two histidines and a glutamate facially coordinated, and two water molecules occupying the remaining sites.23 The two imidazoles, one carboxylate motif is a recurring theme in nonheme iron centers that react directly with O2.24 In tyrosine hydroxylase as well as other amino acid hydoxylases, a pterin cofactor is required ˚ from the iron center becomes for activity. During catalysis a tetrahydrobiopterin bound , 6 A hydroxylated at the C(4a) position, presumably via a C(4a) peroxide intermediate. Coupling pterin oxidation and reduction of O2 likely facilitates the formation of a high valent iron oxidant. In the absence of the pterin cofactor no O2 uptake occurs, whereas in the absence of the protein the pterin reacts with O2 via its deprotonated form.25 Kinetic studies have led to a mechanism involving pre-equilibrium binding of pterin followed by O2 and then tyrosine prior to the first irreversible step in O2 activation. Oxygen-18 isotope effect measurements have indicated 18 ½kcat =KM ðO2 Þ ¼ 1:0175 ^ 0:0019: The isotope effect is relatively unchanged when modified pterins and substrate analogues that do not undergo hydroxylation are employed. The mechanisms of Scheme 24.2 and Scheme 24.3 have been proposed to account for

H

H H N

H

N H H

N

NH2 NH

H

H H N

H

N H H

O2

O

FeIII

FeII

N

NH2 NH Tyr

O

(O •−) 2

H

H H N

H

N H H

N

NH2 NH

O FeIII (O2•−) Tyr

H

H H N

H

N H H

N

NH O

FeII (O2−• ) Tyr

• + NH2

H H

H H N N H H O O

N

+

NH2 NH

O

H

H H N

H

N H H O

FeII

FeIV

Tyr

Tyr

N

NH NH

O OH

SCHEME 24.2 Mechanism of metal-mediated O2 activation in tyrosine hydroxylase.

654

Isotope Effects in Chemistry and Biology

H

H H N

H

N H H

N

NH2 NH

H

H H N

H

N H H

O2

O

FeII

II

Fe

H

H H N

H

N H H FeII Tyr

N

NH2 N

O

•

• − O2

H H

H H N

N

NH Tyr,−H+

O

H

H H N

H

N H H

N

NH2 N H+

N

O

NH2 N −

FeII

O2

N H H O O O FeII Tyr

NH2

Tyr

O2

H

H H N

N

H

N H H O

NH2 N

O OH

FeIV Tyr

SCHEME 24.3 Alternative mechanism of outer-sphere electron transfer from deprotonated pterin to O2 in tyrosine hydroxylase.

the kinetic order and the oxygen-18 isotope effects. In both proposals, rapid and reversible formation of an “O2 complex” occurs prior to the rate-determining change in oxygen bond order.26 Although there is little evidence to rule out a rate-limiting redox step downstream of the first electron transfer from pterin to O2, this would be difficult to reconcile with the absence of oxygenated intermediates at levels detectable by Mo¨ssbuaer or Raman spectroscopy. One ambiguity in the chemical mechanism concerns whether binding of O2 to the iron occurs prior to the transfer of an electron equivalent from the pterin. In Scheme 24.2, the pterin reduces † Fe(III)(O2 2 ) in the first irreversible step, which leads to a peroxide intermediate that is the origin of the iron (IV) intermediate thought to be responsible for tyrosine hydroxylation. As an alternative, the mechanism in Scheme 24.3 has been proposed where O2 partitions into a nonmetal site on the † protein and directly accepts an electron equivalent forming O2 2 as an intermediate. That the amino acid substrate must be bound to the protein prior to the activation of O2 may be explained by a perturbation of the cofactor reduction potentials. The protonation state of the bound pterin during catalysis is an additional ambiguity. From the reduction potentials of the neutral pterin analogues in aqueous solution, DG8 , 18 kcal mol21 is estimated for the electron transfer reaction with O2, Scheme 24.3.25 The thermodynamic barrier actually exceeds the DG‡ , 11 kcal mol 21 derived from the observed kcat =KM ðO2 Þ ¼ 6:0 £ 104 M21 s21.27 This analysis favors a reaction mechanism involving the singly deprotonated form of the pterin, which is possibly stabilized in the enzyme active site. By analogy with flavoprotein oxidases, the anionic pterin, because of its negative reduction potential, could react with O2 via a more thermodynamically favorable electron transfer. Scheme 24.2 and Scheme 24.3 may be viewed as a mechanistic continuum from innersphere to outer-sphere reactions with the magnitude of oxygen-18 KIEs revealing the extent to which bond formation to the iron center accompanies the electron transfer. Taking the isotope effect {18 ½kcat =KM ðO2 Þ , 1:028} exhibited by various forms of glucose oxidase as a benchmark for outer-sphere electron transfer, the experimental oxygen-18 KIE for tyrosine hydroxylase reacting by the mechanism shown in Scheme 24.3 would be expected to be larger than the observed value of 1.0175. In the event of concomitant bond formation to iron upon O2 reduction (Scheme 24.2), a much smaller 18 ½kcat =KM ðO2 Þ to ca. 1.01 would be expected because reduction

Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen

655

of the O – O bond would be partially offset by the formation of a Fe – O bond, as introduced above (Section IV.A); within the context of a semiclassical correction to transition state theory this value would be inflated somewhat by the value of n‡16;16 =n‡18;16 : In neither case, however, does the measured O-18 effect for tyrosine hydroxylase compare to the anticipated limits. † Pre-equilibrium formation of the superoxo FeIII(O2 2 ) intermediate would be expected to exhibit an equilibrium oxygen-18 isotope effect of ca. 1.004, comparable to that seen for the oxygen binding proteins, myoglobin, and hemoglobin. A further one-electron reduction to the peroxo FeIII(O22 2 ) intermediate in the rate-determining step would cause the observed isotope effect to be the product of an EIE and a KIE on the microscopic rate constant for the irreversible step. The latter process has been compared to the reduction of O2 by two electrons and a proton to HO2 2 and the equilibrium effect of ca. 1.034 calculated. However, as noted in the original paper by Francisco et al., a ferric-peroxo species is incapable of direct coupling with the pterin semiquinone, requiring considerable electron delocalization onto the iron center to approximate a ferrous-superoxo species. Such electron delocalization would be expected to reduce 18 ½kcat =KM ðO2 Þ below 1.034, possibly placing it in the range of the measured value. To summarize, the oxygen-18 isotope effect observed for tyrosine hydroxylase is reduced relative to the equilibrium limits calculated for mechanisms involving rate-determining outer† sphere electron transfer to either O2 or FeIII(O2 2 ). This property will be encountered again in additional enzyme systems described below. The discrimination between inner-sphere and outersphere mechanisms is currently an obstacle to understanding O2 reactions with metalloenzymes and free radicals. A better understanding of 18 ½kcat =KM ðO2 Þ in tyrosine hydroxylase awaits (i) the determination of the protonation state and consequently the reduction potential of the pterin, (ii) the † estimation of the reduction potential of FeIII(O2 2 ) using model systems, and (iii) the ability to detect and characterize oxygenated intermediates formed during catalytic turnover.

C. SOYBEAN L IPOXYGENASE Lipoxygenases catalyze the regio- and stereospecific oxidation of 1, 4 (Z, Z) pentadiene-containing fatty acids to hydroperoxides.28a The crystal structure of the soybean enzyme reveals one face of an octahedral iron occupied by three histidine residues and the remaining sites taken up by the carboxyl of an isoleucine at the C terminus, the amide of an asparagine, and a water molecule.28b Mammalian lipoxygenases exhibit similar structures with a single iron center tightly bound in a buried active site. Recently, a manganese enzyme from fungi has been characterized as having 23 to 28% sequence identity to the iron lipoxygenases and similar active site geometry.28c The manganese (III) and iron (III) enzymes are believed to exhibit similar electrochemical reduction potentials. Studies of O2 binding and substrate oxidation have corroborated the mechanism, shown in Scheme 24.4, where hydrogen atom abstraction from bound linoleate occurs prior to O2 entering the catalytic cycle.29 Consistent with this proposal, hydrocarbyl and peroxyl radicals have been detected in EPR studies under anaerobic and aerobic conditions, respectively.30 Soybean lipoxygenase forms the 13-S product with high specificity presumably because of conformational constraints imposed by the protein on the bound substrate.31 Lipoxygenases are unique in that the resting-enzyme is unreactive towards O2 and requires the product hydroperoxide for activation. This contrasts with the behavior of most nonheme iron enzymes where the metal center initially binds O2 and then reacts in the presence of a coreductant to form a ferryl species competent for substrate oxidation.32 The inertness of the ferrous ion in lipoxygenase to O2 can be understood in terms of its large positive one-electron reduction potential of 1.1 V vs. NHE at pH 0.33 Because of its similar reduction potential, the peroxyl radical is thermodynamically capable of oxidizing FeII to FeIII by either electron or hydrogen atom transfer.

656

Isotope Effects in Chemistry and Biology

−

OOC(CH2)7

−

OOC(CH2)7

C5H11 FeIII(OH−)

+ O2

•

C5H11

FeII(H2O)

OO• −

OOC(CH2)7

C5H11 FeII(H2O)

OOH −

OOC(CH2)7

C5H11 FeIII(OH−)

SCHEME 24.4 Proposed mechanism for the oxidation of linoleate by soybean lipoxygenase.

Kinetic studies indicate that soybean lipoxygenase reacts with kcat =KM ðO2 Þ ¼ 2:1 £ 107 M21 s21. The rate constant corresponds to DG‡ , 5 kcal mol21.34 This small barrier precludes the occurrence of significantly endothermic steps subsequent to O2 entering the catalytic cycle and leading up to the first irreversible step. In addition, a moderate KIE on the second-order rate constant 18 ½kcat =KM ðO2 Þ ¼ 1.0116 ^ 0.0016 has been observed indicating a contribution from bond order changes at oxygen in the rate-determining step.35 Together the results of a number of experiments have suggested that the rate-determining step reflects the combination of the pentadienyl radical with O2. On the basis of viscosity studies, this reaction does not appear to be controlled by the diffusion of O2 to the protein. We note that viscosogen studies are sometimes ambiguous because of the deviations from Stokes –Einstein behavior exhibited by small solutes like O2.36 In lipoxygenase, an anomalous viscosogen effect is observed, with kcat =KM ðO2 Þ increasing slightly with increasing solvent viscosity. To explain this result, some contribution from a prebinding step, where O2 partitions into discreet sites in the protein, has been invoked. It is interesting to consider why O2 trapping of linoleate radical in solution37 is a reversible process with radical combination occurring close to the diffusion limit, yet the enzymatic O2 trapping reaction is apparently slower and kinetically irreversible under turnover conditions. Rate-determining electron and hydrogen atom transfer steps for the oxidation FeII(OH2) by the linoleate peroxyl radical have also been considered. The absence of a kinetic solvent isotope effect appears to eliminate rate-determining proton transfer or hydrogen atom transfer from a solvent accessible site. It is more difficult to rule out a rate-determining electron transfer to the peroxyl radical. Arguments have been presented based on the expectation of an oxygen-18 KIE for the reaction that is close to the 18 K ¼ 1:034 calculated for the reduction of O2 to HO2 2 (Table 24.2). It is always possible that a KIE significantly less than expected from the equilibrium value may reflect partial rate limitation by nonisotopic steps, such as a conformational change. The mechanism of rate-determining trapping of O2 by linoleate radical was tested in early studies through the examination of magnetic isotope effects. Comparative oxygen-17 and oxygen-18 KIEs were measured using natural abundance O2 and determining the enrichment of 18O and 17O isotopes. Because 17O possesses nuclear spin it was expected that 17O would react faster than either 16 O or 18O in simple combination reactions with free radicals. Such effects, which derive from the magnetic-field dependence of intersystem crossing between quartet and doublet states, have been observed previously for reactions of O2.38 The lipoxygenase experiments revealed a linear correlation between oxygen-18 and oxygen-17 isotope effects indicating that the reaction exhibits only the mass-dependent isotope effect.35 The absence of a detectable magnetic isotope effect has been rationalized by the involvement of the active-site ferrous ion. The magnetic fields associated with the unpaired electrons on the high-spin iron center are believed to override the magnetic field of 17O. The importance of transition metals with unpaired spins in overcoming spin-barriers during O2 activation processes has been discussed

Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen

657

extensively by Siegbahn and coworkers.39 We note that, at present, the importance of such effects awaits experimental verification.

D. METHANE M ONOOXYGENASE Methane monooxygenases (MMOs) catalyze the oxidation of methane to methanol by exploiting the chemical driving-force associated with the reduction of O2 to H2O. The activation of O2 and oxidation of hydrocarbon substrate is known to occur at a carboxylate-bridged diiron (II) center. Two intermediates, a diiron (III) peroxo and a diiron (IV) oxo with one- or two-oxo bridges, have been spectroscopically characterized. Evidence exists for the formation of additional species during catalysis.40 Oxygen-18 KIEs have been measured using the soluble methane monooxygenase proteins from Methylococcus capsulatus (Bath).41 Experiments were conducted using all MMO proteins, the hydroxylase (MMOH), the reductase (MMOR) which utilizes NADH as a cofactor and the regulatory protein (MMOB). Acetonitrile (CH3CN) a liquid substrate was employed in place of CH4 to simplify the experimental manipulations. The experiment compared oxygen-18 KIEs during the oxidation of CH3CN to HOCH2CN and the unproductive reduction of MMOH by NADH in the absence of substrate. The minimal mechanism shown in Scheme 24.5 accommodates these two processes. The measured oxygen-18 KIEs indicate that similar bond order changes occur during the rate determining steps of O2 activation in the presence and absence of substrate. The 18 ½kcat =KM ðO2 Þ ¼ 1:0152ð^0:0007Þ determined during CH3CN oxidation is within the experimental error of the 18 ½kcat =KM ðO2 Þ ¼ 1:0167ð^0:0010Þ observed during NADH oxidation.41 The observations suggest a common intermediate for the two pathways. In comparison to the structurally related oxygen carrier hemerthryin, which exhibits an EIE of 18 K ¼ 1:0113; the intermediate formed in the rate-determining step of O2 activation by MMO appears to have a weaker O – O bond. See the discussion of O2 binding proteins presented in Section III. Comparison with the calculated EIE in Table 24.2 suggests that the rate-determining step may involve the transfer of either one electron to

FeII

O2

FeII

FeII

Kd ≥ 1.0 mM

O FeIV

O

FeIV

FeII FeIII(O2 −• )

FeII

O O

FeIII

O2

+ CH4, H+

H O FeIII

FeIII

CH3OH + NADH, 2H+

H O

FeIII

+ NADH

FeIII FeIII

+ NADH, H+

FeII

FeII

+ NAD+, H2O

+ NAD+, H2O

SCHEME 24.5 Proposed mechanism accounting for the oxygenase and oxidase activities exhibited by the hydroxylase component of soluble MMO.

658

Isotope Effects in Chemistry and Biology

O2 or electron transfer partially compensated by increased bonding to iron. These explanations have been discussed above for tyrosine hydroxylase, which exhibits a KIE quite close to that seen with MMO. To further characterize the potential oxygenated intermediates, O2 binding studies were undertaken using a Clark electrode which is capable of discerning binding constants of Kd , 1:0 mM. Using all protein components in the MMO system the Kd was determined to be $ 1.0 mM on the basis of no detectable O2 binding. These studies argue against pre-equilibrium formation of a complex between oxidized MMOH and O2. Unfortunately, because of the irreversible nature of O2 consumption, the binding experiments could not be performed with the reduced MMOH, the actual species proposed to bind O2 in the catalytic cycle. To reconcile the steady-state results with the earlier single-turnover studies that indicated oxidation rates independent of the concentration of O2,42 it was argued that O2 binding must be kinetically irreversible. The data may be explained by the formation of a superoxo complex in the rate-determining step, a possibility only in the presence of reduced MMOH. As discussed above, the formation of a superoxo intermediate may occur by either inner-sphere or outer-sphere electron transfer. Due to the mechanistic permutations that are possible, studies of model systems where a single redox process may be isolated hold promise for shedding light on the precise O2 activation mechanism.

E. CYTOCHROME P - 450 P-450 catalyzes the hydroxylation of a wide range of xenobiotic compounds via the production of a highly reactive O2 derived species. While the precise nature of this O2 species is still under debate, it is most often written as a high valent, ferryl intermediate.43 The early steps in O2 activation have been investigated using the well-characterized prokaryotic P-450cam.44 This is comprised of three protein components, viz., the heme iron containing hydroxylase (P-450cam), an iron – sulfur protein that functions as a reduced one-electron carrier (putidaredoxin, Pdxr), and a flavin-dependent putidaredoxin reductase (Pdr). In a detailed kinetic study, Purdy et al.45 have shown that the three proteins behave according to a shuttle mechanism, wherein Pdr catalyzes a one-electron reduction of Pdx to generate Pdxr, which then binds twice to the hydroxylase to deliver a single electron before and after O2 binding (Scheme 24.6). Analogous to hemoglobin (Hb) and myoglobin (Mb), the binding of O2 to the one-electron reduced form of P-450 is expected to produce the equivalent of a ferric –superoxide intermediate. Two important differences between P-450s and the O2 carrier proteins are first, the prerequisite for bound substrate prior to the interaction of O2 with P-450 and second, the replacement of a histidine by a cysteine side chain in the proximal, axial position of the heme iron center of P-450. This both confers further reactivity to the heme iron bound O2 in P-450 and destabilizes the initial O2 complex. The lifetime of oxygenated P-450 is relatively short, leading to oxidized enzyme and free superoxide ion. For this reason, it has been necessary to study the interaction of ferrous P-450 with O2 at low temperature. Measurement of binding of O2 to P-450cam indicates an EIE at 48C of 1.0048 (0.0003) (Figure 24.4) that is remarkably close to the values for hemoglobin and myoglobin at 258C (cf., Table 24.1). Examination of the EIE for myoglobiin at 20 and 48C indicates numbers that are within experimental error, such that a direct comparison can be made of P-450cam to Hb and Mb at the higher temperatures. Although the axial sulfur ligand is clearly essential to P-450 reactivity, it does not influence very greatly the bond order at the initially bound O2.46 The magnitude of the KIE resulting from further activation of O2 was examined under steadystate conditions in the presence of an excess of Pdxr. This provides 18 ½kcat =KM ðO2 Þ ¼ 1:0147ð0:0007Þ which is shown with the EIE in Figure 24.4, illustrating the significant difference between the two measurements. Although the most general kinetic scheme for the interaction of Pdxr and O2 with the ferrous form of P-450 [P(450)r] is a random one, comparison of kcat =KM values for Pdxr and O2 to their rates of binding to P(450)r strongly suggests a preferred ordered

Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen

k

2

(r) dx

f,P

Fe3+

ket,2

,2

x(

r),

k' [Pd on xr ,P ] d

−

(O2• )

(O22−)

of

Pdxox + products

Fe3+ • Pdxr

k'

x( r), of 2 f,P dx (r) ,2

Pdxox

2

Fe2+

] 2 [O O 2 n, k' o O2 ff, k' o

Fe3+ • Pdxr

koff,Pdx(r),1

−

O2•

f,O

kon,Pdx(r),1

Fe3+

ket,1

k of

[Pdxr]

Fe2+ • Pdxr

] 2 [O ,O 2 k on

(2)

Pdxr + PdRs/ox

k [Pd on ,P x ]r d

(1) Pdxox + PdRr/s

659

Fe3+ • Pdxox

SCHEME 24.6 Shuttle mechanism for P-450cam. PdRr/s and PdRs/ox indicate putidaredoxin reductase in the reduced (r), semiquinone (s), and oxidized state (ox), respectively. In (2), the P-450 has already bound its substrate. (From Purdy, M. M., Koo, L. S., Ortiz de Montellano, P. R., and Klinman, J. P., Biochemistry, 43, 271– 281, 2004. With permission.)

pathway with Pdxr binding first: k1

k3

k2

k4

½Cam-Pð450Þr þ Pdxr O Pdxr ½Cam-Pð450Þr þ O2 O k5

r

ox

Pdx ½Cam-Pð450Þ O2 ! ½Cam-Pð450Þ

ð24:4Þ

þ H2 O þ Hydroxycamphor þ Pdxox

Interpretation of the magnitude of O-18 KIE can be made in the context of the kinetic scheme of Equation 24.4, which leads to the expressions for kcat =KM ðO2 Þ and 18 ½kcat =KM ðO2 Þ in Equation 24.5 and Equation 24.6: kcat =KM ðO2 Þ ¼ k3 ðO2 Þk5 =ðk4 þ k5 Þ 18

½kcat =KM ðO2 Þ ¼ ð18 k518 KO2 þ 18 k3 CfÞ=ð1 þ CfÞ

1.016 O/18O Isotope Effect

ð24:6Þ

Kinetic Isotope Effect

Equilibrium Isotope Effect

16

ð24:5Þ

Avg

1.014

18(k /K (O )) = 1.0147 ± 0.0007 cat m 2

1.012 1.01 1.008 1.006 1.004

Avg 18K

1.002 1 (a)

90

O2

100

= 1.0048 ± 0.0003 110 120 [P-450] (mM)

0.3

130 (b)

0.4

0.5

0.6

0.7

Fraction of Conversion

FIGURE 24.4 Comparison of equilibrium isotope effect and kinetic isotope effect in P-450cam.46

660

Isotope Effects in Chemistry and Biology

where Cf is k5 =k4 and 18KO2 is the EIE for O2 binding to Pdxr [Cam-P(450)r]. In contrast to primary hydrogen isotope effects and more similar to that of secondary hydrogen isotope effects, the impact of isotopic labeling contributes to multiple terms in Equation 24.6. In addition to 18KO2, which has been measured, there are three unknowns in Equation 24.6, 18k5, 18 k3, and Cf. The parameter of the greatest interest in this study was 18k5, since this reflects the extent to which electron transfer from Pdxr to the active site of P-450cam is limited by the electron transfer itself, as opposed to a conformational change that “gates” the long range electron transfer. An estimate of Cf , 1 was possible from a variety of experimental probes that included a comparison of kcat =KM ðO2 Þ to k3 :46 This results in the magnitude of 18k5 being directly dependent on the magnitude of 18k3. Inserting values of 18 ½kcat =KM ðO2 Þ ¼ 1:015; 18 KO2 ¼ 1:005, and Cf ¼ 1 into Equation 24.6 yields the final expression, Equation 24.7: 18

k3 ¼ 2:030 2 1:005ð18 k5 Þ

ð24:7Þ

Since both KIEs are expected to be $ 1, the value of 18 k5 can be calculated to range from 1 to 1.025 when 18k3 varies from 1.025 to 1. Significantly, according to Equation 24.7, the value of 18k3 would have to be 1.025 to invoke an isotope effect of unity in the long-range electron transfer step. In the context of the O-18 KIEs available for tyrosine hydroxylase and methane monooxygenase (see above), this value appears outside the range of possibilities. Thus, analysis of P-450cam implicates a nongated electron transfer, where the actual chemical event of transferring an electron limits the rate.46 Unlike hydrogen tunneling, which can be probed with isotopes of hydrogen, the demonstration that long-range electron transfer is a rate-limiting process requires manipulation of reaction driving force and investigation of rates as a function of distance between electron donor and acceptor.47 The recent O-18 isotope effect studies of P-450cam offer a potentially much simpler and more direct method for probing rate-limiting electron transfer to O2.

F. DOPAMINE b-M ONOOXYGENASE AND P EPTIDYLGLYCINE a-H YDROXYLATING M ONOOXYGENASE Dopamine b-monooxygenase (DbM) and peptidylglycine a-hydroxylating monooxygenase (PHM) belong to a small family of copper proteins that catalyze the production of hormones and neurotransmitters. These proteins, while of different overall atomic mass (PHM is 35 kDa and DbM a tetramer of 75 kDa subunits), contain a region of 28% sequence identity that corresponds to the catalytic core responsible for their respective monooxygenase reaction. Within this core are the conserved ligands to the two coppers per subunit, as well as highly conserved cysteine and tyrosine side chains.48 Despite great differences in substrate specificity, with DbM converting catecholamines to their b-hydroxylated analogs and PHM catalyzing a hydroxylation at the a-carbon of C terminally extended peptides, the net reaction and two electron donors are identical for DbM and PHM: R – CH2 – R0 þ O2 þ ASC ! R – CHOH – R0 þ H2 O þ DHASC

ð24:8Þ

where ASC and DHASC are ascorbate and dehydroascorbate, respectively. Early EPR studies of DbM, showed that the two Cu2þ ions in resting enzyme exist in a nonspincoupled state. However, it was not until the solution of an x-ray structure for PHM that the spatial relationship of the two copper ions per subunit became apparent. As shown in Figure 24.5,49 the two ˚ with the intervening region between them occupied by copper centers lie at a distance of 10 to 12 A solvent. Additionally, there does not appear to be a structural motif joining the separate domains of the metal centers that will allow these metals to approach one another at a closer distance. Since the electrons stored in both copper sites are used during the catalytic turnover, the question of how electrons are transferred between the metal sites has received considerable debate.

Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen

661

Ac-DiI-YG

Y318 R240 Q170

N316 H242 M314

H108

CuM

H172 Y79

H244

CuH

H107

FIGURE 24.5 Representation of PHM-active site. (From Prigge, S. T., Kolhekar, A. S., Eipper, B. A., Mains, R. E., and Amzel, L. M., Nat. Struct. Biol., 6, 976– 983, 1999. With permission.)

Another unique aspect of the PHM structure is the location of the active site within a solventexposed pocket. In general, enzyme-catalyzed oxygen-dependent chemistry takes place within a pocket that is sequestered from solvent. The mechanism for DbM and PHM has been probed using a combination of deuterium and oxygen KIEs.50 The deuterium KIEs show that C – H cleavage from substrate is partially ratelimiting for the second-order reaction between reduced, substrate-bound enzyme and oxygen, represented by kcat =KM ðO2 Þ: By contrast, there is no significant measurable KIE for kcat ; indicating rate limitation by product release and, possibly, a long-range electron transfer between metal centers. The intrinsic primary deuterium isotope effect for PHM, calculated from experimental deuterium and tritium KIEs on kcat =KM ; has been found to be fairly independent of temperature, leading to a mechanism in which a hydrogen atom moves from its donor (substrate) to acceptor (activated oxygen) by quantum-mechanical tunneling. The size of the intrinsic deuterium KIE for PHM and DbM (Table 24.3), and the extent of its temperature dependence with PHM can be rationalized by a modified Marcus treatment where low-frequency protein motions are the origin of the reaction barrier.51 Oxygen-18 KIEs have been measured in an effort to understand the nature of the activated oxygen species that catalyzes hydrogen atom abstraction from substrate. In the case of DbM and PHM, it was possible to determine the magnitude of the O-18 KIE using unlabeled and deuterated substrate (Table 24.3). These studies show that the magnitude of the O-18 KIE increases upon substrate deuteration and that it is quite large under these conditions (the KIE approaches 3%). This important result indicates that changes in bond order at O2 are linked in a reversible manner to the C –H bond cleavage step.10,50 If dioxygen activation occurred in an irreversible manner prior to

662

Isotope Effects in Chemistry and Biology

TABLE 24.3 Intrinsic Deuterium KIEs and O-18 KIEs as a Function of a Substrate Deuteration with the Enzymes PHM and DbM Measurement

PHM

DbM

Intrinsic primary deuterium KIE Intrinsic secondary deuterium KIE O-18 KIE (protiated substrate) O-18 KIE (deuterated substrate)

10.7 (0.4) 1.21 (0.01) 1.0173 (0.0009) 1.0212 (0.0018)

10.9 (1.9) 1.19 (0.06) 1.0197 (0.0003) 1.0256 (0.0003)

the involvement of substrate, the magnitude of the O-18 KIE would have to be independent of substrate deuteration. In fact of all the O2-activating systems studied thus far, DbM is the only one to show a definitive “link of reversibility” between the activation of O2 and substrate. The magnitude of the O-18 KIE was further studied as a function of substrate reactivity using DbM, showing a value that decreases in a regular manner from the fastest substrate, dopamine (kC – H ¼ 680 s21), to p-CF3-phenethylamine (kC – H ¼ 2 s21). This trend was initially interpreted in the context of the Hammond postulate, which posits that the position of the transition state for H transfer will be more product-like for the less thermodynamically favorable reaction. Since hydrogen atom abstraction from a preformed CuII(2OOH) was expected to give increased O –O bond cleavage for the slower substrate and, hence, a larger O-18 KIE, an alternative mechanism was proposed. This utilized an active-site tyrosine, to reductively cleave a metal-hydroperoxide prior to C – H activation. This interpretation has now been shown to be incorrect for two reasons. First, mutation of the implicated homologous tyrosine in PHM to phenylalanine did little to change the rate of C –H activation.50 Second, and perhaps most importantly, inferences regarding transition state structure cannot be made from the Hammond postulate when the hydrogen transfer step is occurring via quantum-mechanical tunneling. The current mechanistic framework for DbM and PHM invokes activated oxygen at a single copper site, presumably as a cupric-superoxo-species52 (Scheme 24.7). The precise alignment between the donor (substrate C – H) and the acceptor (activated oxygen) is expected to vary significantly among substrates of varying reactivity. The actual distance at which tunneling occurs almost certainly changes as the structure of substrate varies (note that the intrinsic deuterium KIE for DbM increases from 11 for dopamine to 17 for p-CF3-phenethylamine, suggesting a longer tunneling distance for the latter substrate). Future goals must include a full picture of the hydrogen tunneling process that incorporates nuclear changes at the activated oxygen center and the resulting impact on the magnitude of the experimentally observed O-18 KIEs as a function of substrate structure.

G. COPPER A MINE O XIDASES One of the most extensively studied systems with regard to activation of O2 is the copper amine oxidases.53 These enzymes contain an active site organic cofactor (the protein-derived trihydroxyphenylalanine quinone, topa quinone or TPQ) in close proximity to a CuII center. The overall reaction can be treated as two irreversible half reactions, referred to as a ping pong reaction and analogous in form to glucose oxidase above: E-TPQox þ RCH2 NH2 þ H2 O ! E-TPQred þ RCHO

ð24:9Þ

E-TPQred þ O2 ! E-TPQox þ H2 O2

ð24:10Þ

Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen

NHis SMet

CuA(I)

11Å

NHis

CuB(I) NHis

Ar O2,Substrate

SMet

NHis NHis

NHis

663

NHis

R

H

CuA(I)

H

NHis

CuB(I)(O2)

NHis

NHis

Product H+,Asc NHis SMet

CuA(II)

SMet

NHis

Ar NHis

Ar

CuB(II)− O

H

CuA(I)

H

NHis

NHis NHis

R

NHis

CuB(II)–O2 NHis

H

NHis

NHis

R

kC–H

Ar SMet

NHis

NHis

R

H

CuB(II)−O

Ar

NHis

HO H 2O

H 2O H 2O

SMet

k et

R

O

H H

CuA(II) NHis

NHis

NHis

CuB(II)–OOH NHis

NHis

H

O H

H

CuA(I) NHis

NHis

H O H

SCHEME 24.7 Working chemical mechanism for DbM and PHM. (From Evans, J. P., Ahn, K., and Klinman, J. P., J. Biol. Chem., 278, 49691– 49698, 2003. With permission.)

The first half reaction, summarized in Equation 24.1, has been thoroughly characterized and shown to involve a series of covalent intermediates between the substrate and oxidized cofactor. This reaction leads to the transfer of an amino group from substrate to cofactor, releasing product aldehyde and transforming TPQ into a reduced aminoquinol. The mechanism whereby the aminoquinol is recycled back to TPQ has been a subject of controversy. It seemed reasonable to expect that the proximal CuII center would play a direct role in O2 half reaction, as originally proposed by Dooley and coworkers.54 According to the early mechanism, TPQred would first transfer an electron to copper, producing CuI and the semiquinone † form of TQP (TPQsq). In a subsequent step, O2 bound at the metal site to generate a CuII(O2 2 ) intermediate, with the oxidative cycle being completed by the further loss of a proton and electron from TPQsq to yield CuII(2O – OH) and TPQ. The first evidence that the above scenario may not be correct came from extensive studies of bovine serum amine oxidase that included measurements of O-18 KIEs.55 At the outset, the absence of a viscosogen effect on kcat =KM ðO2 Þ eliminated mechanisms that involve rate-limiting binding of O2 or release of H2O2. The magnitude of the O-18 KIE 1.0101 (^ 0.0009) at pH 8.5, with little or no change with pH, showed that a decrease in bond order was involved in the steps described by kcat =KM ðO2 Þ: Measurement of a solvent KIE that was indistinguishable from unity indicated that proton transfer to O2 was not part of the rate-limiting step(s). Comparison of the size of the O-18 effect to the values in Table 24.2 led to two mechanistic proposals, a kinetically irreversible electron transfer to O2 or a desorption of product H2O2 that was limited by a proteinconformational change rather than simple diffusion away from enzyme. These mechanisms make very different predictions with regard to the nature of the enzyme species that accumulate during

664

Isotope Effects in Chemistry and Biology

catalytic turnover, with rate-limiting loss of H2O2 leading to the accumulation of an oxidized form of enzyme. By contrast, the major species detected during catalytic turnover was the reduced aminoquinol, as expected for a single-rate-limiting electron transfer from TPQred to O2 (Scheme 24.8). Several aspects of the above mechanism were unexpected and unorthodox at the time of its proposal. The first is that copper does not change valence during the reaction and second, as a corollary, that O2 initially binds off the metal at a discrete protein site. Each of these aspects has been examined in greater detail. Substitution of the copper ion by cobaltous ion allowed a direct test of the lack of a need for redox cycling at the metal center during catalytic turnover. The unfavorable potential for reduction of CoII to CoI (e.g., 2 400 to 2 500 mV vs. SCE in methionine synthase)56 argues against the possibility of transfer of an electron from TPQred to metal as an intermediate in O2 reduction. Although the replacement by cobalt ion was found to increase the apparent KM for O2, the limiting rate constant at saturating O2 was unaltered for native- and cobalt-substituted enzyme at pH 7.57 A subsequent study showed that the origin of the increased KM for O2 was likely to lie with the elevation in pKa of the metal-bound water for cobalt substituted enzyme, generating a net increase in charge of plus one within the active site.57 Importantly, the O-18 KIE for the cobaltsubstituted enzyme was the same as for native enzyme, in strong support of a similar mechanism of direct electron transfer from TPQred to O2.58 The manner in which the copper amine oxidases are able to accommodate O2 at a nonmetal site has been investigated using site-specific mutagenesis. Alignment of the amino acid sequence of the bovine serum copper amine oxidase (BSAO) with the copper amine oxidase from Hansenula polymorpha (HPAO), together with the x-ray structure for HPAO, identified a plausible O2 binding site involving Y407, L425, and M634 (using the numbering system in HPAO); the latter two residues are altered to A490 and T695 in BSAO. To determine which of these residues plays a greater role in O2 chemistry, kcat =KM ðO2 Þ was determined for HPAO using the altered enzyme forms, L425A and M634T. Whereas L425A had practically no impact on kcat =KM ðO2 Þ; the M634T mutant created a catalyst that was very similar to BSAO. This focused attention on M634 as a key determinant of O2 reactivity and led to the production of a series of mutants at position 634. Quite unexpectedly, structure reactivity correlations indicated that the size of the 634 side chain, rather than its hydrophobicity, was a major determinant of O2 reactivity.59 A second unexpected finding was the failure to observe any competition between O2 and N2 at this site in the course of O2 reduction to H2O2.59 E

E

HO

O2 NH2

Cu2+

1

2

HO

Cu2+

_•

O O

4

Cu2+

OH

++• .

_• Cu2+

NH2

E H2O

O

H2O2

_

Cu2+

O

NH2

OH

E

HO ++• .

O2

OH

E

3

HO

NH2

O2

OH

E

+ NH2

5

NH4+

O

_

Cu2+

O

SCHEME 24.8 Mechanism for the oxidative half reaction of copper amine oxidase.

O

Oxygen-18 Isotope Effects as a Probe of Enzymatic Activation of Molecular Oxygen

665

In the context of the above findings, it is valuable to compare the O-18 KIE measured with the copper amine oxidases (1.010) to that for glucose oxidase (1.028). For both reactions, the O2 is believed to be reduced via an outer-sphere single-electron transfer to produce superoxide anion and the one-electron oxidized cofactor (flavin semiquinone in glucose oxidase and TPQ semiquinone in copper amine oxidase). Initially, the difference in observed O-18 isotope effect was suggested to arise from possible differences in driving force for O2 reduction between the two enzyme systems. However, in recent studies of glucose oxidase (see Figure 24.2, Section IV.A), the magnitude of 18 ½kcat =KM ðO2 Þ is found to be almost independent of reaction driving force, implicating the importance of outer-sphere reorganization in determining the rate of reaction. If a similar feature pertains to the copper amine oxidases, then we need to look elsewhere for the reason for the reduced O-18 KIE. It is possible that an electrostatic interaction develops between the active-site copper and superoxide anion in the course of O2 reduction, since ultimately reduced O2 is expected to bind to copper (cf., Scheme 24.8). It has been argued that electrostatic factors play a major role in O2 reactivity in glucose oxidase and copper amine oxidase via the proximity of a protonated histidine and CuII center, respectively.18,60 A major difference between these two systems, however, is the degree of potential interaction between the counter ions to superoxide; only the copper amine oxidase is capable of bond formation in the course of negative-charge development on the reduced O2 intermediate.

V. OVERVIEW AND PERSPECTIVES FOR THE FUTURE The studies indicate the enormous progress that has been made in our understanding of O2 reactivity in enzymes during the last 5 to 10 years. In the larger part, this is due to the development of methodology for the precise determination of small O-18 isotope effects that can be readily distinguished from one another (cf., Table 24.4). Several important features have begun to emerge from the aggregate studies that include the recurring themes of: (1) Rate limitation by the initial chemical step during O2 reduction, which avoids the accumulation of reactive oxygen species (2) An important role for electrostatics in reducing the protein reorganization that generally must accompany charge transfer to O2 (3) The observation of discrete pockets for O2 binding within proteins, as well as channels that allow the entry of O2 to these pockets

TABLE 24.4 Summary of O-18 KIEs in Enzymatic Systems that Use O2 as Substrate Enzymatic Systems

Cosubstrate

Rubisco Glucose oxidase Tyrosine hydroxylase Lipoxygenase Methane monooxygenase P-450cam Copper amine oxidase

Ribulose bisphosphate Deoxyglucose Tyrosine Linoleic acid Acetonitrile Camphor Methylamine

18

[kcat/KM(O2)] 1.021 1.028 1.0175 1.0116 1.0152 1.0147 1.010

Ref. 7 18,19 26 29 41 46 55,58

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Isotope Effects in Chemistry and Biology

For the future, it will be important to develop theoretical models that relate the magnitude of observed O-18 isotope effects to specific mechanisms, in particular those that range from outersphere to inner-sphere electron transfer processes. Currently the former can be treated by quantummechanical electron transfer theory whereas the latter must take into account the formation of new bonds in the course of the charge transfer process. Even within the seemingly more straightforward electron transfer theories, the origin of the isotope effect may range from isotopic ZPE differences that impact the driving force ðDG8Þ and reorganization energy ðlÞ to the incorporation of nuclear tunneling contributions. Another promising area is the use of experimental model systems and oxygen isotope effects to probe the structures and reactivity patterns of activated oxygen intermediates. Many synthetic compounds are known which serve as structural models for metalloenzyme active sites and have been shown to reversibly bind dioxygen. Most importantly vibrational spectroscopy data are available allowing measurement and calculation of O2 binding isotope effects on a wide range of compounds. Clearly, a synergy between theoretical and experimental studies is needed to promote these recent and exciting advances regarding a fundamental and widespread process in biology.

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30 Nelson, M. J., Cowling, R. A., and Seitz, S. P., Structural characterization of alkyl and peroxyl radicals in solutions of purple lipoxygenase, Biochemistry, 33, 4966– 4973, 1994. 31 Knapp, M. J., Seebeck, F. P., and Klinman, J. P., Steric control of oxygenation regio-chemistry in soybean lipoxygenase-1, J. Am. Chem. Soc., 123, 2931– 2932, 2001. 32 Rohde, J.-U., In, J.-H., Lim, M. H., Brennessel, W. W., Bukowski, M. R., Stubna, A., Muenck, E., Nam, W., and Que, L. Jr., Crystallographic and spectroscopic characterization of a nonheme Fe(IV) ¼ 0 complex, Science, 299, 1037– 1039, 2003. 33 Hatcher, E., Soudackov, A. V., and Hammes-Schiffer, S., Proton-coupled electron transfer in soybean lipoxygenase, J. Am. Chem. Soc., 126, 5763– 5775, 2004. 34 The rate is derived from extrapolation to high concentrations of peroxide where dissociation of the substrate radical does not cause enzyme deactivation: see Knapp, M. J. and Klinman, J. P., Biochemistry, 42, 11466– 11475, 2003. 35 Glickman, M. H., Cliff, S., Thiemans, M., and Klinman, J. P., Comparative study of 17O and 18O isotope effects as a probe for dioxygen activation: application to the soybean lipoxygenase reaction, J. Am. Chem. Soc., 119, 11357– 11361, 1997. 36 Dunford, H. B. and Hasinoff, B. B., On the rates of enzymatic, protein, and model compound reactions: the importance of diffusion control, J. Inorg. Biochem., 28, 263– 269, 1986. 37 Porter, N. A. and Wujek, J. S., Allylic hydroperoxide rearrangement: beta-scission of concerted pathway? J. Org. Chem., 52, 5085– 5089, 1987. 38 Grissom, C. B., Magnetic field effects in biology: a survey of possible mechanisms with emphasis on radical-pair recombination, Chem. Rev., 95, 3 – 24, 1995. 39 Prabhakar, R., Siegbahn, P. E. M., and Minaev, B. F., A theoretical study of the dioxygen activation by glucose oxidase and copper amine oxidase, Biochim. Biophys. Acta, 1647, 173– 178, 2003. 40 Dunietz, B. D., Beachy, M. D., Cao, Y., Whittington, D. A., Lippard, S. J., and Fiesner, R. A., Large scale ab initio quantum chemical calculation of the intermediates in the soluble methane monooxygenase catalytic cycle, J. Am. Chem. Soc., 122, 2828– 2839, 2000. 41 Stahl, S. S., Francisco, W. A., Merkx, M., Klinman, J. P., and Lippard, S. J., Oxygen kinetic isotope effects in soluble methane monooxygenase, J. Biol. Chem., 276, 4549– 4553, 2001. 42 Liu, K. E., Wang, D., Huynh, B. H., Edmondson, D. E., Salifoglou, A., and Lippard, S. J., Spectroscopic detection of intermediates in the reaction of dioxygen with the reduced methane monooxygenase/hydroxylase from Methylococcus capsulatus (Bath), J. Am. Chem. Soc., 116, 7465– 7466, 1994. 43 Vaz, A. D., Pernecky, S. J., Raner, G. M., and Coon, M. J., Peroxo-iron and oxenoid-iron species as alternative oxygenating agents in cytochrome P450 catalyzed reactions: switching by threonine-302 to alanine mutagenesis of cyctochrome P450 2BA, Proc. Natl. Acad. Sci. U.S.A., 93, 4644– 4648, 1996. 44 (a) Davydov, R., MacDonald, I. D. G., Makris, T. M., Sligar, S. G., and Hoffman, B. M., EPR and ENDOR of catalysis intermediates in cryoreduced native and mutant oxy-cytochrome P450cam: mutation-induced changes in the proton delivery system, J. Am. Chem. Soc., 121, 10654 –10655, 1999; (b) Roitberg, A. E., Holden, M. J., Mayhew, M. P., Kurnikov, I. V., Beratan, D. N., and Vilker, V. L., Binding and electron transfer between putidaroxin and cytochrome P450cam: theory and experiments, J. Am. Chem. Soc., 120, 8927– 8932, 1998. 45 Purdy, M. M., Koo, L. S., Ortiz de Montellano, P. R., and Klinman, J. P., Steady state kinetic investigation of cytochrome P450cam: interaction with redox partners and reaction with molecular oxygen, Biochemistry, 43, 271– 281, 2004. 46 Purdy, M. M., Mechanism of Dioxygen Activation and Electron Transfer in Cytochrome P450cam Studied by O-16/O-18 Isotope Effects and Steady State Kinetics, Ph.D. dissertation, University of California, Berkeley, 2003. 47 Gray, H. B. and Winkler, J. R., Electron transfer in proteins, Ann. Rev. Biochem., 65, 537– 561, 1996. 48 (a) Stewart, L. C. and Klinman, J. P., Dopamine b-hydroxylase of chromaffin granules: structure and function, Ann. Rev. Biochem., 57, 551– 592, 1988; (b) Eipper, B. A., Stoffers, D. A., and Mains, R. E., The biosynthesis of neuropeptides: peptide alpha-amidation, Ann. Rev. Neurosci., 15, 57 – 85, 1992. 49 Prigge, S. T., Kolhekar, A. S., Eipper, B. A., Mains, R. E., and Amzel, L. M., Substrate-mediated electron transfer in peptidylglycine-alpha-hydroxylating monooxygenase, Nat. Struct. Biol., 6, 976– 983, 1999.

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50 Francisco, W. A., Blackburn, N. J., and Klinman, J. P., Oxygen and hydrogen isotope effects in an active site tyrosine to phenylalanine mutant of peptidylglycine a-hydroxylating monooxygenase: mechanistic implications, Biochemistry, 42, 1813– 1819, 2003. 51 Francisco, W. A., Knapp, M. J., Blackburn, N. J., and Klinman, J. P., Hydrogen tunneling in peptidylglycine a-hydroxylating monooxygenase, J. Am. Chem. Soc., 124, 8194– 8195, 2002. 52 Evans, J. P., Ahn, K., and Klinman, J. P., Evidence that dioxygen and substrate activation are tightly coupled in dopamine b-monooxygenase: implications for the reactive oxygen species, J. Biol. Chem., 278, 49691– 49698, 2003. 53 (a) Mure, M., Mills, S. A., and Klinman, J. P., Catalytic mechanism of the topa quinone containing copper amine oxidases, Biochemistry, 41, 9269– 9278, 2002; (b) Dawkes, H. C. and Phillips, S. E. V., Copper amine oxidase: cunning cofactor and controversial copper, Curr. Opin. Struct. Biol., 11, 666– 673, 2001. 54 Dooley, D. M., McGuirl, M. A., Brown, D. E., Turowski, P. N., McIntire, W. S., and Knowles, P. F., A copper (I)-semiquinone state in substrate-reduced amine oxidases, Nature, 349, 262– 264, 1991. 55 Su, Q. and Klinman, P., Probing the mechanism of proton coupled electron transfer to dioxygen: the oxidative half-reaction of bovine serum amine oxidase, Biochemistry, 37, 12513– 12525, 1998. 56 Banerjee, R. V., Harder, S. R., Ragsdale, S. W., and Matthews, R. G., Mechanism of reductive activation of cobalamin-dependent methionine synthase: an electron paramagnetic resonance spectroelectrochemical study, Biochemistry, 29, 1129– 1135, 1990. 57 Mills, S. A. and Klinman, J. P., Evidence against reduction of Cu2þ to Cuþ during dioxygen activation in a copper amine oxidase from yeast, J. Am. Chem. Soc., 122, 9897– 9904, 2000. 58 Mills, S. A., Goto, Y., Su, Q., Plastino, J., and Klinman, J. P., Mechanistic comparison of the cobaltsubstituted and wild-type copper amine oxidase from Hansenula polymorpha, Biochemistry, 41, 10577– 10584, 2002. 59 Goto, Y. and Klinman, J. P., Binding of dioxygen to nonmetal sites in proteins: exploration of the importance of binding site size vs. hydrophobicity in the copper amine oxidase from Hansenula polymorpha, Biochemistry, 41, 13637– 13643, 2002. 60 Klinman, J. P., Life as Aerobes: Are There Simple Rules for Activation of Dioxygen by Enzymes? J. Biol. Inorg. Chem., 6, 1 – 13, 2001.

25

Solution and Computational Studies of Kinetic Isotope Effects in Flavoprotein and Quinoprotein Catalyzed Substrate Oxidations as Probes of Enzymic Hydrogen Tunneling and Mechanism Jaswir Basran, Laura Masgrau, Michael J. Sutcliffe, and Nigel S. Scrutton

CONTENTS I.

Enzymic H-Tunneling and Kinetic Isotope Effects ........................................................ 671 A. Stopped-Flow Methods to Access the Half-Reactions of Flavoenzymes and Quinoproteins ....................................................................... 672 II. Interpreting Temperature Dependence of Isotope Effects in Terms of H-Tunneling................................................................................................................. 673 III. H-Tunneling in Flavoenzymes PETN Reductase and MR ............................................. 675 IV. H-Tunneling in TTQ-Dependent MADH and AADH .................................................... 678 V. Computational Studies of Substrate Oxidation in TTQ-Dependent Amine Dehydrogenases ................................................................................................... 679 VI. H-Tunneling in Flavoprotein Amine Dehydrogenases: TSOX and Engineering Gated Motion in TMADH .......................................................................... 682 VII. Concluding Remarks........................................................................................................ 685 Acknowledgments ........................................................................................................................ 685 References..................................................................................................................................... 685

I. ENZYMIC H-TUNNELING AND KINETIC ISOTOPE EFFECTS Kinetic isotope effects (KIEs) are powerful probes of H-transfer reactions and have provided evidence for nonclassical transfer of the H nucleus in enzymes (see Chapter 28 by Kohen in this volume for a detailed discussion of the use of KIEs to identify tunneling regimes). Early studies of H-transfer by quantum tunneling focused on deviations from values predicted by semi-classical models (in which zero point energies, but not tunneling, have been taken into 671

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H =kT Þ account): KIEs, Swain – Schaad relationships1 [exp lnðk lnðkD =kT Þ . 3:26; where kH, kD, and kT are the rates of transfer for protium, deuterium and tritium, respectively] or Arrhenius prefactor ratios (q1 for a reaction proceeding purely by tunneling, , 1 for moderate tunneling). For a more detailed discussion see, for example, Chapter 28 by Kohen in this volume. Early examples in which H-tunneling was inferred from measurements of KIEs include yeast alcohol dehydrogenase,2 bovine serum amine oxidase,3 horse liver alcohol dehydrogenase,4 and monoamine oxidase.5 These studies were shown to be consistent with the tunnel correction model of semi-classical transfer, which invokes tunneling (through the energy barrier separating reactant from product) just below the classical transition state.6 This so-called Bell correction model6 accommodates small corrections to the rate of a reaction and predicts inflated KIEs and Arrhenius prefactor ratios less than unity (i.e., the so-called Kreevoy criteria for tunneling7) when KIEs are measured as a function of temperature. Nonclassical behavior is expected for a ˚ for protium and 0.45 A ˚ light particle such as the H-nucleus: the de Broglie wavelength is 0.63 A 21 for deuterium (assuming an energy of 20 kJ mol ), and this positional uncertainty gives rise to a significant probability of H-transfer by tunneling. Recent studies from our own group8 – 12 and that of Klinman13,14 have now indicated that the simple Bell-correction model cannot adequately account for observed KIEs in a number of enzyme systems. This has led to full tunneling models, akin to the established models for electron transfer, in which protein and/or substrate fluctuations are required to generate a configuration compatible with tunneling (see e.g., Refs. 14 –16). These full tunneling models are consistent with the strong temperature dependence of reaction rates, the variable temperature dependence of KIEs and the observed range of the Arrhenius prefactor ratio. In this chapter we review our own studies of (i) H-tunneling in flavoprotein and quinoprotein enzymes in which we have provided evidence consistent with H-transfer by quantum tunneling from the vibrational ground state of the reactive C – H bond of the substrate, and either H-tunneling in which the KIE is temperature independent — we interpret this to correspond to the absence of gated motion (i.e., no “compression” of the transfer distance by substrate and/or protein fluctuations) or (ii) H-transfer in which the KIE is temperature dependent — we interpret this to correspond to the involvement of gated motion. Our work10 has also highlighted the importance of energy barrier shape in determining the rates of H-transfer, and the concomitant values of KIEs, obtained in experimental studies.

A. STOPPED- FLOW M ETHODS TO ACCESS THE H ALF- R EACTIONS OF F LAVOENZYMES AND Q UINOPROTEINS The quinoprotein and flavoprotein enzymes are ideally suited to studies of H-transfer during substrate oxidation using stopped-flow methods. Analysis using the steady-state approach is often compromised by the inability to focus on a single chemical step, owing to the existence of multiple barriers for binding, product release and a number of chemical steps, each of which may contribute to the overall catalytic rate. Using the stopped-flow method, the chemical step can often be isolated and the true kinetics of C – H bond breakage determined without complications arising from other events in the catalytic sequence. With flavoprotein and quinoprotein enzymes, the reactions catalysed are conveniently divided into reductive and oxidative half-reactions. Enzyme reduction occurs by breakage of substrate or coenzyme C– H bonds. The kinetics of bond breakage are conveniently followed by absorbance spectrophotometry since the reaction is concomitant with reduction of the redox center. Thus, the alternative redox states of flavins and quinoprotein centres provide a readily available spectroscopic probe for following the kinetics of C – H bond breakage. The oxidative half-reaction usually involves long-range electron transfer to acceptor proteins (e.g., cytochromes, copper proteins, or other flavoproteins), but with flavoproteins can also involve H-transfer to a second substrate. Again, the absorbance changes associated with oxidation of the flavin provide a readily available signal for monitoring H-transfer to the oxidising substrate. The ability to interrogate

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each half-reaction by stopped-flow methods simplifies substantially the kinetic analysis and this makes these enzymes attractive targets in studies of H-transfer employing KIEs as probes of enzymic H-tunneling. Our work has focused on (i) the tryptophan tryptophylquinone (TTQ)-dependent quinoprotein amine oxidases methylamine dehydrogenase (MADH), and aromatic amine dehydrogenase (AADH), and (ii) the flavoenzymes trimethylamine dehydrogenase (TMADH), heterotetrameric sarcosine oxidase (TSOX), morphinone reductase (MR), and pentaerythritol tetranitrate (PETN), reductase. We have also determined high-resolution crystallographic structures for MR,17 PETN reductase,18 TMADH,19 and most recently AADH (Masgrau, L., Roujeinikova, A., Johannissen, L. O., Basran, J., Ranaghan, K., Hothi, P., Mulholland, A., Sutcliffe, M. J., Scrutton, N. S., and Leys, D., submitted), and a crystal structure for MADH is available from the work of others (Ref. 20 and Mathews, F. S., personal communication). In some cases, this has allowed us21 and others22 – 25 to gain additional insight into H-tunneling using computational methods, providing further support for tunneling as the key component in H-transfer reactions in these enzymes.

II. INTERPRETING TEMPERATURE DEPENDENCE OF ISOTOPE EFFECTS IN TERMS OF H-TUNNELING As mentioned in Section I, the temperature (in)dependence of KIEs is a key experimental result when considering the nature of the tunneling, with the proviso that (i) the experimentally accessible temperature range is rather narrow, thus it is not always possible to show unambiguously that the KIE is temperature independent, and (ii) recent studies on a model system (Mincer, J. S., and Schwartz, S. D., unpublished, data) have suggested that it is possible to have a temperature-independent KIE in the presence of gating motion. In particular, our view, based on currently available models and data, is that: (i) H-tunneling in which the KIE is temperature dependent corresponds to the involvement of gated motion (motion along HC in Figure 25.1), and (ii) H-tunneling in which the KIE is temperature independent corresponds to either the absence of, or at least no detectable contributions from, gated motion (i.e., no significant motion along HC in Figure 25.1). In other words, two types of motion are important in enzymic tunneling: (i) those which facilitate attaining a nuclear configuration compatible with tunneling (i.e., a configuration with degenerate quantum states) — termed “passive dynamics,” and (ii) those which enhance the probability of tunneling once (i) has been attained — termed “active dynamics.” It is, however, not possible to completely decouple active and passive dynamics, since a given motion can contribute to both types of H-tunneling. Thus, it is neither possible to map directly from the kinetic data to a detailed picture of the concomitant changes (motions) at the atomic level, nor hence the nature of the (free) energy barrier separating reactants from products. Moreover, in cases where the KIE is temperature dependent (of which, from our own work, there are only two to date10,12) a more traditional explanation is that the reaction takes place partly via the over-the-barrier route and partly by tunneling. In principle, this over/through the barrier explanation and/or the dominance of active gating is consistent with a temperature-dependent KIE, and current methods for studying the KIE cannot unequivocally disentangle the contribution of each to the tunneling reaction. The realization that tunneling might be driven by thermally induced vibrations26 in the protein scaffold (i.e., a thermally fluctuating energy surface), as described by the theoretical model of Kuznestov and Ulstrup15 (analogous to electron transfer theory27) and illustrated in Figure 25.1, has been a major step forward in recent years. This model has been adopted by Knapp and Klinman,14,16 and is of the form: "

ktunnel

(

2ðDG8 þ lÞ2 ¼ ðconst:Þ £ exp 4lRT

)# £ ðF:C: TermÞ £ ðActive Dynamics TermÞ ð25:1Þ

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Isotope Effects in Chemistry and Biology

FIGURE 25.1 Representation of the model for the hydrogen transfer reaction used to interpret the experimental data (see text and Refs. 48,72,73 for more details); some of the parameters in Equation 25.1 and Equation 25.2 are shown. The three axes are: E, energy; q, environmental coordinate (from which the transferred hydrogen atom is excluded); HC, hydrogen coordinate. The four vertical panels show the potential energy curve as a function of the hydrogen coordinate for three values of the environmental coordinate: q R is for the reactant, q p is for the transition state, and q P is for the product. The grey spheres represent the groundstate vibrational wavefunction of the hydrogen nucleus. The panel labeled M shows a Marcus-like view of the free energy curves as functions of this environmental coordinate. The motions of the environment (related to the first exponential in Equation 25.1) modulate the symmetry of the double well, thus allowing the system to reach a configuration with (nearly) degenerate quantum states ðq ¼ qp Þ, from which the hydrogen is able to tunnel (F.C. Term in Equation 25.1 and Equation 25.2). The difference between panels a and b is a gating motion that reduces the distance between the two wells along the HC axis (rH) away from its equilibrium value (r0). This motion increases the probability of tunneling at the (nearly) degenerate configuration q p (active dynamics term in Equation 25.1 and Equation 25.2).

Here, ktunnel is the tunneling rate constant; const. is an isotope-independent term; the term in square brackets is an environmental energy term relating the driving force of the reaction, DG8, and the reorganizational energy, l (R is the gas constant and T the temperature in K); the F.C. Term is the Frank – Condon nuclear overlap along the hydrogen coordinate and arises from the overlap between the initial and the final states of the hydrogen’s wavefunction. In the simplest limit, when only the lowest vibrational level is occupied for the nuclear wavefunction of the hydrogen, the F.C. Term is independent of temperature; otherwise, the F.C. Term will be temperature dependent. Temperature-dependent gating (or active) dynamics, which can be likened to a squeezing of the potential energy barrier, can modulate the F.C. Term. In this case, the KIE is dependent on the energetic cost of gating and the KIE (Equation 25.2) was derived from Equation 25.116: Ð F:C: TermH £ Active Dynamics TermH KIE ¼ Ð F:C: TermD £ Active Dynamics TermD ðr0 ¼

r1 ðr0 r1

expð2mH vH rH2 =2"Þ expð2EX =kB TÞdX expð2mD vD rD2 =2"Þ expð2EX =kB TÞdX

ð25:2Þ

Solution and Computational Studies of Kinetic Isotope Effects

675

Here, kB is Boltzmann’s constant, r0 is the equilibrium, and r1 is the final separation of the reactant and product potential wells along the hydrogen coordinate, vH and vD are the frequencies of the reacting bond, and mH and mD are the masses of the transferred particle for protium and deuterium, respectively. The H/D transfer distance, rH =rD ; is reduced by the distance the gating unit moves, rX ðrH=D ¼ r0 2 rX Þ. The energetic cost of gating ðEX Þ is given by Ref. 16: EX ¼

1 1 "v X 2 ¼ mX v2X rX2 2 X 2

ð25:3Þ

Here, the gating coordinate (X ) is related to the gating oscillation ðvX Þ; the mass of the gating unit ðmX Þ; and rX as follows: qffiffiffiffiffiffiffiffiffiffiffi X ¼ rX mX vX =" ð25:4Þ This model predicts that if the gating term dominates (i.e., "vX , kBT ), the observed KIE can be temperature dependent, since this leads to different transfer distances for the heavy and light isotope.16 In this regime the AH/AD value is predicted to be less than unity. Alternatively, if the Frank – Condon term dominates (i.e., "vX . kB T) the KIE will be temperature independent. In this latter scenario, occupation of excited vibrational levels could result in some temperature dependence.14 However, the Boltzmann distribution at 298 K suggests that tunneling should be predominantly from the vibrational ground state of the nuclear wavefunction of hydrogen. In the regime where "vX , kB T gating plays some role in modulating the tunneling probability, temperature-dependent KIEs are observed and the AH/AD values decrease (compared with the regime where the Frank – Condon term dominates), and may approach unity.16 However, studies of the temperature dependence of the KIE cannot disentangle unequivocally the contribution of each term to the tunneling reaction. Compounding the conceptual problem further, Warshel recently claimed28 his work demonstrates that dynamics does not enhance enzyme catalysis over the equivalent reaction in solution, and he has suggested that the main contribution to catalysis comes from the fact that the barrier is lowered by electrostatic effects.29 A key role for dynamics has, however, been advocated by others from theoretical studies and experimental observations, e.g., Bruno and Bialek,30 Benkovic31 and Schwartz.26,32 – 37 Schwartz’s work (see Chapter 18 by Schwartz in this volume) appears to have established an important role for protein dynamics in enzyme catalysis. Thus, the jury is still very much out as to exactly what properties of the enzyme give rise to the temperature (in)dependence of the KIE — the quest for a detailed understanding of what, at the atomic level, gives rise to these is currently a “holy grail” for quantum enzymologists.

III. H-TUNNELING IN FLAVOENZYMES PETN REDUCTASE AND MR We have used the formulism of Knapp and Klinman (see Section III) to interpret the anomalous temperature dependencies for H-transfer in reactions of (i) PETN reductase with NADPH,12 (ii) MR with NADH,12 and (iii) MR in the oxidative half-reaction with 2-cyclohexenone.12 We have studied flavin reduction in MR and PETN reductase, and flavin oxidation in MR, using stoppedflow and steady-state kinetic methods with protiated and deuterated nicotinamide coenzymes. The temperature (in)dependence of the primary KIE for flavin reduction in MR and PETN reductase by nicotinamide coenzyme indicates that quantum mechanical tunneling plays a major role in hydride transfer. In PETN reductase, the KIE is essentially independent of temperature in the experimentally accessible range, and this contrasts with strongly temperature-dependent reaction rates (Table 25.1). The data are consistent with a tunneling mechanism governed by passive dynamics and from the vibrational ground state of the reactive C – H/D bond. In MR, both the reaction rates with NADH and the KIE are dependent on temperature, and analysis using the Eyring equation suggests that hydride transfer has a major tunneling component, which unlike PETN

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TABLE 25.1 Tunneling Regimes and Associated Parameters in Various Quinoprotein and Flavoprotein Enzymes Enzyme

Substrate

A H/AD

DH H (kJ mol21)

DH D (kJ mol21)

KIEa

Passiveb

MADH MADH AADH AADH AADH TSOX PETNR MR MR TMADH (H172Q) TMADH (Y169F)

Methylamine Ethanolamine Tryptamine Dopamine Benzylamine Sarcosine NADPH NADH 2-Cyclohexenone Trimethylamine Trimethylamine

13.3 0.57

44.6 ^ 0.5 43.5 ^ 0.6

45.0 ^ 0.5 51.9 ^ 1.1 53.5 ^ 1.2 51.6 ^ 0.7 67.1 ^ 0.9 40.0 ^ 1.2 36.6 ^ 0.9 43.5 ^ 0.8 17.1 ^ 0.9 41.7 ^ 2.6 45.1 ^ 1.6

16.8 TI 14.7 TD 54.7 TI 12.9 TI 4.8 TI 7.3 TI 4.1 TI 3.9 TD 3.5 TI 4.6 TI ?

U

9.4 3.7 5.8 4.1 0.126 3.7 7.8 2.5

50.9 ^ 0.7 68.1 ^ 1.4 39.4 ^ 0.9 36.4 ^ 0.9 35.3 ^ 0.5 17.6 ^ 0.9 41.2 ^ 2.6 42.1 ^ 0.9

Gatedb

U U U U U U U U U U

Reference 8 10 10 10 10 11 12 12 12 69 69

a

TI ¼ temperature-independent KIE, TD ¼ temperature-dependent KIE; KIE values for enzyme–substrate combinations displaying a temperature-dependent (TD) KIE (i.e., reactions involving gated motion14,16) are given at 298 K. b The terms “passive” (i.e., the KIE is [almost] temperature independent) and “gated” (i.e., the KIE is temperature dependent) dynamics are taken from the work of Knapp and Klinman.14,16 See text for a discussion of the current limitations of this and other interpretations of factors affecting the temperature (in)dependence of KIEs.

reductase, is gated by thermally induced vibrations in the protein (Table 25.1). We have suggested that PETN reductase is relatively more rigid compared with MR, consistent with gating being less dominant in PETN reductase, which in turn predicts that the KIE would be more temperature dependent in MR than in PETN reductase. In addition, the active site of PETN reductase might be more optimally configured for hydride transfer than that of MR, thus requiring little (or no) vibrational assistance through gated motion. In other words, the active site of PETN reductase is ideally set up to transfer a hydride ion from NADPH to FMN, and nuclear reorganization associated with H-tuneling (i.e., passive dynamics) is the major dynamic component. We have compared the high-resolution crystal structures of MR17 and PETN reductase18 in an attempt to provide insight into why gating is potentially more important in MR. Analysis of the structures of each enzyme suggest a key factor could be double stranded antiparallel b-sheet D, against which the NAD(P)H coenzyme is thought to bind.17 This region harbours arginine residues important in the recognition of the 20 -phosphate of NADPH (PETN reductase) and a glutamate residue required to form a H-bond with the 20 -OH group of NADH (MR). The position of this sheet diverges at Leu-133 (PETN reductase)/Val-138 (MR) and converges again at Ile-141 (PETN reductase)/Gly-146 (MR). There is also an insertion of a glycine residue (Gly-133) in MR immediately before the start of b-sheet D. These differences are consistent with MR being more mobile at physiological temperatures in this region than PETN reductase, which in turn might assist in a squeezing or compression of the potential energy barrier in MR. This suggestion is consistent with the temperature factors for MR (all Ca temperature factors . 40; PDB38 accession code 1GWJ) and PETN reductase (all Ca temperature factors , 20; PDB accession code 1GVQ) in this region. The next stage of our work to test this hypothesis is to obtain structural information for the coenzyme complexes at high resolution and to perform a more detailed theoretical analysis involving QM/MM, variational transition state theory (VTST) with multidimensional tunneling and molecular dynamics studies.

Solution and Computational Studies of Kinetic Isotope Effects R

H

H3C

N

N

H3C

N H

677

O N

H

O H

A O

δ+ δ−

O

NH

Asn189 H2N His 186

+ NH

FIGURE 25.2 Proposed scheme for the oxidative half-reaction of morphinone reductase. The identity of the proton donor in the oxidative half-reaction is not known.

The oxidative half-reaction of MR with the substrate 2-cyclohexenone and NADH at saturating concentrations is fully rate-limiting in steady-state turnover; this has enabled us to investigate potential tunneling regimes in this part of the reaction cycle. Reduction of 2-cyclohexenone involves hydride transfer from FMNH2 and protonation, and thus two H-transfer reactions are involved (Figure 25.2). We have demonstrated that the KIE for hydride transfer from reduced flavin to the a/b unsaturated bond of 2-cyclohexenone is independent of temperature, contrasting with strongly temperature-dependent reaction rates. A large solvent isotope effect (SIE) accompanies the oxidative half-reaction, which is also independent of temperature in the experimentally accessible range and double isotope effects indicate that hydride transfer from the flavin N5 atom to 2-cyclohexenone, and the protonation of 2-cyclohexenone, are coupled. Both the temperatureindependent KIE and SIE suggest that (i) gated motion is not required to compress the energy barrier, and (ii) this reaction proceeds by ground state quantum tunneling. Our work with MR is, to our knowledge, the first to show that both passive and active dynamics are a feature of H-tunneling within the same native enzyme; in the reductive half-reaction we suggest barrier compression is required to facilitate hydride transfer from NADH to FMN, whereas in the oxidative half-reaction the active site is configured to catalyze hydride and proton transfer in a coupled fashion without vibrational assistance through gated motion. The notion that enzymes have evolved to optimize H-tunneling by acquiring strategies during evolution to increase the probability of transfer remains controversial, for example Refs. 39– 41. In one study employing an adenosylcobalamin-dependent diol dehydratase model reaction it is argued39,40 that the B12-dependent enzyme exploits the same level of quantum mechanical tunneling that is available in the reaction occurring in the absence of enzyme (i.e., there is no compressive motion that preferentially enhances H-tunneling in the enzyme over the reaction in solvent). Moreover, Siebrand and Smedarchina42 have questioned the statistical significance of data reported over a relatively narrow temperature range for reactions of wild-type and mutant lipoxygenase — these data were originally presented as evidence for gated motion in this enzyme.14 On theoretical grounds, the same authors also argue that flexible proteins are “ill-equipped to cause strong local compression.” One needs to be mindful of these issues, but our view is that it is not logical to generalize on the basis of a small number of studies and that a case-by-case analysis is appropriate.

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IV. H-TUNNELING IN TTQ-DEPENDENT MADH AND AADH With our work on H-tunneling in quinoprotein enzymes we have focused on TTQ-dependent MADH8,10 and AADH.10 Earlier studies on the quinoprotein bovine serum amine oxidase (a topaquinone-dependent amine oxidase) also demonstrated deviation from semi-classical behavior, suggesting a role for H-tunneling. In this case, the data were interpreted in terms of the tunneling correction models of Bell. In MADH, TTQ reduction is concerted with breakage of the C – H bond from an iminiquinone intermediate that forms rapidly in the reductive half-reaction (Figure 25.3). The rate of breakage of this bond is accessible using stopped-flow spectrophotometry. Reduction of the TTQ cofactor is associated with a large KIE (KIE ¼ 16.8 ^ 0.5 at 298 K) when methylamine and

1 C O

3

2 H 2O

C

C HO

O

.. HNH

C +

C O

HNH

O

+ NH

CH3

CH3

C

H

CH2

B

−

k3 6 C

5 + H 2O

C

C

OH

NH2

(a)

C −

H

CH2

CH2 B

C O

+ NH

O

4

+ NH

B

CH2

C O−

HB

−

NH O

CH CH2

C NH

NH

H2C HN

(b)

CH

O C

O

O

FIGURE 25.3 (a) Reaction mechanism for the reductive half-reaction of MADH. Steps enclosed in the hatched box represent binding steps. A similar scheme has been proposed for the reaction of AADH with aromatic primary amines. (b) The structure of the tryptophan tryptophylquinone (TTQ) cofactor.

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deuterated methylamine are used as substrates. The value of the KIE effect is larger than the upper value expected for reactions described by semi-classical transition state theory (i.e., including zeropoint energy but with no tunneling correction), and is clearly suggestive of tunneling. Our studies of TTQ reduction in MADH indicated that the value of the KIE was independent of temperature in the experimentally accessible region, but significantly the reaction rate was strongly dependent on temperature. These findings are consistent with the dissipative tunneling models14 – 16 as described above — in which the reaction is dominated by environmental reorganization associated with tunneling rather than any gated motion. It is interesting to note that an earlier study43 had observed temperature-independent KIE values (, 2 to 3) in steady-state reactions catalyzed by serine proteases performed in deuterated solvent, and these were suggested to indicate tunneling (note, however, that the effect of D2O on the reaction dynamics is potentially complicated owing to the exchange of protons throughout the protein scaffold). The data were modeled on earlier theoretical treatments of H-tunneling propounded by Dogonadzhe and coworkers, in which thermal vibrations bring the solvent into a configuration favorable to tunneling. At the time of our own work with MADH, similar observations were made with the thermophilic alcohol dehydrogenase of Bacillus stearothermophilus, also indicating that the Bell correction models for H-tunneling were inappropriate for these enzyme-catalyzed reactions.13 We extended our work with MADH to study tunneling behavior with nonphysiological substrates. In our work with ethanolamine, we find evidence for quantum tunneling (Table 25.1); in this case, however, the KIE was dependent on temperature, inconsistent with a reaction dominated by passive dynamics, but instead consistent with gating. Our more recent studies have focused on the related TTQ-dependent enzyme aromatic amine dehydrogenase (AADH). Using similar stopped-flow approaches to those reported for studies with MADH, we have shown that the rate of TTQ reduction by dopamine in AADH has a large, temperature-independent KIE (KIE ¼ 12.9 ^ 0.2), consistent with tunneling involving passive, but not gated, dynamics. We have also demonstrated that H-transfer is compromised with benzylamine as substrate and that the KIE is deflated (4.8 ^ 0.2), although the KIE remains temperature independent. As with dopamine, however, reaction rates are strongly dependent on temperature. Tryptamine is a fast substrate for AADH and this allows determination of the rate of TTQ reduction at low temperatures using the stopped-flow method with protiated substrate. That said, an exceptionally large KIE (54.7 ^ 1.0; single measurement at 48C) is observed for breakage of the substrate C –H/D bond, and the reaction rate (for C – D bond breakage) is strongly dependent on temperature. The exceptionally large KIE for this substrate suggests a major role for H-tunneling in substrate oxidation, and by comparison with studies using dopamine and benzylamine, we inferred that only passive dynamics is involved in H-tunneling. More recent and as yet unpublished studies (Masgrau, L., Roujeinikova, A., Johannissen, L. O., Basran, J., Ranaghan, K., Hothi, P., Mulholland, A., Sutcliffe, M. J., Scrutton, N. S., and Leys, D., submitted) have utilized a stopped-flow instrument with a smaller dead time (0.5 ms), and confirmed that the KIE with tryptamine is indeed independent of temperature in the experimentally accessible range. We have used the KIE as a key indicator of tunneling. However, as mentioned above, it is not possible to map directly from the KIE to either the nature (at the atomic level) of the dynamics or the nature (shape) of the barrier through which tunneling occurs. It is clear, however, in the case of both AADH10 and MADH,21 that the shape of the barrier, unlike the rectangular barrier used to model electron transfer, is complex in nature. It is not possible to map the shape of the barrier using experimental methods; computational methods are used in their stead.

V. COMPUTATIONAL STUDIES OF SUBSTRATE OXIDATION IN TTQ-DEPENDENT AMINE DEHYDROGENASES From the discussion above, it is clear that the factors which give rise to the observed kinetic data for these system are far from trivial, and that computational chemistry can be used to help elucidate

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how these enzymes work and facilitate tunneling at the atomic level. In particular, hybrid quantum mechanical/molecular mechanics (QM/MM) methods have become the method of choice for studying the reactivity of enzyme systems (see Refs. 44,45 and for a recent review see Ref. 46). Within the QM/MM approach most of the system is modeled by a molecular mechanics force field, allowing all the (solvated) enzyme to be included in the calculations; a small region in the active site is treated by quantum mechanics, including the atoms involved in the breaking and forming of bonds (and their close surrounds). The QM/MM method can be used, for example, to explore the potential energy surface (PES), localize the reactant and product enzyme –substrate complexes and even run molecular dynamics based on the combined QM/MM potential. Moreover, with special methods (e.g., umbrella sampling47,48 or free energy perturbation49) a free energy profile along a chosen reaction coordinate can be obtained. These calculations can also be carried out based on empirical valence bond (EVB50) PESs. However, when quantum nuclear effects such as tunneling are thought to contribute to the reaction, further calculations (or significant modifications to those mentioned above) are required. There are several methodologies available51 which allow quantum nuclear effects to be included in enzyme kinetics, of which only variational transition state theory plus multidimensional tunneling corrections (VTST/MT51 – 53) has been applied to the TTQdependent amine dehydrogenase family. The rate constant for a unimolecular reaction is given by: ! kB T 2DGTS ðTÞ VTST=MT exp ð25:5Þ k ðTÞ ¼ gðTÞ h RT Where the free energy barrier (DG TS(T )) is calculated with the reaction coordinate motion separated from all other degrees of freedom normal to it, and g ðT Þ is a transmission coefficient that includes dynamical recrossing and quantum nuclear effects on the reaction coordinate, including the effects of nonseparability. To our knowledge, four computational studies have been published21 – 25 on rate constants in these TTQ-dependent amine dehydrogenases — specifically MADH, the only crystal structure of this class of enzymes available at the time. In all of these a QM/MM approach was used together with VTST/MT rate constant calculations; however, there are significant differences in the way the terms in equation 25.5 were obtained (see below). The natural substrate methylamine was used in all of these studies; the reaction with ethanolamine was also included in the most recent study and results for both substrates compared. In the oxidation of these two amines by MADH the ratelimiting step involves the abstraction by the active site base (Asp 428) of a proton (C –H bond breakage) from the iminoquinone (Figure 25.3), a step shown experimentally to involve H-tunneling, and it is this hydrogen tunneling step that has been studied computationally. Our (preliminary) work on MADH – methylamine21 used Gaussian9454 and AMBER55 for the QM/MM calculations, with the semiempirical PM3 method56 and the link atom approach44,56 – 58 (for interfacing this QM region with the rest of the protein modeled by MM). By optimizing the QM region embedded in the frozen MM environment a reactant complex was found, and a saddle point and product complex were identified. From this saddle point the minimum energy path (MEP) was calculated by following the Page –McIver algorithm59 using POLYRATE 7.4;60 for the QM region, the first and second derivatives of the energy were also calculated along this MEP. Thus, the rate constant was calculated with multidimensional tunneling contributions. The results suggested that approximately 96% of the reaction proceeds by tunneling, with the remaining , 4% proceeding via the classical over-the-barrier route. It was also shown that tunneling needs to be included to obtain a KIE value comparable to the experimental one, and that a multidimensional (i.e., more comprehensive) treatment of tunneling also improved the result: a value of 6.1 was obtained when tunneling was omitted, 9.6 when tunneling was included by a simple one-dimensional model (i.e., the Wigner correction52,61) and 11.1 when the small-curvature tunneling (SCT62) was used; the corresponding experimental value8 is 16.8 ^ 0.5 (all at 298 K). Within the framework of the methodology used to incorporate H-tunneling into calculation of the reaction rate, both this

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study and the studies of others have shown that the reaction proceeds as follows. First, the two heavy atoms (C on the iminoquinone and O on the Asp) approaching each other until the distance is reduced sufficiently for (a significant amount of) tunneling to take place through the PES (note that tunneling probability is related inversely to the tunneling distance). Tunneling is further enhanced by a corner-cutting mechanism as the C –H (and later the O –H) stretching motions become coupled to the reaction coordinate. The induced curvature, together with a reaction coordinate corresponding to the motion of the light particle (H-nucleus), results in a shorter effective tunneling path and, thus, a larger tunneling probability. This is why a one-dimensional treatment of tunneling underestimates the contribution of H-tunneling to the reaction. In the same year, an independent study of the MADH – methylamine system was published.22,23 The authors determined an even larger degree of tunneling for the H-transfer step — increasing the rate of reaction 100-fold, i.e., 99% of the H-transfer reactions proceeding via tunneling and 1% of the reactions proceeding via the classical over-the-barrier route (again at 298 K). Their predicted KIE at 298 K, 18.3, is in good agreement with the experimental result8 and also larger than our calculated value. It is important to note that, although both studies were based on VTST/SCT and QM/MM calculations, the details of the way in which these methodologies were applied differ significantly. First, Alhambra et al.22,23 used the PM3 semiempirical method which included specific reaction parameters (PM3-SRP63) to improve the energetics of the PES. Second, for the dynamical calculations they employed the so-called ensemble average variational transition state approach with multidimensional tunneling corrections (EA-VTST/MT51,53). In this approach, comprising several steps which have been fully described in the respective references, the basic idea is to expand the calculation to a larger number of protein configurations. To this end, the thermal averaged motions of the protein are first included in the calculation of a classical free energy barrier, then an ensemble of enzyme configurations is considered in the subsequent steps. In these steps, most of the protein is fixed (the movable region is usually not larger than 60 atoms), but the magnitudes of interest (e.g., the tunneling correction) are calculated averaging over the different protein configurations considered. The final rate constant is obtained by combining the information obtained in each step. This strategy, developed and implemented in Truhlar’s and Gao’s groups during recent years, is clearly more computationally demanding than calculating only a single MEP with most of the system frozen. However, it has the advantage of corresponding more closely to the enzyme in biological systems. It therefore gives a more detailed picture of the reaction and allows better analysis of what is happening at the atomic level (see, for example, Refs. 64– 66). The third computational study of the MADH – methylamine system24 involves an in silico comparison of three enzymes that have been shown to catalyze H-tunneling — liver alcohol dehydrogenase, methylamine dehydrogenase, and soybean lipoxygenase. Although the potential energy barriers obtained for these three hydrogen transfer reactions were very similar, each system exhibited a different degree of tunneling. The authors rationalized this in terms of different donor – acceptor distance behavior during the reaction, and linked this to corner-cutting and different curvature of the reaction paths. The same MADH –methylamine results were presented in their most recent publication,25 alongside corresponding results for the MADH – ethanolamine system. The methodology they used was again a single MEP calculated with most of the system frozen, but in this instance the semiempirical PM3 quantum mechanics method was adjusted with specific reaction parameters. When used for the MADH – methylamine system, the PM3-SRP-derived KIE and activation energy were closer to the experimental values than the standard PM3 ones; their interpretation of these results remained essentially unchanged. The MADH – ethanolamine system turned out to be more complicated. A number of substrate conformers were found depending on the position of the hydroxyl group of ethanolamine. At first, only the most stable one (denoted by them EA(A), which had the hydroxyl group hydrogen bonded to a water molecule, which was in turn hydrogen bonded to another residue) was investigated. However, the results did not agree completely with

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Isotope Effects in Chemistry and Biology

the experimental observations; indeed, contrary to experiment more tunneling was obtained for ethanolamine than for methylamine. The authors attributed this result to the reactant-side behavior of the ethanolamine reaction path curvature, which was considered to be responsible for a cornercutting tunneling effect taking place over a larger region of the reaction path. In view of the disagreement with the experimental results, the authors decided to consider a second conformer of the substrate. In this conformer, which they termed EA(B), the hydroxyl group of ethanolamine was hydrogen bonded to the nonreactive oxygen atom of the Asp428 (Figure 25.3). The energy profiles and the kinetic results for the two conformers turned out to be very different, the latter having a higher barrier, a more positive energy of reaction, and (concomitantly) much smaller tunneling effects. The authors speculated that, as the two conformers were close in energy, they both could be contributing to the observed kinetic data, in particular the temperature dependence of the KIE. Thus, they suggest that the less stable structure, EA(B), with a smaller KIE, would become more important at higher temperatures. The possible contribution to the observed kinetic data of different reactant conformers is certainly an interesting observation and an issue to be considered when studying the reactivity of any system.67 It would therefore be helpful to have additional information, such as the energy difference between the two ethanolamine conformers or an estimate of the relative populations (even though the authors remarked that they were just speculating). Likewise, there are some aspects that may be interesting to explore further; for example, the origin of the striking reaction path curvature obtained is not discussed. In our view, this could be an artifact of the chosen configuration of the active site, or simply a numerical problem during the calculations. Moreover, the overestimation of tunneling for the ethanolamine substrate may simply be due to the PM3-SRP method not performing equally well for the two systems. On the other hand, it should be noted that ethanolamine has a larger range of available tunneling energies than methylamine because its reaction is calculated to be nearly thermoneutral, whereas that for methylamine is considerably endothermic. This implies that more methylamine tunneling paths will be dismissed because they will not have an exit turning point. This fact alone, without the need of special curvature graphs, could lead to a bigger tunneling correction for ethanolamine, even though its adiabatic barrier height is somehow lower than the methylamine one. Again, the use of only one protein configuration makes it difficult to predict how much the chosen fixed environment is altering these energies of reaction and, therefore, the tunneling contributions. In conclusion, these computational studies suggest that over-the-barrier processes are not the only mechanism employed by MADH, particularly with methylamine. In fact, inclusion of multidimensional tunneling in the calculation of the rate constants is necessary to reproduce the high experimental KIE values, although with the ethanolamine some points still remain unclear. Thus, these computational studies are in accord with the experimental observation that these H-transfer reactions invoke tunneling. Indeed, the degree of tunneling calculated for MADH – methylamine is larger than that for other protein systems, the next largest one being 85% in xylose isomerase.65,68 An on-going challenge for the theoretical calculations is to reproduce “from first principles” the temperature (in)dependence of the KIEs (particularly the independence). Nevertheless, we have to bear in mind that the overall aim is to obtain a better understanding of the mode of action of these enzymes. We are currently at the stage where a range of computational studies using different approaches, enzyme systems, and methods will provide invaluable insight.

VI. H-TUNNELING IN FLAVOPROTEIN AMINE DEHYDROGENASES: TSOX AND ENGINEERING GATED MOTION IN TMADH Like the quinoprotein amine dehydrogenases, the flavoprotein amine dehydrogenases have also proven to be good model systems for studies of enzymic H-tunneling. Klinman and Edmondson provided early evidence for tunneling in mammalian monoamine oxidase, and the data were

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interpreted in terms of the Bell tunneling correction model.5 More recent studies with heterotetrameric sarcosine oxidase (TSOX11) and trimethylamine dehydrogenase (TMADH69) indicate that the Bell tunneling correction model is inappropriate for these enzymes and that KIE data are consistent with the more recent full tunneling models. With TSOX — a diflavin enzyme containing FAD (the site of substrate oxidation) and 8a-(N 3-histidyl)-FMN (the site of oxygen reduction) — treatment of the enzyme with sulfite provides the means for selective formation of a flavin-sulfite adduct with the covalent 8a-(N 3-histidyl)-FMN.11 Formation of the sulfite-flavin adduct suppresses internal electron transfer between the noncovalent FAD and the covalent FMN and thus enables detailed characterization of the kinetics of FAD reduction by sarcosine using stopped-flow methods. The rate of FAD reduction was found to display a simple hyperbolic dependence on sarcosine concentration, and studies in the pH range 6.5 to 10 indicate there are no kinetically influential ionizations in the enzyme– substrate complex. A plot of the limiting rate of flavin reduction and the enzyme –substrate dissociation constant (klim/Kd) vs. pH is bell-shaped and characterized by two macroscopic pKa values of 7.4 ^ 0.1 and 10.4 ^ 0.2, indicating two kinetically influential ionizations in the free enzyme or free substrate which remain to be assigned. The KIE for breakage of the substrate C –H bond is 7.3, and the value is independent of temperature in the experimentally accessible range; in contrast, reaction rates are strongly dependent on temperature (Table 25.1). The lack of a temperature dependence on the KIE suggests gated motion is not dominant in this reaction. We have investigated the effects of compromising mutations on tunneling in the enzyme trimethylamine dehydrogenase (TMADH). Evidence from isotope studies (see below) supports the view that catalysis by TMADH proceeds from a Michaelis complex involving trimethylamine base and not, as thought previously, trimethylammonium cation (Figure 25.4a). Stopped-flow studies and analysis of tunneling regimes in this enzyme are not straightforward owing to the presence of four kinetically influential ionizations in the reduction of the 6-S-cysteinyl FMN of TMADH by substrate (two in the enzyme – substrate complex [pKa 6.5 and 8.2], one attributable to free trimethylamine [pKa 9.8] and one attributed to the free enzyme [pKa , 10] which remains unassigned). In native TMADH, reduction of the flavin by substrate (perdeuterated trimethylamine) is influenced by two ionizations in the Michaelis complex with pKa values of 6.5 and 8.2 and maximal activity is realized in the alkaline region.70 The latter ionization has been attributed to residue His-172 (through studies of the H172Q mutant enzyme71) and, more recently, the former to the ionization of substrate itself.70 Stopped-flow kinetic studies with trimethylamine as substrate have indicated that mutation of His-172 to Gln reduces the limiting rate constant for flavin reduction approximately 10-fold.71 A KIE accompanies flavin reduction by H172Q TMADH, the magnitude of which varies significantly with solution pH. With trimethylamine, flavin reduction by H172Q TMADH is controlled by a single macroscopic ionization (pKa 6.8 ^ 0.1). This ionization is perturbed (pKa 7.4 ^ 0.1) in reactions with perdeuterated trimethylamine and is responsible for the apparent variation in the KIE with solution pH. The isotope dependence of this pKa value is of interest. The evidence suggests that this pKa represents the deprotonation of the substrate molecule itself ([CH3]3NHþ ! [CH3]3N) on moving from low to high pH. It is anticipated that perdeuteration of the substrate will affect this ionization since: (i) the shorter C –D bond results in a larger charge density, and thus it is electron supplying (i.e., stabilizing the N – H bond) relative to C –H; (ii) the perdeuterated substrate has a greater reduced mass for the (CD3)3N –H stretching vibration, and therefore lies lower in the asymmetric potential energy well. Thus, the (CH3)3N –H bond dissociates more readily than the (CD3)3N –H bond, accounting for the elevated macroscopic pKa value seen with perdeuterated substrate in our kinetic studies. Figure 25.4b summarizes the prototropic control on flavin reduction in the enzyme – substrate complex. In Figure 25.4b it is assumed that the rate of breakdown of the ES complex to EP is slow relative to the dissociation steps, so that the dissociation steps remain in thermodynamic equilibrium. Clearly, as a result of the elevated pKa value seen with perdeuterated substrate there is a greater concentration of the ESHþ (unreactive)

684

Isotope Effects in Chemistry and Biology R

R

N

O

N

NH

N Enz

S

O

H

S

O = CH2

(a)

N H

O

N − CH3 CH2 CH3

H2O

O

Enz

H

− N

N

NH S

O

H2C N

CH3 CH3

k1 k2

KAS E + SH+

ES

k3

O NH

N H

+

N

E + S

(b)

O

+

R

_ N

O NH

N H

S

C CH3 H2 CH3

N

Enz

Enz

N

R

N

N

CH3 CH3

EP

KAES k ′1 k ′2

ESH+

FIGURE 25.4 (a) Proposed mechanism for the oxidation of trimethylamine by TMADH.70 (b) Kinetic scheme for the reaction of H172Q mutant TMADH with trimethylamine.

complex (i.e., the lower branch of Figure 25.4b). The effect of this partitioning between ES and ESHþ forms of the enzyme –substrate complex is that the observed KIE is inflated over the intrinsic value that would be realized if the concentration of the ES species were equivalent (at a given pH value) for both perdeuterated and protiated substrate. Only at pH values of 9.5 and above (where the group identified in the plot of k3 vs. pH is fully ionized, and where the rate of flavin reduction is maximal), is the intrinsic isotope effect realized, owing to the enzyme being in the ES form for both protiated and perdeuterated substrate. In this regime, the KIE approaches a constant value of , 4.5. In the enzyme – substrate complex, the pKa for the ionization of trimethylamine (6.8) is more acidic than that of free trimethylamine (9.8). Consequently, in the Michaelis complex, the ionisation of substrate is substantially perturbed leading to a stabilisation of trimethylamine base by , 10 kJ mol21. We have shown by targeted mutagenesis and stoppedflow studies that this reduction of the pKa is a consequence of electronic interaction with residues Tyr-60 and His-172 and these two residues are therefore key for optimizing catalysis in the physiological pH range. Formation of a Michaelis complex with trimethylamine base is consistent with a mechanism of amine oxidation that we advanced in our previous computational and kinetic studies, which involves nucleophilic attack by the substrate nitrogen atom on the electrophilic C4a atom of the flavin isoalloxazine ring (Figure 25.4).

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Substrate bond breakage by wild-type TMADH is too fast to be followed using the stoppedflow method in the regime where both His-172 and trimethylamine in the enzyme– substrate complex are deprotonated (i.e., , pH 10). Consequently, our tunneling studies have focused on the compromised mutant enzymes H172Q and Y169F (, 10-fold and 40-fold reduction in the limiting rate constant for flavin reduction compared with wild-type enzyme, respectively). With H172Q TMADH, flavin reduction is controlled by a single macroscopic ionization for substrate ionization (pKa 6.8), which is slightly elevated compared with the value obtained for wildtype owing to the stabilizing electronic effects of the His-172 side-chain in the latter. With H172Q TMADH, this ionization is perturbed (pKa 7.4) in reactions with perdeuterated trimethylamine and is responsible for the apparent variation in the KIE with solution pH. At pH 9.5, where substrate is fully ionized in the Michaelis complex, the KIE is independent of temperature in the range 277 to 297 K, whilst reaction rates are still strongly dependent on temperature (Table 25.1); this is consistent with H-transfer by tunneling from the vibrational ground state of the reactive bond in a mechanism that is not dependent on gated motion (a possible alternative explanation is that there is still some gating, but this is not visible in the KIE temperature dependence as seen, for example, by Mincer and Schwartz [unpublished results]). With Y169F TMADH, the situation is different: the rate of flavin reduction is , 4-fold more compromised than in H172Q TMADH, and in the case of Y169F TMADH, the KIE is dependent on temperature (Table 25.1). In this case, the temperature dependence of the KIE is consistent with the need for gated motion to facilitate the tunneling process.

VII. CONCLUDING REMARKS KIEs in flavoproteins and quinoproteins have revealed that enzymes catalyze reactions by “pure” quantum tunneling. KIEs give insight at the macroscopic level, but not at the atomic level. Computational studies are used to give insight into the atomic details of the mechanisms used by enzymes that invoke quantum tunneling. Protein dynamics (both active and passive) drives these tunneling reactions, and a picture of how enzymes achieve this is beginning to emerge. However, many of the details at the atomic level remain to be discovered, and these will be probed in greater detail as more refined kinetic (e.g., at cryogenic temperatures) and computational techniques advance.

ACKNOWLEDGMENTS The authors are very grateful to Linus Johannissen, Kamaldeep Chohan, Richard Harris, Parvinder Hothi, Shila Patel, Adrian Mulholland, and Kara Ranaghan for their valuable contributions to, and discussions about, the work presented. The BBSRC, EPSRC, University of Leicester, and Wellcome Trust are thanked for providing financial support.

REFERENCES 1 Swain, C. G., Stivers, E. C., Reuwer, J. F., and Schaad, L. J., J. Am. Chem. Soc., 80, 5885– 5893, 1958. 2 Cha, Y., Murray, C. J., and Klinman, J. P., Hydrogen tunneling in enzyme reactions, Science, 243, 1325– 1330, 1989. 3 Grant, K. L. and Klinman, J. P., Evidence that protium and deuterium undergo significant tunneling in the reaction catalyzed by bovine serum amine oxidase, Biochemistry, 28, 6597– 6605, 1989. 4 Bahnson, B. J., Park, D. H., Kim, K., Plapp, B. V., and Klinman, J. P., Unmasking of hydrogen tunneling in the horse liver alcohol dehydrogenase reaction by site directed mutagenesis, Biochemistry, 32, 5503– 5507, 1993. 5 Jonsson, T., Edmondson, D. E., and Klinman, J. P., Hydrogen tunneling in the flavoenzyme monoamine oxidase B, Biochemistry, 33, 14871– 14878, 1994.

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26 Antoniou, D. and Schwartz, S. D., Activated chemistry in the presence of a strongly symmetrically coupled vibration, J. Chem. Phys., 108, 3620– 3625, 1998. 27 Marcus, R. A. and Sutin, N., Electron transfers in chemistry and biology, Biochim. Biophys. Acta, 811, 265– 322, 1985. 28 Warshel, A. and Villa-Freixa, J., Comment on “effect of active site mutation Phe93- . Trp in the horse liver alcohol dehydrogenase enzyme on catalysis: a molecular dynamics study”, J. Phys. Chem. B, 107, 12370– 12371, 2003. 29 Villa, J. and Warshel, A., Energetics and dynamics of enzymatic reactions, J. Phys. Chem. B, 105, 7887– 7907, 2001. 30 Bruno, W. J. and Bialek, W., Vibrationally enhanced tunneling as a mechanism for enzymatic hydrogen transfer, Biophys. J., 63, 689– 699, 1992. 31 Cameron, C. E. and Benkovic, S. J., Evidence for a functional role of the dynamics of glycine-121 of Escherichia coli dihydrofolate reductase obtained from kinetic analysis of a site-directed mutant, Biochemistry, 36, 15792– 15800, 1997. 32 Caratzoulas, S. and Schwartz, S. D., A computational method to discover the existence of promoting vibrations for chemical reactions in condensed phases, J. Chem. Phys., 114, 2910– 2918, 2001. 33 Antoniou, D. and Schwartz, S. D., Internal enzyme motions as a source of catalytic activity: ratepromoting vibrations and hydrogen tunneling, J. Phys. Chem. B, 105, 5553– 5558, 2001. 34 Antoniou, D., Caratzoulas, S., Kalyanaraman, C., Mincer, J. S., and Schwartz, S. D., Barrier passage and protein dynamics in enzymatically catalyzed reactions, Eur. J. Biochem., 269, 3103– 3112, 2002. 35 Caratzoulas, S., Mincer, J. S., and Schwartz, S. D., Identification of a protein-promoting vibration in the reaction catalyzed by horse liver alcohol dehydrogenase, J. Am. Chem. Soc., 124, 3270– 3276, 2002. 36 Mincer, J. S. and Schwartz, S. D., A computational method to identify residues important in creating a protein promoting vibration in enzymes, J. Phys. Chem. B, 107, 366– 371, 2003. 37 Schwartz, S. D., Response to comment on “effect of active site mutation Phe93 - . Trp in the horse liver alcohol dehydrogenase enzyme on catalysis: a molecular dynamics study”, J. Phys. Chem. B, 107, 12372, 2003. 38 Berman, H. M., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T. N., Weissig, H., Shindyalov, I. N., and Bourne, P. E., The protein data bank, Nucleic Acids Res., 28, 235–242, 2000. 39 Doll, K. M., Bender, B. R., and Finke, R. G., The first experimental test of the hypothesis that enzymes have evolved to enhance hydrogen tunneling, J. Am. Chem. Soc., 125, 10877 –10884, 2003. 40 Doll, K. M. and Finke, R. G., A compelling experimental test of the hypothesis that enzymes have evolved to enhance quantum mechanical tunneling in hydrogen transfer reactions: the betaneopentylcobalamin system combined with prior adocobalamin data, Inorg. Chem., 42, 4849– 4856, 2003. 41 Warshel, A. and Parson, W. W., Dynamics of biochemical and biophysical reactions: insight from computer simulations, Q. Rev. Biophys., 34, 563– 679, 2001. 42 Siebrand, W. and Smedarchina, Z., Temperature dependence of kinetic isotope effects for enzymatic carbon-hydrogen bond cleavage, J. Phys. Chem., 108, 4185– 4195, 2004. 43 Krishtalik, L. I., Charge Transfer Reactions in Electrochemical and Chemical Processes, Consultants Bureau, New York, 1986, pp. 244–298. 44 Field, M. J., Bash, P. A., and Karplus, M., A combined quantum-mechanical and molecular mechanical potential for molecular-dynamics simulations, J. Comput. Chem., 11, 700– 733, 1990. 45 Warshel, A. and Levitt, M., Theoretical studies of enzymatic reactions: dielectric electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme, J. Mol. Biol., 227– 249, 1976. 46 Ranaghan, K. E., Ridder, L., Szefczyk, B., Sokalski, W. A., Hermann, J. C., and Mulholland, A. J., Insights into enzyme catalysis from QM/MM modeling: transition state stabilization in chorismate mutase, Mol. Phys., 101, 2695– 2714, 2003. 47 Torrie, G. M. and Valleau, J. P., Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling, J. Comput. Phys., 23, 187– 199, 1977. 48 Kohen, A. and Klinman, J. P., Hydrogen tunneling in biology, Chem. Biol., 6, R191– R198, 1999. 49 Zwanzig, R. W., High-temperature equation of state by a perturbation method. I. Nonpolar gases, J. Chem. Phys., 1420– 1426, 1954.

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Isotope Effects in Chemistry and Biology 50 Warshel, A., Computer Modeling of Chemical Reactions in Enzymes and Solutions, Wiley, New York, 1991. 51 Gao, J. and Truhlar, D. G., Quantum mechanical methods for enzyme kinetics, Annu. Rev. Phys. Chem., 467– 505, 2002. 52 Truhlar, D. G., Isaacson, A. D., and Garret, B. C., Generalized transition state theory, In Theory of Chemical Reaction Dynamics, Vol. IV, Baer, M., Ed., CRC Press, Boca Raton, FL, pp. 65 – 136, 1985. 53 Truhlar, D. G., Gao, J., Alhambra, C., Garcia-Viloca, M., Corchado, J., Sa´nchez, M. L., and Villa`, J., The incorporation of quantum effects in enzyme kinetics modeling, Acc. Chem. Res., 35, 341– 349, 2001. 54 Frisch, M. J, Trucks, G. W, Schlegel, H. B, Gill, P. M. W, Johnson, B. G., Robb, M. A., Cheeseman, J. R., Keith, T., Petersson, G. A., Montgomery, J. A., Raghavachari, K., Al-Laham, M. A., Zakrzewski, V. G., Ortiz, J. V., Foresman, J. B., Cioslowski, J., Stefanov, B. B., Nanayakkara, A., Challacombe, M., Peng, C. Y., Ayala, P. Y., Chen, W., Wong, M. W., Andres, J. L., Replogle, E. S., Gomperts, R., Martin, R. L., Fox, D. J., Binkley, J. S., Defrees, D. J., Baker, J., Stewart, J. P., Head-Gordon, M., Gonzalez, C., and Pople, J. A., GAUSSIAN 94, Gaussian Inc., Pittsburgh, 1995. 55 Pearlman, D. A., Case, D. A., Caldwell, J. W., Ross, W. S., Cheatham, T. E., Ferguson, D. M., Seibel, G. L., Singh, U. C., Weiner, P. K., and Kollman, P. A., AMBER, University of California, San Francisco, 1995. 56 Stewart, J. J. P., Optimization of parameters for semiempirical methods I. Method, J. Comput. Phys., 10, 209–220, 1989. 57 Singh, U. C. and Kollman, P. A., A combined ab initio quantum mechanical and molecular mechanical method for carrying out simulations on complex molecular systems: applications to the CH3Cl þ Cl2 exchange reaction and gas phase protonation of polyethers, J. Comput. Chem., 7, 718– 730, 1986. 58 Maseras, F. and Morokuma, K., A new ab initio þ molecular mechanics geometry optimization scheme of equilibrium structures and transition states, J. Comput. Chem., 16, 1170– 1179, 1995. 59 Page, M. and McIver, J. W., On evaluating the reaction-path hamiltonian, J. Chem. Phys., 88, 922– 935, 1988. 60 Steckler, R., Chuang, Y. Y., Fast, P. L., Coitino, E. L., Corchado, J., Hu, W. P., Liu, Y. P., Lynch, G. C., Nguyen, K. A., Jackels, C. F., Gu, M. Z., Rossi, I., Clayton, S., Melissas, V. S., Steckler, R., Garrett, B. C., Isaacson, A. D., and Truhlar, D. G., POLYRATE, University of Minnesota, Minneapolis, 1997. ¨ ber das u¨bershrieten von Potentialschwellen bei Chemischen Reaktionen, Z. Phys. 61 Wigner, E. P., U Chem. Abt. B, 19, 203, 1932. 62 Liu, Y. P., Lynch, G. C., Truong, T. N., Lu, D. H., Truhlar, D. G., and Garrett, B. C., Molecular modeling of the kinetic isotope effect for the [1,5] sigmatropic rearrangement of cis-1,3-pentadiene, J. Am. Chem. Soc., 115, 2408– 2415, 1993. 63 Gonza´lez-Lafont, A., Truong, T. N., and Truhlar, D. G., Direct dynamics calculations with neglect of diatomic differential overlap molecular orbital theory with specific reaction parameters, J. Phys. Chem., 95, 4618– 4627, 1991. 64 Alhambra, C., Corchado, J., Sanchez, M. L., Garcia-Viloca, M., Gao, J., and Truhlar, D. G., Canonical variational theory for enzyme kinetics with the protein mean force and multi-dimensional quantum mechanical tunneling dynamics. Theory and application to liver alcohol dehydrogenase, J. Phys. Chem. B, 105, 11326– 11340, 2001. 65 Garcia-Viloca, M., Alhambra, C., Truhlar, D. G., and Gao, J. L., Hydride transfer catalyzed by xylose isomerase: mechanism and quantum effects, J. Comput. Chem., 24, 177– 190, 2003. 66 Garcia-Viloca, M., Truhlar, D. G., and Gao, J., Reaction-path energetics and kinetics of the hydride transfer reaction catalyzed by dihydrofolate reductase, Biochemistry, 42, 13558– 13575, 2003. 67 Zhang, Y. K., Kua, J., and McCammon, J. A., Influence of structural fluctuation on enzyme reaction energy barriers in combined quantum mechanical/molecular mechanical studies, J. Phys. Chem. B, 107, 4459– 4463, 2003. 68 Alhambra, C., Corchado, J., Sanchez, M., Gao, J., and Truhlar, D., Quantum dynamics of hydride transfer in enzyme catalysis, J. Am. Chem. Soc., 122, 8197– 8203, 2000.

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26

Proton Transfer and Proton Conductivity in Condensed Matter Environment Alexander M. Kuznetsov and Jens Ulstrup

CONTENTS I. II.

Introduction ...................................................................................................................... 691 Mechanisms of Elementary Proton Transfer between Molecular Fragments ................ 693 A. Basic PT Model at Fixed Donor/Acceptor Distance............................................... 693 B. The Born –Oppenheimer Approximation and Potential Free-Energy Surfaces ............................................................................................... 695 C. Totally Diabatic Proton Transfer............................................................................. 696 D. Partially Adiabatic Proton Transfer......................................................................... 697 E. Totally Adiabatic Proton Transfer........................................................................... 698 F. General Expressions for the Tunnel Transmission Coefficient and Transition Probability. The Environmental Medium Dynamics...................... 698 G. Fluctuations of the Interreactant Distance and Gated Proton Transfer .................. 700 H. Free Energy Relations and Kinetic Isotope Effects ................................................ 701 III. Proton Transfer in Hydrogen-Bonded Systems............................................................... 702 A. Hydrogen Bonds with Double-Well Proton Potentials ........................................... 703 B. Excess Aqueous Proton Conductivity and Proton Transfer.................................... 705 1. Proton Hops between Two Water Molecules ................................................... 706 2. Short-Range Proton Transfer via Adjacent Zundel Complexes....................... 706 3. Long-Range Proton Transfer via Remote Zundel Complexes ......................... 707 B. Proton Transfer in Single-Well Proton Potentials................................................... 709 IV. Electron-Coupled Proton Transfer................................................................................... 710 A. Mechanisms of Dynamic and Step-Wise Coupling ................................................ 711 B. A View on Coherent Two-Proton Transfer in Zundel Complexes......................... 714 C. Models and Mechanisms of Electron-Coupled Proton Transfer (ECPT) ............... 715 1. Diabatic States................................................................................................... 716 2. Mechanisms of Transitions and Rate Constants............................................... 718 D. Synchronous Electron and Proton Transfer............................................................. 720 V. Concluding Remarks........................................................................................................ 720 Acknowledgments ........................................................................................................................ 722 References..................................................................................................................................... 722

I. INTRODUCTION The ubiquity and importance of proton transfer (PT) as an “elementary” reaction step are equaled only by those of electron transfer (ET). PT and ET show both close physical and formal analogies, 691

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and important differences. Light particles, with conspicuous quantum mechanical (tunneling) features at room temperature are transferred between much heavier molecular fragments in both cases. Both particles are also strongly coupled to the polar or apolar environment (solvent, protein). The view of vibrationally assisted proton tunneling was introduced early.1 – 4 Although not always explicitly recognized, most later approaches to condensed matter proton tunneling rest crucially on these early views. An important difference between ET and PT is that PT involves synchronous bond breaking and formation, whereas the ET donor and acceptor preserve their integrity throughout the ET process. ET can, moreover, involve long distances, significantly exceeding the structural donor and acceptor extension. PT is only feasible over much shorter distances, say a ˚. fraction of an Angstrom or so, even though the equilibrium PT distances are longer, say up to 1 A The short-range nature of PT and hydrogen atom transfer is reflected in other physical differences. “Gating” of the central particle “hop”, i.e., fluctuational shortening of the transfer distance by hindered translational donor – acceptor motion, is much more important for PT than for ET. Thermal activation along this mode is commonly a contributor to the overall activation free energy, and is reflected in the temperature coefficient of the kinetic isotope effect (KIE), caused by different optimum transfer distances of the light (longer transfer distance) and heavy isotopes (shorter transfer distance). A consequence of the short PT distances is, finally, that the physical donor – acceptor interaction is much stronger than in ET processes. This implies that PT processes mostly belong to the adiabatic limit of strong interaction between the donor and acceptor (Section II and Section III). The energy barrier in the double-well proton potential is also strongly distorted (lowered), concealing to an extent the quantum mechanical proton behavior. In the limit of very strong hydrogen bonding the proton barrier vanishes altogether, and PT is entirely controlled by environmental dynamics. This limit may prevail in one of the most central chemical PT processes, excess proton conductivity in aqueous solution.5,6 PT is the central physical event in acid-base reactions and broadly in general acid-base catalysis.7 – 11 PT has a role in the history of solution chemical reactions as the basis for introduction of the Brønsted relation.12 PT is also central in a variety of biological processes, particularly hydrolytic enzyme processes (peptidases and ligases13 – 20), and mechanisms of proton transport through membrane-spanning protein complexes.21 – 23 PT also accompanies ET in coupled patterns in redox enzyme function such as oxidase and dehydrogenase function.24 – 27 If extended to hydrogen atom and hydride transfer, the vibrationally assisted proton/hydrogen atom tunneling phenomenon appears in other, largely metal-containing (Fe, Zn, Ni) oxidases and dehydrogenases.28 – 32 PT or hydrogen atom transfer is, finally, a probe for nuclear tunneling in “physical” processes. Hydrogen diffusion in metals 33,34 is a prime example. Correlated proton tunneling in solid-state hydrogen-bonded networks of amorphous solids,35 – 37 reflected in heat capacity and infrared spectral broadening, are other cases. The view of chemical PT as a fundamental environmentally induced quantum-mechanical phenomenon 1 – 4 could also be bridged early with experimental support from free energy relations (the Brønsted relation) and KIE.38 – 40 Environmental nuclear reorganization was, for example, noted to control free energy relations, and the KIE to display a volcano-like free energy dependence, with maximum close to, but not necessarily coinciding with, thermoneutrality. It was also recognized that PT involving strongly hydrogen-bonded O- and N-acids and bases is characterized by small nuclear reorganization and weakly pronounced tunneling.38 – 40 Strong intramolecular reorganization accompanies PT between weakly hydrogen-bonded C-acids and -bases, which also commonly display much larger KIE. Hydrogen atom transfer at cryogenic temperatures has offered other cases for vibrationally assisted nuclear tunneling.41 – 43 Common patterns are large KIE and a temperature pattern, where the rate constant follows Arrhenius-like behavior at high and temperature independence at low temperatures. The large KIE, sometimes orders of magnitude, reflects clearly tunneling of the hydrogen atom dynamics, caused by freezing of the solvent matrix, with large hydrogen atom transfer distances. The transition temperature between Arrhenius and temperature independent

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behavior around 100 K,41 – 43 however, points to another dynamic feature, i.e., quantum-mechanical “freezing” of the “gating mode” of the hydrogen atom transfer process. These observations thus reflect heavy atom group tunneling.3,7,41 Analysis of other hydrogen and hydride transfer systems based, for example, in transition metal-based homogeneous catalysis has also been reported.44,45 Comprehensive recent studies of kinetic deuterium and tritium isotope effects in enzymecatalyzed hydrogen atom transfer processes, particularly by J.P. Klinman and associates, have disclosed a wealth of novel detail. These studies are reviewed in other chapters but the following summaries key notions: 30,31 (1) The studies cover several metallo- and nonmetallo-enzymes (lipoxygenase, alcohol dehydrogenase, amine-oxidase, and -dehydrogenase), as well as mutant enzymes. (2) Hydrogen atom transfer is rate determining, enabling to associate hydrogen tunneling directly with experimental rate constants and KIE. (3) The kinetic deuterium isotope effects are large, sometimes up to two orders of magnitude. (4) Kinetic deuterium and tritium isotope effects were systematically applied together with mutant proteins, disclosing details in the correlation between thermally activated environmental gating and the central hydrogen tunneling event. Along the same lines, the temperature dependence of the KIE disclosed different tunneling distances of hydrogen, deuterium, and tritium, reflecting the strong tunneling mass dependence. Focus in the present chapter is on parallel recent development in another area of PT science, namely PT in strongly hydrogen-bonded systems. The novel radical-based enzyme processes thus involve “weak” proton/hydrogen atom donor/acceptor interactions giving particularly large values of the KIE. Strongly hydrogen-bonded PT systems are encountered no less ubiquitously. A primary case addressed is excess proton conductivity in aqueous solution.5,6 Quantum-mechanical tunneling features are much less conspicuous in these systems, which represent, however, theoretical challenges competitive with those in weakly interacting hydrogen atom transfer enzyme systems.

II. MECHANISMS OF ELEMENTARY PROTON TRANSFER BETWEEN MOLECULAR FRAGMENTS A. BASIC PT M ODEL AT F IXED D ONOR /ACCEPTOR D ISTANCE The ubiquitous occurrence of multifarious PT processes can be grouped into a small number of physically different “elementary” PT patterns. These are largely based on distinction between “weak” and “strong” interactions between the proton donating and accepting molecular species, and their translational mobility, which also control the gated nature of the PT event. Focus in this chapter is on “relatively strongly” interacting proton donor and acceptor entities, but the basic features are first illuminated by the opposite limit of “very weak” interactions. We thus address first PT between two spatially fixed molecular entities, with no hydrogen bonds neither with each other nor with the environment. This view is heuristic but perhaps representative of hydrogen atom transfer in enzyme systems where the protein imposes a rigid structure at the active site, cf., above. One of the central features of this type of system is the existence of two separate proton potential wells, one representing the donor, the other the acceptor molecule (Figure 26.1). A “high” potential barrier, i.e., significantly above the vibrational zero-point energies along the proton stretching mode, separates the potential wells, even at small donor –acceptor distance. Variation of the interreactant distance (PT gating) affects the barrier height and shape but does not change the double-well feature. Two proton location sites can therefore clearly be distinguished. The first consistent quantum-mechanical model of PT in polar media was for systems of this type,1 – 4 concepts and notions of which are discussed below. Two points are crucial: (1) The protons as well

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∆E E0(q)

E0(q)

rp

FIGURE 26.1 Double-well potential free-energy projections along the proton coordinate, rp ; at different values of the environmental coordinates, represented by q. The proton is trapped at the donor when q is near initial equilibrium, at the acceptor when q is near final equilibrium.

as the electrons, which provide the chemical bonding of the proton to the molecular donor and acceptor are quantum particles. (2) The charge distribution of the donor – acceptor complex (located on the proton) interacts strongly with the environment behaving largely as a classical subsystem. These observations have the consequences: 1. The quantum particle motion is much faster than the low-frequency medium polarization fluctuations. The two subsystems can therefore be separated in a Born – Oppenheimer scheme. 2. Polaron-like effects, i.e., self-trapping of the proton in the donor (initial, i) and acceptor (final, f ) state, are crucial. The charge distribution for given proton localization polarizes the medium and the latter in turn stabilizes the proton state, lowering the energy of the latter. A further consequence is that, the average medium polarization corresponding to proton localization at the donor destabilizes the vacant proton state at the acceptor, and vice versa. 3. The proton can only occupy discrete quantized states mi and nf in the donor and acceptor potential, respectively. Transitions from any initial, mi to any final proton vibrational state, nf are therefore possible only if the state energies match (the Franck –Condon principle, Figure 26.2) ð26:1Þ

1m i ¼ 1n f

U

Em

i

∆Emn

En

f

rp

FIGURE 26.2 Double-well proton potential and proton vibrational resonance splitting at the saddle point with respect to the environmental nuclear coordinates.

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4. Due to strong interactions with the molecular environment the proton energy levels depend strongly on configurational fluctuations of the medium molecules. The proton energy levels are brought to resonance by suitable (multitudinous), thermally activated medium-polarization fluctuations away from initial equilibrium (Figure 26.1). Environmental polarization thus creates the energy barrier separating the initial and final states (Franck –Condon barrier). Thermal fluctuations overcome the barrier and determine the activation free energy. 5. In the limit of weak interaction, double-well PT between initial and final vibrational states (while at resonance) is by tunneling along the proton coordinate, most conspicuously reflected in the preexponential factor (transmission coefficient) of the rate constant.

B. THE B ORN – OPPENHEIMER A PPROXIMATION AND P OTENTIAL F REE- E NERGY S URFACES The chemical bonds between the proton, donor, and acceptor molecules arise by interaction with the electronic subsystems of the molecules. The electron density distribution thus differs in the initial and final states, with the electrons remaining localized within the PT complex. The electron subsystem is thus crucial in all facets of the PT process. One of the most effective theoretical approaches to composite molecular systems is, secondly, the Born –Oppenheimer approximation (BOA), which separates the faster subsystem from the slower ones, and is widely used in ET and PT theory. The BOA is broadly applicable, but deviations from the BOA do cause a slight calculated rate-constant decrease.46 The BOA in PT involves two steps: The electron subsystem is first separated from the proton and molecular environment. In a second step the proton is separated from the heavy nuclear medium molecular dynamics. Two approaches, based on adiabatic and diabatic electronicvibrational basis sets, can be applied on equal footing.47 The diabatic approach is conceptually the simpler and convenient for discussion of the elementary PT scenario. In the first step of the BOA two electronic wave functions, wi ðx; rp ,qÞ and wf ðx; rp ,qÞ; and energies, 1ei ðrp ,qÞ and 1ef ðrp ,qÞ, are introduced for the initial (the proton bound to the donor) and final (the proton bound to the acceptor) states, at fixed proton coordinate rp and solvent polarization fluctuation coordinates q ¼ {qk }: The electronic energies 1ei ðrp ,qÞ and 1ef ðrp ,qÞ added to the free energy of the molecular environment, U0 ðrp ,qÞ, and the energy of proton interaction with the molecular fragment and solvent, constitute the diabatic free energy surfaces (FES) Ui ðrp ,qÞ and Uf ðrp , qÞ: PT is a transition from the neighborhood of the minimum rp0i, q0i of the FES of the initial state Ui ðrp , qÞ to the relaxed state near the minimum rp0f, q0f of the final state Uf ðrp , qÞ (Figure 26.1 and Figure 26.3). Motion on the FES is composite as the path involves both quantum mechanical tunneling along the proton coordinate rp, and classical motion along the coordinates q.1 – 3 Crossing of the diabatic FES Ui ðrp , qÞ ¼ Uf ðrp , qÞ corresponds to electronic energy level resonance. 1ei ðrpp ; qp Þ ¼ 1ef ðrpp ; qp Þ

ð26:2Þ

Equation 26.2 applies when the classical solvent configurational fluctuations have taken the coordinates, q, to the transition configuration, qp, and the proton mode initiated subbarrier tunneling to reach the value rpp : Electronic reorganization can only occur here. The second step in the BOA is the separation of the proton and heavy medium molecules. Diabatic proton wave functions, xmi ðrp , qÞ and xnf ðrp , qÞ and vibrational energies, Emi ðqÞ and Enf ðqÞ, which depend on the polarization coordinates q are then introduced, at fixed q-coordinates. Together with U0 ðqÞ; Emi ðqÞ; and Enf ðqÞ constitute the “reduced” diabatic free energy surfaces (RFES), Umr i ðqÞ, and Unrf ðqÞ, which describe the system free energy dependence solely on the classical coordinates q. Each RFES corresponds to a different pair of proton vibrational states, mi and nf, in the initial and final state, respectively.

696

Isotope Effects in Chemistry and Biology U Ui

Uf ∆E Uad

q

FIGURE 26.3 Reduced partially or totally adiabatic free-energy surface along the environmental coordinate(s) q.

Motion along the RFES is purely classical. Crossing at Umr i ðqÞ ¼ Unr f ðqÞ

ð26:3Þ

corresponds to matching, or resonance between the proton energies, Figure 26.1 and Figure 26.2 Emi ðqÞ ¼ Enf ðqÞ

ð26:4Þ

The BOA and Franck –Condon principle have thus been generalized and used twice. The electronic motion is first separated from the totality of (proton and heavy) nuclear motion, by the “normal” BOA and Franck –Condon principle. The fast, quantum-mechanical nuclear proton motion is next separated from the classical environmental nuclear motion. A general expression for the probability of proton transition from a given initial, mi, to a given final vibrational state, nf has the form 1 – 3,7 Wmn ¼

mn veff k e2Ga =kB T 2p mn

ð26:5Þ

veff is the effective frequency of the whole medium polarization spectrum, Gami nf the activation free energy, and kminf the electron – proton transmission coefficient, both addressed below. The transition path usually passes the saddle point qs at the crossing of the reduced free energy surfaces (Equation 26.3). Gmn a is the difference between the free energy at the saddle point and at the initial equilibrium configuration q0i. The proton zero-point energies in the initial and final states are included in the diabatic free energies. kmn determines the probability of electron and proton reorganization, i.e., both fast subsystems, when the system passes the transition configuration qs. kmn is determined by the resonance splitting DEmn of the proton energy levels Emi and Enf (Figure 26.2). Three cases can be distinguished.3,7,48

C. TOTALLY D IABATIC P ROTON T RANSFER If proton binding to a molecule leads to significant electron localization changes, the overlap of the electron wave functions of the initial and final states is small. The electron exchange matrix element, Vif, coupling the initial and final states (electron resonance integral) is therefore also small. As noted, this limit may have a heuristic character, but hydrogen atom transfer in enzymes or frozen glasses 28 – 32,41 – 43 may approach this limit such as suggested by the huge KIE. Electronic

Proton Transfer and Proton Conductivity in Condensed Matter Environment

697

reorganization in the PT process is then hampered. The resonance splitting of the proton energy levels is 1 – 3 diab mn DEmn < 2Vep ¼ 2Vfi kxmi lxnf l

ð26:6Þ

mn where Vep is the electron – proton resonance integral represented approximately as the product of the electron resonance integral Vif and the overlap of the proton wave functions. This equation emerges out of quantum-mechanical perturbation theory. diab If DEmn is small (cf., below) the reaction is denoted as totally diabatic (or “nonadiabatic”), because the electrons cannot follow the tunnel motion of the proton, and the proton cannot follow the molecular environment through the transition configuration. Gmn a;diab and kmn in this limit have the form 1 – 3

Gmn a;diab

¼ Er þ

DG0mn

2

=4Er ;

kdiab mn

¼

diab DEmn 2

!2

4p3 2 2 " v kB TEr

!1=2 ð26:7Þ

Er is the reorganization free energy of the environment, and DG0mn the reaction free energy, DG0mn ¼ DG000 þ ðEnf 2 Emi Þ

ð26:8Þ

where Emi and Enf are the proton vibrational energies counted from the ground states. The latter are included in the reaction free energy between the ground states, DG000 : The validity criteria for the totally diabatic limit are 3,7 2pVif2 ="lvp k›Ui ðr; qp Þ=›r 2 ›Uf ðr; qp Þ=›rl ! 1 diab DEmn 2

!2

4p3 2 2 " v kB TEr

ð26:9Þ

!1=2 !1

ð26:10Þ

The first inequality states that the electrons cannot follow the proton tunnel motion, the second that both fast subsystems, electrons and the proton, cannot follow the environmental motion. vp ¼ ½Erp =2mp 1=2 is the (imaginary) proton velocity in the tunneling barrier region, mp the proton mass and Erp the proton reorganization energy along the proton mode rp : Erp is determined by the proton vibrational frequency along rp ; i.e., Vp ; and by the equilibrium values of rp in the initial, rpi0 ; and final state, rpf0, Erp ¼ 12 mp Vp ½rpf0 2 rpi0 2 when the proton double-well potential is represented by two displaced harmonic modes, and Equation 26.9 reduces to ðVif Þ2 ! "Vp Erp

ð26:11Þ

D. PARTIALLY A DIABATIC P ROTON T RANSFER This limit accords with small electron density redistribution on proton chemical bonding. The electron resonance integral is then no longer small. The inequality opposite to Equation 26.9 applies, while Equation 26.5 still applies with another meaning of the parameters. The PT process is still conspicuously a tunneling process and follows Equation 26.10 due to small overlap of the proton vibrational wave functions. The electrons thus follow adiabatically the proton, whereas the latter cannot follow the motion of the medium polarization. Hence, the notion “partially adiabatic”.

698

Isotope Effects in Chemistry and Biology

The resonance splitting DEmn remains small and takes the form 3,7,48 ad DEmn ¼

"Vp 1 ð drp ½2mp ðUmn ðrp ; qp Þ 2 Em Þ exp 2 " p

1=2

ð26:12Þ

where Umn ðrp ; qÞ is the lower adiabatic free energy surface constructed from the diabatic surfaces, ad cf., below. As Equation 26.10 still holds, Equation 26.7 remains valid with the substitution of DEmn of Equation 26.12 for DEmn : The partially adiabatic limit prevails commonly to chemical and biological proton and hydrogen atom transfer processes with KIE, significantly exceeding the square root of the isotope mass ratio.40

E. TOTALLY A DIABATIC P ROTON T RANSFER If also the inequality opposite to Equation 26.10 applies, the transition is denoted as totally adiabatic, because now the electrons follow the tunnel motion of the proton and the latter the classical medium polarization. The system is displaced along the lower adiabatic free energy surface (Figure 26.3) constructed from the diabatic surfaces Ur ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ad 2 Umr i þ Unr f 2 ðUmr i 2 Unrf Þ2 þ ½DEmn 2

ð26:13Þ

The transmission coefficient is unity

kad mn ¼ 1

ð26:14Þ

The activation free energy is approximately 3,7 pffiffiffiffiffiffiffiffiffiffiffiffi mn ad Gmn a;ad ¼ Ga;diab 2 DEmn að1 2 aÞ

ð26:15Þ

where a is the symmetry factor

a¼

›Umr i ›qk

›Umr i ›Unr f 2 › qk ›qk

ð26:16Þ

The activation free energy is lower than for diabatic and partially adiabatic transitions by a quantity determined by the resonance splitting of the proton levels. The totally adiabatic limit applies to many hydrogen-bonded PT systems in general acid-base reactions or hydrolytic enzyme processes with PT between mobile O- and N-donor and acceptor groups. In addition to inertial mass effects reflected in the effective solvent vibrational frequencies the KIE is, interestingly, determined by the activation free energy difference caused by different tunneling splittings of the light and heavy isotopes. PT is thus still a tunneling process, even though a preexponential tunneling factor is absent in the rate constant.

F. GENERAL E XPRESSIONS FOR THE T UNNEL T RANSMISSION C OEFFICIENT AND T RANSITION P ROBABILITY. T HE E NVIRONMENTAL M EDIUM DYNAMICS The transmission coefficient is in general the averaged Landau – Zener probability, Pmn LZ ; of the transition from the initial to the final reduced diabatic free energy surface on passage through

Proton Transfer and Proton Conductivity in Condensed Matter Environment

699

the transition region. Classical ballistic surface motion gives Pmn LZ ¼

1 2 expð22pgmn Þ ; 1 1 2 expð22pgmn Þ 2

gmn ¼ ðDEmn =2Þ2 =2"v½kB TEr =p

1=2

ð26:17Þ

DEmn is given by either Equation 26.6 or Equation 26.12. Equation 26.17 extends to both the adiabatic and diabatic limits. The overall transition probability including transitions between arbitrary vibrational proton states has a simple form for the totally diabatic and partially adiabatic cases 1 – 3 W¼

1 X 2Em =kB T e i Wmn Zi m,n

ð26:18Þ

Zi is the vibrational proton partition function for the initial potential well. Summation is over all vibrational states. Transitions between the ground states dominate for thermoneutral reactions. Excited vibrational states gain importance toward the activationless and barrierless free energy regions.2,3,7,49 When PT between two vibrational states is fast, the rate may be limited by the diffusion flux in the wells along q. The rate constant then follows Kramers-type equations.50 – 52 The rate constant is formally identical to the corresponding equation in the totally adiabatic limit (see Equation 26.5 with k ¼ 1 and Ga:ad: of Equation 26.15 for Gmn a ). The following interpolation formula was suggested to cover both the diabatic and Kramers limits for transitions between the ground vibrational states 52 – 54 k¼

n:ad W00 WKramers n:ad W00 þ WKramers

ð26:19Þ

WKramers is the probability of stochastic diffusion-like transition in the double-well potential with the following form for a parabolic barrier with the frequency v p 7,50,52 parab WKramers

1 1 vp Gad ¼ exp 2 a t 2p v kB T

! ð26:20Þ

where t is the relaxation time for motion along the reaction coordinate. Extension to all vibrational states is complicated and does not reduce to a simple summation of terms for each pair of vibrational states. An algorithm has been suggested 55 and the rate constant obtained in the form of continued fractions for transition from the initial ground vibrational state to all final vibrational states. The algorithm takes into account the exhaustion of the initial potential well population due to transitions to low-lying vibrational states when the system moves towards the free energy surface crossing between the ground state and a given excited state. The continued fraction ends when one of the transitions reaches the adiabatic limit. Equation 26.6 and Equation 26.12 describe the proton level splitting DE in the limits of small and large electron exchange factor Vif : The equations can be generalized. For example, DE can be related to the Landau –Zener tunnel probability between two free energy surfaces along the proton coordinate as 56,57 qffiffiffiffiffiffiffi DE < ð"Vp =pÞ Ptunn: LZ 58,59 Ptunn: LZ for arbitrary values of the electron exchange factor is also available.

ð26:21Þ

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Isotope Effects in Chemistry and Biology

G. FLUCTUATIONS OF THE I NTERREACTANT D ISTANCE AND G ATED P ROTON T RANSFER A feature of chemical and biological PT reactions is the “gated” nature of the transitions, rooted in the fast decrease of the proton transition probability with increasing interreactant distance, R, between the proton donor and acceptor. The PT probability is therefore very sensitive to fluctuations in R. The PT distance corresponding to the equilibrium positions of the proton donor and acceptor is far too long for proton tunneling or reasonable KIE. When the reaction is slow, gating can be incorporated by averaging the transition probability over R,40,41 using various models for the interreactant potential. Parabolic and Morse-like potentials have been used.4,38 – 40 The harmonic potential enables, further averaging in the whole temperature range.3,7,41 Variation of the interreactant distance, R, gives roughly the following PT distance Dr DrðQÞ ¼ Dr0 þ gQ

ð26:22Þ

where: Q ¼ ðR 2 R0 Þðmv="Þ1=2 is the dimensionless interreactant distance counted from equilibrium R0 ; g ¼ ð"=mvÞ1=2 : Dr0 ¼ R0 2 rpD 2 rpA ; and rpD and rpA are the equilibrium bond lengths in D and A, respectively. If the interreactant potential has the form UðRÞ ¼

1 "vQ2 2

ð26:23Þ

the average transition probability between the ground vibrational states in the totally diabatic or partially adiabatic limit is 7 W¼

ðcþi1 c2i1

duGðuÞSðuÞ

ð26:24Þ

where GðuÞ alone after integration over u gives the transition probability in the absence of gating. SðuÞ represents the gating effect ð pffiffi SðuÞ ¼ 1= p gðuÞ dQ exp½2g2 ðuÞQ2 n o n o exp 2 12 ½mp Vp =" ½DrðQÞ 2 =exp 2 12 ½mp Vp =" ½DrðQ ¼ 0Þ 2 gðuÞ ¼ sinh GðuÞ ¼

"vs 2kB T

cosh

"vs ð1 2 uÞ "vs u cosh 2kB T 2kB T

1=2

ðDE=2Þ2 uDF00 E uð1 2 uÞ exp 2 þ r i"kB T kB T kB T

ð26:25Þ ð26:26Þ ð26:27Þ

In this form SðuÞ also extends to tunneling along the R-mode. In the high-temperature limit SðuÞ ¼ S ¼

ð"vs =2kB TÞ1=2 ½ð"vs =2kB TÞ þ ðmp Vp =2"Þg2 1=2 " # ð"vs =2kB TÞðmp Vp =2"ÞðDr0 Þ2 =exp½2ðmp Vp =2"ÞðDr0 Þ2 exp 2 ð"vs =2kB TÞ þ ðmp Vp =2"Þg2 W00 ¼ W00 ðR ¼ R0 ÞS

ð26:28Þ ð26:29Þ

Proton Transfer and Proton Conductivity in Condensed Matter Environment

701

This appealing form shows that gating involves thermally activated motion along the gating coordinate from equilibrium to a smaller value, combined with the ratio between the proton tunneling factor at this lower R-value and at the equilibrium value R ¼ R0 : The overall effect is favorable to proton tunneling, i.e., S . 1. The higher the temperature, the larger the gating effect, due to more facile interreactant distance variation. This seems crucial and much more important in PT than in ET since facile ET can proceed over much longer distances.

H. FREE E NERGY R ELATIONS AND K INETIC I SOTOPE E FFECTS PT reactions enjoy a historical status as the first basis for the notion of general acid-base catalysis and the dependence of chemical rate constants on the reaction free energy.7,12 Free energy relations and the KIE of chemical and biological PT processes are discussed extensively elsewhere [see Ref. 38 to Ref. 40, and other chapters]. We provide a brief summary here. Free energy relations are inherent in the formalism above, for the totally diabatic and partially adiabatic limits particularly in Equation 26.7, Equation 26.8, and Equation 26.18. Each term in Equation 26.18 gives a quadratic dependence of the logarithm of the transition probability on the reaction free energy. The variation of the overall transition probability is weaker due to contributions from different transitions, but with the proton ground vibrational states dominating for approximately thermoneutral reactions ðDG0 < 0Þ: The symmetry factor, cf., Equation 26.16 is defined as

a00 ¼ 2kB T

d ln W00 1 DG0 ¼ þ 2 2Er d DG0

ð26:30Þ

a00 ¼ 12 at DG0 ¼ 0 and tends linearly to zero and 1 as DG0 ! 2Er and Er ; respectively. The variation of the observed symmetry factor a is slower due to contributions from initial ða00 . 1=2Þ and final ða00 , 1=2Þ excited proton vibrational states when a00 deviates from 1/2. In totally adiabatic reactions the system moves along the adiabatic free energy surface of the ground state. Equation 26.15 with m ¼ n ¼ 0 may be used approximately in certain ranges of DG0 : More exact expressions are discussed in Section III.A. PT and hydrogen atom transfer are unique cases for direct observation of nuclear tunneling at room temperature, in the form of the kinetic deuterium and tritium isotope effects. The KIE is rooted in the mass sensitivity of particle tunneling. The particular importance in PT reactions is that the isotope mass ratios are larger than for any other element. As deuteron and triton tunneling is hampered relative to proton tunneling the “gated” hydron features emerge as the heavier isotopes tunnel over shorter distances, achieved at the cost of additional activation free energy along the gating mode. These effects reach a subtle balance and provide a major contribution to the temperature dependence of the KIE. The origin of the KIE is different in adiabatic and diabatic processes. The dominating effect of isotope substitution for totally diabatic and partially adiabatic reactions is best illustrated for approximately thermoneutral reactions dominated by the proton ground vibrational states. The transition probability is then kH < FðRpH ÞW00 ðRpH ÞDRH

ð26:31Þ

FðRH Þ is the probability of approach (binary distribution function) of the proton donor and acceptor along the gating mode R to a given distance RH ; and DRH the effective R-range. All the terms in Equation 26.31 are sensitive to isotope substitution. The effective PT distance RpH corresponds to the maximum of the product of the first two factors on the right-hand side of Equation 26.31. FðRH Þ at small RH usually decreases with decreasing RH ; whereas W00 decreases rapidly with increasing RH due to the sharp decrease of the proton wave function overlap in the

702

Isotope Effects in Chemistry and Biology

transmission coefficient kH 00 ; Equation 26.6 and Equation 26.7. The decrease is faster for the heavier isotopes, resulting in smaller effective interreactant distances R p. This effect is reflected in the isotope dependence of the activation free energy which is otherwise less isotope sensitive since the hydrogen transfer properties are already included as the zero-point energy difference in the initial and final states. The kinetic isotope effect for the 0 ! 0 transition is then KH=D <

vH FðRpH Þ kH00 DRH vD FðRpD Þ kD00 DRD

ð26:32Þ

p D Since in general RpD , RpH ; then FðRpD Þ , FðRpH Þ: However, kD 00 and the product FðRD Þk00 are larger than for deuterium and PT at the same interreactant distance, giving smaller KIE than for fixed R. R p can be estimated from specific intermolecular potential. Parabolic 4,38,40 and Morse potentials 38 have been used. Transitions involving excited vibrational proton states are less sensitive to isotope substitution due to the larger overlap and slower transfer distance dependence of the hydron wave functions. The KIE therefore decreases with increasing lDG0 l due to increasing contributions of excited state terms, Equation 26.18.38 – 40 The KIE for totally diabatic and partially adiabatic transitions is rooted in the transmission coefficient and the interreactant potential, but appears differently in totally adiabatic reactions. The transmission coefficient is here unity, and the KIE determined by the isotope sensitivity of the effective frequency v1ff and the activation free energy Gad: a ; say, in a deuterated medium. The activation free energy of the 0 ! 0 transition involves, further, the resonance splitting of the proton energy levels DE00 ; Equation 26.15, which is smaller for the heavier isotope, giving a higher activation barrier. The KIE in totally adiabatic transitions is thus

KH=D

" # vHeff dDE{að1 2 aÞ}1=2 ¼ D exp ; kB T veff

H D dDE ¼ DE00 2 DE00

ð26:33Þ

The symmetry factor depends on the reaction free energy DG0 according to Equation 26.30. The dependence of KH=D on DG0 therefore has a volcano-type form 3,7,38,40

KH=D

" ( vH dDE 12 ¼ D exp 2kB T v

DG0 Er

!2 )#1=2 ð26:34Þ

with a maximum at DG0 ¼ 0: Unlike diabatic transitions the KIE, further, decreases with increasing D H : and DE00 transfer distance due to the decrease of DE00

III. PROTON TRANSFER IN HYDROGEN-BONDED SYSTEMS The microscopic PT mechanisms discussed in Section II are general. Hydrogen-bonded donor and acceptor molecules, mutually or with the molecular environment, however, display other features not immediately covered by this formalism. A prime example is the excess proton conductivity in aqueous solution. The following observations are important for hydrogen-bonded systems with intermediate or strong hydrogen bonds:

Proton Transfer and Proton Conductivity in Condensed Matter Environment

703

1. Hydrogen bond formation between the donor and acceptor gives a strong decrease of the potential barrier along the proton coordinate. 2. Strong hydrogen bonds lead the barrier to disappear altogether, accommodating the proton in a single potential well. 3. The state of the hydrogen bond depends strongly on the interactions with the environment. Interaction with the nearest solvent molecules is thus crucial.

A. HYDROGEN B ONDS WITH D OUBLE- W ELL P ROTON P OTENTIALS We discuss first finite double-well PT systems. With some exceptions60 PT between O- and Nhydrogen-bonding molecules are mostly totally adiabatic. If the hydrogen bond is not strong, two proton potential wells separated by a barrier remain. The potential shape is strongly affected by the molecular environment, and in some cases this interaction alone creates the second potential well.61 PT in important cases is associated with the breaking/formation of the hydrogen bonds between the molecules of the PT complex and the nearest solvent molecules S1 þ D – Hþ … A … S2 ! S1 … D … Hþ – A þ S2

ð26:35Þ

This pattern resembles nucleophilic substitution.62 The PT scenario can be viewed as in Figure 26.4. The proton is initially located near the donor molecule in a deeper potential well than near the acceptor A. The latter forms a hydrogen bond with the solvent molecule S2 while S1 does not form a hydrogen bond with D. Synchronous shift of the solvent molecules (S1 toward D and S2 away from A) deforms the proton potential towards approximate symmetry at some (transition) configuration. The diabatic proton vibrational energy levels match in this configuration ðE0i ¼ E0f Þ and the proton density is equally distributed between D and A (Figure 26.2). Further motion of S1 and S2 in the same direction results in adiabatic shift of the proton toward A, formation of a hydrogen bond S1 … D, and breaking of the A … S2 bond. Other solvent molecules are included as an effective medium in the reorganization free energy similar as for ET reactions.3,7,40 The intrinsic properties of the proton appear in the shape of the lower adiabatic free energy surface spanned by the classical nuclear modes, Equation 26.12, through the resonance splitting DE of the diabatic proton energy levels E0i and E0f : A double-well adiabatic free energy surface Uad. can be constructed from the diabatic surfaces Uir ðx; y; R; {qk }Þ and Ufr ðx; y; R; {qk }Þ; Equation 26.13 since the notion of two diabatic proton states (representing each potential well) approximately applies. (i) S1

D

(f) S1

D

H

+

H

+

A

S2

A

S2

FIGURE 26.4 “Elementary” PT scenario. The solvent molecule S2 is initially (i) hydrogen bonded to the proton acceptor A, while the solvent molecule S1 near the donor D is not hydrogen bonded. PT (f) is triggered by synchronous displacement of S1 and S2. S1 moves towards hydrogen bonding with D and S2 away from A, breaking the hydrogen bond to A.

704

Isotope Effects in Chemistry and Biology

The reactive nuclear modes for the mechanism represented by Equation 26.35 and Figure 26.4 include: (1) (2) (3) (4) (5)

The D – A distance R of the donor and acceptor molecules D and A The coordinate of the transferable proton rp along the line connecting D and A The distance x between S1 and D The distance y between S2 and A The coordinates {qk } of the effective oscillators describing the medium polarization outside the reaction complex.

The reduced initial state diabatic free energy surface is constructed as 6,62,63 Uir ðx; y; R; {qk }Þ ¼ Epi ðx; y; {qk }; RÞ þ Uisolv ð{qk }; x; yÞ þ Vi ðx; RÞ þ u2 ðy 2 y0i ðx; RÞÞ ð26:36Þ where Epi is the proton ground state eigenvalue in the donor molecule counted from the proton potential minimum. Vi ðx; RÞ includes the mutual donor/acceptor interaction and the interaction between S1 and the donor molecule at the initial equilibrium position y ¼ y0i ðx; RÞ of S2. u2 ðy 2 y0i ðx; RÞÞ is the vibrational potential of the hydrogen bond A … S2, and Uisolv ð{qk }; x; yÞ the free energy of the medium polarization. The reduced final state diabatic free energy surface Ufr ðx; y; R; {qk }Þ is, similarly Uir ðx; y; R; {qk }Þ ¼ Epf ðx; y; {qk }; RÞ þ Ufsolv ð{qk }; x; yÞ þ Vf ðy; RÞ þ u1 ðx 2 x0f ðy; RÞÞ ð26:37Þ The activation free energy is62,63 2 Gad: a ¼ a Er ðxs ; ys ; Rs Þ þ Vi ðxs ; Rs Þ 2 Vi ðx0 ; R0 Þ þ u2 ðys 2 y0i ðxs ; Rs ÞÞ

2

1 a DEðxs ; ys ; Rs Þ 2 12a

1=2

ð26:38Þ

where a is the symmetry factor, Er the outer-sphere solvent reorganization free energy, and xs, ys, Rs the saddle point coordinates on the adiabatic free energy surface. Equation 26.36 to Equation 26.38 are completely general for the mechanism of Equation 26.35. We need next to specify the potentials. We note first that Er depends on the charge transfer distance, i.e., on R. The potential Vi ðx; RÞ can, secondly, be approximately separated as Vi ðx; RÞ < Vi ðxÞ þ VDA ðRÞ

ð26:39Þ

The resonance splitting DE depends strongly (exponentially) on R while the dependence on x and y may be neglected. a depends on the reaction free energy (driving force) DG0 and varies between 0 and 1, but the quantity ½að1 2 aÞ 1=2 only varies weakly (by < 20% in the a-interval between 0.2 and 0.8) and may be taken approximately as 0.5. With these approximations Rs is independent of DG0 and determined by 62,63

›VDA ðRÞ 1 ›DE ¼ ›R 2 ›R

ð26:40Þ

Proton Transfer and Proton Conductivity in Condensed Matter Environment

705

Leaving only leading terms in the equations for xs and ys, we obtain ð1 2 aÞ

›Vi ðxÞ ›u þa 1 ¼0 ›x ›x

ð26:41Þ

ð1 2 aÞ

›u2 ›V ðyÞ þa f ¼0 ›y ›y

ð26:42Þ

DG0 ¼ ð2a 2 1ÞEr ðxs ; ys ; Rs Þ þ Vi ðxs Þ 2 Vf ðys Þ þ u2 ðys 2 y0i ðxs ; Rs ÞÞ ð2a 2 1Þ 2 u1 ðxs 2 x0f ðys ; Rs ÞÞ 2 DEðxs ; ys ; Rs Þ 2½að1 2 aÞ 1=2

ð26:43Þ

where the potential VDA ðRÞ is taken as the same in the initial and final states. Equation 26.38 and Equation 26.41 to Equation 26.43 give the dependence of the activation free energy on the driving force. The equations are parametric with a as running variable. Equation 26.41 and Equation 26.42 are solved for xs and ys at each a-value, and the values of xs and ys inserted in Equation 26.38 and Equation 26.43. We consider this approach for a particular PT case, i.e., excess proton conductivity in pure water.

B. EXCESS AQUEOUS P ROTON C ONDUCTIVITY AND P ROTON T RANSFER Excess proton conductivity in bulk water has a long history.5,6,64,65 Studies in modern times began in the early 20th century,66 – 71 comprehensively reviewed.71 – 78 Considerable theoretical progress based on molecular dynamics simulations at both classical and quantum mechanical levels, and on condensed matter charge transfer theory has been achieved.6,79 – 87 A key notion is that excess proton conductivity is carried by PT between a small number of distinguishable molecular entities. In a certain way excess aqueous proton conductivity then comes to stand forward as a “prototype” chemical PT process between strongly hydrogen-bonded donor and acceptor groups. In comparison with general acid-base reactions and other PT reactions, excess aqueous proton conductivity, however, displays distinctive features. The donor groups, i.e., the hydroxonium ion, H3Oþ, and the Zundel ion, H5Oþ 2 , cf., below represent distributions of variably hydrated, rapidly interconverting species. The distribution, and ultimately the nature of the species change as the temperature is varied. With a view on both classical excess proton mobility studies 66 – 77 and molecular dynamics and other simulations,80 – 88 single-PT between distinct molecular entities, however, appear to remain as an adequate view over limited temperature ranges. The individual PT steps are, however, fast and approach the relaxation times of solvent water molecules or molecular clusters, which may therefore not relax fully between the individual PT steps. This issue is addressed in Section IV.B. A final distinctive feature is that, the molecular donor and acceptor, i.e., the hydroxonium or Zundel complex and water, respectively, can be so strongly coupled that the PT potential is converted to a single-well potential in the transition configuration with respect to the local and bulk solvent molecules. The PT process is then entirely controlled by environmental solvent dynamics, and proton tunneling features vanish altogether. Taken together, molecular dynamics simulations, condensed matter charge transfer theory, and numerous experimental studies distinguish the following (groups of) mechanisms: 1. Classical, “vehicular” diffusion of the hydroxonium ion H3Oþ as a whole. We shall not consider this mechanism further. 2. Proton hops from the hydroxonium ion, H3Oþ, to a water molecule.

706

Isotope Effects in Chemistry and Biology

W1

W1

WD WD WA WA

W2 (i)

W2 (f)

FIGURE 26.5 PT in the H3Oþ mechanism of excess aqueous proton conductivity. This pattern follows the “generic” pattern in Figure 26.4.

3. “Short-range” PT via adjacent Zundel complexes. 4. “Long-range” proton transitions via remote Zundel complexes. 1. Proton Hops between Two Water Molecules An SN2 mechanism for PT between two water molecules was introduced and discussed recently6 as an extension of views of Voth et al.81 and Agmon5: ðH2 OÞW1 þ ðH2 OÞD 2 Hþ … ðH2 OÞA … ðH2 OÞW2 ) ðH2 OÞW1 … ðH2 OÞD … Hþ 2 ðH2 OÞA þ ðH2 OÞW2

ð26:44Þ

resembling the general mechanism of Equation 26.35. The transition is viewed as synchronous motion of two water molecules nearest to the donor – acceptor complex. Breaking/formation of two hydrogen bonds is accompanied by adiabatic proton tunnel shift from H3Oþ ion to an adjacent H2O molecule (Figure 26.5). Equation 26.37 and Equation 26.38 apply, with a ¼ 1/2 (Equation 26.42). Ga can be estimated using Morse-exponential potentials and Lennard– Jones potentials for the hydrogen bond A … W2 and nonbonded interaction W1/D, and for the O – O donor/acceptor interaction, respectively.6 Depending on the parameter values Ga varies between 0 and 0.27 eV. The experimental value is 0.11 eV.5,74,75 The analysis also points to significant gating along the O – O mode towards the transition configuration Rps prior to proton tunneling. 2. Short-Range Proton Transfer via Adjacent Zundel Complexes The second important structural entity is the Zundel complex with the proton located symmetrically between two water molecules and a considerably shorter O – O distance than in bulk water.78,79,83 PT via Zundel complexes implies that a given complex disappears and a new one with the nearest water molecule is formed (Figure 26.6), by synchronous motion of two water molecules, and displacement of two protons. The detailed dynamic behavior of other nearest water molecules is of minor importance and may be included in the outer-sphere medium polarization. The reduced

Proton Transfer and Proton Conductivity in Condensed Matter Environment

707

3 3 2

2

1 1 (i)

(f)

FIGURE 26.6 Structural diffusion via adjacent Zundel complexes. The initial Zundel complex (i) is composed of the proton located symmetrically between the water molecules 1 and 2. In the final-state Zundel complex (f) the proton is located symmetrically between the water molecules 2 and 3.

diabatic free energy surfaces can be written as i solv Uir ðR12 ; R23 ; {qk }Þ ¼ V23 ðR23 Þ þ u12 ðR12 2 R0i ð{qk }; R12 ; R23 Þ 12 ðR23 ÞÞ þ Ui

ð26:45Þ

f solv Ufr ðR12 ; R23 ; {qk }Þ ¼ V12 ðR12 Þ þ u23 ðR23 2 R0f 23 ðR12 ÞÞ þ Uf ð{qk }; R12 ; R23 Þ

ð26:46Þ

R12 and R23 are the distances between the oxygen atoms O(1), O(2), and O(3) of the water molecules of the PT complex (Figure 26.6). The first term in Equation 26.45 represents the interaction between the Zundel complex and nearest water molecule in the solvation shell, the second term the O(1) – O(2) vibrational potential, and the last term the inertial bulk medium polarization free energy. Analogous terms apply to the final state. The adiabatic free energy surface is determined by Equation 26.13 with DEmn replaced by the resonance splitting of two proton vibrational energy levels in the two-dimensional potential spanned by the coordinates of both protons, DEZ;short ðR12 ; R23 Þ: The resonance splitting depends exponentially on both R12 and R23. The activation barrier is calculated from6 GaZ;short ¼

1 1 i s Er ðRs12 ; Rs23 Þ þ V23 DEZ;short ðRs12 ; Rs23 Þ ðRs23 Þ þ u12 ðRs12 2 R0i 12 ðR23 ÞÞ 2 4 2

ð26:47Þ

where “s” denotes the saddle point. Morse-exponential forms give GaZ;short ¼

1 Z;short 1 E þ DZ;short 2 4 r 2

short DE0Z

short DE0Z 8 12 4DZ ; short

!

ð26:48Þ

short where: DZ,short is the dissociation energy of the short-range Zundel complex, DE0Z the maximum resonance splitting in this two-PT mechanism. The charge transfer distance a is longer for this mechanism than for PT mediated by the hydroxonium ion. The activation free energy is also higher, particularly due to the larger outersphere reorganization energy. This issue is addressed below.

3. Long-Range Proton Transfer via Remote Zundel Complexes Structural diffusion of the Zundel complex can also be mediated by PT in a competitive mechanism in which a given Zundel complex disappears and a new complex with the next nearest neighbor

708

Isotope Effects in Chemistry and Biology

3

3 2

2

(i)

4

4

(f)

1

1

FIGURE 26.7 Structural diffusion via remote, second-nearest Zundel complexes. The initial Zundel complex (i) consists of the proton and the water molecules 1 and 2. The hydrogen-bonded water molecular network is indicated by the water molecules 3 and 4. The final complex (f) is the proton between the water molecules 3 and 4.

water molecules is formed (Figure 26.7). We denote this mechanism as “PT via remote Zundel complexes”, or “long-range” PT via Zundel complexes. Three protons are transferred and three water molecules displaced synchronously. The reduced diabatic free energy surfaces including only nearest-neighbor interactions have the form i i Ui < ui12 ðr12 Þ þ V23 ðr23 Þ þ V34 ðr34 Þ

f f Uf < V12 ðr12 Þ þ V23 ðr23 Þ þ uf34 ðr34 Þ

i f V23 ðr23 Þ ; V23 ðr23 Þ

ð26:49Þ

With Morse-exponential molecular potentials u12 ¼ u34 ¼ DZ;long {1 2 exp½2gZ;long ðr 2 r0 Þ }2 V12 ¼ V34 ¼ DZ;long exp½22gZ;long ðr 2 r0 Þ

ð26:50Þ

long 0 0 Þ exp½2gZ;long ðr34 2 r34 Þ DEZ;long ¼ DE0Z exp½2gZ;long ðr12 2 r12 0 exp½2gZ;long ðr23 2 r23 Þ

V23 ¼ DZ;long exp{1 2 exp½2gZ;long ðr23 2

ð26:50aÞ 0 r23 Þ

2

}

the activation barrier becomes GaZ;long <

1 1 1 1 1 2 DEZ;long E þ DZ;long þ DZ;long D2Z;long þ D d2 4 r 2 8 16 23;long Z;long 2

ð26:51Þ

long long where DZ;long ¼ DE0Z =DZ;long ! 1 and dZ;long ¼ DE0Z =D23;long ! 1. The activation free energy for this transition is higher than both for PT via the hydroxonium ion and for short-range PT via Zundel complexes. The importance of the various mechanisms to the observable proton mobility, however, depends on several factors, of which we have focused on the elementary PT transitions. Other factors are:

1. The contribution to the diffusion coefficient of a given mechanism with PT distance an, is Dn ¼ Pn Wn a2n

ð26:52Þ

Proton Transfer and Proton Conductivity in Condensed Matter Environment

709

Pn is the fraction of the species of a given type “n” (hydroxonium ion, Zundel complex) and Wn the transition probability per unit time carried by mechanism “n”. This equation is valid if, following a given proton hop the appropriate structure for the next hop exists; otherwise reverse hopping occurs. Molecular dynamics simulations support this.85 Hops of this type are then determined by the probability that the corresponding structure is present. 2. Pn for the H3Oþ ions and Zundel complexes are of the same order at room temperature. 3. The factor a2n is considerably larger for long-distance than for short-distance hops. This compensates the smaller values of the transition probability Wn for the long-distance hops.

B. PROTON T RANSFER IN S INGLE- W ELL P ROTON P OTENTIALS The double-well proton potential along the PT coordinate reduces to a single-well potential in donor – acceptor systems with strong and short hydrogen bonds. PT in single-well potentials holds many of the features of double-well systems but differs in some respects. Most conspicuous is that the mere PT concept now preserves its meaning solely due to the molecular interactions with the solvent.88 Two different proton localization sites can still be distinguished. Formation of the A … S2 hydrogen bond and interaction of the nonbonded S1 molecule with D –Hþ produces a shift of the proton potential minimum toward D. Synchronous shift of S1 toward D and of S2 away from A results in adiabatic shift of the proton potential minimum from D to A, Figure 26.8. This is accompanied by S1 … D hydrogen bond formation and A … S2 hydrogen bond breaking. A potential barrier along the coordinates of the classical subsystem thus still separates the initial and final states. Here, the adiabatic approach should be used from the very beginning to construct the adiabatic free energy surfaces. The proton wave function x and ground-state energy E0 in the single-well potential are calculated at fixed positions of the molecules constituting the classical subsystem, using the BOA. The zero-point energy E0 counted from the single-well proton potential minimum energy, contributes to the effective potential Uad: ðx; y; R; {qk }Þ determining the classical subsystem

U D

p

Ep(s0i)

A

Ep(s∗)

0i rp

rp

FIGURE 26.8 Single-well potentials for the proton in a complex with a short hydrogen bond at the initial equilibrium (s0i) and at the transition configuration (sp) for a classical reactive mode. The proton remains in its ground vibrational state Ep, counted from the bottom of the potential well, in the course of the transition.

710

Isotope Effects in Chemistry and Biology

motion, with the minimum potential energy included. The adiabatic free energy surface takes the form Uad: ðx; y; R; {qk }Þ ¼ U0 ðx; y; R; {qk }Þ þ E0 ðx; y; R; {qk }Þ

ð26:53Þ

where U0 ðx; y; R; {qk }Þ is the potential for the bare classical subsystem. The solvent reorganization energy vanishes since there is now only a single proton state at each value of the environmental coordinates, and the adiabatic free energy surface Uad: ðx; y; R; {qk }Þ can r be replaced by a reduced free energy surface, Uad: ðx; y; R; {qk0 ðx; y; RÞ}Þ: The latter differs from Uad: ðx; y; RÞ by the substitution of the equilibrium medium coordinates qk0 ðx; y; RÞ for qk for each r value of the other reactive coordinates. The extrema of Uad: ðx; y; R; {qk0 ðx; y; RÞ}Þ and the activation free energy are determined by r ›Uad: ›E 0 þ ¼ 0; ›s l ›s l

sl ¼ {x; y; R}

ð26:54Þ

Gad a ¼ U0 ðxs ; ys ; Rs {qk }Þ 2 U0 ðx0i ; y0i ; R0i {qk0i }Þ þ E0 ðxs ; ys ; Rs {qk }Þ 2 E0 ðx0i ; y0i ; R0i {qk0i }Þ

ð26:55Þ

where the subscript “s” denotes the saddle point and i refers to the initial equilibrium configuration of the classical subsystem. As noted in Section II.H, the KIE for PT in hydrogen-bonded systems is determined by the effective frequencies and activation free energies for the different isotopes. In strongly hydrogenbonded systems with single-well proton potentials the KIE is entirely determined by the effective frequency and the proton vibrational ground state energy difference.

IV. ELECTRON-COUPLED PROTON TRANSFER ET and PT, in a wide sense “coupled” over broad time windows, is a notion in broad chemical and biological contexts. ET and PT are separate events (but still “coupled”), for example, in proton transport and pumping through large membrane-spanning protein complexes such as photosynthetic reaction centers,23,89,90 bacteriorhodopsin,91 or cytochrome c oxidase.21,22 “Coupling” here implies that electric field changes, which accompany ET induce local pKa changes at temporary proton donor and acceptor sites, triggering the proton transport. This applies also to photochemical processes in molecular triads,92,93 electrochemical discharge of some ions,94 and broadly in intraand inter-molecular PT following photo excitation.95 This limit is represented by the model in Figure 26.9, which shows ET between a donor and an acceptor molecule separated by hydrogen bonds with nonsymmetric location of one or two protons. ET results in a shift of one or both protons and switching of the hydrogen bonds. Electron-coupled PT (ECPT) represents a particular case of a more general class of coupled chemical reactions3,7,61,95 – 97 addressed theoretically in a number of reports.57,97 – 101 ECPT can be treated within the general theory of PT reactions.57 The main concepts of this view are summarized below. The other limit of the ET/PT window corresponds to completely “synchronous” ET/PT, merging into hydrogen atom transfer, i.e., a radical reaction. As noted, this limit is encountered in

D−

H+

A

D

H+

A−

FIGURE 26.9 Schematic view of PT and hydrogen-bond switching induced by ET in an ECPT scheme.

Proton Transfer and Proton Conductivity in Condensed Matter Environment

711

other photo-excited hydrogen atom transfer processes of organic molecules in frozen glasses 42,43,102 and a range of enzyme and metalloenzyme processes.28 – 32 The spectacular nuclear tunneling features have made these systems attractive targets for detailed analysis.40 Physically and formally hydrogen atom transfer follows closely general condensed matter PT theory and will be addressed briefly in Section IV.C.

A. MECHANISMS OF DYNAMIC AND S TEP- W ISE C OUPLING We consider two coupled reactions of the type De ðeÞ þ Ae ! De þ Ae ðeÞ;

Dp ðpÞ þ Ap ! Dp þ Ap ðpÞ

ð26:56Þ

in general with different donor, De and Dp, and acceptor, Ae and Ap, species for ET and PT. The two systems considered separately are characterized by their own reactive mode sets {Q1} and {Q2}, including both molecular coordinates and coordinates characterizing the dynamic inertial solvent polarization. Some modes can be common for both sets. In general the potential energy surfaces are multidimensional. In the simplest case each reaction corresponds to motion along the reaction coordinate from the initial to the final equilibrium configuration over a potential barrier, Figure 26.10. In classical potential barrier crossing induced by a suitable fluctuation from the initial equilibrium configuration sufficient energy is provided to exceed the barrier. The reaction is completed when the excess energy is lost and the system is trapped near the final equilibrium configuration. Two microscopic coupling mechanisms can be envisaged.3,7,57,61,97 In a step-wise mechanism, the reactions proceed sequentially. The energy is fully dissipated after the first step and renewed fluctuations required to overcome the barrier for the next step (Figure 26.11).3 Coupling of two reactions is caused by: (1) The equilibrium configuration of the complex for the second reaction is distorted by the shift of atoms and charges after the first reaction. (2) The initial or final state free energies for the second reaction are changed by the first reaction. Both effects are caused by the interaction of the two complexes, and can lead to a decrease of the activation barrier for the second step. Two other points are important: 1. The overall reaction can be exothermic even if the second step alone is endothermic;

q2

e

f 2 1

i

p

q1

FIGURE 26.10 Iso-energetic potential surface cross sections of ECPT spanned by two environmental solvent coordinates q1 and q2 : Coupled ET and PT may proceed synchronously (curve 1) or along a curvilinear trajectory through a vibrationally unrelaxed intermediate state (curve 2).

712

Isotope Effects in Chemistry and Biology U Ui

UB Uf

q

FIGURE 26.11 Reduced potential surfaces spanned by the solvent coordinate(s) q in step-wise, or quantumstatistical ECPT through a vibrationally-equilibrated intermediate state.

2. the overall reaction can be less exothermic compared to the reaction free energies evolved in the two individual steps. This phenomenon has been denoted as energy “transduction” or “recuperation”.3,103 The step-wise mechanism has also been denoted as quantum-statistical.97 The steady-state rate constant is described by the well-known expression k¼

k~1 k~2 kz1 þ k~2

ð26:57Þ

where: k~1 is the forward and kz1 the reverse rate constant of the first step, and k~2 the rate constant of the second step. Two routes are feasible for the overall ET/PT reaction with an intermediate state B(e) or B(p) corresponding to either ET or PT i ! BðeÞ ! f

and

i ! BðpÞ ! f

ð26:58Þ

The rate constants in Equation 26.57 are obviously different for each route. A different coupling mechanism is the dynamic coupling of the two reactions.3,7,97 This mechanism can operate in two ways: 1. Particles of both reaction complexes proceed synchronously, leading to direct transition from their initial to their final equilibrium configuration of both reaction complexes (curve 1, Figure 26.10).3 In particular, synchronous breaking and formation of several chemical bonds can be involved.104 This is different from hydrogen atom transfer where joint ET and PT involves a single reactant and product. 2. The mechanisms may operate as a coherent transition through dynamically populated intermediate states.7,97,105 Early examples are bridge-assisted ET, the theory of which is developed in detail.106 – 115Coherent transitions can in turn operate in two ways: (i) Friction along the reactive modes can be small. The motion is then close to ballistic, and having overcome the first potential barrier the system also overcomes the second one. This mechanism is selective, as to the free energy surface organization.113 In quasione-dimensional motion the activation free energy is determined by the highest potential barrier107,111 but the activation free energy is higher than both barriers in multidimensional space,113 cf., Section IV.B, caused by curvilinear system trajectories

Proton Transfer and Proton Conductivity in Condensed Matter Environment

713

U UB

Ui

Uf

q

FIGURE 26.12 Coherent ECPT through a dynamically populated intermediate state. Strong friction. Transition from intermediate to final state feasible only during intermediate state relaxation.

(curve 2 in Figure 26.10). Notably, the energy which evolves in the first step is not dissipated or dephased but used directly to cross the second potential barrier.113 (ii) If friction is strong, the system relaxes towards intermediate state equilibrium after crossing the first barrier. Transition to the final state is possible only if the trajectory can cross the free energy surfaces of the intermediate, UB ; into the final, Uf ; state whilst relaxing (Figure 26.10, Figure 26.11, and Figure 26.12). The probability of switching between the two free energy surfaces, PLZ ; must also satisfy the Landau –Zener adiabaticity criterion. The further scenario depends on the mutual organization of the free energy surfaces, Figure 26.12 and Figure 26.13. If the free energy surfaces are located as in Figure 26.12 and the transition is adiabatic, the system switches smoothly to the final state equilibrium. The energy is lowered all along the way, i.e., the transition is purely relaxational. The probabilities to switch to the free energy surface

U Ui

UB

Uf

q

FIGURE 26.13 Coherent ECPT through a dynamically populated intermediate state. Temporary system trapping in the upper well between intermediate and final states in the “nearly” fully adiabatic limit. This configuration also represents in situ STM of redox molecules or molecular three-electrode transistor function.116

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Isotope Effects in Chemistry and Biology

Uf ; or continue along UB ; are, respectively, WBf ¼ PLZ ;

WBB ¼ 1 2 PLZ

ð26:59Þ

A different case is shown in Figure 26.13. Due to the nearly adiabatic character of the transition between UB and Uf the system is temporarily trapped in the upper potential well formed by the crossing of UB and Uf. Since the Landau –Zener probability, PLZ ; is close to unity a small probability for exit either to the final or intermediate state through the crossing remains. As the number of these passages is infinite the total probabilities WBF and WBB ; are PLZ 1 ; WBB ¼ ; PLZ , 1 ð26:60Þ WBf ¼ 1 þ PLZ 1 þ PLZ As PLZ is close to unity, both WBf < WBB < 1=2: The flux thus splits equally between the intermediate and final states and the final state yield is 1/2. The transition for the second half of the flux to the final state is sequential, step-wise. The total rate constant is 7,105 k ¼ WBf kiB þ WBB

kiB kBf kBi þ kBf

ð26:61Þ

where WBf and WBB are described by Equation 26.59 or Equation 26.60 for the cases in Figure 26.11 or Figure 26.12, respectively. (For more general expressions, including the case (2)(i), see Refs. 61,105). As a note of observation, Figure 26.13 is also representative of two-step ET in electrochemical scanning tunneling microscopy (in situ STM) or the equivalent three-electrode configuration in recently reported single-molecule molecular transistors.116

B. A V IEW ON C OHERENT T WO- P ROTON T RANSFER IN Z UNDEL C OMPLEXES Multiple-PT steps in the Zundel complexes were viewed above as elementary PT events. The highly ordered structure required, in PT via Zundel complexes, however, makes successive “shortrange” structural diffusion events competitive. The rapid interconversion between H3Oþ and H5Oþ 2 in time ranges comparable to those of the proton conductivity itself further implies that the vibrational local solvent molecular modes may not relax fully between successive proton hops. The intermediate states corresponding to single- or double-PT then comply with the notion of dynamically populated intermediate states. We consider briefly the implications of such a view for two-PT. Two (sets of) independent, weakly damped, local solvent modes or two weakly damped gating modes R12 and R23 span the potential surfaces. Since each mode set is reorganized independently in the two consecutive single-PT steps the potential surfaces are fundamentally multidimensional, with the three potential minima in a two-dimensional representation located at the corners of a right-angled triangle (Figure 26.14). Both transitions are, moreover, fully adiabatic. As a further simplifying notion we assume that only a single mode is reorganized in each step, representative of the two gating modes. For the symmetric two successive PT steps the potential surfaces in the initial (i), intermediate (d), and final (f) state are 1 1 2 Ui ðR12 ; R23 Þ ¼ mv2 ðR12 2 R0i mv2 R223 12 Þ þ 2 2 Ud ðR12 ; R23 Þ ¼

1 1 mv2 R212 þ mv2 R223 2 2

Uf ðR12 ; R23 Þ ¼

1 1 2 mv2 ðR12 Þ2 þ mv2 ðR23 2 R0f 23 Þ 2 2

ð26:62Þ

where m is the effective mass and v the frequency of each of the symmetric reactive modes. R0i 12 and R0f 23 are the displacements along R12 and R23 : The identical minimum free energies have been omitted.

Proton Transfer and Proton Conductivity in Condensed Matter Environment

715

R12

R°12

R°23

R23

FIGURE 26.14 Iso-energetic potential surface cross sections of coherent two-PT through a dynamically populated single-PT intermediate state. The arrow shows a (dominating) curvilinear trajectory crossing the transition between the intermediate and final state above the saddle point.

The activation (free) energy and preexponential factor are sffiffiffiffiffiffiffi 1 1 Er Ga ¼ Er ; A¼ v 2 2p3=2 kB T

ð26:63Þ

where Er is the single-PT reorganization energy. Equation 26.62 and Equation 26.63 hold the implications: 1. The two protons are transferred consecutively through an intermediate state corresponding to single-PT, but coherently, i.e., weakly damped and without relaxation in the gating or local environmental modes in this state. 2. The activation free energy is higher than for single-step PT, for which Ga ¼ 14 Er : This reflects the curvilinear trajectory (Figure 26.10 and Figure 26.14). As noted, the coherent “long-range” mechanism is, however, competitive because of the generally small values of Er and the longer effective PT distance. 3. This view rests on weak friction. Strong friction would trap the system in the intermediate state and destroy the coherence. By the curvilinear character of the trajectory dephasing is, however, included in this coherence view and traps the system in the final state. 4. The view represented byEquation 26.62 and Figure 26.14 is directly reprentative of structural diffusion via remote undel complexes. The intermediate state in structural diffusion of two hydroxomium ions of the adjacent Zundel complex is modified by a finite energy gap relative to the symmetric initial and final states.

C. MODELS AND M ECHANISMS OF E LECTRON- C OUPLED P ROTON T RANSFER (ECPT ) Electron-coupled PT follows the patterns of elementary PT reactions, especially in the limit of diabatic synchronous ET and PT.1,2,38 – 40 The main difference is that two different intermediate and vibrationally relaxed or unrelaxed states are involved, where either the electron or the proton is

716

Isotope Effects in Chemistry and Biology

relocated between the corresponding donor and acceptor. This four-state model was studied recently.98 – 101 A simple approach was also suggested,57 which offers a straightforward, transparent basis for discussion of a given mechanism, and how the schemes can operate in ET-coupled PT reactions. We note first that transitions between these four states can be diabatic or adiabatic. This does not imply that the description must rest on different basis sets.47 Correct results can be obtained in either set. The diabatic basis set, however, offers the simplest view. 1. Diabatic States Diabatic states corresponding to different electron and proton localization are 57,98 – 100 De

Ae

Dp

Ap

i

1

0

1

0

e

0

1

1

0

p

1

0

0

1

f

0

1

0

1

ð26:64Þ

0 and 1 denote the occupation of the corresponding sites. The states i and f are initial and final states, respectively, e corresponds to ET, and p to PT. The diabatic electron and proton wave functions are denoted as wg ðx; rp ; qÞ and xg ðrp ; qÞ (g ¼ i,e,p,f), respectively, the corresponding ground state energies as 1eg ðrp ; qÞ and 1pr ðqÞ where x; rp ; q are the electron and proton coordinates, and the coordinates describing the inertial medium polarization. The dependence of the diabatic wave functions on the coordinates of the inertial subsystem can be neglected. The Born –Oppenheimer proton energies can be written as 1gp ðqÞ ¼ 1gp0 ðqÞ þ 1ge ðqÞ;

1ge ðqÞ ¼ k1eg ðrp ; qlg ¼ kxg l1eg ðrp ; qÞlxg l

ð26:65Þ

1ge ðqÞ is the electronic energy, including the solvent polarization, averaged over the proton state, and 1gp0 ðqÞ the energy of the proton itself at the “isolated” solvated proton donor (g ¼ i) or acceptor (g ¼ p,f). The diabatic free energy surfaces are given by Ur ðqÞ ¼ U0 ðqÞ þ 1gp ðqÞ

ð26:66Þ

where U0 ðqÞ is the free energy of the inertial polarization PðrÞ itself. In the continuum approximation U0 ðqÞ ¼ 1pr ðqÞ ¼ 1^p0 r 2

ð

2p ð 3 2 d r P ðrÞ c

d3 r PðrÞDpr ðrÞ þ 1^e0 r 2

ð26:67Þ ð

d3 rPðrÞDer ðrÞ

ð26:68Þ

where Dgp and Deg are the electric inductions due to the proton and the electron in the state g, respectively. 1^ge0 and 1^p0 g are the energy of the isolated states, in the absence of solvent polarization. The first two terms on the right-hand side of Equation 26.68 describe the dependence of the proton energies on the solvent polarization, the last two this dependence for the electron energies.

Proton Transfer and Proton Conductivity in Condensed Matter Environment

717

A dimensionless quantity j varying from 0 to 1 is useful as the reactive mode to describe the variation of the medium polarization117,118 P¼

c ½De þ jðDee 2 Dei Þ ; 4p i

c¼

1 1 2 11 10

ð26:69Þ

where 11 and 10 are the high- and low-frequency solvent dielectric constants. It is, further, convenient119 to consider polarization variations at different charge localization sites by introducing two independent variables, P¼

c ½De þ Dpi þ jðDee 2 Dei Þ þ hðDpp 2 Dpi Þ 4p i

ð26:70Þ

where j and h vary between zero and unity. Using Equation 26.67, Equation 26.68, and Equation 26.70 we can write the diabatic free energy surfaces in the form Ui ðqÞ ¼ j 2 Ee þ h2 Ep þ 2jhEep

ð26:71Þ

Ue ðqÞ ¼ ð1 2 j Þ2 Ee þ h2 Ep 2 2ð1 2 j ÞhEep þ DG0e

ð26:72Þ

Up ðqÞ ¼ j 2 Ee þ ð1 2 hÞ2 Ep 2 2j ð1 2 hÞEep þ DG0p

ð26:73Þ

Uf ðqÞ ¼ ð1 2 j Þ2 Ee þ ð1 2 hÞ2 Ep 2 2ð1 2 j Þð1 2 hÞEep þ DG0

ð26:74Þ

where DG0e ; DG0p ; and DG0 are the free energies of the transitions from the initial state to the states indicated by the subscripts. Further, Ee ¼

c ð 3 d rðDee 2 Dei Þ2 ; 8p

Ep ¼

c ð 3 d rðDpp 2 Dpi Þ2 8p

ð26:75Þ

are the solvent reorganization free energies for the ET and PT steps, and Eep ¼

c ð 3 d rðDee 2 Dei ÞðDpp 2 Dpi Þ 8p

ð26:76Þ

the coupling constant between the j and h modes. The value and sign of Eep depends on the structural organization of the electron and proton donors and acceptors. Coupling facilitates transitions when ET and PT are approximately in the same direction, i.e., Eep , 0: Eep ¼ 0 when the two directions are perpendicular, with no direct coupling in the free energy surfaces. The free energy surfaces of Equation 26.71 to Equation 26.74 represent paraboloids. The transitions between different states occur at the crossings of the corresponding free energy surfaces. Crossing of Ui and Ue corresponds to matching of the electron energy levels, (cf., Equation 26.66, Equation 26.68, and Equation 26.70) 1ei ðj; hÞ ¼ 1ee ðj; hÞ

ð26:77Þ

Coupling is determined by the resonance splitting of the electron levels119 D1e < 2Vie where Vie is the electron resonance integral.

ð26:78Þ

718

Isotope Effects in Chemistry and Biology

Similarly, crossing of the free energy surfaces Ui and Up corresponds to matching of the proton energy levels p0 1p0 i ð j ; hÞ ¼ 1 e ð j ; hÞ

ð26:79Þ

and coupling of these states is determined by the resonance splitting of the proton energy levels DEp0 :120 In the limit of electronically diabatic reactions this quantity is DEp0 < 2Vip kxi lxp l

ð26:80Þ

where the electron resonance integral Vip is determined only by the electronic properties of the PT complex. DEp0 in the electronically adiabatic case was discussed in Section II.D. The resonance splitting is now determined by the properties of the given PT complex and determines the coupling of the states e and p. A formally similar quantity, the resonance splitting of the total proton energy levels, DEp ; Equation 26.65, couples the states i and f. This quantity is determined by the properties of both PT and ET complexes (see Section IV). Statistical and dynamical limits for the coupling can be distinguished. We consider first transitions via relaxed intermediate states, then dynamically coupled transitions. 2. Mechanisms of Transitions and Rate Constants Statistical coupling. Motion along the reactive modes is stochastic, diffusion-like in a polar solvent medium and the variables j and h the only reactive modes. Mechanism (2)(i) (Section IVA) is then excluded. The mechanism is sequential step-wise transitions, provided that each step belongs to the normal reaction free energy region. Two independent routes, i.e., i ! e ! f and i ! p ! f are possible. The rate constant is the sum of the rate constants for both routes, Equation 26.57 kseq: ¼

kip kpf kie kef þ kpi þ kpf kei þ kef

ð26:81Þ

The rate constants are given by ET and PT theory 3,7,40,121,122 mg

kmg

G v 2 a ¼ eff kmg e kB T 2p

ð26:82Þ

where kmg is the transmission coefficient and veff the effective frequency. kmg is small for diabatic transitions and the activation free energy Gmg a determined by the saddle point of the free energy surface crossing. In adiabatic transitions the transmission coefficient is kmg ¼ 1 and the activation barrier reduced by a quantity dF related to the resonance splitting D13 pffiffiffiffiffiffiffiffiffiffiffi dF < D1 uð1 2 uÞ

ð26:83Þ

where u is the symmetry factor (see below). We discuss below the diabatic activation free energies. Extension to the adiabatic limit is straightforward using Equation 26.83. Electron transfer step i ! e. The coordinates of the saddle point on the crossing between Ui and Ue are js ¼ uie ; hs ¼ 0 where ! 1 DG0e ð26:84Þ uie ¼ 1þ Ee 2

Proton Transfer and Proton Conductivity in Condensed Matter Environment

719

The activation free energy and electronic transmission coefficient are given by 3,120 – 122 Gie a

¼

u2ie Ee ;

kie ¼

4p3 "2 v2 kB TEe

Vie2

!1=2 ð26:85Þ

Proton transfer step e ! f. The coordinates of the saddle point on the crossing of Ue and Uf are js ¼ 1; hs ¼ uef where 1 DG0 2 DG0e uef ¼ 1þ 2 Ee

! ð26:86Þ

Further, Gef a

¼

u2ef Ep ;

kip ¼

DEp0 2

!2

4p3 2 2 " v kB TEp

!1=2 ð26:87Þ

Proton transfer step i ! p. The coordinates of the saddle point on the crossing of Ui and Up are js ¼ 0; hs ¼ uip with ! DG0p 1 uip ¼ 1þ ; 2 Ep

2 Gip a ¼ uip Ep

ð26:88Þ

The transmission coefficient is given by Equation 26.87. Electron transfer step p ! f. The coordinates of the saddle point on the crossing of Up and Uf are js ¼ upf ; hs ¼ 1 where DG0 2 DG0p 1 upf ¼ 1þ 2 Ee

! ð26:89Þ

The activation free energy is 2 Gpf a ¼ upf Ep

ð26:90Þ

The transmission coefficient is again given by Equation 26.85. The activation free energies for the reverse transitions e ! i and p ! i are 2 Gei a ¼ uei Ee ;

! 1 DG0e 12 uei ¼ ; 2 Ee

2 Gpi a ¼ upi Ep

DG0p 1 12 upi ¼ 2 Ep

ð26:91Þ ! ð26:92Þ

Dynamic coupling. This limit only prevails by the relaxational mechanism (2)(ii) (Section IVA). If the second step is sufficiently exothermic the system can thus switch to the surface Uf during relaxation to the equilibrium intermediate configuration. Such a mechanism was discussed early123 and more recently101 for the route i ! e ! f. Dynamic coupling is feasible for either of the routes, i ! e ! f and i ! p ! f, with the rate constant given by Equation 26.59 to Equation 26.61.

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Isotope Effects in Chemistry and Biology

D. SYNCHRONOUS E LECTRON AND P ROTON T RANSFER Synchronous ET and PT includes for example direct hydrogen atom transfer from the initial, i, to the final state, f. Provided that the environmental reorganization free energy is large, i.e., the “strong-coupling” limit prevails, hydrogen atom transfer resembles closely PT processes,1 – 3,38 – 40, 42 and virtually the whole formalism, summarized in Section II.A to Section II.F, applies to hydrogen atom transfer as well.3,40 The activation free energy for the totally diabatic and partially adiabatic limit is, for example, determined by the saddle point of the crossing of the free energy surfaces Ui and Uf ; js ¼ hs ¼ uif where ! 1 DG0 uif ¼ 1þ ; 2 Eif

Eif ¼ Ee þ Ep þ Eep

ð26:93Þ

cf., Equation 26.75 and Equation 26.76. The activation free energy is Gifa ¼ u2if Eif

ð26:94Þ

We note that as j ¼ h at the saddle point, a single coordinate is sufficient for the description of synchronous electron – proton transfer. The transmission coefficient in the totally adiabatic limit is unity and the activation barrier reduced by a quantity of the type in Equation 26.83 with the resonance splitting of the proton vibrational energy levels DEp for D1:48 Both electron and proton follow adiabatically the inertial solvent polarization in this limit. The validity criteria separating these limits, and the relationships between the resonance splitting and the Landau – Zener transition probabilities are the same as discussed in Section II. The overview of the ECPT processes above offers precise formal frames for coupled ET/PT processes representing time windows from complete local and bulk environmental relaxation between the ET and PT events to completely synchronous ET and PT. The latter can either represent single-hydrogen atom transfer or spatially separate ET and PT. ET and PT via dynamically-populated intermediate states have also been addressed. ET-induced PT and proton conduction through large membrane-spanning protein complexes is likely to accord with sequential, or statistical modes, many hydrogen atom transfer processes with synchronous ECPT. Photo-induced proton transfer, proton exchange in organic dimers,124 and excess aqueous proton conductivity are, finally, possible cases for partial relaxation of dynamically-populated intermediate states in two-step ECPT or two-PT mechanisms.

V. CONCLUDING REMARKS Proton (“hydron”) and hydrogen atom transfer has been among the most important classes of target systems in chemical rate theory. Proton/hydrogen atom transfer has offered recent challenges in experimental innovation and theoretical framing. Novel areas, addressed in the present and other chapters have been hydrogen atom transfer in redox metalloenzymes (metallo-oxidases and -dehydrogenases) as one limit of molecular structural complexity, and proton transfer in strongly hydrogen-bonded systems as the other limit, with aqueous proton mobility and excess proton conductivity in focus. The theoretical concepts and formalism of condensed matter charge transfer theory, particularly the notion of environmentally (vibrationally) assisted proton/hydron tunneling are common frames for these widely different chemical and biological systems. Experimental approaches are, however, different. Wider perspectives are also different but with common denominators. Proton tunneling and kinetic isotope effects have been visible concepts in hydrolytic enzyme processes for many years [see Ref. 15 to Ref. 20, Ref. 125, and references there]. The tunneling

Proton Transfer and Proton Conductivity in Condensed Matter Environment

721

features and KIE have, however, mostly been weakly pronounced. The hydrogen atom-based oxidase and dehydrogenase systems alluded to 28 – 32 offer novel cases for strongly pronounced room temperature hydrogen atom tunneling and large kinetic deuterium isotope effects, induced by strong coupling to the environmental dynamics. An interesting observation is that high structural complexity has come to be a prerequisite for controlling the weak donor – acceptor interactions and long particle distance in the totally diabatic or partially adiabatic frames needed for pronounced quantum-mechanical features. Use of mutant proteins offers identification of amino acid residues and other structural elements in the environmental vibrational assistance and gating of the hydrogen atom transfer events. In these ways perspectives emerge for the mapping of the complete hydrogen tunneling events in enzyme systems close to the molecular level. These perspectives could extend to other system classes including technologically interesting processes in homogeneous and heterogeneous catalysis. The different nature of strongly hydrogen-bonded systems, with a particular view on excess aqueous proton conductivity, has prompted a different experimental and theoretical focus. Excess aqueous proton conductivity has, for example, rested on ionic mobility studies 73 – 75,77 and on NMR spectroscopy [see Ref. 5, and references there in]. The immediate outcome is a diffusion coefficient, an apparent activation energy, and a kinetic deuterium isotope effect of the order of the square root of the mass ratio. Insight has also come from mobility studies over (very) broad temperature ranges.73 – 75 The relative simplicity of either excess proton conductivity or acid-base reactions of other small solute molecules in water molecular assemblies also enables molecular dynamics and other computational studies at very sophisticated levels.79 – 87 Condensed matter charge transfer theory with its long-time established and currently evolving status 6,124 offers a powerful and transparent bridge to the computational results, such as also shown in the present chapter. The discussion above has, finally, focused solely on PT in homogeneous, isotropic media. Important elements of both general PT theory and specific models, say for excess proton conductivity in bulk solution carry over to PT and proton conduction in confined media. Both the theory broadly and the relevant models suggest strongly that notions such as proton conduction through membrane channels with “long-range” character of the proton transport must involve sequences of individual proton hops or correlated two-proton hops between localized proton donor and acceptor sites. Multiple-PT, i.e., coherent transfer of several or many protons is thus highly unlikely events in the strongly coupled polar reaction medium. The correlation between the fundamental PT act and observable proton mobility patterns in the heterogeneous environment of proton conducting Nafion membranes has, for example, been analyzed in detail.126 The PT act involves proton exchange between stationary but flexible sulfonate groups and mobile water molecules, i.e., between strongly hydrogen-bonded donor and acceptor groups in approximately thermoneutral single- or two-PT steps. These resemble the proton conductivity mechanisms based on Zundel complexes. The observed proton mobility is, however, also determined by the whole set of membrane pore distribution parameters both along a given pore and across the membrane surface, as well as by the membrane swelling characteristics. Proton transport through biological membrane-spanning large proton complexes would also follow the patterns of sequential single- or two-PT events between localized residues in the protein structure and water molecules in the proton transport channels. As noted, these are coupled to separate ET events. Biological proton transport complexes could be more suitable targets for theoretical condensed matter PT science than, for example, amorphous synthetic membrane systems such as Nafion. This is because both the protein structures and the aqueous proton transport channels are becoming much better characterized.21,22 A primary condition in both areas is, however, that the vibrationally assisted tunneling patterns in bulk hydrogen-bonded systems is first understood.

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Isotope Effects in Chemistry and Biology

ACKNOWLEDGMENTS Financial support from the Russian Foundation for Basic Research (Contract No. 03-03-32935) and the Danish Technical Science Research Council (Contract No. 26-00-0034) is acknowledged.

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27

Mechanisms of CH-Bond Cleavage Catalyzed by Enzymes Willem Siebrand and Zorka Smedarchina

CONTENTS I. II.

Introduction ...................................................................................................................... 725 Observations ..................................................................................................................... 727 A. Rate Constants.......................................................................................................... 727 B. Kinetic Isotope Effects............................................................................................. 728 C. Temperature Dependence ........................................................................................ 728 D. Systems without Proteins......................................................................................... 729 III. Theoretical Models .......................................................................................................... 730 A. Two-Oscillator Models ............................................................................................ 730 B. Golden Rule Treatment............................................................................................ 731 C. Semiclassical Instanton Approach ........................................................................... 734 D. Model Parameters .................................................................................................... 734 IV. Applications...................................................................................................................... 735 A. Coenzyme B12 .......................................................................................................... 735 B. Lipoxygenase ........................................................................................................... 736 C. Primary Amine Dehydrogenases ............................................................................. 737 D. Dicopper Complexes................................................................................................ 738 V. Discussion ........................................................................................................................ 738 Acknowledgments ........................................................................................................................ 739 References..................................................................................................................................... 739

I. INTRODUCTION Deuterium-labeling of substrates is an effective way to gather information on enzymatic reactions involving hydrogen transfer. If deuterium substitution reduces the rate of the reaction substantially, it implies that hydrogen transfer is a rate-determining step and that this step proceeds by quantummechanical tunneling. Large kinetic effects of deuterium substitution are observed in reactions in which a carbon – hydrogen bond is broken.1 – 8 This is to be expected since these strong nonpolar bonds are difficult to cleave. As a result, the transfer barrier will be high and/or the tunneling path long. An increase in temperature will generally decrease the tunneling contribution but, paradoxically, it has been observed that in several cases the deuterium effect, although very large, seems insensitive to such an increase.3 – 7 In this contribution, we consider both the theoretical and the experimental aspects of this paradox. To initiate hydrogen transfer, enzyme and substrate typically form a complex in which the hydrogen donor and acceptor groups are brought close together. The instantaneous separation of these groups will oscillate with frequencies determined by the forces that hold the complex together, 725

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Isotope Effects in Chemistry and Biology

which normally are (weak) van der Waals forces and hydrogen bonding. The corresponding vibrations, together with the tunneling vibration along the CH-stretch coordinate, govern the temperature dependence of the deuterium effect, the quantitative aspects of which we explored earlier.9 At temperatures about which enzymes normally operate, say 310 ^ 30 K, the CH-stretch vibration and even the CD-stretch vibration is unlikely to be thermally excited; however, the donor – acceptor vibrations governed by van der Waals potentials will be sensitive to temperature and thus will be a cause, and probably the main cause, of the temperature dependence of the deuterium effect. A temperature-insensitive deuterium effect would seem to imply that these vibrations have frequencies outside the range where they can be thermally excited. Such high frequencies are normally restricted to systems bound by valence rather than van der Waals forces, the exception being systems bound by strong hydrogen bonds. However, CH bonds are nonpolar and therefore reluctant participants in hydrogen-bond formation. Although there have been several attempts to account for the large kinetic isotope effects observed in these reactions,4,10,11 it remains unclear why these effects show such a weak temperature dependence at the relatively elevated temperatures where enzymes operate. Studies that consider a single system at one temperature are unlikely to shed much light on the problem. It is therefore timely to adopt a broader point of view and consider all available information on temperature-dependent isotope effects or at least a representative sample of it. Our approach will be based on a critical examination of the reported observations followed by theoretical modeling of the deuterium effect and its temperature dependence. The aim will be to find out whether the observations are compatible with these models and, if not, to probe whether the models or the data are defective. The problem with theoretical approaches is that their results depend on many parameters that cannot be calculated with the required accuracy for these complex systems. Structural parameters used as input for quantum-chemical calculations are usually not available for the enzyme– substrate complex and thus must be inferred solely from the enzyme structure. Even when these methods can deal adequately with the active-site structure, the necessarily very approximate treatment of the surrounding protein will introduce substantial uncertainty. Therefore we will adopt a more qualitative approach and confine ourselves to methods that focus on general trends without requiring input parameters that are difficult to obtain. Inspection of the KIE data available for enzymatic CH cleavage indicates that not all of these data show a weak temperature dependence. For instance, for the hydrogen abstraction step in the reaction catalyzed by methylmalonyl-coenzyme A mutase (MMCoAM), a coenzyme B12-dependent enzyme,12 Chowdhury and Banerjee13,14 found that the KIE increased from 36 at 293 K to 50 at 278 K, values roughly reproduced by semiempirical calculations.15 Doll, Finke and Bender16,17 showed that the same abstraction step occurs in solution with similar temperaturedependent KIEs in the absence of a protein. The actual abstracting is known to be done by a carboncentered radical formed by homolysis of a cobalt –carbon bond of coenzyme B12. From a chemical point of view, abstraction via a radical mechanism is indeed a most plausible approach to breaking a nonpolar CH bond.18 The apparent parallel evolution of an ionic cleavage mechanism may be due to the difficulty of controlling agressive free radicals. This ionic mechanism appears to be associated with a much weaker temperature dependence of the KIE. A typical example is the oxidation of linoleic acid by lipoxygenase-1 (SLO1) as reported recently by Klinman’s group,3,4 in which the KIE increases from 64 at 318 K to 81 at 278 K. In this reaction a C-11 methylenic hydrogen is transferred to the OH ligand of an iron atom that is part of a Fe(III) $ Fe(II) redox center, which abstracts an electron from the methylene group. This motif, hydrogen abstraction whereby the proton is bound by an oxygen lone pair accompanied by electron transfer to a redox system, which may be metallic as in SLO1 or organic as in methylamine dehydrogenase (MADH),5 – 7 is a common feature of many such enzymes. Those for which temperature-dependent KIEs have been measured show large to very large KIEs (10 – 100) with a weak to very weak temperature dependence. The authors of the SLO1 data4 have analyzed their results in terms of ˚. a harmonic two-oscillator model9,10 and found a tunneling distance in the range 0.6 –0.7 A

Mechanisms of CH-Bond Cleavage Catalyzed by Enzymes

727

r ξ r R

FIGURE 27.1 Relationship between the proton donor – acceptor distance R, tunneling distance r, and van der Waals radius r for C –H· · ·(C or O) proton transfer.

˚ for CH2 and O, The tunneling distance estimated from van der Waals radii (2.0 and 1.4 A ˚ respectively) and bond lengths (1.1 and 1.0 A for CH and OH, respectively) amounts to about ˚ .19 The dilemma is illustrated in Figure 27.1. Inside the van der Waals radius r, the potential 1.3 A V(R), where R is the C· · ·O distance, becomes repulsive and increases as ðr=RÞn where n , 9– 12. The possibility that an external force can overcome this strong repulsion in a biological environment seems remote. This indicates that either the tunneling distance deduced from the kinetic data is too short or there is an additional attractive force, such as hydrogen bonding, that is strong enough to overcome the repulsion. Several quantum-chemical calculations have been reported11,20 – 22 for these systems. An attempt to calculate rate constants for neutral hydrogen and deuterium atom transfer for the SLO1 reaction, using transition state theory with semiclassical tunneling corrections (TST/ST) in the smallcurvature approximation, based on a PM3/d potential with linoleic acid replaced by 1,4-pentadiene, could not reproduce the observed large KIEs and did not address the problem of the weak temperature dependence.11 Very recently, Lehnert and Solomon20 carried out a density functional calculation on a model of the SLO1-linoleic acid complex at the B3LYP level with a LanL2DZ basis ˚ , leading to set. Although the C· · ·O distance was not fully optimized, they found that a value of 3.0 A ˚ , reproduced the observed energetics of the reaction reasonably a tunneling distance of about 1.0 A well. It was also concluded that the electron and the proton move independently but simultaneously. However, the problem of the large KIE and its weak temperature dependence was not addressed. The MADH reaction has been studied by several groups.11,21,22 The rate constant for proton abstraction from methylamine was calculated by TST/ST. Although the reported KIE at fixed temperature was generally well reproduced, the calculations did not consider the problem of its very weak temperature dependence, nor did they consider the observation that the kinetic analysis led to an anomalously large KIE for the enzyme –substrate formation and dissociation rate constants.

II. OBSERVATIONS A. RATE C ONSTANTS Carbon –hydrogen bond cleavage reactions catalyzed by enzymes normally consist of a series of consecutive steps, which may or may not be reversible. If the rate decreases sharply when hydrogens of the substrate are replaced by deuterium, one of these steps must be hydrogen transfer. To measure the rate constant of this step, it will be necessary to monitor the rate of disappearance of a reactant or appearance of a product, whose signal may or may not be easily distinguished from that of other compounds in the reaction vessel. Since in the reactions to be considered this rate depends on steps prior to the transfer step, in particular the rate of formation and dissociation of

728

Isotope Effects in Chemistry and Biology

the enzyme– substrate complex, a kinetic analysis will be required based on the nature of these steps. The observed overall rate is then decomposed into the rates of the three or more reactions involved by varying the substrate concentration. Necessary, but not necessarily sufficient, conditions for accepting the results of such a procedure are that the data points fit the kinetic curves and that significant deuterium isotope effects are obtained only for those steps that involve hydrogen transfer. Unfortunately, the reported data do not always indicate whether the results meet these minimum consistency requirements. Further tests of the validity of the reported kinetic analysis may be possible when rate contants obtained for closely related enzymes or closely related substrates can be compared, since relative rate constants are often easier to interpret than absolute rate constants. Enzymes can be systematically varied by generating mutants based on the replacement of single amino acids. Such information is available for reactions catalyzed by SLO1.4 Many enzymes, e.g., methylamine and aromatic amine dehydrogenase (AADH), catalyze the reaction of a variety of substrates, which can thus be compared.5 – 7 An interesting way to study enzymatic reaction mechanisms is to synthesize model systems that mimic the reactive site without the protein. Such model systems are available for MMCoAM16,17 and for several monooxygenases whose active site contains a dicopper complex.23 In the following sections this information will be used to examine the corresponding hydrogen transfer mechanisms.

B. KINETIC I SOTOPE E FFECTS As usual the (temperature-dependent) deuterium effect will be expressed as the ratio of the transfer rate constants of hydrogen and deuterium: hðTÞ ¼ kH ðTÞ=kD ðTÞ and h will be referred to as the kinetic isotope effect or KIE. Although our present ability to calculate absolute rate constants for something as complex as an enzymatic reaction is obviously limited, KIEs are relative-rate constants that depend on far fewer parameters than the corresponding absolute-rate constants and should therefore be easier to interpret by theoretical models. Within the Born – Oppenheimer approximation, deuterium substitution does not affect the equilibrium structure and thus can influence the transfer dynamics only through its effect on the vibrational force field. This means that our knowledge of molecular vibrations can be directly applied to the interpretation of KIEs. If a proton is transferred from a CH to an OH bond, the transfer coordinate will, of necessity, contain CH- and OH-stretch (and possibly -bend) components; these are responsible for the KIE. The magnitude of the KIE will depend on the length of the tunneling path, i.e., the part of the path under the barrier as distinguished from the part in the well. The relation between these two parts will depend on the relative positions of the C and O donor and acceptor atoms and thus on the instantaneous vibrational separation of these atoms. This separation will oscillate with frequencies of all the normal modes with components along C· · ·O, but for illustrative purposes, we may restrict ourselves to the C· · ·O stretching mode. Any parameter that influences the amplitude of this mode, hereafter referred to as the promoting mode, will affect the tunneling. Hence some knowledge of this vibration is essential for our ability to understand KIEs. Similar arguments apply if a hydrogen atom is transferred from a CH bond to a carbon-centered radical.

C. TEMPERATURE D EPENDENCE The temperature dependence of hydrogen transfer rate constants is usually of the curved-Arrhenius form kðTÞ ¼ Ae2Ea ðTÞ=kB T

ð27:1Þ

The curvature arises because the activation energy Ea varies from zero for T ¼ 0, where the reaction proceeds entirely by zero-level tunneling, to the barrier height for very high temperatures, where classical transfer prevails. However, within the small temperature intervals for which kinetic data of

Mechanisms of CH-Bond Cleavage Catalyzed by Enzymes

729

2.5

4−6

2

log KIE

1

11

3

1.5

8 7 9

2

1

10

0.5 2.5

2.7

2.9

3.1

3.3

3.5 3.7 3.9 1000/T

4.1

4.3

4.5

4.7

FIGURE 27.2 Arrhenius plots of the KIEs of the systems listed in Table 27.1.

the enzymatic reactions are available, the curvature is usually negligible, so that the standard Arrhenius equation with constant Ea holds. Among the factors contributing to the apparent activation energy in that temperature interval are the endothermicity, if any, and thermal excitation of the vibrations involved in the tunneling, namely the tunneling vibration, the promoting mode, which modulates the tunneling distance, and vibrations other than the tunneling vibrations that are displaced between the initial and final state. Due to their high frequencies, CH and CD stretching vibrations are unlikely to be thermally excited at room temperature. The other displaced vibrations generally contribute little to the KIE. The temperature dependence of the KIE follows directly from Equation D H ; where T is now a parameter rather 2 Ea;T 27.1 with A replaced by ðA H =A D Þ and Ea(T ) by Ea;T than a variable. In Figure 27.2 we depict Arrhenius plots of the KIEs to be discussed in Section IV, each of which is reported as being linear in the narrow temperature interval where data are available. Relevant parameter values are listed in Table 27.1. Since the resulting slopes and intercepts have no clear physical meaning, it is not surprising that this graph shows little or no regularity. To obtain a more meaningful representation of the dependence of the KIE on temperature, we propose the alternative form D H Ea;T 2 Ea;T d ln hðTÞ d ln hðTÞ AH ¼ ln ; ¼ h ðTÞ 2 ln kB T AD T d T 21 d ln T 21

ð27:2Þ

This removes the apparent activation energies in favor of the observed KIE, but not the frequency factors. To replace these by quantities that are either observable or calculable, we will need a model. This will be introduced in the following section.

D. SYSTEMS WITHOUT P ROTEINS In the enzymes under discussion the acceptor groups that abstract a hydrogen or proton from the substrate do not belong to an amino acid residue but to an active group that may or may not contain a metal ion and is specific for the enzyme. The question thus arises to what extent the unusual features associated with deuterium substitution are due to the protein. For instance, it is conceivable that the protein around the active site assumes a structure that is particularly favorable to tunneling by bringing the donor and acceptor groups closer together than expected on the basis of the normal

730

Isotope Effects in Chemistry and Biology

TABLE 27.1 Kinetic Parameters (T in K, k in sec21, Ea, the Apparent Activation Energy, in kcal mol21) used in the Analysis of Temperature-Dependent Kinetic Isotope Effects in (model) Enzymes No. 1 2 3 4a 4b 4 5 6 7 8 9 10 10a 11

Reactant(s)

T

DT

MMCoAM (MeO)AdoCbl NeopentylCbl Linoleic/SLO1 Mutant I553A 4a þ 4b Mutant L546A Mutant L754A EtOHNH2/MADH MeNH2/MADH Dopam./AADH BenzylNH2/AADH Tryptam./AADH DiCu(Bn3)complex

293 373 298 298 298 298 298 298 298 298 298 298 277 233

15 40 30 40 40 40 40 40 35 35 35 35 — 40

kH 206

401 302 293 1.28 0.18 14 175 132 1.81 503 3.55(23)

EH a

ln KIE(T )

d ln KIE T d T 21

Ref.

18.8 ^ 0.8

3.8 2.3 3.4 4.4 4.6 4.5 4.7 4.7 2.8 2.8 2.6 1.6 4.0 3.7

6.9 3.5 5.3 2.4 5.7 3.2 2.7 2.8 2.6 0.6 0.4 0.3 — 2.7

14 17 17 4 4 4 4 4 6 5 6 6 6 23

1.7 ^ 0.3 1.9 ^ 0.4 1.9 ^ 0.2 4.0 ^ 0.2 3.7 ^ 0.4 10.4 ^ 0.1 10.6 ^ 0.1 12.1 ^ 0.2 16.3 ^ 0.3 — 14.2 ^ 0.3

The SLO1 parameters are derived from least-square Arrhenius fits to the data reported in the Supplementary Material of Ref. 4 and differ slightly from the data listed in Table 1 of that paper.

attraction and repulsion between these groups.4 This issue can be studied by preparing systems that model the active site without the protein. Apart from allowing a comparison of the rate of transfer with and without the protein, these model systems have two other advantages. First, they allow direct observation of the transfer kinetics without interference from prior reactions such as formation and dissociation of the enzyme –substrate complex. Second, they are more amenable to theoretical calculations than the actual enzyme and thus offer better prospects for determining the structure of the active site of the enzyme –substrate complex. Two examples of such model systems are part of the present investigation. In the case of MMCoAM,13,14 the model system16,17 is identical with the coenzyme that catalyzes the reaction or a simple derivative thereof. In that case a direct comparison between the enzymatic and the model reaction is possible. The other example is that of monooxygenases that contain a dicopper complex, which activates dioxygen.23 In that case temperature-dependent KIEs are available only for the model system, so that the comparison is limited to enzymatic KIEs in which the hydrogen transfer is catalyzed by different active sites.

III. THEORETICAL MODELS A. TWO- OSCILLATOR M ODELS To analyze KIEs and their temperature dependence we consider three types of molecular vibrations. First, we include vibrations that contribute to the tunneling coordinate and take their potential to be a double-minimum potential with two wells representing the initial and final configuration and a barrier whose top is the transition state. This potential we represent either by two crossing antiparallel Morse potentials corresponding to CH and OH stretching vibrations, or by a quartic potential containing only terms that are quartic and quadratic in the tunneling coordinate. Second, we include vibrations that modulate the transfer distance. Based on an earlier analysis,24 we represent these vibrations by a single promoting mode, which we treat as a harmonic oscillator.

Mechanisms of CH-Bond Cleavage Catalyzed by Enzymes

731

Its coordinate is taken to be collinear with the tunneling coordinate. Third, there are vibrations that are displaced between the initial and final configuration. These we represent classically by an isotope-independent reorganization energy, so that they do not enter directly into the expressions for the KIE. We are thus left with a two-dimensional tunneling model. To calculate the temperaturedependent KIE for this model, we use two complementary methods. We use the Golden Rule of time-dependent perturbation theory to treat the transfer as a nonadiabatic transition between the two Morse potentials.9 In addition, we use semiclassical Instanton theory25 to treat it as an adiabatic transition along a quartic potential. The vibrational parameters required we take from spectroscopic observations.

B. GOLDEN R ULE T REATMENT The Golden Rule states that the transfer rate constant between an initial state labeled i and a contiuum of final states {f} is proportional to the square of the matrix element coupling the states, subject to energy conservation: kfi ¼ ð2p="ÞlVfi l2 dðe {f } 2 e i Þ

ð27:3Þ

To generate the continuum, we make the usual assumption that the discrete levels of the model are coupled to a high density of levels of the real system and implement this coupling by multiplying each level by a lineshape function. Using the Born –Oppenheimer and the Condon approximation, we find that the matrix element Vfi will basically consist of a squared vibrational overlap integral multiplied by an electronic factor which is the same for hydrogen and deuterium transfer and cancels out in the calculation of the KIE. To obtain the thermal rate constant k(T), we must carry out this calculation for all vibrational levels of i that are thermally populated, as well as all final levels modulated by their lineshape function, and sum the contributions. For the calculation of the KIE the lineshape function also cancels out within the narrow temperature interval of the observations, so that we are left with Boltzmann-weighted squared overlap factors. The advantage of this approach is that we know a priori a good deal about the relevant vibrations. The CH and OH vibrations can be represented by Morse oscillators with parameters taken from standard spectroscopic observations. The vibrations modulating the C · · · C and C · · · O separation for hydrogen and proton transfer, respectively, are the true focus of our investigation since they help elucidating the structure of the active site. Our strategy will be to calculate the KIE for parameters varying from situations where the force field is determined by van der Waals interactions to those in which there is strong hydrogen bonding. The former situation will be characterized by low frequencies and relatively large effective masses and the latter by higher frequencies and smaller masses. This model was used many years ago to study hydrogen tunneling in simple organic systems.9 It allows us to obtain accurate numerical values for the required overlap integrals and thus the KIE and its temperature dependence for a given choice of vibrational parameters. These parameters must, of course, form self-consistent sets, depending on the assumed strength of the enzyme– substrate force field. Numerical results obtained by this method are depicted by solid lines in Figure 27.3 in the form of a plot of the temperature dependence of the KIE, expressed as in Equation 27.2, against the logarithm of the KIE. The parameters characterizing the curves are listed in Table 27.2. The curves can be represented by an equation of the form d ln h d ln h ; . D1 ln h 2 D2 ln2 h 21 T dT d ln T 21

ð27:4Þ

732

Isotope Effects in Chemistry and Biology 8

A

B

1

7 6 d ln KIE / d ln T −1

2

C

5 4

3

4

3

6 5

11 8

2

E G

1 0

D

10 0

1

F

H

2

9

7

3 ln KIE

4

5

6

FIGURE 27.3 Comparison of experimental data points for enzymatic (model) reactions with a general model relation, represented by curves that relate the temperature dependence of the KIE to the KIE. The curves are calculated for a two-oscillator model, in which the tunneling potential is represented by two crossing Morse potentials or a quartic potential, and the promoting-mode potential is harmonic, for the parameter values listed in Table 27.2. The data points are derived from reported KIEs and are listed in Table 27.1.

This equation represents the desired generalization of Equation 27.2 in which the ratio of frequency factors A H =A D is replaced by ln hðTÞ; which is observed, and the factors D1,2, which can be calculated. The factor D1 is close to 2 and becomes equal to 2 for the simplified model discussed below; hence it contains no information that is specific to the system. Such information is limited to the factor D2, which controls the curvature. Comparison with Equation 27.2 shows that this factor relates the KIE to the ratio of frequency factors A H =A D and thus represents the dynamic properties of the system. TABLE 27.2 Fixed Parameters used to Calculate the Curves Displayed in Figure 27.3 Letter B C D E F G H

Type

r0

v H/v D

2X H/2X D

lDEl

Morse Morse Morse Morse Morse Quartic Quartic

1.7 1.3 1.0 1.0 1.0 1.0 0.6

3000/2150 3000/2150 2850/2000 2850/2000 2850/2000 2850/2000 2700/1900

70/35 60/30 120/60 120/60 120/60

4 10

˚ , the frequency v H/v D and the (negative) anharmonicity X H/X D in cm21, and the The transfer distance r0 is expressed in A 21 exo(endo)thermicity ^DE in kcal mol . Curve A is the limiting result if the tunneling distance is very large.

Mechanisms of CH-Bond Cleavage Catalyzed by Enzymes

733

It follows from Figure 27.3 and Table 27.1 that D2 is small if the donor – acceptor distance and the promoting-mode force field are governed by van der Waals distances; comparison of the solid curves of Figure 27.3 with the parameter values in Table 27.2 shows that it increases rapidly if the distance is reduced and the force field correspondingly increased. Large KIEs with a weak temperature dependence occur only for large D2 values, which imply short tunneling distances along with high-frequency vibrations between the donor and acceptor groups. To gain qualitative insight into these properties of the model, we derive Equation 27.4 for a simplified version that allows an approximate analytical evaluation. To this end we neglect anharmonicity and assume a symmetric, i.e., thermoneutral, tunneling potential. We also assume that the tunneling vibrations have a frequency high enough to neglect contributions from thermally excited levels, so that the temperature dependence of the KIE will be governed solely by the promoting mode. The one-dimensional squared zero-point overlap integral along the tunneling coordinate is given by s200 ðrÞ ¼ e2r

2

=2a20

¼ e2r

2

mv=2"

ð27:5Þ

where r is the instantaneous transfer distance, and m and v are the effective mass and frequency of the tunneling mode, so that a0 is the zero-point amplitude. We will use the notation aH 0 ¼ a0 for the 1 24 light isotope and aD ¼ 2 a for the heavy isotope. As illustrated in Figure 27.1, for the collinear 0 0 model of Ref. 9, the relation between the transfer distance r and the promoting-mode coordinate R is given by r ¼ R 2 2j; where 2j is the sum of the donor –H and acceptor – H bond lengths. To obtain the thermal squared overlap integral S200 ðTÞ; wepaverage s200 ðrÞ over the equilibrium ffiffi 21 distribution of donor– acceptor separations PðR; TÞ ¼ ½ pAðTÞ exp{ 2 ½ðR 2 R0 Þ=AðTÞ 2 }; where R0 is the equilibrium donor – acceptor distance. This yields S200 ðTÞ

ð 2 2 2 1 ¼ pffiffi dR e2ðR22jÞ =2a0 2½ðR2R0 Þ=AðTÞ ¼ pAðTÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2a20 e2r0 =½2a0 þA ðTÞ 2 2a0 þ A2 ðTÞ

ð27:6Þ

where r0 ¼ R0 2 2j is the equilibrium transfer distance, and A2 ðTÞ ¼ A20 coth

"V 2kB T

ð27:7Þ

pffiffiffiffiffiffiffiffiffi M and V being the effective mass and frequency of the promoting mode, so that A0 ¼ "=M V is the zero-point amplitude. For this simplified model the KIE can be expressed as19 ln C h ¼

pffiffi 2a20 r02 ð 2 2 1Þ pffiffi ½2a20 þ A2 ðTÞ ½2a20 þ A2 ðTÞ 2

ð27:8Þ

where vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u 2a2 þ A2 ðTÞ C ¼ t 2 0 2 pffiffi 2a0 þ A ðTÞ 2

ð27:9Þ

varies between 1 if 2a20 q A2 ðTÞ and 0.84 if 2a20 p A2 ðTÞ; for qualitative purposes it can be set equal to unity. Approximating the hyperbolic cotangent in Equation 27.7 by its inverse argument, which

734

Isotope Effects in Chemistry and Biology

is justified if "V , kB T; we obtain pffiffi 1 d ln C h 4a20 þ ð 2 þ 1ÞA2 ðTÞ 2 pffiffi ln C h ¼ 2 ln Ch 2 T d T 21 ð 2 2 1Þr02

ð27:10Þ

which is of the same form as Equation 27.4. For small anharmonicities we can generalize these equations by replacing the zero-point amplitude a0 by an effective value aeff . a0.19 Equation 27.10 predicts that for large tunneling distances D2 vanishes, so that d lnh=d lnT 21 ¼ 2 ln h. This limiting curve is represented by the dashed line in Figure 27.3.

C. SEMICLASSICAL I NSTANTON A PPROACH Equations of the form of Equation 27.4 are not limited to the Golden Rule treatment but can also be derived if we replace the crossing Morse functions of the tunneling vibration by a smooth adiabatic potential and calculate tunneling rate constants with Instanton techniques.25,26 In this approach the set of all possible tunneling paths is represented by its dominant component, i.e., the path that minimizes the action. The Instanton action for our two-dimensional model potential can be evaluated accurately if the double-minimum potential along the tunneling coordinate contains only quartic and quadratic terms. Such a potential is defined by two parameters, for which we choose the tunneling distance r0 and the curvature at the bottom of the wells v. This leads to a potential of the form VðxÞ ¼

1 2 2 2 vx ð2x =r0 2 1Þ 4

ð27:11Þ

where x is the tunneling coordinate, which in turn is a function of the promoting-mode coordinate. Details of this calculation will be reported elsewhere.27 Numerical evaluation of the KIE and its temperature dependence leads to the dot-dash curves in Figure 27.3 for the parameter values listed in Table 27.2. For the same tunneling distance, they are quite similar to the curves calculated with the Golden Rule and crossing Morse potentials. When AðTÞ is approximated by its high-temperature value, this approach leads to equations of the same form as Equation 27.4, Equation 27.8, and Equation 27.10 but with a different coefficient for D2, namely19,27 pffiffi a20 r02 ð 2 2 1Þ ih ln h ¼ h pffiffii 3 a20 þ 14 A2 ðTÞ a20 þ 14 A2 ðTÞ 2

ð27:12Þ

pffiffi 2a20 þ 14 ð 2 þ 1ÞA2 ðTÞ 2 1 d lnh pffiffi ln h ¼ 2 ln h 2 3 T d T 21 ð 2 2 1Þr02

ð27:13Þ

for C ¼ 1. Just as Equation 27.8 and Equation 27.10, Equation 27.12 and Equation 27.13 represent results that are more approximate than the plotted curves.

D. MODEL PARAMETERS The values chosen for the model parameters reflect our desire to keep the number of different parameters to a minimum and to use only sets that are physically reasonable. Thus we do not distinguish between CH and OH stretching modes; also, we set the effective masses of the tunneling vibration for hydrogen pffiffi and deuterium equal to 1 and 2, and the ratio of the corresponding frequencies equal to 2: Tunneling distances are chosen so as to represent values ranging from that ˚ , to that derived from van der Waals radii for methyl and methylene groups, viz. 1.7 A

Mechanisms of CH-Bond Cleavage Catalyzed by Enzymes

735

˚ . If the tunneling distance r0 $ 1.3 A ˚ , the corresponding to very strong hydrogen bonding, viz. 0.6 A tunneling-mode frequencies are taken to be 3000 and 2150 cm21, since the oscillators will be essentially unperturbed. CH- and OH-stretch anharmonicities of isolated unperturbed CH and OH molecules are in the range 60 –80 cm21;28 for their deuterated equivalents they are half these values. We shall use values X H/X D ¼ 2 60/2 30 and 2 70/2 35 cm21; note that the notation X common for anharmonicity constants in polyatomics corresponds to vexe commonly used for diatomics. The curves in Figure 27.3 are calculated by varying the promoting-mode frequency V for chosen values of the other parameters. ˚ , the relation between the KIE and its temperature For tunneling distances r0 $ 1.3 A dependence is determined mostly by the linear term of Equation 27.4, so that curves B and C of ˚ , respectively, are insensitive to the values of the Figure 27.3, calculated for r0 ¼ 1.7 and 1.3 A reduced mass of the promoting mode. In other words, a change in its value simply shifts the KIE and its temperature dependence along the curve. The van der Waals force field implied by the chosen tunneling distance calls for relatively large masses and low frequencies. Curve B is close to curve A, the limiting value for very large tunneling distances. If withdrawal of electron density by the redox system causes hydrogen bonding, the tunneling distance will be reduced. Correspondingly, there should be a decrease of the tunneling-mode frequency and an increase in anharmonicity. The values chosen for the calculation are a tunneling ˚ , frequencies vH =vD ¼ 2850/2000 cm21 and corresponding anharmonicities distance r0 ¼ 1.0 A H D X =X ¼ 2 120/2 60 cm21. The reduced tunneling distance will be accompanied by a strengthening of the promoting-mode force field and will bring its reduced mass closer to that of a CO molecule. Using a value M ¼ 10, we obtain curve D in Figure 27.3. If the transfer is exo- or endothermic, transition from excited levels of the tunneling mode may become significant. The effect on the KIE becomes larger the smaller the tunneling distance. Curves E and F are calculated for exo(endo)thermicities of 4 and 10 kcal mol21, respectively, the other parameters being the same as those of curve D. The resulting reduction in the temperature dependence of the KIE increases with increasing KIE and decreasing tunneling distance. Curve G is calculated for the same tunneling distance, tunneling-mode frequency, and promoting-mode mass as curve D, but for a quartic potential of the form of Equation 27.11. The two curves are very similar in the range of interest. ˚ is realistic for a hydrogen It seems doubtful whether a tunneling distance lower than about 1.0 A bond involving a CH fragment. In view of the large distortion of the CH and OH stretching potentials caused by such strong hydrogen bonding, the quartic potential seems to be more realistic than crossing Morse potentials. For illustrative purposes, we show curve H calculated with ˚ , a tunneling-mode frequency a quartic potential for a tunneling distance r 0 ¼ 0.6 A 21 H D v =v ¼ 2700/1900 cm , and a promoting-mode mass M ¼ 8.

IV. APPLICATIONS A. COENZYME B 12 Coenzyme B12 is a cobalamine (cobalt-centered complex, Cbl) that, under appropriate stimulation by the substrate or solvent, releases a ligand in the form of a free radical.12 Normally this ligand is 50 -deoxyadenosyl (Ado). From experimental and theoretical studies of cobalamines with other ligands, it is found that the rate of homolysis increases rapidly with the size of the ligand.29 This suggests that the thermal homolysis is spontaneous and aided by repulsive interactions. The radical formed in this manner subsequently abstracts a hydrogen from a CH bond of the substrate or solvent. This process shows a strong, temperature-dependent KIE. Chowdhury and Banerjee14 have measured the KIE of this hydrogen abstraction step in the reaction catalyzed by methylmalonyl-coenzyme A mutase (MMCoAM). More recently, Doll and Finke17 studied the same step in three different cobalamines, including coenzyme B12

736

Isotope Effects in Chemistry and Biology

(AdoCbl), dissolved in ethylene glycol, no protein being present. They showed that the KIEs of all three tested ligands fit the same linear Arrhenius plot as the KIEs of Chowdhury and Banerjee,14 from which they concluded that the presence of the protein has no significant effect on the hydrogen transfer rate. Our analysis of the same data is summarized in Figure 27.3 in the form of a comparison of the observed KIEs and their temperature dependence with curve B, calculated for a tunneling distance ˚ . The point labeled 1 represents the Chowdhury and Banerjee data for MMCoAM and of 1.7 A the points 2 and 3 the Doll and Finke data for AdoCbl (as well as for MeOAdoCbl, which was kinetically indistinguishable from AdoCbl) and neopentyl-Cbl, respectively. The plotted data are listed in Table 27.1. The minimum van der Waals separation between methyl and/or methylene ˚ ; earlier theoretical and experimental studies of hydrogen transfer between a groups is about 4 A ˚ ,30 supported by direct methyl radical and the methyl group of methanol led to an estimate of 3.9 A 31 measurements based on electron spin resonance methods. Hence, the expected minimum ˚ , the value represented by curve B in Figure 27.3. tunneling distance is about 1.7 A This curve provides an excellent fit to the solution data of Doll and Finke and hence agrees with the value for the tunneling distance derived from van der Waals radii and supported by direct measurements on the methyl – methanol system. Point 1, which represents the enzymatic reaction, shows a larger temperature dependence than predicted on the basis of this tunneling distance. This may indicate a larger tunneling distance in the enzyme, which is not unreasonable since the methylmalonyl-CoA substrate, from which the hydrogen is abstracted, is much bulkier than the ethylene glycol that replaces it in the experiments of Doll and Finke. However, since an increase by ˚ , which would not be enough for a good fit, reduces the rate by more than an order of 0.1 A magnitude, the relatively fast rates observed by Chowdhury and Banerjee do not support such an interpretation. In fact, these authors caution that the KIEs they report may represent a lower limit, due to the possibility of a rapid equilibrium between coenzyme B12 and cobalt(II)alamin in the active site. A comparison of Figure 3 of Ref. 13 with Figure 2 and Table 1 of Ref. 14 suggests that the enzyme – substrate equilibrium constant Kd derived along with the tunneling rate constant by13 kf and14 kþ2H=D is not isotope-independent, but shows a KIE of about 3, coincidentally the same amount that separates the reported KIE of the proton transfer step from the prediction of curve B. In view of these uncertainties, it seems advisable to assume that the same tunneling distance of about ˚ applies to all these experiments, which supports the conclusion of Doll and Finke,17 drawn 1.7 A from a plot similar to Figure 27.2, that the effect of the enzyme on the hydrogen transfer rate is minimal.

B. LIPOXYGENASE Lipoxygenase is a metallo-enzyme containing a six-coordinated iron ion that alternates between Fe(III) and Fe(II) during the catalytic cycle. The proposed first step in the cycle is one-electron oxidation by Fe(III)OH accompanied by proton transfer to yield a delocalized radical and ˚ x-ray structure is available for the enzyme without the substrate. The probable Fe(II)OH2. A 1.4 A location of the substrate (in this case linoleic acid) in one of the available pockets inside the protein has been identified, but the exact position of the methylene group (C-11) remains unknown. The analysis of the electron –proton transfer step is based on data obtained from a study of soybean lipoxygenase-1 by Klinman’s group.3,4 These authors measured the rate of hydrogen abstraction from the C-11 methylenic group in the temperature range 278– 323 K for linoleic acid and two isotopomers, namely 11-(S)-[2H] and 11,11-[2H2] linoleic acid and reported values kcat ¼ 3 –300 sec21 and KIEs of about 80. The rate constants show a weak temperature dependence in this range, the apparent activation energy being about 2 kcal mol21. The study also included SLO1 mutants in which a bulky amino acid near the active site is replaced by alanine. From the data reported in (the Supplementary Material of) Ref. 4, it follows that the difference in the rate constants of the wild-type enzyme and the I553A mutant has no

Mechanisms of CH-Bond Cleavage Catalyzed by Enzymes

737

statistical significance,19 which indicates that the mutation concerns a nonessential amino acid. To reduce the statistical error, we therefore combine these two data sets. The combined data set is represented by point 4 in Figure 27.3. The other mutants, L546A and L754A, show a strongly reduced activity relative to the wild-type enzyme, although their temperature-dependent KIEs remain virtually unchanged. This suggest that the proton-transfer step is not significantly affected by the mutation. The corresponding, nearly coincident points 5 and 6 in Figure 27.3 are close to point 4 of the wild-type enzyme. The data points used are listed in Table 27.1. Early attempts to model the KIEs and their temperature dependence4 based on a two-oscillator model similar to that of Ref. 9, but with harmonic rather than Morse oscillators, led to an estimate of ˚ for the tunneling distance, a value half of that deduced from van der Waals radii. No 0.6 –0.7 A convincing reason could be given for this small value. The curves displayed in Figure 27.3 suggest that the introduction of an attractive interaction into the tunneling potential, to reduce the tunneling ˚. frequency and introduce strong anharmonicity, can increase this estimate to approximately 1.0 A ˚ Points 4 –6 roughly fit curves D and G, calculated for a tunneling distance r0 ¼ 1.0 A and a frequency v H/v D ¼ 2850/2000 cm21. Curve D represents the tunneling mode Morse potentials with an anharmonicity X H/X D ¼ 2 120/2 60 cm21, while curve G represents a quartic potential. The tunneling potentials used to calculate curves D and G imply hydrogen bonding between the methylene group of linoleic acid and the OH ligand of the iron ion. The strong hydrogen bond in, ˚ .32 The present result of about e.g., malonaldehyde gives rise to a tunneling distances of 0.74 A ˚ ˚ 1.0 A lies between this value and that of 1.3 A based on van der Waals radii. It thus represents moderate hydrogen bonding. Normally, CH bonds are not involved in hydrogen bonding of this strength. The present case is special, however, because of the proximity of the redox system, which is strong enough to abstract an electron during the hydrogen transfer. Hence this interpretation not only leads to a tunneling potential based on a physically acceptable relation between the tunneling distance and the shape of the potential, but also assigns a specific role in the transfer dynamics to the redox system.

C. PRIMARY A MINE D EHYDROGENASES Methylamine dehydrogenase catalyzes the oxidative demethylation of primary amines to form aldehydes and ammonia. In these reactions, studied by the groups of Davidson7 and Scrutton,5,6 proton abstraction by an active-site base from an iminoquinone intermediate is accompanied by electron transfer to a tryptophylquinone cofactor. The basic mechanism is very similar to that of lipoxygenase, although the enzyme has no metal in the active site. The rate constants reported for two substrates, methylamine and ethanolamine, are also in a similar range, but are subject to activation energies of 10 –15 kcal mol21. The KIEs reported are somewhat smaller than those for SLO1 and show a weak to very weak temperature dependence, as illustrated in Figure 27.2. However, the reaction with methylamine was found not to follow standard Michaelis– Menten kinetics. Specifically, the kinetic analysis led to rate constants for complex formation kform and H D H D dissociation kdiss that are isotope-dependent: kform =kform ¼ 6:4 and kdiss =kdiss ¼ 19; values comparable to the KIE for the hydrogen transfer step. This is not realistic and indicates that the kinetic scheme used to separate the hydrogen transfer step from the other reaction steps was not successful. We therefore make no attempt to interpret the temperature dependence of the KIE of methylamine (point 7 in Figure 27.3). It does not fit curve F, which combines a tunneling distance of ˚ with the maximum endothermicity compatible with the apparent activation energy; although 1.0 A ˚ we consider to be unrealistic. it roughly fits curve H, the corresponding tunneling distance of 0.6 A The data for ethanolamine, represented by point 8 in Figure 27.3, do not show this anomalous behavior. This point fits curves D and G, i.e., the curves used to fit the SLO1 data, as well as curve E, which includes a small endothermicity. Hence to interpret this result, the same arguments apply as those presented in the preceding subsection.

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Aromatic amine dehydrogenase (AADH) catalyzes the same reaction as MADH except the substrates are aromatic rather than aliphatic amines. The same group6 measured the KIE for three substrates and found that it increases in the order benzylamine , dopamine , tryptamine. This was surprising since the rate constants increase in the reverse order. A larger KIE implies a higher and wider barrier, which in turn implies a lower rate of transfer. The anomalous result suggests that the kinetic scheme used did not succeed in separating the tunneling step from other steps in the reaction mechanism, a problem similar to that encountered in the case of methylamine discussed in the preceding subsection. This would not be surprising since this scheme assumes that there are two separate and reversible steps after complex formation and before proton transfer. In the kinetic analysis the steps prior to tunneling are represented by a single equilibrium constant, which exhibits some isotope dependence but less so than that of methylamine. The temperature dependence of the KIE could only be measured for benzylamine and dopamine. As in the case of methylamine, the corresponding points 9 and 10 in Figure 27.3 correspond to tunneling distances that we consider to be unphysically small.

D. DICOPPER C OMPLEXES There are several enzymes (monooxygenases) in which the active site contains a copper complex that binds and activates dioxygen such that it abstracts a hydrogen from an alkyl group. To model the active site of the enzyme, bis(m-oxo)dicopper complexes have been synthesized in which the hydrogen is abstracted from an N-alkyl copper ligand.23 Prior to or simultaneous with the proton abstraction, an electron transfers from this ligand to a copper ion, which is thereby reduced from CuIII to CuII. The proton transfers to one of the two oxygen atoms that bridge the copper ions. This reaction does not require the presence of a protein and also avoids the kinetic complications inherent in enzyme – substrate complexation. Otherwise the mechanism is similar to that of SLO1 and shows similar behavior when the ligands are deuterated, namely KIEs that are large but only weakly temperature dependent, as illustrated in Figure 27.2. The corresponding data point is labeled 11 in Figure 27.3. The data fit the same model curve found appropriate for the SLO1 reaction and for enzymatic demethylation of ethanolamine. As in these reactions, we propose that the dicopper center acts as a redox system that withdraws electron density from the ligand so as to give rise to a moderately strong C – H· · ·O bond, which serves as a proton conduit.

V. DISCUSSION An easy way to cleave a CH bond is to abstract the hydrogen by means of a free radical. However, such radicals tend to be agressive and nonselective, which is probably the main reason for the emergence of ionic proton abstraction as an alternative mechanism in biology. Proton abstraction from a CH bond is very difficult, however, since this bond is essentially nonpolar. The approach of a functional group with a large proton affinity is hindered by a wall of electronic repulsion. An obvious way to reduce this repulsion is withdrawal of electron density. The presence of a strong redox system at the reaction site suggests that evolution has indeed taken this route. The AdoCHz2 radical formed in the MMCoAM reaction remains harmless since it does not escape but immediately abstracts a hydrogen from the substrate, which subsequently undergoes the 1,2-rearrangement that is the purpose of the process. The rearranged radical then reverses the cycle by abstracting a hydrogen from the AdoCH3 moiety leading to reformation of AdoCbl. The efficacy of this process clearly hinges on the Co – ligand dissociation energy, which is apparently tuned by the protein and the substrate, and on the rearrangement of the radicalized substrate. The strength of the interaction driving the carbon-centered free radical reaction33 allows the hydrogen transfer to proceed efficiently despite the relatively large tunneling distance. The observed KIEs and their temperature dependence both for the enzyme13,14 and the model16,17 are in line with theoretical predictions and with data derived earlier for a simple organic free radical reaction.30

Mechanisms of CH-Bond Cleavage Catalyzed by Enzymes

739

The situation is quite different for proton transfer from a CH to an OH bond. Here the tunneling distance may be smaller but the interaction is intrinsically very weak and would give rise to high barriers. However, a strong redox system is present in the active sites of these systems and abstracts an electron from the CH bond in the course of the hydrogen transfer. We can visualize this effect by assuming that the electron and proton move separately, namely towards different destinations, but simultaneously, namely within the same time interval, so that in the transition state of the proton transfer reaction roughly half an electron is withdrawn. This will drastically lower the width and the energy of the barrier and correspondingly increase the anharmonicity of the CH and OH stretching vibrations, a condition normally referred to as hydrogen bonding. In this situation proton tunneling at a biologically acceptable rate is made possible by the contribution of the C· · ·O stretching vibration. The relatively high frequency of this vibration causes the weak temperature dependence of the KIE. The KIE remains large because the redox assistance induces only moderate strength into the C – H· · ·O hydrogen bond. The experimental evidence supporting this interpretation is not without ambiguity. While all ˚ derived from van der Waals the data1 – 8 support a tunneling distance smaller than the value of 1.3 A 3 – 5,23 ˚ radii, some are consistent with a value of about 1.0 A indicative of moderate hydrogen bonding, while others suggest much smaller distances5,6 indicative of very strong hydrogen bonding. However, the latter data show internal inconsistencies and are considered less reliable ˚ will prove to be than the former. We therefore suspect that a tunneling distance of about 1.0 A typical for all these systems. A value of this magnitude was also proposed in a recent theoretical study of the SLO1 reaction by Lehnert and Solomon.20 Although this result was not based on full optimization of the C· · ·O distance, it was found to be favored by the energetics of the reaction.

ACKNOWLEDGMENTS Helpful discussions with Rick Finke are gratefully acknowledged.

REFERENCES 1 Bahnson, B. J. and Klinman, J. P., Hydrogen tunneling in enzyme catalysis, Methods Enzymol., 249, 374– 398, 1995. 2 Kohen, A. and Klinman, J. P., Enzyme catalysis: beyond classical paradigms, Acc. Chem. Res., 31, 397– 404, 1998; Kohen, A., and Klinman, J. P., Hydrogen tunneling in biology, Chem. Biol., 6, R191– R198, 1999; Rickert, K. W. and Klinman, J. P., Computational study of tunneling and coupled motion in alcohol dehydrogenase-catalyzed reactions: implication for measured hydrogen and carbon isotope effects, J. Am. Chem. Soc., 121, 1997– 2004, 1999; Kohen, A., Cannio, R., Bartolucchi, S., and Klinman, J. P., Enzyme dynamics and hydrogen tunneling in a thermophilic alcohol dehydrogenase, Nature, 399, 496– 499, 1999; Cha, Y., Murray, C. J., and Klinman, J. P., Hydrogen tunneling in enzyme reactions, Science, 243, 1325– 1330, 1989. 3 Rickert, K. W. and Klinman, J. P., Nature of hydrogen transfer in soybean lipoxygenase 1: separation of primary and secondary isotope effects, Biochemistry, 38, 12218 –12228, 1999. 4 Knapp, M. J., Rickert, K. W., and Klinman, J. P., Temperature-dependent isotope effects in soybean lipoxygenase-1: correlating hydrogen tunneling with protein dynamics, J. Am. Chem. Soc., 124, 3865– 3875, 2002. 5 Basran, J. B., Sutcliffe, M. J., and Scrutton, N. S., Enzymatic H-transfer requires vibration-driven extreme tunneling, Biochemistry, 38, 3218– 3222, 1999. 6 Basran, J. B., Patel, S., Sutcliffe, M. J., and Scrutton, N. S., Importance of barrier shape in enzymecatalyzed reactions, J. Biol. Chem., 276, 6234– 6242, 2001. 7 Brooks, H. B., Jones, L. H., and Davidson, V. L., Deuterium kinetic isotope effect and stopped-flow kinetic studies of the quinoprotein methylamine dehydrogenase, Biochemistry, 32, 2725– 2729, 1993;

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8

9 10 11 12 13

14

15 16 17

18 19 20 21

22 23

24 25

Hyun, Y. L. and Davidson, V. L., Unusually large isotope effect for the reaction of aromatic amine dehydrogenase. A common feature of quinoproteins?, Biochim. Biophys. Acta., 1251, 198– 200, 1995. Nesheim, J. C. and Lipscomb, J. D., Large kinetic isotope effects in methane oxidation catalyzed by methane monooxygenase: evidence for C– H bond cleavage in a reaction cycle intermediate, Biochemistry, 35, 10240– 10247, 1996; Brazeau, B. J., Wallar, B. J., and Lipscomb, J. D., Unmasking of deuterium kinetic isotope effects on the methane monooxygenase compound Q reaction by sitedirected mutagenesis of component B, J. Am. Chem. Soc., 123, 10421 –10422, 2001; Brazeau, B. J., Austin, R. N., Tarr, C., Groves, J. T., and Lipscomb, J. D., Intermediate Q from soluble methane monooxygenase hydroxylates the mechanistic substrate probe norcarane: evidence for a stepwise reaction, J. Am. Chem. Soc., 123, 11831– 11837, 2001. Siebrand, W., Wildman, T. A., and Zgierski, M. Z., Golden Rule treatment of hydrogen-transfer reactions. 1. Principles, J. Am. Chem. Soc., 106, 4083– 4089, 1984; 2. Applications. J. Am. Chem. Soc., 106, 4089– 4096, 1984. Kuznetsov, A. M. and Ulstrup, J., Proton and hydrogen atom tunneling in hydrolytic and redox enzyme catalysis, Can. J. Chem., 77, 1085– 1096, 1999. Tresadern, G., McNamara, P., Mohr, M., Wang, H., Burton, N. A., and Hillier, I. H., Calculations of hydrogen tunneling and enzyme catalysis: a comparison of liver alcohol dehydrogenase, methylamine dehydrogenase and soybean lipoxygenase, Chem. Phys. Lett., 358, 489– 494, 2002. Halpern, J., Mechanism of coenzyme B12-dependent rearrangements, Science, 227, 869– 875, 1985. Chowdhury, S. and Banerjee, R., Thermodynamic and kinetic characterization of Co – C bond homolysis catalyzed by coenzyme B12-dependent methylmalonyl-CoA mutase, Biochemistry, 39, 7998– 8006, 2000. Chowdhury, S. and Banerjee, R., Evidence of quantum mechanical tunneling in the coupled cobalt – carbon bond homolysis – substrate radical generation reaction catalyzed by methylmalonyl-CoA mutase, J. Am. Chem. Soc., 122, 5417– 5418, 2000. DybaŁa-Defratyka, A. and Paneth, P., Theoretical evaluation of the hydrogen kinetic isotope effect on the first step of the methylmalonyl-CoA mutase reaction, J. Inorg. Biochem., 86, 681– 689, 2001. Doll, K. M., Bender, B. R., and Finke, R. G., The first experimental test of the hypothesis that enzymes have evolved to enhance hydrogen tunneling, J. Am. Chem. Soc., 125, 10877– 10885, 2003. Doll, K. M. and Finke, R. G., A compelling experimental test of the hypothesis that enzymes have evolved to enhance quantum mechanical tunneling in hydrogen transfer reactions: the b-neopentylcobalamin system combined with prior adocobalamin data, Inorg. Chem., 42, 4849– 4857, 2003. Frey, P. A., Importance of organic radicals in enzymatic cleavage of unactivated C– H bonds, Chem. Rev., 90, 1343– 1357, 1990. Siebrand, W. and Smedarchina, Z., Temperature dependence of kinetic isotope effects for enzymatic carbon– hydrogen bond cleavage, J. Phys. Chem. B, 108, 4185– 4195, 2004. Lehnert, N. and Solomon, E. I., Density-functional investigation on the mechanism of H-atom abstraction by lipoxygenase, J. Biol. Inorg. Chem., 8, 294– 305, 2003. Fauler, P. F., Tresadern, G., Chohan, K. K., Scrutton, N. S., Sutcliffe, M. J., and Burton, N. A., QM/ MM studies show substantial tunneling for the hydrogen-transfer reaction in methylamine dehydrogenase, J. Am. Chem. Soc., 123, 8604 –8605, 2001. Alhambra, C., Sa´nchez, M. L., Corchado, J. C., Gao, J., and Truhlar, D. J., Erratum to: quantum mechanical tunneling in methylamine dehydrogenase [Chem. Phys. Lett. 347, 512– 518, 2001], Chem. Phys. Lett., 355, 388– 394, 2002. Mahapatra, S., Halfen, J. A., and Tolman, W. B., Mechanistic study of the oxidative N-dealkylation reactions of bis(m-oxo)dicopper complexes, J. Am. Chem. Soc., 118, 11575– 11586, 1996. Only the complex with benzyl ligands is used in our analysis, since a peroxo isomer may contribute to the complex with isopropyl ligands, Tolman, W.B., Personal communication, 2003. Smedarchina, Z. and Siebrand, W., Multimode approach to hydrogen tunneling, Chem. Phys., 170, 347– 359, 1993. Benderskii, V. A., Makarov, D. E., and Wight, C. H., Chemical dynamics at low temperatures, Adv. Chem. Phys., 88, 1 – 385, 1994.

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26 Siebrand, W., Smedarchina, Z., Zgierski, M. Z., and Ferna´ndez-Ramos, A., Proton tunneling in polyatomic molecules: a direct dynamics instanton approach, Int. Rev. Phys. Chem., 18, 5 – 41, 1999; Smedarchina, Z., Ferna´ndez-Ramos, A., and Siebrand, W., DOIT: a program to calculate thermal rate constants and mode-specific tunneling splittings directly from quantum-chemical calculations, J. Comp. Chem., 22, 787– 801, 2001. 27 Smedarchina, Z. and Siebrand, W., Generalized Swain-Schaad relations including tunneling and temperature dependence, Chem. Phys. Lett., 410, 370– 376, 2005. 28 Herzberg, G., Molecular Spectra and Molecular Structure. I. Diatomic Molecules, 2nd ed., D. van Nostrand Company, Princeton, 1950. 29 Andruniow, T., Zgierski, M. Z., and Kozlowski, P., Density functional theory analysis of stereoelectronic properties of cobalamins, J. Phys. Chem. B, 104, 10921– 10929, 2000; New light on the Co– C bond activation in B12-dependent enzymes from density functional theory. Chem. Phys. Lett., 331, 509– 513, 2000; Theoretical determination of the Co – C bond energy dissociation in cobalamins. J. Am. Chem. Soc., 123, 2679 –2690, 2001. 30 Doba, T., Ingold, K. U., Siebrand, W., and Wildman, T. A., Hydrogen abstraction by methyl radicals in glasses, Faraday Discuss. Chem. Soc., 78, 175– 191, 1984. 31 Doba, T., Ingold, K. U., Reddoch, A. H., Siebrand, W., and Wildman, T. A., Electron spin resonance (ESR) investigation of the structure of methyl radical trapping sites in methanol glass, J. Chem. Phys., 86, 6622– 6630, 1987. 32 Baughcum, S. L., Duerst, R. W., Rowe, W. F., Smith, Z., and Wilson, E. B., Microwave spectroscopic study of malonaldehyde (3-hydroxy-2-propenal) 2. Structure, dipole moment and tunneling, J. Am. Chem. Soc., 103, 6296– 6303, 1981. 33 Wildman, T. A., An ab initio quantum chemical study of hydrogen abstraction from methane by methyl, Chem. Phys. Lett., 126, 325– 329, 1986.

28

Kinetic Isotope Effects as Probes for Hydrogen Tunneling in Enzyme Catalysis Amnon Kohen

CONTENTS I.

II.

III.

IV.

Introduction ...................................................................................................................... 744 A. Enzyme Catalysis ..................................................................................................... 744 B. The Chemical Step: Contributions of Quantum Mechanical Tunneling, Equilibrium Fluctuations, and Dynamics ................................................................ 744 Kinetic Isotope Effects as Probes of the Chemical Step................................................. 745 A. Semiclassical Relationship of Reaction Rates of H, D, and T ............................... 746 B. Primary (18) Swain – Schaad Relationship............................................................... 746 1. Intrinsic 18 KIE ................................................................................................. 746 2. Experimental Examples Using Intrinsic 18 KIE............................................... 748 a. Peptidylglycine a-Hydroxylating Monooxygenase .................................... 748 b. Thymidylate Synthase ................................................................................. 748 c. Dihydrofolate Reductase ............................................................................. 748 C. Secondary (28) Swain– Schaad Relationship........................................................... 748 1. Mixed Labeling Experiments as Probes for Tunneling and 18– 28 Coupled Motion ............................................................................... 749 2. Upper Semiclassical Limit for 28 Swain – Schaad Relationship....................... 750 a. Zero-Point Energy and Reduced Mass Considerations .............................. 750 b. Vibrational Analysis.................................................................................... 751 c. Effect of Kinetic Complexity ...................................................................... 751 d. The New Effective Upper Limit ................................................................. 752 3. Experimental Examples Using 28 Swain – Schaad Exponents.......................... 753 a. Horse Liver Alcohol Dehydrogenase.......................................................... 753 b. Thermophilic ADH from Bacillus stearothermophilus (ADH-hT)............ 753 Temperature Dependence of KIEs .................................................................................. 753 A. Temperature Dependence of Reaction Rates and KIEs .......................................... 753 B. KIEs on Arrhenius Activation Factors .................................................................... 754 C. Experimental Examples Using Isotope Effects on Arrhenius Activation Factors .................................................................................................... 755 1. Soybean Lipoxygenase-1 .................................................................................. 755 2. Thermophilic ADH (ADH-hT) ......................................................................... 756 Theoretical Approaches ................................................................................................... 757 A. Phenomenological “Marcus-Like” Models ............................................................. 757 B. QM/MM Models and Simulations........................................................................... 758

743

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Isotope Effects in Chemistry and Biology

V. Comparison to Studies of Nonenzymatic Reactions ....................................................... 759 VI. Conclusions ...................................................................................................................... 760 References..................................................................................................................................... 760

I. INTRODUCTION A. ENZYME C ATALYSIS In biological systems, enzymes are the catalysts that direct, control and enhance chemical transformations. Enzymes evolved to accomplish two almost contradictory tasks: one is catalyzing a reaction at a rate most suitable for organism function and the other is preventing alternative sideprocesses that would commonly occur in nonenzymatic reactions. In other words, an enzyme not only catalyzes the reaction of interest, it also inhibits side reactions and the formation of byproducts. Commonly, the first effect is denoted as catalysis and the second is denoted as specificity. The rate enhancement is often many orders of magnitude greater than the reaction in solution, and of substantial biological importance. Catalysis, or “catalytic power,” is the ratio between the reaction rate of the catalyzed reaction and that of the uncatalyzed reaction. It is defined as kcat =kun where kcat is the rate of the catalyzed reaction and kun is the rate of the uncatalyzed reaction. By definition, catalysis should be unit-less (a ratio of rate constants), thus care must be practiced while determining catalytic power that kcat and kun have the same units. Enzyme catalysis (also see Chapter 30 by Paul Cook) can be studied from various points of view: regulation, structural aspects, order of reactant binding and product release, the role of functional residues (e.g., general bases or acids), etc. This chapter presents the use of kinetic isotope effects (KIEs) as tools for studying the physical nature of enzyme catalyzed C – H bond activation.

B. THE C HEMICAL S TEP: C ONTRIBUTIONS OF Q UANTUM M ECHANICAL T UNNELING, E QUILIBRIUM F LUCTUATIONS, AND DYNAMICS Many experimental and theoretical studies have attempted to assess the specific contributions of physical phenomena to enzymatic rate enhancement. These contributions include: the relations between the reactive complex’s structure and function; the location and structure of the transition state (TS) along the reaction coordinate; TS stabilization; ground state (GS) destabilization; the significance of quantum mechanical (QM) phenomena; the dynamics of the system; energy distribution through the normal modes of an enzyme; and contributions of entropy and enthalpy. An attempt to “break” catalysis into additive contributions is always a somewhat artificial process. The quantitative degree of each contribution is inherently model dependent. For example, the contributions of electrostatics and dynamics are nonadditive as the latter includes variations in the electrostatic field. Nevertheless, once such an aspect is defined it is of interest to assess its importance to catalysis. Three relevant terms need to be clearly defined, namely, QM tunneling, equilibrium fluctuations and dynamics: Quantum mechanical tunneling: Tunneling is the phenomenon by which a particle transfers through a reaction barrier by means of its wave-like properties.1 Figure 28.1 graphically illustrates this phenomenon for a symmetric double well system such as the C – H –C hydrogen transfer. It is important to note that the tunneling probability is affected by both the distance between the R and the P wells and their symmetry. A lighter isotope has a higher tunneling probability than a heavier one, since a heavy isotope has a lower zero-point energy (ZPE) and its probability function is more localized in its well. Consequently, KIEs are effective tools for studying tunneling. Two practical applications are described below: the Swain– Schaad exponential relationship (Section II) and temperature dependency of KIEs (Section III). Dynamics vs. equilibrium fluctuations: The definition of dynamics, and the distinction between dynamic motion and entropy effects, is controversial.2 – 5 In a previous review, we considered any

Kinetic Isotope Effects as Probes for Hydrogen Tunneling

745

E

R P Reaction Coordinate

FIGURE 28.1 An example of ground-state nuclear tunneling. The reactant well (R) is on the left and the product well (P) is on the right. The fine lines represent the probability (nuclear c 2) of finding the nuclei in the reactant or the product wells. More overlap between the probability functions of the R and P results in higher tunneling probability.

nuclear motion as a dynamic phenomenon regardless of whether or not in thermal equilibrium with the environment (Boltzman distribution).6 Here we will use the terminology of Benkovic, HammesSchiffer, Warshel and others, in which dynamics are only nonequilibrium motions along the reaction coordinate.3,7 – 9 Yet, this rule cannot be strictly followed since terms like “protein dynamics” or “molecular dynamics” are commonly used to denote nuclear motion that is in thermal equilibrium with the environment.

II. KINETIC ISOTOPE EFFECTS AS PROBES OF THE CHEMICAL STEP KIE is the ratio of rates of two isotopologue reactants (molecules that only differ in their isotopic composition). For H-transfer reactions, this ratio of rates between light and heavy isotopes is characteristic of the reaction coordinate and the nature of the transition state. KIE results from energy of activation differences for the different isotopologue reactants, and much of its magnitude is due to the differences of ZPE between the ground state and the TS of the reaction:10 ‡

‡

kH =kD < eðDGD 2DGH Þ=RT

ð28:1Þ

where DG‡D 2 DG‡H < ZPE‡D 2 ZPERD 2 ZPE‡H þ ZPERH ; R is the gas constant and T the absolute temperature. The KIE measured for a bond cleavage or formation is depicted as primary (18) KIE. The KIE measured with isotopologues that are labeled on a position other than the one that is being cleaved, is denoted as secondary (28) KIE. 28 KIEs result from a change in bonding force constants and vibrational frequencies during the reaction (e.g., s (s – sp3) to s (s –sp2)). Secondary KIEs are normally smaller or equal to the reaction’s equilibrium isotope effect (EIE). The 28 EIE, results from the change in bond order from reactants to products, while the KIE is only affected by the change from the GS to the TS. In the investigation of KIEs and their temperature dependence, explicit care must be exercised regarding several questions: (i) Is the KIE measured on a single kinetic step (e.g., internal KIE) or on a kinetically complex rate constant (e.g., V=K or kcat)? (ii) Is there an isotope effect on steps other than the one under investigation (e.g., binding, see Chapter 42 by Vern L. Schramm and Brett Lewis)? (iii) What is the effect of kinetic steps that are not isotopically sensitive but still mask the isotope effect?

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(iv) Are the KIEs measured for the same rate constant throughout the whole temperature range (when examined)? (v) Are all the experimental conditions (e.g., pH, ionic strength) the same at all temperatures? The first three questions are related to “kinetic complexity,” which reflects the fact that the observed KIE (KIEobs) is often smaller than the intrinsic KIE (KIEint). This is due to the ratio between the isotopically sensitive step and the isotopically nonsensitive steps that lead to the decomposition of the same reactive complex. The mathematical treatment is rigorously described in several published reviews11,12 and Chapter 37 (by Wallace Cleland) in this volume. KIEobs ¼

KIEint þ Cf þ Cr ·EIE 1 þ Cf þ Cr

ð28:2Þ

where EIE is the equilibrium isotope effect and Cf and Cr are the forward and reverse commitments to catalysis, respectively. Cf is the ratio between the rate of the isotopically sensitive step forward (e.g., kH-transfer) and the rates of the preceding isotopically nonsensitive steps backward. Cr is the ratio between the rate of the isotopically sensitive step backward and the rates of the succeeding isotopically nonsensitive steps forward. Techniques that allow estimation of the intrinsic effect from the observed one are discussed in Section II.B.1 and in Chapter 37 (by Wallace Cleland). The importance of intrinsic KIE is that it imposes a strict constraint on any mechanism, theoretical model, analysis, or simulation addressing the chemistry under study. As described in the following section, intrinsic KIEs are unique as they are directly affected by the reaction potential surface and other physical features.

A. SEMICLASSICAL R ELATIONSHIP OF R EACTION R ATES OF H , D , AND T The kinetic relationship among the three isotopes of hydrogen has wide usage as a mechanistic tool in organic and physical chemistry. The Swain – Schaad exponential relationship (EXP, as defined in Equation 28.3 to Equation 28.5) is the semiclassical (no tunneling) correlation among the rates of the three isotopes of hydrogen, and was first defined by Swain et al. in 1958.13 This relationship can be predicted using the masses of the isotopes under examination.14 The EXP results primarily from ZPE differences between the ground state and the transition state. Since H has a higher GS ZPE than D and T, it should react faster than D and T (Figure 28.2). Several investigators examined this relationship under extreme temperatures (20 to 1000 K), and as a probe for tunneling.15 – 17 This isotopic relationship was also used in experimental and theoretical studies to suggest a coupled motion between primary and secondary hydrogens for hydride transfer reactions, such as elimination in the gas phase, and in organic solvents.18 – 20 Two common uses of the Swain – Schaad relationship in enzymology are described in the following sections.

B. PRIMARY (18) S WAIN – S CHAAD R ELATIONSHIP 1. Intrinsic 18 KIE The Swain –Schaad exponential relationship was originally defined for primary (18) KIEs:13 kH ¼ kT

kH kD

EXP

or EXP ¼

lnðkH =kT Þ lnðkH =kD Þ

ð28:3Þ

where ki is the reaction rate constant for isotope i. EXP can be calculated from14 EXP ¼

pffiffiffiffi pffiffiffiffi 1= mH 2 1 mT lnðkH =kT Þ ¼ pffiffiffiffi pffiffiffiffi lnðkH =kD Þ 1= mH 2 1 mD

ð28:4Þ

Kinetic Isotope Effects as Probes for Hydrogen Tunneling

747

H D T

Ea semiclassical

H

E

D T

Oth

Reaction er

Mo

te

Coordina

de

s

FIGURE 28.2 Different energies of activation ðDEa Þ for H, D, and T resulting from their different zero-point energies (ZPE) at the ground state and transition state. The GS-ZPE is constituted by all degrees of freedom but mostly by the C– H stretching frequency, and the TS-ZPE is constituted by all degrees of freedom orthogonal to the reaction coordinate. This type of consideration is depicted as “semiclassical.”

where mi is the reduced mass for isotope i. The original EXP was calculated for H/T vs. H/D KIEs and yielded a value of 1.44 (using atomic masses). Some labeling patterns use T as a frame of reference and to compare H and D. Their EXP follows: pffiffiffiffi pffiffiffiffi 1= mH 2 1 mT lnðkH =kT Þ ¼ pffiffiffiffi EXP ¼ pffiffiffiffi lnðkD =kT Þ 1= mD 2 1 mT

ð28:5Þ

Equation 28.5 defines the relationship of H/T to D/T KIEs, for which the semiclassical EXP is 3.26 (for atomic masses: mi ¼ mi Þ: Importantly, these relationships are almost independent of the shape of the reaction potential surface.14 Both measurements (Equation 28.4 and Equation 28.5) establish the relationship among the three isotopes for a one step reaction (single barrier). If the chemical step is masked by kinetic complexity (Equation 28.2) the observed KIE (KIEobs) will be smaller than the intrinsic one. In the case of Equation 28.5, this will affect H/T KIE more than D/T KIE and the observed EXP will be smaller than the intrinsic one. For the experiment pertinent to Equation 28.4, such kinetic complexity will have the opposite effect and the observed EXP will be larger than the intrinsic one. Measuring the H/D/T relationship, either as H/T vs. H/D or H/T vs. D/T KIEobss, together with the relevant Swain – Schaad relationship, affords a system with only one unknown, namely, the intrinsic KIE. Northrop12,21,22 developed a method for calculating the intrinsic KIE from the observed KIEs. This method assumes no significant deviation of the intrinsic 18 KIE from their semiclassically predicted values, p and is described in detail in Chapter 37 (by Wallace Cleland) in this volume. p It is important to note that for 18 KIEs and the resulting 18 Swain–Schaad exponents, no EXP values larger than 3.6 were found in the literature. Even in cases in which tunneling was evident from mixed labeling experiments (Section II.C) or from the temperature dependency of KIEs (Section III), the 18 EXP did not significantly differ from the semiclassically predicted value of 3.3 for lnðkH =kT Þ=lnðkD =kT Þ or 1.4 for lnðkH =kT Þ=lnðkH =kD Þ:

748

Isotope Effects in Chemistry and Biology

2. Experimental Examples Using Intrinsic 18 KIE a. Peptidylglycine a-Hydroxylating Monooxygenase Peptidylglycine a-hydroxylating monooxygenase (PHM) initiates the oxidative cleavage of C-terminal, glycine-extended peptides by Hz abstraction from the a-carbon of glycine. In the PHM reaction, substrate binding and product release contribute significantly to rate limitation under the conditions of steady-state turnover.23 Francisco et al.24 studied PHM using H/T and H/D competitive KIEs. The observed KIEs increased with temperature, but the intrinsic KIEs, that were calculated using the Northrop method, were all close to 10. These KIEs exhibited a very small temperature dependency, leading to AH =AD of 5.9 ^ 3.2. These values, together with a large energy of activation (Ea , 13 kcal mol21) suggested “environmentally enhanced tunneling.”24 This model is described in more detail below (Section IV.A). b. Thymidylate Synthase Thymidylate synthase catalyzes the reductive methylation of 20 -deoxyuridine-50 -monophosphate (dUMP) to 20 -deoxythymidine-50 -monophosphate (dTMP). The cofactor N 5,N 10-methylene5,6,7,8-tetrahydrofolate (CH2H4folate) serves as a donor of both methylene and hydride.25 We have recently studied the hydride transfer step using competitive H/T and D/T KIEs.26 The observed KIEs were used to calculate their intrinsic values. These intrinsic KIEs were temperature independent with H/T KIEs close to 7 and AH/AT ¼ 6.8 ^ 2.8. As discussed below (Section IV.A) these results served as evidence for QM tunneling, and together with the reaction’s small energy of activation (Ea ¼ 4.0 ^ 0.1 kcal mol21) suggested a model in which the temperature dependence of the rate results from the reorganization of the system (isotopically insensitive), and the isotopically sensitive step is temperature independent (e.g., H-tunneling). The observed and intrinsic KIEs were used to calculate the commitment to catalysis (Equation 28.2, and it was found that between 20 and 308C the hydride transfer is fully rate determining while at elevated and reduced temperatures the commitment increases. c. Dihydrofolate Reductase Dihydrofolate reductase (DFHR) catalyzes the stereospecific reduction of 7,8-dihydrofolate (H2F) to 5,6,7,8-tetrahydrofolate (H4F), using nicotinamide adenine dinucleotide phosphate (NADPH) as the hydride donor. Specifically, the pro-R hydride is transferred from the C-4 of NADPH to C-6 of H2F. The complete kinetic scheme for DHFR is complex and the H-transfer step is partly rate determining only at high a pH.27 Presteady state stopped flow measurements resulted in H/D KIE between 2.8 and 3.0.28 In a recent study we measured H/T and D/T competitive KIEs and calculated an intrinsic H/D KIE of 3.5 ^ 0.2.29 This KIE is in excellent agreement with the calculated KIE of 3.4.30 This is significant because hitherto the presteady state rate was denoted as “the H-transfer rate.” A KIE of 3.5 also exposed a commitment of 0.25 on the presteady state rate. This commitment suggested that the presteady state rate contains an additional step, most likely the reorganization of the nicotinamide ring in and out of the active site. The commitment was temperature dependent and so were the observed KIEs. Nevertheless, the calculated intrinsic KIEs were temperature independent with AH =AT ¼ 7:2 ^ 3:5; which served as evidence of H-tunneling.29

C. SECONDARY (28) S WAIN – S CHAAD R ELATIONSHIP Secondary (28) Swain – Schaad relationship is calculated from 28 KIEs, i.e., not the hydrogen whose bond is being cleaved but its geminal neighbor. In several cases a breakdown of this 28 Swain – Schaad relationship was used as evidence of tunneling contribution. A number of these reported studies used mixed labeling experiments, as described below. In experiments of this type, the

Kinetic Isotope Effects as Probes for Hydrogen Tunneling

749

breakdown of the Swain –Schaad relationship indicates both tunneling and coupled motion between the primary and secondary hydrogens.31,32 1. Mixed Labeling Experiments as Probes for Tunneling and 18– 28 Coupled Motion Mixed labeling experiments consist of an isotopic labeling pattern that is more complex than the one considered in the original Swain – Schaad relationship. Several theoretical studies in the 1980s suggested that mixed labeling experiments would be the most sensitive indicators of H-tunneling.18,19 In a mixed labeling experiment, the 18 H/T KIE is measured with H in the 28 position and is denoted as kHH/kTH, where kij is the rate constant for H-transfer with isotope i in the 18 position and isotope j in the 28 position. The 28 H/T KIE is measured with H at the R position and is denoted as kHH/kHT. The 18 and 28 D/T KIE measurements, on the other hand, are conducted with D in the geminal position, and are denoted as kDD/kTD and kDD/kDT, respectively (Figure 28.3 and Equation 28.6 below): 28M EXP ¼

lnðkHH =kHT Þ lnðkDD =kDT Þ

ð28:6Þ

The exponential relationship resulting from such mixed labeling experiments is denoted as MEXP. If the 18 and 28 hydrogens are independent of each other, the isotopic labeling of one should not affect the isotope effect of the other. This is denoted as the rule of geometrical mean (RGM33): r¼

lnðkHi =kHT Þ ¼1 lnðkDi =kDT Þ

ð28:7Þ

where i is H or D. The RGM predicts that the isotopic label at the geminal position should not affect the MEXP: 28EXP ¼

lnðkHH =kHT Þ lnðkHH =kHT Þ ¼ ¼ 28M EXP lnðkHD =kHT Þ lnðkDD =kDT Þ

ð28:8Þ

If the motions of the 18 and 28 hydrogens are coupled along the reaction coordinate a breakdown of the RGM will result in an inflated 28 MEXP. The 18 KIE will have a secondary component, and will be deflated, but since the 28 H/D KIE is very small (, 1.2), the expected deflation of the 18 MEXP is very small. The 28 KIE on the other hand, will have a primary component and will be significantly inflated. Tunneling of the 18 H will induce a large 28 H/T KIE ðkHH =kHT Þ relative to the more semiclassical 28 D/T KIE (kDD/kDT), due to the reduced effect of tunneling from D in the primary position. In the mixed labeling experiment, when there is coupled motion between the 18 and 28 hydrogens, tunneling along the reaction coordinate will result in the inflation of the 28 MEXP because H tunneling is more significant than D tunneling. T2°

H2° R

H1° R'

H1°

R

T2°

D2°

R'

Labeling for 2° (H/T)H

R

D1° R'

R

D1° R'

Labeling for 2° (D/T)D

FIGURE 28.3 The isotopic labeling pattern for a mixed-labeling experiment.

750

Isotope Effects in Chemistry and Biology H'2°

H2° H1°

R R' sp3

H'2°

H2°

+

+ R'' R''' sp2

H1°

R R' sp2

R''' R'' sp3

FIGURE 28.4 The general H-transfer reaction discussed in the text.

The MEXP is a product of the original Swain – Schaad EXP and RGM (r): r·EXP ¼

lnðkHH =kHT Þ lnðkDH =kDT Þ lnðkHH =kHT Þ · ¼ ¼ M EXP lnðkDH =kDT Þ lnðkDD =kDT Þ lnðkDD =kDT Þ

ð28:9Þ

A mathematically rigorous explanation of the high sensitivity of the mixed labeling experiment to H-tunneling can be found in Refs. 34 and 35. Both Huskey35 and Saunders36,37 have shown independently that exceptionally large values of MEXP are only computed for 28 KIEs resulting from coupled motion and tunneling. They both concluded that the extra isotopic substitution is an essential feature of the experimental design. 2. Upper Semiclassical Limit for 28 Swain – Schaad Relationship For EXP as defined in Equation 28.5, values smaller than its semiclassical lower limit can be explained by kinetic complexity and values larger than its upper limit serve as evidence of tunneling. Until recently, the upper semiclassical limit used was 3.34.19,38 An upper limit that is more realistic and relevant to the commonly used mixed labeling experiment is calculated below.39 This limit for EXP with no tunneling contribution is calculated using three different approaches. a. Zero-Point Energy and Reduced Mass Considerations The common upper limit EXP ¼ lnðkH =kT Þ=lnðkD =kT Þ ¼ 3:34 was calculated using Equation 28.5 and the reduced masses of C –H, C – D and C –T.19,31,32,38 This limit was calculated with the reduced masses of only the two atoms whose covalent bond is being cleaved in the reaction. If the motion of atoms other than the hydrogen and the carbon whose bond is being cleaved are part of the reaction coordinate, a different reduced mass has to be considered. MEXPs were calculated from Equation 28.4 and Equation 28.5 using the reduced masses of a variety of coupled modes for the general reaction illustrated in Figure 28.4. It is possible to consider coupling of another coordinate to the cleaved C –H bond using the secular equation:40 lFG 2 lIl ¼ 0

ð28:10Þ

Here F is the force constant matrix, G defines the reduced mass and coupling for each internal coordinate, and I is the identity matrix. The unknown ðlÞ is related to a vibrational frequency of the system and can be used to calculate the ZPE and, hence, MEXP. pp Equation 28.10 can be used to examine a slightly more complex case, such as the coupling of two bonds in a CH2 group. In this case the frequencies are a function not only of the three masses but also of the H –C – H angle ðwÞ: Both the 18 and the 28 MEXP(w)s reach a maximum at MEXP(1808) ¼ 3.34. A similar analysis of the coupling between the H – C –H bend and the C – H stretch yields 18 and 28 MEXPs that are functions of the C –H bond length, w; and the ratio of the force constants for the pp l ¼ 4p2 n 2 ; if F is of dimension N £ N; there will be N values of l that satisfy Equation 28.10, i.e., N vibrational frequencies.

Kinetic Isotope Effects as Probes for Hydrogen Tunneling

751

stretch and bend. However, an exhaustive search of the parameter space yielded only a few M EXP . 3.34. The maximum MEXP found in these studies was 4.25.39 b. Vibrational Analysis A system in which more than two coordinates are coupled cannot be solved analytically. In an attempt to define a maximum for a semiclassical MEXP (SC MEXP) for such a system, a numerical simulation was employed. Yeast ADH was chosen as a model for the general reaction in Figure 28.4. The experimental KIEs and EXPs of Cha et al.41 for yeast ADH were reproduced42 and used to parameterize the coupling constants of a truncated model of the alcoholate substrate and the cofactor (NADþ). Following in the footsteps of Rucker and Klinman43 we used vibrational analysis and the Bigeleisen –Mayer equation to calculate isotope effects for the ADH-catalyzed oxidation of benzyl alcohol by NADþ. In the context of the Bigeleisen – Mayer equation, KIEs can be calculated from:44,45 KIE ¼ MMI·EXC·ZPE

ð28:11Þ

where MMI is a mass-moment of inertia term, EXC is a vibrational excitation term and ZPE is a zero-point energy term. A truncated system was used with empirical force constants and geometric parameters for the cutoff model of reactants and proposed transition states. The Bell tunneling correction46 was applied to calculate KIEs and the experimental results of Cha et al.41 were used to parameterize the potential surface and coupling constants. In accordance with previous studies, we found that the 28 D/T KIE is more sensitive to changes in reaction coordinate properties than other parameters. Rucker and Klinman studied a linear ðu ¼ 1808Þ C – H –C transfer and a dihedral angle between the two 28 hydrogens ðfÞ equal to 08 (i.e., 28 hydrogens and the C– H – C system were in one plane). We39 examined a conformational space of 2408 . u . 1208 and 3608 for f was studied. u and f define the relative conformation of the reactants. Other geometric parameters and the force constants, excluding the off-diagonal coupling constants, were set as described in Ref. 43. The program BEBOVIB IV47 was used to solve the vibrational secular equations for the molecules of interest. At each conformation, the coupling constants were adjusted while fitting the calculated KIE to the experimental value reported by Cha et al.41 Additionally, the enthalpy of formation for each conformation was calculated at the AM1 level of theory to avoid conformations that were not physically relevant (enthalpy of formation, DHf, higher by 2 kcal mol21 than the average one: DDHf $ DHf 2 DHf ). Conformations that exhibit DDHf close to 2 kcal mol21 were further examined at the MP3 and AM1-SM2 levels of theory, but no significant DDHf, that might suggest significant steric repulsion, was found. The semiclassical KIE (SC KIE) was used to calculate the semiclassical MEXP (SC MEXP) from Equation 28.4 and Equation 28.5. Most of the calculated values were between 3.3 and 3.4 for 18 SC MEXP and between 3.3 and 3.6 for 28 SC MEXPs. Yet, several SC MEXPs exceeded those values. The 28 SC MEXP reached a maximum value of 4.60 at u ¼ 2008 and f ¼ 1208:39 Practically, this value establishes an upper limit above which tunneling has to be implied to explain the experimental data. c. Effect of Kinetic Complexity In contrast to the single step kinetics considered above, a complex reaction kinetics may result in an inflated 28 MEXP. In such a case the intrinsic MEXP (MEXPint) may follow the Swain – Schaad prediction but an inflated observed MEXP (MEXPobs) can result from reversible reaction with a large EIE. This kinetic complexity may alter the observed KIE as described above Equation 28.2.12 It is apparent from Equation 28.2 that KIEobs can range from KIEint to EIE as a function of Cr and from KIEint to unity as a function of Cf : The effect of the commitment to catalysis on MEXPobs is not

752

Isotope Effects in Chemistry and Biology

FIGURE 28.5 Secondary MEXPobs as a function of 28 H/T KIEint and Cr ; where Cf ¼ 0 and 28 H/T EIE ¼ 1.35 (Equation 28.12). (Reproduced from Kohen, A. and Jensen, J. H., Boundary conditions for the Swain – Schaad relationship as a criterion for hydrogen tunneling, J. Am. Chem. Soc., 124, 3858– 3864, 2002. With permission.)

immediately apparent. In order to study this effect, Equation 28.6 was rewritten for KIEobs using Equation 28.2 to calculate the 28 MEXPobs as a function of Cr and KIEint: M

EXPobs ¼ ¼

lnðH=T KIEobs Þ lnðD=T KIEobs Þ ln½ðKIEint þ EIE·Cr Þ=ð1 þ Cr Þ 1=3:3

ln{½KIEint

ð28:12Þ

þ EIE1=3:3 ·Cr =ðkH =kD Þr =½1 þ Cr =ðkH =kD Þr }

where ðkH =kD Þr is the 18 KIE for the reverse reaction and is used to calculate the reverse commitment ðCr Þ for D from that of H. The Cr is the ratio between product release and the reverse H-transfer. When using the mixed labeling method, the Cr for D is the Cr for H divided by the 18 H/D KIE for the reverse reaction (e.g., 1.6 for the aldehyde to alcohol conversion catalyzed by ADH48). Since this is a semiclassical model (no tunneling) all the parameters in the denominator are dependent on the corresponding parameters in the numerator. D/T KIEint and EIE are related to the H/T isotope effects in accord with the Swain– Schaad relationship. Cf . 0 will always decrease the KIEobs and is insignificant in a search for maximal SC MEXP, so Cf ¼ 0 was used to calculate M EXPobs. Partial derivatives of this function do not result in maxima, hence a numerical method was employed to examine Equation 28.12. For most of the parameter space the observed 28 MEXP is close to 3.3. Yet, inflated observed MEXPs result from a combination of a small but finite Cr ; a large 28 EIE, and a small intrinsic 28 KIE. Figure 28.5 graphically illustrates Equation 28.12. Figure 28.5 presents MEXPobs as a function of both 28 H/T KIEint and Cr at the narrow parameter range where inflated MEXP values arise (EIE ¼ 1.35, 0 , Cr , 3; 1:01 , KIEint , 1:1; and 3.3 , M EXPobs , 5:0). The MEXPobs is again close to 3.3 for a wide range of values, but if Cr ! 0.225 and KIEint ! 1.01, then MEXPobs ! 4.8. Special attention should be paid to the values for which the 28 KIEobs is small (, 1.06). This last analysis results in an upper limit of 4.8. This analysis is valid for every H-transfer in a reaction that involves 18 and 28 hydrogens (Figure 28.4). The only “system specific” parameter used was the 18 H/T KIE on the reverse reaction (by which Cr for D is related to that for H in Equation 28.3). This 18 KIE should be evaluated for each system under investigation. d. The New Effective Upper Limit Following the work described above, it is suggested that for the mixed labeling method ðkHH =kHT vs. kDD =kDT Þ an experimental 28 MEXP larger than 4.8 (within statistical experimental error) may

Kinetic Isotope Effects as Probes for Hydrogen Tunneling

753

serve as a reliable indication of H-tunneling.39 In the case of experimental 28 MEXP between 3.3 and 4.8 additional evidence is needed to indicate H-tunneling. Such additional examination consists of simple analytical or numerical solutions such as those described above and, in more detail, in Ref. 39. Alternatively, a higher level of calculation could also be employed as discussed below (Section IV.B). For ADHs, for example, several state of the art theoretical examinations have recently supported the tunneling contribution suggested from the inflated 28 MEXPs.49 – 52 Recent theoretical studies by Hirschi and Singleton calculated 28 Swain –Schaad exponents for a wide range of simple reactions.53 In accordance with the findings and conclusions of this chapter, they found that for significant 28 KIEs (cf. larger than 1.1) deviation of 28 Exp from the 30 to 4 range is indication for tunneling and/or coupled motion. 3. Experimental Examples Using 28 Swain – Schaad Exponents a. Horse Liver Alcohol Dehydrogenase Alcohol dehydrogenases (ADHs) catalyze the reversible oxidation of alcohols to aldehydes with NADþ as the oxidative reagent. Horse liver alcohol dehydrogenase (HLADH) has been extensively studied by means of 28 Swain – Schaad relationships.54 – 56 Two interesting conclusions of these studies were that, (i) For two mutants (F93T and F93T;V203G), a longer donor – acceptor distance (measured by x-ray crystallography) led to a smaller 28 exponent,56 and (ii) For a series of mutants a correlation exists between the catalytic efficiency (V/K ) and the 28 exponent.55 These findings seem to be in accordance with tunneling models in which the barrier width plays a critical role, and with the contribution of coupled motion to catalysis. b. Thermophilic ADH from Bacillus stearothermophilus (ADH-hT) At the physiological temperature of this thermophilic ADH (, 658C) inflated 28 Swain – Schaad exponents (, 15)57 indicated a signature of H-tunneling similar to that of the mesophilic yeast ADH at 258C.41 At temperatures below 308C these exponents declined toward the semiclassical region and the enthalpy of activation increased significantly (14.6 to 23.6 kcal mol21 for H-transfer and 15.1 to 31.4 kcal mol21 for D-transfer). This phenomenon was interpreted as indicating a decreased tunneling contribution at reduced temperature due to different environmental sampling at high (physiological) and low temperatures. These findings were then correlated to the increased rigidity of the enzyme at lower temperatures.58,59 These studies suggested that similar enzymes that catalyze the same reaction at very different temperatures evolved to have similar rigidities in their respective physiological conditions and similar tunneling contributions to the H-transfer process. Interestingly, these results suggested possible relationships between protein rigidity and the degree of tunneling. Together with temperature dependency studies that are described below (Section III.C.2), a model was suggested in which the enzyme’s fluctuations are coupled to the reaction coordinate.32,51,57,60

III. TEMPERATURE DEPENDENCE OF KIEs A. TEMPERATURE D EPENDENCE OF R EACTION R ATES AND KIE S Traditional literature treats enzyme catalyzed reactions, including hydrogen transfer, in terms of transition state theory (TST).32,61,62 TST assumes that the reaction coordinate may be described by a free-energy minimum (the reactant well) and a free-energy maximum that is the saddle point leading to product. The distribution of states between the ground state (GS, at the minimum) and the transition state (TS, at the top of the barrier) is assumed to be an equilibrium process that follows the Boltzmann distribution. Consequently, the reaction’s rate is exponentially dependent on the

754

Isotope Effects in Chemistry and Biology

reciprocal absolute temperature ð1=TÞ as reflected by the Arrhenius equation: k ¼ A e2Ea =RT

ð28:13Þ

where A is the Arrhenius preexponential factor, Ea is the activation energy and R is the gas constant. Since KIE is the ratio of the reactions’ rates, its temperature dependency will follow: kl A DEaðh2lÞ ¼ l e RT kh Ah

ð28:14Þ

where h and l are the heavy and light isotopes, respectively. This equation is useful as long as the reaction is thermally activated. At low temperatures, the contribution of tunneling becomes significant, as no thermal energy is available for activation. This causes a curvature in the Arrhenius plot as illustrated in Figure 28.6. Conventional tunneling, through a single, rigid barrier is temperature independent and may affect both the preexponential and the exponential factors.

B. KIE S ON A RRHENIUS ACTIVATION FACTORS Following Equation 28.14, with tunneling correction,14,46 the KIE’s temperature dependency will reflect the differences in the energy of activation for the two isotopes, and the KIE on the preexponential factors ðAl =Ah Þ should be close to unity (no tunneling, region I in Figure 28.6). Deviation from unity with no tunneling seems to be confined pffiffiffiffiffiffiffiffito a limited range, as extensively discussed in the literature.16,31,63 – 66 These limits follow mh =ml . Al =Ah . ml =mh where m is the reduced mass. These limits for hydrogen KIEs are summarized in Table 28.1. At a very low temperature, where only tunneling contributes significantly to rates, it is predicted that the KIEs will be very large (over six orders of magnitude67) and AH/AD will be much larger than unity (extensive tunneling, region III in Figure 28.6). Between high and low temperature extremes, the Arrhenius plot of the KIEs will be curved, as the light isotope tunnels at a higher temperature than the heavy one. At this region, the Arrhenius slope will be very steep and AH/AD will be smaller than unity (moderate tunneling region II in Figure 28.6). This has been deliberated in several previous reviews,31,32 and has been used as a probe for tunneling in a wide variety of enzymatic II

ln(k )

I

III i1

(a)

i2

ln(k1/k2)

i1/i 2

ln(1)

(b) 1/T

FIGURE 28.6 An Arrhenius plot of a hydrogen transfer that is consistent with a tunneling correction to TST. (a) Arrhenius plot of a light isotope ði1Þ and heavy isotope ði2Þ: (b) Arrhenius plot of their KIE ði1=i2Þ: Highlighted are experimental temperature ranges for three systems: I, a system with no tunneling contribution, II, a system with moderate tunneling, and III, a system with extensive tunneling contribution. The dashed lines are the tangents to the plot at each region. This illustration is similar to several schemes we and others have suggested in the past.31,32,94,112

Kinetic Isotope Effects as Probes for Hydrogen Tunneling

755

TABLE 28.1 Semiclassical Limits for the KIE on Arrhenius Preexponential Factors14,46,66

Upper limit Lower limit

AH/AT

AH/AD

AD/AT

1.7 0.3

1.4 0.5

1.2 0.7

systems (Table 28.2). According to this model, an AH/AD smaller than the semiclassical lower limit (Table 28.1) indicates tunneling of only the light isotope (“moderate tunneling region”32). Whereas AH/AD larger than unity indicates tunneling of both isotopes (“extensive tunneling region”32). Table 28.2 summarizes several reports of H-tunneling based on preexponential Arrhenius factors that were outside the semiclassical range (Table 28.1). Several experimental AH/ADs, that do not match the criteria set by the above model are discussed in Section III.C and alternative models are presented in Section IV.

C. EXPERIMENTAL E XAMPLES U SING I SOTOPE E FFECTS ON A RRHENIUS ACTIVATION FACTORS 1. Soybean Lipoxygenase-1 Lipoxygenases catalyze the oxidation of linoleic acid (LA) to 13-(S)-hydroperoxy-9,11-(Z,E )octadecadienoic acid (13-(S)-HPOD).68 This reaction proceeds via an initial, rate-limiting abstraction of the pro-S hydrogen radical from C11 of LA by the Fe3þ-OH cofactor, forming both a substrate-derived radical intermediate and Fe2þ-OH2. Molecular oxygen rapidly reacts with this radical, eventually forming 13-(S)-HPOD and regenerating the resting enzyme. The abstraction of H or D from the pro-S C11 position of LA by the wild type Soybean Lipoxygenase-1 (SBL-1) has

TABLE 28.2 Enzymatic Systems for which Tunneling Was Suggested from Temperature Dependencies Enzyme

kH/kD

AH/AD

Ref.

Soybean lipoxygenase, wt. Soybean lipoxygenase, mutants Methane monooxygenase Galactose oxidase Methylamine dehydrogenase Methylamine dehydrogenase (TTQ-dependent) Trimethylamine dehydrogenase Sarcosine oxidase Methyl malonyl CoA mutase Acyl CoA desaturase Peptidylglycine a-hydroxylating monooxygenase

81 93– 112 50– 100 16 17 12.9 4.6 7.3 36 23 10

18 4– 0.12

68 68 101 102 103 104 105 106 107 108 24

kH/kT

AH/AT

Bovine serum amine oxidase Monoamine oxidase Thymidylate synthase Dihydrofolate reductase

35 22 7 6

0.25 13 9.0 7.8 5.8 0.08 2.2 5.9

0.12 0.13 7 6

109 110 26 29

756

Isotope Effects in Chemistry and Biology 1000

kcat (s−1)

100

10

1 3

3.1

3.2

3.3 3

10 /T

3.4

3.5

3.6

3.7

(K−1)

FIGURE 28.7 Arrhenius plot of kinetic data for WT-SLO (filled symbols) and Ile553Ala (open symbols) using protio-LA (circles) and deutero-LA (diamonds). Nonlinear fits to the Arrhenius equation are shown as solid lines. (Reproduced from Knapp, M. J., Rickert, K., and Klinman, J. P., Temperature-dependent isotope effects in soybean lipoxygenase-1: correlating hydrogen tunneling with protein dynamics, J. Am. Chem. Soc., 124, 3865– 3874, 2002. With permission from American Chemical Society.)

very large KIEs (, 80) and a large AH/AD (, 20),68 – 70 which would suggest it fits region III in Figure 28.6 (extensive tunneling). Yet, its KIEs are “only” around 80, while the above model (Section III.B) would predict much larger KIEs.67 For the wild type SBL-1, the Ea for the H-transfer was small (, 2 kcal mol21) and the DEa was , 1 kcal mol21. Several mutants of SBL-1 were also studied and they exhibited Arrhenius plots that ranged between regions II to III in Figure 28.6. A typical Arrhenius plot for both the wild type and mutant SBL-1 is presented in Figure 28.7. Additionally, a phenomenological model that can better describe the SBL-1 reaction is discussed below (Section IV.A). 2. Thermophilic ADH (ADH-hT) Another example for studies of temperature independent KIEs is taken from our work with thermophilic ADH from Bacillus stereothermophilus which is demonstrated in Figure 28.8.57 – 59 Under physiological conditions (30 to 658C), this enzyme had AH =AT and AD =AT larger than the semiclassical limits. Yet, its KIEs were relatively small (, 3) and the enthalpy of activation for H and D was rather large (14.6 and 15.1 kcal mol21, respectively). As discussed below (Section IV.A), this can only be explained by a “Marcus-like” model in which the temperature dependencies of the reaction rate and of the KIE are separated. Under 308C, both isotopes had a much larger energy of activation and a large temperature dependence for KIEs. This would fit region II in Figure 28.6, and was interpreted as “activity phase transition” due to the increased rigidity of this thermophilic enzyme at reduced temperatures.57 – 59 Several studies by Sutcliffe, Scrutton and coworkers71 – 73 also resulted in temperature independent KIEs, with large AH =AD : Those works are described in detail in Chapter 25 (by Sutcliffe and Scrutton). The systems studied by Sutcliffe and Scrutton had enthalpies of activation much larger than the semiclassical, rigid model prediction. As discussed below in Section IV, such findings have led to the development of many theoretical models attempting to explain the experimental results. It must be emphasized that the semiclassical limits for the energy of activation (the slope of the Arrhenius plot) are not well defined. Consequently, in order to establish nonclassical features from temperature-independent KIEs, the preexponential Arrhenius factor must be outside its semiclassical limits. For example, a recent paper misinterpreted “nearly temperature-independent”

757 10 9 8 7 6

kcat (s−1)

100

5

10

4

H/D KIE (Dkcat)

Kinetic Isotope Effects as Probes for Hydrogen Tunneling

3

1

3.0

3.2 3.4 1/T [K −1]×10−3

3.6

2

FIGURE 28. 8 A thermophilic ADH (ADH-hT) with benzyl alcohol ðXÞ; [7-2H2] benzyl alcohol (B) and their KIEs (V). (Reproduced using data published in Kohen, A., Cannio, R., Bartolucci, S., and Klinman, J. P., Enzyme dynamics and hydrogen tunneling in a thermophilic alcohol dehydrogenase, Nature, 399, 496– 499, 1999. With permission.)

KIEs (actually DEa ¼ 3:0 ^ 0:7 kJ mol21) with AH =AD close to unity as “Evidence for environmentally coupled hydrogen tunneling during dihydrofolate reductase catalysis.”74 Actually, the temperature dependence of the KIEs in that study (above 208C) was well within the semiclassical range. Over the years, TST has been modified and corrected for the kinetic effects of tunneling, barrier recrossing and medium viscosity. Yet, none of these can explain temperature-independent KIE accompanied by a large enthalpy of activation. Developing a theory that will explain such a phenomenon is an ongoing challenge. The next section describes attempts to lay the general foundation for such a theory.

IV. THEORETICAL APPROACHES A. PHENOMENOLOGICAL "M ARCUS-L IKE " MODELS The question of whether enzymatic complex dynamics have a direct role in catalyzing the reaction’s chemical step is still a controversial one. The need to consider “dynamic” models was raised following experimental findings that were not in accordance with “rigid” models. Models that attempted to reproduce temperature independent KIEs with significant enthalpy of activation required the isotopically sensitive step to have little or no enthalpy of activation (e.g., transfer via QM tunneling), while the large temperature dependency of the reactions results from classically activated rearrangements of the potential surface. Several such theoretical models have been recently proposed.51,60,69,73,75 The common theme in these models involves two requirements for efficient tunneling; degeneracy of the reactant and product energy levels, and narrow barrier width. Following the works of Marcus,76 several theoretical models were developed in an effort to rationalize H-transfer in the condensed phase and particularly in enzymes (e.g., Burgis and Hynes,77,78 Kuznetzov and Ulstrup,79 Benkovic and Bruce,80 Warshel,3,81 and Schwartz60). Two common features of these models are direct effects of the potential surface fluctuations on the reaction rate, and separation of the temperature dependence of the rate and the KIE. An example of a “Marcus-like” model is illustrated in Figure 28.9. Environmentally coupled hydrogen tunneling models can accommodate the composite kinetic data for WT-SLO and its mutants.68,69 The model of Kuznetsov and Ulstrup79 was used to account for the variable temperature KIE data. In this model, the rate for Hz transfer is governed by an isotope-independent term (constant), a Marcus-like term, and a “gating” term (the F.C. Term in

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Isotope Effects in Chemistry and Biology G

R

P λ

∆r

++

∆G°

g

tin

ga

Qe

nv

FIGURE 28. 9 Illustration of “Marcus-like” models: energy surface of environmentally coupled hydrogen tunneling. The coordinates presented are: Qenv ; environmental free energy with the free energy of reaction ðDG8Þ and reorganization energy ðlÞ; and gating, hydrogen potential energy surface, at different environmental configurations. R (black) is the reactant configuration and P (gray) is the product configuration. Gating alters the distance ðDrÞ of hydrogen transfer.69 The probability of finding the hydrogen at the reactant or the product states is illustrated for the reactive conformation (the symmetric double well indicated by ‡). For alternative graphic illustrations of such models see Refs. 31,32.

Equation 28.15). The Marcus term relates l; the reorganization energy, to DG8; the driving force for the reaction (Equation 28.15), where R and T are the gas constant and absolute temperature, respectively. This term has only weak isotopic dependency when tunneling takes place from vibrationally excited states. The isotopically sensitive term is the Franck– Condon nuclear overlap along the hydrogen coordinate (F.C. Term), which is the weighted hydrogen tunneling probability. This term arises from the overlap between the initial and the final states of the hydrogen’s wave function and, consequently, depends on the thermal population of each vibration level. The temperature dependence of KIEs arises from the thermal population of excited vibration levels. This model was developed for nonadiabatic radical-transfer (Hz) reactions and the full scope of its applications is yet to be explored: k ¼ ðconstantÞe2ðDG8þlÞ

2

=ð4lRTÞ

ðF:C:TermÞ

ð28:15Þ

The fact that studies of enzymes at this level are so interdisciplinary can results in misunderstandings and disagreements between disciplines. Most Biochemists consider any motion of the protein or the enzymatic complex to be “dynamics.” Most Physical Chemists on the other hand, will use that term only for motions along the reaction coordinate that are not in thermal (Boltzmann) equilibrium with their environment.7 By their nomenclature, fluctuations that are in thermal equilibrium (like environmental rearrangement, gating motion, etc.) do not constitute a dynamical effect. For example, in Refs. 31,32 we suggested models in which dynamic rearrangement of the reaction’s potential surface plays a key role in the enhancement of the reaction’s rate (e.g., Figure 28.5 in Ref. 32). Vila´ and Warshel3 on the other hand, offer a similar graphic (Ref. 3, Figure 2) to argue against such a role. Apparently, differences in terminology have resulted in contradictory wording in the conclusions. Up to date, temperature dependency of rates and KIEs is still a major challenge for explicit theoretical models or simulations and (to the best of our knowledge) only two studies have reproduced such phenomenon.51,82

B. QM/MM M ODELS AND S IMULATIONS Recently, several computational studies were conducted in an attempt to reproduce and explain experimental findings such as the breakdown of the Swain – Schaad relationship and the nonclassical temperature dependency of KIEs. Those studies employed a molecular mechanics

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(MM) based simulation of the exterior of a protein with high level ab initio calculations along the reaction coordinate and in the vicinity of the reacting atoms. Various methods were used to “buffer” the transition between these two regions. The general name for this kind of calculation is QuantumMechanics/Molecular-Mechanics (QM/MM). Most of these studies investigated enzymatic systems for which ample kinetic, structural and other data were available (e.g., TIM,4,83 carbonic anhydrase,84 ADH,3,50,52 LDH,8 methylamine dehydrogenase,85 – 87 SBL-1,82,88 and DHFR30). These state of the art simulations were able to reproduce rates and H/D KIEs but had little success in addressing temperature dependencies (with the exception of Ref. 82) and secondary KIEs (with the exception of Refs. 50 and 89). High-level calculations of this kind are of critical importance as there is no other way to bring together all the molecular and kinetic data. While direct molecular dynamic simulations are limited to the nanosecond range, free-energy perturbation/umbrella sampling calculations,9,30 allow one to explore the effect of a much larger conformational space by forcing the system to move to different regions upon changing the charge distribution of the reacting system. Currently, the main limitations of QM/MM models are the short time scale of the simulation and the inherent compromise between accuracy (high level of theory) and conformational flexibility (large conformational space). This prohibits the coverage of a more substantial range of motion in the duration of an entire catalytic cycle while investigating quantum mechanical phenomena such as tunneling.

V. COMPARISON TO STUDIES OF NONENZYMATIC REACTIONS Temperature independent KIEs were also reported for nonenzymatic H-transfer reactions in solution and in solid state, although most could not be considered as models for any known enzymatic reaction.6,90 – 93 A recent example came from the NMR studies of intramolecular H-transfer in a porphyrin and its anion.94 Another recent study of a solid state reaction also invoked vibration-assisted hydrogen tunneling.95 Additionally, theoretical studies by Warshel and coworkers suggested that the contribution of the environmental fluctuations to the reaction rate in the enzyme’s active site are similar to those of solvent fluctuations in a solvent cage.9,96 Whether phenomena such as H-tunneling, vibrationally enhanced tunneling, and environmentally enhanced reaction rates contribute to enzyme catalysis is an open question. To answer this question the catalyzed and uncatalyzed reactions must be compared. Unfortunately, it is not commonly possible to measure the rate or KIE of uncatalyzed reactions that are relevant to enzymecatalyzed reactions. Most uncatalyzed, or nonenzymatically catalyzed, reactions lead to many byproducts and may proceed through a different transition state than the enzymatic one. A couple of studies that addressed this issue were recently published. Wolfenden and coworkers97 studied hydrolytic reaction and could extrapolate the Arrhenius plots of the uncatalyzed reaction from 700 to 8008C to the physiological temperature. Their measurements were compared to enzymecatalyzed hydrolysis, and led to the conclusion that the catalytic effect resulted from differences in enthalpy rather than entropy. Finke and coworkers,98,99 studied a model system that might be similar to the methylmalonyl – CoA mutase catalyzed reaction.100 In this study, moderate tunneling (tunneling of only the light particle) was suggested for a model reaction from AH =AD , 1: This result is qualitatively similar to the one reported for the methylmalonyl – CoA mutase catalyzed reaction.100 Finke and coworkers suggested that these findings are in accordance with a similar degree of tunneling for both enzymatic and nonenzymatic systems. In our minds, since the H-transfer in the methylmalonyl –CoA mutase reaction is not rate determining, and since the uncatalyzed reaction was initiated by photolysis (which may lead to a vibrationally excited state for the H-transfer), it is not yet clear how general that conclusion is. Furthermore, AH/AD is a qualitative parameter, which makes it hard to quantify the degree of the tunneling contribution. Inconsistent with the suggestion that enzymes do not affect tunneling is the fact that a single mutation in an enzyme can control tunneling, as demonstrated for HLADH mutants F93T and F93T-V203A55 discussed above (Section II.C.3.a).

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VI. CONCLUSIONS Kinetic isotope effects, their temperature dependence, and the internal relationships between them are important tools in studying the nature of H-transfer. KIEs are one of the only measurable effects that directly address the transition state of a reaction. Changing the atom of interest into one of its isotopes imposes a minimal effect on the reaction’s potential surface being studied. In contrast to some alternative methods (e.g., linear free-energy relationships) the experimental effect of isotopic substitution is localized to the reaction coordinate of interest. Consequently, KIEs provide information about the reaction coordinate without imposing major side effects on its environment. However, care must be exercised to assure proper analysis, error propagation, and interpretation of observed KIEs. In summary, studying KIEs affords a variety of benefits, including: It facilitates examination of physical phenomena such as quantum-mechanical tunneling, vibrationally-coupled tunneling, and 18–28 coupled motion. † Intrinsic KIEs can be studied even for kinetically complex systems such as most enzymatic reactions. † KIE studies can examine whether a model that excludes a phenomenon like tunneling is adequate for a certain system. † KIEs are useful in providing experimental restriction to high level calculations along the reaction coordinate. †

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92 Anhede, B. and Bergman, N. A., Transition-state structure and the temperature dependence of the kinetic isotope effect, J. Am. Chem. Soc., 106, 7634– 7636, 1984. 93 Koch, H. F., Dahlberg, D. B., McEntee, M. F., and Klecha, C. J., Use of kinetic isotope effects in mechanism studies. Anomalous Arrhenius parameters in the study of elimination reactions, J. Am. Chem. Soc., 98, 1060– 1061, 1976. 94 Braun, J., Schwesinger, R., Williams, P. G., Morimoto, H., Wemmer, D. E., and Limbach, H. H., Kinetic H/D/T isotope and solid state effects on the tautomerism of the conjugate porphyrin monoanion, J. Am. Chem. Soc., 118, 11101– 11110, 1996. 95 Prass, B., Stehlik, D., Chan, I. Y., Trakhtenberg, L. I., and Klochikhin, V. L., Vibration-assisted intermolecular hydrogen tunneling in photoreactive doped molecular crystals: effect of temperature and pressure, Ber. Bunsenges. Phys. Chem., 102, 498– 503, 1998. 96 Shurki, A., Strajbl, M., Villa, J., and Warshel, A., How much do enzymes really gain by restraining their reacting fragments?, J. Am. Chem. Soc., 124, 4097– 4107, 2002. 97 Wolfenden, R., Snider, M., Ridgway, C., and Miller, B., The temperature dependence of enzyme rate enhancement, J. Am. Chem. Soc., 121, 7419– 7420, 1999. 98 Doll, K. M., Bender, B. R., and Finke, R. G., The first experimental test of the hypothesis that enzymes have evolved to enhance hydrogen tunneling, J. Am. Chem. Soc., 125, 10877– 10884, 2003. 99 Doll, K. M. and Finke, R. G., A compelling experimental test of the hypothesis that enzymes have evolved to enhance quantum mechanical tunneling in hydrogen transfer reactions: The b-eopentylcobalamin system combined with prior adocobalamin data, Inorg. Chem., 42, 4849 –4856, 2003. 100 Padmakumar, R., Padmakumar, R., and Banerjee, R., Evidence that cobalt– carbon bond homolysis is coupled to hydrogen atom abstraction from substrate in methylmalonyl-CoA mutase, Biochemistry, 36, 3713– 3718, 1997. 101 Nesheim, J. C. and Lipscomb, J. D., Large kinetic isotope effects in methane oxidation catalyzed by methane monooxygenase: evidence for C –H bond cleavage in a reaction cycle intermediate, Biochemistry, 35, 10240– 10247, 1996. 102 Whittaker, M. M., Ballou, D. P., and Whittaker, J. W., Kinetic isotope effects as probes of the mechanism of galactose oxidase, Biochemistry, 37, 8426– 8436, 1998. 103 Basran, J., Sutcliffe, M. J., and Scrutton, N. S., Enzymatic H-transfer requires vibration-driven extreme tunneling, Biochemistry, 38, 3218– 3222, 1999. 104 Basran, J., Patel, S., Sutcliffe, M. J., and Scrutton, N. S., Importance of barrier shape in enzymecatalyzed reactions — vibrationally assisted tunneling in tryptophan tryptophylquinone-dependent amine dehydrogenase, J. Biol. Chem., 276, 6234– 6242, 2001. 105 Basran, J., Sutcliffe, M. J., and Scrutton, N. S., Deuterium isotope effects during carbon– hydrogen cleavage by trimethylamine dehydrogenase, J. Biol. Chem., 276, 24581 –24587, 2001. 106 Harris, R. J., Meskys, R., Sutcliffe, M. J., and Scrutton, N. S., Kinetic studies of the mechanism of carbon – hydrogen bond breakage by the heterotetrameric sarcosine oxidase of Arthrobacter sp. 1-IN, Biochemistry, 39, 1189– 1198, 2000. 107 Chowdhury, S. and Banerjee, R., Evidence for quantum mechanical tunneling in the coupled cobalt– carbon bond homolysis-substrate radical generation reaction catalyzed by methylmalonyl-CoA mutase, J. Am. Chem. Soc., 122, 5417– 5418, 2000. 108 Abad, J. L., Camps, F., and Fabrias, G., Is hydrogen tunneling involved in acylCoA desaturase reactions? The case of a D9 desaturase that transforms (E)-11-tetradecenoic acid into (Z,E)-9,11tetradeienoic acid, Angew. Chem. Int. Ed., 122, 3279– 3281, 2000. 109 Grant, K. L. and Klinman, J. P., Evidence that both protium and deuterium undergo significant tunneling in the reaction catalyzed by bovine serum amine oxidase, Biochemistry, 28, 6597– 6605, 1989. 110 Jonsson, T., Edmondson, D. E., and Klinman, J. P., Hydrogen tunneling in the flavoenzyme monoamine oxidase B, Biochemistry, 33, 14871– 14878, 1994. 111 Jonsson, T., Glickman, M. H., Sun, S. J., and Klinman, J. P., Experimental evidence for extensive tunneling of hydrogen in the lipoxygenase reaction — implications for enzyme catalysis, J. Am. Chem. Soc., 118, 10319– 10320, 1996.

29

Hydrogen Bonds, Transition-State Stabilization, and Enzyme Catalysis Richard L. Schowen

CONTENTS I.

The Problem of Enzyme Catalysis .................................................................................. 766 A. Magnitudes of Catalytic Accelerations by Enzymes .............................................. 766 B. Transition-State Stabilization and Catalysis............................................................ 767 C. H-Bonds as a Means of Transition-State Stabilization ........................................... 768 D. Beyond the Transition-State Theory of Catalysis ................................................... 770 II. Structure and Strength of H-Bonds ................................................................................. 771 A. The Concept of H-Bond Strength............................................................................ 771 B. Categorization of H-Bonds ...................................................................................... 772 C. Some Probes of Hydrogen Bonds............................................................................ 772 1. NMR Approaches.............................................................................................. 773 2. Theoretical Studies of H-Bonds........................................................................ 773 3. Thermochemical, Spectroscopic, and Structural Approaches .......................... 774 III. Isotope Effects in Hydrogen Bonding ............................................................................. 775 A. Simple H-Bonds ....................................................................................................... 775 B. Unusual H-Bonds ..................................................................................................... 776 C. Primary Catalytic H-Bonds...................................................................................... 776 D. Secondary Catalytic H-Bonds.................................................................................. 777 IV. Issues in H-Bonding and Enzyme Catalysis ................................................................... 777 A. Cautionary Notes on Mutations at H-Bonding Sites in Enzymes .......................... 777 1. H-Bonds in the Orientation of Ligands for Optimal Catalysis ........................ 777 2. The Catalytic Triad of Serine Hydrolases ........................................................ 779 B. Primary Catalytic H-Bonds...................................................................................... 781 C. Secondary Catalytic H-Bonds.................................................................................. 781 1. The Catalytic Triad of Serine Hydrolases ........................................................ 781 2. The Oxyanion Hole of Serine Hydrolases........................................................ 785 D. Conformational Changes Signaled by Proton Inventories ...................................... 787 V. Summary .......................................................................................................................... 788 References..................................................................................................................................... 789

765

766

Isotope Effects in Chemistry and Biology O

O CO2

HN O H2O3PO

N

CO2H

O

HO

HN H

O

H2O3PO OH

N

O

HO

OH

SCHEME 29.1 Decarboxylation of OMP.

I. THE PROBLEM OF ENZYME CATALYSIS A. MAGNITUDES OF C ATALYTIC ACCELERATIONS BY E NZYMES The kinetics of enzyme-catalyzed reactions (Lescovac;1 Segel;2 see also the chapter in this volume by Cleland) are characterized for the most part by two kinds of rate constants, customarily given the notation kcat (a first-order rate constant, dimensions sec21, corresponding to the circumstance in which the enzyme is fully saturated by substrates, reaction intermediates, or products as ligands) and kcat =KM (a second-order rate constant, dimensions M 21 sec21, corresponding to the circumstance in which some form of the enzyme is reacting with a free substrate molecule). To measure the magnitude of the catalytic acceleration produced by an enzyme, each of these rate constants can be compared to the rate constant of a reaction not involving an enzyme, often called the “uncatalyzed reaction” or sometimes the “standard reaction”.3 The choice of this reaction is completely arbitrary, although a desire for convenience and simple interpretation of the calculated accelerations suggests the choice of a reaction closely related in its chemistry to the processes occurring through the action of the enzyme. If the standard reaction chosen in a given case should be kinetically first order with rate constant kunc (/sec), then the two possible comparisons to enzymic rate constants are:4 rate enhancement ¼ RE ¼ kcat =kunc ðdimensionlessÞ

ð29:1Þ

catalytic proficiency ¼ CP ¼ ðkcat =KM Þ=kunc ðdimensions=MÞ

ð29:2Þ

Obviously, the choice of standard reaction is critical to the numerical values of RE and CP, and a realistic or convenient choice is made difficult by the fact that organisms have evolved toward a high signal to noise ratio in the chemical reactions that constitute their physiology. This end is achieved through selection of intrinsically slow reactions for use in physiology, followed by the evolution of enzymes with high catalytic accelerations. Thus high reaction rates can be attained, yet the rates are precisely controlled by the amount of the enzyme and by its properties, such as susceptibility to inhibition. Wolfenden and his coworkers5 have made heroic efforts to measure the very slow rates of “bestchoice” uncatalyzed reactions, providing thereby a collection of the most meaningful values of RE and CP. For example, the decarboxylation in free aqueous solution (pH 6.8) of 1-methylorotate, a standard reaction for catalysis by OMP decarboxylase (Scheme 29.1), was studied kinetically in sealed vials between 100 and 2008C and the rate constant extrapolated to 258C to yield a value of kunc ¼ 2.8 £ 10216/sec. This value can then be compared with kcat ¼ 50/sec to give RE ¼ 1.7 £ 1017, or with kcat/KM ¼ 7 £ 106/M/sec to give CP ¼ 2.5 £ 1022/M. These are large values, indicators of the high catalytic power of OMP decarboxylase and suggesting that the development of mechanistic explanations for the catalytic power of effective enzymes should be a challenging problem.

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B. TRANSITION- S TATE S TABILIZATION AND C ATALYSIS The basis for much of the discussion of enzyme catalysis is the “ultrasimple” version of the transition-state theory (the terminology of Johnston6); in which any rate constant k is described by the following form of the Eyring equation: k ¼ ðkB T=hÞexpð2DG‡ =RTÞ

ð29:3Þ

where: kB is the Boltzmann constant, h the Planck constant, R the gas constant, T the Kelvin temperature, and DG ‡ is the standard Gibbs free energy of activation. (Far more sophisticated and complete accounts of transition-state theories are found in the chapters in this volume by Bigeleisen and by Wolfsberg.) The free energy of activation is: DG‡ ¼ GoTS 2 GoRS

ð29:4Þ

GoTS

where is the standard Gibbs free energy of the transition state (with one special feature) and GoRS is the standard Gibbs free energy of the reactant state. The special feature is the omission from the transition-state free energy GoTS of the contribution of the single vibrational motion that carries the system through the transition state, the reaction – coordinate motion. This omission is given little notice in most cases, but can be indicated by referring to DG ‡ and the corresponding DH‡ and DS‡ as quasithermodynamic functions. As Equation 29.4 suggests, DG‡ ¼ 2RT ln K ‡ , where K ‡ is a quasiequilibrium constant (contribution of the reaction coordinate motion of the transition state omitted) for the activation reaction RS ! TS. These matters are unproblematic for first-order reactions, since the dimensions of both the rate constant and kB T=h are s21, but for second-order reactions, the rate constant must be multiplied by a standard-state concentration chosen for one of the reactants in order to make Equation 29.3 dimensionally correct.p If the activation equilibria (i.e., the quasiequilibria for conversion of the reactant state to the transition state) for kcat =KM and kunc are written as: E þ S ! ETk=K S ! Tunc

DG‡k ¼ GoETk=K 2 GoE 2 GoS

ð29:5Þ

DG‡unc ¼ GoTunc 2 GoS

ð29:6Þ

where: E and S are free enzyme and free substrate, respectively, ETk=K is the enzyme-bound transition state corresponding to the parameter kcat =KM and Tunc is the transition state for the

p The choice of standard states has caused a certain amount of concern.71,72 The choice corresponds to the conversion of concentrations to thermodynamic activities so that equilibrium constants and thus thermodynamic parameters are properly expressed (see Pitzer and Brewer’s revision of Lewis and Randall,7 Chapter 20, and also the discussion by Gurney9 of cratic and unitary contributions to thermodynamic quantities). It is sometimes thought of also as the use of pseudo first-order rate constants to make Equation 29.3 dimensionally correct, but the concept must be carried a bit further. If kcat/KM ¼ 7 £ 106/M 21/sec21 for OMP-decarboxylase is converted to a pseudo first order rate constant at 1 M OMP, (kcat/KM)1M ¼ 7 £ 106/sec21, then the free energy of activation can be calculated correctly from the expression DG‡ ¼ 2RT ln½ðkcat =KM Þ1M =ðkB T=hÞ only if one also chooses standard-state concentrations for free enzyme and for enzyme-bound transition state such that these are equal to each other (other than its equality, these need not be specified). If these were not equal to each other, (kcat/KM)1M would have to be multiplied by its ratio before calculation of DG ‡. If DG ‡ is calculated from DG‡ ¼ 2RT ln½ðkcat =KM Þ1M =ðkB T=hÞ ; this restriction on standard states is implicitly adopted whether one recognizes it or not. Similarly, if one calculates DG ‡ for the uncatalyzed reaction from the expression DG‡ ¼ 2RT ln½ðkunc Þ=ðkB T=hÞ ; there is an implicit choice of equal standard-state concentrations for reactant state and transition state. Finally, if one calculates a dimensionless value of CP ¼ 2.5 £ 1022 from (kcat/KM)1M/kunc, one should carry the same reactant-state and transition-state standard states throughout. A choice of 1 M as standard-state concentration for substrate in addressing kcat/KM then translates into implicit choices of 1 M for substrate in the uncatalyzed reaction and for the transition state of the uncatalyzed reaction as well. Enzyme and enzyme-bound transition state should have equal standard-state concentrations but the value need not be further specified. These matters are not limited to the use of diagrams of free energy vs. reaction progress, and cannot be obviated by restricting attention only to unimolecular reaction steps. The choice of standard states is inherent in the calculation from rate constants of the thermodynamic or quasithermodynamic parameters of activation.

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Isotope Effects in Chemistry and Biology

uncatalyzed reaction in free solution, then CP becomes the equilibrium constant for the reaction: E þ Tunc ! ETk=K

ð29:7Þ

and the value of DGoCP ¼ 2RT ln CP; the free energy released when the transition state for the uncatalyzed reaction binds into the enzyme active site and forms the enzyme-transition state complex, therefore gives a direct measure of the stabilization by the enzyme of the transition state. Similarly, if the activation equilibria for kcat and kunc are written as: ES ! ETk

DG‡k ¼ GoET 2 GoES

ð29:8Þ

S ! Tunc

DG‡unc ¼ GoTunc 2 GoS

ð29:9Þ

where ES and ETk are, respectively, the enzyme-bound substrate and the enzyme-bound transition state, that is, rate limiting for the parameter kcat ; S, and Tunc again the substrate and transition state in free solution, then RE becomes the equilibrium constant for the reaction: ES þ Tunc ! ETk þ S

ð29:10Þ

Note that the transition states ETk and ETk=K may or may not be identical; that point must be established by other experiments. Now the quantity DGoRE ¼ 2RT ln RE is the free energy liberated by the displacement from the active site of the enzyme of substrate S by Tunc to form the enzyme-bound transition state ETk and the substrate in free solution. Since DGoRE ¼ ðGoETk þ GoS Þ 2 ðGoES þ GoTunc Þ

ð29:11Þ

DGoRE ¼ ðGoETk 2 ½GoE þ GoTunc Þ 2 ðGoES 2 ½GoE þ GoS Þ

ð29:12Þ

can also be written: The value of DGoRE measures the stabilization of the transition state by the enzyme (i.e., 2 [GoE þ GTunco]) diminished by the stabilization of the reactant state by the enzyme 2 [GoE þ GoS]), or the net transition-state stabilization by the enzyme.

GoETk (GoES

C. H- B ONDS AS

A

M EANS OF T RANSITION- S TATE S TABILIZATION

Let us consider the transition-state stabilization energy that is required to account for typical enzyme activities, in order to ask whether the energies for typical H-bonding interactions are sufficient to be regarded as important contributors to either catalytic proficiencies CP or rate enhancements RE. The value of CP for OMP-decarboxylase (OMP-D), calculated above as about 1022 M 21 yields o DGCP ¼ 2 31 kcal/mol (1 M standard state). This is the free energy liberated by the reaction: OMP-D þ Tunc ! OMP-D : Tk=K

ð29:13Þ

at a standard-state concentration for Tunc of 1 M (and at standard-state concentrations for OMP-D and OMP-D:Tk/K which are equal to each other, and most simply also 1 M), where Tk/K refers to the transition state that is rate limiting for the constant kcat =KM : The formal standard-state choice is arbitrary, corresponding to the choice of a standard-state concentration for substrate in the enzymic reaction,p but the choice will affect the value of DGoCP. Common standard-state choices for substrate are 1 mM, 1 mM, and 1 M and will generate values for DGoCP of 2 22.6, 2 26.8, and 2 31 kcal/mol; perhaps it is obvious that a choice of 10222 M would yield DGoCP ¼ 0. The reason for these variations is the free-energy cost of bringing the substrate from its standard-state concentration into the enzyme active site. Concerning this cost of concentrating the substrate, Pitzer and Brewer7 wrote: “Of all the applications of thermodynamics to chemistry, none has in the past presented greater difficulties, or been the subject of more misunderstanding, than the one involved in the

Hydrogen Bonds, Transition-State Stabilization, and Enzyme Catalysis

769

calculation of what has rather loosely been called the free energy of dilution, namely, the difference in the chemical potential or partial molal free energy of a dissolved substance at two concentrations.” As Jencks8 noted in describing the Circe effect, the energy of some of the attractive interactions in the complex between enzyme and ligand must be utilized to offset the entropic penalty for concentrating a dilute free ligand into the active site. The more dilute the ligand, the greater the cost and the smaller the remaining energy that can be “expressed as catalysis.” (See also Gurney9 on “cratic” contributions to thermodynamic quantities). So the question then becomes: How much energy is there to be accounted for in terms of active-site interactions such as H-bonding between enzyme and transition-state molecule? To answer this question, we must remove from DGoCP the free energy required to concentrate the transition-state molecules from free solution at their standard-state concentration into the active site. The result should then be the total free energy of interaction between enzyme and transitionstate ligand. One approach goes back to discussions of orbital steering and how to correct for the propinquity effect in substrate binding. There the idea was to change the standard-state concentration of the free ligand to such a high concentration that the active site of every enzyme molecule in solution would automatically be occupied by a ligand molecule. At least for substrates about the size of a water molecule, conversion to a standard-state concentration of 55.5 M would guarantee that every site in the solution would be surrounded by substrate; this argument suggested thus a value of the order of 100 M. Another argument held this value to be too high because a complete surrounding of the active site was unnecessary, and a value of 10 M was favored. Thus if the standard-state range of 10 to 100 M were held to be an adequate correction, the energy to be ascribed to the net favorable interactions of the transition state with the active site of OMP-D would be DGoCP ¼ 2 32.4 kcal/mol (100 M standard state) to 2 33.8 kcal/mol (10 M standard state). The upshot is that the precise energy to be accounted for in explaining the value of CP for any enzyme will be uncertain to some extent, but for highly accelerating enzymes like OMP-decarboxylase, the problem is not very serious, only about 1 to 3 kcal/mol in about 30 kcal/mol. Even for the less accelerating carbonic anhydrase, with CP ¼ 1010/M, around 14 kcal/mol is indicated and an uncertainty of 1 to 3 kcal/mol is not extraordinarily severe. Can H-bonds contribute substantially to stabilization energies of the order of, say, 15 to 30 kcal/mol? The answer will depend on a number of factors, some more obvious than others, many of which have been addressed with particular care and clarity by Fersht10 and by Warshel and Papazyan:11 1. The strength of an individual H-bond, say from an active-site residue to an atom of the transition state in the active site, must not be considered as the energy required for dissociation of the partner groups into free and unattached donor and acceptor. Instead, as the transition state dissociates from the active site of the enzyme it enters the aqueous solution and may find donor and acceptor sites within its structure now participating in H-bonds to water molecules. Thus the energetic contribution to transition-state stabilization and thus to catalysis of an H-bond between a ligand group and an enzyme group cannot be as large as the energy of dissociation to the free, unattached donor and acceptor. The actual contribution will be this energy of dissociation diminished by the energy of formation of any H-bonds that form in aqueous solution at the same position (see particularly the discussion in Fersht,10 pp. 337 to 339, of H-bond inventories). 2. The “strengths” of various kinds of hydrogen bonds, in this sense of relative strengths, have been the subject of considerable dispute, but there may be an approaching consensus that the effective strength of an active-site H-bond relative to aqueous solvation rarely, if ever, exceeds around 10 kcal/mol. If this value is provisionally adopted, then transition-state stabilization energies of 15 to 30 kcal/mol, if wholly due to

770

Isotope Effects in Chemistry and Biology

H-bonds, would correspond to around two to four H-bonds of relative strength 10 kcal/mol, and more than two to four H-bonds of lower relative strength. 3. There are a large number of potential H-bond donors and acceptors typically found in active sites. Thus, it seems likely that a considerable part of transition-state stabilization energies in the range of 15 to 30 kcal/mol could be effected by H-bonding interactions. If some or all of such interactions were of unusual strength, say 10 kcal/mol, then a few H-bonds would account for much of the catalytic power, but no unreasonable number of interactions would be needed even if each were of quite modest strength. An important point emphasized by Warshel and Papazyan11 is that, if the enzymic H-bond participants in the active site are pre-organized to fit well the available H-bond partners in a transition state derived from a specific substrate, then there will be little entropic cost for the formation of the H-bonds once the ligand is located in the active site. In free aqueous solution, on the other hand, assembly and orientation of the water molecules that will replace the enzymic groups as H-bond partners will possibly be costly in loss of entropy. This effect would increase the stabilizing effect of the H-bonds in the complex of enzyme and transition state.

D. BEYOND THE T RANSITION -STATE T HEORY OF C ATALYSIS The “ultrasimple” transition-state theoretical formulation employed here, in which the Eyring equation (Equation 29.3) has a universal preexponential factor of kB T=h; has served for many years as a vehicle for presenting and discussing experimental findings in enzyme catalysis and related fields of solution kinetics, and with reasonable frequency for theoretical treatments as well. The customary formulation omits a transmission coefficient k, which can be inserted if needed and used for simple accounts of quantum tunneling.6,12 Several developments in the last few years have suggested that more complex treatments are required to account for new results. Among the new results felt by various authors not to be compatible with the ultrasimple transition-state theory are: 1. Evidence for extensive tunneling, particularly that which results in large kinetic isotope effects (kH/kD . 20 at room temperature) insensitive to temperature (see the chapters by Klinman and Kohen in this volume). 2. Evidence that vibrational motions of the enzyme, some remote from the active site, may be coupled in some sense to the reaction-coordinate motion so as to enhance the rate of reaction13,14 (see also the chapter by Schwartz in this volume). 3. Evidence from molecular-dynamics studies using the “near-attack conformation” (NAC) approach that indicated in some particular cases large populations of reactive conformations of the substrate in the active site, it then being conceivable that these reactive species could be carried into the product state by thermal motions of the enzyme.15 Garcia-Viloca et al.16 have emphasized the power of modern versions of the transition-state theory, particularly combined with simulational approaches, to address many of these phenomena, while others have advocated reaction-rate formulations less familiar than the transition-state theory (see the chapter in this volume by Schwartz). Such evidence as just described may eventually indicate that novel roles for hydrogen bonding must be devised to account for enzyme catalysis (e.g., networks of hydrogen bonds as the machinery of transmitting “promoting” vibrational motions). The situation at this point is, however, fluid and unclear. We shall refrain then from further speculation along these lines, and restrict the treatment in the present article to cases understandable in terms of the traditional solution chemist’s transition-state theory.

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II. STRUCTURE AND STRENGTH OF H-BONDS A. THE C ONCEPT OF H-BOND S TRENGTH The simplest definition of the strength of an H-bond is the energy change that accompanies fission of the bond. For example, the gas-phase dissociation energy of the H-bonded complex of fluoride ion with HF requires 39 ^ 1 kcal/mol:17 FHF2 ! F2 þ HF DHo ¼ HoF þ HoHF 2 HoFHF ¼ 39 kcal=mol

ð29:14Þ

A problem arises if one interprets the value in terms of properties of the individual species, i.e., if one attempts to interpret DHo in Equation 29.14 in terms of ideas about HoF, HoHF, or HoFHF. For example, one idea is that the large value of DHo could arise because of the concentrated negative charge on fluoride ion, which is expected to make it unstable in the gas phase (large value of HoF), with no unusual features to be attributed to the bifluoride ion itself other than some stabilization of the negative charge by its dispersal in the complex (HoHF 2 HoFHF, only a little different from zero). A different idea is that the large value of DHo arises because the hydrogen bond in the FHF2 complex is a bond of unusual strength, the stability of the negative charge being about the same in the complex or with the bare fluoride ion (a large negative value of HoFHF makes HoF þ HoHF relatively negligible). Evidence other than the energetics of dissociation must be brought to bear in order to discriminate between an explanation for a large bond dissociation energy that emphasizes an unusually high energy for the dissociated product state and the alternative explanation that emphasizes an unusually low energy for the associated reactant state.† When the particular question about H-bond strength has to do with the contributions of transitionstate or reactant-state H-bonds to enzyme catalysis, then it becomes particularly important to remain aware that ligand-binding equilibria such as Equation 29.7 or Equation 29.10 above involve a comparison of substrate or transition state in the enzyme active site with substrate or transition state in aqueous solution.11 The ligand in solution will have H-bonds from and to water molecules at the same sites in its structure where enzyme active-site proton acceptor or donors are likely to be interacting with the bound ligand. The specification of the energetic contribution of particular H-bonds to the binding equilibria and thus to catalysis necessarily therefore involves not merely estimates of the energy of dissociation of enzyme –ligand H-bonds. Also required are estimates of the energy of dissociation of the corresponding bonds to water, and in both cases the effects of preexisting structures must be considered (such as preorganized H-bond networks in enzyme active sites, or water structure in free solution).

† The same point obviously applies to the energetics of any dissociation process. When the dissociation phenomenon is that of an enzyme-transition state complex to yield free enzyme and free transition state, as in Equation 29.7 and Equation 29.10, one cannot distinguish from the energetic data alone between the idea that a large value of the dissociation energy reflects powerful attractive forces in the enzyme-transition state complex and the idea that a large value of the dissociation energy reflects the instability of the transition state in free aqueous solution.14 A certain amount of acrimony in the discussion of enzyme-transition-state affinities seems to have had as its basis the feeling that the statement “catalysis derives from transition-state stabilization” must mean “catalysis derives from attractive interactions between enzyme and transition state in their complex.” Of course the meaning “catalysis derives from the high energy of the transition state in free solution with no particular attractive interactions being present in the enzyme-transition-state complex” is equally compatible with the original statement. One should not, however, conclude in a case where this latter meaning is correct that the “enzyme does not stabilize the transition state.” In common thermodynamic usage, a more stable substance is one with a lower standard free energy of formation. If catalysis by the enzyme is occurring, then within the confines of transition-state theory, the enzyme-transition-state complex must have a lower free energy of formation that the sum of the free energies of formation of free enzyme and free transition state. The combination of the free enzyme with free transition state leads to a more stable substance, the enzyme-transition-state complex: the enzyme therefore stabilizes the transition state.

772

Isotope Effects in Chemistry and Biology

B. CATEGORIZATION OF H - B ONDS A thorough typology of H-bonds was presented by Hibbert and Emsley,18 and is frequently cited in discussion of H-bond contributions to enzyme catalysis. Here we shall make use of a very much simplified threefold taxonomy: 1. By simple H-bonds we refer to the familiar H-bond interactions with effective bond ˚ or more, and strengths of a few kcal/mol, distances between the end-atoms of 2.5 A isotopic fractionation factors within 10 to 20% of unity. 2. By unusual H-bonds we mean interactions that may have effective bond strengths greater than those of simple H-bonds, sometimes tens of kcal/mol, distances between ˚ , and isotopic fractionation factors that may the end atoms that may be less than 2.5 A be substantially lower than unity (e.g., 0.2 to 0.5). One subset of unusual H-bonds19 is formed by the so called “low barrier H-bonds” or LBHBs, to which we will return below. 3. By catalytic H-bonds we mean interactions found only in transition states, such that vibrations of the linkage may be, but need not be, components of the reactioncoordinate motion of the transition state. Catalytic H-bonds will be divided into primary catalytic H-bonds, in which one or both of the end atoms X or Y of the Hbond X · · · H · · · Y definitely moves in the reaction-coordinate motion, and secondary catalytic H-bonds, in which neither end atom need necessarily move in the reactioncoordinate motion, although these may so move. Primary catalytic H-bonds are preeminently present in transition states for general acid-base catalysis. When an active-site base removes a proton from a position at which positive charge is increasing in the reacting substrate molecule, that interaction creates a primary catalytic H-bond. Similarly, when an active-site acid donates a proton to a position at which negative charge is increasing in the reacting substrate molecule, that interaction creates a primary catalytic H-bond. If the motions of the proton in these bonds are largely components of the reaction-coordinate motion, then the isotope effects that will be observed are primary isotope effects and should be understood within the well developed theoretical framework for primary hydrogen isotope effects. These kinds of interactions have been called “in-flight” protons20 and the corresponding X · · · H · · · Y system has not usually been thought of as an H-bond. If the motions of the proton in these bonds are not largely components of the reaction-coordinate motion, but are instead occurring within a stable potential, then the observed isotope effects should be understood in terms of H-bond isotope effects and the interactions should be understandable in terms of the properties of H-bonds. This point is discussed in greater detail below. Secondary catalytic H-bonds arise if the primary catalytic H-bond is linked to a second H-bond, as in the catalytic triads B2 · · · H · · · B1 · · · H · · · X seen in the active sites of hydrolases, or in lengthier chains of H-bonds leading away from reacting species as in the active sites of carbonic anhydrase and other enzymes (see the chapter in this volume by Silverman). In principle, the motions of protons in secondary catalytic H-bonds may also constitute components of the reaction-coordinate motion, in which case isotope effects should be treated as primary isotope effects for “in-flight” protons, or the protons may be chiefly in stable potentials, in which case these fall into the H-bond categories.

C. SOME P ROBES OF H YDROGEN B ONDS Several approaches to H-bonds and their properties merit brief consideration here. Most of the methods are also considered at length in subsequent discussions.

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1. NMR Approaches The power of sophisticated NMR approaches to illuminate H-bond characteristics has been employed with excellent effect by Limbach and his collaborators, as particularly exemplified by their recent study21 of collidine:HF complexes. By the use of 1H, 19F, and 15N spectra at temperatures from 97 to 150K for solutions at varying ratios of collidine and HF in CD3F and CDF2Cl mixtures where the dielectric constant can be controlled by the solvent composition and temperature, it was possible to observe and characterize the chemical shifts and coupling constants for H-bonded complexes of fluoride ion with one to four HF molecules, of collidine with HF and of collidinium ion with bifluoride ion and with bifluoride ion bearing one and two HF molecules. Interpretation of the NMR parameters led to a complete geometrical structure for each of the complexes. Of special interest was the temperature dependent structure of the collidine:HF complex (the structure was varied chiefly as a result of the temperature-dependent dielectric constant of the solvent mixtures, higher dielectric constants favoring a collidium – fluoride-like complex and lower dielectric ˚, constants the electrically neutral complex). At a temperature of 180K, the NF distance was 2.41 A ˚ , and the HF distance 1.17 A ˚ . As the temperature was lowered to 150K, the the NH distance 1.24 A ˚ and the HF distance increasing to H-bond “symmetrized” with the NH distance decreasing to 1.20 A ˚ , the NF distance remaining equal to their sum as all these complexes formed linear H-bonds. 1.21 A As the temperature was further lowered, the ionic asymmetrical structure came to dominate so that at ˚ , the NH distance 1.14 A ˚ , and the HF distance 1.31 A ˚ . These 103K, the NF distance was 2.45 A studies are establishing the entire systematics of relationships among NMR observables and H-bond characteristics, which will render future NMR studies of H-bonds in enzyme –ligand complexes far more informative. Bachovchin and his collaborators (see references in Ash et al.22) have been exceedingly effective in the detailed characterization of H-bond properties in enzyme active sites. Much of their work has been focused on the catalytic H-bonds in the active sites of hydrolases, and is discussed with other work on this subject below. Perrin and his collaborators23 – 27 have made a series of contributions to the NMR study of Hbonds, including the use of isotope-perturbation techniques. Perrin and Ohta26 measured the 13C spectra of protonated forms of the “proton sponges” 1-8-bis(dimethylamino)naphthalene and its 2,7-dimethoxy derivative when the molecules had been asymmetrically labeled with deuterium in their methyl groups. The H-bonds in both cases exhibited double minimum potentials in spite of the fact that the H-bonds are stronger than H-bonds to dimethylanilinium cations by around 10 kcal/mol in the case of the parent compound and 16 kcal/mol in the case of the dimethoxy derivative. Perrin and Ohta concluded that proton sponges do not possess H-bonds of unusual intrinsic strength but instead that dissociation of the proton leads to an exceedingly unstable conjugate base in which both steric and electronic repulsions favor conversion to the protonated form. 2. Theoretical Studies of H-Bonds Theoretical approaches are a particularly powerful method for determining reliably and examining in detail many of the features of H-bonds that are of the most intense concern in enzyme catalysis. These properties include the geometry of H-bonds, the degree of covalent character, the energy of dissociation, and the sensitivity of H-bond properties to environment. Theoretical studies by Karplus, McCammon, Warshel, and other theorists who have focused particular attention on questions of enzyme catalysis are referred to in Section IV below. In the present section, we mention several contributions of general and methodological importance. Scheiner28,29 has long been active in the ab initio quantum theory of H-bonding and has published an influential review and book. His recent contributions have included characterizations of H-bonds of O –H donors to aromatic amino acids,30 the strongest such interactions being simple H-bonds (e.g., O · · · H · · · O), H-bonds to the aromatic p-cloud being of intermediate strength,

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Isotope Effects in Chemistry and Biology

and the weakest interactions H-bonds donated by CH bonds. The latter, somewhat controversial interactions have also been explored in detail.31 Del Bene and her collaborators32 have reported ab initio calculations on NMR coupling constants as a function of H-bond geometry that are strongly complementary to the NMR studies of Limbach and his coworkers described in Section II.C.1 above. Her studies have elucidated the properties of the two-bond spin – spin coupling constant across H-bonds in neutral F – H · · · N systems and ion pair systems (F2· · · NHþ), where the 19F– 15N coupling constant was at issue, and C · · · H · · · N systems where the 13C – 15N coupling constant was considered. In all cases, coupling was dominated by the Fermi contact term and was a sensitive measure of the end atom distance (F– N or C –N) across the H-bond, being insensitive to small deviations of the angle at H from linearity. The coupling constants were larger in the ion-pair H-bonds than in the neutral-pair H-bonds at the same end atom distance, reflecting a greater degree of “shared-proton” character in the ion-pair system. When the systems were segregated into classes with similar hybridization at the acceptor nitrogen atom, then within each class, the coupling constant was found to be a good quadratic function of the end atom distance. This work is likely to be very important in the justification and calibration of experimental NMR determinations of H-bond geometry. McAllister and his coworkers have reported a series of ab initio studies, in many cases supporting a role for unusual H-bonds in enzyme catalysis. One problem that has plagued NMR spectrsocopists has been the relationship, if any, between the proton NMR chemical shift for an H-bonding proton and the dissociation energy of the bond. Kumar and McAllister33 reported a linear relationship with a slope of about 1.5 kcal/mol-ppm. 3. Thermochemical, Spectroscopic, and Structural Approaches Reinhardt et al.34 made use of calorimetric techniques to determine the enthalpies of formation for complexes in chloroform solution between carboxylic acids of pK (aqueous) from 0.23 to 4.76 with 1-methyl, 1-butyl, and 1-t-butyl imidazole. These systems involve the same functional groups as those that could form a secondary catalytic H-bond in enzymes that have a catalytic triad (see below). The measured enthalpies of complex formation are not greatly different for the three substituted imidazoles and rise steadily from 2 13 to 2 14 kcal/mol for the trifluoroacetic acid complex to about 2 4 to 2 5 kcal/mol for the acetic acid complex. When the stronger base nonylamine was used, much more negative enthalpies, from 2 24 to 2 13 kcal/mol were measured. Infrared spectra in the carboxyl-stretching region 1600 to 1800/cm were used to try to sort out the contributions within the complexes of simple H-bonds between electrically neutral partners, unusual H-bonds, and ion pairs. The nonylamine complexes exhibited single carboxyl-group absorptions similar in frequency to the signals for tetrabutylammonium salts of the carboxylates, and were thus assumed to be ion-pair complexes. Absorptions at a similar frequency in the imidazole complexes were then taken to indicate ion-pair contributions. A signal at the absorption frequency of the free acid was taken as diagnostic of a simple H-bond. On this basis, the authors conclude that trifluoroacetic acid (aqueous pK 0.23) forms an ion-pair complex with the imidazoles (a single infrared maxium at about 1670/cm) and that the acids with aqueous pK from 2.86 to 4.76 form simple H-bond complexes (an infrared maximum for the acetic-acid complex was observed at 1709/cm, the same as for free acetic acid). Complexes with dichloracetic (aqueous pK 1.29) and dichloropropionic (aqueous pK 2.06) acids exhibit two infrared maxima, one below 1700/cm, reminiscent of that in the ion-pair complexes, and another around 1700, which was taken to correspond to a LBHB rather than to the simple H-bond. The extinction coefficient for the LBHB signal was taken as the average of the extinction coefficients for the ion-pair species and the simple H-bond species, leading to an estimated population of LBHB species in the 45 to 68% range. Thus the enthalpies of formation of 12 to 15 kcal/mol were taken to arise in about equal measure from the formation of ion pairs and LBHBs, giving both an enthalpy of formation of 12 to 15 kcal/mol.

Hydrogen Bonds, Transition-State Stabilization, and Enzyme Catalysis

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Perhaps the most surprising aspect of the formation of LBHBs specifically for acids of pK 1 to 2 in water with imidazole bases with pKs around 6 to 8 in water would be the large mismatch in pK of the bonding partners. The results of this study should be compared with those of Shan et al.35 discussed in Section IV.C.1 below. Over decades, extraordinary studies by infrared spectroscopic methods of unusual H-bonding in various systems have appeared from the group of G. Zundel. These findings, exemplified by the work of Bartl et al.36 on cases in enzymes, merit much attention from those who wish to relate H-bonding to enzyme catalysis. In the recent past, crystalline salts of (S · · · H · · · S)þ cations have been prepared with carborane anions (e.g., [CH6B11Cl6]2) that are weakly basic such that the cations are stable even when the two end groups of the H-bond are very weakly basic.37 The small S –S distances observed ˚ when S is diethyl ether, 2.35 A ˚ crystallographically suggest that these are unusual H-bonds: 2.40 A ˚ ˚ when S is tetrahydrofuran, 2.47 A when S is benzophenone, 2.48 A when S is nitrobenzene. The opportunity to conduct further studies of these materials should illuminate greatly the question of unusual H-bonds. Isotopic techniques have of course played a vital role in studies of the structure and strength of H-bonds. these are separately considered in the following section.

III. ISOTOPE EFFECTS IN HYDROGEN BONDING A. SIMPLE H-BONDS The formation and dissociation of simple H-bonds, as are formed at most backbone positions in folded proteins, appears to be associated with small isotope effects. The folding experiments of Krantz et al.38 suggest that average backbone H-bonds in helices are more stable by a factor around 1.01 (per site) when deuterated than when protiated, while backbone sites in coils or sheets exhibit, on the average, still smaller isotope effects. In NMR experiments where site specific equilibrium isotopic fractionation factors (defined as wi ¼ [D/H]i/[D/H]water for the backbone N-H of the ith residue; for complete information see the chapter by Quinn in this volume) were determined, Loh, and Markley39 found isotopic preferences in staphylococcal nuclease to range from w; ¼ 0.3 to w ¼ 1.5, the value of 0.3 representing a 3.3-fold preference for H over D, as expected for an isotopic center in a loose potential, and the value of 1.5 a preference for D over H that would signal a stiff potential. The average values of w for various protein substructures spanned a range from 0.8 for helices to 1.0 for solvent exposed sites, all with errors of 0.1 to 0.3. By use of a different technique that corrects for the isotope dependence of magnetization exchange between solvent and protein (a problem that can render the apparent w values too small), LiWang, and Bax40 found for human ubiquitin that all w values were between 1.0 and 1.2, average value 1.11. Bowers and Klevit41 repeated the ubiquitin determination with different controls for the magnetization-transfer problem and obtained values within the range 0.9 to 1.3 for 75% of the sites with an average value of 1.07. However, extreme values were found at a few sites (1.5 for Ser 20, 0.3 for Thr 9). Theoretical work on peptide clusters by Edison et al.42 generally reproduced most of the values observed up to that point, and led to the suggestion that low fractionation factors, suggesting a loose binding potential that favors protium over deuterium, might arise either from an H-bond involving a charged residue or involvement of the site in a network of H-bonds. Bowers and Klevit41 suggested that fractionation factors greater than unity may arise from restrictions to bending motions of bonds to the isotopic center produced by formation of H-bonding interactions. It seems reasonable to conclude that simple H-bonds, typified by backbone interactions such as N – H · · · OyC bonds, will produce scarcely any isotope effect with the small detectable effects corresponding to f values between about 0.8 and 1.2.

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Isotope Effects in Chemistry and Biology

B. UNUSUAL H-BONDS Centers involved in unusual H-bonds may have larger isotope fractionations, and in the limit of short, strong, or low barrier H-bonds (19, LBHBs: see Section IV.C.1). the fractionation factor may be as low as 0.3. It is now reasonably clear that other unusual H-bonds are also capable of substantial isotope fractionation. For example, in the painstaking study of Shan, Loh, and Herschlag,35 ring substitution in phthalic acids was used to modulate the difference in pK of the two adjacent carboxyl groups on the ring. For phthalic acid itself in DMSO, the proton chemical shift of the fully protonated acid was 13.2 ppm, not within the canonical 17 to 21 ppm range suggested for LBHBs (Section IV.C.1). The monoanion, where the pKs of the two carboxyl groups are necessarily equal, shows the proton chemical shift moved to 20.8 ppm, within the LBHB range, and the fractionation factor was 0.56 ^ 0.06, also consistent with an LBHB; three of the criteria enumerated in Section IV.C.1 for an LBHB were thus met. However, when substituents are introduced that remove the pK equality, moving the DpK from 0 to 0.46, the proton chemical shift of the monoanion remained at 20.4 to 20.5, showing the NMR signal not to be diagnostic of a canonical LBHB. Also, for the case of 4-chlorophthalate monoanion (DpK ¼ 0.38, proton chemical shift 20.5), the fractionation factor was 0.51 ^ 0.07, indicating neither the proton chemical shift nor substantial isotope fractionation to be diagnostic of LBHBs. This report and others since have supported the view that a continuous range of H-bond properties, including isotope fractionation, is connected to the variation of such features as the structure of the bonding partners, the polarity of the medium, etc., and that there seems to be no qualitative distinction between weaker and stronger unusual H-bonds.

C. PRIMARY C ATALYTIC H-BONDS As discussed in Section II.B., the primary catalytic H-bond that is seen in the abstraction of a proton by a general base catalyst or the donation of a proton by a general acid catalyst has commonly not been thought of as an H-bond. Instead, the proton tends to be regarded by many as an “in-flight” participant in the reaction-coordinate motion of the transition state, with transfer of the proton and whatever process is being catalyzed (typically the rearrangement of an array of atoms heavier than hydrogen so that the process is called “heavy-atom reorganization”) occurring concertedly. The concept that the general acid-base catalysis in some cases might not involve transfer of the proton as part of the reaction-coordinate motion for the reaction being catalyzed arose in a discussion of Swain et al.43 that isotope effects are smaller than expected in such reactions, when the atoms between which proton transfer appears to be occurring are electronegative atoms such as O, N, or S. When proton transfer to or from carbon is involved, isotope effects kH/kD are commonly 4 to 7, while with electronegative atoms, the usual observation is in the 2 to 4 range or even smaller. The suggestion was therefore made that, instead of participating in the reaction-coordinate motion, the proton in general catalysis was in a stable potential at the transition state. The transfer of the proton was imagined to occur either just before or just after the transition state for the heavy-atom reorganization. A remarkably prescient improvement on this suggestion was made by Cordes44 and Kreevoy, as illustrated in Figure 29.1. They envisioned proton motion to be coordinated with the reactioncoordinate motion but not a component of it, in fact orthogonal to it so that the proton is always in a stable potential. The illustration suggests that as the proton potential shifts from an asymmetric potential in the reactant state to an oppositely asymmetric potential in the product state it passes through a symmetrical single minimum potential at the point in the transition state where the affinity of both partners is roughly equal. Some years later, Kreevoy and his coworkers (reviewed in Kreevoy and Liang19) formed very similar hydrogen bonds in aprotic media, and showed them to have the needed isotope fractionation. Eliason and Kreevoy45 suggested that the hydrogen bridges

Hydrogen Bonds, Transition-State Stabilization, and Enzyme Catalysis

C

Ö:

δ+ C

H B

Ö:

777 δ+ C

δ− H B

C

δ+ Ö:

H

δ− B

H

δ− B

R

R

R

Ö:

C

+

R

Ö

H

− B

R

FIGURE 29.1 An illustration reproduced with permission from Cordes,44 who acknowledges a suggestion by M. M. Kreevoy, showing how the small isotope effects observed in general acid catalyzed C–O fission can be accounted for by a proton transfer process simultaneous with the C–O fission process such that the proton remains in a stable potential well throughout. Note especially the single minimum potential function at upper right, portraying a low-barrier or no-barrier hydrogen bond for the primary catalytic H-bond at the transition state.

between electronegative atoms (O, N, and S) in general acid-base catalysis were exactly these relatively strong H-bonds. Further evidence on this point will be discussed below. Overall, the evidence is most consistent with the view that primary catalytic H-bonds without substantial reaction-coordinate participation are characteristic of general acid-base catalysis when the proton is located between electronegative atoms, but that when the proton is being abstracted from or donated to carbon, then reactioncoordinate participation and larger isotope effects are the rule.

D. SECONDARY C ATALYTIC H-BONDS Isotope effects associated with secondary catalytic H-bonds is a far less well developed subject than the picture for primary catalytic H-bonds. Much of the attention in this area has centered on the catalytic machinery of the serine hydrolases and is discussed below (Sections IV.C.1 and IV.C.2) and in the chapter in this volume by Quinn.

IV. ISSUES IN H-BONDING AND ENZYME CATALYSIS A. CAUTIONARY N OTES ON M UTATIONS AT H-B ONDING S ITES IN E NZYMES 1. H-Bonds in the Orientation of Ligands for Optimal Catalysis The examination of H-bonds and other interactions in the stabilization of transition states and other ligands by enzymes was the subject of a pioneering set of investigations by Fersht and his coworkers, focused on tyrosyl-t-RNA synthetase (see the review in Fersht,10 pp. 420– 449). The main technique employed was the careful construction and characterization of a series of mutant enzymes, in which side chain structures for residues surrounding the ligands in the active site were mutated to other residues such that suspected interactions would be removed or modified. Then the kinetic consequences of the mutations were evaluated to place a quantitative value on the energy of the lost interactions. An emphasis of this work was the potential complexity of the effects of a mutation, and the extraordinary caution that is necessary in drawing even semiquantitative

778

Isotope Effects in Chemistry and Biology

RS

RS

Ad

O

oxidation

H HO HO

O H O

OH HO

Ad

O H

O O

OH

elimination of RSH

OH

Ad

O O

Ad

OH

addition of HOH

H

reduction HO

Ad

OH

SCHEME 29.2 The action of S-adenosylhomocysteine hydrolase proceeds by a redox partial reaction involving a tightly bound NAD/NADH (first and fourth steps) that spans an elimination or addition partial reaction (second and third steps).

conclusions from the kinetic effects of a mutation. The necessary protocols for assuring that the structural consequences of a mutation, and their energetic and kinetic implications were given an adequate account were laid out and have set the standard for subsequent research. An example of a study46 far less thorough than the canonical work by Fersht and his coworkers may serve to illustrate some features of the approach. The enzyme S-adenosylhomocysteine hydrolase (AdoHcyase) catalyzes the succession of reactions portrayed in Scheme 29.2. Overall, S-adenosylhomocysteine (AdoHcy) is hydrolytically converted to adenosine (Ado) and homocyteine (Hcy). The mammalian enzymes possess a tightly bound NADþ, and the initial step after binding of AdoHcy is conversion of AdoHcy to the 30 -keto derivative with concurrent reduction of the bound NADþ to NADH. The keto group activates the adjacent 40 -CH bond, permitting elimination of Hcy to form the a, b-unsaturated ketone, which now serves as a Michael acceptor in the addition of water. In a final series of steps, the 30 -ketoAdo is reduced by the bound NADH to Ado, which is then released. Figure 29.2 shows a schematic depiction of several groups thought to orient the substrate or a reactive side-chain of the enzyme by means of simple H-bonds (Asp 190 [D190]; Asn 191 [N191], Asn 181 [N181]), and the side chain of Lys 186 [K186], thought to act as general acid-base catalyst hydride transfer in redox reaction Ad RS

O H

H O

O H H

D190

O C O H2N C NH2

proton transfer in el-ad reaction

H2N

C

N191 O K186 O

kcat /KM

kcat

redox

el-ad

+3.1

+1.9

+4.3

+2.2

+3.3

+2.9

+3.8

+3.3

+6.0

+4.2

+7.4

+3.7

+3.1

+2.1

+4.5

+3.3

N181

FIGURE 29.2 Effects (DDG ‡, kcal/mol) of mutations to Ala of four residues proposed for H-bonding roles in transition-state stabilization in the action of S-adenosylhomocysteine hydrolase.46 In addition to effects on kcat/KM and on kcat, effects are also shown for irreversible-inhibitor reactions that mimic the redox partial reaction and the elimination-addition (el-ad) partial reaction. D190 is thought to orient the substrate properly for hydride-transfer and proton-transfer reactions by H-bonding to the 20 -OH. N191 and N181 are thought to orient K186 for a role as proton donor and acceptor, by H-bonding to the amino group. K186 as a general acidbase catalyst would form primary catalytic H-bonds to the 30 -OH and the 30 -keto group.

Hydrogen Bonds, Transition-State Stabilization, and Enzyme Catalysis

779

and thus to stabilize the transition states by primary catalytic H-bonds (the 20 -oxygen atom becomes more electrically positive as the 20 -OH group is oxidized and more negative as the 40 -C –H bond is cleaved, the Michael addition of water occurs and the reduction of the 20 -keto group occurs). Each of these candidate residues was mutated to Ala, thus replacing the side chains in each case by a methyl group. The mutations were shown not to affect the secondary or tertiary structures in any manner spectrsocopically detectable, nor to disrupt the tetrameric quaternary structure. Also shown in Figure 29.2 are the effects DDG ‡ of mutation of each of the residues to Ala on the magnitude of DG ‡ for four different rate constants: kcat/KM for conversion of AdoHcy to Ado and Hcy, which should reflect effects on total transition-state stabilization relative to the wild type enzyme; kcat for the same conversion, which should reflect effects on net transition-state stabilization (transition-state stabilization diminished by reactant-state stabilization); the rate constant for an irreversible-inhibition reaction that mimics the redox partial reaction; and the rate constant for an irreversible-inhibition that mimics the elimination-addition partial reactions. The data for the partial reactions are most easily addressed. Loss of the side chains of the residues suspected of simple H-bond interactions (D190, N191, and N181) has in each case increased reaction free energy barriers by 3.8 to 4.5 kcal/mol in the redox model reaction and by 2.2 to 3.7 kcal/mol in the elimination-addition model reaction. These energies are just in the range expected for simple H-bond interactions. Furthermore, these represent upper limits to the H-bond interaction energy, because removal of the H-bond may in each case produce further secondary consequences, such as failure to maintain proper alignment for hydride transfer at C-30 or proton transfer at C-40 for general acid-base catalysis by K186. Each of the secondary effects could produce free-energy changes larger than those expected for H-bond interactions. Mutation of K186 to Ala produces larger effects on the free-energy barriers, an increase of 7.4 kcal/mol in the redox model reaction and an effect of 3.7 kcal/mol in the elimination-addition model reaction. This result is in agreement with the expectation that primary catalytic H-bonds will be stronger than simple H-bonds. The fact that the effect is twice as large in the redox reaction as in the elimination-addition reaction indicates that general acid-base catalysis is more important for the redox reaction, suggesting that perhaps the elimination of Hcy and addition of water involve C –H bond fission and formation at C-40 concerted with C –S fission and C – O formation at C-50 so that little negative charge arises on the 30 -keto group in the elimination or addition transition states. The energetic effects of each mutation on the overall kinetic parameters emerge as weighted averages of the effects on the two partial reactions. The weighting factors for each partial reaction are given by the fractional degree to which that partial reaction determines the rate for the overall kinetic parameter in question.46 2. The Catalytic Triad of Serine Hydrolases The active site of many hydrolases contains a “catalytic triad,” a chain of three H-bonded (or potentially H-bonded) amino-acid side chains of which the final member will function as a nucleophile toward the substrate in one or more catalytic transition states: the paradigm is the Asp · · · H · · · His · · · H · · · Ser of such serine proteases as trypsin, chymotrypsin, a-lytic protease, elastase, and subtilisin (see Figure 29.3A and references in Hedstrom’s review47). The chain was originally proposed as catalytically important, the concept being that as the serine hydroxyl group attacks the substrate amide-carbonyl, proton transfer from Ser to His could be coupled to proton transfer from His to Asp. The positive charge of the proton would therefore be removed from the neighborhood of the nucleophilic oxygen to an unusually long distance, generating presumably a heightened form of general-base catalysis (“charge relay”). Robillard and Shulman48 then observed that the Asp – H – His proton exhibited a far downfield resonance sensitive to pH (15 ppm at pH 9, 18 ppm at pH 4), consistent with an unusual character for this H-bond. Later, very extensive work by Bachovchin and his coworkers (extending over many years but reviewed in Ash et al.22) showed from carefully calibrated 15N NMR chemical-shift and spin –spin

780

Isotope Effects in Chemistry and Biology 57 N

57 N 56 N

O

Asp104

(a)

56 N

H H

Ser214

H H

O

His57

− O H O

H N

N

Asn102 N H

O

H

H

O H

Ser195

(b)

Ser214

His57 N N

Ser195

H H

O

FIGURE 29.3 An illustration, reproduced with permission from Sprang, S., Standing, T., Fletterick, R. J., Stround, R. M., Finer-Moore, J., Xuong, N. H., Hamlin, R., Rutter, W. J., and Craik, C. S., Science, 237, 905– 909, 1987, showing the structural effect of the D102N mutation in trypsin (a), the wild-type structure showing the catalytic triad, with D102, H57, and S195 forming a chain of two H-bonds such that the H57-S195 bond is a candidate for a primary catalytic H-bond and the D102-H57 bond is a candidate for a secondary catalytic H-bond in the transition state for nucleophilic attack by the HO group of S195 (b), the active site array in the D102N mutant. N102 is now not an H-bond acceptor from H57, but instead an H-bond donor; H57 is no longer an H-bond acceptor from S195 but instead an H-bond donor. H57 has reverted to the more stable tautomer. For the mutant enzyme to undergo acylation at S195, the active site H-bond network must be disrupted, and H57 must undergo an endergonic tautomerization.

coupling measurements that there is no circumstance in which net proton transfer from the His residue to the Asp residue occurs. The overall result of these studies was an extremely detailed and reliable picture, not merely of the Asp – His interaction, but of many of the important features of catalysis in quantitative detail. These findings cast great doubt on the original form of the charge relay hypothesis. Nevertheless it was generally recognized that some structural or functional role for the Asp residue (such as electrostatic stabilization of a protonated His residue) must exist to account for its evolutionary conservation. Craik and his coworkers49 produced the Asp-102-Asn mutant of trypsin, in which the proton accepting capability of the Asp carboxylate group has been removed by conversion to the essentially non basic amide group of an asparagine residue. With a Z-lysyl-S-benzyl ester as substrate at neutral pH, the mutant enzyme has the value of kcat/KM reduced from that of the wild type enzyme by 11,300-fold, corresponding to an increase in the free energy of activation of about 6 kcal/mol. In addition, the value of kcat is reduced from that of the wild type enzyme by 4400-fold, corresponding to an increase in the free energy of activation of 5 kcal/mol. It is tempting to interpret this result to mean that the increases in free energy of activation represent changes in the free energy of the enzyme-bound transition state with no effect of the mutation on the free energy of the reactant states. This would then indicate that conversion of the basic Asp to nonbasic Asn removes a transition-state stabilizing interaction with a strength of 5 to 6 kcal/mol, most logically an H-bond between Asp and His. An H-bond of this strength would be marginally in the category of an unusual H-bond. Craik and his coworkers were far too cautious to make such a claim. They examined the x-ray structure of the mutant enzyme in comparison to the structure of the wild type enzyme50 and found the result shown in Figure 29.3. The mutant enzyme, in order to undergo acylation by a substrate, must break up a novel network of H-bonds created by the mutation and, in addition the His residue must tautomerize to the less stable isomer held in place in the wild-type active site by its own, different H-bonding network. If each of the three “incorrect” H-bonds present in the mutant active site were to be disrupted at an energy cost of 2 kcal/mol, then the entire loss of activity in the mutant would be explained. The energetics of any possible secondary catalytic H-bond between His-57 and Asp-102 cannot then be deduced from these observations.

Hydrogen Bonds, Transition-State Stabilization, and Enzyme Catalysis

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B. PRIMARY C ATALYTIC H - B ONDS As described in Section II.B and Section III.C, the isotope effects associated with primary catalytic H-bonds between electronegative atoms, such as occur in general acid-base catalysis of heavy-atom reorganization reactions (e.g., acyl transfer, phosphoryl transfer) are commonly so small as to suggest that the interactions do not involve participation of proton motion in the reaction coordinate but instead may represent unusual H-bonds with considerable isotope fractionation. The bond strength to the acid-base catalytic partner in a primary catalytic H-bond would have to be substantial relative to the similar bond to water to account for the Brønsted dependence of rate on base strength. For catalysis by a base of pK 7 in a reaction with Brønsted b ¼ 0.5, log (kB/kW) ¼ 0.5(15.72 7) ¼ 4.4. The ratio kB/kW is the quasiequilibrium constant for reaction of the base B with the transition state for water catalysis to displace the water molecule W from the primary catalytic H-bond and leave B in its place. The quasiequilibrium constant is 104 – 5, which measures the strength of the B · · · H · · · X bond relative to the W · · · H · · · X bond. The bond strength for the complex with B is larger by 5 to 7 kcal/mol than that for the complex with W. This would be consistent with an unusual H-bond of moderate relative strength. It was possible recently51 to relate two bodies of information to evaluate the unusual H-bond model without reaction-coordinate participation for general catalysis in this class of reactions. The result indicates that primary catalytic H-bonds indeed possess stable potentials and respond correspondingly to changes in reaction conditions, favoring the unusual H-bond model. First, Limbach, Denisov, and their coworkers had presented experimental evidence of the probable trajectory for proton motion between two H-bonding partners as the conditions were varied (the temperature and solvent composition of Freon mixtures were varied, with NMR chemical shifts and coupling constants across the H-bond being used to determine H-bond structural features). They showed that as conditions were changed for H-bonded systems A · · · H · · · B so that the proton gradually moved from being fully associated with partner A to being fully associated with partner B, the bond lengths A –H and B –H always summed to the A –B distance and the Pauling bond orders of the two bonds summed to unity. These results therefore provided a model for the behavior of primary catalytic H-bonds that justifies the use of Brønsted slopes to derive the degree of proton transfer at the transition state, the necessary assumption of conserved bond order at the transferring proton being confirmed by the NMR findings. The proton in these equilibrium experiments was in a fully stable potential at all times. Second, Jencks and his coworkers had earlier studied general acid-base catalysis of a ring opening reaction in which general catalysis involved electronegative partners across the catalytic proton. They varied the pK of both the external catalyst and the substrate group with which the primary catalytic H-bond is formed and determined the rates for the various combinations. From the rate constants and pKs, they determined how the Brønsted coefficients on one side of the H-bond vary as the pK of the other partner is varied. The Limbach–Denisov results suggest that the Brønsted coefficients will correctly yield the location of the proton between the two bonding partners. The result is that an increase in the basicity of either bonding partner causes the proton to move closer to the more basic center. This is the behavior predicted for catalytic H-bonds with stable potentials, while the Marcus theory predicts an opposite behavior for catalytic H-bonds with unstable potentials. These results are then most consistent with the idea of primary catalytic H-bonds between electronegative atoms being unusual H-bonds of perhaps 5 to 7 kcal/mol strength relative to a similar bond to water and with isotope fractionations corresponding to isotope effects of 2 to 4 (w about 0.3 to 0.5).

C. SECONDARY C ATALYTIC H - B ONDS 1. The Catalytic Triad of Serine Hydrolases The concept that primary catalytic H-bonds are not “in-flight protons” but rather unusual H-bonds of moderate strength and moderate isotope fractionation is readily extended to many enzymic

782

Isotope Effects in Chemistry and Biology

examples of general catalysis involving catalytic H-bonds between electronegative atoms. These include general acid-base catalysis of the acylation and deacylation of the active site serine of serine hydrolases by the neighboring histidine (Figure 29.3A). If the Asp – His interaction in these cases were a secondary catalytic H-bond of similar character to the primary catalytic His – Ser bond, then similar isotope fractionation might be expected. In 1984, Venkatasubban and Schowen52 reviewed solvent isotope-effect studies (proton inventories; see the article in this volume by Quinn) for various serine hydrolases. In these studies, the rate constant is determined as a function of the atom fraction of deuterium n in mixtures of H2O and D2O. The point of such experiments is to deduce whether the observed solvent isotope effect kHOH/kDOD arises from a single protonic site in the transition state, as would be expected if isotope fractionation were occurring only at the His – Ser primary catalytic H-bond, or at two sites, as would be expected if isotope fractionation were occurring at both the Asp – His secondary catalytic H-bond and the primary His – Ser site. The essence of the approach is expressed in Equation 29.15 to Equation 29.17. The rate constants in isotopic mixtures kn are averages of the rate constants for the H-substituted species reacting with rate constant kHOH and the D-substituted species reacting with rate constant kDOD. If the isotope effect comes from a single site that in the reactant-state enzyme is isotopically labeled statistically, i.e., to the same extent as the water, then as Equation 29.15 shows, kn is a kn ¼ ð1 2 nÞkHOH þ nkDOD ¼ kHOH ð1 2 n þ n½kDOD =kHOH Þ

ð29:15Þ

linear function of n. If the isotope effect comes from two sites, so that kDOD/kHOH is a product of the inverse isotope effects wa at site a and wb at site b; and both sites are labeled statistically, then kn as a function of n will be given by Equation 29.16 to Equation 29.17, i.e., kn is a quadratic function of n: Both kinds of relationships were observed, the outcome being generally similar for various hydrolases and dependent mainly on substrate structure. kn ¼ ð1 2 nÞ2 kHOH þ nð1 2 nÞkHOH ðwa þ wb Þ þ n2 kHOH ðwa wb Þ

ð29:16Þ

kn ¼ kHOH ð1 2 n þ nwa Þð1 2 n þ nwb Þ

ð29:17Þ

When the substrate had little capacity for specific interaction with the enzyme, as in the case of p-nitrophenyl esters of nonamino acids or of amino acids that did not correspond to the hydrolase specificity, linear relationships were observed, consistent with isotope fractionation in the transition only at the primary catalytic (Asp – Ser) H-bond. If derivatives of amino acids specific for a particular enzyme were employed and if further residues were provided N-terminally to generate an oligopeptide substrate with considerable capacity for interaction with the enzyme active site, then often (not universally) the isotope effect became larger and arose from two sites. This was consistent with the idea that the increased interaction with substrate structure in the transition state was “activating” the Asp – His secondary catalytic H-bond, leading it to become a fractionating site. Venkatasubban and Schowen52 concluded that there was insufficient information to estimate reliably a level of transition-state stabilization that accompanied the introduction of a second isotope-fractionating site, but an unreliable estimate suggested 2 to 3 kcal/mol or a factor of several hundreds in the rate. Similar results were again reviewed by Schowen53 in 1988. The most systematic investigation then was that of Stein and Strimpler,54 who determined kinetic parameters and proton inventories (for kcat in cases where it represents deacylation of the acyl enzyme) for substrates of porcine pancreatic elastase, human leukocyte elastase, and a-chymotrypsin. For the pancreatic elastase, introduction of the second fractionating site was accompanied by an increase in kcat of 25%, suggesting no highly effective new interaction. For the leukocyte elastase, the second site entered with an increase in kcat of fourfold (less than 1 kcal/mol). For a-chymotrypsin, the effect was

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sixfold, amounting to around 1 kcal/mol. Similar conclusions were later reached by Bibbs et al.,53 who compared the proton-inventory results for 27 substrates of trypsin. For 12 cases in which the leaving group was identical, six cases showed one fractionating site and values of kcat from 5/sec to 169/sec with a mean value of 66/sec. The remaining six cases showed two fractionating sites and values of kcat from 16/sec to 170/sec with a mean value of 85/sec. The introduction of the fractionating site in these cases had no detectable effect on the rate. The net result of the isotope-effect studies is therefore that an interaction is activated by substrates sufficiently capable of interaction with the enzyme that, as expected for an unusual H-bond, produces isotope fractionation of the order of up to a factor of 2 (w of 0.5). The increase in rate associated with the appearance of this interaction is, however, negligible on the scale of overall reductions in energy barriers by enzyme catalysis. It should be noted that nothing in the data requires that the secondary fractionating site be the Asp – His H-bond. Another candidate is the H-bonds of the oxyanion hole,55 as discussed in Section IV.C.2 below. In 1994, discussion of a secondary catalytic H-bond at the Asp – His site of serine hydrolases had arisen again with the proposal of Frey and his collaborators that a LBHB at this position could account for a considerable part of the catalytic acceleration (the publications are referred to in more recent papers by Neidhart et al.56 and by Westler et al.57). Much criticism ensued, targeted not only on the Frey proposal but also on the suggestion, summarized by Cleland, Frey, and Gerlt,58 that LBHBs are unusual H-bonds of a highly specific nature that may contribute very consequentially to many examples of enzyme catalysis. LBHBs were said by these authors to exist under the following set of circumstances: An LBHB should exist ˚ , should between groups of equal pK, with a distance between the partner end atoms of , 2.5 A possess a DH of formation of 15 to 20 kcal/mol, should be largely covalent, and should exhibit as experimentally accessible signatures a proton NMR chemical shift that is far downfield (17 to 21 ppm), a deuterium fractionation factor as low as 0.3 for symmetrical LBHBs, and perturbations of “IR stretching frequency and differences in proton and deuterium NMR chemical shifts.” It was also noted that strong H-bonds can be formed in nonaqueous solvents and that because “the active site of an enzyme is no longer aqueous once it has closed around a substrate, the properties of hydrogen bonds in organic solvents (strongly isotope-fractionating H-bonds between identical bases were observed by Kreevoy and Liang19 in aprotic solvents) are highly pertinent to enzymatic catalysis.” It was widely believed, while probably not intended by the authors, that this article was suggesting that identification of an LBHB by any one of the experimental signatures enumerated would mean that the interaction under observation must possess all properties listed for an LBHB, including a short end-atom distance, bonding partners of equal pK, and — most important for enzyme catalysis — an enthalpy and possibly a free energy of formation in the range of 15 to 20 kcal/mol. If a free energy contribution of this magnitude were made to catalysis by a single LBHB, it would translate into an acceleration factor of 1010 to 1015, corresponding to most or all of the acceleration factors of many enzymes. It would not be useful to rehearse every aspect of the exchanges that followed the various publications in which the LBHB proposal was initially elaborated (the most direct exchange of views can be found in Warshel et al.,59 which contains responses from Cleland and Kreevoy and from Frey). Instead a summary of the main points should suffice. The chief arguments advanced by the critics were: (1) that the experimental criteria advanced (or apparently advanced) as signals of the presence of an LBHB were in fact not uniquely diagnostic of an H-bond that possessed all the properties listed in the last paragraph; (2) that H-bonds that exhibited one or several of the properties proposed for LBHBs often did not exhibit some of the other proposed properties; (3) that the existence of H-bonds with all of the proposed properties in enzymes was unlikely; (4) that if such H-bonds were present in enzymes, their presence would be unlikely to produce large catalytic accelerations.

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Isotope Effects in Chemistry and Biology

The very thorough study from Bachovchin and his coworkers,22 for example, offered data and cited published results to support a number of these contentions. The following account gives a sampling of their findings. In the 1980s, Bachovchin and his coworkers had prepared and characterized complexes of serine proteases with boro-peptide inhibitors of which the boron center binds to the active-site Ser, producing an analog of the tetrahedral intermediate in the catalytic reaction, and causing the addition of a proton to the active-site His so that both His – Ser and Asp-His H-bonds exist. The latter exhibits a proton chemical shift of 16 ppm, similar to findings of Frey and his coworkers for closely related cases. However, contrary to the LBHB model, the 15N – 1H coupling constant shows that the Asp – His proton is essentially completely localized on the His nitrogen rather than being centrally located and shared equally with Asp. The problem of whether a low-field proton NMR signal is an absolute diagnostic indicator of a LBHB has also been studied theoretically by Garcia-Viloca et al.,60 leading them to conclude that LBHBs “will always have an unusually downfield 1H NMR chemical shift but the opposite statement is not necessarily true.” This same transition-state-analog complex of Bachovchin and his coworkers, in which the dissociation and reassociation of the inhibitor occurs only over a period of minutes, undergoes hydrogen exchange with the external water solution at the Asp –His protonic site and at the His –Ser protonic site (which also shows a proton chemical shift of 16 to 17 ppm downfield) over a period of milliseconds, as measured by lineshape analysis. Thus in the presence of the tightly bound transition-state analog, hydrogen exchange at the two H-bonds is so rapid as to indicate high accessibility of the H-bonds to the aqueous solvent. This is inconsistent with the proposition that substrate binding renders the environment of the H-bonds similar to that of an aprotic solvent. Novel results were also presented for an organic-chemical model for the active-site Asp – His interaction, leading to the overall conclusion that a transition-state H-bond at the Asp – His site might have a dissociation energy of 5 kcal/mol (if wholly available for catalysis, an acceleration factor of about 4000). This value is substantially below the 15 to 20 kcal/mol (catalytic acceleration factors of 1010 to 1015) apparently implied in the proposal described above. Warshel and his coworkers11,61 – 63,65 have been leaders in the application of state-of-the-art quantum-mechanical theory to questions in enzyme catalysis. In their critique of the role of LBHBs in enzyme catalysis, Warshel and Papazyan11 were especially clear in their exposition of the special roles that theory can play in discussions of catalysis. Their approach is centered around empirical valence bond (EVB) theory in which an H-bond X · · · H · · · Y is represented as a resonance hybrid of contributing structures {X – H Y}, {X2 H –Y}, and {X2 Hþ Y2}. Ab initio methods are used to obtain gas-phase free energy vs. structure functions for each of the three contributors. The three functions are then projected onto a two-dimensional representation for the H-bonding system. The two projected (“impure”) functions in effect correspond to the two free-energy functions, one for the X – H species, the other for the H –Y species, that contribute to a double-minimum potential for the H-bond. The barrier for exchange of the proton between the two minima would be formed according to Marcus theory by the intersection of the two free-energy functions. This picture is improved in EVB theory by solutions for the ground state free-energy function that include an exchange integral which measures covalent character; the method also takes account of polarizability of the basic sites and of environmental effects. A highly informative feature of Warshel and Papazyan’s treatment11 is their consideration of the HO · · · H · · · OH2 system in the gas phase and in aqueous solution. In the gas phase, a single minimum H-bond is formed with a free energy of formation from hydroxide ion and a water ˚ . When the system is immersed in water molecule of 2 35 kcal/mol. The O – O distance is 2.4 A solvent, the energy at all points on the energy curve is decreased but the energy of asymmetric structures with concentrated charge is more greatly decreased than the energy of symmetric structures with dispersed charge. The result is conversion in aqueous solution to a double-minimum H-bond (the free energy of formation from gaseous hydroxide ion and water is 2 100 kcal/mol) ˚ . This with a barrier of 20 kcal/mol separating the minima. The O – O distance has increased to 2.8 A calculation is the origin of the much remarked conclusion of Warshel and Papazyan that the

Hydrogen Bonds, Transition-State Stabilization, and Enzyme Catalysis

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conversion in an enzyme active site of an ordinary (simple in our terminology) H-bond to an LBHB would have an anticatalytic effect. Indeed, such a conversion would increase the free energy by 65 kcal/mol, slowing the action of the enzyme by 46 orders of magnitude. As Warshel and Papazyan note, there are stabilizing features to short H-bonds such as increased covalent character, but to the extent that desolvation is required in order to convert a simple H-bond to a short H-bond, the overall energy will rise, particularly in ionic systems. Scheiner and Kar64 produced a careful study by ab initio methods in which the end atom distance, the relative proton affinities of the H-bond partners, the dielectric constant of the environment, and other features of H-bond systems were systematically varied to address the question, what sort of conditions might be required to generate H-bonding interactions that would fulfill the LBHB criteria cited above. Their conclusion was that there is “little possibility that interactions between neutral partners can be strengthened to the extent” that would be needed, the maximum dissociation energy for such a case probably being around 10 kcal/mol. On the other hand, McAllister and his coworkers, as described in Section II.C.2 above, found results in specific cases not inconsistent with the LBHB proposal. The work of Shan, Loh, and Herschlag35 has already been discussed in small part in Section III.B. Much additional work was reported and discussed in this paper, among it a correlation of the equilibrium constant for formation of unusual H-bonds and the difference in pK of the bonding partners. The free-energy relationship for the reaction in tetrahydrofuran is linear and includes the point for equal pKs of the bonding partners. This shows the strength of H-bonds at this point not to be enhanced over the expected value in any way. In the discussion of other findings from these authors in Section III.B above, it was noted that fractionation factors and proton chemical shifts were also immune to some change in the relative pKs of the bonding partners. Other aspects of this work also helped in interpreting the observations of Mulholland et al.,66 whose theoretical studies showed that two H-bonds are formed by a neutral His and a water molecule to an enolate intermediate (and enolate like transition state) in the action of citrate synthase. The two H-bonds were unusually strong (averaging over 9 kcal/mol each) but the protons remained nearer the donors rather than being centrally located, so the H-bonds were certainly stronger than simple H-bonds but not LBHBs by the proposed criteria. These are examples of a growing number of indications that there is not a threshold of properties that separates unusual H-bonds of varying strength from simple H-bonds, but instead that as H-bonds become shorter and have greater covalent character, these undergo changes in other properties, including dissociation energy, in a gradual, continuous manner. These unusual H-bonds can become strong enough to make important contributions to catalysis, conceivably amounting to net transition-state stabilization of the order of 10 kcal/mol, corresponding to an acceleration factor of up to 107. This is a contribution of consequence, although often insufficient to account fully for most cases of enzyme catalysis. In addition, these unusual H-bonds very rarely, if ever, seem to fulfill the full set of expectations for an LBHB, as listed by Cleland, Frey, and Gerlt.58 Nevertheless, if the requirements are made looser and vaguer so as to encompass H-bonds that are asymmetric in the location of the proton, with longer end-atom distances, with bonding partners with unequal pKs, and with smaller H-bond energies, then such H-bonds can be important contributors to enzyme catalysis (Ref. 66; see also the thorough discussion of Stratton et al.67). It seems likely although insufficiently established that many such H-bonds will show isotope fractionation and thus contribute to isotope effects. 2. The Oxyanion Hole of Serine Hydrolases In addition to the catalytic triad, which functions to remove and donate protons at the nucleophilic Ser and the group that departs from the tetrahedral intermediate, serine hydrolases commonly possess an H-bond array known as the oxyanion hole.68 This structure consists of two or three N –H bonds (sometimes an O –H bond); in chymotrypsin the oxyanion hole is formed by the backbone

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NH bonds of Asp 196 and Gly 197. The oxyanion center formed by Ser addition to the substrate carbonyl group is perfectly located in the tetrahedral intermediate and transition states to be stabilized by H-bonds from these quite weakly acidic sites. The degree of stabilization provided by these interactions, and whether the H-bonds are isotope fractionating and thus contribute to observed solvent isotope effects are significant questions. An important experiment was reported by Scholten et al.69 and followed up by Chang et al.70 Scholten et al. prepared a derivative of a-chymotrypsin (methyl-chymotrypsin) in which the activesite His residue was methylated so that a secondary catalytic H-bond at the Asp –His site was impossible. The catalytic activity of the modified enzyme is reduced by three to four orders of magnitude. With the ethyl ester of N-acetyltyrosine, a solvent isotope effect of 3.7 was measured for the deacylation reaction of the acyl-enzyme of methyl-chymotrypsin and shown from the curvature of the proton inventory to arise from two or three sites. The authors favored as a mechanism for methyl chymotrypsin either Me-His acting as a general base with a chain of two water molecules, or the participation of fractionating sites in the oxyanion hole. As the oxyanion hole of chymotrypsin involves backbone N – H bonds, it was not possible to decide between these mechanisms by mutation of mehyl-chymotrypsin to remove the interactions. Chang et al.70 made use of subtilisin, where the oxyanion hole is formed in part by the side chains of Asn 155 and Thr 220. They mutated Asn 155, the more important participant, to glycine, thus removing one of the oxyanion-hole interactions. The mutation reduced the values of both kcat and kcat/KM by a factor of about 100. With Suc-Ala-Ala-Pro-Phe-p-nitroanilide as a substrate, Chang et al. found the solvent isotope effects for the mutant enzyme to be identical with those for the wild-type enzyme, the proton inventory for kcat being curved in consistency with a multiplesite origin for the isotope effect. They found one site to have a fractionation factor of about 0.4 (kH/kD ¼ 2.5) and a further isotope effect of about 1.2 could be fit either as a secondary catalytic H-bond or as arising from many sites. There was also a large many site contribution to the isotope effect on kcat/KM, as is often the case (see the discussion below of the work of Stein72). The simplest interpretation of the findings of Chang et al. is that the oxyanion-hole site at Asn 155 contributes modestly to catalysis (100-fold, corresponding to 2 to 3 kcal/mol of net transitionstate stabilization) and that the H-bond is not isotope fractionating. Furthermore, the second fractionating site in both wild-type and mutant enzymes, if a single site, is most reasonably the secondary catalytic H-bond at the Asp –His site; alternatively, the contribution could be a multi-site effect, arising from many positions in the enzyme or surrounding solvent. The most surprising result in the work of Scholten et al. was perhaps that curvature in the proton inventory, corresponding to two or three contributing sites, was observed with methylchymotrypsin for the simple substrate Ac-Tyr ethyl ester. The discussion of this work by Chang et al. assumed that such curvature would also have been observed with wild-type chymotrypsin, so that the two studies of Scholten et al. and Chang et al., taken together, excluded both the Asp – His site and the oxyanion hole as the second fractionating site. However, the substrate Ac-Tyr ethyl ester is incapable of the extensive interaction with the enzyme that is necessarily to generate more than one contributing site to the isotope effect (see Section IV.C.1 above). Ac-Tyr ethyl ester would have been expected to generate a linear proton inventory with the wild-type enzyme, and this is in fact the case (J. Tan, unpublished data). Thus methylation at His has introduced a novel second fractionating site that was not present with the wild-type enzyme, and as Scholten et al. concluded, the mechanism for the methylated enzyme may well be different from that for the wild-type enzyme. These suggestions that the H-bond interactions in oxyanion holes of serine hydrolases are commonly simple H-bonds that make nonnegligible but not very large contributions to catalysis are reinforced by the theoretical work of Zhang et al.71 on acetylcholinesterase. This enzyme possesses both a catalytic triad and an oxyanion hole, where the H-bonds are donated by the backbone N –H groups of Gly 121 and Gly 122. Combined ab initio quantum mechanical and molecular mechanics (QM/MM) simulations were carried out of the acylation pathway for acetylcholine as substrate.

Hydrogen Bonds, Transition-State Stabilization, and Enzyme Catalysis

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The calculations showed simple H-bonds to the substrate carbonyl in the enzyme– substrate ˚ . As the reaction then proceeded through the complex, the N · · · H · · · O distances being 2.8 to 2.9 A transition state to the tetrahedral intermediate, these distances remained long, the average values ˚. over the entire simulation being 3.0 ^ 0.2 (Gly 121) and 2.9 ^ 0.2 (Gly 122) A Thus the properties of the transition-state N · · · H · · · O interactions at the oxyanion hole of serine hydrolases appear to be most consistent with the view that the interactions have long endatom distances, are of the modest strength, and lack isotope fractionation, all properties of simple H-bonds.

D. CONFORMATIONAL C HANGES S IGNALED BY P ROTON I NVENTORIES As emphasized above, the simple H-bonds typical of those that maintain in part the threedimensional conformational structure of enzymes tend to exhibit very small isotope effects on formation or dissociation of the H-bond. It might be expected, however, that changes involving large numbers of simple H-bond reorganizations might exhibit large isotope effects by multiplication of many small ones. Probable examples of such behavior have been noted by several workers (see the chapter in this volume by Quinn). We will illustrate the point here by an important study of Stein.72 Stein employed the proton inventory method to examine isotope effects in the action of human leukocyte elastase on hydrolysis of a “good” substrate MeO2Suc-Ala-Ala-Pro-Val-p-nitroanilide (kcat/KM ¼ 1.8 £ 105 M/sec). For the parameter kcat/KM, the rate may be determined either by events during binding of the substrate into the active site or by events during acyl transfer from substrate to the active-site Ser residue. The proton-inventory expression for such a serial, two-step process is given by: 21 ðkcat =KM Þ21 þ ðan Þ21 n ¼ ðbn Þ

ð29:18Þ

where n is the atom fraction of deuterium in the solvent, b is the rate constant for binding and a the rate constant for acylation. It was suspected, on the basis of experience with substrates of this high reactivity, that two isotopically fractionating sites would participate in the acylation step, presumably one for the primary His –Ser catalytic H-bond and one for the secondary Asp – His catalytic H-bond. Approximating its isotopic fractionation factors as equal, it was assumed that an ¼ ao ð1 2 n þ nwÞ2 Z n

ð29:19Þ

where ao is the value of a in pure H2O, w is the fractionation factor for each of the two catalytic H-bonds, and Z is a factor that allows for an isotope effect arising from many small changes in the protein structure, as in a conformational change induced by binding and necessary for catalysis. The factor Z has the form shown because if a large number m of sites each generates a small fractionation p then the corresponding proton-inventory expression ð1 2 n þ npÞm ¼ ð1 2 n½1 2 p Þm becomes, because 1 2 p is very small, ½expð2n½1 2 p Þ m ¼ ½expð2m½1 2 p Þ n which becomes because m and p are constants Z n : The expression for the binding rate constant, assuming the same conformation change, is: bn ¼ bo Z n

ð29:20Þ

The combination of Equation 29.18 to Equation 29.20 was then fitted to the data satisfactorily, as is shown in Figure 29.4. The fit gives ao ¼ bo ¼ 3.6 £ 105 M/sec, consistent with the two steps occurring at equal rates, w ¼ 0.54, consistent with isotope effects kH/kD around two at each of two sites, presumably the primary and secondary catalytic H-bonds, and Z ¼ 1.5 from a possible conformation change. The inverse isotope effect of 1.5 for the conformation change is indicative of a slight net tightening of the binding at many exchangeable sites; if the tightening corresponded

788

Isotope Effects in Chemistry and Biology 1.05 n



Z C1 +

0.95



C2



1.00

−1

(1 − n + n∅)2

0.90 (k2/km)n (kc /km)n−0

0.85 0.80 0.75 0.70 0.65 0.60

0.0

0.2

0.4 0.6 nD2O

0.8

1.0

FIGURE 29.4 An illustration, reproduced with permission, from Stein, R. L., J. Am. Chem. Soc., 107, 7768– 7769, 1985. Proton inventory for elastase with a “good” substrate, consistent with a model in which kcat/KM is determined about equally by substrating binding and by acylation of the active-site Ser residue. In the acylation step, two sites (presumably a primary and a secondary catalytic H-bond) each contribute normal isotope effects of about 2. In both binding and acylation, a conformation change generates an inverse isotope effect of 1.5.

to an average fractionation factor of 1.01, then around 40 sites would generate the observed factor. For the case of subtilisin with a very similar substrate, studied by Chang et al.,70 the conformational change on binding generates an inverse isotope effect of about 1.7 so the situation there appears very much as it is with elastase. As Figure 29.4 shows, the proton inventory is strongly indicative of these multi-site effects.

V. SUMMARY Probably the most common role of H-bonding in enzyme catalysis is in the formation of simple H-bonds between enzyme groups and substrate groups that serve to orient substrates and intermediates for favorable interaction with catalytic centers in the active site. These interactions commonly generate no isotope effects, although small changes in isotope fractionation at many sites, which may well be involved in simple H-bonds, may be able to generate observable multi-site isotope effects, such as are seen with some hydrolases. Unusual H-bonds may generate larger isotope effects, most commonly in primary catalytic H-bonds formed in general acid-base catalysis and less commonly in secondary catalytic H-bonds. Primary catalytic H-bonds may perhaps account for four or five orders of magnitude in catalytic acceleration (up to, say, 7 kcal/mol in transition-state stabilization). Most evidence for secondary catalytic H-bonds is consistent only with smaller levels of acceleration, perhaps factors of 10 to 1000 and transition-state stabilizations of up to 5 kcal/mol. The evidence, however, does not support the presence in enzyme-active sites of unusual H-bonds that simultaneously meet each of the very restrictive criteria for the designation and LBHBs. However, unusual H-bonds that meet some but not all of the criteria may occur with some frequency and may lead to modest acceleration factors of a few thousand or less.

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REFERENCES 1 Lescovac, V., Comprehensive Enzyme Kinetics, Kluwer Academic/Plenum, New York, 2003. 2 Segel, I. H., Enzyme Kinetics, Wiley Interscience, New York, 1975. 3 Schowen, R. L., Catalytic power and transition-state stabilization, In Transition States of Biochemical Processes, Gandour, R. D. and Schowen, R. L., Eds., Plenum, New York, pp. 77 – 114, 1978. 4 Radzicka, A. and Wolfenden, R., A proficient enzyme, Science, 267, 90– 93, 1995. 5 Wolfenden, R. and Snider, M. J., The depth of chemical time and the power of enzymes as catalysts, Acc. Chem. Res., 34, 938– 945, 2001. Miller, B. G. and Wolfenden, R., Catalytic proficiency: the unusual case of OMP decarboxylase, Annu. Rev. Biochem., 71, 847– 885, 2002. 6 Johnston, H. S., Gas Phase Reaction Rate Theory, Ronald Press Co., New York, p. 128, 1966. 7 Pitzer, K. S. and Brewer, L., Thermodynamics, Lewis, G. N. and Randall, M., Eds., McGraw-Hill, New York, Revision of, chap. 20, p. 242, 1961. 8 Jencks, W. P., Binding energy, specificity, and enzymic catalysis: the Circe effect, Adv. Enzymol. Relat. Areas Mol. Biol., 43, 219– 410, 1975. 9 Gurney, R. W., Ionic Processes in Solution, McGraw-Hill, New York, 1953, Reprinted by New York: Dover Publications, 1963. 10 Fersht, A., Structure and Mechanism in Protein Science, W.H. Freeman, New York, 1999. 11 Warshel, A. and Papazyan, A., Energy considerations show that low barrier hydrogen bonds do not offer a catalytic advantage over ordinary hydrogen bonds, Proc. Natl Acad. Sci. U.S.A., 93, 13665– 13670, 1996. 12 Bell, R. P., The Tunnel Effect in Chemistry, Chapman & Hall, London, 1980. 13 Antoniou, D. and Schwartz, S. D., Large kinetic isotope effects in enzymatic proton transfer and the role of substrate oscillations, Proc. Natl Acad. Sci. U.S.A., 94, 12360– 12365, 1997. 14 Benkovic, S. J. and Hammes-Schiffer, S., A perspective on enzyme catalysis, Science, 301, 1196– 1202, 2003. 15 Bruice, T. C., A view at the millennium: the efficiency of enzymatic catalysis, Acc. Chem. Res., 35, 139– 148, 2002. 16 Garcia-Viloca, M., Gao, J., Karplus, M., and Truhlar, D. G., How enzymes work: analysis by modern rate theory and computer simulations, Science, 303, 186–195, 2004. 17 Larson, J. W. and McMahon, T. B., Gas-phase bifluoride ion. An ion cyclotron resonance determination of the hydrogen bond energy in FHF- from gas-phase fluoride transfer equilibrium measurements, J. Am. Chem. Soc., 104, 5848– 5849, 1982. 18 Hibbert, F. and Emsley, J., Hydrogen bonding and chemical reactivity, Adv. Phys. Org. Chem., 26, 255– 379, 1990. 19 Kreevoy, M. M. and Liang, T. M., Structures and isotopic fractionation factors of complexes, AlHA2 2, J. Am. Chem. Soc., 102, 3315– 3322, 1980. 20 Albery, W. J., Solvent isotope effects, In Proton-Transfer Reactions, Caldin, E. and Gold, V., Eds., Chapman & Hall, London, pp. 263– 315, 1975. 21 Shenderovich, I. G., Tolstoy, P. M., Golubev, N. S., Smirnov, S. N., Denisov, G. S., and Limbach, H. H., Low-temperature NMR studies of the structure and dynamics of a novel series of acid-base complexes of HF with collidine exhibiting scalar couplings across hydrogen bonds, J. Am. Chem. Soc., 125, 11710– 11720, 2003. 22 Ash, E. L., Sudmeier, J. L., De Fabo, E. C., and Bachovchin, W. W., A low-barrier hydrogen bond in the catalytic triad of serine proteases? Theory versus experiment, Science, 278, 1128– 1132, 1997. 23 Perrin, C. L., Symmetries of hydrogen bonds in solution, Science, 266, 1665– 1668, 1994. 24 Perrin, C. L. and Nielson, J. B., “Strong” hydrogen bonds in chemistry and biology, Annu. Rev. Phys. Chem., 48, 511– 544, 1997. 25 Perrin, C. L. and Kim, Y. J., Symmetry of the hydrogen bond in malonaldehyde enol in solution, J. Am. Chem. Soc., 120, 12641 –12645, 1998. 26 Perrin, C. L. and Ohta, K. K., Symmetry of NHN hydrogen bonds in 1,8-bis(dimethylamino)naphthalene-Hþ and 2,7-dimethoxy-1,8-bis(dimethylamino)naphthalene-Hþ, J. Am. Chem. Soc., 123, 6520– 6526, 2001. 27 Perrin, C. L. and Ohta, B. K., Symmetry of NHN hydrogen bonds in solution, J. Mol. Struct., 644, 1 – 12, 2003.

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28 Scheiner, S., Ab initio studies of hydrogen bonds: the water dimer paradigm, Annu. Rev. Phys. Chem., 45, 23 – 56, 1994. 29 Scheiner, S., Hydrogen Bonding: a Theoretical Perspective, Oxford University Press, Oxford, 1997. 30 Scheiner, S., Kar, T., and Pattanayak, J., Comparison of various types of hydrogen bonds involving aromatic amino acids, J. Am. Chem. Soc., 124, 13257– 13264, 2002. 31 Scheiner, S., Grabowski, S. J., and Kar, T., Influence of hybridization and substitution on the properties of the C H · · · O hydrogen bond, J. Phys. Chem. A, 105, 10607 – 10612, 2001. 32 Del Bene, J. E., Perera, S. A., Bartlett, R. J., Yanez, M., Mo, O., Elguero, J., and Alkorta, I., Two-bond 19 15 F- N spin –spin coupling constants (2hJF-N) across F-H · · · N hydrogen bonds, J. Phys. Chem. A, 107, 3121– 3125, 2003; Del Bene, J. E., Perera, S. A., Bartlett, R. J., Yanez, M., Mo, O., Elguero, J., and Alkorta, I., Two-bond 19F-15N spin– spin coupling constants (2hJF-N) across N-Hþ· · · F hydrogen bonds, J. Phys. Chem. A, 107, 3126– 3131, 2003; Del Bene, J. E., Perera, S. A., Bartlett, R. J., Yanez, M., Mo, O., Elguero, J., and Alkorta, I., Two-bond 13C-15N spin– spin coupling constants (2hJC-N) across C– H– N hydrogen bonds, J. Phys. Chem. A, 107, 3222– 3227, 2003. 33 Kumar, G. A. and McAllister, M. A., Theoretical investigation of the relationship between proton NMR chemical shift and hydrogen bond strength, J. Org. Chem., 63, 6968– 6972, 1998, See also Chen, J., McAllister, M. A., Lee, J. K., and Houk, K. N. Short, strong hydrogen bonds in the gas phase and solution: Theoretical exploration of pKa matching and environmental effects on the strengths of hydrogen bonds and their potential roles in enzymatic catalysis, J. Org. Chem., 63, 4611 –4619, 1998. 34 Reinhardt, L. A., Sacksteder, K. A., and Cleland, W. W., Enthalpic studies of complex formation between carboxylic acids and 1-alkylimidazoles, J. Am. Chem. Soc., 120, 13366– 13369, 1998. 35 Shan, S. O., Loh, S., and Herschlag, D., The energetics of hydrogen bonds in model systems: implications for enzymatic catalysis, Science, 272, 97– 101, 1996, See also Shan, S.O. and Herschlag, D., The change in hydrogen bond strength accompanying charge rearrangement: implications for enzymatic catalysis, Proc. Natl Acad. Sci. USA, 93, 14474– 14479, 1996. 36 Bartl, F., Palm, D., Schinzel, R., and Zundel, G., Proton relay system in the active site of maltodextrinphosphorylase via hydrogen bonds with large proton polarizability: an FT-IR difference spectroscopy study, Eur. Biophys. J., 28, 200– 207, 1999. 37 Stasko, D., Hoffmann, S. P., Kim, K. C., Fackler, N. L., Larsen, A. S., Drovetskaya, T., Tham, F. S., Reed, C. A., Rickard, C. E., Boyd, P. D., and Stoyanov, E. S., Molecular structure of the solvated proton in isolated salts. Short, strong, low barrier (SSLB) H-bonds, J. Am. Chem. Soc., 124, 13869– 13876, 2002. 38 Krantz, B. A., Srivastava, A. K., Nauli, S., Baker, D., Sauer, R. T., and Sosnick, T. R., Understanding protein hydrogen bond formation with kinetic H/D amide isotope effects, Nat. Struct. Biol., 9, 458– 463, 2002. 39 Loh, S. N. and Markley, J. L., Hydrogen bonding in proteins as studied by amide hydrogen H/D fractionation factors: application to staphylococcal nuclease, Biochemistry, 33, 1029– 1036, 1994. 40 LiWang, A. C. and Bax, A., Equilibrium protium or deuterium fractionation of backbone amides in U-13C/15N labeled human ubiquitin by triple resonance NMR, J. Am. Chem. Soc., 118, 12864– 12865, 1996. 41 Bowers, P. M. and Klevit, R. E., Hydrogen bonding and equilibrium isotope enrichment in histidinecontaining proteins, Nat. Struct. Biol., 3, 522– 531, 1996. 42 Edison, A. S., Weinhold, F., and Markley, J. L., Theoretical studies of protium or deuterium fractionation factors and cooperative hydrogen bonding in peptides, J. Am. Chem. Soc., 117, 9619– 9624, 1995. 43 Swain, C. G., Kuhn, D. A., and Schowen, R. L., Effect of structural changes in reactants on the position of hydrogen-bonding hydrogens and solvating molecules in transition states. The mechanism of tetrahydrofuran formation from 4-chlorobutanol, J. Am. Chem. Soc., 87, 1553– 1561, 1965. 44 Cordes, E. H., Mechanism and catalysis for the hydrolysis of acetals, ketals, and ortho esters, Prog. Phys. Org. Chem., 4, 1 – 44, 1967. 45 Eliason, R. and Kreevoy, M. M., Kinetic hydrogen isotope effects in the concerted mechanism for the hydrolysis of acetals, ketals, and ortho esters, J. Am. Chem. Soc., 100, 7037– 7041, 1978. 46 Elrod, P., Zhang, J., Yang, X., Yin, D., Hu, Y., Borchardt, R. T., and Schowen, R. L., Contributions of active site residues to the partial and overall catalytic activities of human S-adenosylhomocysteine hydrolase, Biochemistry., 41, 8134– 8142, 2002.

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47 Hedstrom, L., Serine protease mechanism and specificity, Chem. Rev., 102, 4501– 4523, 2002. 48 Robillard, G. and Shulman, R.G., High resolution nuclear magnetic resonance study of hte histidine – aspartate hydrogen bond in chymotrypsin and chymotrypsinogen, J. Mol. Biol., 71, 507– 511, 1972. 49 Craik, C. S., Roczniak, S., Largman, C., and Rutter, W. J., The catalytic role of the active site aspartic acid in serine proteases, Science, 237, 909– 913, 1987. 50 Sprang, S., Standing, T., Fletterick, R. J., Stroud, R. M., Finer-Moore, J., Xuong, N. H., Hamlin, R., Rutter, W. J., and Craik, C. S., The three-dimensional structure of Asn102 mutant of trypsin: role of Asp102 in serine protease catalysis, Science, 237, 905– 909, 1987. 51 Schowen, K. B., Limbach, H. H., Denisov, G. S., and Schowen, R. L., Hydrogen bonds and proton transfer in general-catalytic transition-state stabilization in enzyme catalysis, Biochim. Biophys. Acta, 1458, 43 – 62, 2000. 52 Venkatasubban, K. S. and Schowen, R. L., The proton inventory technique, CRC Crit. Rev. Biochem., 17, 1 – 44, 1984. 53 Schowen, R. L., Structural and energetic aspects of protolytic catalysis by enzymes: charge-relay catalysis in the function of serine proteases, In Molecular Structure and Energetics, Liebman, J. F. and Greenberg, A., Eds., VCH, New York, pp. 119– 168, 1988, See also Bibbs, J.A., Garoutte, M. P., Wang, B., Tittel, P. D., Schowen, K. B., and Schowen, R. L., The contribution of secondary proton bridges to the catalytic power of the serine proteases, Ber Bunsenges Phys. Chem., 102, 573– 579, 1997. 54 Sten, R. L. and Strimpler, A. M., P1 residue determines the operation of the catalytic triad of serine proteases during hydrolyses of acyl enzymes, J. Am. Chem. Soc., 109, 4387– 4390, 1987. 55 Fink, A. L., Enzyme Mechanisms, Williams, A. and Page, M. I., Eds., R.S.C., London, 1987, chap. 10. 56 Neidhart, D., Wei, Y., Cassidy, C., Lin, J., Cleland, W. W., and Frey, P. A., Correlation of low-barrier hydrogen bonding and oxyanion binding in transition state analogue complexes of chymotrypsin, Biochemistry, 40, 2439– 2447, 2001. 57 Westler, W. M., Frey, P. A., Lin, J., Wemmer, D. E., Morimoto, H., Williams, P. G., and Markley, J. L., Evidence for a strong hydrogen bond in the catalytic dyad of transition-state analogue inhibitor complexes of chymotrypsin from proton-triton NMR isotope shifts, J. Am. Chem. Soc., 124, 4196– 4197, 2002. 58 Cleland, W. W., Frey, P. A., and Gerlt, J. A., The low barrier hydrogen bond in enzymatic catalysis, J. Biol. Chem., 273, 25529 –25532, 1998. 59 Warshel, A., Papazyan, A., and Kollman, P. A., On low barrier hydrogen bonds and enzyme catalysis, Science, 269, 102– 106, 1995. 60 Garcia-Viloca, M., Gelabert, R., Gonza´lez-Lafont, A., Moreno, M., and Lluch, J. M., Is an extremely low-field proton signal in the NMR spectrum conclusive evidence for a low-barrier hydrogen bond?, J. Phys. Chem. A, 101, 8727– 8733, 1997. 61 Warshel, A., Computer Modeling of Chemical Reactions in Enzymes and Solutions, WileyInterscience, New York, 1991. 62 Warshel, A., Molecular dynamics simulations of biological reactions, Acc. Chem. Res., 35, 385– 395, 2002. 63 Warshel, A., Computer simulations of enzyme catalysis: methods, progress, and insights, Annu. Rev. Biophys. Biomol. Struct., 32, 425– 443, 2003. 64 Scheiner, S. and Kar, T., The nonexistence of specially stabilized hydrogen bonds in enzymes, J. Am. Chem. Soc., 117, 6970– 6975, 1995. 65 Shurki, A. and Warshel, A., Structure or function correlations of proteins using MM, QM/MM, and related approaches: methods, concepts, pitfalls, and current progress, Adv. Protein Chem., 66, 249– 313, 2003. 66 Mulholland, A. J., Lyne, P. D., and Karplus, M., Ab initio QM/MM study of the citrate synthase mechanism. A low-barrier hydrogen bond is not involved, J. Am. Chem. Soc., 122, 534– 535, 2000. 67 Stratton, J. R., Pelton, J. G., and Kirsch, J. F., A novel engineered subtilisin BPN’ lacking a low-barrier hydrogen bond in the catalytic triad, Biochemistry, 40, 10411– 10416, 2001. 68 Menard, R. and Storer, A. C., Oxyanion hole interactions in serine and cysteine proteases, Biol. Chem. Hoppe Seyler, 373, 393– 400, 1992. 69 Scholten, J. D., Hogg, J. L., and Raushel, F. M., Methyl chymotrypsin catalyzed hydrolysis of specific substrate esters indicate multiple proton catalysis is possible with a modified charge relay triad, J. Am. Chem. Soc., 110, 8246– 8247, 1988.

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70 Chang, T. K., Chiang, Y., Guo, H. X., Kresge, A. J., Mathew, L., Powell, M. F., and Wells, J. A., Solvent isotope effects in H2O – D2O mixtures (proton inventories) on serine-protease-catalyzed hydrolysis reactions. Influence of oxyanion hole interactions and medium effects, J. Am. Chem. Soc., 118, 8802– 9907, 1996. 71 Zhang, Y., Kua, J., and McCammon, J. A., Role of the catalytic triad and oxyanion hole in acetylcholinesterase catalysis: an ab initio QM/MM study, J. Am. Chem. Soc., 124, 10572– 10577, 2002. 72 Stein, R. L., Catalysis by human leukocyte elastase. 5. Structural features of the virtual transition state for acylation, J. Am. Chem. Soc., 107, 7768– 7769, 1985. 73 Cleland, W. W. and Northrop, D. B., Energetics of substrate binding, catalysis, and product release, Methods Enzymol., 308(part E), 3– 27, 1999. 74 Northrop, D. B., Rethinking fundamentals of enzyme action, Adv. Enzymol. Relat. Areas Mol. Biol., 73, 25 – 55, 1999.

30

Substrate and pH Dependence of Isotope Effects in Enzyme Catalyzed Reactions William E. Karsten and Paul F. Cook

CONTENTS I. II.

III.

IV.

Introduction ...................................................................................................................... 794 A. Nomenclature ........................................................................................................... 794 B. Types of Isotope Effects .......................................................................................... 794 Enzyme-Catalyzed vs. Nonenzymatic Reactions ............................................................ 794 A. Physical vs. Chemical Steps .................................................................................... 794 B. Commitment Factors................................................................................................ 795 C. Substrate Stickiness.................................................................................................. 795 Substrate Dependence of Isotope Effects ........................................................................ 796 A. Sequential Mechanisms............................................................................................ 796 1. Ordered Mechanisms (k5, k6, k7, and k8 ¼ 0)................................................... 797 a. Formate Dehydrogenase.............................................................................. 797 b. Mannitol Dehydrogenase ............................................................................ 798 2. Random Mechanisms (All Rate Constants of Mechanism 3 Apply)......................................................................................... 798 a. NAD-Malic Enzyme.................................................................................... 799 b. Ketopantoate Reductase .............................................................................. 799 B. Ping Pong Kinetic Mechanisms............................................................................... 800 1. Dihydroorotate Dehydrogenase......................................................................... 801 2. p-Cresol Methylhydroxylase ............................................................................. 801 C. Substrate Dependence of Isotope Effects in Terreactant and Higher Order Mechanisms................................................................................ 801 1. NAD-Malic Enzyme.......................................................................................... 802 2. Alanine Dehydrogenase .................................................................................... 802 pH Dependence of Isotope Effects .................................................................................. 802 A. Proton Transfer and Chemistry are Concerted........................................................ 802 1. Random Addition of Proton and Substrate to Enzyme .................................... 803 a. NADP-Malic Enzyme ................................................................................. 804 b. Nitroalkane Oxidase .................................................................................... 804 2. Dead-End Protonation of Enzyme .................................................................... 804 a. NAD-Malic Enzyme.................................................................................... 805 3. Dead-End Protonation of Enzyme and the Enzyme-Reactant Complex ......... 805 a. Ketpantoate Reductase ................................................................................ 805 4. Dead-End Formation of a Protonated Enzyme-Reactant Complex ................. 806

793

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Isotope Effects in Chemistry and Biology

B.

Proton Transfer and Chemistry not Concerted........................................................ 806 1. Equine Liver Alcohol Dehydrogenase.............................................................. 807 V. Closing Remarks .............................................................................................................. 807 References..................................................................................................................................... 808

I. INTRODUCTION Isotope effects are perhaps one of the most powerful tools available to the mechanistic investigator based on the amount and different types of information one can obtain. In this short treatise, an overview of isotope effects applied to the determination of the kinetic mechanism of enzymatic reactions via the substrate and pH dependence of isotope effects will be provided. An isotope effect can be defined as a change in the rate or equilibrium constant of a reaction upon substitution of a heavy atom for a light one at, or adjacent to, the position of bond cleavage in a molecule undergoing reaction. Isotopes provide an excellent tool for the study of reaction because they are isosteric and isoelectronic, and thus nonperturbing. Isotope effects reflect changes in vibrational frequencies of reactants as they are converted to products in the rate-determining transition states. The theory of the substrate and pH dependence of isotope effects will be considered briefly in this chapter, and applications from the literature of the theory to specific enzyme-catalyzed reactions will be presented. More in-depth treatment of the theory has been published previously,1 – 6 and this chapter is meant to give the reader a more user friendly, semiquantitative feel for the theory.

A. NOMENCLATURE Isotope effects are given as the ratio of the rates with light and heavy atom substitutions, e.g., kH/kD. In the following discussion, isotope effects will be abbreviated using a leading superscript to identify the heavy atom.1,7 For example, kH/kD, kC-12/kC-13 are Dk and 13k, respectively, while secondary deuterium kinetic isotope effects are a-Dk and b-Dk, respectively. Isotope effects on equilibrium constants are likewise given as DKeq, 13Keq, etc. In the case of observed isotope effects in enzymecatalyzed reactions, the limiting macroscopic rate constants are used, e.g., DV and D(V/K ).

B. TYPES o F I SOTOPE E FFECTS There are two types of isotope effects termed primary and secondary. If a heavy atom is substituted for a light one in the bond undergoing cleavage, a primary kinetic isotope effect is observed on the rate of the reaction, expressed as the ratio of the rates with light and heavy atoms, kL/kH or KL/KH, where L and H refer to light and heavy, respectively. If a heavy atom is substituted for a light one in a bond that does not undergo bond cleavage, but is bonded to or in a position remote to an atom that does undergo bond cleavage, a secondary kinetic isotope effect is observed on the rate of the reaction. If the heavy atom is attached to the one undergoing bond cleavage it is termed a, while if it is in a position attached to an atom adjacent to the one undergoing bond cleavage it is termed b. Isotope effects can also be observed on the equilibrium constant for a given reaction, and these may be primary or secondary. In this chapter only primary kinetic deuterium isotope effects will be discussed, because of their magnitude and ease of measurement. The theory, however, applies to any measurable isotope effect.

II. ENZYME-CATALYZED VS. NONENZYMATIC REACTIONS A. PHYSICAL v S. C HEMICAL S TEPS Application of isotope effects to the study of uncatalyzed and enzyme-catalyzed reactions differs in one major respect. In the case of the uncatalyzed reaction, the chemical (isotope sensitive) step is

Substrate and pH Dependence of Isotope Effects in Enzyme Catalyzed Reactions

795

usually rate-limiting, Equation 30.1, while in the case of the enzyme-catalyzed reaction, the isotope sensitive step is seldom the only rate-limiting step along the reaction pathway, Equation 30.2. kchem

A ! B

ð30:1Þ

k1 A

k3 B

k5

kchem

k7

k2

k4

k6

k2chem

k8

k9

k11

E O EA O EAB O E0 AB O E0 PQ O EPQ ! EQ ! E

ð30:2Þ

The physical steps, substrate binding and product release represented by rate constants k1 – k4 and k9, k11, and enzyme structural changes represented by rate constants k5 and k6, and k7 and k8, often limit or contribute to rate limitation overall. Thus, although the measured isotope effect, Dk, can be directly interpreted in terms of transition state structure for the uncatalyzed reaction, the rate expressions for enzyme-catalyzed reactions are usually much more complex, and may preclude interpretation of the observed isotope effects in terms of transition state structure, unless it is equal to the intrinsic isotope effect, or the intrinsic isotope effect can be calculated. The intrinsic isotope effect is the isotope effect on the bond-breaking step. However, the observed isotope effect in an enzyme-catalyzed reaction will depend on which substrate is varied and on the kind of kinetic mechanism. Application of isotope effects to kinetic mechanism is based on the theory presented below. For a discussion of the measurement of isotope effects see Chapter 40 by Cleland.

B. COMMITMENT FACTORS The term “commitment to catalysis” was first coined by Northrop7 to describe ratios of rate constants found in the expressions for isotope effect rate equations. Three terms were defined, the forward and reverse commitments to catalysis, cf and cr, and the catalytic ratio depicted by R but changed to cVf to be consistent with the term commitment.1 A commitment to catalysis is a ratio of net rate constants that provides an indication of the partitioning of reactant (substrate, cf, or product, cr) toward product (cf) or toward substrate (cr) through the catalytic steps once bound to enzyme compared to dissociation to give free reactant. These commitment factors are comprised of an internal and external component, see Chapter 40, but the internal component is generally devoid of substrate and pH dependence. The present treatment will focus on the external component of the commitment, cf-ex. The catalytic ratio is the sum of the ratios of a reduced rate constant for the catalytic step to a number of net rate constants for steps that limit V. The reduced catalytic rate constant is the actual rate constant for the catalytic step multiplied by the fraction of enzyme forms, at saturating substrates, prior to the step that at equilibrium are ready to undergo the catalytic step. The net rate constants that are compared to the reduced catalytic rate constant are those for product formation that follow the catalytic step, as well as ones for formation of the complex that undergoes the catalytic step, and any other unimolecular steps in the mechanism, such as isomerization of binary complexes. For mechanism 2 above, cf, and cr are kchem/k60 and k-chem/k70 , respectively, where k60 is for release of either A or B, while k70 is for release of P. The catalytic ratio, in this case, is [kchem/(1 þ k6/k5)], the rate constant for the chemical step corrected for the pre-equilibrium conformational change, multiplied by the sum of the reciprocal of the net rate constants present at saturating substrates, k5, k70 , k9, and k11.

C. SUBSTRATE S TICKINESS A sticky substrate is one that reacts through the first irreversible step (usually product release) faster than it dissociates from the enzyme. The stickiness ratio, Sr, is thus the ratio of the net rate constant for reaction of the initial ES complex through the first irreversible step and the rate constant for substrate dissociation. Stickiness affects pH profiles and the degree to which isotope effects on V=K are expressed, and this aspect will be discussed in detail below.

796

Isotope Effects in Chemistry and Biology

III. SUBSTRATE DEPENDENCE OF ISOTOPE EFFECTS A. SEQUENTIAL M ECHANISMS In an ordered kinetic mechanism, the V=K for the first substrate bound is the on-rate k1, which will not be sensitive to substitution of deuterium for protium and thus the isotope effect on V=Ka will be unity. An exception is when an isotope effect is observed on substrate binding (see Chapter 45 by Schramm). The lack of an isotope effect on V=Ka is the basis for the theory for determination of kinetic mechanism, specifically the order of addition of substrates. Consider the following minimal Scheme for a sequential mechanism. k1A E k8B

EA k2

k4 k5

k7

k 3B

k9

EAB

EQ

k11

E

ð30:3Þ

k6A

EB

In mechanism 30.3, only k9 is sensitive to isotopic substitution. The rate constant k9 is not a simple microscopic rate constant but is the net rate constant for conversion of EAB to EQ and can contain contributions from conformation changes, the bond-breaking step(s) and release of the first product. Unless the bond-breaking step is the slowest step, the isotope effect on k9 will be lower than that on the bond-breaking step. Mechanism 30.3 is sufficient for a consideration of kinetic mechanism since k9 will not depend on the concentration of substrates. Thus, the equations used for the isotope effects on V and V=K will include only an external forward commitment. The expression for Dk9 will include the internal part of the commitment as well as all of the reverse commitment.1 The expressions for V and DV, based on mechanism 3 are V¼

k9 k11 Et k9 þ k11

D

V¼

D

k9 þ k9 =k11 : 1 þ k9 =k11

ð30:4Þ

V is obtained at saturating concentrations of substrates, and the isotope effect on V will be independent of whether a random or ordered kinetic mechanism is considered. The lower the ratio of k9/k11, the larger the isotope effect on V. When product release is rapid compared to the net rate constant for the bond-breaking step (k9), DV will be equal to Dk9. (It should be noted that Dk9 is not necessarily equal to the intrinsic deuterium isotope effect on the chemical step.) The ratio k9 =k11 is, as discussed above, the ratio to catalysis, cVf.7 Information on the order of addition of reactants will be obtained from the isotope effects on the V=K values for the individual substrates, as can be seen from the expressions for the forward commitment factor for the varied substrate for deuterium isotope effects. It would be for the labeled substrate in the case of a tritium isotope effect, i.e., when the competitive method is used (Chapter 40). A general expression for the V=K isotope effect is given in Equation 30.5, where cf is the external part of the forward commitment. D

ðV=KÞ ¼

D

k9 þ cf 1 þ cf

ð30:5Þ

The forward commitment factor is the ratio of the net rate constant of the catalytic step to the net off-rate constant for the varied substrate (deuterium isotope effect) or the labeled substrate (tritium isotope effect). For mechanism 30.3, cf is given in Equation 30.6 and Equation 30.7 when A and B are

Substrate and pH Dependence of Isotope Effects in Enzyme Catalyzed Reactions

797

the varied substrate, respectively. cf ¼

cf ¼

k9 k2 k4 k5 þ k2 þ k3 B k9 k5 k7 k4 þ k7 þ k6 A

ð30:6Þ

ð30:7Þ

Differences in sequential kinetic mechanisms are defined by the presence or absence of rate constants in mechanism 30.3. Several of the commonly encountered sequential kinetic mechanisms are discussed below with respect to their discrimination by isotope effects obtained on V=Ka and V=Kb : 1. Ordered Mechanisms (k5, k6, k7, and k850) In a steady state ordered kinetic mechanism, the off-rate constant for the first substrate, k2, is not much greater than V=Et : The expression for cf with B varied, Equation 30.7, reduces to k9/k4, and is independent of the concentration of A. The expression for cf with B varied, Equation 30.6, reduces to k ðk þ k3 BÞ ð30:8Þ cf ¼ 9 2 k2 k4 The value of D(V/Ka) depends on B, Equation 30.8. As B approaches zero cf is k9/k4 and D(V/Ka) will become equal to D(V/Kb), while as B tends to infinity, k3B will become much greater than k2, cf approaches infinity and the value of D(V/Ka) will be unity at infinite B. The value of unity results from A becoming trapped on enzyme by B at high concentrations. As a result, once A is bound it is committed to form product rather than dissociate to regenerate free reactant and enzyme. The level of B giving a value of [D(V/Kb) þ 1]/2 is KiaKb/Ka. In an equilibrium ordered kinetic mechanism, k2 is much greater than k3B at most reasonable concentrations of B. Under these conditions, the expression for cf with A varied reduces to k9/k4, independent of B, as for the steady state mechanism. With B varied, cf is also k9/k4 and the isotope effect will be constant, i.e., D(V/Ka) ¼ D(V/Kb) whatever substrate is varied and at any concentration of the fixed substrate (as long as the condition k2 q k3 B is satisfied). Using isotope effects alone it is not possible to distinguish an equilibrium ordered mechanism from a rapid equilibrium random one. The equilibrium ordered mechanism, however, gives a distinctive initial velocity pattern that intersects on the ordinate when B is varied and intersects to the left of the abscissa when A is varied with the replot of slope vs. 1/B passing through the origin. If a dead-end EB complex is present, the isotope effects will not be affected, but the initial velocity pattern will no longer intersect on the ordinate with B varied nor will the slope replot pass through the origin when A is varied. As a result the equilibrium ordered mechanism with a dead-end EB complex cannot be distinguished from a random mechanism in which k2 ; k7 . V=Et : (These mechanisms can be distinguished using initial velocity studies in the presence of inhibitors.) There are a number of examples of isotope effects in ordered kinetic mechanism in the literature and a few are discussed below, to give the reader an idea of the kinds of qualitative and quantitative information that can be obtained. a. Formate Dehydrogenase Formate dehydrogenase catalyzes the oxidation of formic acid to CO2 using NADþ as the oxidant. The measured primary deuterium isotope effects are D(V/KNAD) ¼ 1, D(V/Kformate) ¼ 3.42, and

798

Isotope Effects in Chemistry and Biology

D

V ¼ 2.33.8 The finite isotope effect on V/Kformate and the isotope effect of 1 on V/KNAD are indicative of an ordered kinetic mechanism with NAD binding prior to formate. The deuterium isotope effect on V/Kformate is thought to be the intrinsic isotope effect. The smaller isotope effect on V compared to DV/Kformate suggests that some step not included in the expression of V=K; such as release of NADH or isomerization of the E:NAD complex, is partially rate limiting and serves to partially mask the observed isotope effect on V: The difference in the isotope effects on V and V=K can be used to estimate the relative rate of, for example, last product release to release of reactant from the Michaelis complex. From a consideration of Equation 30.4 to Equation 30.6 with A at infinite concentration, if 1 is subtracted from both sides of the equations and the ratio taken, the expression (D V 2 1)/( DV/ K 2 1) ¼ (1 þ k9/k4)/(1 þ k9/k11) is obtained. The estimate will be most accurate for the stickiest substrates and when the isotope effects are small. For example, the estimate cannot be made for formate dehydrogenase because the isotope effect on V/Kformate represents the intrinsic isotope effect and thus formate is not a sticky substrate. However, an estimate of the catalytic ratio, k9/k11, can be obtained by substituting a value of 3.4 for Dk9 into Equation 30.4, and solving for the ratio using the observed value of 2.3. A value of about 1 is obtained for the ratio, suggesting the rate of release of NADH is about the same as the net rate constant for catalysis. b. Mannitol Dehydrogenase Another enzyme that appears to conform to an ordered mechanism based on the primary deuterium isotope effects is the mannitol dehydrogenase from Pseudomonas fluorescens.9 Mannitol dehydrogenase catalyzes the conversion of D-mannitol and NADþ to D-fructose and NADH. The enzyme, when assayed at pH 8.2 in the direction of reduction of fructose using NADH(D), gives values of 1.4 ^ 0.2, 2.5 ^ 0.1 and 1.0 ^ 0.1, for DV, D(V/Kfructose), and D(V/KNADH). The isotope effects are consistent with a steady state ordered mechanism with NADH binding before mannitol and NAD released last. Again the smaller values of DV compared to D(V/Kfructose) suggest rate limitation by product release that reduces the value of the isotope effect on V: For mannitol dehydrogenase the calculated value for k11/k4 is approximately 3.5 and is probably only fairly accurate given the size of the isotope effects. Thus, the release of NADþ from the E:NAD complex is probably about 3.5-times faster than release of mannitol from the E:NAD:mannitol complex. 2. Random Mechanisms (All Rate Constants of Mechanism 3 Apply) In the case of a steady-state random mechanism k2, k4, k5, and k7 are not much greater than k9. With A or B varied and the fixed substrate maintained at low levels (B/Kib or A/Kia approach zero) such that k3 B , k2 and k6 A , k7 in Equation 30.6 and Equation 30.7, respectively, the expression for the forward commitment factor is given by Equation 30.9. cf ¼

k9 k4 þ k5

ð30:9Þ

Free enzyme predominates and both pathways for formation of EAB are operative. The magnitude of the observed D(V/K) for either of the substrates will be identical under these conditions and will depend on the values of k4 and k5 relative to k9. As B approaches infinity Equation 30.6 reduces to k9/k5, and the observed isotope effect will depend on the values of k9 and k5, that is the off-rate constant for B from EAB, and k9. If B is sticky in the EAB complex (k4 , k5), the isotope effect will be independent of B since release of A from EAB dominates, while if B is not sticky in EAB (k4 $ k5), the isotope effect will increase as B increases and the release of A from EAB is favored. In this symmetric mechanism the same logic can be used with B varied and A as the fixed varying substrate. As A approaches infinity Equation

Substrate and pH Dependence of Isotope Effects in Enzyme Catalyzed Reactions

799

30.6 reduces to k9/k4. The observed isotope effect will be independent of A as it is increased if k5 , k4 (A is sticky in EAB), while the isotope effect will increase if k5 . k4 (A is not sticky in EAB). Qualitative and semi-quantitative information are obtained from the measured isotope effects. A random kinetic mechanism is indicated by finite values of isotope effects on V/Ka (B saturating) and V/Kb (A saturating). The magnitude of the observed isotope effects provides information on the preference of the two pathways, EAB to EA or EAB to EB. The largest DV/K reflects the favored pathway, e.g., if D(V/Ka) . D(V/Kb) the pathway for EAB to EB is preferred. If, on the other hand, D (V/Ka) ¼ D(V/Kb), there are two possibilities, either neither substrate is sticky and the kinetic mechanism is rapid equilibrium random (k4, k5 . k9), or both substrates are equally sticky and the kinetic mechanism is steady state random (k4 ¼ k5 # k9), unless an equilibrium ordered initial velocity pattern is observed (see above). Both possibilities have been reported. A modification of the steady state random mechanism is one in which substrates are sticky in the ternary complex but bind in rapid equilibrium to the binary complexes, i.e., k2 and k7 q V=Et but k4 and k5 are less than or not much greater than V/Et. When A is varied at any finite B, k2 will exceed k3B and when B is varied at any finite A, k7 will exceed k6A, and Equation 30.6 and Equation 30.7 are k9/(k4 þ k5). The full value of Dk9 will not be seen unless A or B are not sticky from the ternary complex. Additional information is needed, for example dissociation constants for EA and EB, to distinguish between this mechanism and those discussed above. Nonetheless, the conclusions concerning ordered vs. random kinetic mechanisms still apply. a. NAD-Malic Enzyme Primary kinetic deuterium isotope effects have been determined for the NAD-malic enzyme from Ascaris suum.10,11 The enzyme catalyzes the divalent metal ion dependent oxidative decarboxylation of L-malate using NADþ to produce pyruvate, CO2, and NADH. The primary deuterium isotope effects on V/Kmalate and V/KNAD are both about 1.5 and the effect on V is 2. The finite isotope effects on the V=K for both reactants indicate a random kinetic mechanism. Earlier studies of the closely related NADP-malic enzyme suggested an ordered kinetic mechanism,12 but later work indicated a random kinetic mechanism substantiated by the isotope effect results.13 Equality of the isotope effects on the reactant V=Ks indicates either a rapid equilibrium random mechanism or a steady state random mechanism in which there is no preference for release of NADþ and malate from the E:NAD:Mg:malate complex. Later isotope partitioning results demonstrated the mechanism was steady state random.14 The slightly larger isotope effect on V suggests isomerization of the E:NAD complex16 is about twice as fast as release of reactant from the central complex. b. Ketopantoate Reductase Ketopantoate reductase catalyzes the NADPH dependent reduction of ketopantoate to give NADPþ and hydroxypantoate.15 At pH 7.5 the primary deuterium isotope effects have been determined for the natural substrates and several alternative substrates. For the natural substrates the istotope effects are 1.8 for D(V/Kketopantoate) and 1.5 for D(V/KNADPH). The D(V/KNADPH) value increases to 2.7 with the alternative substrate a-ketoisovalerate. Compared to the natural substrate, the isotope effect increases to 1.9 with the NADPH substrate analog 3-acetylpyridineADPH (APADPH) and ketopantoate. On the basis of product inhibition and direct binding studies with ketopantoate, the authors suggested an ordered mechanism with NADPH binding prior to ketopantoate. However, the finite isotope effects observed with both of the natural substrates, NADPH and ketopantoate, suggest a random kinetic mechanism, and the significantly larger isotope effects observed with a-ketoisovalerate and APADPH clearly support a random mechanism with the alternative substrates.

800

Isotope Effects in Chemistry and Biology

B. PING P ONG K INETIC M ECHANISMS A ping pong kinetic mechanism is one in which a product is released prior to binding one of the substrates. Although initial velocity and isotope exchange studies will usually already have been used to show that the mechanism appears ping pong, isotope effects are also useful. Consider the following mechanism. k1 A

k3

k5 B

k7

E O EA ! F O FB ! E k2

ð30:10Þ

k6

In mechanism 30.10, only the rate constant k3 is sensitive to isotopic substitution. It will usually be true that only one of the two half reactions in a ping pong mechanism is sensitive to isotopic substitution unless: (1) there is a significant amount of internal return of deuterium to a nonexchangeable covalent position (which may be true for some of the aminotransferases where the a-proton could be transferred to C40 of pyridoxamine 50 -phosphate); or (2) a position is monitored in which the label is not potentially eliminated in the first half reaction (as is the case for secondary isotope effects or some heavier atom effects). Only the case where one of the two half reactions is isotope sensitive will be discussed; this is the most common case encountered. In mechanism 30.10 k3 and k7 are not simple microscopic rate constants but are the net rate constants for conversion of either EA to F or FB to E and contain contributions from conformation changes, the bond-breaking step(s) and release of the product. Only the bond-breaking step is sensitive to isotopic substitution; and unless the bond-breaking step is the slowest step, the isotope effect on k3 will be lower than the intrinsic isotope effect on the bond-breaking step. The following discussion applies to classical onesite or non-classical two-site ping pong kinetic mechanisms. The expressions for V; V=Ka ; and V=Kb are given V¼

k3 k7 k3 þ k7

V k1 k3 ¼ Ka k2 þ k3

V k5 k7 ¼ : Kb k6 þ k7

ð30:11Þ

The expressions for the isotope effects on V and V=Ka are given in Equation 30.12. The isotope effect on V=Kb will be equal to unity, since it does not include k3. D D

V¼

k3 þ

k3 k7

k 1þ 3 k7

D D

V=Ka ¼

k3 þ

k3 k2

k 1þ 3 k2

ð30:12Þ

If it is not known whether the kinetic mechanism is sequential or ping pong, the isotope effects on the V=Ks for A and B will not distinguish between a steady state ordered or ping pong mechanism. A parallel initial velocity pattern would suggest a ping pong kinetic mechanism, but it is possible to obtain near parallel lines for an ordered mechanism. The isotope effects will provide information on the relative rates of the two half reactions. If the first half reaction completely limits the overall reaction ðk7 q k3 Þ; then DV will exhibit the full isotope effect on k3 (Dk3) and D(V/Ka) will be equal to DV if A is not sticky, i.e., if k2 . k3 : If the value of DV , D(V/Ka), the second half reaction contributes to rate limitation (if it is unity the second half reaction is rate-limiting). A lower limit for k3 =k7 can be obtained using D(V/Ka) as a lower limit of Dk3 and substituting into the equation for DV. If DV . D(V/Ka), A is sticky, and an estimate of k3 =k2 can be obtained using DV as an estimate of Dk3 and substituting into the equation for D(V/Ka).

Substrate and pH Dependence of Isotope Effects in Enzyme Catalyzed Reactions

801

1. Dihydroorotate Dehydrogenase Dihydroorotate dehydrogenase from Clostridium oroticum is a flavin-containing enzyme that catalyzes the conversion of L-dihydroorotate to orotate using NADþ as the ultimate electron acceptor. A parallel initial velocity pattern and isotope exchange studies support a ping pong kinetic mechanism for this enzyme, and in fact it has a two site ping pong kinetic mechanism.17 The first half reaction involves conversion of dihydroorotate to orotate leading to reduced enzyme, and the second half-reaction is reoxidation of the enzyme-bound flavin by NADþ. Primary deuterium isotope effects determined at pH 6.5 using dihydroorotate dideuterated at the C5 position gave isotope effects of D (V/K dihydroorotate) ¼ 1.51 ^ 0.07, D (V/K NAD ) ¼ 1.1 ^ 0.07 and D V ¼ 1.34 ^ 0.02. The absence of an isotope effect on D(V/KNAD) is consistent with an ordered or ping pong mechanism, but the additional evidence from initial velocity and isotope exchange studies confirm the ping pong mechanism. The somewhat smaller value of DV compared to D (V/Kdihydroorotate) suggests some rate limitation by the second half-reaction, reduction of NADþ by the enzyme-bound flavin at site 2, in the overall reaction. The errors, in this case, preclude a quantitative assessment. 2. p-Cresol Methylhydroxylase The enzyme p-cresol methylhydroxylase flavocytochrome is another flavin containing enzyme that has been shown to display a parallel initial velocity pattern suggesting a ping pong kinetic mechanism.18 The reaction involves reduction of the enzyme-bound flavin by cresol, release of the oxidized substrate, and reoxidation of the reduced flavin by an oxidant (the artificial electron acceptor, phenazine methosulfate was used by the authors). The isotope effects are DV ¼ 2.6 ^ 0.2, D (V/Kcresol) ¼ 7.0 ^ 0.4, and D(V/Kphenazine methosulfate) ¼ 1.00 ^ 0.03. The isotope effects support the assignment of a ping pong mechanism to the enzyme as suggested by the parallel initial velocity pattern. The isotope effect on the V=K for cresol is thought to be an intrinsic value suggesting that reduction of the flavin completely limits the first half reaction. The significantly lower value for the isotope effect on V indicates that the second half-reaction contributes to limitation of the overall reaction, and an estimate of about 3 is obtained, from the expression for DV in Equation 30.12 using 7 as an estimate of Dk3, for the relative rates of the first and second half reactions.

C. SUBSTRATE D EPENDENCE o F I SOTOPE E FFECTS IN T ERREACTANT AND H IGHER O RDER M ECHANISMS The simplest way to treat a terreactant mechanism is to measure the initial rate varying one reactant with the other two maintained at saturating concentration. Consider the following random terreactant kinetic mechanism.

EA

EAB k5C

k6 E

EB

EAC

k 7B k8 k 9A

EC

EBC

k10

EABC

k 11

Products

ð30:13Þ

802

Isotope Effects in Chemistry and Biology

In Equation 30.13, only the rate constants for formation of the Michaelis complex, EABC, and the net rate constant for product formation, k11, are given. Three experiments can thus be carried out with A and B saturating and C varied, A and C saturating and B varied, and B and C saturating and A varied. Under these conditions, the mechanism reduces to the following: k5 C

k11

EAB O EABC ! Products k6

k7 B

k11

EAC O EABC ! Products k8

k9 A

k11

EBC O EABC ! Products k10

ð30:14Þ ð30:15Þ ð30:16Þ

The deuterium isotope effect on V/Kc, V/Kb and V/Ka would then be measured. The cf terms for Equation 30.14 to Equation 30.16 are k11 =k6 ; k11 =k8 ; and k11 =k10 ; respectively. If finite values are obtained for all three V/K values, evidence is obtained for a random mechanism and an estimate of the relative rates of the three pathways can be obtained. If any of the isotope effects are equal to unity, cf is large, data suggest the pathway does not exist. For example, a finite isotope effect on V/Kc, but values of unity for the other two suggest a mechanism in which A and B must bind before C. A quadreactant or higher order mechanism can be studied using a similar approach. 1. NAD-Malic Enzyme In this way isotope effects were measured in the reverse, terreactant, reaction direction for the NAD-malic enzyme22 reaction. Malic enzyme catalyzes the reductive carboxylation of pyruvate to give malate with NADH as the reductant. At saturating concentrations of two of the reactants and varying the third, the following isotope effects were measured: DV ¼ D(V/KNADH) ¼ 1.47 ^ 0.06, D V ¼ D(V/Kpyruvate) ¼ 1.64 ^ 0.08, and DV ¼ D(V/KCO2) ¼ 1.65 ^ 0.08. Data were indicative of a rapid equilibrium random mechanism in the slow reaction direction for the NAD-malic enzyme. 2. Alanine Dehydrogenase In the non-physiologic direction of the alanine dehydrogenase reaction, the enzyme catalyzes a reductive amination of pyruvate using NADH and ammonia to give L-alanine. A value of 1.7 was measured for D(V/KNH3) in the ordered terreactant reaction at saturating concentrations of the other two reactants.23 Much higher values, 2.5 to 3.5, were measured for the same parameter using alternative dinucleotide substrates or hydroxypuruvate. These data allowed an estimate of the ratelimitation of hydride transfer with these reactants so that 15N kinetic isotope effects could be measured.

IV. P H DEPENDENCE OF ISOTOPE EFFECTS A. PROTON T RANSFER A ND C HEMISTRY A RE C ONCERTED Kinetic parameters depend on pH as a result of titration of groups responsible for reactant binding, catalysis, or maintaining the correct conformation of the enzyme. The parameters of interest are the limiting rate constants V; (measured as the substrate concentration(s) approach(es) infinity) and V=K (measured as the substrate concentration(s) approach(es) zero). Free enzyme and reactant predominate under V=K conditions, while enzyme-reactant and/or enzyme-product complexes predominate under V conditions. Information on kinetic and chemical mechanism can be obtained

Substrate and pH Dependence of Isotope Effects in Enzyme Catalyzed Reactions

803

if the isotope effects are measured at the optimum pH and at a pH where the kinetic parameter is decreasing by a factor of 10 per pH unit. Consider the following model for a bireactant mechanism at saturating concentrations of A. k7B

EAH H /K1

k8

EAHB

H /K 2

EA

k3B k4

EAB

ð30:17Þ

Q

P k9

EQ

k11

E

In the above mechanism, k9 represents the chemical step(s), where k9 is not a microscopic rate constant but a net rate constant for conversion of the EAB complex to EQ and P. The rate constants k3 and k7 are for binding to EA and EAH, respectively, while k4 and k8 are for release of B from EAB and EAHB, respectively, and k11 is for release of the last product, Q. K1 and K2 are acid dissociation constants for EAH and EAHB, respectively, and H is hydrogen ion concentration. Mechanism 30.17 indicates a decrease in activity as the pH decreases, but the treatment presented below will be identical if the EAHB form of the enzyme is the active one and activity decreases as the pH increases, or if two groups, one protonated and another unprotonated, were required for activity and the activity decreases as the pH decreases below the pK of one group or above the pK of the second group. At pH values above pK1 and pK2, i.e., HpK1, K2, the following expressions are obtained for V; V/Kb, DV, and DV/Kb. The V=K for the second reactant to bind in an ordered mechanism or for either reactant at saturating concentration of the other in a random mechanism is measured. An isotope effect will always be observed on this parameter as long as the chemical step contributes to rate limitation (see discussion of kinetic mechanism above). V¼

k9 k11 k9 þ k11

V k3 k9 ¼ Kb k4 þ k9

D

V¼

D

k9 þ k9 =k11 1 þ k9 =k11

D

V Kb

¼

D

k9 þ k9 =k4 1 þ k9 =k4

ð30:18Þ

The net rate constant k9 is common to all of the mechanisms that will be discussed below, and includes any reverse commitment factor that might be expressed. Thus, only the forward commitment factor (cf), k9 =k4 ; and the catalytic ratio (cVf), k9 =k11 will determine differences in the parameters of interest, Equation 30.18. There are four possibilities for the pH dependence of kinetic parameters and isotope effects based on mechanism 30.17, and each gives, theoretically, different results for at least one of the parameters in Equation 30.18. The four possibilities are obtained as a result of some of the processes in mechanism 30.17 being absent. 1. Random Addition of Proton and Substrate to Enzyme If all of the rate and equilibrium constants in mechanism 30.17 are finite, the following expressions are obtained for the parameters in Equation 30.18 at pH values below pK1 and pK2. V¼

k9 K2 H

V k k K ¼ 3 9 1 Kb k4 H

D

V ¼ D k9

D

V Kb

¼ D k9

ð30:19Þ

Although the values for pK1 and pK2 may differ, measurement of the isotope effects on V and V/Kb at pH values below them will generate the largest observable isotope effect on both parameters. The measured value will not be the intrinsic isotope effect unless the chemical step is completely limiting for the overall reaction under these conditions. The isotope effects, if not equal to one another at high pH, k4 – k11 ; will become equal at low pH, i.e., approximately 1 pH unit below the pK values. (If the chemical step limits at all pH values, the isotope effects on V and V=Kb will be equal to one

804

Isotope Effects in Chemistry and Biology

another at all pH values.) Given a value for Dk9, one can estimate values for cf and cVf, and thus the ratio k4 =k11 : The kinetic parameters decrease at low pH as a result of a decrease in the concentration of EA and EAB. A number of examples of this behavior have been published and some of these are discussed below. a. NADP-Malic Enzyme The pigeon liver NADP-malic enzyme catalyzes the divalent metal ion dependent oxidative decarboxylation of L-malate to give pyruvate, CO2 and NADPH. The pH dependence of the kinetic parameter V decreases at low and high pH with pK values of 4.5 and 9.5. The V=K for malate also decreases at low and high pH with pK values of 6 and 8.12 The primary deuterium isotope effect on V=Kmalate is pH independent and equal to 1.5. In contrast, DV is unity at pH 7, but increases to 1.4 and 1.29 at pH 4 and 9.5, respectively.2 The V isotope effect becomes nearly equal to that on V=K at pH 4, which is below the pK of 4.5. The V isotope effect does not become equal to the one on V=K at high pH, but was determined at the pK value of 9.5 rather than one pH unit above the pK. At the pK value the prediction is the isotope effect should be half the value between the minimum seen at lower pH and the maximum at pH value above the pK. The isotope effect on V=K is pH independent and suggests that cf is equal to zero for L-malate over the entire pH range. With the exception of low and high pH, where the catalytic pathway limits and an isotope effect on V is observed, the absence of an isotope effect of V suggests the NADPH release limits the reaction at neutral pH values. b. Nitroalkane Oxidase Another enzyme that conforms to the full mechanism is the nitroalkane oxidase from Fusarium oxysporum. The enzyme is an FAD-containing oxidase that catalyzes the oxidation of nitroalkanes to aldehydes or ketones with the production of nitrite and hydrogen peroxide. Nitroalkane oxidase is proposed to proceed by a ping pong Iso-mechanism.19 After binding to the oxidized form of the enzyme, a proton is abstracted from the a-carbon of the nitroalkane substrate. Substitution of deuterium at the a-carbon leads to the primary deuterium isotope effect. The pH dependence of the reaction with nitroethane as the substrate shows two pK values of 6.9 and 9.5 in the log V=K pH profile indicating that one enzymatic group must be protonated and a second group unprotonated for activity. The primary deuterium isotope effect on V/Knitroethane is pH independent. The value of DV increases from a value of 1.4 at pH 8 –9 to a value of 7.4 at pH 5, equal to D(V/Knitroethane). The commitments to catalysis must be small since D(V/K) is nearly equal to the maximum value of the isotope effect of 9.3 for deprotonation of nitroethane in solution. In contrast the DV value of 1.4 at pH 8 indicates that steps subsequent to hydrogen bond cleavage are slow, likely in the second-half reaction. 2. Dead-End Protonation of Enzyme If reactants bind only to the correctly protonated form of the enzyme, and vice versa, k7, k8, and K2 are absent in mechanism 30.17. For this mechanism, the following expressions are obtained for the parameters in Equation 30.18 at low pH values. V¼ D

k9 k11 k9 þ k11 D

k9 þ k9 =k11 V¼ 1 þ k9 =k11

V k3 k9 K1 ¼ ðk4 þ k9 ÞH Kb D

V Kb

D

k þ k9 =k4 ¼ 9 1 þ k9 =k4

ð30:20Þ

Substrate and pH Dependence of Isotope Effects in Enzyme Catalyzed Reactions

805

In this case V and DV will be pH independent since EAB cannot be protonated once reactant binds and locks the protonation state of all reactant and enzyme functional groups in their optimum protonation state for catalysis. Reactant competes with the proton at low reactant concentration, but once bound a productive Michaelis complex is formed. As a result, although V=K is pH dependent, D V/Kb will be independent of pH. This mechanism is easily distinguished from the full mechanism (see above) and those discussed below by the pH independence of V. This is also a common mechanism and an example is discussed below. a. NAD-Malic Enzyme The NAD-malic enzyme from Ascaris suum has been shown to conform to the mechanism for which reactants bind to only the correctly protonated enzyme form.10 The kinetic parameters V=Kmalate ; V=KNAD and V decrease below a pK of about 5. The primary deuterium isotope effect on V=K for malate and NAD is 1.5 and is pH independent, but DV decreases to a value of one at low pH. The decrease in DV suggests a nonisotope sensitive step is becoming slower at low pH leading to the decrease in the isotope effect. (This step was shown to be isomerization of the E:NAD complex.16) However, with the slow alternative substrate thio-NAD, V=Kmalate decreases below a pK of about 5, but V does not decrease at pH values below pH 5 suggesting substrate binds to only the correctly protonated form of the enzyme. The DV values for thio-NAD are pH independent as expected for a mechanism involving dead-end protonation of the enzyme. 3. Dead-End Protonation of Enzyme and the Enzyme-Reactant Complex If reactants bind only to the correctly protonated form of the enzyme, B can compete with a proton for the EA complex. However, the EAB complex can be protonated to give a dead-end EAHB complex, that is k7 and k8, are absent in mechanism 30.17. For this mechanism, the following expressions are obtained for the parameters in Equation 30.18 at low pH values. V¼

k9 k4 H

V k3 k9 K1 ¼ Kb ðk4 þ k9 ÞH

D

V ¼ D k9

D

V Kb

¼

D

k9 þ k9 =k4 1 þ k9 =k4

ð30:21Þ

In this case, V and V=Kb will be pH dependent since both EA and EAB can be protonated. Once B is bound, the isotope effect on V/Kb is pH independent as for case 2 above. However, as EAB builds up at saturating B, the decreasing pH results in an increase in EAHB, a decrease in rate and an increase in DV. This case can be distinguished from the others by the pH independence of D V/Kb. An example of this behavior is discussed below. a. Ketopantoate Reductase Ketopantoate reductase conforms to this mechanism with the exception that deprotonation of the enzyme-substrate complex forms the dead-end complex. The log V/KNADPH pH profile displays two pK values at pH 6.2 and 8.7, while V/Kketopantoate is pH independent at low pH with a decrease above a single pK of 8.1. The V pH profile decreases above a single pK of pH 8.4. The pK at low pH in the V/KNADPH profile was attributed to the 20 -phosphate of NADPH, which indicates preferential binding of the dianionic form of the phosphate monoester of NADPH. A general acid-base mechanism in which a single group is required to be in opposite protonation states for catalysis in the forward and reverse reaction directions was proposed.20 In the reductive reaction direction the group would be required to be protonated to donate a proton to the ketooxygen of the substrate ketopantoate during hydride transfer to yield the hydroxypantoate product. The primary deuterium isotope effects are 1.8 on V/Kketopantoate and 1.5 on V/KNADPH and are pH independent from pH 6.5 to 9.5. By contrast DV increases from a value of approximately 1.1 at pH 6.5 to a maximum value

806

Isotope Effects in Chemistry and Biology

of 2.5 at pH 9.6. The isotope effect on DV appears to increase above a pK similar to the pK in the log V profile. Formation of a dead-end unprotonated enzyme substrate complex will show the pH dependence of the isotope effects displayed by ketopantoate reductase where DV increases above the value of the DV/K isotope effects. It is unclear at this point why ketopantoate is unable to dissociate from the unprotonated E-NADPH-ketopantoate complex. 4. Dead-End Formation of a Protonated Enzyme-Reactant Complex If the EA complex can be protonated and substrate can bind to the EAH complex, but locks the proton on enzyme, a dead end complex is formed. For this case the equilibrium constant K2 is absent. The following expressions are obtained for the parameters in Equation 30.18 at low pH values. V¼ D

k3 k9 k8 K1 ðk4 þ k9 Þk7 H

V Kb

V k3 k9 K 1 ¼ Kb ðk4 þ k9 ÞH

D

V¼

D

k þ k9 =k4 ¼ 9 1 þ k9 =k4

D

k9 þ k9 =k4 1 þ k9 =k4

ð30:22Þ

Once B is bound, the partitioning of EAB is unaffected by pH, so that DV/Kb is pH independent, although V=Kb does depend on pH. This case differs from 2 above in the behavior of V; which is pH dependent as a result of the ability of B to bind to EAH to give a dead-end EAHB in constant ratio with EA. Although B is saturating, it is never saturating for the EA to EAB pathway, and conditions are those for V=K: To date, no examples of this case have been reported.

B. PROTON T RANSFER A ND C HEMISTRY N OT C ONCERTED Proton transfer can occur in a step separate from a second chemical step. In this case the behavior differs from that discussed for the concerted cases discussed above. Consider the general mechanism below. k1 A

k3

k5

k2

k4 H

k6

k7

E O EA O EX O EQ ! E

ð30:23Þ

In mechanism 30.23, all rate constants have their usual meaning. Two chemical steps are depicted, a proton transfer, k3 and k4H, generating an intermediate EX, and a second chemical step to give the final product EQ, k5 and k6. There will also, of course, be pH dependence as shown above in mechanism 30.17 as a result of protonation of E and EA, but for simplicity, these pH dependencies are not considered in this mechanism. Expressions for kinetic parameters V and V=Ka are shown in Equation 30.24 and Equation 30.25. V¼

H 1þ K3

V ¼ Ka

H K3

k5 k6 1 1 1þ þ k5 þ k7 k3 k7

1þ

k6 k7

k1 k5 k2 þ k5

1 1 þ k2 k3

ð30:24Þ

ð30:25Þ

In Equation 30.24 and Equation 30.25, K3 is equal to k3 =k4 ; the acid dissociation constant for the pH dependent step. V and V/Ka exhibit pH dependence as a result of enzyme being maintained

Substrate and pH Dependence of Isotope Effects in Enzyme Catalyzed Reactions

807

in the EA form. From these equations, expressions for the isotope effects on V and V/Ka are generated. D D

V¼

1=k3 þ 1=k7 k þ 6 D Keq 1 þ H=K3 k7 1=k3 þ 1=k7 k 1 þ k5 þ 6 1 þ H=K3 k7

k5 þ k5

D D

V=Ka ¼

k5 k k 1 þ 3 þ 6 D Keq k4 H k2 k7 k5 k3 k 1þ þ 6 1þ k4 H k2 k7

k5 þ

ð30:26Þ

ð30:27Þ

The isotope effect on V will be highest at pH values below pK3 as cf, {k5[(1/k3 þ 1/k7)/(1 þ H/ K3)]}, tends to zero, and will go to a constant value at high pH dependent on the value of cf þ cr ; k5 ð1=k3 þ 1=k7 Þ þ k6 =k7 : The pH dependence of DV/K is diagnostic for this mechanism. Its value is predicted to be identical to that for DV at low pH as cf ; ðk5 =k4 HÞð1 þ k3 =k2 Þ; tends to zero, and will decrease to a value of 1 as the pH increases and cf tends to infinity. The decrease to unity can occur in a pH range where there is no change in the value of the kinetic parameter, and results from the loss of the proton produced in the proton transfer step, committing the reaction to go toward products. In the reverse reaction direction cr is identical to cf in the forward reaction direction, and the isotope effect on V=Kq will decrease at high pH to a value equal to DKeq. The prototypical example of this behavior is exhibited by alcohol dehydrogenase and this will be discussed below. 1. Equine Liver Alcohol Dehydrogenase The equine liver alcohol dehydrogenase catalyzes the NAD-dependent oxidation of a number of primary and secondary alcohols. With cyclohexanol as a substrate, a complex pH-rate profile is observed as a result of substrate activation by the alcohol substrate.3 The V and V=K profiles for cyclohexanol decrease at low pH with pK values of 6.2 and 7.1, respectively. In the reductive direction, V decreases above a single pK of 8.4 and V/Kcyclohexanone decreases to a limiting slope of 2 with pK values of 8.8 and 9.7. The primary deuterium isotope effect, obtained using cyclohexanol1-D in the direction of alcohol oxidation, is 2.5 and pH independent below pH 8.5, but decreases to a value of 1 at high pH. The isotope effect, obtained with NADD in the direction of ketone reduction, is 2.2 and pH independent below pH 8.5, and decreases to a value of 0.85 at high pH. The equilibrium isotope effect has been determined to be 0.85.21 The pK values determined for the isotope effect data are both 9.4. The isotope effect on V is 1.15 in both reaction directions at pH 8.5 and decreases to a value of 1 at high pH. The data suggest a nonisotope dependent pH dependent step precedes hydride transfer from alcohol to NAD. It has been proposed the reaction proceeds via an alkoxide mechanism in which inner-sphere coordination to the active site Zn allows abstraction of the alcoholic proton by an active-site proton relay consisting of a serine and histidine.3

V. CLOSING REMARKS The substrate and pH dependence of isotope effects is arguably one of the most powerful techniques available for the determination of kinetic mechanism, and the location of rate-determining steps along the reaction pathway. The limitation of the method is that an isotope effect must be observed. Even with an absence of a finite isotope effect with the natural substrates, one can substitute slower substrates that provide a higher likelihood of observing an isotope effect. It should be mentioned, however, that the substrate dependence of isotope effects does not replace initial rate studies in the

808

Isotope Effects in Chemistry and Biology

absence and presence of inhibitors, since the amount of information obtained is much more substantial with the latter technique. Isotope effects should be used whenever possible, however, to confirm kinetic mechanism, especially when one is trying to distinguish between two or more possibilities. Isotope effects have already proven an invaluable tool in characterization of site directed mutant enzymes.

REFERENCES 1 Cook, P. F. and Cleland, W. W., Mechanistic deductions from isotope effects in multireactant enzyme mechanisms, Biochemistry, 20, 1790– 1796, 1981. 2 Cook, P. F. and Cleland, W. W., pH variation of isotope effects in enzyme-catalyzed reactions. 1. Isotope-dependent step pH dependent, Biochemistry, 20, 1797 –1804, 1981. 3 Cook, P. F. and Cleland, W. W., pH variation of isotope effects in enzyme-catalyzed reactions. 2. Isotope-dependent step not pH dependent. Kinetic mechanism of alcohol dehydrogenase, Biochemistry, 20, 1805– 1816, 1981. 4 Cook, P. F., Kinetic and regulatory mechanisms from isotope effects, In Enzyme Mechanism from Isotope Effects, Cook, P. F., Ed., CRC Press, Inc., Boca Raton, pp. 203– 230, 1991. 5 Cook, P. F., pH dependence of isotope effects, In Enzyme Mechanism from Isotope Effects, Cook, P. F., Ed., CRC Press, Inc., Boca Raton, pp. 231– 245, 1991. 6 Cook, P. F., Mechanism from isotope effects, Isotopes in Environmental and Health Sciences, 34, 3 – 17, 1998. 7 Northrop, D. B., Steady state analysis of kinetic isotope effects in enzymic reactions, Biochemistry, 14, 2644– 2651, 1975. 8 Blanchard, J. S. and Cleland, W. W., Kinetic and chemical mechanisms of yeast formate dehydrogenase, Biochemistry, 19, 3543– 3550, 1980. 9 Slatner, M., Nidetzky, B., and Kulbe, K. D., Kinetic study of the catalytic mechanism of mannitol dehydrogenase from Pseudomonas fluorescens, Biochemistry, 38, 10489– 10498, 1999. 10 Kiick, D. M., Harris, B. G., and Cook, P. F., Protonation mechanism and location of rate determining steps for the Ascaris suum nicotinamide adenine dinucleotide-malic enzyme reaction from isotope effects and pH studies, Biochemistry, 25, 227–246, 1986. 11 Karsten, W. E. and Cook, P. F., Stepwise versus concerted oxidative decarboxylation catalyzed by malic enzyme: a reinvestigation, Biochemistry, 33, 2096 –2103, 1994. 12 Schimerlik, M. I. and Cleland, W. W., pH variation of the kinetic parameters and the catalytic mechanism of malic enzyme, Biochemistry, 16, 576– 583, 1977. 13 Weiss, P. M., Gavva, S. R., Harris, B. G., Urbauer, J., Cleland, W. W., and Cook, P. F., Multiple isotope effects with alternative dinucleotide substrates as a probe of the malic enzyme reaction, Biochemistry, 30, 5755– 5763, 1991. 14 Chen, C. Y., Harris, B. G., and Cook, P. F., Isotope partitioning for NAD-malic enzyme from Ascaris suum confirms a steady-state random mechanism, Biochemistry, 27, 212– 219, 1988. 15 Zheng, R. and Blanchard, J. S., Substrate specificity and kinetic isotope effect analysis of the Escherichia coli ketopantoate reductase, Biochemistry, 42, 11289– 11296, 2003. 16 Rajapaksa, R., Abu-Soud, H., Raushel, F. M., Harris, B. H., and Cook, P. F., Pre-steady state kinetics reveal a slow isomerization of the enzyme-NAD complex in the NAD-malic enzyme reaction, Biochemistry, 32, 1928– 1934, 1993. 17 Argyrou, A., Washabaugh, M. W., and Pickart, C. W., Dihydroorotate dehydrogenase from Clostridium oroticum is a class 1B enzyme and utilizes a concerted mechanism of catalysis, Biochemistry, 39, 10373– 10384, 2003. 18 McIntire, W. S., Hopper, D. J., and Singer, T. P., Steady-state and stopped flow kinetic measurements of the primary deuterium isotope effect catalyzed by p-cresol methylhydroxylase, Biochemistry, 26, 4107– 4117, 1987. 19 Gadda, G. and Fitzpatrick, P. F., Mechanism of nitroalkane oxidase: 2. pH and kinetic isotope effects, Biochemistry, 39, 1406– 1410, 2000. 20 Zheng, R. and Blanchard, J. S., Kinetic and mechanistic analysis of the Escherichia coli panE-encoded ketopantoate reductase, Biochemistry, 39, 3708– 3717, 2000.

Substrate and pH Dependence of Isotope Effects in Enzyme Catalyzed Reactions

809

21 Cook, P. F., Blanchard, J. S., and Cleland, W. W., Primary and secondary deuterium isotope effects on equilibrium constants for enzyme-catalyzed reactions, Biochemistry, 19, 4853– 4858, 1980. 22 Mallick, S., Harris, B. G., and Cook, P. F., Kinetic mechanism of the Ascaris suum NAD-malic enzyme in the direction of reductive carboxylation of pyruvate, J. Biol. Chem., 266, 2732– 2738, 1991. 23 Weiss, P. M., Chen, C. Y., Cleland, W. W., and Cook, P. F., Primary deuterium and 15N isotope effects as a mechanistic probe of alanine and glutamate dehydrogenases, Biochemistry, 27, 4814– 4822, 1988.

31

Catalysis by Alcohol Dehydrogenases Bryce V. Plapp

CONTENTS I. II.

Introduction ...................................................................................................................... 811 Mechanism and Structure of Alcohol Dehydrogenases .................................................. 812 A. Kinetic Mechanism from Isotope Effects ................................................................ 812 B. Structural Studies of Ternary Complexes ............................................................... 813 III. Transient Kinetics and Simulation of Complete Mechanisms........................................ 814 IV. Unmasking Chemistry for Mechanistic Studies .............................................................. 816 A. Poor Substrates and Chemical Modification ........................................................... 816 B. Site-Directed Mutagenesis ....................................................................................... 816 C. Isotope Effects and Altered Rate Constants ............................................................ 818 V. Dynamics of Hydrogen Transfer ..................................................................................... 822 A. Tunneling Detected by Comparison of H/D/T Isotope Effects .............................. 822 B. Temperature Dependence and Isotope Effects ........................................................ 822 C. Pressure and Isotope Effects .................................................................................... 825 D. Protein Motions and Computational Studies........................................................... 826 VI. Solvent (D2O) Isotope Effects and Proton Transfer........................................................ 827 A. Steady-State Kinetic Studies.................................................................................... 827 B. Transient Kinetic Studies and a Low-Barrier Hydrogen Bond............................... 827 VII. Future Studies................................................................................................................... 828 Acknowledgments ........................................................................................................................ 830 References..................................................................................................................................... 830

I. INTRODUCTION Determination of kinetic (substrate) and solvent isotope effects can provide important information about which steps are rate limiting in an enzyme mechanism and about the chemistry of the catalytic reaction. Interpretation of the isotope effects usually requires a comprehensive study of the steady-state and transient kinetics, including pH dependences and substrate specificity, and may be facilitated by the study of site-directed mutations that make the steps involving chemistry rate limiting. Knowledge of three-dimensional structures of the enzyme and its complexes is useful for interpretation of results and the design of particular mutations. In this chapter, applications of isotope effects for study of catalysis by horse liver and yeast alcohol dehydrogenases (ADH1) will be considered as an example of the approaches that have led to insights into the mechanisms of Abbreviations: ADH, alcohol dehydrogenase; V1, V2, turnover numbers for alcohol oxidation and reduction of aldehyde or ketone, respectively; V/K, catalytic efficiency for the indicated varied substrate, e.g., V1/Kb, catalytic efficiency for alcohol oxidation; DV, D(V/K ), deuterium isotope effect on the indicated kinetic parameter; V203A, the substitution of Val-203 with Ala, etc.

811

812

Isotope Effects in Chemistry and Biology

proton and hydride transfer. The results suggest that a low-barrier hydrogen bond stabilizes the ground state of the Michaelis complex for the oxidation of alcohols and that hydride is transferred with quantum mechanical tunneling involving protein motions. Many articles and reviews have been written about this topic.1 – 3 This chapter cannot cover all of the literature, but rather will consider some critical experimental results that are relevant to the mechanism of alcohol dehydrogenases and identify topics for further research.

II. MECHANISM AND STRUCTURE OF ALCOHOL DEHYDROGENASES A. KINETIC M ECHANISM FROM I SOTOPE E FFECTS Alcohol dehydrogenases catalyze the reversible reactions of oxidation of alcohols using NADþ and the reduction of aldehydes or ketones using NADH.4,5 The mechanism of the wild-type enzyme isolated from horse liver was established as Ordered Bi Bi with release of product coenzyme the major rate-limiting step in either the forward or reverse reactions from pH 6.0 to 9.9.6,7 Early transient kinetic studies had suggested that release of NADH is rate limiting for turnover in alcohol oxidation, leading to the classic “Theorell – Chance” mechanism,8 but the kinetic significance of ternary complexes was established by product and dead-end inhibition studies.6 Further transient kinetics studies led to the conclusion that the binding of NADþ is a two-step process involving an isomerization of the enzyme –NADþ complex.9 X-ray crystallography demonstrated that apoenzyme and complexes with coenzyme have different conformations, due to rotation of the catalytic domains toward the coenzyme binding domains upon complex formation.10 Evidence for a kinetically significant isomerization of the enzyme – NADH complex is controversial. Thus, the mechanism shown in Scheme 31.1 is considered the minimal mechanism for the enzyme acting on ethanol, benzyl alcohol, and other good substrates. Although the mechanisms of the horse and yeast enzymes are predominantly ordered with ethanol as substrate, as determined by steady-state and transient kinetics, random pathways may be more or less significant with various substrates. Isotope exchange studies show that the horse enzyme has a partly random mechanism, with alcohol binding before NADþ and coenzymes dissociating from central complexes.11,12 Quantitative conclusions about the fraction of alternative pathways as determined from the isotope exchange studies are compromised because abortive complexes, such as enzyme –NADH –alcohol, may form and provide an alternative pathway for dissociation of NADH.13 – 15 Human ADH1Cp2 (g2g2) acts predominantly through the alternative pathway at moderate (social drinking) levels of ethanol.16 The use of kinetic isotope effects to distinguish among possible mechanisms has been developed in a series of papers with alcohol dehydrogenases as an informative example.17 The small isotope effects (1.1 or less) with cyclohexanol for DV1 and D(V1/K NADþ) and the significant effects on D(V1/Kcyclohexanol) ¼ 3.1 suggest a predominantly ordered reaction in the forward reaction, whereas the larger effects with cyclohexanone for DV2 ¼ 1.46, D(V2/KNADH) ¼ 1.2, and D (V2/Kcyclohexanone) ¼ 2.2 indicate some randomness for the reverse reaction. For yeast alcohol dehydrogenase, isotope effects for 2-propanol oxidation on DV1, D(V1/KNADþ), and D(V1/K2-propanol) of about 2.1 fit with a random mechanism with some degree of stickiness of substrates.

E-NAD+ k−1

k1 E

SCHEME 31.1

k2 k−2 k−6 k6

E∗-NAD+

E-NADH

k3 k−3 k−5 k5

E-NAD+ -RCH2OH k−4

k4

E-NADH-RCHO

Catalysis by Alcohol Dehydrogenases

813

However, 2-propanol is a poor substrate for yeast ADH, so that hydride transfer is a major rate-limiting step.18 Nevertheless, with ethanol as substrate, steady-state kinetics also suggest a random mechanism with a preferred pathway with NADþ binding first,18 which is consistent with significant isotope effects (DV1 ¼ 1.8 –2.2, D(V1/Kethanol) ¼ 3.1 –3.2, and D(V1/K þ NAD) ¼ 1.8– 2.1) where V1/K is larger for alcohol than for coenzyme.19 The isotope effects with acetaldehyde and NADH(D) are close to 1,17 suggesting that the mechanism may be ordered in the reverse reaction. An objective of the kinetic and isotope studies is to obtain quantitative information about the magnitudes of the rate constants for each step in the reaction mechanism and the intrinsic isotope effects, which are decreased to the observed value by the commitments to catalysis.17 By determination of D(V1/K2-propanol) and T(V1/K2-propanol), it is possible to estimate by Northrop’s method20,21 (see also Chapter 37 by Cleland in this volume) the intrinsic H/D isotope effect of 5.7 for oxidation of 2-propanol by yeast ADH,22 and similarly an intrinsic H/D isotope effect of 6.3 for oxidation of cyclohexanol by horse liver ADH.23 By the use of multiple isotope effects (primary 13C and primary and secondary 2H), the intrinsic H/D effect for oxidation of benzyl alcohol was estimated to be 4.0 for horse liver ADH and 7.0 for yeast ADH. It was also concluded from the magnitude of the 12C/13C effects that motions of the carbons are involved in the hydride transfer.24 An important tool for exposing the maximum observable isotope effect is to vary the pH of the reaction, as commitments are highest at optimal pH and decreased at pH values where the rate of reaction is slower.23 For horse liver and yeast ADH, V1/Kalcohol typically increases with increasing pH, approaching optimal values at pH of about 8, whereas V2/Kcarbonyl increases with decreasing pH, approaching optimal values below pH 8.7,22,23,25 These effects fit with the proposal that alcohol oxidation requires deprotonation of the alcohol to form the alkoxide, which readily transfers a hydride ion to the NADþ, whereas carbonyl reduction requires protonation of the intermediate alkoxide so that alcohol can dissociate from the enzyme.

B. STRUCTURAL S TUDIES OF T ERNARY C OMPLEXES Structural evidence for the mechanisms by which horse liver ADH catalyzes the proton and hydride transfers became apparent when the structure of the first active enzyme-substrate complex was determined by x-ray crystallography.26 Studies at high resolution provide structures that define the binding of NADþ and alcohols or NADH and aldehyde analogs.27 – 31 Figure 31.1 provides a stereo view of the complex with NADþ and 2, 3, 4, 5, 6-pentafluorobenzyl alcohol (unreactive due to the electron withdrawing fluorines), which appears to mimic closely the Michaelis complex. V203

V203

T178

V292 NAD

V292 NAD

I318

Zn S48

V294 H51

F93

PFB

H51

F93

L116

L116 L57

I318

Zn S48

V294 PFB

T178

L141

L57

L141

FIGURE 31.1 Stereo view of active site of horse liver alcohol dehydrogenase. The complex with NADþ and 2,3,4,5,6-pentafluorobenzyl alcohol from 1HLD.PDB is shown.27

814

Isotope Effects in Chemistry and Biology

˚ from C4 of the nicotinamide ring The methylene carbon of the benzyl alcohol is positioned 3.4 A and oriented so that direct transfer of hydride could occur. Since the transfer occurs with quantum mechanical tunneling, it appears that the distance between the reacting carbons should approach ˚ and that motions of the protein allow or may even facilitate the transfer.32 – 34 about 2.7 A The nicotinamide ring may also pucker (to a distorted boat conformation) in the transition state, moving C4 toward the reacting carbon of the substrate.33,35 Computations indicate that the energy barrier for hydride transfer significantly decreases when the carbons move closer, associated with C4 moving about 78 out of the plane of the nicotinamide ring in the transition state.33 In the structure of the enzyme complexes with NADþ and pentafluorobenzyl alcohol, the nicotinamide ring is planar.27 In contrast, the nicotinamide ring in the complex with NADH and (R)-N-1methylhexylformamide (an aldehyde analogue) is puckered, with C4 approximately 188 out of the plane in the direction of the carbonyl carbon.31 This is the first structure of a relevant complex of an NAD-dependent dehydrogenase to show a puckered, reduced nicotinamide ring. We do not know if the puckering represents a typical ground state or is a result of interactions with the protein. The structure in the transition state may be similar to that in the complex with NADþ and pyrazole, ˚ ) with a nitrogen where C4 is about 218 out of the plane and forms a partial covalent bond (1.7 A 36 of the pyrazole. The x-ray crystallography shows that puckering is structurally allowed and may be relevant to the mechanism. Secondary 15N isotope effects with N1-labeled 3-acetylpyridine adenine dinucleotide are close to unity, suggesting that the pyridine ring is not distorted in the transition state.37 However, computations suggest that a puckered pyridine ring retains its aromatic character, and thus a significant 15N isotope effect need not be expected.38 The oxygen of the substrate is ligated to the catalytic zinc and hydrogen bonded to the hydroxyl group of the nearby Ser-48, which is part of a potential proton relay system connected to His-51, which is on the surface of the enzyme and can act as a base to facilitate deprotonation of the alcohol (Scheme 31.2). His-51 participates in the reaction, as substitution with Gln decreases activity and alters the pH dependence of the reaction, but His-51 is not required.39 – 41 It appears that ligation of the alcohol to the zinc is also a major contributing factor for the formation of the alkoxide, which would readily transfer hydride ion to the oxidized nicotinamide ring.42

III. TRANSIENT KINETICS AND SIMULATION OF COMPLETE MECHANISMS After a kinetic mechanism is established by steady-state studies, rate constants for some steps in the mechanism can be estimated from analysis of the kinetic constants.43 However, isomerizations of transitory complexes (for instance steps 2 and 4 in Scheme 31.1) do not affect the overall form of the kinetic equations, and thus it becomes important to apply transient kinetics to investigate these steps. Steady-state kinetics studies suggest that the complexes of horse liver ADH with coenzyme

SCHEME 31.2

Catalysis by Alcohol Dehydrogenases

815

isomerize (probably related to the conformational change observed by x-ray crystallography), but rate constants for each step could not be calculated.6 The binding of coenzymes can be studied independently of the catalytic reaction, and the rate constants can be used to calculate the dissociation constants, which are then compared to those determined from steady-state kinetics.44,45 For horse liver ADH, the binding of NADþ involves the two-step process, which becomes apparent because the observed first-order rate constant for binding approximates a limiting value at high concentrations of NADþ.9 Furthermore, because the dissociation of the enzyme– NADH complex is rate limiting for turnover with the wild-type enzyme, oxidation of alcohols exhibits a burst phase in the transient kinetics when the reaction is studied by stopped-flow techniques with high concentrations of enzyme and substrates.46 Analysis of the progress curves by established methods can lead to estimates of some or all of the rate constants in the process.47 Alternatively, and more conveniently, a kinetic simulation program can be used to analyze simultaneously progress curves obtained with different concentrations of substrates.48 Combination of studies from coenzyme binding and transient reactions for oxidation of several alcohols and reduction of the corresponding aldehydes has led to estimates for rate constants for overall mechanisms that include interconversion of ternary complexes.49 The use of deuterated substrates is helpful in assessing if the mechanisms are complete (all kinetically significant steps identified) and which steps contribute to rate limitation. For instance, the observed transient reaction with NADþ and ethanol or benzyl alcohol shows an H/D isotope effect of 3.8 or 3.6, respectively, but 1-propanol or 1-butanol have values less than 1.5. Likewise, the transient reactions for reaction of NADH with benzaldehyde and cyclohexanone have H/D isotope effects of 1.9 and 2.4, respectively, but the reactions with acetaldehyde, n-propanal, and n-butanal have values close to 1. For the longer aliphatic substrates, one may conclude that commitments to catalysis decrease the apparent isotope effect. The observed rate constant for the transient reaction (exponential burst phase for NADH formation or utilization) is not simply the rate constant for hydride transfer, but is in principle a relaxation constant for the process that includes the chemical reaction.50 Furthermore, the equations defining the isotope effect for the burst reaction in terms of commitment factors are not the same as the equations for the steady-state reactions.51 Thus, simulations are useful for analyzing the magnitudes of the isotope effects on the transient phase.51,52 Conversely, knowledge of the intrinsic isotope effects can facilitate the interpretation of transient kinetic data that are used to characterize the steps in the reaction. Because the minimal mechanism describing the reaction becomes more complicated as new techniques and experiments expose additional steps, simulation of the overall mechanism to fit the transient results and to obtain estimates of the microscopic constants is a continuing process. For liver alcohol dehydrogenase, simulation of the mechanism given in Scheme 31.1 for the reactions of four different alcohol/aldehyde pairs and its corresponding deuterated substrates (alcohol or NADD) leads to apparent intrinsic isotope effects.49 It is important to note that we use “simulation” to describe the procedure because fitting of the data requires that a mechanism be defined, and the magnitudes of the rate constants that are obtained will change when the mechanism is changed. The data are actually fitted to the mechanism, with the FITSIM program,48 and standard errors for good fits are less than 10% of the values. Values for H/D effects of 3.2 for oxidation of ethanol and of 1.7 for reduction of acetaldehyde, and 6.1 for benzyl alcohol and 2.8 for benzaldehyde were estimated from fitting of the transient progress curves for the deuterated substrates while fixing the rate constants for the steps involving binding and dissociation, which should have small isotope effects. By contrast, the results with n-propanol and n-butanol yield values for the isotope effect on the chemical step of about 1. Since the isotope effects for alcohol oxidation should match the estimated intrinsic effects (4 –7), and the ratio of the isotope effects for the forward and reverse reactions should match the equilibrium isotope effect (1.07 for ethanol or 1.35 with secondary effects, Ref. 53), the simulations are not sufficiently robust or the mechanism is not complete.

816

Isotope Effects in Chemistry and Biology E-NAD+-alcohol

E-NAD+-alkoxide

SCHEME 31.3

Inclusion of additional steps, however, requires methods of observation of the species in those steps. For instance, if the ionization of the alcohol to form the alkoxide (Scheme 31.3) is to be included in the mechanism in Scheme 31.1, pH-dependence or proton-release experiments are required. The pK for this ionization was estimated at 7.2, and rate constants appropriate for diffusion-limited proton transfers can be obtained from the simulation. With the inclusion of this step, calculation of the rate constants expected for an intrinsic effect of six gives values that are close to the experimentally determined values for ethanol oxidation, but not for acetaldehyde reduction.49 It appears that mechanisms with at least three isomerizations of central complexes must be considered.22,49 Of course, standard errors increase when additional steps are included unless data that measure those steps are included in the fitting. Thus, data for the proton release upon formation of the E– NADþ – alcohol complex should be included in the fitting. Proton transfers are not necessarily faster than other steps in an enzymatic mechanism, and may be rate limiting. Global fitting of data from various experiments that measure different species can in principle provide a complete analysis. At this stage, the isotope effects provide a test of the understanding of the mechanism, and the estimated rate constants provide values for calculating commitment factors.

IV. UNMASKING CHEMISTRY FOR MECHANISTIC STUDIES A. POOR S UBSTRATES AND C HEMICAL M ODIFICATION As discussed above, the rate-limiting steps in the forward and reverse reactions catalyzed by wildtype horse enzyme are release of the product coenzyme when good substrates such as ethanol or benzyl alcohol are used. In order to study the chemical steps, one can use “poor” substrates to make chemistry limiting (decreasing “external commitments”), as was exploited by Cook and Cleland in studies on the reactions of 2-propanol and acetone with yeast ADH,22,23 or by Klinman54 – 56 in studies on reactions of p-substituted benzyl alcohols and benzaldehydes with yeast ADH. One can also determine V/Ksubstrate, which is independent of the coenzyme binding steps, but may still be limited by slow dissociation of sticky products or isomerizations. Coenzyme analogs, chemical modification, and site-directed mutagenesis can also be used to increase the rate of coenzyme release or to decrease the rate of hydrogen transfer. An early example of this approach for the horse liver enzyme used picolinimidylation of amino groups to unmask an H/D isotope effect of 4.8 on the steady-state turnover of ethanol oxidation.57 Modification of Lys-228 in the coenzyme-binding site increases the rate constant for dissociation of NADH from 5.5 sec21 to more than 200 sec21 and decreases the apparent rate constant for hydride transfer for ethanol oxidation from 180 to 32 sec21. Another modification, which hydroxybutyrimidylated the amino groups, was used to unmask hydrogen transfer for reactions of p-substituted benzyl alcohols and benzaldehydes, increasing DV from about 1.1 to 2.4 and allowing studies on the substrate structure –activity relationships.58 However, such chemical modifications may not be complete or specific, and a better approach is to use site-directed mutations.

B. SITE-D IRECTED M UTAGENESIS Based on the three-dimensional structure of horse ADH, it is possible to target amino acid residues that may participate in different steps of the mechanism. Thus, residues in the substrate binding site (Leu-57, Phe-93), in the coenzyme binding site (Val-203, Val-292), and in a loop that

Catalysis by Alcohol Dehydrogenases

817

rearranges during the conformational change that occurs upon binding of coenzyme (Gly-293, Pro-295) were substituted. Table 31.1 summarizes the steady-state kinetic constants for the enzymes with substitutions at these positions, and the results illustrate the quite varied effects. The steady-state kinetic constants provide the foundation for understanding the effects of the substitutions. The L57F and F93A substitutions were made in order to begin to explain the differences in substrate specificity of the human ADH2 and ADH1C isoenzymes, respectively, TABLE 31.1 Steady-State Kinetic Constants for Wild-Type and Mutated Liver Alcohol Dehydrogenases Acting on Benzyl Alcohol and Benzaldehyde Kinetic Constant Ka (mM)

Wild-Typea 3.4

Kb (mM)

11

Kp (mM)

31

Kq (mM) Kia (mM) Kiq (mM) 21

V1 (sec ) V2 (sec21)

L57Fb 4.6 3.9 13

1.7

1.0

26

65

440

480

4900

10

80

41

950

29

1600

930

2000

30

19

280

2.1

0.31

2.2

510 850

45

84

Ki cyclohexylformamide (mM)

8100

1100

2.0

680

Ki pentafluorobenzyl alcohol (mM)

2800

2.0 180

Ki AMP (mM)

92 270

0.036b

V2/Kp (mM 21 sec21)

1.6 44h

18 1.8 70 —

3.0i 8.0

2.0

1.1

j

0.13, 308C —

G293A/P295T e

220

0.21

V1/Kb (mM 21 sec21)

V292Td

140

0.26

6.8

V292Sa

94

0.44

11

Turnover number (sec21)g

4.6

V203Ad

18

44b

15

21

Keq (pM)f

F93Ac

0.24 — 51b —

30

6.3 160

9.3 190

1.3 51

23

28

35

460

360

400

10

37

32

18

21

34

49

86

40

4.6 — 1.9 —

2.0 11

5.5 23

0.16

7.7 — 580 100

Kinetic constants were determined at 258C in 33 mM sodium phosphate and 0.25 mM EDTA buffer, pH 8.0. Ka, Kb, Kp, Kq are the Michaelis constants for NADþ, benzyl alcohol, benzaldehyde, and NADH, respectively. Kia and Kiq are the inhibition constants for NADþ and NADH, respectively. V1 is the turnover number for benzyl alcohol oxidation, and V2 is the turnover number for benzaldehyde reduction. The Michaelis constants were typically determined by varying substrates in a systematic manner and fitting the data with SEQUEN,120 and the inhibition constants for coenzymes were determined by product inhibition by one coenzyme against varied concentrations of the other coenzyme with fitting by COMP.120 The standard errors for the fitted values were 10 to 25% of the values. Adapted with permission from Rubach, J. K., Ramaswamy, S., and Plapp, B. V., Biochemistry, 40, 12686–19694, 2001, Rubach, J. K. and Plapp, B. V., Biochemistry, 42, 2907–2915, 2003, Rubach, J. K. and Plapp, B. V., Biochemistry, 41, 15770– 15779, 2002, Ramaswamy, S., Park, D.-H., and Plapp, B. V., Biochemistry, 38, 13951– 13959, 1999. Copyright (1999, 2001, 2002, 2003) American Chemical Society. a Recombinant wild-type enzyme from Rubach, J. K., Ramaswamy, S., and Plapp, B. V., Biochemistry, 40, 12686–12694, 2001, and unpublished data for commercial enzyme by K. B. Berst. b From Kim, K., Ph.D. Thesis, The University of Iowa, 1994, and K. B. Berst, unpublished data. c From Rubach, J. K. and Plapp, B. V., Biochemistry, 41, 15770–15779, 2002. d From Rubach, J. K. and Plapp, B. V., Biochemistry, 42, 2907–2915, 2003. e From Ramaswamy, S., Park, D.-H., and Plapp, B. V., Biochemistry, 38, 13951–13959, 1999. At pH 8, 308C. f Keq is the Haldane relationship calculated from V1KpKiq[Hþ]/V2KbKia. Values for the equilibrium constant have been estimated to be 35 to 70 pM.54,122,123 The agreement of the Keq values calculated from the Haldane relationship with the known values indicates good internal consistency. g Turnover number determined in a standard enzyme assay at 258C,124 based on titration of active sites with NADþ in the presence of pyrazole. h From Fan, F. and Plapp, B. V., Biochemistry, 34, 4709–4713, 1995. i From Shearer, G. L., Kim, K., Lee, K. M., Wang, C. K., and Plapp, B. V., Biochemistry, 32, 11186–11194, 1993. j From Ramaswamy, S., Scholze, M., and Plapp, B. V., Biochemistry, 36, 3522–3527, 1997.

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Isotope Effects in Chemistry and Biology

which have the substituted residues. Catalytic efficiencies (V1/Kb or V2/Kp) for substrates and binding (Kd) for pentafluorobenzyl alcohol are affected substantially. It is surprising that NADH binds about ten fold more tightly to the F93A enzyme (Kiq), but the removal of the benzene ring near the binding site for the nicotinamide ring allows alternative binding modes. The threedimensional structure shows that the nicotinamide ring of NADþ binds in the same location in the F93A and wild-type enzymes, but binding of NADH has not been determined.59 The V203A enzyme has kinetic constants that are similar to those for wild-type ADH, but V/Kb is somewhat lower and Kiq is somewhat higher. The structure of V203A enzyme complexed with NADþ and 2,2,2-trifluoroethanol shows that the catalytic and coenzyme binding domains are slightly closer together than in wild-type enzyme, but the distance between C1 of the alcohol and C4 of the nicotinamide ring is slightly longer than in the Val-203 enzyme.60 The differences may account for the decreases in V1/Kb and hydrogen tunneling (see later). The V292S and V292T enzymes have substantially increased kinetic constants, in particular showing decreased affinities for coenzymes and increased turnover numbers. Since the affinity for AMP is not altered significantly with these enzymes, the binding of the adenosine portion of the coenzyme does not appear to be affected. (This is in contrast to the effects of the I224G and I269S substitutions in the adenine binding site where affinity for AMP and coenzymes is decreased.61) It is remarkable that the small structural changes have such large effects on activity. X-ray crystallography of the V292S enzyme shows that the enzyme crystallizes in the open conformation, very similar to the wild-type apoenzyme, but with an additional water molecule fitting into the space created by the V292S substitution.34 In contrast, the V292T enzyme crystallized in the closed conformation, very similar to the ternary complexes of wild-type enzyme, again with a new water molecule to interact with Thr-292.36 Examination of the structures provides no simple explanation for the different conformations, but the equilibrium position between conformations may be delicately balanced and affected by the particular ligands present. There appears to be no structural reason that would prevent the V292S enzyme from changing conformation, as Ser-292 should be easily be accommodated in the closed conformation. As binding of coenzyme is coupled to the conformational change, altering the energetics of the change also affects the affinities for coenzymes. The doubly-mutated G293A/P295T enzyme was selected from enzymes with various substitutions at these positions because this enzyme appeared to have much higher activity than wild-type enzyme. Michaelis and inhibition constants are greatly increased, but the turnover numbers are not much different from wild-type enzyme. Significantly, catalytic efficiencies and affinities for dead-end inhibitors, pentafluorobenzyl alcohol and N-cyclohexylformamide, are decreased. These changes indicate that the structures of the enzyme –coenzyme complexes have been affected by the substitutions. X-ray crystallography shows that the enzyme crystallizes in the open conformation, even when NADþ and 2,2,2-trifluoroalcohol are present.62 The substitutions are accommodated readily in the apoenzyme structure, but model building shows that Ala-293 and Thr-295 would cause steric hindrance in the holo-enzyme (closed conformation) that could only be relieved by local structural changes. Thus, it appears that the G293A/P295T enzyme is “locked” in the open conformation. Nevertheless, the enzyme may change conformation infrequently to catalyze the reaction. The G293A/P295T, V292S, and V292T enzymes are examples of enzymes with decreased commitments to catalysis, thus making them useful for studies on hydride transfer reactions.

C. ISOTOPE E FFECTS AND A LTERED R ATE C ONSTANTS The substrate kinetic isotope effects for these various enzymes show that hydride transfer for oxidation of benzyl alcohol has become more rate limiting as compared to wild-type enzyme for the V203A, V292S, and V292T enzymes (Table 31.2, and the G293A/P295T enzyme in Table 31.7). Values of DV1 approaching 4 may be approximating intrinsic H/D effects. The mechanisms may

Catalysis by Alcohol Dehydrogenases

819

TABLE 31.2 Substrate Isotope Effects on Benzyl Alcohol Oxidation and Benzaldehyde Reduction Catalyzed by Alcohol Dehydrogenases Isotope Effecta D

V1 V1/Kb D V1/Ka D V2 D V2/Kp D V2/Kq D

Wild-Typeb

L57Fb

F93Ac

V203Ad

V292Sd

V292Td

1.4 ^ 0.1 1.8 ^ 0.2 1.0 ^ 0.4 1.1 ^ 0.1 2.6 ^ 0.2 1.0 ^ 0.1

1.6 ^ 0.1 2.9 ^ 0.3 1.0 ^ 0.3 1.0 ^ 0.1 3.2 ^ 0.3 1.2 ^ 0.2

1.1 ^ 0.1 2.5 ^ 0.4 N.D 2.9 ^ 0.2 3.2 ^ 0.2 1.8 ^ 0.7

3.4 ^ 0.2 4.1 ^ 0.6 2.6 ^ 0.7 1.1 ^ 0.1 3.1 ^ 0.5 0.75 ^ 0.12

4.3 ^ 0.7 3.4 ^ 1.2 1.3 ^ 0.4 1.3 ^ 0.1 3.1 ^ 0.2 1.8 ^ 0.1

3.6 ^ 0.3 4.1 ^ 0.6 3.5 ^ 0.4 1.7 ^ 0.3 3.4 ^ 0.3 1.7 ^ 0.2

Adapted with permission from Rubach, J. K. and Plapp, B. V., Biochemistry, 42, 2907–2915, 2003, Rubach, J. K. and Plapp, B. V., Biochemistry, 41, 15770–15779, 2002. Copyright (2002, 2003) American Chemical Society. a The superscript D represents the ratio of kinetic constants with protio and deuterio substrates. Kinetic parameters were determined by initial velocity studies where concentrations of substrate and coenzyme were varied systematically and the data were fitted with SEQUEN,120 or with fixed coenzyme concentration and varied substrate concentrations and the data were fitted with HYPER.120 The second design permitted experiments with protio and deuterio substrates on the same day, and the results were more consistent. Benzyl alcohol-a, a-d2 and NADD were used. b From K. B. Berst, unpublished data. c From Rubach, J. K. and Plapp, B. V., Biochemistry, 41, 15770–15779, 2002. d From Rubach, J. K. and Plapp, B. V., Biochemistry, 42, 2907–2915, 2003.

have also become more random for binding of substrates, as DV1/Ka values have increased above the value of one expected for an ordered mechanism. For the reduction of benzaldehyde, hydride transfer (indicated by DV2) has become more rate limiting for the V203A and V292T enzymes, and the mechanisms show evidence of randomness. These mutated enzymes are useful for studies on the mechanism of hydride transfer. Further insights into the steps that are affected by the amino acid substitutions are obtained from transient kinetic studies, which determine the overall rate constants of binding and dissociation of coenzymes and the rate constants for the transient oxidation of benzyl alcohol and the reduction of benzaldehyde. The data in Table 31.3 show that L57F and wild-type enzymes are similar, as was also determined from the steady-state results in Table 31.1 and Table 31.2. The F93A enzyme is remarkable because the rate constant for dissociation of NADH is decreased by 18-fold, consistent with decreases in the dissociation constant (Kiq) for NADH and in the turnover number (V1) with benzyl alcohol (Table 31.1). The rate constant for the transient phase for oxidation of alcohol is faster than V1, but sevenfold slower than the rate constant for wild-type enzyme, and the isotope effect of four indicates that hydride transfer is a major rate-limitation for this step. Likewise, the reduction of benzaldehyde shows a large isotope effect of 3.4, but the magnitude of the rate constant is similar to the turnover number (V2), indicating the chemical step is limiting the reduction. The rate constants for binding of NADþ are significantly decreased by the substitutions in the nicotinamide binding site, V203A, V292S, and V292T. Since the rate constants are lower than expected for a diffusion-limited reaction, the overall binding of NADþ is probably decreased because the rate constants for isomerization of the enzyme– NADþ complex are altered. The rate constants for association of NADH with these enzymes are only modestly affected, but since the equilibrium dissociation constants for NADH (Kiq) are significantly increased with the V292S and V292T enzymes, the rate constants for dissociation must also be increased. The calculated values for koff,NADH are much faster than the turnover numbers for benzyl alcohol oxidation, and the large isotope effects on V1 (Table 31.2) indicate that hydride transfer, not dissociation of NADH, is a major rate-limiting step for these enzymes. The transient studies on benzyl alcohol oxidation

820

Isotope Effects in Chemistry and Biology

TABLE 31.3 Observed Rate Constants and Isotope Effects for Transient Reactions of Alcohol Dehydrogenases Rate Constant

Wild-Typea

L57Fb

F93Ab

V203Ac

V292Sc

V292Tc

kon, NADþ (mM 21 sec21)d kon, NADH (mM 21 sec21) koff, NADH (sec21) kmax, oxidation (sec21)f D kmax, oxidationi kmax, reduction (sec21) D kmax, reduction

1.2 ^ 0.1 11 ^ 1 5.5 ^ 0.5 24 ^ 3 3.6 ^ 0.5 320 ^ 50 1.9 ^ 0.3

1.8 ^ 0.1 15 ^ 2 6.4 ^ 0.2 12 ^ 1 — .61b —

1.9 ^ 0.2 13 ^ 1 0.34 ^ 0.01 3.3 ^ 0.6g 4 ^ 1g 2.2 ^ 0.2g 3.4 ^ 0.2g

0.27 ^ 0.03 3.7 ^ 1 20 ^ 3 1.5 ^ 0.2h 3.8 ^ 0.4 58 ^ 9h 1.1 ^ 0.1

0.11 ^ 0.03 7^2 210e 6.3 ^ 0.6hh 4.3 ^ 0.3 .160 (koff)j —

0.25 ^ 0.06 13 ^ 2 250e 9 ^ 1h 3.6 ^ 0.2 .190 (koff)j —

Experiments were performed at 258C in 33 mM sodium phosphate and 0.25 mM EDTA buffer, pH 8. Overall rate constants for binding of NADþ were monitored by the formation of the ternary complex with pyrazole, those for binding of NADH by protein fluorescence, and those for dissociation of NADH by trapping free enzyme as the complex with NADþ and pyrazole. The transient rates of oxidation were determined with varied concentrations of benzyl alcohol at a fixed concentration of NADþ(2 to 10 mM depending upon the enzyme), and the rates for reduction with 0.1 mM NADH(D) with varied concentrations of benzaldehyde. The maximum rate constant was determined by fitting the observed rate constants with the program HYPER (1990, 2002).120 Adapted with permission from Rubach, J. K. and Plapp, B. V., Biochemistry, 42, 2907–2915, 2003, Sekhar, V. C. and Plapp,2 B. V., Biochemistry, 29, 4289–4295, 1990, Rubach, J. K. and Plapp, B. V., Biochemistry, 41, 15770–15779, 2002. Copyright (1990, 2002, 2003) American Chemical Society. a From Sekhar, V. C. and Plapp, B. V., Biochemistry, 29, 4289–4295, 1990. b From Kim, K., Ph.D. Thesis, The University of Iowa, 1994. For the transient reduction of benzaldehyde, the concentration of NADH was 11 mM and not saturating. c From Rubach, J. K. and Plapp, B. V., Biochemistry, 42, 2907–2915, 2003. d 1 mM 21 sec21 ¼ 1 £ 106 M21 sec21. e Calculated from koff ¼ kon(Kiq) using Kiq from Table 31.1. f Maximum rate constant for the transient oxidation of benzyl alcohol at saturating concentrations. g From Rubach, J. K. and Plapp, B. V., Biochemistry, 41, 15770–15779, 2002. h These values for kmax are equivalent to the Vmax for the steady-state reaction. i The superscript D represents the ratio of kinetic constants with protio and deuterio substrates. j Not measurable, but at least as fast as V2 .

showed no exponential burst phase, and the observed reaction corresponded to the turnover. The transient phase for reduction of benzaldehyde was too fast to measure with the stopped-flow instrument used. As a step toward the description of complete mechanisms, data from the transient reactions for binding of coenzymes and reactions of substrates were combined for simulations summarized in Table 31.4. In general, the fittings give good simulations with relatively small standard errors. However, the estimated rate constants may differ by twofold between different experiments, and calculation of values for comparison with steady-state kinetic constants also can exhibit differences of twofold. Of course, differences of this magnitude are commonly accepted by enzymologists as experimental variation. Therefore, we focus on larger changes for discussion of the effects of the mutations. Although the dissociation constant for NADþ is similar for the three enzymes (Kia, Table 31.1), the simulations suggest substantially different effects on the isomerization, in particular k22. The increases suggest that the isomerization of the enzyme –NADþ complex would favor the initial (“open”) complex. Substantial effects are also apparent for the binding of alcohol (k3), the forward and reverse chemical steps (k4 and k24), and the binding of aldehyde (k25). Thus, it appears that the L57F enzyme binds the substrates faster, whereas the F93A enzyme binds the substrates slower than wild-type enzyme does. The binding

Catalysis by Alcohol Dehydrogenases

821

TABLE 31.4 Estimated Rate Constants for the Oxidation of Benzyl Alcohol and Reduction of Benzaldehyde Catalyzed by Native, L57F, and F93A Alcohol Dehydrogenases. Rate constants are defined in Scheme 31.1 Rate Constant

Wild-Typea

L57Fb

F93A

k1 (mM 21sec21) k21 (sec21) k2 (sec21) k22 (sec21) k3 (mM 21sec21) k23 (sec21) k4 (sec21) k24 (sec21) k5 (sec21) k25 (mM 21sec21) k6 (sec21) k26 (mM 21sec21)

45 ^ 3 23,000 ^ 1,400 620 ^ 30 64 ^ 3 3.7 ^ 0.4 58 ^ 5 38 ^ 3 310 ^ 30 66 ^ 6 0.83 ^ 0.08 5.5 ^ 0.5 11 ^ 1

97 ^ 3 5,200 ^ 130 370 ^ 40 1300 ^ 100 35 ^ 1 110 ^ 3 60 ^ 10 1700 ^ 300 160 ^ 20 4.2 ^ 0.1 6.4 ^ 0.2 15 ^ 2

130 ^ 4b 22,000 ^ 600b 320 ^ 8b 730 ^ 20b 0.87 ^ 0.41c 110 ^ 50c 5.1 ^ 0.2c 2.3 ^ 0.1c 360 ^ 40c 0.25 ^ 0.03c 0.34 ^ 0.01b 13 ^ 1b

Determined from the simulation and fitting of the progress curves for the transient oxidation of benzyl alcohol and the reduction of benzaldehyde at pH 8 and 258C. The data were fitted with FITSIM48 to the mechanism shown. The progress curves for NADþ binding in the presence of pyrazole were simulated for the two-step reaction, and the binding and dissociation of NADH were determined in separate experiments (Table 31.3). Rate constants for coenzyme binding were fixed for the simulations of the transient progress curves for benzyl alcohol oxidation and benzaldehyde reduction. Typical reactions used 10 mN (N, concentration of active sites) enzyme, 2 mM NADþ, and four or more concentrations of benzyl alcohol (8 to 25 or 20 to 240 mM). Benzaldehyde reduction used 10 mN enzyme, 10 to 100 mM NADH and 10 to 50 or 40 to 5300 mM benzaldehyde. NADH formation or utilization was monitored at 328 nm in the stopped-flow instrument. All progress curves were fitted simultaneously. Standard errors for the values are listed, but it underestimate the accuracy. Adapted with permission from From Rubach, J. K. and Plapp, B. V., Biochemistry, 41, 15770– 15779, 2002, Sekhar, V. C. and Plapp, B. V., Biochemistry, 29, 4289–4295, 1990. Copyright (1990, 2003) American Chemical Society. a From Sekhar, V. C. and Plapp, B. V., Biochemistry, 29, 4289– 4295, 1990. b From Kim K., Ph.D. Thesis, The University of Iowa, 1994. c From Rubach, J. K. and Plapp, B. V., Biochemistry, 41, 15770–15779, 2002.

constants for benzyl alcohol calculated from k23/k3 also parallel the experimentally determined Ki values for pentafluorobenzyl alcohol (Table 31.1). These effects might be expected from knowledge of the structure of the enzyme, but the results provide a quantitative analysis. Of particular interest are the effects of the substitutions on the apparent rate constants for hydride transfer. L57F ADH has faster rates, and F93A ADH has much slower rates than the wild-type enzyme. The L57F substitution apparently improves the geometry, perhaps by restricting the binding modes of the substrates or dynamics of hydride transfer, which will be discussed below. By contrast, the F93A substitution increases the size of the substrate binding pocket, apparently allowing for multiple binding modes as also shown by NMR and x-ray crystallography, and substantially decreases rates of hydride transfer.59 This enzyme is unusual in that the reduction of benzaldehyde is rate limiting for the overall reaction and shows a large isotope effect on turnover (Table 31.2). The simulations represent a step forward in providing a comprehensive mechanism, but it remains to be determined how well the estimated rate constants fit with commitment factors and explain the observed isotope effects. Eventually, we expect that mechanisms based on global analysis of all of the results from different kinetic methods (steady state, transient, and isotopic) should explain the data.

822

Isotope Effects in Chemistry and Biology

V. DYNAMICS OF HYDROGEN TRANSFER A. TUNNELING D ETECTED BY C OMPARISON OF H/D/T I SOTOPE E FFECTS In pioneering work, Klinman and coworkers exploited multiple isotope effects for studying hydrogen tunneling and coupled motion in the oxidation of benzyl alcohol catalyzed by yeast and horse alcohol dehydrogenases. The approach determines the primary and secondary isotope effects for substitution at the alpha carbon and predicts that the exponent EXP ¼ lnðkH =kT Þ=lnðkD =kT Þ is larger than the value of 3.26 to 3.34 expected from semiclassical reduced mass considerations (Swain – Schaad relationship) when tunneling occurs, because H is more likely to tunnel than D or T.2,63,64 Exponents for secondary isotope effects exceeding this limit are diagnostic of tunneling and coupled motion.65 Recent computations suggest that the limiting value of the exponent should be 4.8 when coupled motion is included in the calculation66 (see also Chapter 28 by Kohen in this volume). Values lower than these limits may arise because of kinetic complexity, due to commitment factors, and make the analysis uncertain. Cha et al. studied the oxidation catalyzed by yeast alcohol dehydrogenase of benzyl alcohol, which is a poor substrate and has very little commitment to catalysis.56,67 Values for the exponent of 3.58 (^ 0.08) for the primary isotope effect and 10.2 (^ 2.4) for the secondary effect clearly support a mechanism that involves hydrogen tunneling and coupled motion. The extension of these studies to horse liver alcohol dehydrogenase, for which x-ray crystallography of the complexes with NADþ and p-bromobenzyl or pentafluorobenzyl alcohols27 provides a structural basis for understanding hydrogen transfer, required that the commitments to catalysis be altered so that the chemical step was unmasked for oxidation of benzyl alcohol, which is a good substrate. By inspection of the structure of the ternary complex, we thought that increasing or decreasing the size of the substrate binding pocket would decrease the commitments.32 Indeed, the L57F and F93W substitutions produce secondary exponents of 8.50 and 6.13, consistent with tunneling and coupled motion, whereas the exponent for the wild-type enzyme is below 4.10. ˚ As the distance between C4 of the nicotinamide ring and the a-C of the benzyl alcohol is 3.4 A in the complex (Figure 31.1), and hydrogen tunneling occurs when the distance between reacting ˚ ,33 we assume that the substrates must move closer together for carbons becomes about 2.7 A tunneling to occur. The enzyme binds and orients the substrates, and it may participate in required motions. The role of amino acid residues in the nicotinamide binding site was first tested with the V203A substitution, which increases the size of the site and diminishes the catalytic efficiency by eightfold at pH 8 (Table 31.1). In the pH 7 buffer used for isotope effects, catalytic efficiency is decreased about 40-fold, and the exponents relating the H/D/T effects show no evidence of tunneling.60 The structure of the V203A/F93W enzyme complexed with NADþ and 2,2,2trifluoroethanol suggests that the distance between C1 of the alcohol and C4 of the nicotinamide ˚ relative to the distance observed in the F93W enzyme (for which there ring is increased by 0.4 A is evidence of tunneling) because the nicotinamde ring can move closer to residue 203.68 The diminished rates of hydrogen transfer, catalytic efficiency, and tunneling because of the V203A substitution are consistent with the increased distance between the carbons. It is gratifying that computational studies also correlate distance effects and motions of Val-203 with hydride transfer.33,69,70

B. TEMPERATURE D EPENDENCE AND I SOTOPE E FFECTS A second approach for investigating hydrogen tunneling determines the temperature dependence of the reaction and the temperature dependence on the isotope effect.2,64,71 – 74 In principle, hydrogen tunneling through a rigid energy barrier is temperature independent and would be significant at low temperature. As the reaction temperature increases, tunneling contributes less to the reaction rate. Thus, Arrhenius plots will be nonlinear (concave), and the temperature dependence of the isotope effect will be diagnostic for tunneling. In particular, the ratio of Arrhenius preexponential factors

Catalysis by Alcohol Dehydrogenases

823

(A in lnk ¼ lnA þ Ea =RT) for the different isotopically labeled substrates should be either significantly less than or larger than 1.0 depending upon which temperature range is studied for the enzyme (see Chapter 28 by Kohen in this volume). For yeast ADH, AH/AD was 1.1 (^ 0.1), which does not provide evidence of tunneling, as the theoretical semi classical lower limit for reactions without tunneling is 0.75. In contrast, for two mutated horse liver ADHs, the ESE and L57F enzymes, the AH/AT and AD/AT ratios are less than their semi classical lower limits of 0.6 and 0.9.32 The ESE enzyme is a chimeric enzyme with substitutions of four residues from the horse ADH E isoenzyme by those of the ADH S isoenzyme, including deletion of Asp-115 in the substrate binding pocket, which confers activity on steroids.75 The ESE enzyme did not show evidence of tunneling from the breakdown of the Swain – Schaad relationship.32 Although the different approaches did not provide consistent evidence of tunneling for the yeast and liver ESE ADHs, the different methods are complementary. Changes in commitment factors and rate-limiting steps as a function of temperature could complicate the interpretation. Extension of the two approaches to a thermophilic ADH permitted studies over a wider temperature range and exposed a potential role for protein motions in the hydrogen transfer reaction.76 The ADH from Bacillus stearothermophilus exhibits exponents for the secondary isotope effect that exceed the semiclassical limit and increase as temperature is raised to 658C, where the enzyme normally functions. These data are consistent with tunneling. The Arrhenius plots determined for V1 (kcat) in “noncompetitive” steady-state experiments with varied concentrations of NADþ and protio or dideuterio benzyl alcohol show convex behavior with an increased slope (higher enthalpy of activation, DH ‡) below 308C. In the temperature range between 5 and 308C, there appears to be less tunneling, in contrast to expectations. This is ascribed to reduced mobility of the protein. In the temperature range from 30 to 658C, the Arrhenius plots are parallel, and the values of DH ‡ are large and indistinguishable for the H and D substrates, with an AH/AD ratio of 2.2 (^ 1.1). These results were explained by invoking protein dynamics that provided “vibrationally enhanced tunneling”. At high temperatures, the protein motions could facilitate tunneling, but at low temperature, the protein was more rigid and less able to promote tunneling. Physical studies on the H/D exchange of amide backbone support a decrease in protein mobility at lower temperature.77 Kinetic studies in methanol/water systems were used to explore for tunneling at low temperatures with the mesophilic horse liver alcohol dehydrogenase, especially the F93W enzyme, which exhibits tunneling at 258C.78 Evidence for increased tunneling was obtained as temperature was lowered from 3 to 2358C, but the Arrhenius plots were linear. Following the lead of Klinman, Kohen, Scrutton and coworkers, we34 studied the temperature and isotope effects on catalysis by the horse V292S ADH, chosen because it has decreased affinity for coenzymes, higher turnover numbers and large isotope effects (Table 31.1 and Table 31.2). For the oxidation of benzyl alcohol, the H/ D isotope effect on V1 is 4.3, and the transient reaction exhibits no burst phase, indicating that hydride transfer appears to be rate limiting. For the reduction of benzaldehyde, the isotope effect on turnover is only 1.3, but the effect on V2/Kp is 3.1, indicating rate-limiting hydride transfer. The ratio of the isotope effects for the forward and reverse reactions is 1.39, close to the expected equilibrium isotope effect of 1.35. In order to minimize effects of temperature due to changes in pK values of groups involved in catalysis, the reactions were studied at pH values where the rates were maximal and pH independent. Thus, oxidation of benzyl alcohol (V1) was studied at pH 9.2, above the pK of 6.7 at 258C controlling the forward reaction, and reduction of benzaldehyde (V2/Kp) was studied at pH 7.0, below the pK of 9.5 at 258C. The pK for V1 is temperature dependent, with an enthalpy of ionization of 9.4 kcal/mol, whereas the pK of 9.5 for V2/Kp is temperature independent. Isotope effects for benzyl alcohol oxidation (V1) were independent of pH over the range from pH 6.5 to 10.5, whereas the isotope effect for benzaldehyde reduction (V2/Kp) increased from 2.6 at pH 9.2 to 3.1 at pH 7. Thus, it appears that the isotope effects approach intrinsic values. At each temperature, V1 was determined by varying the concentrations of NADþ and benzyl alcohol

824

Isotope Effects in Chemistry and Biology

in a fixed ratio and fitting the data to the polynomial function for a sequential reaction. V2/Kp was determined with a concentration of benzaldehyde that is less than 5% of the Km value (examined at different temperatures) with varied concentrations of NADH and fitting the data to the Michaelis –Menten equation. The data are presented in Figure 31.2, using the Eyring plot in preference to the Arrhenius plot. The oxidation of benzyl alcohol shows a large enthalpy of activation and essentially temperature-independent isotope effects. Since the experiments are noncompetitive, in that separate solutions are used to determine the reaction rates, the variation of the rate constants and isotope effects is larger than those determined by the competitive methods for determining H/D/T effects. Even with precise data, however, one cannot prove that the isotope effects are temperature independent. The temperature dependence for the reverse reaction is interesting because previous studies only considered the forward reaction and because the Eyring plots are obviously convex. The enthalpy of activation was calculated with the van’t Hoff equation, because it is theoretically more rigorous than attempting to dissect out dependences at high and low temperature. The isotope effects are apparently temperature independent in the reverse reaction, also. The results from the fitting of the data are given in Table 31.5. For the forward reaction, the values of DH ‡ are large and not significantly different for the protio and deuterio substrates. We think that these results are best explained by vibrationally enhanced tunneling.34 When the same data are fitted with the Arrhenius equation, the ratio of preexponential factors (AH/AD) is not experimentally distinguishable from 1 (error of ^ 0.5), and therefore this value cannot be used as a criterion for tunneling. For the reverse reaction,

−1

55

40

T°C 30

15

5

7.5

−2 In(V2 /Kp / T)

In(V1/ T)

−4 −5 −6

T°C 30

15

6

6.5 6.0 5.5

−7 −8

5.0

1.6

1.6

1.4

1.4

InDV2 /Kp

InDV1

40

7.0

−3

1.2 1.0 0.8 0.6 3.0

55

1.2 1.0 0.8

3.2 3.4 1000/T

3.6

0.6 3.0

3.2 3.4 1000/T

3.6

FIGURE 31.2 Eyring plots for the temperature dependence and deuterium isotope effects for V1 and V2/Kp for V292S ADH. The buffers were 10 mM Na4P2O7 and 1.7 mM NaH2PO4, pH 9.2 for V1 and 10 mM Na4P2O7 and 16 mM NaH2PO4, pH 7.0, for V2/Kp. (upper left) Oxidation of benzyl alcohol. V1 for oxidation of benzyl alcohol (B), V1 for oxidation of benzyl alcohol-a-a-d2 (X). (lower left) Kinetic isotope effect on V1 for oxidation of benzyl alcohol (O). (upper right) Reduction of benzaldehyde. V2/Kp for reduction of benzaldehyde with NADH (B). V2/Kp for reduction of benzaldehyde with NADD (X). (lower right) Kinetic isotope effect on V2/Kp for reduction of benzaldehyde (O). Reprinted with permission from Rubach, J. K., Ramaswamy, S., and Plapp, B. V., Biochemistry, 40, 12686– 12694, 2001. Copyright (2001) American Chemical Society.

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TABLE 31.5 Activation Parameters from Temperature Dependence of Reactions Catalyzed by V292S Alcohol Dehydrogenase Eyringa V1 (H) V1 (D) Arrheniusb V1 (H) V1 (D) van’t Hoffc V2/Kp (H) V2/Kp (D)

DH ‡ (kcal/mol) 12.8 ^ 0.5 13.6 ^ 0.9

DS ‡ (cal/mol K) 212.2 ^ 1.8 212.2 ^ 3.0 A (s21)

Ea (kcal/mol) 13.4 ^ 0.2 14.2 ^ 0.9 DH‡ (kcal/mol) 3.2 ^ 0.8 3.2 ^ 0.9

3.8 (^ 1.0) £ 1010 3.6 (^ 1.2) £ 1010 DS‡ (cal/mol K) 222.5 ^ 2.7 224.6 ^ 3.1

DG ‡ (kcal/mol), 258C 16.39 ^ 0.02 17.23 ^ 0.04 AH/AD 1.05 ^ 0.46 — DG‡ (kcal/mol), 258C

DCp‡ (cal/mol K)

9.89 ^ 0.03 10.58 ^ 0.04

2273 ^ 89 2236 ^ 94

Reprinted with permission from Rubach, J. K., Ramaswamy, S., and Plapp, B. V., Biochemistry, 40, 12686–12694, 2001. Copyright (2001) American Chemical Society. a Fitted to the equation lnðk=TÞ ¼ lnðkB =hÞ þ DS‡ =R 2 DH‡ =RT. b Fitted to the equation ln k ¼ ln A 2 Ea =RT. DH ‡ DS‡ 1 T 1 T c þ þ DCp 2 þ R þ ln ; TR, reference temperature, Fitted to the equation lnðk=TÞ ¼ lnðkB =hÞ 2 RT R R RT R TR 258C.

the fits to the van’t Hoff equation show a small, but significant DH ‡ and a significant heat capacity term, which suggests that protein fluctuations or conformational changes contribute to catalysis. We do not think that the results can be explained by changes in commitment factors because the isotope effect is independent of temperature, and rate constants would have to change in a fortuitously complementary manner. The nonlinear temperature dependence is consistent with vibrationally assisted tunneling,79 but rigorous theoretical treatment for convex Eyring plots needs to be developed.80 These experiments provide challenging data for computational studies. The significant enthalpies of activation and essentially temperature independent isotope effects for the reactions of thermophilic and horse V292S ADHs have not been explained by transition state theory with rigid energy barriers. Instead, the concept that protein motions provide vibrationally enhanced tunneling with fluctuating potential energy barriers is a reasonable explanation.71,72,81 – 86 That protein motions are involved in enzymatic reactions is obvious to enzymologists and crystallographers, but the words that are used to describe the phenomenon need to be defined in physical terms that are testable experimentally and theoretically. It remains to be determined that protein motions are coupled to the reaction coordinate and contribute to tunneling.70,87 – 89 Indeed, some would argue that enzyme motions do not contribute to catalysis.90,91

C. PRESSURE AND I SOTOPE E FFECTS A third experimental approach that provides evidence for the mechanism of hydrogen transfer determines the effects of pressure on catalysis by yeast ADH. Pressure (up to 2 kbar) increases the catalytic efficiency (V1/Kb) for benzyl alcohol oxidation by yeast ADH by 5.6-fold and decreases the observed isotope effect from 5 to 2.792 (see also Chapter 32 by Northop in this volume). A global analysis of the data suggested that the isotope effect arose from transition-state

826

Isotope Effects in Chemistry and Biology

phenomena, rather than ground-state differences, and would be consistent with hydrogen tunneling and coupled motion with protein promoting motions. The large volume of activation, DV ‡, of 2 38 ml/mol became even more negative with dideuteriobenzyl alcohol, 2 49 ml/mol, and may be interpreted as a compression in the enzyme that is sensitive to the motions of the substrates during hydrogen transfer.

D. PROTEIN M OTIONS AND C OMPUTATIONAL S TUDIES The conformational change that brings the catalytic and coenzyme binding domains closer together in horse ADH is a potential source of motion that could bring the substrates closer together in the active site. Preliminary analysis using TLS (Translation, Libration, S — the correlation matrix) analysis of x-ray data suggests that the open and closed enzymes have different motions, but it is not clear that the motions are relevant to catalysis.93 After formation of the closed ternary complex, the substrates and amino acid residues could have further motions that are coupled to the reaction coordinate. Molecular dynamics studies94 should complement x-ray results for the identification of motions relevant for catalysis. Since enzymes are large molecules that have delicate intra molecular interactions, it is reasonable to suppose that motions near the active site would be affected by motions and structural changes that are distant from the active site. Thus, it is relevant that substitutions of distal amino acid residues in dihydrofolate reductase have significant effects on hydride transfer rates.87,95,96 In this regard, the studies on ADH with substitutions remote from the nicotinamide or substrate binding sites have not shown changes in tunneling as measured by the secondary exponent relating H/D/ T isotope effects.97 The substitutions were made at Ile-224, which participates in binding of the adenine ring of the coenzyme and is a residue affecting maximum velocities. However, the catalytic efficiencies (V1/Kb) of the wild-type and mutated enzymes are similar, suggesting that when the NADþ is bound in the active site it has an environment similar to that in wild-type enzyme. Perhaps other residues that have been identified as participating in promoting vibrations89 would have significant effects. Since the extent of tunneling in ADH may be moderate as compared to other enzymes and is correlated with catalytic efficiency, substitutions in distant sites that have larger effects on V/K may affect tunneling.60,97 The studies that detect hydrogen tunneling in the ADHs are important for understanding the enzymatic mechanism. It is significant that computations can reproduce the experimentally determined isotope effects and the breakdown of the Swain– Schaad exponent only when tunneling is included.98,99 Nevertheless, we expect that hydrogen will transfer with tunneling because of its intrinsic wave-like nature, and we ask how much quantum mechanical tunneling contributes to the rate enhancement in the catalytic reaction. Estimates depend upon the theoretical models and computational methods used. Factors of about twofold have been reported.99,100 Other estimates suggest that the free-energy barrier for benzyl alcoholate oxidation is decreased to the experimental value of 15.6 kcal/mol by about 1.8 kcal/mol due to the quantum effects.33,90,98,101 Thus, the rate constant of 24 sec21 (Table 31.3) for wild-type enzyme would be 1.3 sec21 without the quantum effects. In this regard, it is useful to consider the kinetic data for mutated ADHs that have been studied with the multiple isotope effects. The V203A mutant of horse ADH has diminished catalytic efficiency (Table 31.1) and hydrogen tunneling,60 and the rate constant for hydrogen transfer is 1.5 sec21 (Table 31.3). Substitutions of other residues near the nicotinamide binding site, such as V292S, decrease the rate of hydrogen transfer to a lesser extent (Table 31.3, Ref. 36). It appears that hydrogen tunneling is a feature of the hydrogen transfer, but that the overall catalytic efficiency depends on many amino acid residues, rather than a few critical residues. Residues distant from the active site may contribute, but it is the cumulative and cooperative effects that provide the orders of magnitude of catalytic efficiency characteristic of the enzyme.

Catalysis by Alcohol Dehydrogenases

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VI. SOLVENT (D2O) ISOTOPE EFFECTS AND PROTON TRANSFER A. STEADY- S TATE K INETIC S TUDIES Alcohol dehydrogenases catalyze hydride and proton transfer, and it is of interest to know at what step in the mechanism the proton is transferred. It has been generally assumed that binding of the alcohol to the zinc promotes ionization of the alcohol so that the alkoxide is the species that transfers hydride ion most rapidly.4,25,55 Computations support the proposal that proton transfer occurs before hydride transfer in the reactions of ADH and malate dehydrogenase.99,102,103 Solvent isotope effects can provide additional information about the mechanism, especially when combined with pH dependence and substrate isotope effects.104 Proton inventory studies can provide information about the reactant and transition states, when the enzymatic mechanism is sufficiently well understood to postulate structures for the species involved105,106 (see also Chapter 41 by Quinn in this volume). For yeast ADH, Welsh et al.107 determined that oxidation of p-methoxybenzyl alcohol had a normal solvent isotope effect (kH2O/kD2O) of 1.2 and that reduction of p-methoxybenzaldehyde had an inverse effect of 0.5 at the respective pH maxima. Oxidation was most rapid above a pK value of 8.1 –8.3, and reduction was most rapid below the pK of 8.3. The primary substrate isotope effects of 3.2 for oxidation and reduction indicate that the hydride transfer is a major rate-limiting step for turnover in each direction. The solvent effects were interpreted as reflecting reactions at the catalytic zinc water, with at least a two-step mechanism. Studies on the effects of pressure on the solvent isotope effect on V1/Kb for benzyl alcohol oxidation yielded a small inverse effect of 0.9, which was explained by a mechanism in which the zinc –H2O (with a fractionation factor of 0.56) in the reactant state was replaced by alcohol that was partially charged in transition state.108 For horse liver ADH, Taylor did a complete steady-state study of the pH dependence of the solvent isotope effects and found large normal effects (3.7) for maximum velocities for cyclohexanol oxidation and cyclohexanone reduction.109 These reactions are controlled by the dissociation of the respective enzyme– coenzyme complexes. The solvent isotope effect for V1/Kb for cyclohexanol is 1.7, and for V2/Kp with cyclohexanone is 1.2. Since these reactions are at least partially controlled by the hydride transfer step,23 mechanisms involving displacement of the water bound to zinc can also be invoked. However, it is not clear why the solvent isotope effects are different for yeast and liver ADHs.

B. TRANSIENT K INETIC S TUDIES AND

A

L OW-B ARRIER H YDROGEN B OND

Transient kinetic studies with horse liver ADH revealed a small inverse solvent effect (0.83 at pH 8.75) for the oxidation of benzyl alcohol.110 No effect was observed on the rapid transient (controlled by hydride transfer) for reduction of aromatic aldehydes, but a large normal effect was observed on the slow transient, which is presumably related to protonation of the alkoxide and dissociation of the alcohol from the catalytic zinc. We studied the solvent isotope effects on transient oxidation of ethanol by horse liver ADH, using the proton inventory approach.49 The apparent rate constant for the exponential burst phase of the reaction shows an inverse isotope effect of 0.50; in other words the reaction proceeds twice as fast in D2O as in H2O (Figure 31.3). The convex proton inventory is fitted well by the Gross – Butler equation,104,106 kn ¼ k0 ð1 2 n þ nw T Þ=ð1 2 n þ nw R Þ with values of 0.37 for the fractionation factor in the reactant state (wR) and 0.73 for the fractionation factor in the transition state (w T). Solvent isotope effects can arise for many reasons, and as with any kinetics experiment, the simplest interpretation that is consistent with the results should be chosen. The results fit the proposed mechanism for ADH shown in the scheme inset in Figure 31.3 where the reactant state has the alkoxide hydrogen bonded to the hydroxyl group of Ser-48 and the transition state has a diminished charge as the alkoxide is oxidized by NADþ. Fractionation factors of 0.5 are reasonable for a protonic site hydrogen bonded to hydroxide or alkoxide ions in aqueous medium,

828

Isotope Effects in Chemistry and Biology 300 −

Zn

S48

O H

O

+ H C H NAD

k obs, s−1

250

R fR = 0.37

Zn − δ O H

O

H C H NAD δ R f T = 0.73

Zn

S48 +

O H C

S48 H

O

NADH

R fP = 1.0

200

150 0.0

0.2

0.4 0.6 Mole Fraction D2O

0.8

1.0

FIGURE 31.3 Proton inventory for the transient oxidation of ethanol by wild-type horse liver alcohol dehydrogenase. Buffers contained 33 mM sodium phosphate at pL 8.0 and 258C and different mole ratios of D2O.104 Rate constants for the exponential burst phase of NADH formation (monitored at 328 nm in a Kinetic Instruments stopped-flow instrument, dead time of 1.3 msec) were determined with 2 mM NADþ and 50 mM ethanol. Use of ethanol-d5 gave an isotope effect of 3.8 on this rate constant. Adapted with permission from Sekhar, V.C. and Plapp, B.V., Biochemistry, 29, 4289– 4295, 1990. Copyright (1990) American Chemical Society.

and in nonaqueous media, such as acetonitrile, a value of 0.31 was measured.111 The low fractionation factor in the reactant state is consistent with a short, strong, or “low-barrier” hydrogen bond,112,113 and the distance between the oxygen of the hydroxyl groups of 2,3,4,5,6˚ , as determined by x-ray crystallography at 1.1 A ˚ pentafluorobenzyl alcohol and Ser-48 is 2.51 A resolution (B. V. Plapp and S. Ramaswamy, unpublished results). It appears that the interaction with the zinc and Ser-48 hydroxyl group stabilizes the alkoxide and facilitates the subsequent hydride transfer. It is important to note that the solvent isotope effect is inverse, indicating a reactant state effect, in contrast to a normal effect, which would indicate a transition state effect. Moreover, the environment of the hydrogen-bonded hydrogen and the transfer of the hydride ion to the NADþ change synchronously. We presume that the proton on the hydroxyl group of the alcohol is transferred through the proton relay system shown in Scheme 31.2 before the transient oxidation of the alcohol is measured in the stopped-flow instrument, as also indicated by computations.99,102 Further evidence supporting a synchronous transfer comes from a consideration of the substrate and solvent isotope affects together (Table 31.6). For the transient reaction of ethanol and benzyl alcohol oxidation, the effect of solvent on the substrate isotope effect is small, consistent with a “concerted” mechanism where each isotope effect is independent of the other isotope effect.114 For the alcohols that have faster rates for the transient reaction, the substrate isotope effect is diminished because of increased commitment factors, and the solvent isotope effect is diminished. These results support the idea that the solvent and substrate effects are reporting on the same step. Likewise, the steady-state oxidation of benzyl alcohol catalyzed by the G293A/P295T enzyme, where hydride transfer is rate limiting, shows large normal substrate and large inverse solvent isotope effects that are essentially independent of each other (Table 31.7). Similar results were obtained with steady-state kinetics studies on flavocytochrome b2.115

VII. FUTURE STUDIES Although the mechanisms of alcohol dehydrogenases have been extensively studied by a variety of techniques, with isotope effects an important tool, there is much to be done. The minimal kinetic

Catalysis by Alcohol Dehydrogenases

829

TABLE 31.6 Substrate and Solvent Isotope Effects on the Transient Oxidation of Alcohols Catalyzed by Horse Liver Alcohol Dehydrogenase Substrate Benzyl alcohol Ethanol 1-Propanol 1-Butanol Cyclohexanol

Deuterio Label

k, sec21

(a,a-d2) (d5) (1,1-d2) (d9) (d11)

25 ^ 1 180 ^ 10 360 ^ 10 320 ^ 10 270 ^ 10

D

kH2O

4.6 ^ 0.1 3.5 ^ 0.2 1.7 ^ 0.1 1.2 ^ 0.1 2.1 ^ 0.1

H

D2O

kD2O

4.3 ^ 0.2 3.1 ^ 0.1 1.3 ^ 0.04 1.1 ^ 0.04 2.0 ^ 0.1

kH

0.57 ^ 0.02 0.62 ^ 0.03 0.85 ^ 0.03 0.93 ^ 0.04 0.83 ^ 0.03

D2O

kD

0.54 ^ 0.02 0.56 ^ 0.02 0.64 ^ 0.02 0.86 ^ 0.03 0.79 ^ 0.03

The rate constants for the transient oxidation of the protio (H) and deuterio (D) substrates catalyzed by wild-type natural enzyme were determined separately with 2 mM NADþ and varied concentrations of alcohol in 33 mM sodium phosphate buffer, pL 8 and 258C in a BioLogic SFM3 stopped flow instrument. The pL (or pH) of 8 was chosen because the transient reaction shows maximum values above the pK of 6.4.46,125 The reaction was monitored by the formation of NADH at 331 nm, near the isosbestic point for the absorption of enzyme-bound and free NADH. The apparent rate constants for the exponential phase were fitted to the Michaelis– Menten equation to obtain the maximum velocities,120 given as k above. The enzyme was dialyzed into the buffer containing 99.5% D2O. The buffer in D2O was prepared from salts that were lyophilized from D2O in order to exchange protons and had the same ratio of buffering species as the buffer in H2O, producing an equivalent pL.104 The superscript represents the H/D or H2O/D2O isotope effect, whereas the subscript represents the medium or the substrate. Dr. Doo-Hong Park contributed these results.

mechanism for the liver ADH shown in Scheme 31.1 is incomplete, and studies that monitor proton release and probable isomerizations of transitory complexes are required to identify additional intermediates and to determine the rate constants for the steps. Estimates of commitment factors and the intrinsic isotope effects need to be refined. Multiple isotope effects need to be determined, extending the study in Ref. 24, so that the transition state structure is better defined. The binding of coenzymes and substrates need to be studied further in order to detect ground state effects. Does the enzyme affect the vibrational states of NADH, as shown by studies with lactate and malate dehydrogenases?116 Since the nicotinamide ring in NADH bound in a ternary complex of liver ADH is puckered,31 the binary complex might show some effects in Raman spectra. Likewise, the binding of aldehydes and aldehyde analogs show shifts in the Raman spectra, indicating some ground state effects.117,118 Isotopically labeled ligands are required for the characterization of these

TABLE 31.7 Substrate and Solvent Isotope Effects on Steady-State Oxidation of Benzyl Alcohol Catalyzed by the G293A/P295T Alcohol Dehydrogenase k V V/K

D

kH2O

4.5 ^ 0.3 6.0 ^ 0.4

H

kD2O

4.0 ^ 0.3 6.1 ^ 0.4

D2O

kH

0.51 ^ 0.04 0.40 ^ 0.03

D2O

kD

0.46 ^ 0.03 0.40 ^ 0.03

The kinetic parameters were determined with 10 mM NADþ and varied concentrations of benzyl alcohol in 10 mM Na4P2O7 with 10 mM sodium carbonate buffer, pL 10, at 308C The pL was chosen because the reactions approximate maximal values above a pK of 8.4 for V1 and 9.3 for V1/Kb.62 Initial velocity data were measured at 340 nm due to formation of NADH, and the data were fitted to the Michaelis–Menten equation.120 Buffers with 99.5% D2O and benzyl alcohol-a,a-d2 were used to determine the isotope effects. V1 was 3.4 sec21 and V1/Kb was 2.1 mM 21 sec21 in H2O with protio benzyl alcohol. Adapted with permission From Ramaswamy, S., Park, D.-H., and Plapp, B.V., Biochemistry, 38, 13951–13959, 1999. Copyright (1999) American Chemical Society.

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Isotope Effects in Chemistry and Biology

effects. Since the equilibrium position for the oxidation of benzyl alcohol is shifted by about 100fold toward aldehyde formation when the substrates are bound to the enzyme (Kint), binding interactions are affecting the energetics.119 Perhaps a major contributor to the shift in equilibrium is the formation of the zinc alkoxide complex. Studies, such as x-ray or NMR, that can provide direct evidence for protein motions are needed. Computational studies that take advantage of more powerful computers and better theories and that include more atoms and extend the simulation times are needed so that the energetic factors can be evaluated. Computations have not yet reproduced the experimental values for hydride transfer for different substrates and various mutated enzymes. Data are required so that computations can be tested rigorously. The rate constant for the nonenzymatic reaction of NADþ and an alcohol needs to be determined in order to evaluate the contribution of the enzyme to catalytic rate enhancement. ADH catalyzes a simple reaction that provides a good model for understanding enzyme catalysis, and future studies should explore the roles of protein motion in the whole enzyme.

ACKNOWLEDGMENTS I thank the many members of my laboratory who initiated and carried out the studies described here and the National Institute on Alcohol Abuse and Alcoholism and the National Science Foundation for support of our research.

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62 Ramaswamy, S., Park, D.-H., and Plapp, B. V., Substitutions in a flexible loop of horse liver alcohol dehydrogenase hinder the conformational change and unmask hydrogen transfer, Biochemistry, 38, 13951– 13959, 1999. 63 Klinman, J. P., Quantum mechanical effects in enzyme-catalyzed hydrogen transfer reactions, Trends Biochem. Sci., 14, 368– 373, 1989. 64 Bahnson, B. J. and Klinman, J. P., Hydrogen tunneling in enzyme catalysis, Methods Enzymol., 249, 373– 397, 1995. 65 Huskey, W. P. and Schowen, R. L., Reaction-coordinate tunneling in hydride-transfer reactions, J. Am. Chem. Soc., 105, 5704 –5706, 1983. 66 Kohen, A. and Jensen, J. H., Boundary conditions for the Swain – Schaad relationship as a criterion for hydrogen tunneling, J. Am. Chem. Soc., 124, 3858– 3864, 2002. 67 Cha, Y., Murray, C. J., and Klinman, J. P., Hydrogen tunneling in enzyme reactions, Science, 243, 1325– 1330, 1989. 68 Colby, T. D., Bahnson, B. J., Chin, J. K., Klinman, J. P., and Goldstein, B. M., Active site modifications in a double mutant of liver alcohol dehydrogenase: structural studies of two enzyme – ligand complexes, Biochemistry, 37, 9295– 9304, 1998. 69 Luo, J., Kahn, K., and Bruice, T. C., The linear dependence of logðkcat =Km Þ for reduction of NADþ by PhCH2OH on the distance between reactants when catalyzed by horse liver alcohol dehydrogenase and 203 single point mutants, Bioorg. Chem., 27, 289–296, 1999. 70 Caratzoulas, S., Mincer, J. S., and Schwartz, S. D., Identification of a protein-promoting vibration in the reaction catalyzed by horse liver alcohol dehydrogenase, J. Am. Chem. Soc., 124, 3270– 3276, 2002. 71 Kohen, A. and Klinman, J. P., Hydrogen tunneling in biology, Chem. Biol., 6, R191–R198, 1999. 72 Kohen, A., Kinetic isotope effects as probes for hydrogen tunneling, coupled motion and dynamics contributions to enzyme catalysis, Prog. React. Kinet. Mech., 28, 119– 156, 2003. 73 Basran, J., Sutcliffe, M. J., and Scrutton, N. S., Enzymatic H2 transfer requires vibration-driven extreme tunneling, Biochemistry, 38, 3218– 3222, 1999. 74 Scrutton, N. S., Basran, J., and Sutcliffe, M. J., New insights into enzyme catalysis. Ground state tunneling driven by protein dynamics, Eur. J. Biochem., 264, 666– 671, 1999. 75 Park, D.-H. and Plapp, B. V., Inter-conversion of E and S isoenzymes of horse liver alcohol dehydrogenase. Several residues contribute indirectly to catalysis, J. Biol. Chem., 267, 5527– 5533, 1992. 76 Kohen, A., Cannio, R., Bartolucci, S., and Klinman, J. P., Enzyme dynamics and hydrogen tunneling in a thermophilic alcohol dehydrogenase, Nature, 399, 496– 499, 1999. 77 Kohen, A. and Klinman, J., Protein flexibility correlates with degree of hydrogen tunneling in thermophilic and mesophilic alcohol dehydrogenases, J. Am. Chem. Soc., 122, 10738– 10739, 2000. 78 Tsai, S. and Klinman, J. P., Probes of hydrogen tunneling with horse liver alcohol dehydrogenase at subzero temperatures, Biochemistry, 40, 2303– 2311, 2001. 79 Antoniou, D. and Schwartz, S. D., Internal enzyme motions as a source of catalytic activity: rate-promoting vibrations and hydrogen tunneling, J. Phys. Chem. B, 105, 5553– 5558, 2001. 80 Truhlar, D. G. and Kohen, A., Convex Arrhenius plots and their interpretation, Proc. Natl. Acad. Sci. U.S.A., 98, 848– 851, 2001. 81 Borgis, D. and Hynes, J. T., Molecular-dynamics simulation for a model nonadiabatic proton transfer reaction in solution, J. Chem. Phys., 94, 3619– 3628, 1991. 82 Bruno, W. J. and Bialek, W., Vibrationally enhanced tunneling as a mechanism for enzymatic hydrogen transfer, Biophys. J., 63, 689–699, 1992. 83 Antoniou, D. and Schwartz, S. D., Activated chemistry in the presence of a strongly symmetrically coupled vibration, J. Chem. Phys., 108, 3620– 3625, 1998. 84 Kuznetsov, A. M. and Ulstrup, J., Proton and hydrogen atom tunneling in hydrolytic and redox enzyme catalysis, Canadian J. Chem. Revue Canadienne de Chimie, 77, 1085– 1096, 1999. 85 Sutcliffe, M. J. and Scrutton, N. S., A new conceptual framework for enzyme catalysis. Hydrogen tunneling coupled to enzyme dynamics in flavoprotein and quinoprotein enzymes, Eur. J. Biochem., 269, 3096– 3102, 2002. 86 Knapp, M. J. and Klinman, J. P., Environmentally coupled hydrogen tunneling. Linking catalysis to dynamics, Eur. J. Biochem., 269, 3113– 3121, 2002.

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32

Effects of High Hydrostatic Pressure on Isotope Effectsq Dexter B. Northrop

CONTENTS I. II. III.

Introduction ...................................................................................................................... 837 Theory .............................................................................................................................. 837 Experimental Examples ................................................................................................... 841 A. Hydrogen Tunneling ................................................................................................ 841 B. Yeast Alcohol Dehydrogenase................................................................................. 842 C. Yeast Formate Dehydrogenase ................................................................................ 844 IV. Conclusion........................................................................................................................ 844 References..................................................................................................................................... 844

I. INTRODUCTION Pressure can affect kinetic isotope effects (KIEs) in either of two ways: it can alter an intrinsic isotope effect directly by perturbing transition-state (TS) phenomena, such as hydrogen tunneling, or it can alter the expression of an otherwise stable intrinsic isotope effect by perturbing the kinetic complexity of reactions having multiple steps, as in an enzymatic reaction. At present there are few examples of either of the two kinds of effects, so making generalizations is rather tenuous. Nevertheless, at this juncture it appears that both ways are accessible experimentally (below 3 kbar), both can be characterized with considerable precision when present alone, and because the two follow different functions, it is sometimes possible to separate and characterize both when they appear together. Indeed, the primary purpose of undertaking pressure kinetics at present is not to find out “what happens” at high pressure, but rather to use pressure as a perturbant to deconstruct overlapping kinetic phenomena and tease out individual components, defined in terms relating to their occurrence at ambient pressure.

II. THEORY Most enzymologists, indeed most biologists, are unfamiliar with the physical chemistry of high hydrostatic pressure. In a first approximation, it is tempting to think of high pressure as “squeezing” the enzyme protein and causing it to assume new conformations. This is incorrect. High pressures do not actually squeeze reactant molecules, but rather they simply change the distribution between equilibria that are already present at atmospheric pressure, according to Le Chatelier’s principle,

q

This project was supported by NSF Grant MCB-0211290. Phone: (608) 263-2519, Fax: (608) 262-3397, E-mail: [email protected].

837

838

Isotope Effects in Chemistry and Biology

in the direction of the smaller volumes of the system. These changes in equilibria obey the function: Kp ¼ K0 e2DVK p=RT

ð32:1Þ

where K0 is an equilibrium constant under zero or atmospheric conditions (the difference is insignificant), DVK is the volume difference between the two states at equilibrium, p is the pressure, R is the gas constant, and T is the temperature. When multiple equilibria are present, data as a function of pressure are fitted to the appropriate algorithm consisting of multiple representations of Equation 32.1 combined in different ways. The final fit is, in effect, an extrapolation back to atmospheric pressure, and the resulting parameters describe the system under ambient conditions.1 It turns out that, for most proteins subjected to pressure in an aqueous system, most of the volume change occurs not in the protein itself but in the solvent. For example, breaking salt bridges exposes positive and negative ions to water and the water molecules undergo electrostriction as they solvate ions, a process that occupies less space than found in bulk water. Absolute rate theory2 includes a quasi-equilibrium constant between the reactant state and the TS, K ‡, multiplied by some physical constants. Hence, Equation 32.1 may be recast for kinetics: kp ¼ kT

‡ kB T ‡2DV ‡ p=RT K ¼ k0 e2DV p=RT h

ð32:2Þ

where kT is the transmission coefficient, kB is Boltzman’s constant, h is Plank’s constant, DV ‡ is the activation volume or volume difference between the reactant state and TS, and k0 is the rate constant for the chemical transformation under atmospheric conditions. Because of the Born –Oppenheimer approximation, one might expect activation volumes to be independent of isotopic mass differences, and this turns out to be the case as verified by Isaacs.3 The origin of normal isotope effects lies in changes in vibrational frequencies within the Bigeleisen Equation:4 0 !1 D Q H D 2Dmi =2 1 2 emi B C n i =n i e H B 1 2 e mi C n^ kH C H B ! ¼ ^B ð32:3Þ C D m C kD nD B j @ Q H D 2Dmj =2 1 2 e A n j =n j e H 1 2 e mj where n represents a vibrational frequency and m a reduced mass of hydrogen or deuterium, as indicated by superscripts. Isaacs examined whether or not these vibrational frequencies were sensitive to pressure and concluded from infrared and Raman spectra at high pressure that the sensitivities are small and likely to be insignificant within the experimental range of pressure available to kinetic studies. This conclusion was verified experimentally in a variety of reactions displaying normal isotope effects, but not so with respect to large isotope effects suspected of hydrogen tunneling. Hydrogen tunneling has been described by Bell5 in the following formulation: kH kD

obs

¼

kH QH ¼ kD QD

D D

k Q

ð32:4Þ

where kH =kD (or D k) is the semi-classical isotope effect and QH =QD (or D Q) is the Bell tunneling correction. Because of the lack of pressure effects on normal isotope effects as demonstrated by Isaacs, Northrop6 derived an expression that divides an observed isotope effect into a reactant state function that is independent of pressure (Dk) and a TS function that is dependent (DQ 2 1): D

k p¼ Dk

D

Q 2 1 e2DVQ p=RT þ D k

ð32:5Þ

where DVQ is the apparent volume difference between a hydride and deuteride ion accompanying the tunneling correction. According to Equation 32.5, as pressure is raised the (DQ 2 1) function

Effects of High Hydrostatic Pressure on Isotope Effects

839

goes to zero leaving behind Dk. Regression analysis of the curve returns a precise estimate of D Q which makes it becomes possible to fit experimental data directly to the Bell Tunneling Correction Equation. Bell assumed a truncated parabolic energy barrier and derived the following series function:5 1 X

1=2mt 2 ð21Þn sinð1=2mt Þ n¼1

Qt ¼

exp

mt 2 2np a mt mt 2 2np mt

ð32:6Þ

where ut equals hn ‡ c=kB T; a equals E=kB TN; n ‡ is the imaginary frequency, c is the speed of light, E is the activation energy, and N is Avogadro’s number. According to Hook’s law, the 1 imaginary frequencies of hydride and deuteride transfer are related by the ratios n ‡H : n ‡D ¼ 12 2 : D 2 12 2 ¼ 1:414: Therefore, combining this ratio with an experimental value of Q and a pair of Bell Tunneling Correction Equations can return estimates of its components, QH and QD (see Ref. 6 for a short computer program to do this). The kinetic complexity of enzymatic reactions has been a challenge for chemists interested in pressure effects, as well as isotope effects, for a very long time. Laidler and Bunting7 correctly separated the former into effects on V (Vmax or kcat) and V/K (kcat/Km or substrate capture) and attempted to deal with further complexities by combining them in an obscure composite volume, which did not come into wide usage but was later deconstructed.8 Northrop9 formulated the expression of an intrinsic isotope effect within V=K as follows: D

V=K ¼

D

k þ Cf þ Cr Keq 1 þ Cf þ Cr

ð32:7Þ

where Cf and Cr represent the forward and reverse Commitments to Catalysis, respectively. In order to incorporate the components of Equation 32.6 into a global expression for pressure effects, it was necessary to address the possibility of high pressures causing inactivation or denaturation of enzyme proteins. This was done by adopting the formalism of an iso-mechanism as illustrated in Cleland line diagram of Scheme 32.1:10 This iso-mechanism has two steps in the isomerization segment following the formation of a product form, F: the conversion of F to an intermediate form, G (which could be the release of a second product or a second half reaction), and isomerization of the intermediate back to the substrate form, E. The conversion of G to E is shown twice: first, as a reversible process in the beginning of the diagram which is how it appears to substrate capture at low [S], and second as an irreversible process at the end which is how it appears in kcat at high or infinite [S]. At high pressure, G may stand for an obligate intermediate, as in Scheme 32.1, or any other form or forms that may be coexist with E, including partially unfolded or dissociated enzyme. In this scheme, the forward and reverse commitments to catalysis are equal to k3 =k2 and k4 =k5 ; respectively. Kinetics of pressure effects, a single KIE and the availability of active enzyme within an iso-mechanism

P k5

S k1

k2

KG/E G

E

k3 ES

FP k4

SCHEME 32.1

k7

k9

F → G →E

840

Isotope Effects in Chemistry and Biology

were combined in the following global kinetic expression for substrate capture:10 ! ! ‡ k1 R0 e2DV p=RT lV=Klp ¼ ‡ ‡ 1 þ KG=E e2DVG=E p=RT 1 þ Ci D k 2 1 þ Cf e2DV p=RT þ Cr e2DV p=RT

ð32:8Þ

The first expression in parentheses contains the diffusion-controlled rate constant for collisions between substrate and enzyme, k1, which is least affected by changes in viscosity due to changes in pressure at temperatures near 308C.11 The expression also contains the iso-mechanism equilibrium constant, KG/E, between the form of enzyme, G, that is unable to participate in substrate capture and that which can, form E, governed by the volume change, DVG/E. The numerator of the second term contains a series of ratios of rate constants, R0, equal to k3 k5 =k2 k4 in Scheme 32.1, governed by a single activation volume, DV ‡. The nature of this volume change is illustrated by the displacement of the vertical arrow in the activation energy diagram in Figure 32.1, governed by k1 R0 : It is independent of intermittent changes that occur, in ES for example, because it is a state function, like entropy or equilibria, i.e., twists and turns in a garden hose do not affect the water levels at its two ends. This is the volume difference between free enzyme (or an enzyme –coenzyme complex, see below) and substrate, E þ S, and the TS of an isotopically sensitive step, here EX‡. The denominator of the second term contains the decimal fraction of isotopic labeling, Ci (equal to zero when non-isotopic reactions are being addressed), combined with the intrinsic isotope effect-minusone, together with the forward and reverse commitments. The commitments may change with pressure, most likely as a simple function of DV †, as shown in Equation 32.8, which keeps the number of variables to a minimum despite kinetic complexity. (More volume changes are possible, such as in differences between substrates and products, but are not likely to be resolved within experimental data.) When commitments are very small and the intrinsic isotope effect is fully expressed on substrate capture (which sometimes obtains with poor substrates), Equation 32.8 reduces to: ! ! ‡ k1 R0 e2DV p=RT ð32:9Þ lV=Klp ¼ 1 þ Ci D k 2 1 f 1 þ KG=E e2DVG=E p=RT

DV

Energy

k1Ro EX G+S

FY

ES E+S

FP F+P

Reaction Coordinate

FIGURE 32.1 Activation energy diagram for substrate capture within an iso-mechanism with small commitments to catalysis. The volume change represented by DV ‡ encompasses the portion of the mechanism represented by the vertical arrow. As such it includes any and all volume changes that occur between the limits of E þ S and EX‡ and, being a system function, any volume changes in the solvent associated with the binding of substrate.

Effects of High Hydrostatic Pressure on Isotope Effects

841

Combining hydrogen tunneling with the kinetic complexity of an enzymatic reaction yields the following multiphasic pressure function for hydrogen tunneling in an enzyme reaction:12 lV=Klp ¼

k1 1 þ KG=E e2DVG=E p=RT

!

‡

R0 e2DV p=RT ‡ 1 þ Ci D k D Q 2 1 e2DVQ p=RT þ D k 2 1 þ ðCf þ Cr Þe2DV p=RT

! ð32:10Þ

which in the absence of significant commitments reduces to the triphasic function: lV=Klp ¼

k1 1 þ KG=E e2DVG=E p=RT

!

‡

R0 e2DV p=RT 1 þ Ci D k D Q 2 1 e2DVQ p=RT þ D k 2 1

! ð32:11Þ

Extending this same approach to solvent isotope effects generates the following Equation 32.13: 0

k1 B B 1 þ 0:24C i lV=Klp ¼ B B @ KG=E e2DVG=E p=RT 1þ 1 þ Ci ½D2 O KG=E 2 1

1 ! ‡ C C R0 e2DV p=RT C ‡ C A 1 þ Ci ½D2 O k 2 1 þ ðCf þ Cr Þe2DV p=RT

ð32:12Þ

where the 0.24 term corrects for medium viscosity effects of D2O,13 D2 O K is the solvent isotope effect on an iso-mechanism equilibrium, and D2 O k is an intrinsic solvent isotope effect. When commitments are very small and the intrinsic solvent isotope effect is fully expressed on substrate capture, Equation 32.12 reduces to: 0

k1 B B 1 þ 0:24C i lV=Klp ¼ B B @ KG=E e2DVG=E p=RT 1þ 1 þ Ci ½D2 O KG=E 2 1

1 ! ‡ C C R0 e2DV p=RT C C 1 þ C ½D2 O k 2 1 A i

ð32:13Þ

These equations for substrate capture may look very complex and forbidding, but equivalent expressions for maximal velocities would be much worse. That is because reactive steps beyond the first irreversible step would be added to the kinetic expression and changes in concentrations of intermediates such as ES and EP will contribute to the function. Moreover, the relative contributions of rate constants and intermediates varies within a sequential kinetic mechanism with later steps contributing more quantitatively than earlier ones, and at present we have no way to address this discrepancy.15

III. EXPERIMENTAL EXAMPLES A. HYDROGEN T UNNELING Of the eight reactions with unusually large deuterium isotope effects that were examined under pressure by Isaacs and coworkers, the oxidation of leuco-crystal violet by chloranil (tetrachlorop-benzoquinone) was the most thoroughly documented.14 Their measured reaction rates obtained at 298C in acetonitrile at pressures up to 2 kbar are reproduced graphically in Figure 32.2. The data were fitted to Equation 32.5 to obtain the solid curved line and dashed asymptote, which clearly separates the observed isotope effect into two components, one sensitive to pressure (the curved

842

Isotope Effects in Chemistry and Biology

12

kH/ KD

11 10 9 8 7 0.0

0.5

1.0 1.5 Pressure (kbar)

2.0

2.5

FIGURE 32.2 Deuterium isotope effects on the hydride transfer from chloranil to leuco-crystal violet as a function of hydrostatic pressure. The solid line is a fit to Equation 32.5. The short-dashed line is the asymptote to the fitted curve and represents the semi-classical isotope effect.

line) and one not (the asymptote). The former has a fitted value of DQ ¼ 1.44 ^ 0.02 and the latter D k ¼ 7.8 ^ 0.1. The curve is governed by a volume change of DVQ ¼ 36.5 ^ 3.0 mL/mol, whose origins are a mystery inviting further investigation. Whatever the origin and whether or not this reaction involves true hydrogen tunneling, this exercise shows that the nonclassical component of the isotope effect can not only be detected by pressure studies, it can also be quantified with great precision. Here it accounts for 33 ^ 1% of the observed deuterium isotope effect at atmospheric pressure. Assuming it is in fact hydrogen tunneling, then incorporating the TS effect into a pair of Bell Tunneling Correction Equations allows the precise estimation of the reaction frequency as n ‡H ¼ 797 ^ 12 cm21 with QH ¼ 1.97 ^ 0.05.

B. YEAST A LCOHOL D EHYDROGENASE The oxidation of benzyl alcohol by yeast alcohol dehydrogenase fully expresses a deuterium isotope effect on substrate capture that is suspected of hydrogen tunneling,16 which made it a desirable candidate for high pressure kinetics. Figure 32.3a shows the biphasic effect of pressure expressed on V/K, for both protium and deuterium transfers, and a fit of the data to Equation 32.9.17 Moderate pressure increases the substrate capture of benzyl alcohol by activating the hydride transfer step.18 This means that the TS for hydride transfer has a smaller volume than the free alcohol plus the capturing form of enzyme, Ep-NADþ, with a DV‡H ¼ 2 38 ^ 1 mL/mol. Pressures above 1.5 kbar decrease substrate capture of benzyl alcohol by favoring a conformation of enzyme which binds NADþ less tightly (a G form of enzyme in Scheme 32.1). This means that the ground state for tighter binding, E*-NADþ, has a larger volume than the collision complex, E-NADþ, with a DV * ¼ þ 73 ^ 2 mL/mol. The equilibrium constant of the conformational change is K *eq ¼ 75 ^ 13 at one atmosphere of pressure. Consistent with these assignments are more recent data for the oxidation of isopropanol by YADH which return values for DV * ¼ 2 74 ^ 2 mL/mol and K *eq ¼ 91 ^ 25, respectively, which are not significantly different, nor should they be as they represent nucleotide binding in the absence of an alcohol substrate.19 The activation volume for the hydride transfer step, however, was significantly smaller at DV‡H ¼ 2 29.6 ^ 2.0 mL/mol. In addition, a pressure effect on the spectral properties of the enzyme – NADH complex confirms the conformational assignment.20 Figure 32.3b shows the ratio of pressure effects on protium and deuterium substrate capture of benzyl alcohol by YADH, expressed as monophasic decrease in the deuterium isotope effect. It follows, therefore, that the volume of activation for the TS of deuteride transfer must be even more negative, at DV‡D ¼ 2 49 ^ 1 mL/mol.

Effects of High Hydrostatic Pressure on Isotope Effects

843

1000

V/K M−1sec−1)

800

(a)

600 400 200 0 5

3

D

(V/K )

4

2 1 (b)

0.0

0.5

1.0 1.5 2.0 Pressure (kbar)

2.5

FIGURE 32.3 Effect of pressure on the capture of benzyl alcohol (†) and dideutero benzyl alcohol (B) by

yeast alcohol dehydrogenase and on deuterium isotope effects (O).

The fit of the data in Figure 32.3 to Equation 32.9 also returned a value for the semiclassical isotope effect arising from vibrational differences in a reactant state not significantly different from one. p Instead, all of the isotope effect apparently arises from a TS phenomenon, expressed as DQ ¼ 4.99 ^ 0.37 with an apparent volume change of DVQ ¼ 10.4 ^ 1.5 mL/mol. These numbers match a QH¼11.7 ^ 1.1 with a reaction frequency of v‡H ¼ 1220 ^ 11.19 Recently, the effect of pressure on the oxidation of 13C-labeled benzyl alcohol has been measured using the competitive method (unpublished results). The result is strikingly similar to the data in Figure 32.3b, namely a monophasic decrease in the isotope with increasing pressure approaching an asymptotic value not significantly different from one, denoting the absence of a semiclassical isotope effect originating in differences in zero-point vibrational frequencies, with values of 13 Q12 ¼ 1.028 ^ 0.001 and DVQ ¼ 13.1 ^ 0.7 mL/ mol. The similarities in the values of DVQ when an extra mass unit to either end of the scissile C –H bond strongly suggest that the two measured volume changes have a common origin. Given that carbon can undergo very little tunneling, if any, these data argue against tunneling as being the origin of the pressure effect on deuterium isotopes effect of YADH. The effect of pressure on the substrate capture of benzyl alcohol by yeast alcohol dehydrogenase was measured also in D2O and compared to the protium data in Figure 32.3a using Equation 32.12.21 The result is a smooth sigmoidal transition from a small inverse kinetic solvent p Whether the semi-classical isotope effect was really insignificant had been questioned because the regression of data in Figure 32.3B required, in effect, a rather large extrapolation to abolish D(V/K). However, similar experiments with ethanol as a substrate produced D(V/K) ¼ 2.67 at ambient pressure and attained an experimental value of D(V/K) ¼ 1.03 at less than 1 kbar.20

844

Isotope Effects in Chemistry and Biology

isotope effect of D2 O k ¼ 0.901 ^ 0.075 to a larger equilibrium effect of D2 O K eq ¼ 0.38 ^ 0.04. The former provides evidence in support of a proposed mechanism in which the breaking of a low barrier hydrogen bond is concerted with hydride transfer22 while the latter is an effect on the E*-NADþ O E-NADþ equilibrium that provides evidence in support of the proposal that torsional motions of exchangeable protons are restricted when enzymes “crunch” down on substrates.23

C. YEAST F ORMATE D EHYDROGENASE Experiments similar to those in Figure 32.3a were performed with yeast formate dehydrogenase and produced similar biphasic curves, characterized by DV‡H ¼ 2 9.3 ^ 1.1 mL/mol, DV * ¼ 2 84 ^ 6 mL/mol, K *eq ¼ 1000 ^ 500 mL/mol and Dk ¼ 1.02 ^ 0.03.24 The isotope effect on substrate capture was again solely derived from a TS phenomonon, with DQ ¼ 2.73 ^ 0.20 and D k ¼ 1.02 ^ 0.03; however, in this instance it increased slightly with increasing pressure, dependent upon an apparent volume change of DVQ ¼ 2 2.5 ^ 2.2 mL/mol. This TS phenomenon is inconsistent will all current models of hydrogen tunneling.

IV. CONCLUSION Apparent isotope effects can have multiple origins. High hydrostatic pressures can in some instances separate these origins from one another and quantify them. Combining pressure effects with isotope effects in the study of reactions with multiple steps, as in enzymatic reactions, makes it possible to assign measured activation volumes to the isotopically sensitive step. Apparent pressure effects can also have multiple origins, and the presence of an isotope effect can distinguish between volume changes associated with isotopic versus non-isotopic steps. The few examples presented show that it is possible to study isotope effects as a function of pressures that are readily accessible in the laboratory, and to fit the resulting data to reasonable models of kinetic mechanisms with considerable precision. Retrieving quantitative physical data from TS, in the form of activation volumes, opens the door to new forms of structure/activity relationships.

REFERENCES 1 Northrop, D. B., Effects of high pressure on enzymatic activity, Biochem. Biophys. Acta, 1595, 71 – 79, 2002. 2 Glastone, S., Laidler, K. J., and Eyring, H., The Theory of Rate Processes, McGraw-Hill, New York, 1941. 3 Isaacs, N. S., The effect of pressure on kinetic isotope effects, In Isotope Effects in Organic Chemistry, Vol. 6, Buncel, E. and Lee, C. C., Eds., Elsevier, Amsterdam, pp. 67 – 105, 1984. 4 Bigeleisen, J. and Wolfsberg, M., Theoretical and experimental aspects of isotope effects in chemical kinetics, Adv. Phys. Chem., 1, 15 –76, 1958. 5 Bell, R. P., The Tunnel Effect in Chemistry, Chapman and Hall, London and New York, 1980. 6 Northrop, D. B., Effects of high pressure on isotope effects and hydrogen tunneling, J. Am. Chem. Soc., 121, 3521– 3524, 1999. 7 Laidler, K. L. and Bunting, P. S., The Chemical Kinetics of Enzyme Action, 2nd ed., Clarendon, Oxford, pp. 220– 232, 1973. 8 Northrop, D. B., Steady-state kinetics at high pressure, In Effects of High Pressure on Molecular Biology and Enzymology, Markely, J. L., Royer, C. A., and Northrop, D. B., Eds., Oxford University Press, New York, pp. 122– 241, 1996. 9 Northrop, D. B., Determining the absolute magnitude of hydrogen isotope effects, In Isotope Effects on Enzyme-Catalyzed Reactions, Cleland, W. W., O’Leary, M. H., and Northrop, D. B., Eds., University Park Press, Baltimore, pp. 122– 152, 1977. 10 Cho, Y.-K. and Northrop, D. B., Effects of pressure on the kinetics of capture by yeast alcohol dehydrogenase, Biochemistry, 38, 7470– 7475, 1999.

Effects of High Hydrostatic Pressure on Isotope Effects

845

11 Bett, K. E. and Cappi, J. B., Effect of pressure on the viscosity of water, Nature, 207, 620– 621, 1965. 12 Cho, Y.-K. and Northrop, D. B., Effect of pressure on deuterium isotope effects of yeast alcohol dehydrogenase: evidence for mechanical models of catalysis, Biochemistry, 39, 2406– 2412, 2000. 13 Northrop, D. B. and Cho, Y.-K., Effects of high pressure on solvent isotope effects of yeast alcohol dehydrogenase, Biophys. J., 79, 1621– 1628, 2000. 14 Isaacs, N. S., Javaid, K., and Rannala, E., Reactions at high pressure. Part 5. The effect of pressure on some primary kinetic isotope effects, J. Chem. Soc., Perkin, II, 709– 711, 1978. 15 Northrop, D. B., Uses of isotope effects in the study of enzymes, Methods: A Companion to Methods in Enzymology, Vol. 24, pp. 117– 124, 2001. 16 Cha, Y., Murray, C. J., and Klinman, J. P., Hydrogen tunneling in enzymic reactions, Science, 243, 1325– 1330, 1989. 17 Northrop, D. B. and Cho, Y.-K., Effect of pressure on deuterium isotope effects of yeast alcohol dehydrogenase: evidence for mechanical models of catalysis, Biochemistry, 39, 2406– 2412, 2000. 18 Cho, Y.-K. and Northrop, D. B., Effects of pressure on the kinetics of capture by yeast alcohol dehydrogenase, Biochemistry, 38, 7470– 7475, 1999. 19 Northrop, D. B. and Cho, Y.-K., Effects of high pressure on solvent isotope effects of yeast alcohol dehydrogenase, Biophys. J., 79, 1621– 1628, 2000. 20 Park, H., Kidman, G., and Northrop, D. B., Effects of pressure on deuterium isotope effects of yeast alcohol dehydrogenase using alternative substrates, Arch. Biochem. Biophys, 433, 335–340, 2005. 21 Kidman, G., Northrop, D.B., Effect of pressure on nucleotide binding to yeast alcohol dehydrogenase, Prot. Pept. Lett., 12, 495–497, 2005. 22 Ramaswamy, S., Park, D. H., and Plapp, B. V., Substitutions in a flexible loop of horse liver alcohol dehydrogenase hinder the conformational change and unmask hydrogen transfer, Biochemistry, 38, 13951– 13959, 1999. 23 Cleland, W. W., The use of isotope effects in the detailed analysis of catalytic mechanisms of enzymes, Bioorg. Chem., 15, 283– 302, 1987. 24 Quirk, D. J. and Northrop, D. B., Effect of pressure on deuterium isotope effects of yeast formate dehydrogenase, Biochemistry, 40, 847– 851, 2001.

33

Solvent Hydrogen Isotope Effects in Catalysis by Carbonic Anhydrase: Proton Transfer through Intervening Water Molecules David N. Silverman and Ileana Elder

CONTENTS I.

Introduction ...................................................................................................................... 847 A. Catalytic Mechanism................................................................................................ 848 B. Structure ................................................................................................................... 849 C. Relevance of Ordered Water in Crystal Structures of Carbonic Anhydrase ............................................................................................ 850 II. Isotope Effects on First Stage of Catalysis — The Hydration of CO2 ........................... 850 III. Solvent Hydrogen Isotope Effects on the Proton Transfer Steps ................................... 850 A. Intramolecular Proton Transfer in Catalysis by Carbonic Anhydrase.................... 850 1. Proton inventory ................................................................................................ 850 2. Interpreting the Isotope Effects......................................................................... 851 3. Theorists View of Isotope Effects in Catalysis by Carbonic Anhydrase......... 851 4. Use of 18O Exchange ........................................................................................ 852 5. Marcus Rate Theory Allows Enhanced Interpretation of Proton-Transfer Rates ................................................................................... 853 6. Marcus Plot for Intramolecular Proton Transfer .............................................. 854 7. Marcus Plot for Solvent Hydrogen Isotope Effects.......................................... 855 B. Intermolecular Proton Transfer in Catalysis by Carbonic Anhydrase.................... 856 1. Marcus Plot for Solvent Hydrogen Isotope Effects.......................................... 856 2. The Marcus Formalism Extended to the b and g Classes ............................... 857 VI. Conclusions ...................................................................................................................... 857 References..................................................................................................................................... 857

I. INTRODUCTION The seminal study of isotope effects in the catalysis of CO2 hydration by carbonic anhydrase laid the framework not only for many future studies of isotope effects in this pathway but also for an isotopic approach to understanding proton transfer through water bridges. This was the 1975 report of Steiner, Jonsson, and Lindskog1 demonstrating the solvent hydrogen isotope effect of 3.8 on the maximal velocity of catalysis by carbonic anhydrase, an isotope effect that was correctly interpreted 847

848

Isotope Effects in Chemistry and Biology

as arising from an intramolecular proton transfer. The current review expands on this and related investigations and is centered on understanding proton transfers through intervening water molecules in a protein environment. Detailed crystal structures are now available for very complex systems in which proton transport through distances plays a significant role, such as bacteriorhodopsin, cytochrome c oxidase, and the bacterial reaction center. In these cases the ˚ and involve intervening water and amino-acid proton-transfer pathway may extend up to 15 or 20 A residues, a tortuous pathway in which often the identity of the proton acceptor and donor may be in doubt. Catalysis by carbonic anhydrase is a very useful model to study such processes since the identity of the proton donor and acceptor is clear, the rate constant for proton transfer limits maximal velocity, and the pathway involves intervening water. The crystal structures of carbonic anhydrase show ordered water between the proton donor and acceptor, often in hydrogen-bonded chains although the relevance of this to proton transfer is not known. Moreover, there is a very broad range of isozymes of the carbonic anhydrases (CAs) p to work with including three classes of structurally unrelated carbonic anhydrases, examples of convergent evolution. The enzymes in these classes are all zinc-metalloenzymes but bear no amino-acid homologies between classes2: the a class which includes the animal and human CAs, the b class which includes the plant and many bacterial CAs, and the g class which includes an archaeal CA. The catalytic mechanism of the carbonic anhydrases has been the subject of many reviews.3 – 8 Carbonic anhydrases carry out several major physiological functions including: (1) rapid 2 conversion of stored HCO2 3 into CO2, as in red cells and in photosynthesis; (2) production of HCO3 in formation of secretory fluids such as ocular and cerebrospinal fluids; (3) production of Hþ from water and CO2 in renal acidification of urine and gastric acid secretion; and (4) facilitation of diffusion of CO2 across membranes as in the lens of the eye. These topics are reviewed in Ref. 9.

A. CATALYTIC M ECHANISM There is a wide body of evidence supporting a two-stage catalysis (ping-pong or iso-mechanism) for the a CA isozymes,3 – 10 as well as for the b CA from Arabidopsis thaliana11 and the g CA from Methanosarcina thermophila.12 The first stage shown in Equation 33.1 is the interconversion of CO2 and HCO2 3 , occurring most likely by nucleophilic attack of zinc-bound hydroxide on CO2: ½H2 O

2 CO2 þ EZnOH2 O EZnHCO2 3 O EZnH2 O þ HCO3

ð33:1Þ

The second stage of catalysis is the proton-transfer sequence that converts the enzyme back to the active zinc-hydroxide form, as shown in Equation 33.2: ½B

EZnH2 O O Hþ EZnOH2 Oþ EZnOH2 ½BH

ð33:2Þ

Here B is a proton acceptor (buffer) in solution and Hþ written before E indicates a protonated shuttle residue. The intramolecular proton transfer step from EZnH2O to HþEZnOH2, close to 106 s21 for human carbonic anhydrase II (HCA II), is rate limiting for the maximal velocity of catalysis.1,3 The steady-state rate constant kcat =Km is close to diffusion controlled at 108 M21 s21. There is a convincing set of data spanning over 20 years to indicate that His 64 shuttles protons between solution and the zinc-bound water in HCA II.3 This was confirmed by site-directed mutagenesis.13 The rate of proton transfer in Equation 33.2 was decreased as much as 50-fold for the mutant H64A HCA II while the rate of interconversion of CO2 and HCO2 3 (Equation 33.1) was the same for mutant and wild-type enzymes.13 The shuttle role of His 64 was further supported in p Abbreviations: CA, carbonic anhydrase; HCA II, isozyme II of human carbonic anhydrase; SHIE, solvent hydrogen isotope effect; ATCA, Arabidopsis thaliana carbonic anhydrase; MTCA, Methanosarcina thermophila carbonic anhydrase.

Solvent Hydrogen Isotope Effects in Catalysis by Carbonic Anhydrase

849

the same work by chemical rescue of the mutant H64A by the addition of imidazole in solution. Moreover, when the naturally occurring Lys 64 in the slower isozyme III was replaced by His there was an activation as large as ten-fold in activity; this and related mutants of HCA III then had properties such as pH-rate profile and solvent hydrogen isotope effects similar to HCA II.14,15

B. STRUCTURE The crystal structures of the isozymes of the carbonic anhydrases in the a class reported to date (isozymes I, II, III, IV, V, XII) are very similar.16 – 18 Among CA isozymes in the a class, the work ˚ resolution for H64A HCA II.19 of McKenna and colleagues has refined crystal structures to 1.05 A These isozymes have molecular mass near 30 kDa and contain a dominant 10-stranded, twisted b pleated sheet from which emerge three histidine ligands of the zinc (Figure 33.1). The active-site ˚ and with the zinc located cavity is conical with a diameter at the surface of the protein of about 15 A ˚ at the bottom about 15 A from the surface. The geometry about the zinc is tetrahedral with three histidine ligands and a fourth solvent ligand, which has a pKa from about 5 to 7 depending on the isozyme. This aqueous ligand of the zinc is a hydrogen-bond donor to the hydroxyl side chain of Thr 199 which in turn is a hydrogen bond donor to the carboxylate of Glu 106.16 This conformation has been suggested to orient the unpaired, reactive orbitals of the zinc-bound hydroxide for nucleophilic attack on CO2.20 The proton shuttle residue His 64 in isozyme II has its side chain in two orientations, as determined by crystallography;21 one is oriented with the imidazole ring pointed toward the metal and the second pointed out into the opening of the active-site cavity (Figure 33.1). It has been suggested that conformational flexibility is a requirement for the side chain of an effective proton shuttle residue, as also demonstrated in a chemically modified CA V.22 It is interesting that the glutamate residue identified as a proton shuttle in the carbonic anhydrase of the g class23 also shows two orientations in crystal structures.24 A crystal structure for the methanoarchaeon CA in the g class Methanosarcina thermophila at ˚ is reported.24 Although this archaeal CA has no sequence homology with the CAs in the 1.46 A a class, it does have its zinc coordinated to three histidine ligands in an approximately tetrahedral geometry, which is similar to the a CAs, although other features of its active site are very different.

T199

T200

W1

H64 "in" "out"

OH−

W3A Zn

W2 W3B

H94

H96

N62 N67

FIGURE 33.1 Active site of human carbonic anhydrase II showing the zinc ion (large sphere) coordinated to His 94, His 96, and His 119. Proton shuttle residue His 64 is shown in the “in” and “out” conformations. Shown as small spheres are four water molecules, W1, W2, W3A, and W3B extending from the zinc-bound water to the His 64. Residues Thr 199 and Thr 200 are involved in hydrogen bonding to W1 and the zinc-bound water.

850

Isotope Effects in Chemistry and Biology

There are currently four crystal structures for b CAs reported, all very similar to each other;25 for the b CAs the ligands of the zinc are two cysteines and a histidine.

C. RELEVANCE OF O RDERED WATER IN C RYSTAL S TRUCTURES OF C ARBONIC A NHYDRASE The N 1 2 of the imidazole side chain of His 64 when oriented toward the zinc in HCA II is too far ˚ ; Ref. 21) for direct proton transfer. The proton transfers that regenerate from the zinc (about 7.5 A active enzyme (Equation 33.2) must occur by proton transfer through intervening hydrogen-bonded water bridges. The crystal structures of CA show up to 15 water molecules apparently ordered within the active-site cavity. In the crystal structures of HCA II, there are two or three ordered water molecules between the zinc-bound solvent and the side chain of His 64 (Ref. 16; Fisher, Hernandez, and McKenna, personal communication) (Figure 33.1). Although these water molecules are hydrogen-bonded to each other, the distance between the distal H2O (W2, W3A, or W3B of ˚ to 3.6 A ˚ is too distant for substantial Figure 33.1) and the imidazole ring of His 64 at about 3.3 A hydrogen bonding. The relevance of these ordered water structures to the kinetics of proton transfer in the catalysis is uncertain.

II. ISOTOPE EFFECTS ON FIRST STAGE OF CATALYSIS — THE HYDRATION OF CO2 The solvent hydrogen isotope effect (SHIE) determined for the steps of Equation 33.1 is unity, measured for kcat =Km for hydration of CO2 by stopped-flow spectrophotometry,1,26 for 18O exchange 13 28 between CO2 and water,27 and for the interconversion of CO2 and HCO2 3 by C NMR. Also, the SHIE on the carbon isotope effect for the carbonic anhydrase catalyzed dehydration of bicarbonate is unity.29 These results strongly suggest that the zinc-bound hydroxide is the catalytic entity in a direct nucleophilic attack rather than a general base mechanism. This conclusion also relies on the structure of a bicarbonate – CA complex showing bicarbonate bound to the metal.30,31

III. SOLVENT HYDROGEN ISOTOPE EFFECTS ON THE PROTON TRANSFER STEPS A. INTRAMOLECULAR P ROTON T RANSFER IN C ATALYSIS BY C ARBONIC A NHYDRASE The SHIE on the proton-transfer steps in catalysis by CA II is near 3.8 measured by stopped flow.1,26 Since this was not a buffer-dependent effect, Steiner et al.1 suggested an intramolecular proton transfer in the second stage of catalysis (Equation 33.2) that is rate limiting for kcat. The extent to which the intramolecular proton transfer is rate limiting for overall catalytic turnover is estimated to be about 43% based on inhibition of catalysis,32 about 64% based on isotopic induced transport in the Britton experiment,33 and 95% based on simulations of the catalysis.34 1. Proton inventory Venkatasubban and Silverman35 measured the SHIE on kcat for CO2 hydration catalyzed by bovine CA II as a function of the deuterium content of the solvent. These experiments were done in the presence of excess buffer to assure that the intramolecular proton transfer step was rate limiting. Plots of kcat against deuterium content of solvent were clearly nonlinear and bulging down; moreover, there was an exponential dependence of kcat on the atom fraction of deuterium in solvent. The most straightforward interpretation of this result is that more than one proton is in motion in the transition state. The Gross – Butler equation was applied to conclude that the data were marginally consistent with two protons transferring, and were fit best with the transfer of three or

Solvent Hydrogen Isotope Effects in Catalysis by Carbonic Anhydrase

851

more protons. This was the first case of an enzymic process for which there was a logarithmic relationship between kcat and the atom fraction of deuterium in solvent. Such behavior is expected for a multiproton mechanism such as proton transfer through water bridges in which many protons each contribute a small but approximately equal normal isotope effect.36 It is interesting that identical results of a SHIE on kcat near 3.8 and an exponential dependence of kcat on atom fraction of deuterium in solvent was obtained in the case of the mutant H64A HCA II activated for proton transfer by the presence of 100 mM imidazole.37 Moreover, the activation of H64A HCA II catalyzed hydration of CO2 by imidazole resulted in values of kcat nearly as great as in the wild-type enzyme. Clearly, the mechanism of proton transfer is similar in wild-type HCA II and in H64A HCA II activated by the exogenous proton acceptor imidazole. It should be noted that these proton inventory results are also consistent with the possibility that the observed isotope effects are caused entirely or in part by solution changes in the active site which occur in the rate-determining step; that is, a nonlinear proton inventory can be due to accumulated medium effects.38 However, the nonlinear proton inventory was initially interpreted within the context of the proposal of Steiner et al.1 and is also consistent with rate-limiting proton transfer through water bridges. 2. Interpreting the Isotope Effects Kassebaum and Silverman39 estimated the H/D fractionation factor of the aqueous ligand of cobalt in Co(II)-substituted bovine CA II; this allows further comment on the SHIE observed in catalysis by carbonic anhydrase. This fractionation factor was measured for the rapidly exchanging aqueous ligands of the cobalt determined from the 1H NMR relaxation rates of water in solutions of enzyme in H2O and D2O. The fractionation factor f is 0.77 ^ 0.01 for the high pH form of the aqueous ligand of the cobalt in CA II. Assuming a stepwise proton transfer and a transition state that involves formation of a hydronium ion (fractionation factor f ¼ 0:69; Ref. 36) and using the above estimate for the aqueous ligand of the metal in its high pH form, we can obtain a value of the overall isotope effect for the proton transfer by use of the Gross – Butler equation36: ðkcat Þn ¼ ðkcat Þ0 ð1 2 n þ 0:77nÞð1 2 n þ 0:69nÞ3

ð33:3Þ

Here n is the atom fraction of deuterium in solvent. This approach assumes that the fractionation factor of water in the active-site cavity is unity, which is the fractionation factor of water in bulk solvent. The result of ðkcat Þ0 =ðkcat Þ1 ¼ 3:9 for n ¼ 1:0 in Equation 33.3 is in good agreement with the observed value near 3.8. Such a calculation is also consistent with the exponential dependence of the SHIE on n. This agreement may be entirely fortuitous, however. Cui and Karplus40 have made density functional theory calculations on a CA II model system to show that proton transfer through a hydronium ion intermediate is a much less favorable event than a concerted proton transfer with no such intermediate. 3. Theorists View of Isotope Effects in Catalysis by Carbonic Anhydrase There have been many publications by theorists on the mechanism of catalysis of carbonic anhydrase. These have included work on the function of residues in the active site and the 20,41,42 mechanism of the CO2/HCO2 and on proton transfer in the catalysis.43,44 3 interconversion 45 The theoretical study of Smedarchina et al. has approached an interpretation of the solvent hydrogen isotope effects in catalysis by carbonic anhydrase. This is a density functional theory approach to the rate-limiting proton transfer steps based on a model of the active site including the zinc and its ligands (modeled by three methylimidazole molecules), a methylimidazole molecule representing His 64 the position of which is based on the “in” conformation,21 and a bridge of two water molecules connecting the zinc-bound water through hydrogen bonds to His 64. The proton transfer from the zinc-bound water to His 64 was found to be concerted and accompanied by

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quantum mechanical tunneling through a single barrier.45 Work of Cui and Karplus40 showed that introducing a third or fourth water molecule in the bridge promoted stepwise proton transfer rather than concerted, and that such stepwise transfers proceeded through intermediates with hydronium ion character representing pathways of much larger energy barriers than the concerted pathway involving two water molecules. The calculated SHIE associated with the concerted process involving tunneling was small mainly because of strong coupling to the skeletal motions of the water bridge.45 That is, phase coherence of the heavy atoms in the water bridge provides the coupling component that promotes tunneling and reduces the kinetic isotope effect. In contrast, the SHIE calculated by classical transition-state theory was predicted to be much larger due to the cumulative effect of zero-point energies on the three protons in motion. In the quantum mechanical approach, the calculated rate constant for proton transfer near 109 s21 is three orders of magnitude greater than that observed. This indicates that the calculated proton transfer is preceded by an unfavorable pre-equilibrium; that is, the configuration of water molecules effective in proton transfer is a high energy structure, present at very low concentrations.45 This approach simulates rather accurately not only the maximal SHIE of 3.8 on the maximal velocity of catalyzed CO2 hydration but also the exponential dependence of the SHIE on the atom fraction of deuterium in solvent reported by Venkatasubban and Silverman.35 4. Use of

18

O Exchange

The main advantage of measuring the exchange of 18O between CO2 and water (compared with stopped-flow, for example) is that the technique is performed at chemical equilibrium where maintaining constant pH is not a problem; therefore, 18O-exchange experiments can be run in the absence of buffers or at very large concentrations of buffers. This is an advantage because buffers can act as proton donors and acceptors in the catalysis46 and can mask the intramolecular proton transfer.13 Another advantage is that this method allows the measurement of the rate of protontransfer steps in catalysis. In this method, the rate of exchange of 18O between CO2 and water and the rate of exchange of 18O between 12C- and 13C-containing CO2 is measured by mass spectrometry.47 Two rates can be determined by this method. The first is the rate of interconversion of CO2 and HCO2 3 at chemical equilibrium (Equation 33.4). The second we designate RH2 O , the rate of release of H18 2 O from the enzyme (Equation 33.5). RH2 O is limited in rate by the proton transfer steps from donor groups to the zinc-bound labeled hydroxide. HCOO18 O2 þ EZnH2 O O EZn18 OH2 þ CO2 þ H2 O þ

H2 O

H – His64· · ·EZn18 OH2 O His64· · ·EZn18 OH2 O His64· · ·EZnH2 O þ H2 18 O

ð33:4Þ ð33:5Þ

Jewell et al.14 replaced Lys 64 with His in HCA III and observed enhanced 18O-exchange activity through proton transfer. Moreover, this activation by His 64 showed an apparent pKa of 7 with a maximum at low pH consistent with proton transfer from the imidazolium ring of His 64 to the zincbound hydroxide. Experiments also showed that replacements of Phe 198, located on a side of the active-site cavity opposite His 64, had a significant influence on the pKa of the zinc-bound water.48 Moreover, observing the catalysis of CO2 hydration, Tu et al.49 determined that the changes in kcat =Km caused by the replacement of Phe 198 with other residues had, in most cases, a simple additive effect on the changes in kcat =Km caused by replacement of Lys 64. That is, these are noninteracting sites. Silverman et al.15 used a series of mutants containing His 64 with additional replacements at Arg 67 and Phe 198 to measure the rate constant for intramolecular proton transfer; these mutants had values of the pKa of the zinc-bound water from 5 to 9. The rate constants for proton transfer from His 64 to the zinc-bound hydroxide obeyed a Brønsted correlation showing sharp curvature characteristic of a facile proton transfer (Figure 33.2). This suggests an unfavorable

Solvent Hydrogen Isotope Effects in Catalysis by Carbonic Anhydrase

853

6 6

4

9

8 2

log (k B)

5

7,1

5

Intramolecular 1 K64H 2 K64H-R67N 3 K64H-F198L 4 K64H-R67N-F198L 5 K64H-F198D 6 K64H-R67N-F198D

3

4

Intermolecular 7 wild-type + imidazole 8 R67N + imidazole 9 F198D + imidazole

a b 3 −6

a wild-type b K64A

−4

−2 ∆pk a

0

2

FIGURE 33.2 The logarithm of the rate constant for intramolecular proton transfer kB (s21) from His 64 or exogenous imidazole to the zinc-bound hydroxide in the mutants of human carbonic anhydrase III listed on the right. The abscissa is DpKa ¼ pKaðZnH2 OÞ 2 pKaðdonorÞ in which the donor group is His 64 or imidazole added to solution at 258C. The solid line is a least squares fit to the Marcus theory of Equation 33.6 with the intrinsic kinetic barrier and work functions given in Table 33.1. (From Silverman, D. N., Tu, C. K., Chen, X., Tanhauser, S. M., Kresge, A. J., and Laipis, P. J., Biochemistry, 32, 10757– 10762, 1993. With permission.)

pre-equilibrium to explain why the observed rate constant for proton transfer is not greater than 106 s21. Marcus rate theory was used to quantitate this effect. 5. Marcus Rate Theory Allows Enhanced Interpretation of Proton-Transfer Rates The value of Marcus theory is that it describes proton transfer for a series of homologous reactions that can be compared with other proton-transfer systems, and that it allows the separation of the kinetic and thermodynamic parts of the observed activation energy. Marcus theory was originally applied to electron transfers but appears to be widely applicable to proton transfer reactions between small molecules in solution.50,51 The standard free energy of reaction with the active-site conformation optimal for proton transfer is DGoR ; with the measured overall free energy for the reaction given by DGo ¼ wr þ DGoR 2 wp : Here wr is the work of bringing the active site into the configuration that allows proton transfer; this work term for dehydration is possibly related to the energy required to align acceptor, donor, and intervening hydrogen-bonded water for facile proton transfer. The energy wp is this work term for the corresponding reorganization for the hydration direction. The observed overall activation barrier for the proton transfer DG‡ is given in Marcus theory by Equation 33.6 which relates this to an intrinsic energy barrier DG‡o ; which is the value of DG‡ 2 wr when DGoR ¼ 0:7,51 This intrinsic barrier DG‡o in other notation is referred to as the reorganization energy l=4: DG‡ ¼ wr þ ð1 þ DGoR =4 DG‡o Þ2 DG‡o

ð33:6Þ

Utilization of Equation 33.6 assumes that the work terms w r and w p as well as the intrinsic energy barrier DG‡o do not vary for proton transfer between the series of homologous proton donors and acceptors to which the equation is fit. Despite much work by both experimentalists and theorists, the significance of the term w r is not clear for proton-transfer reactions between small molecules.7,52 A major challenge is to determine what factors contribute to the intrinsic kinetic barrier and work terms in an enzymatic proton

854

Isotope Effects in Chemistry and Biology

transfer. The application of the Marcus theory to intramolecular proton transfer in CAs (see for example, Kresge and Silverman7) is the most complete application to date of Marcus theory to an enzymic proton transfer. 6. Marcus Plot for Intramolecular Proton Transfer The free-energy plot of the rate constants for proton transfer kB from His 64 to the zinc-bound hydroxide for a series of mutants of HCA III containing His 64 as well as mutations at residues 67 and 198 showed rather sharp curvature (Figure 33.2). The rate constant kB was determined from the 18 O exchange rate constant RH2 O /[E]. The plot is extended in the region of low DpKa by the inclusion of points representing wild type CA III and K64A CA III; these mutants have no apparent proton donors in the active site cavity but appear to lie on the curve which is a fit of all of the data to the Marcus rate theory (Equation 33.6). This free-energy plot represents proton transfer between nitrogen and oxygen acids and bases (that is, His 64 and zinc-bound hydroxide), and the value of the intrinsic kinetic barrier DG‡o ¼ 1.4 ^ 0.3 kcal/mol obtained from this plot15 is similar to the value near 2 kcal/mol obtained for nonenzymic, bimolecular proton transfers between nitrogen and oxygen acids and bases in solution.50 It is clear that the intramolecular proton transfer in CA III is dominated by a large work function w r ¼ 10.0 ^ 0.2 kcal/mol for the dehydration direction and 5.9 ^ 1.1 in the hydration direction (Table 33.1). These results were not altered significantly by the omission in Figure 33.2 of the data for wild-type and K64A CA III nor were they altered by the omission of the three points for the intermolecular proton transfer to imidazole. The interpretation of Figure 33.2 and the data in Table 33.1 were obtained by application of the Marcus formalism of Equation 33.6 obtained for a two-state system described by two intersecting parabolas representing the energetics of the reactants, products, and transition state51. A different

TABLE 33.1 Marcus Theory Parameters for Proton Transfer in Isozymes of Carbonic Anhydrase System

Proton Donor

DG ‡o (kcal/mol)

w r (kcal/mol)

w p (kcal/mol)

CA III

His 64a His 67b Glu or Asp 64c His 64 (from SHIE)d

Intramolecular 1.4 ^ 0.3 1.3 ^ 0.3 2.2 ^ 0.5 1.3 ^ 0.3

10.0 ^ 0.2 5.9 ^ 1.1 10.9 ^ 0.1 5.9 ^ 1.1 10.8 ^ 0.1 4.0 ^ 1.6 ðwr 2 wp ¼ 0:6 ^ 0:5Þ

CA V CA II H216N ATCA E84A MTCA Nonenzymic

Bufferse Buffers (from SHIE)f Buffersg Buffersg Buffer to Bufferh

Intermolecular 0.8 ^ 0.5 0.6 ^ 0.5 0.34 ^ 0.06 0.3 ^ 0.1 2.0

10.0 ^ 0.2 8.2 ^ 1.0 ðwr 2 wp ¼ 0:9 ^ 0:5Þ 8.4 ^ 0.4 10.1 ^ 0.1 10 ^ 1 11 ^ 1 3.0

a

Silverman et al.15 Ren et al.66 c Tu et al.13 d Silverman et al.15 SHIE is the solvent hydrogen isotope effect. Independent measurements of w r and w p cannot be made by this method, but the results yield the difference wr 2 wp ¼ 0:6 ^ 0:5 kcal=mol: e Earnhardt et al.63 f Taoka et al.37 g Tu et al.64 h Kresge.50 b

Solvent Hydrogen Isotope Effects in Catalysis by Carbonic Anhydrase

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approach has been taken by Warshel and colleagues42,53 considering a three-state model treated by empirical valence bond calculations for the proton transfer from His 64 to a bridging water molecule to the metal-bound hydroxide. The results of this analysis are very different than those using the Marcus relationship of Equation 33.6. This three-state model and valence bond approach resulted in an intrinsic kinetic barrier near 20 kcal/mol.53 Moreover the work terms w r and w p are negligible, perhaps more consistent with an enzyme in which the proton donor and acceptor are rather fixed in the active-site cavity. Another feature of this valence bond approach is that the apparent flattening of the free-energy plot of Figure 33.2 at DpKa . 0 is not due to approaching the Marcus inverted region, but due to changes in the free energy of intermediate states in the threestate model.53 7. Marcus Plot for Solvent Hydrogen Isotope Effects It was recognized by Melander54 and Westheimer55 that a maximal SHIE in proton-transfer limited reactions within a series of homologous acceptors and donors would be observed between sites for which DpKa is close to zero. This has been verified in nonbiological systems in which a plot of the isotope effect against DpKa is bell shaped and narrow with a maximum near DpKa zero.56,57 This observation is attributed to the expected position of the proton in the transition state midway between proton acceptor and donor for DpKa at zero, and hence most susceptible to slower motion by deuteron substitution. This would be expected to extend to intramolecular transfer in a protein, and was observed in the case of proton transfer from His 64 in mutants of human CA III measured by 18O exchange (Figure 33.3). The Marcus theory describes the properties of a deuteron transfer as well as a proton transfer, and hence is applicable to the interpretation of the SHIE.58,59 Application of the Marcus theory to the SHIE for the proton transfer is based on the assumption that the SHIE appears in the intrinsic kinetic barrier rather than in the work functions, since the work functions presumably do not describe the proton transfer but the processes of bringing the reactants together into a reaction complex.58 Application of Marcus theory to the SHIE of Figure 33.3 yields DGo‡ of 1.3 ^ 0.3 kcal/mol (Table 33.1).15 Although using the SHIE cannot give independently both wr and wp , it does give their difference.58 For the data of Figure 33.3, wr 2 wp ¼ 0:6 ^ 0:5 kcal/mol, which is not in good agreement with the value of this difference near 4 kcal/mol determined by the rate constant 5 4 4 (k B)H2O (k B)D2O

3 3 6 2

9

7

5

1 0 −4

−2

0 ∆pK a

2

FIGURE 33.3 Solvent hydrogen isotope effects on the rate constants for proton transfer in the mutants of human carbonic anhydrase III numbered in Figure 33.2 as a function of DpKa ¼ pKaðZnH2 OÞ 2 pKaðdonorÞ : The solid line is a fit to the Marcus equation describing isotope effects, Equation 10 of Kresge et al.,58 with parameters given in Table 33.1. (From Silverman, D. N., Tu, C. K., Chen, X., Tanhauser, S. M., Kresge, A. J., and Laipis, P. J., Biochemistry, 32, 10757– 10762, 1993. With permission.)

856

Isotope Effects in Chemistry and Biology

for proton transfer (Table 33.1). This may indicate a failure in the approach to explaining these data, or may result from accumulated experimental uncertainties. The observation that the maximum in the observed SHIEs of Figure 33.3 does not appear to occur exactly at DpKa ¼ 0 is also related to the value of wr 2 wp .15 Perhaps the assumption that there is no isotope effect in the work terms wr and wp affects the application of Marcus theory to the SHIE data. It is a tenet from much research in physical organic chemistry that proton transfer between electronegative atoms such as oxygen and nitrogen are seldom if ever rate determining, and that isotope effects on such transfers are not primary but secondary in nature. That is, the isotope effects are not due to the transferred proton itself but to various reorganizations of solvation shells. This idea is stated succinctly in a “Solvation Rule” by Swain, Kuhn, and Schowen60 and later by Albery.61 If this is correct, then a substantial SHIE should appear in what has to be done to prepare the system for the proton transfer, i.e. in the work terms wr and wp . There still may be a small SHIE in DG‡o associated with reorganization of solvent as the proton transfers from donor to acceptor. These ideas are contrary to the current approach that there is no isotope effect in the work terms,15,58 an assumption that seems to apply to proton transfers to and from carbon, but perhaps does not apply to proton transfers between electronegative ions.

B. INTERMOLECULAR P ROTON T RANSFER IN C ATALYSIS BY C ARBONIC A NHYDRASE Chemical rescue experiments utilize derivatives of imidazole and pyridine (the rescue agents) to replace the function of His 64. Moreover, rescue agents used are limited mostly to mono- and dimethyl derivatives, avoiding significant electrostatic effects of halogenated derivatives, for example. In almost every rescue experiment, saturable activation of the catalytic activity of H64A HCA II is achieved. A very significant observation is that the maximal proton transfer rate from certain rescue agents to the zinc-bound hydroxide is very similar in magnitude and pH dependence to wild-type HCA II with its proton shuttle His 64 (see, for example, Figure 6 of Ref. 62). This suggests at least two interpretations: either the rescue agent binds to H64A at a site occupied by His 64 in wild type, or the rescue agent binds elsewhere and the proton transfer is not specifically dependent on the location of the proton donor because of the flexibility of water structures or formation of hydronium-like intermediates. These buffers enhance the proton transfer components of the catalysis (Equation 33.2 and Equation 33.5) with relatively little effect on kcat =Km ; which describes the conversion of CO2 into HCO2 3 . A small inhibition of kcat =Km was observed for some buffers that was taken into account in the analysis of proton transfer.63 The Brønsted plots resulting from these experiments were very similar in form to those obtained for intramolecular proton transfer. The parameters of the Marcus equation for these cases of intermolecular proton transfer are compared in Table 33.1 with the parameters for intramolecular proton transfer. 1. Marcus Plot for Solvent Hydrogen Isotope Effects Taoka et al.37 reported the dependence on the DpKa between proton donor and acceptor of the SHIE on kcat for CA catalyzed hydration of CO2. In these experiments, the mutant H64A HCA II lacking the proton shuttle was activated by exogenous proton acceptors, derivatives of imidazole and pyridine, which provided the change in the pKa. A plot of the SHIE versus the pKa of these exogenous proton acceptors was bell-shaped in appearance (see Figure 2 of Ref. 37), very similar to that of Figure 33.3. A fit to Marcus rate theory, as described above, resulted in values of the Marcus parameters similar to the case in which the proton transfer was intramolecular. The intrinsic kinetic barrier DG‡o was small at 0.6 ^ 0.5 kcal/mol and the work terms wr 2 wp ¼ 0:9 ^ 0:5 kcal/mol (Table 33.1). These results emphasize the similarity of proton transfer processes between wild-type HCA II and H64A HCA II activated by exogenous proton acceptors.

Solvent Hydrogen Isotope Effects in Catalysis by Carbonic Anhydrase

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2. The Marcus Formalism Extended to the b and g Classes The similarity in Marcus parameters for several cases of catalysis by CAs from the a class (Table 33.1) led us to seek other systems to determine how the Marcus parameters for proton transfer would differ. This approach was applied to a b-class CA from Arabidopsis thaliana (H216N ATCA), and to a g-class CA from Methanosarcina thermophila (E84A MTCA), both of which have proton-transfer residues replaced with residues that do not support proton transfer. Again we observed saturable enhancement of both the b and g CAs in chemical rescue by derivatives of imidazole and pyridine.64 The rate constants for proton transfer were determined from the maximal velocity of catalyzed hydration of CO2 measured by stopped flow and from the catalyzed exchange of 18O between CO2 and water measured by mass spectrometry. The freeenergy plots for H216N ATCA and E84A MTCA are similar to that of Figure 33.2 for HCA III. Marcus rate theory was used to interpret free-energy plots for intermolecular proton transfer in catalysis by mutants of CAs in the b and g classes. The results were similar in showing that the intrinsic barrier for proton transfer is small, near 0.3 kcal/mol (Table 33.1). Catalysis by CAs in the b and g classes is dominated by a large work function of 8 to 11 kcal/mol as in the a class; this is the component of the observed free energy of activation for proton transfer that does not depend on DpKa (or depends very weakly on DpKa).

VI. CONCLUSIONS Carbonic anhydrase provides a useful scaffold to study proton transfer through intervening water bridges in a protein environment. The maximal SHIE is about 4 and varies with the difference in the pKa of donor and acceptor. Applying the Marcus rate theory shows that the intrinsic kinetic barriers measured for carbonic anhydrase are small with values near or less than 2 kcal/mol, consistent with bimolecular, nonenzymic proton transfers between nitrogen and oxygen acids and bases in solution.50 They show large work functions w r in the range up to 11 kcal/mol. It is this work function that answers the question of why proton transfer in carbonic anhydrase, which is at most 106 s21, is so much slower than the maximal rates of proton transfer near 1011 s21 observed, for example, for proton transfer from naphthol-related photo acids to acetate in solution. A major challenge is to determine what processes contribute to the work functions. The data suggest that the enhanced proton transfer represented by the low values of the intrinsic barrier DG ‡o reflects a significant similarity in the properties of proton translocation and possibly a similarity in the activesite cavities of the a, b, and g classes of CA that have evolved to promote facile proton transfer. The flexibility of water structures through which proton transfer proceeds may explain in part why the Marcus parameters are similar for the activation of H216N ATCA, E84A MTCA, and the CAs of the a class in Table 33.1. A factor that may contribute to the very low values of the Marcus intrinsic barrier is the partially hydrophobic environment of the active site which is sequestered away from bulk solution; Bernasconi65 presents data on nonenzymic reactions to indicate that values of DG ‡o may be decreased as much as an order of magnitude in aprotic compared with aqueous solutions.

REFERENCES 1 Steiner, H., Jonsson, B.H., and Lindskog, S., The catalytic mechanism of carbonic anhydrase, Eur. J. Biochem., 59, 253– 259, 1975. 2 Hewett-Emmett, D. and Tashian, R. E., Functional diversity, conservation, and convergence in the evolution of the a-, b-, and g-carbonic anhydrase gene families, Mol. Phylogen. Evol., 5, 50 – 77, 1996. 3 Silverman, D. N. and Lindskog, S., The catalytic mechanism of carbonic anhydrase: implications of a rate-limiting protolysis of water, Acc. Chem. Res., 21, 30 –36, 1988. 4 Christianson, D. W. and Fierke, C. A., Structural and molecular biology in the dissection of mechanism and the engineering of zinc binding in human carbonic anhydrase II, Acc. Chem. Res., 29, 331–339, 1996.

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5 Lindskog, S. and Silverman, D. N., The catalytic mechanism of mammalian carbonic anydrases, In Carbonic Anhydrases — New Horizons, Chegwidden, W. R., Carter, N. D., and Edward, Y. H., Eds., Birkha¨user Verlag, Basel, pp. 175–195, 2000. 6 Khalifah, R. G. and Silverman, D. N., Carbonic anhydrase kinetics and molecular function, In The Carbonic Anhydrases, Dodgson, S. J., Tashian, R. E., Gros, G., and Carter, N. D., Eds., Plenum press, New York, pp. 40 – 70, 1991. 7 Kresge, A. J. and Silverman, D. N., Application of marcus rate theory to proton transfer in enzymecatalyzed reactions, Methods Enzymol., 308, 276– 297, 1999. 8 Silverman, D. N., Marcus rate theory applied to enzymatic proton transfer, Biochim. Biophys. Acta, 1458, 88 – 103, 2000, (Thematic issue on proton transfer in biological systems). 9 Chegwidden, W. R., Carter, N. D., and Edwards, Y. H., The Carbonic Anhydrases: New Horizons, Birkhauser, Basel, Switz, 2000. 10 Northrop, D. B. and Rebholz, K. L., Kinetics of enzymes with iso-mechanisms: solvent isotope effects, Arch. Biochem. Biophys., 342, 317– 321, 1997. 11 Rowlett, R. S., Tu, C. K., McKay, M. M., Preiss, J. R., Loomis, R. J., Hicks, K. A., Marchione, R. J., Strong, J. A., Donovan, G. S., and Chamberlin, J. E., Kinetic characterization of wild-type and proton transfer-impaired variants of beta-carbonic anhydrase from Arabidopsis thaliana, Arch. Biochem. Biophys., 404, 197– 209, 2002. 12 Alber, B. E., Colangelo, C. M., Dong, J., Stalhandske, C. M. V., Baird, T. T., Tu, C. K., Fierke, C. A., Silverman, D. N., Scott, R. A., and Ferry, J. G., Kinetic and spectroscopic characterization of the zinc and cobalt gamma carbonic anhydrase from the methanoarchaeon Methanosarcina thermophila, Biochemistry, 38, 13119– 13128, 1999. 13 Tu, C. K., Silverman, D. N., Forsman, C., Jonsson, B. H., and Lindskog, S., Role of histidine 64 in the catalytic mechanism of human carbonic anhydrase II studied with a site-specific mutant, Biochemistry, 28, 7913– 7918, 1989. 14 Jewell, D. A., Tu, C. K., Paranawithana, S. R., Tanhauser, S. M., LoGrasso, P. V., Laipis, P. J., and Silverman, D. N., Enhancement of the catalytic properties of human carbonic anhydrase III by sitedirected mutagenesis, Biochemistry, 30, 1484 –1490, 1991. 15 Silverman, D. N., Tu, C. K., Chen, X., Tanhauser, S. M., Kresge, A. J., and Laipis, P. J., Rate-equilibria relationships in intramolecular proton transfer in human carbonic anhydrase III, Biochemistry, 32, 10757– 10762, 1993. 16 Eriksson, A. E., Jones, T. A., and Liljas, A., The refined structure of human carbonic anhydrase II at ˚ resolution, Proteins: Struct. Funct. Genet., 4, 274– 282, 1988. 2.0 A ˚ resolution, 17 Eriksson, E. A. and Liljas, A., Refined structure of bovine carbonic anhydrase III at 2.0 A Proteins: Struct. Funct. Genet., 16, 29 – 42, 1993. 18 Boriack-Sjodin, P. A., Heck, R. W., Laipis, P. J., Silverman, D. N., and Christianson, D. W., Structure ˚ resolution: implications for catalytic determination of mitochondrial carbonic anhydrase V at 2.45 A proton transfer and inhibitor design, Proc. Natl. Acad. Sci. USA, 92, 10949 –10953, 1995. 19 Duda, D., Govindasamy, L., Agbandje-McKenna, M., Tu, C. K., Silverman, D. N., and McKenna, R., ˚ resolution: implications of chemical The refined atomic structure of carbonic anhydrase II at 1.05 A rescue of proton transfer, Acta Crystallogr., D59, 93 – 104, 2003. 20 Merz, K. M., Insights into the function of the zinc hydroxide-Thr 199-Glu 106 hydrogen-bonding network in carbonic anhydrases, J. Mol. Biol., 214, 799– 802, 1990. 21 Nair, S. K. and Christianson, D. W., Unexpected pH-dependent conformation of His 64, the proton shuttle of carbonic anhydrase II, J. Am. Chem. Soc., 113, 9455– 9458, 1991. 22 Jude, K. M., Wright, S. K., Tu, C. K., Silverman, D. N., Viola, R. E., and Christianson, D. W., Crystal structure of F65A/Y131C-methylimidazole carbonic anhydrase V reveals architectural features of an engineered proton shuttle, Biochemistry, 41, 2485– 2491, 2002. 23 Tripp, B. and Ferry, J. A., Proton transfer in catalysis by carbonic anhydrase from Methanosarcina thermophila, Biochemistry, 39, 9232– 9240, 2000. 24 Iverson, T. M., Alber, B. E., Kisker, C., Ferry, J. G., and Rees, D. C., A closer look at the active site of gamma-class carbonic anhydrases: high-resolution crystallographic studies of the carbonic anhydrase from Methanosarcina thermophila, Biochemistry, 39, 9222–9231, 2000. 25 Kimber, M. S. and Pai, E. F., The active site structure of Pisum sativum beta-carbonic anhydrase is a mirror image of that of alpha-carbonic anhydrases, EMBO J., 19, 1407– 1418, 2000.

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26 Pocker, Y. and Bjorkquist, D. W., Comparative studies of bovine carbonic anhydrase in water and water-d2. Stopped-flow studies of the kinetics of interconversion of carbon dioxide and bicarbonate(1 2 ) ion, Biochemistry, 16, 5698– 5707, 1977. 27 Silverman, D. N., Tu, C. K., Lindskog, S., and Wynns, G. C., Rate of exchange of water from the active site of human carbonic anhydrase C, J. Am. Chem. Soc., 101, 6734– 6740, 1979. 28 Simonsson, I., Jonsson, B.-H., and Lindskog, S., A 13C NMR study of CO2-HCO2 3 exchange catalyzed by human carbonic anhydrase C at chemical equilibrium, Eur. J. Biochem., 93, 409– 417, 1979. 29 Paneth, P. and O’Leary, M. H., Isotope effect for the zinc hydroxide mechanism of carbonic anhydrase catalysis, Biochemistry, 26, 1728 –1731, 1987. 30 Xue, Y., Vidgren, Y., Svensson, L. A., Liljas, A., and Lindskog, S., Crystallographic analysis of The 200 ! His human carbonic anhydrase II and its complex with the substrate HCO2 3 , Proteins: Struct. Funct. Genet., 15, 80 – 87, 1993. 31 Ha˚kansson, K. and Wehnert, A., Structure of cobalt carbonic anhydrase complexed with bicarbonate, J. Mol. Biol., 228, 1212– 1218, 1992. 32 Rebholz, K. L. and Northrop, D. B., Kinetics of enzymes with iso-mechanisms: dead-end inhibition of fumarase and carbonic anhydrase II, Arch. Biochem. Biophys., 312, 227– 233, 1994. 33 Northrop, D. B. and Simpson, F. B., Kinetics of enzymes with isomechanisms: Britton induced transport catalyzed by bovine carbonic anhydrase II, measured by rapid-flow mass spectrometry, Arch. Biochem. Biophys., 352, 288– 292, 1998. 34 Rowlett, R. S., The reversible inhibition of carbonic anhydrase II: computer simulations of a proposed mechanism of action, J. Protein Chem., 3, 369– 393, 1984. 35 Venkatasubban, K. S. and Silverman, D. N., Carbon dioxide hydration activity of carbonic anhydrase in mixtures of water and deuterium oxide, Biochemistry, 19, 4984– 4989, 1980. 36 Schowen, K. B. and Schowen, R. L., Solvent isotope effects on enzyme systems, MethodEnzymol., 87, 551– 606, 1982. 37 Taoka, S., Tu, C. K., Kistler, K. A., and Silverman, D. N., Comparison of intra- and intermolecular proton transfer in human carbonic anhydrase II, J. Biol. Chem., 269, 17988– 17992, 1994. 38 Kresge, A. J., Solvent isotope effects and mechanism of chymotrypsing action, J. Am. Chem. Soc., 95, 3065– 3067, 1973. 39 Kassebaum, J. W. and Silverman, D. N., Hydrogen/deuterium fractionation factors of the aqueous ligand of cobalt in Co(II)-substituted carbonic anhydrase, J. Am. Chem. Soc., 111, 2691– 2696, 1989. 40 Cui, Q. and Karplus, M., Is a “proton wire” concerted or stepwise? A model study of proton transfer in carbonic anhydrase, J. Phys. Chem. B, 107, 1071–1078, 2003. 41 Liang, J.-Y. and Lipscomb, W. N., Hydration of carbon dioxide by carbonic anhydrase: internal proton transfer of Znþ2-bound HCO2 3 , Biochemistry, 26, 5293– 5301, 1987. ˚ qvist, J. and Warshel, A., Simulation of enzyme reactions using valence bond force fields and other 42 A hybrid quantum/classical approaches, Chem. Rev., 93, 2523– 2544, 1993. 43 Lu, D. and Voth, G. A., Proton transfer in the enzyme carbonic anhydrase: an ab initio study, J. Am. Chem. Soc., 120, 4006– 4014, 1998. 44 Isaev, A. and Scheiner, S., Proton conduction of water molecules in carbonic anhydrase, J. Phys. Chem. B, 105, 6420– 6426, 2001. 45 Smedarchina, Z., Siebrand, W., Fernandez-Ramos, A., and Cui, Q., Kinetic isotope effects for concerted multiple proton transfer: a direct dynamics study on an active-site model of carbonic anhydrase II, J. Am. Chem. Soc., 125, 243– 251, 2003. 46 Rowlett, R. S. and Silverman, D. N., Kinetics of the protonation of buffer and hydration of CO2 catalyzed by human carbonic anhydrase II, J. Am. Chem. Soc., 104, 6737– 6741, 1982. 47 Silverman, D. N., Carbonic anhydrase: oxygen-18 exchange catalyzed by an enzyme with ratecontributing proton-transfer steps, Methods Enzymol., 87, 732– 752, 1982. 48 LoGrasso, P. V., Tu, C. K., Chen, X., Taoka, S., Laipis, P. J., and Silverman, D. N., Influence of aminoacid replacement at position 198 on catalytic properties of zinc-bound water in human carbonic anhydrase III, Biochemistry, 32, 5786– 5791, 1993. 49 Tu, C. K., Chen, X., Ren, X., LoGrasso, P. V., Jewell, D. A., Laipis, P. J., and Silverman, D. N., Interactions of active-site residues and catalytic activity of human carbonic anhydrase III, J. Biol. Chem., 269, 23002 –23006, 1994.

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50 Kresge, A. J., What makes proton transfer fast?, Acc. Chem. Res, 8, 354– 360, 1975. 51 Marcus, R. A., Theoretical relations among rate constants, barriers, and Brønsted slopes of chemical reactions, J. Phys. Chem., 72, 891– 899, 1968. 52 Kim, Y., Truhlar, D. G., and Kreevoy, M. M., An experimentally based family of potential energy surfaces for hydride transfer between NAD þ analogues, J. Am. Chem. Soc., 113, 7837– 7847, 1991. 53 Braun-Sand, S., Strajbl, M., and Warshel, A., Studies of proton translocations in biological systems: simulating proton transport in carbonic anhydrase by EVB-based models, Biophys. J., 87, 2221– 2239, 2004. 54 Melander, L., Isotope Effects on Reaction Rates, Ronald Press, New York, 1960. 55 Westheimer, F. H., The magnitude of the primary kinetic isotope effect for compounds of hydrogen and deuterium, Chem. Rev., 61, 265– 273, 1961. 56 Bergman, N.-A. and Kresge, A. J., An isotope effect maximum for proton transfer between normal acids and bases, J. Am. Chem. Soc., 5954 –5956, 1978. 57 Cox, M. M. and Jencks, W. P., General acid catalysis of the aminolysis of phenyl acetate by a preassociation mechanism, J. Am. Chem. Soc., 100, 5956– 5957, 1978. 58 Kresge, A. J., Sagatys, D. S., and Chen, H. L., Vinyl ether hydrolysis. 9. Isotope effects on proton transfer from the hydronium ion, J. Am. Chem. Soc., 99, 7228– 7233, 1977. 59 Kreevoy, M. M. and Oh, S.-W., Relation between rate and equilibrium constants for proton transfer reactions, J. Am. Chem. Soc., 95, 4805– 4810, 1973. 60 Swain, C. G., Kuhn, D. A., and Schowen, R. L., Effect of structural changes in reactants on the position of hydrogen-bonding hydrogens and solvating molecules in transition states. The mechanism of tetrahydrofuran formation from 4-chlorobutanol, J. Am. Chem. Soc., 87, 1553– 1561, 1965. 61 Albery, W. J., Solvent isotope effects, In Proton Transfer Reactions, Caldin, E. F. and Gold, V., Eds., Chapman and Hall, London, pp. 263–316, 1974. 62 Duda, D., Tu, C. K., Qian, M., Laipis, P., Agbandje-McKenna, M., Silverman, D. N., and McKenna, R., Structural and kinetic analysis of the chemical rescue of the proton transfer function of carbonic anhydrase II, Biochemistry, 40, 1741– 1748, 2001. 63 Earnhardt, J. N., Tu, C. K., and Silverman, D. N., Intermolecular proton transfer in catalysis by carbonic anhydrase V, Canadian J. Chem., 77, 726– 732, 1999, (Special Issue Honoring A. Jerry Kresge). 64 Tu, C. K., Rowlett, R. S., Tripp, B. C., Ferry, J. G., and Silverman, D. N., Chemical rescue of proton transfer in catalysis by carbonic anhydrase in the beta and gamma class, Biochemistry, 41, 15429– 15435, 2002. 65 Bernasconi, C. F., Intrinsic barriers of reactions and the principle of nonperfect synchronization, Acc. Chem. Rev., 20, 301– 398, 1987. 66 Ren, X., Tu, C. K., Laipis, P. J., and Silverman, D. N., Proton transfer by histidine 67 in site-directed mutants of human carbonic anhydrase III, Biochemistry, 34, 8492– 8498, 1995.

34

Isotope Effects from Partitioning of Intermediates in Enzyme-Catalyzed Hydroxylation Reactions Paul F. Fitzpatrick

CONTENTS I. II. III.

Introduction ...................................................................................................................... 861 Theory .............................................................................................................................. 862 Examples .......................................................................................................................... 866 A. Cytochrome P450..................................................................................................... 866 B. The Aromatic Amino Acid Hydroxylases............................................................... 868 C. Dopamine b-Monooxygenase.................................................................................. 870 IV. Conclusion........................................................................................................................ 871 References..................................................................................................................................... 871

I. INTRODUCTION As discussed elsewhere in this volume, substitution with a heavier isotope of an atom in the substrate for an enzyme-catalyzed reaction will not necessarily result in a detectable change in either the rate of product formation in a single assay or on the Vmax and V/K values determined from more complete kinetic analyses. Expression of intrinsic isotope effects on Vmax values can be masked by slower chemical steps steps which are not isotope-sensitive and by slow product release, while effects on V/K values can be masked by high commitments to catalysis.1 Occasionally, these problems can be minimized by either changing the conditions, e.g., carrying out assays off the pH optimum,2 or by utilizing a slower substrate. However, in the case of enzymes which carry out hydroxylation reactions, it is frequently found that neither of these approaches are successful. The reactions of many hydroxylases can be divided into two discrete chemical steps: the reaction of oxygen with an enzyme bound cofactor such as a metal to form the hydroxylating intermediate and the subsequent reaction of this intermediate with the organic substrate. Formation of the hydroxylating intermediate is generally irreversible; thus, if one is interested in using kinetic isotope effects to probe the mechanism of hydroxylation, V/K isotope effects are ruled out because the isotope-sensitive step occurs after an irreversible step. If formation of the hydroxylating intermediate is much slower than the subsequent hydroxylation or if product release is slow, Vmax isotope effects are significantly smaller than the intrinsic values. This problem cannot necessarily be circumvented by using a slow substrate, in that the hydroxylating intermediate is unstable and will often carry out an alternative reaction if the substrate is too unreactive. This alternative pathway may be simply unproductive breakdown of the hydroxylating intermediate or a reaction at 861

862

Isotope Effects in Chemistry and Biology

a different site on the substrate. As discussed here, either of these alternative pathways can be used to determine kinetic isotope effects which are not reflected in V/K or Vmax values.

II. THEORY The partitioning among different pathways for product formation in an enzyme active site can be more sensitive to isotope substitution than the overall rate of product formation since steps such as substrate binding and product release frequently do not affect the partitioning. Isotope effects determined from product ratios have generally been referred to as intramolecular isotope effects to distinguish them from intermolecular ones, that is, isotope effects measured from the differences in rates of product formation from separate isotopically labeled and unlabeled molecules.3 There have been several different experimental treatments of such intramolecular isotope effects. Most commonly, the isotope effect has been measured by determining the isotopic content by mass spectrometry of a product arising from a substrate deuterated at one of two equivalent positions. For example, intramolecular isotope effects for cytochrome P450-catalyzed reactions have been measured by determining the isotopic content of the 1-octanol formed upon hydroxylation of 1,1,1-2H3-octane or of the N-methylaniline formed upon demethylation of N-methyl-N(2H3-methyl)aniline. The seminal analysis of such experiments is that of Miwa et al.,3 with further elaboration by Jones et al.4 and Korzekwa et al.5 The mechanism used by Miwa et al.3 in their initial treatment of intramolecular isotope effects is shown in Scheme 34.1. When an isotope effect involves a competition between an isotopically substituted position and a separate unsubstituted but chemically equivalent position, one must allow for the possibility that the substrate must reorient in the active site for reaction at the two alternate positions. ESH refers to the substrate positioned for CH bond cleavage with rate constant k5H, while ESD refers to the substrate positioned for CD bond cleavage with rate constant k5D. The orientation of the substrate is symmetrical in ESH and ESD; as a result, the formation of both complexes occurs with the identical rate constants k3 and k4 : The relationship between the measured intramolecular isotope effect and the intrinsic isotope effect for the mechanism of Scheme 34.1 is given by Equation 34.1.3 Here, Dkobs is the ratio of the amount of the product that arises by cleavage of a CH bond to that which arises upon cleavage of a CD bond. In the example of demethylation of Nmethyl-N-(2H3-methyl)aniline noted above, the isotope effect is the ratio of the N-methylaniline containing three deuteriums to that containing none. Equation 34.1 illustrates that an intramolecular isotope effect is not necessarily equal to the intrinsic isotope effect, because of either reversible CH bond cleavage or, more commonly, a lack of equilibration of the two binding modes required for CH versus CD bond cleavage. Fortunately, for hydroxylation reactions CH bond cleavage is generally irreversible, so that the primary concern is nonequilibration of the two binding modes. D

kobs ¼ PH =PD ¼

D

k5 þ k5 =k4 þ D Keq k6 =k7 1 þ k5 =k4 þ k6 =k7

k5H ESH

k3 k1 k2

SCHEME 34.1

k4

ES

E+S

k3 k4

k6H

EPH

k5D ESD

k6D

EPD

k7

ð34:1Þ

E+PH

k7 E+PD

Isotope Effects from Partitioning of Intermediates

863 EP2

E+S ku

ESH

k3 k1

E+S

ES

k4 k3

E'S

k2

k9

k4 ku E+S

ESD k9'

k5H k6H k5D k6D

EP1H

EP1D

k7

k7

E+P1H

E+P1D

EP2

SCHEME 34.2

The mechanism of Scheme 34.1 can be expanded to the more general mechanism of Scheme 34.2 to allow a more complete analysis of hydroxylation reactions. This scheme introduces a step for formation of the hydroxylating intermediate E0 after substrate binding; this may encompass several steps, but the overall reaction is generally irreversible. It also introduces the possibility of the formation of an alternate product, P2 ; by reaction at a different site than the one of interest. This alternate pathway may also exhibit an isotope effect depending on the labeling of the substrate, so that k9 – k09 ; this problem can be avoided by using substrate deuterated only at the site of interest. Finally, there is the possibility that the hydroxylating intermediate can break down unproductively with rate constant ku : The intramolecular isotope effect if there is no isotope effect on k9 is described by Equation 34.2. D

kobs ¼ PH =PD ¼

D

k5 þ k5 =ðk4 þ k9 þ ku Þ þ D Keq k6 =k7 1 þ k5 =ðk4 þ k9 þ ku Þ þ k6 =k7

ð34:2Þ

If CH bond cleavage is irreversible ðk6 ¼ 0Þ; as is typically the case in hydroxylation reactions, Equation 34.2 simplifies to Equation 34.3. D

kobs ¼ PH =PD ¼

D

k5 þ k5 =ðk4 þ k9 þ ku Þ 1 þ k5 =ðk4 þ k9 þ ku Þ

ð34:3Þ

Measurement of an intramolecular isotope effect typically requires a substrate containing two equivalent reactive positions, one of which is deuterated. For an enzyme-catalyzed reaction in which multiple products are formed ðk9 . 0Þ or there is significant unproductive breakdown of the hydroxylating intermediate ðku . 0Þ; an alternative approach is to measure the effect on the product composition with deuterated versus nondeuterated substrate. An advantage of this approach is that there need not be hydrogens at equivalent sites that can be partially replaced with deuterium. This is a noncompetitive experiment, in that one carries out separate reactions with isotopically labeled and unlabelled substrates. Consequently, kinetic isotope effects measured in this fashion have not been routinely referred to as intramolecular isotope effects, although the fundamental theory is the same: a kinetic isotope effect occurs in the partitioning of an enzyme-bound intermediate. Scheme 34.3 depicts a minimal mechanism for such an analysis. In effect, with the nondeuterated substrate the enzyme can only go through the upper pathway of Scheme 34.2, while the deuterated substrate restricts the reaction to the lower pathway. Again, all of the reactions of the hydroxylating intermediate are assumed to be irreversible. If any is not, the relevant rate constant must be replaced with the appropriate net rate constant. Experimentally, one determines the mole fraction of the product formed by CH bond cleavage relative to the amount of total turnover, using both the isotopically labeled and unlabeled substrates. The amount of total turnover is most easily measured from the amount of reducing equivalents consumed. The isotope effect on the mole

864

Isotope Effects in Chemistry and Biology

SCHEME 34.3

fraction of the product arising from the isotope-sensitive step is given by Equation 34.4:

D

kobs ¼

P1 P1 þ P2 þ U P1 P1 þ P2 þ U

H

¼

D

k5 þ k5 =ðk9 þ ku Þ 1 þ k5 =ðk9 þ ku Þ

ð34:4Þ

D

Here U is the amount of unproductive turnover; this is simply the difference between the total reducing equivalents consumed and the amount of hydroxylated products formed. In Equation 34.4 the measured isotope effect is reduced from the intrinsic value by the factor k5 =ðk9 þ ku Þ: This value can readily be calculated from the mole fraction for the product of interest obtained with the unlabeled substrate, since this equals k5 =ðk5 þ k9 þ ku Þ: The disadvantage of this approach to a competitive measurement of the intramolecular isotope effect is of course the same disadvantage attendant upon any noncompetitive approach, the decreased precision resulting from comparison of values from separate experiments. The possibility of alternate reaction pathways for the hydroxylating intermediate can result in partial or complete expression of the intrinsic isotope effect on the Vmax and V=K values. The simplest situation in which this can occur is when the partitioning is between CH bond cleavage to form the hydroxylated product and unproductive decay of the hydroxylating intermediate. This is described by the mechanism of Scheme 34.3 when k9 ¼ 0: If one measures the rate of formation of the hydroxylated product P1 ; the isotope effects on the V and V=K values are described by Equation 34.5 and Equation 34.6. k5 ð1=k3 þ 1=k7 Þ ku k3 þ 1 D V¼ k5 ð1=k3 þ 1=k7 Þ 1þ ku k3 þ 1

ð34:5Þ

k5 k 1þ 3 ku k2 V=K ¼ k5 k3 1þ 1þ ku k2

ð34:6Þ

D

k5 þ

D

D

k5 þ

In both, the magnitude of the observed isotope effect will increase as ku becomes greater than k5 : When ku q k5 ; both DV and DV/K will equal the intrinsic isotope effect. It should be emphasized that application of the equations requires that the rate of formation of P1 is being measured. If one instead measures the rate of consumption of reducing equivalents, e.g., NADPH oxidation, or of oxygen, one may observe no isotope effect or an isotope effect that is decreased from the intrinsic value. The situation becomes more complex if an alternate product can be formed ðk9 – 0Þ; since the rate of P2 release becomes important. In such a case one can even observe an inverse isotope effect if P2 is released from the enzyme more rapidly than P1 and deuteration of P1 results in increased formation of P2 :

Isotope Effects from Partitioning of Intermediates

865

In the discussion so far the isotope effect has been caused by the difference in the rate of CH bond cleavage when the hydrogen is replaced by deuterium at a different but equivalent site in the same molecule or at the same site in a different molecule. If there are multiple hydrogens on the carbon of interest, they are typically all replaced with deuterium, so that the observed isotope effect is a combination of primary and secondary effects. If one instead determines the isotope effect for a carbon containing both deuterium and hydrogen, one has the potential for measurement of both primary and secondary effects. This is best done in the case of hydroxylation of a methyl group, e.g., benzylic hydroxylation, as discussed by Hanzlik,6 where stereochemistry is not a concern. Experimentally, one measures the isotopic composition of the hydroxylated product, e.g., benzyl alcohol, arising from the substrate in which the methyl group has one or two deuterium atoms. When both deuterium and hydrogen are present on the same methyl group, the hydroxylated product can be formed by cleavage of a CH bond or a CD bond. The relative amounts of these can be readily determined by mass spectrometry of the hydroxylated product. In the case of a monodeuterated methyl group, the partitioning among the different products can be described by Scheme 34.4. Formation of the deuterated alcohol by cleavage of a CH bond with rate constant kHD-H will exhibit a secondary isotope effect, while formation of the nondeuterated product will involve cleavage of a CD bond with rate constant kHH-D and will exhibit a primary isotope effect. The relative amounts of the deuterated and nondeuterated products are thus given by the ratio of these rate constants, correcting for the fact that cleavage of a CH bond is twice as likely (Equation 34.7). Since the primary isotope effect Dk ¼ kHH-H/kHH-D and the secondary isotope effect aDk ¼ kHH-H/kHD-H, the isotopic composition of the alcohols yields the ratio of the primary and secondary isotope effects. A similar equation can be derived for substrate with two deuterium atoms on the methyl group (Equation 34.8). If the Rule of the Geometric Mean holds, the isotopic composition of the products from the mono- and dideuterated substrates should differ by a factor of four. Because of this interdependence of Equation 34.7 and Equation 34.8, one additional relationship between the primary and secondary isotope effects is required. Measuring the intrinsic isotope effect with the fully deuterated substrate will yield the necessary value, because Equation 34.9 can then be used: R-CDHOH 2k k ¼ HD-H ¼ 2 HD-H R-CH2 OH kHH-D kHH-H

kHH-H kHH-D

¼

R-CD2 OH k 1 kDD-H ¼ DD-H ¼ R-CDHOH 2kHD-D 2 kHH-H

kHH-H kHD-D

¼

D

k ¼ D kðaD kÞ2

2D k aD k D

k

2a D k

ð34:7Þ ð34:8Þ ð34:9Þ

To date isotope effects on partitioning have been measured only for deuterium isotope effects, but there is no a priori reason other than precision that similar analyses could not be carried out with heavy atom isotope effects. 14C effects in particular should be of sufficient magnitude for measurement, for example, using o-xylene in which only one methyl is labeled. Because

kHD-H H R C D H

kHH-D kHD-H

SCHEME 34.4

OH R C D H H R C OH H H R C D OH

866

Isotope Effects in Chemistry and Biology

the competition is between carbons instead of hydrogens attached to the same carbon, the problem of slow reorientation cannot be ignored.

III. EXAMPLES A. CYTOCHROME P450 The most extensive use of intramolecular isotope effects has been in the study of the cytochrome P450 hydroxylases, and much of the formal description of the theory has been developed with regard to these enzymes.3 – 6 In a typical cytochrome P450 reaction (Scheme 34.5), binding of the substrate to the low spin ferric enzyme is followed by transfer of an electron to the heme, generating the high spin ferrous form which binds oxygen; the subsequent transfer of a second electron to the heme leads to cleavage of the OO bond and hydroxylation of the substrate.7 The slowest step in the overall reaction is generally considered to be the transfer of the second electron, although product release can also be slow.8 The irreversible formation of the hydroxylating intermediate, typically described as a Fe(IV)yO porphyrin cation radical, occurs after substrate binding, so that intrinsic isotope effects on the CH bond cleavage step are not expressed in DV/K values.3 The slow transfer of the second electron and slow product release suppress expression of an isotope effect on the Vmax value if the rate of NADPH oxidation is measured. Early measurements of intramolecular isotope effects played a fundamental part in the understanding of the mechanism of cytochrome P450. Hjelmeland et al.9 determined an intramolecular isotope effect of 11 for benzylic hydroxylation by measuring the relative amounts of hydrogen and deuterium in 1-HO-1,3-diphenylpropane formed from 1,1-2H2-1,3-diphenylpropane by rat microsomes (Scheme 34.6). The large isotope effect was attributed to a mechanism involving radical abstraction and recombination. Groves et al.10 examined the hydroxylation of exo, exo, exo, exo-2,3,5,6-2H4-norbornane by P450LM2; both exo- and endo-2-norborneol are formed in the reaction (Scheme 34.7). The ratio of the trideuterated alcohols to the tetradeuterated alcohols gave a value of 1.67 for the apparent intramolecular isotope. This value had to be corrected for the preference of the enzyme to abstract the exo-hydrogen and for the loss of stereochemistry after CH bond cleavage, yielding an intrinsic isotope effect of 11.5. The combination of this large value and the loss of stereochemistry was taken as evidence for formation of a carbon radical intermediate by hydrogen atom abstraction, followed by oxygen addition to form the alcohol. Since the early work of Hjelmeland9 there have been a number of additional measurements of intramolecular isotope effects on benzylic hydroxylation by cytochrome P450. Hanzlik and Ling11 studied the reaction with phenobarbital induced rat microsomes, which contain predominantly isozymes CYP2B1 and CYP2B2, using o-xylene and p-xylene as substrates. The products are the methylbenzyl alcohols and the various possible dimethylphenols. The product compositions were

SCHEME 34.5

Isotope Effects from Partitioning of Intermediates D

DH H

HO

867 DH H

D

D H OH

+

SCHEME 34.6

determined with xylenes containing one, two or three deuteriums in a single methyl group as well as those with both methyl groups fully deuterated and with the fully deuterated substrates. The observed intramolecular isotope effects for the substrates in which one methyl group was fully deuterated were 9.6 for o-xylene and 7.5 with p-xylene. The smaller value obtained with the latter was consistent with the two methyl groups not fully equilibrating in this case, i.e., k5 , k4 in Scheme 34.1, a conclusion later confirmed with the bacterial enzyme.12 Comparison of the products from o-xylene containing partially deuterated methyl groups allowed the primary and secondary effects to be calculated. The average primary isotope effect of 5.9 and the average secondary effect of 1.14 were consistent with the removal of the hydrogen as a hydrogen atom, as proposed by Groves et al.10 Finally, an isotope effect for ring hydroxylation of 0.94 was determined by comparison of the relative amounts of phenols formed from o-(C2H3)2-xylene and fully deuterated o-xylene, consistent with an electrophilic aromatic substitution reaction. Intramolecular isotope effects for benzylic hydroxylation by cytochrome P450 have also been measured with toluene as substrate, with qualitatively similar results.13 In this case isotope effects were also measured by several other approaches. If the rate of formation of benzyl alcohol was measured, the DV effect was 1.9 for toluene in which the methyl group contained three deuterium atoms. In contrast, the isotope effect on the product composition (Equation 34.4) was 7.4, demonstrating the advantage of measuring intramolecular effects. Finally, measurement of the total amount of hydroxylated products gave a DV value of 0.67; this inverse effect was attributed to faster release of cresol products (P2 in Scheme 34.3) than of benzyl alcohol. Intramolecular isotope effects for the hydroxylation of octanol have been measured as probes of the mechanism of aliphatic hydroxylation by cytochrome P450. Phenobarbital-induced cytochrome P450 catalyzes the formation of 1-octanol and 2-octanol at a ratio of 1:23.4 If 1,1,1-2H3-octanol is used as substrate, the ratio of the amount of dideuterated 1-octanol to the amount of trideuterated 1-octanol formed is 9.4, close to the value obtained by Hjelmeland et al.9 with uninduced microsomes. The relationship of this to the intrinsic isotope effect is given by Equation 34.3 by setting ku to zero; k9 is the rate constant for removal of hydrogen from carbon 2 to form 2-octanol. If k4 is very large, so that the interchange of the two methyl groups in the active site is much more rapid than CH bond cleavage, the intrinsic isotope effect for CH bond cleavage is 9.4. Alternatively, if k4 ¼0; the intrinsic isotope effect can be calculated from Equation 34.3, since k9 =k5 ¼23; giving an intrinsic isotope effect of 9.8. Besides placing a very narrow limit on the intrinsic isotope effect for this cytochrome P450-catalyzed reaction, this example illustrates that an intrinsic isotope effect can be determined from an intramolecular isotope effect if the partitioning among the different pathways is measured. Similar analyses were subsequently carried out using partially deuterated octanes to separate out the primary and secondary isotope effects.14 Comparison of the isotopic composition of the products from 1,1,1-2H3-octanol and from 1,1,8,8-2H4-octane gave a primary isotope effect of 6.8 and a secondary isotope effect of 1.07, while the comparable analysis with 1,8-2H2-octane gave values of 6.5 and 1.10. Both values for the secondary isotope effect are consistent with hydrogen atom abstraction.

SCHEME 34.7

868

Isotope Effects in Chemistry and Biology OH R N CH2 CH3

R N CH2 CH3

kC R N CH3 CH3

kN

I: R =

CH2 C CH3

O R NH + CH2 CH3

R N CH3 CH3 CH3

II: R =

SCHEME 34.8

Cytochrome P450 will also catalyze the demethylation of methylamines, and two principle mechanisms have been proposed for the reaction (Scheme 34.8).7 The first resembles the mechanism proposed for aliphatic and benzylic hydroxylation, abstraction of the hydrogen as a hydrogen atom. In the second, an electron is abstracted from the nitrogen. This is followed by loss of a proton from the methyl carbon and transfer of an electron from the carbon to the nitrogen. Rebound of the oxygen then forms the hydroxymethylamine, which eliminates formaldehyde nonezymatically. Since amine cation formation should show no isotope effect while CH bond cleavage will, a number of efforts have been made to determine intrinsic deuterium isotope effects for demethylation reactions. Miwa et al.3 determined the isotope effects with N,N-dimethylphentermine (I in Scheme 34.8). Isotope effects on the Vmax and V=K values were very close to one when rates of demethylation were compared with substrates in which both methyl groups contained only hydrogen or only deuterium. If, instead, the DV and DV/K values were determined by comparing the rates of formation of the deuterated and nondeuterated products with the substrate in which only one methyl group was deuterated, the values were 1.6 to 2, depending upon the enzyme source. These authors derived Equation 34.1 to describe the results in terms of Scheme 34.1 and concluded that k5 =k4 was significant in this system, so that the measured intramolecular isotope effect was less than the intrinsic isotope effect on k5 : Subsequent analyses have relied on dimethylanilines, measuring the isotopic composition of the product from substrate in which one methyl group is deuterated (II in Scheme 34.8). Using cytochrome P450 from a variety of sources, the intramolecular isotope effects are similar to those with N,N-dimethylphentermine, 1.6– 1.8.15 – 17 However, values as large as 6.9 have been reported using alternative sources of activated oxygen, dimethylanilines containing electron-donating groups, or amides in place of amines.16 – 19 The observation of significant isotope effects would be consistent with the upper path in Scheme 34.8, with the range of values being due either to differences in the intrinsic isotope effect or to differences in the value of k5 =k4 (Scheme 34.1). Alternatively, the lower path of Scheme 34.8 would result in an intramolecular isotope effect if the initial electron abstraction were reversible. Consistent with this interpretation, when these same deuterated dimethylanilines are used as substrates for peroxidases, large (8 –10) values are found;15, 17 the reaction in this case occurs by the lower pathway,7 so that the large observed isotope effects must be due to reversible formation of the amine cation. Finally, the deuterium isotope effect for nonenzymatic deprotonation of the methyl group in di-p-anisylmethylamine cation radical is 7.7, demonstrating that large isotope effects can arise via the lower pathway in Scheme 34.8.20

B. THE A ROMATIC A MINO ACID H YDROXYLASES The three aromatic amino acid hydroxylases phenylalanine hydroxylase (PheH), tyrosine hydroxylase (TyrH) and tryptophan hydroxylase (TrpH) are nonheme iron enzymes that catalyze

Isotope Effects from Partitioning of Intermediates

869

the hydroxylation of the aromatic rings of their respective substrates using a tetrahydropterin as the source of electrons.21 The present understanding of the mechanism of aromatic hydroxylation by these enzymes is given in Scheme 34.9.22 Oxygen addition to the aromatic ring of the substrate in Scheme 34.9 occurs by electrophilic aromatic substitution by an Fe(IV)yO intermediate. Formation of the new carbon –oxygen bond should result in an inverse deuterium isotope effect if the carbon at the site of attack has deuterium bound. Consistent with this expectation, there is an isotope effect of 0.93 on the Vmax value when 5-2H-tryptophan is used as substrate for TrpH.23 This value is comparable with the equilibrium isotope effect calculated for this reaction, suggesting that oxygen addition to the aromatic ring is rate-limiting for turnover by TrpH, a conclusion supported by the variation in the Vmax value with tryptophan analogues.24 In contrast, there is no isotope effect on either the Vmax or V/K tyr value for TyrH when 3,5-2H2-tyrosine is the substrate,25 suggesting that other steps in turnover are slower than oxygen addition in this enzyme. The lack of sensitivity of the Vmax value to changes in the amino acid substrate and the absence of a burst in the first turnover suggest that the slow step occurs in the formation of the Fe(IV)yO species;26 this situation resembles that with cytochrome P450. A number of mutant forms of TyrH have been described in which a substantial fraction of the reducing equivalents from the tetrahydropterin are consumed without concomitant amino acid hydroxylation. This phenomenon is consistent with the kinetic mechanism of Scheme 34.3 if k9 ¼ 0: When the isotope effect on the amount of dihydroxyphenylalanine formed versus tetrahydropterin consumed was measured with several of these mutant enzymes, an average value of 0.93 was obtained.27 The isotope effects on the rate of tyrosine production similarly yielded an average DV value of 0.93 for the same enzymes. Both results are consistent with ku q k5 : Thus, in the case of this enzyme, an isotope effect could be unmasked by introduction of an alternate unproductive pathway. The aromatic amino acid hydroxylases will also catalyze hydroxylation of benzylic carbons.22 In the case of TyrH, the intrinsic primary and secondary isotope effects for the benzylic

H R

H H N N H H O2

N

NH2 NH

H R

O

Fe(II) HO

+ H2 NH3 C CH CO2−

H H N

N H H O O O Fe(II)

NH2 N

HO

HO HO

H H N

N

N H H O H O Fe(II)

H R

H H N

N

NH2

N H H O H O

N

Fe(IV)O

+ H2 NH3 C CH CO2−

H R

SCHEME 34.9

N

NH2 N

+ H2 NH3 C CH CO2−

+ H2 NH3 C CH CO2−

HO

H R

H H N

N H H O H O H

HO

OFe

NH2

N N

+ H2 NH3 C CH CO2−

870

Isotope Effects in Chemistry and Biology

+

HOH2C

H3C

H2 C

+

NH3 CH CO2−

HO TyrH

+

+

H3C H3C HO

H2 NH3 C CH CO2−

+

H2 NH3 C CH CO2− +

H2 NH3 C CH CO2−

SCHEME 34.10

hydroxylation of the methyl group in 4-methylphenylalanine has been determined by measuring isotope effects on product composition. Three hydroxylated products are formed in this case, 4-hydroxy-3-methylphenylalanine, 3-hydroxy-4-methylphenylalanine, and 4-hydroxymethylphenylalanine (Scheme 34.10).28 The mole fraction of the product arising from benzylic hydroxylation was determined with 4-methylphenylalanine containing 0, 1, 2 or 3 deuterium atoms in the methyl group.29 The data were analyzed using Equation 34.4 and the mechanism of Scheme 34.3, to obtain kinetic isotope effects of 1.6, 3.0 and 14.0 for the monodeuterated, dideuterated and trideuterated substrates, respectively. The isotopic compositions of the 4-hydroxyphenylalanine formed from the monodeuterated and dideuterated substrates were determined by mass spectrometry. Application of Equation 34.7 to Equation 34.9 then gave an intrinsic primary effect of 9.6 and a secondary isotope effect of 1.21. The magnitude of the secondary isotope effect is consistent with removal of the methyl hydrogen as a hydrogen atom, a reaction similar to that carried out by cytochrome P450, suggesting that the proposed Fe(IV)yO species in the aromatic amino acid hydroxylases has similar reactivity to that in cytochrome P450.

C. DOPAMINE b-M ONOOXYGENASE In the examples so far, an enzyme-bound intermediate partitions between different catalytic pathways. Mechanism-based inhibitors, or suicide substrates, inactivate their target enzymes by generating an enzyme-bound intermediate that partitions between product formation and enzyme inactivation. Dopamine b-monooxygenase (DbM) provides an example in which the isotope effect on this partitioning has been used to probe the mechanism of inactivation. DbM catalyzes the benzylic hydroxylation of dopamine to form norepinephrine.30 In contrast to the previous two systems, this is a copper-dependent enzyme. While there are two copper atoms per active site, only one is thought to be a component of the hydroxylating intermediate, a copper peroxide. Several lines of evidence are consistent with removal of the benzylic hydrogen as a hydrogen atom to form a carbon-centered radical, a reaction formally analogous to those of cytochrome P450 and TyrH.31 – 33 DbM is inactivated by monosubstituted hydrazines; one, benzylhydrazine, acts as a mechanismbased inhibitor.34 As a probe of the mechanism of inactivation the deuterium isotope effect on the second-order rate constant for inactivation was determined with benzylhydrazine containing deuterium at the benzylic position. Surprisingly, the rate constant for inactivation increased by 3.6fold with the deuterated compound. The number of turnovers per inactivation decreased from 160 with the nondeuterated compound to 14 with the deuterated species for an isotope effect on the partition ratio of 11. This latter value can be considered an isotope effect on partitioning of an intermediate, with the alternative pathway from E0 S in Scheme 34.3 leading not to another hydroxylated substrate but to inactive enzyme. The isotope effect on the rate of oxygen

Isotope Effects from Partitioning of Intermediates

871

H2 H C N NH2 -H

-e

H H C N NH2

H2 H C N NH2

H H C N NH2 OH

+ CH2 H2N NH2

SCHEME 34.11

consumption was also measured; because of the high partition ratio, this is equivalent to the rate of hydroxylation. The DV value was 13 and the D(V/K) value was 3.1. The data were consistent with the mechanism of Scheme 34.11. Here, the partitioning between hydroxylation and enzyme inactivation occurs at the point of the initial reaction between the copper-peroxide hydroxylating intermediate and benzylhydrazine. Removal of the benzylic hydrogen as a hydrogen atom forms a benzylic radical; this step shows a deuterium isotope effect of about 12 with dideuterated benzylhydrazine, similar to the product of the intrinsic primary and secondary deuterium isotope effects for dopamine.31 Inactivation occurs when the electrophilic copper-peroxide instead removes an electron from the nitrogen, a reaction similar to that proposed for amine demethylation by cytochrome P450. Generation of the hydrazine cation radical is followed by cleavage of the carbon – nitrogen bond to generate hydrazine and a benzylic radical that can covalently modify the enzyme. The increased rate constant for inactivation upon deuteration is a combination of slower turnover and an increase in the partitioning to enzyme inactivation. The decrease in the DV/K value relative to the DV value suggests that there is a significant forward commitment with this substrate of about 4.2. The second order rate constant for inactivation is equal to the V=K value multiplied by the partition ratio. The observed 3.6-fold increase in the rate of inactivation upon deuteration is therefore a combination of the 3.1-fold decrease in the V=K value and the 12-fold increase in partitioning to inactive enzyme.

IV. CONCLUSION As demonstrated by the examples here, a branch point in an enzyme-catalyzed reaction that results in more than one product can result in an isotope effect if only one of the outcomes is isotope sensitive. The alternate pathway can be the formation of a different product, an unproductive reaction, or enzyme inactivation. The effect of isotopic substitution on the partitioning at this branch point will be closer to the intrinsic isotope effect for the isotope-sensitive step than will DV and DV/K values. Thus, for enzymes such as hydroxylases where multiple products are formed, measurement of such isotope effects provides a powerful tool for mechanistic analysis.

REFERENCES 1 Northrop, D. B., Steady-state analysis of kinetic isotope effects in enzymic reactions, Biochemistry, 14, 2644– 2651, 1975. 2 Cook, P. F. and Cleland, W. W., pH variation of isotope effects in enzyme-catalyzed reactions. 1. Isotope- and pH-dependent steps the same, Biochemistry, 20, 1797– 1805, 1981.

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3 Miwa, G. T., Garland, W. A., Hodshon, B. J., Lu, A. Y. H., and Northrop, D. B., Kinetic isotope effects in cytochrome P-450-catalyzed oxidation reactions. Intermolecular and intramolecular deuterium isotope effects during the N-demethylation of N,N-dimethylphentermine, J. Biol. Chem., 255, 6049– 6054, 1980. 4 Jones, J. P., Korzekwa, K., Rettie, A. E., and Trager, W., Isotopically sensitive branching and its effect on the observed intramolecular isotope effects in cytochrome P-450 catalyzed reactions: a new method for the estimation of intrinsic isotope effects, J. Am. Chem. Soc., 108, 7074– 7078, 1986. 5 Korzekwa, K. R., Trager, W. F., and Gillette, J. R., Theory for the observed isotope effects from enzymatic systems that form multiple products via branched reaction patyways: Cytochrome P-450, Biochemistry, 28, 9012– 9018, 1989. 6 Hanzlik, R. P., Hogberg, K., Moon, J. B., and Judson, C. M., Intramolecular kinetic deuterium isotope effects on microsomal hydroxylation and chemical chlorination of toluene-a-d1 and toluene-a,a-d2, J. Am. Chem., 107, 7164– 7167, 1985. 7 Sono, M., Roach, M. P., Coulter, E. D., and Dawson, J. H., Heme-containing oxygenases, Chem. Rev., 96, 2841– 2888, 1996. 8 Ling, K.-H. J. and Hanzlik, R. P., Deuterium isotope effects on toluene metabolism. Product release as a rate-limiting step in cytochrome P-450 catalysis, Biochem. Biophys. Res. Commun., 160, 844– 849, 1989. 9 Hjelmeland, L. M., Aronow, L., and Trudell, J. R., Intramolecular determination of primary kinetic isotope effects in hydroxylations catalyzed by cytochrome P-450, Biochem. Biophys. Res. Commun., 76, 541– 549, 1977. 10 Groves, J. T., McClusky, G. A., White, R. E., and Coon, M. J., Aliphatic hydroxylation by highly purified liver microsomal cytochrome P-450. Evidence for a carbon radical intermediate, Biochem. Biophys. Res. Commun., 81, 154– 160, 1978. 11 Hanzlik, R. P. and Ling, K.-H. J., Active site dynamics of xylene hydroxylation by cytochrome P-450 as revealed by kinetic deuterium isotope effects, J. Am. Chem. Soc., 115, 9363 –9370, 1993. 12 Audergon, C., Iyer, K. R., Jones, J. P., Darbyshire, J. F., and Trager, W. F., Experimental theoretical study of the effect of active-site constrained substrate motion on the magnitude of the observed intramolecular isotope effect for the p450 101 catalyzed benzylic hydroxylation of isomeric xylenes and 4,40 -dimethylbiphenyl, J. Am. Chem. Soc., 121, 41 – 47, 1999. 13 Hanzlik, R. P. and Ling, K.-H. J., Active site dynamics of toluene hydroxylation by cytochrome P-450, J. Org. Chem., 55, 3992– 3997, 1990. 14 Jones, J. P. and Trager, W., The separation of the intramolecular isotope effect for the cytochrome P-450 catalyzed hydroxylation of n-octane into its primary and secondary components, J. Am. Chem. Soc., 109, 2171–2173, 1987. 15 Miwa, G. T., Walsh, J. S., Kedderis, G. L., and Hollenberg, P. F., The use of intramolecular isotope effects to distinguish between deprotonation and hydrogen atom abstraction mechanisms in cytochrome P-450-and peroxidase-catalyzed N-demethylation reactions, J. Biol. Chem., 258, 14445– 14449, 1983. 16 Dinnocenzo, J. P., Karki, S. B., and Jones, J. P., On isotope effects for the cytochrome P-450 oxidation of substituted N,N-dimethylanilines, J. Am. Chem. Soc., 115, 7111– 7116, 1993. 17 Guengerich, F. P., Yun, C.-H., and MacDonald, T. L., Evidence for a 1-electron oxidation mechanism in N-dealkylation of N,N-dialkylanilines by cytochrome P450 2B1. Kinetic hydrogen isotope effects, linear free energy relationships, comparisons with horseradish peroxidase, and studies with oxygen surrogates, J. Biol. Chem., 271, 27321– 27329, 1996. 18 Hall, L. R. and Hanzlik, R. P., Kinetic deuterium isotope effects on the N-demethylation of tertiary amides by cytochrome P-450, J. Biol. Chem., 265, 12349– 12355, 1990. 19 Karki, S. B., Dinnocenzo, J. P., Jones, J. P., and Korzekwa, K. R., Mechanism of oxidative amine dealkylation of substituted N,N-dimethylanilines by cytochrome P-450: application of isotope effects profiles, J. Am. Chem. Soc., 117, 3657 –3664, 1995. 20 Dinnocenzo, J. P. and Banach, T. E., Deprotonation of tertiary amine cation radicals. A direct experimental approach, J. Am. Chem. Soc., 111, 8646– 8653, 1989. 21 Fitzpatrick, P. F., The tetrahydropterin-dependent amino acid hydroxylases, Annu. Rev. Biochem., 68, 355– 381, 1999.

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22 Fitzpatrick, P. F., Mechanism of aromatic amino acid hydroxylation, Biochemistry, 42, 14083– 14091, 2003. 23 Moran, G. R., Derecskei-Kovacs, A., Hillas, P. J., and Fitzpatrick, P. F., On the catalytic mechanism of tryptophan hydroxylase, J. Am. Chem. Soc., 122, 4535 –4541, 2000. 24 Moran, G. R., Phillips, R. S., and Fitzpatrick, P. F., The influence of steric bulk and electrostatics on the hydroxylation regiospecificity of tryptophan hydroxylase: characterization of methyltryptophans and azatryptophans as substrates, Biochemistry, 38, 16283– 16289, 1999. 25 Fitzpatrick, P. F., The steady state kinetic mechanism of rat tyrosine hydroxylase, Biochemistry, 30, 3658– 3662, 1991. 26 Fitzpatrick, P. F., Studies of the rate-limiting step in the tyrosine hydroxylase reaction: alternate substrates, solvent isotope effects, and transition state analogs, Biochemistry, 30, 6386– 6391, 1991. 27 Frantom, P. A. and Fitzpatrick, P. F., Uncoupled forms of tyrosine hydroxylase unmask kinetic isotope effects on chemical steps, J. Am. Chem. Soc., 125, 16190– 16191, 2003. 28 Hillas, P. J. and Fitzpatrick, P. F., A mechanism for hydroxylation by tyrosine hydroxylase based on partitioning of substituted phenylalanines, Biochemistry, 35, 6969– 6975, 1996. 29 Frantom, P. A., Pongdee, R., Sulikowski, G. A., and Fitzpatrick, P. F., Intrinsic deuterium isotope effects on benzylic hydroxylation by tyrosine hydroxylase, J. Am. Chem. Soc., 124, 4202– 4203, 2002. 30 Klinman, J. P., Mechanisms whereby mononuclear copper proteins functionalize organic substrates, Chem. Rev., 96, 2541– 2562, 1996. 31 Miller, S. M. and Klinman, J. P., Secondary isotope effects and structure – reactivity correlations in the dopamine b-monooxygenase reaction: evidence for a chemical mechanism, Biochemistry, 24, 2114– 2127, 1985. 32 Miller, S. M. and Klinman, J. P., Magnitude of intrinsic isotope effects in the dopamine b-monooxygenase reaction, Biochemistry, 22, 3091 –3096, 1983. 33 Fitzpatrick, P. F., Flory, D. R. Jr., and Villafranca, J. J., 3-Phenylpropenes as mechanism-based inhibitors of dopamine b-hydroxylase: evidence for a radical mechanism, Biochemistry, 24, 2108– 2114, 1985. 34 Fitzpatrick, P. F. and Villafranca, J. J., The mechanism of inactivation of dopamine b-hydroxylase by hydrazines, J. Biol. Chem., 261, 4510– 4518, 1986.

35

Chlorine Kinetic Isotope Effects on Biological Systems Piotr Paneth

CONTENTS I.

Introduction ...................................................................................................................... 875 A. Dehalogenation — Environmental Perspective....................................................... 875 B. Chlorine Kinetic Isotope Effects.............................................................................. 876 C. Calculations of Chlorine KIEs................................................................................. 876 D. Chlorine Isotopic Ratio Measurements ................................................................... 877 II. Chlorine Isotopic Fractionation in Microbial Processes ................................................. 878 A. Chlorine KIEs on Microbial Degradation of Chlorinated Compounds .................. 878 1. Reduction of Perchlorate................................................................................... 878 2. Reduction of Chlorinated Aliphatic Hydrocarbons .......................................... 879 B. Chlorine KIEs on Reactions Catalyzed by Dehalogenases..................................... 879 1. Haloalkane Dehalogenases................................................................................ 880 2. DL -2-Haloacid Dehalogenase ............................................................................ 883 3. Fluoroacetate Dehalogenase.............................................................................. 885 4. 4-Chlorobenzoil-CoA Dehalogenase ................................................................ 885 C. Chlorine KIEs on Enzymatic Halogenation ............................................................ 886 III. Future Perspectives .......................................................................................................... 888 Acknowledgments ........................................................................................................................ 888 References..................................................................................................................................... 888

I. INTRODUCTION A. DEHALOGENATION — E NVIRONMENTAL P ERSPECTIVE 1 Halogenated compounds constitute the most widespread class of chemicals in the present ecosystem. They originate from anthropogenic sources because of extensive use of halogenated organic compounds in dyes, pesticides, fire retardants, polymeric materials, and solvents. However, they do not come from xenobiotic sources only. For example some fungi are estimated to emit 1,60,000 tons of chloromethane to the atmosphere each year. Nevertheless, industrial activities have significantly disturbed the natural balance between production and consumption of certain halogenated compounds. A number of pathways have evolved that aim at regaining the natural balance by facilitating degradation of halogenated compounds. The key reaction of this degradation is the actual dehalogenation during which the halogen substituent, which is usually responsible for the toxic and xenobiotic character of the compound, is removed. There is a plethora of processes that lead to dehalogenation. The diversity is illustrated in Figure 35.1 using the example of chlorobenzoates for which at least three different dechlorination pathways are found in nature. Among halogenated compounds the most important from an environmental point of view are the chlorinated compounds. Their negative influence spreads throughout the whole ecosystem 875

876

Isotope Effects in Chemistry and Biology CO2− OH

CO2−

CO2− + H+

Cl

+ Cl−

H −

CO2

OH

FIGURE 35.1 Different pathways of dehalogenation of chlorobenzoates. Top; oxygenolytic by Burkholderia cepacia strain 2CBS (requires NADH, Hþ, and O2, yields CO2 and NADþ). Middle: reductive by Desulfomonile tiedjei (requires 2[H]). Bottom; hydrolytic by Pseudomonas sp. Strain CBS3 (requires H2O).

from ozone layer destruction in the atmosphere, through soil pollution by chloropesticides, to underground water contamination that poses health threats (e.g., perchlorite). Understanding of dehalogenation processes becomes, therefore, one of the most vigorously developing subjects of chemistry, biochemistry, and biotechnology.

B. CHLORINE K INETIC I SOTOPE E FFECTS In the 50 year history2 of chlorine kinetic isotope effects (KIEs) a substantial development in methodology has resulted in numerous reports of chlorine KIEs on chemical reactions. They were mostly measured in elimination and substitution reactions because negatively charged chlorine is frequently the leaving group in both of these classes of reaction. Chlorine KIEs seemed most suited to mechanistic applications because both of the factors controlling the magnitude of an KIE, namely the temperature-dependent and -independent factors, favor the light isotope, rendering leaving-group KIEs large and changing in magnitude as the position of the TS changes along the reaction path. The temperature-independent factor (TIF) originates in the isotopic mass difference in the isotopes, while the temperature-dependent factor (TDF) reflects crowding around the isotopic atom as the reactant reactant approaches the transition state. Mass-dependent isotopic fractionation increases with an increase in the mass difference between isotopomers and the decrease of atomic mass of the element. Thus, as described in other chapters KIEs of hydrogen isotopes are large while those of “heavy atoms” (usually carbon and heavier than carbon elements) are much smaller. Reviews,3 including both theoretical aspects and compilation of experimental data, of studies of chlorine KIEs have been published earlier, and thus no attempt is made here to recapitulate. It is worth keeping in mind that the theory predicts4 that the maximum chlorine KIE for an SN2 reaction is about 1.019 although the largest experimentally observed chlorine KIE5 is only 1.0125 and typical values are between 1.006 and 1.008.

C. CALCULATIONS OF C HLORINE KIE S For the purpose of the discussion in this chapter several calculations have been carried out. Recently, detailed analysis of the KIEs and their theoretical predictions have been presented for a model SN2 reaction in which chlorine atom is the leaving group.6 The chlorine KIE was

Chlorine Kinetic Isotope Effects on Biological Systems

877

1.0085

Chlorine KIE

1.0080 1.0075 1.0070 1.0065 1.0060 1.0055 2.0

2.1

2.2 2.3 C−Cl bond length

2.4

FIGURE 35.2 Chlorine KIE in the reaction between cyanide anion and ethyl chloride calculated at different levels of theory. Points from left to right correspond to semiempirical, DFT, post-HF, and ab initio levels, respectively. The error bars correspond to one standard deviation.

modeled over a wide number of levels of theory. In Figure 35.2 the results of these calculations are summarized. Points in this figure, from left to right, correspond to averaged results obtained from semiempirical, DFT, post-HF, and HF calculations, respectively. Single standard deviation values are indicated for both the magnitude of chlorine KIE and the C– Cl bond length. Comparison of these results with the experimental value of 1:0070 ^ 0:0003 indicates that semiempirical methods substantially underestimate the magnitude of the Cl-KIE, while “classical” ab initio methods equally strongly overestimate it. Not surprisingly, the best results are obtained at the most advanced post-HF levels. Good results were also obtained with DFT methods. Among these B3LYP and B1LYP functionals7 together with aug-cc-pVDZ basis set8 proved to give the best overall results. Therefore, calculations made for the discussion presented herein were carried out at the B1LYP/aug-cc-pVDZ level as implemented in the Gaussian program.9 Chlorine isotope effects were calculated using the ISOEFF98 program.10

D. CHLORINE I SOTOPIC R ATIO M EASUREMENTS In nature, the relative abundance of the two stable isotopes of chlorine, 35Cl and 37Cl, is close to 3:1, making natural chlorine isotopic composition very convenient for the analysis. As mentioned above, the most frequent product of a substitution or an elimination reaction is chloride. Thus the initial step is usually precipitation of silver chloride, which is used directly or indirectly for isotopic analysis. Alternatively, chlorinated organic compounds can be isolated (e.g., chromatographically) and then combusted to yield chloride. Procedures for quantitative conversion of chlorinated compounds to chloride are well established and widely used in quantitative analysis of organic compounds. Three different procedures used currently for the chlorine isotopic analysis, which employ methyl chloride, silver chloride, or cesium chloride, are briefly described below. Isotopic analysis is most precise when isotopic ratio mass spectrometry (IRMS) is used. IRMS analysis uses gaseous material and dual inlet systems that allow alternation between a sample and a reference gas. In the case of chlorine isotopic analysis volatile methyl chloride is used. Typically, the deviation from the isotopic composition of a standard is reported in parts-per-thousands (“per-mil,” ‰) — this quantity is usually denoted as d. In the case of chlorine isotopic composition, the reference gas standard mean ocean chloride (SMOC) is used.11 The high precision of the isotopic analysis of this method is offset to some extent by the need for conversion of chlorinated

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Isotope Effects in Chemistry and Biology

material into methyl chloride (tedious and prone to isotopic fractionation), although standardized procedures have been developed.12 Silver chloride can be used directly with fast atom bombardment – isotope ratio mass spectroscopy (FAB – IRMS) technique.13 The advantage of this method is that silver chloride is used directly. It is melted at a temperature below 4808C on a metal support and inserted directly into the mass spectrometer inlet system. The negative ion mass spectra show chlorine peaks at 35 and 37 m=z almost exclusively and can be measured simultaneously with two collectors. Compared to IRMS of gaseous samples, this method is less precise because no standard is introduced simultaneously with the sample. Although the requirement for AgCl is negligible, samples of about 5 mg for a single measurement are used. This amount is necessary for accurate handling of samples. Much smaller samples are necessary for the isotopic analysis that uses the Cs2Clþ signal from CsCl spectra obtained with thermal ionization mass spectrometry.14 As with the methyl chloride technique, conversion of chlorinated material to cesium chloride is necessary. Aqueous solutions of CsCl must be purified to eliminate organic contaminants that would prevent ionization to Cs2Clþ.

II. CHLORINE ISOTOPIC FRACTIONATION IN MICROBIAL PROCESSES Currently described chlorine isotopic studies on systems of biological importance can be divided into two groups. In the first the overall isotopic fractionation caused by whole organisms is studied. These studies usually aim at environmental applications of chlorine isotopic composition (e.g., identification and quantification of a pollutant). The processes involved usually consist of several dehalogenation steps that are not separable under the experimental conditions. In the second group, well defined, purified enzymes are used. These studies aim at learning mechanistic details of different dehalogenation processes. Below all of the studies described thus far in the literature are briefly presented.

A. CHLORINE KIES ON M ICROBIAL D EGRADATION OF C HLORINATED C OMPOUNDS 1. Reduction of Perchlorate Two recent papers on chlorine isotopic fractionation by biological systems deal with the reduction of perchlorate15,16 and are the result of the growing concern about its presence in drinking water because of groundwater contamination. Ammonium perchlorate is used as the primary oxidizer in the solid propellant in many rockets and missiles. Other perchlorate salts are also commonly used in fireworks, munitions, air bags, highway flares, and matches. Perchlorate competitively inhibits iodide transport resulting in Naþ – I2 symporter (NIS) malfunction and reduction of thyroid hormone production that can impair the development of the thyroid gland, posing substantial risk to pregnant women and children. A variety of bacteria (widely distributed in soils, groundwater, and surface water) can degrade perchlorate under anaerobic conditions using the molecule as the terminal electron acceptor. In the isotope effect experiments, Dechlorosoma suillum originating from groundwater was used. Large values of chlorine KIEs of 1:0169 ^ 0:0001 and 1:0148 ^ 0:0007 were obtained in both cases. These values are larger than any chlorine KIE reported on an elementary reaction and undoubtedly represent accumulation of individual chlorine KIEs of sequential, isotope-sensitive steps. It has been postulated that the overall degradation initially involves conversion of perchlorate to chlorate and then to chlorite, both catalyzed by perchlorate reductase.17 Subsequent reaction is catalyzed by chlorite dismutase, which yields chloride and dioxygen.18 The reduction of perchlorate to chloride is an interesting case from the chlorine isotope fractionation point of view. We are accustomed to chlorine KIEs being small. However, calculations indicate that the overall equilibrium isotope effect amounts to 1.06! Equilibrium chlorine isotope effects on individual steps are also very large because substantial loss of bonding to the isotopic atom

Chlorine Kinetic Isotope Effects on Biological Systems

879

occurs at each step: 1:0194

1:0218

1:0176

2 2 2 ClO2 4 ! ClO3 ! ClO2 ! Cl

Because no traces of intermediate reactants (chlorate and chlorite) have been detected it seems most likely that the first step determines the overall reaction rate, otherwise the substantial equilibrium chlorine isotope effect on this step should be expressed in the experimental value of chlorine KIE. Full understanding of the magnitude of the chlorine isotope fractionation for the overall perchlorate reduction and its individual steps evidently requires further investigations. Nevertheless, even at this early stage of studies, stable isotope analysis has proved to be the only technique whereby biodegradation can be successfully distinguished from other nonbiological mechanisms. 2. Reduction of Chlorinated Aliphatic Hydrocarbons Aerobic and anaerobic dechlorination of chlorinated alkanes have been described in the literature. These compounds, used as industrial solvents and degreasing agents, are persistent pollutants of surface soil, aquifers, and groundwater. Their removal from the environment is of particular importance because they are known or suspected to be carcinogenic or mutagenic in humans. Aerobic degradation of dichloromethane (DCM) by MC8b, a gram-negative methylotrophic organism, is characterized by a chlorine KIE19,20 of 1:0038 ^ 0:0003: It is assumed that this isotope effect is associated with the first step of the reaction, glutathione-dependent nucleophilic substitution. The chlorine KIE calculated for the gas phase SN2 reaction between methyl chloride and HS2 is 1.0077. Changing methyl chloride to dichloromethane lowers this value to 1.0059. Further change from HS2 to MeS2 yields a chlorine KIE of 1.0057. Thus considering a more realistic model of the reaction catalyzed by the enzyme changes the calculated chlorine KIE towards the observed isotope effect. However, the observed value seems to be too small to correspond directly to the intrinsic chlorine KIE and thus most probably there is another step partially rate determining that lowers the experimentally observed isotope effect. An interesting aspect of these studies is the proposal that the slope of the correlation between carbon and chlorine isotopic fractionations can be used as a characteristic signature that may be used for identification of dehalogenation processes in field studies at contaminated sites. Anaerobic processes show larger observed chlorine KIEs.21 Dehalogenation of tetrachloroethene (PCE) to cis-dichloroethene (c DCE) studied with two different species, Dehalospirilum multivorans and Dehalobacter restrictus shows chlorine KIEs of 1:0066 ^ 0:0010 and 1:0074 ^ 0:0006; respectively, when hydrogen is used as the electron donor. When pyruvate is used instead of hydrogen with D. multivorans the experimental value is 1:0058 ^ 0:0003; unchanged within the experimental error. It should be kept in mind that the reported values correspond to at least a two-step dehalogenation: Cl2 C ¼ CCl2 ! Cl2 C ¼ CHCl ! ClHC ¼ CHCl When the isotope fractionation of the transient kinetics is resolved the chlorine KIE on the second step can be evaluated.22 It was found to be equal to 1.0056 for three different anaerobic cultures. It is worth noticing that this value agrees very well with the abovementioned theoretical estimate.

B. CHLORINE KIES ON R EACTIONS C ATALYZED BY D EHALOGENASES In the microbial dehalogenations described above it is not possible to study chlorine KIEs on the individual reactions as several isotopically sensitive steps are operating simultaneously. Also, the nature of these individual steps, as well as structures of the active sites of the corresponding enzymes are not known. Thus, although these results are very important, e.g., in tracing the origin

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of particular pollutants, they are not helpful in learning mechanisms of chlorine isotopic fractionation. This goal can be achieved in studies of well-defined reactions catalyzed by isolated enzymes. Several such cases have been reported in the literature and are reviewed below. 1. Haloalkane Dehalogenases Haloalkane dehalogenases catalyze hydrolysis of the C – Cl bond of aliphatic chloroalkanes. The enzyme from Xanthobacter autotrophi-cus GJ10 (DhlA) uses a pollutant, 1,2-dichloroethane (DCE), as the “natural” substrate. Many aspects of the mechanism of this enzyme have been thoroughly studied both experimentally23 and theoretically24 because the production of the carcinogenic DCE used as substrate in the production of vinyl chloride and subsequently PVC is the most abundant of all chloroorganic products.25 It is assumed that the reaction proceeds with two subsequent nuclephilic substitutions with Asp124 acting as the nucleophile. In the first chemical step chloride ions are released and an enzyme –intermediate complex is formed, and in the second step this complex is hydrolyzed to alcohol with the regeneration of the free enzyme as shown in Figure 35.3. The release of chlorine from DCE, which is hydrogen bonded to two tryptophan residues (Trp125 and Trp175), was shown to determine the overall reaction rate. Chlorine KIEs measured for the dehalogenase are 1.0045 for 1,2-dichloroethane and 1.0066 for 1-chlorobutane, a slow substrate. Results were interpreted as indicative of the isotope effect measured for 1-chlorobutane being very close to an intrinsic chlorine KIE, while the lower value obtained with DCE suggests mechanistic complexity. It has been argued that the reversibility of the dehalogenation step nicely explains all kinetic observations. These conclusions were supported by the ONIOM calculations26 of the intrinsic chlorine KIEs in the enzyme active site at the B1LYP/ 6-31G(d):PM3 level.27,28 For both substrates a chlorine KIE of about 1.0065 has been calculated suggesting that the same intrinsic chlorine KIE should be expected in both cases. However, the reversibility of the dehalogenation step does not agree with the energetics obtained in these calculations. In fact, an important problem has been overlooked in the analysis of these isotope effects. For the slow substrate it was impossible to achieve a 100% conversion of substrate to product, and therefore isotopic analysis of the reactant was performed by combusting it and recovering the chloride formed. In the case of DCE, however, isotopic analysis of the product after full conversion was performed. Since the substrate has two equivalent chlorine positions, only one of which is being split during the enzymatic dehalogenation, the isotopic composition of the product after full conversion is not equal to the isotopic composition of the initial substrate. The isotopic composition of chloride originating from molecules containing two different isotopes reflects the intrinsic isotope effect, as illustrated in Figure 35.4. Assuming equal isotopic compositions at both sites and neglecting the secondary chlorine KIE, it can be shown that the isotopic composition of the product at full conversion (R1) will deviate from that of the starting DCE (R0) due to the intramolecular isotope effect on the singly labeled DCE molecules. The magnitude of this deviation depends on the isotopic composition. In the limiting cases, if no singly labeled molecules are present then R1 ¼ R0 : If only singly labeled O− Enz

C

H

R2

C R1 O −OOC

Cl

H Cl−

Enz

O−

R

O

O C

C

R1 COO−

Enz

+ H+ + HO

C O

R2 C

R1 COO−

O

FIGURE 35.3 Enzymatic dehalogenation proceeding through two consecutive SN2 elemental steps and a covalent enzyme – intermediate complex.

Chlorine Kinetic Isotope Effects on Biological Systems

ClL−CH2−CH2−ClL

881 2kL

kL ClH−CH2−CH2−ClL

ClH−CH2−CH2−ClH

ClL− ClL−

kH ClH− 2kH

ClH−

FIGURE 35.4 Isotopic rate constants of different isotopomers of DCE leading to differences in isotopic composition of the reactant before reaction and product after full conversion because of the intramolecular KIE.

DCE molecules are present, on the other hand, then the ratio R0 =R1 will be equal to the intrinsic isotope effect kL/kH. In the case of natural chlorine isotopic composition (L:H about 3:1) the ratio of ClL – CH2 – CH2 –ClL to ClH – CH2 – CH2 – ClL is about 9:6 and therefore only about 40% (6/15) of the intrinsic chlorine KIE should be expressed. Thus the isotope effect calculated using R1 instead of R0 should be smaller by about 0.0025 and this is what is seen within the experimental error in case of DCE dehalogenation by DhlA. Inclusion of different isotopic composition at the two sites introduces a statistical factor but does not change the final result. The difference gets a bit smaller when the secondary Cl-KIE is included but this change is negligible because this isotope effect is small; e.g., for the reaction between DCE and cyanide ion in the gas phase the primary Cl-KIE has been calculated to be equal to 1.0070 while the secondary Cl-KIE is only 1.00015! Combination of the experimental values with modeling of the intrinsic Cl-KIE indicate that the ˚ compared to the length of breaking C – Cl bond in the transition state is elongated by about 0.5 A this bond in the reactant, and the forming O – C bond is longer than the one in the product of ˚ . In the traditional, qualitative description of transition states, this the dehalogenation step by 0.64 A corresponds to the symmetrical, exploded (loose) transition state. The calculations further indicate that there is enough flexibility within the active site that the carboxylic moiety of Asp124 moves ˚ toward the carbon atom of 1,2-dichloroethane and acts as the nucleophile. This aspartate about 1 A is immediately N terminal to tryptophan-125, which participates in the hydrogen bonding of the chlorine atom of the transition state. Thus, the optimal geometry for the hydrogen bonding network has to be compromised to allow nucleophilic attack by the carboxyl group. As a result, the tryptophan rings are rotated out of their optimal positions during the reaction; the dehalogenation step resembles drawing a bow: the high-energy cost is compensated by the increased hydrogen bonding to the chlorine atom. The subsequent hydrolysis triggers release of the strain and the return of the tryptophan residue to the optimal position. The reverse process is energetically unfavorable. This scenario, illustrated in Figure 35.5, has recently been confirmed by molecular dynamic studies of this system29 as well as by whole enzyme quantum mechanical calculations. Recently, the same set of Cl-KIEs has been measured for another enzyme in this family, haloalkane dehalogenase LinB.30 A distinctive difference between DhlA and LinB is a reversal of specificity of these enzymes: LinB catalyzes dehalogenation of chlorobutane a few orders of magnitude faster than DCE. The measured chlorine KIEs are equal to 1:0065 ^ 0:0004 and 1:0066 ^ 0:0004 for DCE and chlorobutane, respectively. In both cases R0 has been used. The magnitude of the observed chlorine KIEs suggests that in the case of LinB the observed values are equal to the intrinsic Cl-KIE for the dehalogenation. The postulated mechanism assumes that the hydrolysis of the enzyme-bound intermediate formed in the dehalogenation is the over-all rate-determining step.31 Observation of the full intrinsic KIE on a step preceeding

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FIGURE 35.5 Optimized structures of the reactant, transition state, and product of the dehalogenation step catalyzed by DhlA. (Reproduced from Lewandowicz, A., Rudzinski, J., Tronstad, L., Widersten, M., Ryberg, P., Matsson, O., and Paneth, P., J. Am. Chem. Soc., 123, 4550– 4555, 2001. With permission.)

the rate-determining step implies that the dehalogenation step is irreversible and the steps preceeding it do not introduce a commitment (see Equation 37.17). We have carried out calculations using a novel program that allows optimization of the whole enzyme, reactants, and crystallographic water at the quantum (semiempirical) level.32 These calculations indicate that the difference in mechanisms of DhlA and LinB lies in the larger stabilization of the enzyme-bound intermediate in the case of the former enzyme. Figure 35.6 illustrates a structure obtained for the DhlA-calatyzed dehalogenation of DCE. Feasibility of performing quantum mechanical calculations on the whole enzymatic systems (in the present case about 5500 atoms!) should allow an increase in our understanding of the interplay between all important factors of enzymatic catalysis of haloalkane dehalogenases in the near future. The results of calculations can be accommodated together with the experimental Cl-KIEs when it is assumed that in both enzymes the dehalogenation step for both reactants is irreversible. The requirement that no commitment is expressed in the observed Cl-KIE does not, however, agree

FIGURE 35.6 Structure of DhlA with reactant and crystallographic water optimized at AM1 level.33

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883

with the recent molecular dynamics results,34 which suggest that the slow step in LinB catalysis of DCE is its transport into the active site. Vigorous studies are underway to clarify this issue. 2.

DL -2-Haloacid

Dehalogenase

DL -2-haloacid

dehalogenase from Pseudomonas sp. 113 (DL -DEX 113) catalyzes the hydrolytic dehalogenation of both D - and L -2-haloalkanoic acids. Site-directed mutagenesis studies35 of 26 catalytically essential residues that are conserved between DL -DEX 113 and D -2-haloacid dehalogenase showed that Thr65, Glu69, and Asp194 are essential for the dehalogenation of both R(þ )-2-haloalkanoic acids and S(2 )-2-haloalkanoic acids. The change in activity of all mutants toward R(þ )- and S(2 )-2-chloropropionate are about equal. Moreover, R(þ )-2-chloropropionate competitively inhibits the enzymatic dehalogenation of S(2 )-2-chloropropionate, and vice versa. Based on these results, it has been proposed that both enantiomers share a common active site of DL -DEX 113. Typically, dehalogenases react via the mechanism depicted in Figure 35.3; an active site carboxylate group attacks the substrate carbon atom bonded to the halogen atom, leading to the formation of an ester intermediate that is subsequently hydrolyzed. 18O trace studies36 indicate, however, that in case of DL -DEX 113 a water molecule directly attacks the a-carbon of 2-haloalkanoic acid to displace the halogen atom (Figure 35.7) and the reaction proceeds without formation of the enzyme-bound intermediate. Different chlorine KIEs have been found for the stereoisomers; 1:0105 ^ 0:0001 for S(2 )-2chloropropionate and 1:0090 ^ 0:0005 for the R(þ )-isomer.37 Two alternative explanations of this result can be proposed. It has been suggested previously that the intrinsic chlorine KIEs for these two stereoisomers are different due to their different orientation within the active site. The orientation illustrated on the right side of Figure 35.8 causes interactions with the leaving group to be missing in case of S(2 )-2-chloropropionate what leads to larger chlorine KIE for this isomer. Small rate differences between the isomers favor, however, an alternative explanation in which the same intrinsic Cl-KIE is assumed for both stereoisomers and the observed isotope effects are different because of the presence of a commitment factor. If the Cl-KIE observed for S(2 )-2chloropropionate is assumed to be equal to an assumed intrinsic isotope effect of 1.0105 then (from Equation 37.17) it follows that the commitment factor for the R-(þ )- reactant is equal to 0.2. In fact, the magnitude of the chlorine KIE observed for S(2 )-isomer of 1.0105 is very reasonable for the intrinsic value. In recent studies of the dependence of Cl-KIEs on the nature of the nucleophile in a model SN2 reaction38 it was found that for most of the typical nucleophiles the chlorine KIE falls within a small range of 1.006 – 1.008. Only in cases of very low or very high C –Cl bond order values in the transition state are Cl-KIEs outside the range are found, as illustrated in Figure 35.9. Circles in Figure 35.9 illustrate the change in the Cl-KIE for methyl chloride with a water molecule acting as the nucleophile (squares correspond to anionic nucleophile NH2 2 ). The values plotted were obtained by using a continuum COSMO solvent model39 in calculations and parameters for different solvents. In order to reach extreme values of Cl-KIE an artificially lowered H O− Enz

C O

H

O

R2 C

R1 − OOC

O−

Cl

Enz Cl−

+ H+ + HO

C O

R2 C

R1 COO−

FIGURE 35.7 Enzymatic dehalogenation proceeding through single SN2 reaction with direct attack of the water molecule.

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FIGURE 35.8 Chlorine atom of R-(þ)-transition state hydrogen hydrogen-bounded to a residue of the DL -DEX 113 active site (left structure). S-(2 )-transition state (right structure) lacking such interactions due to steric hindrance and electrostatic repulsion.

dielectric constant of the solvent had to be used as illustrated in Table 35.1 on the example of water as the nucleophile. As can be seen from Figure 35.9 and Table 35.1, a chlorine KIE of over 1.01 is obtained for water molecules acting as a nucleophile in an environment of very low polarity (consistent with active sites of enzymes). The above results nicely correlate with the chlorine KIEs measured for the DL -DEX 113 catalyzed reactions and explain the differences in Cl-KIE between enzymes reacting via pathways in Figure 35.3 and Figure 35.7. In the case of DL -DEX 113 it is thus possible that the intrinsic chlorine KIE is observed for S-(2 )-isomer and part of it is masked by a commitment in the reaction with the R-(þ )-stereoisomeric reactant. It is, however, also possible that the value of the intrinsic Cl-KIE is even higher and in both cases it is masked by commitments but to a different extent. Presently available data do not allow differentiation between these alternative scenarios.

1.020

Chlorine KIE

1.015

1.010 Typical nuclephiles 1.005

1.000 0.00

0.25

0.50 C–Cl bond order

0.75

1.00

FIGURE 35.9 Dependence of calculated chlorine KIE on the C– Cl bond order in the transition state (see text for details).

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TABLE 35.1 Chlorine KIEs in the SN2 Reaction Between Water and Methyl Chloride Calculated Using a Continuum CPM Solvent Model Solvent Used in Calculations

Dielectric Constant

C –Cl Bond Order

Cl-KIE

78.39 7.58 4.335 3.5 3.0 2.7

0.41 0.37 0.32 0.29 0.25 0.22

1.0080 1.0082 1.0085 1.0089 1.0099 1.0115

Water THF Diethyl ether Diethyl ether Diethyl ether Diethyl ether

3. Fluoroacetate Dehalogenase Fluoroacetate dehalogenase catalyzes the hydrolytic dehalogenation of haloacetates to produce glycolate and a halide ion via the pathway depicted in Figure 35.3. Recent studies40 indicate that the carboxyl moiety of Asp105 acts as the nucleophile attacking the a-carbon of the reactant and leading to the formation of an ester intermediate, which is subsequently hydrolyzed in the ratelimiting step.41 Activity of this enzyme toward fluoroacetate is about five times higher than toward chloroacetate although the dissociation energy of the C –F bond of aliphatic fluorocompounds is among the highest found in bonds of natural products. Fluoroacetate dehalogenase is an example of a successful application of an enzymatic process in practice; it has been applied to the detoxification of poisonous plants containing high concentrations of fluoroacetate ingested by domestic animals.42 Lack of the three-dimensional structure of this enzyme precludes modeling of the processes in the active site. However, both theoretical calculations and the experimental chlorine KIE on the chemical dehalogenation of chloroacetate by direct nucleophilic attack of hydroxide ð1:0079 ^ 0:0004Þ; suggest that the experimental value of the chlorine KIE in the enzyme reaction ð1:0082 ^ 0:0005Þ is close to the intrinsic KIE and that there are no steps preceding the dehalogenation step that contribute to the overall reaction rate. It also implies that the dehalogenation step is irreversible. The overall mechanism is thus very similar to that of LinB haloalkane dehalogenase. 4. 4-Chlorobenzoil-CoA Dehalogenase The hydrolytic dehalogenation reaction of 4-chlorobenzoyl-coenzyme-A (4-CBA-CoA) catalyzed by Pseudomonas sp. strain 4-CBA-CoA dehalogenase43,44 is a step in the 4-CBA degradation pathway operational in bacteria adapted to use this soil pollutant as an energy source.45 The nucleophilic displacement of the chloride ion by an active site carboxylate (Asp145) proceeds via an SNAr mechanism that is a stepwise process with formation of a Meisenheimer complex. The transition state for its formation is illustrated by Figure 35.9. Hydrolysis of the benzoyl ester, assisted by His90, generates 4-HBA-CoA that, along with a proton and the chloride ion, is released from the enzyme active site. An important difference between the mechanism of catalysis of this enzyme and the other dehalogenases described above is that this reaction is reversible. This means that the observed chlorine KIE, 1:0090 ^ 0:0006; is smaller than the intrinsic value for the formation of the Meisenheimer complex. On the basis of rates of all of the individual steps evaluated from kinetic measurements the intrinsic chlorine KIE of 1.0125 has been estimated, matching the largest experimental value of chlorine KIE reported in the literature.46 Two features of the transition state depicted in Figure 35.10 and the corresponding isotope effect are worth noticing. First, unlike other dehalogenases the nucleophilic attack is not in line; the forming O –C bond and the breaking C – Cl

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FIGURE 35.10 Structure of the transition state for the Meisenheimer complex formation calculated at semiemirical level.

bond do not lie on a straight line. Second, analysis of the intrinsic chlorine KIE calculated at the AM1 level indicates that the major contributions to this isotope effect originate predominantly in bending rather than stretching vibrations. This is very uncommon and probably due to the aromatic character of the phenyl ring. Distortion of the chlorine atom from the plane of the ring is in this case so energetically unfavorable that it gives rise to large chlorine KIE.

C. CHLORINE KIES ON E NZYMATIC H ALOGENATION Isotopic fractionation during enzymatic chlorination has been reported only for Fe(III)-hemechloroperoxidase (CPO), an enzyme isolated from the fungus Caldariomyces fumago. This enzyme catalyzes the reaction in the direction opposite to dehalogenation; introduction a chlorine atom that originates as a chloride into organic compounds. Studies were undertaken specifically to address the question of the origin of some chlorinated compounds detectable worldwide, in particular to test whether the chlorine isotope signature can differentiate between natural and anthropogenic sources of these pollutants. The commonly accepted mechanism implies oxidation of heme moiety to O ¼ Fe(IV)þ by H2O2 in the first step. Upon binding of chloride it is then oxidized to OCl2, which in turn chlorinates the organic reactant in subsequent step(s): FeðIIIÞ 2 CPO þ H2 O2 $ FeðIVÞþ 2 CPO þ H2 O FeðIVÞþ 2 CPO þ Cl2

rate-determining step

$

FeðIIIÞ 2 CPO þ OCl2

OCl2 þ TMB=DMP $ chlorinated products It has been postulated that the oxidation of chloride is the overall rate determining step. In case of 1,3,5-trimethoxybenzene (TMB) over 90% of the product was 2,4-dichloro-1,3,5-trimethoxybenzene (with traces of mono- and trichlorinated congeners), while in case of 3,5-dimethylphenol (DMP) trichlorinated and dichlorinated products were obtained in a 3:1 ratio.

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Chlorine KIEs have been found to be equal to 1:0123 ^ 0:0003 for TMB and 1:0107 ^ 0:0002 for DMP, respectively. The interpretation of these KIEs is not easy due to experimental and theoretical complications. First, they were determined from isotopic composition of the starting chloride and chlorinated products, which were combusted and converted to methyl chloride for isotopic composition analysis without separation. Thus, even if equal values of the chlorine KIEs are assumed for the first, second, and third chlorine atom incorporation (and there is no guarantee that these KIEs are equal), the experimental isotope effects represent a complicated function of multiple KIEs.47 Secondly, the enzymatic reaction is complex with more than one step sensitive to isotopic substitution; both oxidation of chloride to hypochlorite and chloride transfer from hypochlorite to the benzene ring should fractionate chlorine isotopes. Furthermore, for the first of these reactions an inverse chlorine isotope effect should be expected because in the course of reaction a new bond to the isotopic atom is being formed. Theoretical evaluation of the equilibrium chlorine isotope effect for this step yielded a value of 0.9942. The KIE should be close to unity as TIF and TDF factors act in opposite directions and mostly cancel out. Normal (larger than unity) values of the experimental Cl-KIEs suggest that the second step is irreversible for both reactants and its intrinsic KIEs are mostly expressed in the observed value. It is not, however, clear why the chlorine KIE on chlorine atom transfer should yield such large isotope effects. Unfortunately, no theoretical or experimental data are available to address the question of the magnitude of the chlorine KIE in such reactions. The experimental chlorine KIEs for the nonenzymatic reactions have been determined but without knowledge of their actual mechanism these values cannot be used as a measure of the intrinsic value. Thus, while reaching the goal of demonstrating the usefulness of chlorine isotopic fractionation in distinguishing between biotic and abiotic formation of chlorinated pollutants, these studies pose more questions regarding the magnitude of chlorine KIEs on elemental processes than they answer.

TABLE 35.2 Library of Chlorine KIEs on Biological Systems Substrate

Enzyme/Organisms

Cl-KIE

Ref.

DCE t ButCl DCE t ButCl (R)-(þ )-2-Chloropropionate (S)-(2)-2-Chloropropionate Cl-Bz-CoA ClCH2COO2 DCM TCE TCE TCE PCE PCE PCE ClO2 4 ClO2 4 TMB DMP

Haloalkane dehalogenase DhlA Haloalkane dehalogenase DhlA Haloalkane dehalogenase LinB Haloalkane dehalogenase LinB DL -2-haloacid dehalogenase DL -2-haloacid dehalogenase 4-Chlorobenzoyl-CoA dehalogenase Fluoroacetate dehalogenase Gram-negative MC8b StrainT Consortium N Consortium F D. multivorans/hydrogen D. multivorans/pyruvate Db. restrictus/hydrogen Dechlorosoma suillum Dechlorosoma suillum Fe(III)-heme-chloroperoxidase Fe(III)-heme-chloroperoxidase

1.0045 ^ 0.0004 1.0066 ^ 0.0004 1.0065 ^ 0.0004 1.0066 ^ 0.0004 1.0090 ^ 0.0005 1.0105 ^ 0.0001 1.0090 ^ 0.0006 1.0082 ^ 0.0005 1.0038 ^ 0.0003 1.0055 ^ 0.0009 1.0056 ^ 0.0007 1.0057 ^ 0.0010 1.0066 ^ 0.0010 1.0058 ^ 0.0003 1.0074 ^ 0.0006 1.0169 ^ 0.0001 1.0148 ^ 0.0007 1.0123 ^ 0.0003 1.0107 ^ 0.0002

48,49 48,49 20 20 27,49 27,49 49,50 49,51 19,20 22 22 22 21 21 21 15 16 52 52

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III. FUTURE PERSPECTIVES It is evident that with halogenated compounds being the largest group of pollutants in air, groundwaters, and soil, understanding chlorine isotopic fractionation is becoming a pressing issue not only from the scientific but mainly from the practical point of view. At the same time, our current understanding of this phenomenon is surprisingly deficient not only at the microbial level but also in cases where (de)halogenating enzymes are isolated and well characterized. With the chlorine isotopic signature emerging as the tool of the future in identification and tracing of chlorinated pollutants it is evident that a vast amount of work is needed to extend our understanding of dehalogenating systems and to build a sufficiently large library of chlorine isotope effects. Table 35.2 summarizes the presently known experimental values of chlorine KIEs for the enzymatic systems. It is author’s hope that it will be the germ for the further exploration of the chlorine isotopic fractionation of biologically relevant systems.

ACKNOWLEDGMENTS This work was supported by a series of grants from the State Committee for Scientific Research (Poland), the most recent being grant 4 T09A 030 25 (2003 –2006).

REFERENCES 1 Ha¨ggblom, M. M. and Bossert, I. D., Eds., In Dehalogenation Microbial Processes and Environmental Applications, Kluwer Academic Publishers, Boston, 2003. 2 Bartholomew, R. M., Brown, F., and Lounsbury, M., Chlorine isotope effect in reactions of tert-butyl chloride, Can. J. Chem., 32, 979– 983, 1954. 3 Shiner, V. J., Jr. and Wilgis, F. P., Heavy atom isotope rate effects in solvolytic nucleophilic reactions at saturated carbon, Heavy Atom Isotope Effects, Isotopes in Organic Chemistry, Vol. 8, Elsevier, Amsterdam, pp. 239–336, Chap. 6, 1992. 4 Paneth, P., Some analytical aspects of the measurements of heavy atom kinetic isotope effects, Talanta, 34, 877– 883, 1987. 5 Koch, H. F., McLennan, D. J., Koch, J. G., Tumas, W., Dobson, B., and Koch, N. H., Use of kinetic isotope effects in mechanistic studies. 4. Chlorine isotope effects associated with alkoxide-promoted dehydrochlorination reactions, J. Am. Chem. Soc., 105, 1930– 1937, 1983. 6 Fang, Y.-r., Gao, Y., Ryberg, P., Eriksson, J., KoŁodziejska-Huben, M., DybaŁa-Defratyka, A., Danielsson, R., Paneth, P., Matsson, O., and Westaway, K. C., Experimental and theoretical multiple kinetic isotope effects for an SN2 reaction. An attempt to determine transition-state structure and the ability of theoretical methods to predict experimental kinetic isotope effects, Chem. Eur. J., 9, 2696– 2709, 2003. 7 Becke, A. D., Density-functional thermochemistry. III. The role of exact exchange, J. Chem. Phys., 98, 5648– 5652, 1993; Becke, A. D., Density-functional thermochemistry. IV. A new dynamical correlation functional and implications for exact-exchange mixing, J. Chem. Phys., 104, 1040– 1046, 1996. 8 Woon, D. E. and Dunning, H. T. Jr., Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminium through argon, J. Chem. Phys., 98, 1358– 1371, 1993. 9 Gaussian 98 (Revision A.9), Frisch, M. J., Trucks, G. W., Schlegel, H. B., Scuseria, G. E., Robb, M. A., Cheeseman, J. R., Zakrzewski, V. G., Montgomery, J. A., Stratmann, R. E., Burant, J. C., Dapprich, S., Millam, J. M., Daniels, A. D., Kudin, K. N., Strain, M. C., Farkas, O., Tomasi, J., Barone, V., Cossi, M., Cammi, R., Mennucci, B., Pomelli, C., Adamo, C., Clifford, S., Ochterski, J., Petersson, G. A., Ayala, P. Y., Cui, Q., Morokuma, K., Malick, D. K., Rabuck, A. D., Raghavachari, K., Foresman, J. B., Cioslowski, J., Ortiz, J. V., Stefanov, B. B., Liu, G., Liashenko, A., Piskorz, P., Komaromi, I., Gomperts, R., Martin, R. L., Fox, D. J., Keith, T., Al-Laham, M. A., Peng, C. Y., Nanayakkara, A.,

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10 11 12

13 14

15 16 17 18 19 20 21 22 23

24

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Gonzalez, C., Challacombe, M., Gill, P. M. W., Johnson, B. G., Chen, W., Wong, W. M., Andres, J. L., Head-Gordon, M., Replogle, E. S., and Pople, J. A., Gaussian, Inc., Pittsburgh, PA, 1998. Anisimov, V. and Paneth, P., ISOEFF98. A program for studies of isotope effects using hessian modifications, J. Math. Chem., 26, 75 –86, 1999. Long, A., Eastoe, C. J., Kaufmann, R. S., Martin, J. G., Wirt, L., and Finley, J. B., High-precision measurement of chlorine stable isotope ratios, Geochim. Cosmochim. Acta, 57, 2907– 2912, 1993. Holt, B. D., Sturchio, C. N., Abrajano, T., and Heraty, L. J., Conversion of chlorinated organic compounds to carbon dioxide and methyl chloride for isotopic analysis of carbon and chlorine, Anal. Chem., 69, 2727– 2733, 1997; Ader, M. L., Coleman, M., Doyle, S. P., Stroud, M., and Wakelin, D., Methods for the stable isotopic analysis of chlorine in chlorate and perchlorate compounds, Anal. Chem., 73, 4946– 4951, 2001. Westaway, K. C., Koerner, T., Fang, Y.-R., Rudzin´ski, J., and Paneth, P., A new method of determining chlorine kinetic isotope effects, Anal. Chem., 70, 3548– 3552, 1998. Numata, M., Nakamura, N., Koshikawa, H., and Terashima, Y., Chlorine stable isotope measurements of chlorinated aliphatic hydrocarbons by thermal ionization mass spectrometry, Anal. Chim. Acta, 455, 1 – 9, 2002; Numata, M., Nakamura, N., and Gamo, T., Precise measurement of chlorine stable isotopic ratios by thermal ionization mass spectrometry, Geochem. J., 35, 89 – 100, 2001. Coleman, M. L., Ader, M., Chaudhuri, S., and Coates, J. D., Microbial isotopic fractionation of perchlorate chlorine, Appl. Environ. Microbiol., 69, 4997– 5000, 2003. Sturchio, N. C., Hatzinger, P. B., Arkins, M. D., Suh, C., and Heraty, L. J., Chlorine isotope fractionation during microbial reduction of perchlorate, Environ. Sci. Technol., 37, 3859– 3863, 2003. Kengen, S. W. M., Rikken, G. B., Hagen, W. R., van Ginkel, C. G., and Stams, A. J. M., Purification and characterization of (per)chlorate reductase from the chlorate-respiring strain GR-1, J. Bacteriol., 181, 6706– 6711, 1999. Coates, J. D., Michaelidou, U., Bruce, R. A., O’Connor, S. M., Crespi, J. N., and Achenbach, L. A., The ubiquity and diversity of dissimilatory (per)chlorate-reducing bacteria, Appl. Environ. Microbiol., 65, 5234– 5241, 1999. Heraty, L. J., Fuller, M. E., Huang, L., Abrajano, T. Jr., and Sturchio, N. C., Isotopic fractionation of carbon and chlorine by microbial degradation of dichloromethane, Org. Geochem., 30, 793– 799, 1999. Sturchio, N. C., Clausen, J. C., Heraty, L. J., Huang, L., Holt, B. D., and Abrajano, T., Stable chlorine isotope investigation of natural attenuation of trichloroethene in an aerobic aquifer, Environ. Sci. Technol., 32, 3037– 3042, 1998. Coleman, M. L., McGenity, T. J., and Isaacs, M. C. P., Stable isotopic fractionation associated with anaerobic degradation of chlorinated hydrocarbons, Nineth Annual V. M. Goldschmidt Conference, Lunar and Planetary Institute, Houston, Abstract #7253, LPI Contribution No. 971, 1999. Numata, M., Nakamura, N., Koshikawa, H., and Terashima, Y., Chlorine isotope fractionation during reductive dechlorination of chlorinated ethenes by anaerobic bacteria, Environ. Sci. Technol., 36, 4389– 4394, 2002. Pikkemaat, M. G., Ridder, I. S., Rozeboom, H. J., Kalk, K. H., Dijkstra, B. W., and Janssen, D. B., Crystallographic and kinetic evidence of a collision complex formed during halide import in haloalkane dehalogenase, Biochemistry, 38, 12052– 12061, 1999; Schindler, J. F., Naranjo, P. A., Honaberger, D. A., Chang, C.-H., Brainard, J. R., Vanderberg, L. A., and Unkefer, C. J., Haloalkane dehalogenases: steady-state kinetics and halide inhibition, Biochemistry, 38, 5772– 5778, 1999; Krooshof, G. H., Ridder, I. S., Tepper, A. W. J. A., Vos, G. J., Rozeboom, H. J., Kalk, K. H., Dijkstra, B. W. and Janssen, D. B., Kinetic characterization and X-ray structure of a mutant of haloalkane dehalogenase with higher catalytic activity and modified substrate range, Biochemistry, 37, 15013– 15023, 1998. Damborsky´, J., Kuty´, M., Ne´mec, M., and Kocˇa, J., A molecular modeling study of the catalytic mechanism of haloalkane dehalogenase: 2. Quantum chemical study of complete reaction mechanism, J. Chem. Inf. Comput. Sci., 37, 562– 568, 1997; Lightstone, F. C., Zheng, Y.-J., and Bruice, T. C., Molecular dynamics simulations of ground and transition states for the SN2 displacement of Cl- from 1,2-dichloroethane at the active site of xanthobacter autotrophicus haloalkane dehalogenase, J. Am. Chem. Soc., 120, 5611–5621, 1998; Robert, D., Girone´s, X., and Carbo-Dorca, R. C., Quantification of the influence of single-point mutations on haloalkane dehalogenase activity: A molecular quantum similarity study, J. Chem. Inf. Comput. Sci., 40, 839– 846, 2000.

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25 Ha¨ggblom, M. M. and Bossert, I. D., Halogenated organic compounds — a global perspective, In Dehalogenation Microbial Processes and Environmental Applications, Ha¨ggblom, M. M. and Bossert, I. D., Eds., Kluwer Academic Publishers, Boston, pp. 3 – 29, 2003. 26 Maseras, F. and Morokuma, K., IMOMM: A new ab Initio þ molecular mechanics geometry optimization scheme of equilibrium structures and transition states, J. Comput. Chem., 16, 1170– 1179, 1995. 27 Hariharan, P. C. and Pople, J. A., The influence of polarization functions on molecular orbital hydrogenation energies, Theor. Chim. Acta, 28, 213– 222, 1973. 28 Stewart, J. J. P., Optimization of parameters for semiempirical methods. I. methods, J. Comput. Chem., 10, 209– 220, 1989. 29 Devi-Kesavan, L. S. and Gao, J., A combined QM/MM study of the mechanism and kinetic isotope effect of the nucleophilic substitution reaction in haloalkane dehalogenase, J. Am. Chem. Soc., 125, 1532– 1540, 2003. 30 Sicinska, D., Kolodziejska-Huben, M., Prokop, Z., Monincova, M., Damborsky, J., and Paneth, P., in preparation. 31 Michal Bohac, M., Nagata, Y., Prokop, Z., Prokop, M., Monincova, M., Tsuda, M., Koca, J., and Damborsky, J., Halide-stabilizing residues of haloalkane dehalogenases studied by quantum mechanic calculations and site-directed mutagenesis, Biochemistry, 41, 14272– 14280, 2002. 32 Anikin, N. A., Anisimov, V. M., Bugaenko, V. L., Bobrikov, V. V., and Andreyev, A. M., LocalSCF Method for semiempirical quantum-chemical calculation of ultra-large bio-molecules, J. Chem. Phys., 121, 1266– 1270, 2004. 33 Dewar, M. J. S., Zoebisch, E. G., Healy, E. F., and Stewart, J. J. P., Development and use of quantum mechanical molecular models. 76. AM1: a new general purpose quantum mechanical molecular model, J. Am. Chem. Soc., 107, 3902– 3909, 1985. 34 Streltsov, V. A., Prokop, Z., Damborsky, J., Nagata, Y., Oakley, A., and Wilce, M. C. J., Haloalkane dehalogenase LinB from Sphingomonas paucimobilis UT26: x-ray crystallographic studies of dehalogenation of brominated substrates, Biochemistry, 42, 12719– 12720, 2003. 35 Motosugi, K., Esaki, N., and Soda, K. J., Purification and properties of a new enzyme, DL -2-haloacid dehalogenase, from Pseudomonas sp, Bacteriology, 150, 522– 527, 1982. 36 Nardi-Dei, V., Kurihara, T., Park, C., Miyagi, M., Tsunasawa, S., Soda, K., and Esaki, N., DL -2-haloacid dehalogenase from Pseudomonas sp. 113 is a new class of dehalogenase catalyzing hydrolytic dehalogenation not involving enzyme – substrate ester intermediate, J. Biol. Chem., 274, 20977– 20981, 1999; Kurihara, T., Esaki, N., and Soda, K., Bacterial 2-haloacid dehalogenases: Structures and reaction mechanisms, J. Mol. Catal. B, 10, 57 – 65, 2000. 37 Sicin´ska, D., Rudzin´ski, J., Kwiecien´, R., Kurihara, T., Esaki, N., and Paneth, P., Mechanism of the Reaction Catalyzed by DL -2-Haloacid Dehalogenase from the Chlorine Kinetic Isotope Effects, submitted for publication. 38 Dybała-Defratyka, A., Rostkowski, M., Matsson, O., Westaway, K.C., and Paneth, P., A new interpretaion of chlorine leaving group kinetic isotope effects; a theoretical approach, J. Org. Chem., 69, 4900– 4905, 2004. 39 Klamt, A. and Schu¨u¨rmann, G. J., COSMO: A new approach to dielectric screening in solvents with explicit expressions for the screening energy and its gradient, J. Chem. Soc. Perkin Trans., 2, 799– 805, 1993. 40 Kurihara, T., Esaki, N., and Soda, K., Bacterial 2-haloacid dehalogenases: structures and reaction mechanisms, J. Mol. Catal. B, 10, 57 – 65, 2000. 41 Kurihara, T., Personal Communication. 42 Gregg, K., Hamdorf, B., Henderson, K., Kopecny, J., and Wong, C., Appl. Environ. Microbiol., 64, 3496, 1998. 43 Benning, M. M., Taylor, K. L., Liu, R. Q., Yang, G., Xiang, H., Wesenberg, G., Dunaway-Mariano, D., and Holden, H. M., Structure of 4-chlorobenzoyl coenzyme a dehalogenase determined to ˚ resolution: An enzyme catalyst generated via adaptive mutation, Biochemistry, 35, 1.8 A 8103– 8109, 1996. 44 Liang, P. H., Yang, G., and Dunaway-Mariano, D., Specificity of 4-chlorobenzoyl coenzyme a dehalogenase catalyzed dehalogenation of halogenated aromatics, Biochemistry, 32, 12245– 12250, 1993.

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45 Dunaway-Mariano, D. and Babbitt, P. C., On the origins and functions of the enzymes of the 4-chlorobenzoate to 4-hydroxybenzoate converting pathway, Biodegradation, 5, 259– 276, 1994. 46 Koch, H. F., McLennan, D. J., Koch, J. G., Tumas, W., Dobson, B., and Koch, N. H., Use of kinetic isotope effects in mechanistic studies. 4. Chlorine isotope effects associated with alkoxide-promoted dehydrochlorination reactions, J. Am. Chem. Soc., 105, 1930– 1937, 1983. 47 Paneth, P., On the application of the steady state to kinetic isotope effects, J. Am. Chem. Soc., 107, 7070– 7071, 1985. 48 Lewandowicz, A., Rudzinski, J., Tronstad, L., Widersten, M., Ryberg, P., Matsson, O., and Paneth, P., Chlorine kinetic isotope effects on the haloalkane dehalogenase reaction, J. Am. Chem. Soc., 123, 4550– 4555, 2001. 49 Paneth, P., Chlorine kinetic isotope effects on enzymatic dehalogenation, Acc. Chem. Res., 36, 120– 126, 2003. 50 Lewandowicz, A., Rudzinski, J., Luo, L., Dunaway-Mariano, D., and Paneth, P., Chlorine kinetic isotope effect on the 4-chlorobenoyl-CoA dehalogenase reaction, Arch. Biochem. Biophys., 398, 249– 252, 2002. 51 Lewandowicz, A., Sicinska, D., Rudzinski, J., Ichiyama, S., Kurihara, T., Esaki, N., and Paneth, P., Chlorine kinetic isotope effects on the fluoroacetate dehalogenase reaction, J. Am. Chem. Soc., 123, 9192– 9193, 2001. 52 Reddy, C. M., Xu, L., Drenzek, N. J., Sturchio, N. C., Heraty, L. J., Kimblin, C., and Butler, A., A chlorine isotope effect for enzyme-catalyzed chlorination, J. Am. Chem. Soc., 124, 14526– 14527, 2002.

36

Nucleophile Isotope Effects Vernon E. Anderson, Adam G. Cassano, and Michael E. Harris

CONTENTS I. II.

Nucleophilic Activation and Reaction Mechanisms ....................................................... 893 18 O Isotope Effects........................................................................................................... 895 A. Activation of Water and Associated Equilibrium Isotope Effects.......................... 895 1. Desolvation and H-Bonding.............................................................................. 895 2. Equilibrium Isotope Effects on Hydroxide Formation ..................................... 898 3. Isotope Effects on Coordination of Water by Metal Ions ................................ 898 B. Kinetic Effects on Reactions.................................................................................... 899 1. Experimental and Theoretical Considerations.................................................. 899 2. Nucleophilic Attack on Electrophilic sp2 Carbon ............................................ 901 3. Attack on Carbon –Oxygen Double Bonds, Addition to Carbonyl Compounds ................................................................................... 903 a. Hydrolysis of Formic Acid Derivatives...................................................... 903 b. Carboxypeptidase Catalyzed Amide and Ester Hydrolysis........................ 905 4. Hydrolysis of Phosphate Esters......................................................................... 905 5. 18knuc Effect on Hexokinase .............................................................................. 907 15 III. N Isotope Effects........................................................................................................... 908 A. Equilibrium Isotope Effects on Protonation and N – C Bond Formation................ 908 B. Kinetic Isotope Effects on Nucleophilic Attack at sp3 Carbon .............................. 908 C. Kinetic Isotope Effects on Enzyme Catalyzed Nucleophilic Attack at sp2 Carbons .............................................................................................. 909 IV. Prospectus......................................................................................................................... 910 Acknowledgments ........................................................................................................................ 911 References..................................................................................................................................... 911

I. NUCLEOPHILIC ACTIVATION AND REACTION MECHANISMS Nucleophilic attack on an electrophilic center is a fundamental process in many organic and biochemical reaction mechanisms. Nucleophiles are characterized by having a lone electron pair, i.e., they are Lewis bases. The characterization of nucleophilic kinetic isotope effects (KIEs) has centered on oxygen and nitrogen, where both informative experimental and theoretical studies have been undertaken. Two fundamentally different nucleophilic processes at the central atom, addition and substitution, can occur as shown in Figure 36.1. Differentiating reactions 2 to 4 in Figure 36.1 plays a major role in characterizing the mechanisms of substitution reactions. Numerous chapters in this volume and previous work have focused on applying isotope effects to characterize these reactions on the basis of primary leaving group KIEs, primary KIEs on the central atom and on secondary KIEs associated with isotopic substitution of atoms attached to the central atom. These KIEs are determined in order to provide a characterization of the bonding of the central atom at the rate determining transition state. Large 893

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Isotope Effects in Chemistry and Biology

X

Nu

Nu

X

Nu

X

addition R2 Nu

R2 Nu

LG R2

R3

LG

R1 Nu

R1

Nu

substitution

R3

R3 X

X LG

Nu

X

+ LG R1

LG

Nu

X

LG

Nu

X

addition

elimination

+ LG

Nu

LG

Nu

X

X LG Nu

X

X

X

LG elimination

Nu

+

LG

addition

Nu

FIGURE 36.1 Nucleophilic reaction types, addition, substitution, addition elimination, elimination addition. The nucleophile (Nu) and leaving groups (LG) are identified. In this chapter the electrophilic atom is either C or P. The wavy line bonds indicate bonds that may or may not be present, e.g., the addition elimination reaction could be applied to a sp2 carbonyl or a tetrahedral phosphate.

primary leaving group KIEs have been observed and are interpreted as being consistent with cleavage of the bond to the central atom making a significant contribution to the reaction coordinate motion of the rate determining transition state.1 Secondary KIEs are taken to reflect changes in bonding that occur as the hybridization state of the central atom changes during the reaction coordinate2 – 4 (Chapter by Hengge). Even with leaving group and secondary KIEs, information on the bonding to the nucleophile is lacking. This gap can be best filled by determining nucleophile KIEs to completely differentiate and characterize the mechanisms of Figure 36.1. Four fundamental chemical processes can affect the bonding of the nucleophile during a chemical reaction and consequently the observed KIE: (1) its protonation state, (2) its coordination to a Lewis acid, (3) its coordination to a Bronsted base, (4) the extent of bond formation in the rate determining transition state (Figure 36.2). Consideration of the importance of each of these effects for different nucleophiles constitutes the major emphasis of this chapter. To permit

Nucleophile Isotope Effects

895

Deprotonation

Nu H

metal coordination

Mn+

H Nu

H H

Nu H

general base

B

H Nu H

bond formation

H Nu

C(P)

H

FIGURE 36.2 Processes which affect the observed nucleophile kinetic isotope effects. Protonation state and activation by deprotonation result in significant normal KIEs. Coordination to Lewis acids contributes an inverse factor to nucleophile KIEs. General base activation presumably contributes a normal factor to the KIE attributable to a decrease in the Nu– H bond order. Bond formation has a normal contribution from the reaction coordinate motion and in inverse effect due to the increased bonding to the nucleophile. Addition to carbon and phosphorous electrophiles will be considered.

discussion of the different contributions to the observed KIEs, 18kwater is used to refer to the overall or observed KIE, 18knuc is the intrinsic KIE on the nucleophilic addition step of the reaction, 18O is used to represent , 50% enriched oxygen, and O is used to identify an oxygen atom whose isotopic composition (18O/16O) will be determined by isotope ratio mass spectrometry. Although nucleophile KIEs have the potential to reveal important aspects of reaction mechanisms, they have not been widely investigated due to significant theoretical and technical difficulties. As discussed below, 15N KIEs have been observed in the range of 0.97 to 1.03 and 18 O KIEs fall in the relatively larger range of 0.93 to 1.05. While direct comparison experiments have been reported,5 competitive methods1 (Bigeleisen and Cleland chapters) are effectively required for the determination of most KIEs. In many nucleophilic reactions, the nucleophile concentration is maintained in large excess over the substrate to either increase the effective reaction rate, or to maintain pseudo first order reaction conditions. This effect also exists in hydrolysis and solvolysis reactions, where the solvent participates as the nucleophile, and consequently is present in extreme excess. The result of holding the isotopically investigated substrate in large excess on the isotopic variation of substrate and product is contrasted with the more common approach of monitoring an irreversible reaction in Figure 36.3. The independence of the substrate and product isotope ratio with the fraction of reaction requires that the isotopic composition of both be determined. This technical requirement that isotope ratios need to be determined for two different chemical species does not superficially present a significant problem, however the measurement of the isotope ratios in two different species increases the potential for systematic errors as discussed in detail below.

II.

18

O ISOTOPE EFFECTS

A. ACTIVATION oF WATER AND A SSOCIATED E QUILIBRIUM I SOTOPE E FFECTS 1. Desolvation and H-Bonding Hydrolysis reactions are one of the most important classes of biological nucleophilic reactions. The degradation of every biomacromolecular molecule including proteins and peptides, DNA, RNA,

896

Isotope Effects in Chemistry and Biology 0.22 Sub 100% cnv Sub <1% cnv P <1% cnv

% 18-O

0.21

P <100% cnv

0.2

0.19

0

0.2

0.4

0.6

0.8

1

FIGURE 36.3 18O/16O ratio as a function of fractional conversion assuming a normal 18O KIE. When the substrate is initially present at substoichiometric amounts it will be completely (100%) converted to product in an irreversible reaction. As shown by the heavy dashed line (- - -) the isotope ratio of the remaining substrate will continuously increase as the reaction progresses. Due to the normal KIE, the 18O/16O isotope ratio of the product as shown by the light dashed line (- - -) is less, but approaches the initial substrate ratio at completion. For reactions where the nucleophile is present in great excess the 18O/16O ratio is invariant as the reaction progresses (heavy solid line, – ). The 18O/16O ratio of the product is also invariant, but decreased due to the assumed normal KIE (thin solid line, – ). In this case, both the substrate and product 18O/16O ratios must be determined, but the fraction of reaction does not enter into the calculation of the KIE.

polysaccharides, and fat occurs through hydrolytic mechanisms. The favorable thermodynamics of hydrolysis would lead to complete degradation of all of the cellular structures to their monomeric constituents if the nucleophilicity of water were not carefully controlled by the responsible enzymatic systems. Yet, one general mechanism of promoting the rapid hydrolyses characteristic of enzyme catalyzed solvolysis is to activate the nucleophilic water molecule at the active site. Three possible mechanisms to enhance the nucleophilicity of water at an enzyme active site are illustrated in Figure 36.4: (1) desolvation, (2) H-bonding to a general base, (3) coordination to metal ions. Each of these factors impacts the bonding environment of the nucleophile in the transition state. In Figure 36.4a the nucleophilic water present in enoyl-CoA hydratase is H-bonded to two active site glutamate residues leaving a nucleophilic lone pair directed towards the electrophilic center.6 – 8 Steric constraints preclude the presence of any additional disordered water molecules in this region of the active site. A similar example is observed with protein tyrosine phosphatase; the nucleophilic water molecule is activated by reduced H-bond acceptance and increased H-bond donation. In the crystal structure, the water molecule proposed to be the nucleophile has become completely desolvated while two protons are inferred to be H-bonded to active site residues.9 The mutation of one of these residues, Asp-356, to alanine results in a large decrease in activity10 implicating it as a general acid/base group. The third general mechanism of activation is coordination to active site metal ions. In Figure 36.4b the nucleophilic water molecule is coordinated to a Mn2þ ion at the active site of Escherichia coli nucleotidase and directed for an inline displacement of the g-phosphate of the nucleotide.11,12 Each of these chemical environments will significantly alter the fractionation factor of the nucleophile. Thus, nucleophile KIEs have the potential to provide a crucial window into the mechanism of nucleophile activation in enzymatic catalysis. To provide an a priori estimate of the EIE introduced by changes in bonding environment of the nucleophilic water at enzyme active sites, simple model reactions may be considered. Contributions to the EIE associated with both desolvation (van Hook chapter) and formation of the H-bond are reflected in Equation 36.1 and

Nucleophile Isotope Effects

897

Nucleophilic water

(a)

(b) FIGURE 36.4 (a) Active site of enoyl-CoA hydratase (adapted from protein database structure 1EY3) shown with CPK spheres at 75% radius, and the substrate present as a ball and stick model; carbon, nitrogen and oxygen are differentiated by progressively darker shades. The nucleophilic water molecule, at center, H-bonds to Glu-144 (behind) and Glu-164 (above) and is oriented for addition to the substrate (left). The nucleophilic water is not an H-bond acceptor as the rest of the active site is comprised of saturated C atoms which exclude any additional solvent molecules. (b) Active site of E. coli nucleotidase (adapted from protein database structure 1HPU) with the putative nucleophilic water coordinated (dark CPK sphere) to one of the two Mn2þ atoms (lighter CPK spheres) and positioned for an inline displacement of the terminal phosphate of the inhibitor, AMPPCP (ball and stick model).

Equation 36.2, respectively. H2 Oaq Y H2 Ogas

18

H2 Oaq þ B : YHOH : B

Keq ¼ 1:008 18

Keq ¼ ??

ð36:1Þ ð36:2Þ

There are 18O EIEs that accompany both processes. The normal 18O EIE on desolvation of water, determined as its vaporization EIE is 1.008,13 reflects the loss of H-bond donors to the nucleophilic electron pairs. The 18O EIE on water forming an H-bond to a general base will also be normal as a reflection of the reduced stretching force constant for the O – H bond. The magnitude of the decrease will depend on the strength of the H-bond and the identity of the H-bond acceptor.

898

Isotope Effects in Chemistry and Biology

o o

o

o

SCHEME 36.1

However, the 18O EIE on forming an H-bond is predicted by DFT quantum chemical calculations to be normal and as large as 1.035. p 2. Equilibrium Isotope Effects on Hydroxide Formation In some aqueous phase reactions, hydroxide has been proposed to be the actual nucleophilic species. Ionization of water to form hydroxide is associated with a normal 18O EIE of 1.0414 as measured by changes in the relative vapor pressure. Exact interpretation of this value depends on assumptions made on the volatile unit, but if the presumption is that the vapor pressure of the hydroxide ion and its first solvation shell are all minimal,14 this value is a composite effect on both the hydroxide and the three first water molecules of the first hydration shell as shown in Scheme 36.1. The 18O EIE on forming hydroxide is normal due to the loss of the O – H bond. Because of the strong H-bond between the hydroxide and the solvating water molecules, these three water molecules are also anticipated to have a smaller, but still normal, 18O EIE. The experimental value of 1.04 is the product of the fractionation factors for all four oxygen atoms. 3. Isotope Effects on Coordination of Water by Metal Ions Metal ions have been shown to catalyze many solvolysis and hydrolysis reactions. Interactions with divalent metal ions are widespread and an important feature of many hydrolytic enzymes. Characterizing the effect of metal ion coordination on water is consequently essential to understanding how these interactions promote the nucleophilicity of the bound water. Coordination 2þ to metal ions decreases the pKa of water. The pKas of Mg(H2O)2þ 6 and Zn(H2O)4 are 11.4 and 15 9.0. This enhancement of the acidity of the bound water molecule can contribute to catalysis by favoring the formation of the nucleophilic hydroxide at an enzyme active site as shown for E. coli nucleotidase in Figure 36.4b. There will be two EIEs required to identify and interpret the role of the metal ion in promoting the nucleophilicity of the coordinated water molecule: the 18O EIE on coordination of H2O (Equation 36.3a) and the 18O EIE on ionization of metal coordinated water (Equation 36.3b). As shown in Equation 36.3c, the product of these two effects can be combined to determine the overall EIE on coordination and subsequent ionization of the bound water molecule. 18 18 nþ MðH2 OÞnþ 6 þ H2 O Y H2 O þ MðH2 OÞ5 ðH2 OÞ 18

KM†H2 Ocoord , 0:998 for n ¼ 1; 0:99 for n ¼ 2 and 0:98 for n ¼ 3

p Calculated at the DFT B3LYP level for water H-bonded to formate as suggested in Equation 36.2.

ð36:3aÞ

Nucleophile Isotope Effects

MðH2 OÞ5 ðH2 18 OÞnþ þ MðH2 OÞ5 ðOHÞðn21Þþ Y MðH2 OÞ5 ð18 OHÞðn21Þþ þ MðH2 OÞnþ 6 18

KM†H2 Oioniz ¼ 1:008 for CoðNH3 Þ5 H2 O3þ MðH2 OÞ5 ðOHÞnþ þ H2 18 O Y H2 O þ MðH2 OÞ5 ð18 OHÞnþ

899

ð36:3bÞ

ð36:3cÞ

18

Kcoord. ioniz are unmeasured, but calculated to be 0.998 for Co(III). Importantly, coordination of a water molecule results in inverse 18O EIEs. These inverse effects were determined by measuring the 16O/18O ratio in water vapor above concentrated solutions of the chloride salts of the metal ions.16 The results indicate that direct coordination to a metal ion results in a stiffer environment than provided by the solvation of bulk water. The only EIE on acid dissociation of metal coordinated water has been measured for Co3þ where the effect was reduced to 1.00817 from 1.04 for uncoordinated water. This comparison suggests that the loss of the O –H bond from the coordinated water must be compensated for by increased interaction with the metal ion. For Co3þ then the overall effect of coordination and ionization is dominated by the inverse coordination factor, and the value is predicted to be inverse. For the divalent ions of most interest in biochemical reactions, such as Mg2þ, Mn2þ, and Zn2þ, the smaller inverse effect on coordination and potentially smaller compensation on ionization suggest that the EIE for coordination and ionization will be normal, but significantly less than the value observed for uncoordinated water.

B. KINETIC E FFECTS oN R EACTIONS 1. Experimental and Theoretical Considerations H18 2 O KIEs have been determined by three different procedures: equilibrium perturbation, isotope ratio determinations, and direct comparison. Although significant results have been obtained, each one of these protocols has its own inherent limitations. The equilibrium perturbation method, as suggested by the name, requires that the equilibrium constant be within an order of magnitude of unity;18 the measurement of water and product isotope ratios has uncertainties associated with the accuracy of the isotope ratio measurements; and direct comparison studies are prone to the errors associated with running parallel reactions. The use of isotope ratio measurements to study hydrolysis reactions was introduced by Breslow19 (see below). In the reaction shown in Equation 36.4, the 18O KIE, 18kwater, is determined by measuring the isotope ratio of the water and the product. H2 18 O=H2 16 O þ Peptide ! RCð18 OÞO2 þ RCð16 O2 Þ2 18

kwater ¼ ðH2 18 O=H2 16 OÞ=ðRCð18 OÞO2 =RCð16 O2 Þ2

ð36:4Þ

Since the water is in vast excess, its isotope ratio will not vary during the reaction as shown in Figure 36.3, so the isotope ratio in the product will be similarly invariant with the fraction of reaction. This simplifies the calculation of the isotope effect, as the fraction of reaction need not be determined and the reaction can be run to completion. In their initial determination of an 18O KIE on hydrolysis, Breslow and coworkers19 16 determined the H18 2 O/H2 O by equilibration with CO2. Then both the water equilibrated CO2 and 18 the product RC( O)O2/RC(16O2)2 isotope ratios were determined by GC-quadrupole mass spectrometry. This protocol hopes to limit the instrumental bias by using the same instrument, if not the same analyte, to determine both the water and product isotope ratios. The development of electrospray and MALDI ionization techniques has made it possible to determine isotope ratios with a precision of better than 0.5% in relatively large biological molecules ranging from amino acids20 to cofactors and nucleotides.21 This capability invited the use of H18 2 O to determine 18 kwater values by determining the water isotope ratio by CO2 equilibration and the hydrolysis

900

Isotope Effects in Chemistry and Biology

product isotope ratio by either of these whole molecules, soft ionization mass spectrometric methods. Success in this regard is reported in the determination of the isotope effect on hydrolysis of nitrophenyl thymidine-50 -monophosphate.21 The application of this methodology, however, requires the recognition of the potential different instrumental biases and application of the best procedures to detect and minimize this source of possible systematic error. Due to their mass granularity quadrupole mass spectrometers have the potential to generate systematic errors in isotope ratio measurements.22 We have demonstrated that the best method to eliminate this source of inaccuracy is to fit the limited range mass profile scan as shown in Figure 36.5 assuming a sequence of identically shaped peaks.23 Isotope ratio mass spectrometers are also subject to instrumental bias which is corrected by the routine use of external calibration standards. However, if the instrumental bias is identical for both the product and substrate isotope measurements, it will cancel out. The standard 16 determination of H18 2 O/H2 O has been by equilibration with CO2 and subsequent determination of the isotope ratio in CO2. To determine the product 18O/16O ratio by the same method would require that the O of interest must be directly converted to CO2. This approach was taken by Marlier24 for the study of formate ester and amide hydrolysis, but has not been subsequently employed. Recent developments in continuous flow isotope ratio mass spectrometry have made it possible to convert many O-containing molecules, including water, to C ; O where the C ; 16O/C ; 18O isotope ratio is determined at natural abundance online with an isotope ratio mass spectrometer.25 By determining the 18O nucleophile isotope ratio and the product isotope ratio in the same analyte with the same instrument as shown in Scheme 36.2 any instrumental bias will not contribute systematic error to the isotope effect calculation. If the conversion of water or product to C ; O is incomplete, isotopic fractionation that occurs during the chemical reaction will contribute to the error. This approach has been applied to the determination of the 18 ðV=KÞ for the water nucleophile in adenosine deaminase.23 The ability to quantitatively convert many different molecules to C ; O and determine the oxygen isotope ratio online should provide a practical method of determining many nucleophile KIEs.

4e+5

Intensity

3e+5

2e+5

1e+5

0 320

321

322

323 m/z

324

325

326

FIGURE 36.5 Profile scan mass spectrum of [50 -18O1]TMP obtained by hydrolysis of nitrophenyl thymidine50 -monophosphate. The solid line was obtained by a nonlinear least squares fit to a series of functions that describe a spectroscopic peak with asymmetric leading and tailing contributions. The individual peaks were constrained to differ only in amplitude and to be separated by 1 mm. The contribution of each mass isotopomer to the spectrum is determined by analytical integration of the fitted equation.

Nucleophile Isotope Effects

901 R1

R O H

R3 R 2

C

18

R1

reaction

LG

RO

C

LG

R2

R3

high temperature carbon reduction

O

+

O

isotope ratio mass spectrometry

knuc = Nuc(18O)/Nuc(16O) Product(18O)/Product(16O)

SCHEME 36.2

Unfortunately, minimal theoretical work has been undertaken to characterize isotope effects on nucleophilic additions and this area is ripe for such exploration. One important exception is the study of hydroxide addition to acetaldehyde by Hogg et al.26 In this systematic theoretical study done with a BEBOVIB27 force field, the extent of bond formation between the nucleophilic O and the carbonyl C was systematically varied from a bond order of 0.1 to 1.0. The 18knuc decreased monotonically from 1.031 to 0.966. In the early “loose” transition states, the imaginary reaction coordinate vibration contributes a significant normal effect while the zero point energy term is minimally inverse. As the bond order in the transition state increases, the largest effect is to decrease the zero point energy contribution. The decrease in the zero point energy term is attributed to the stiffening of the C – O stretch and angle bending modes that include the nucleophile O. At a bond order of , 0.5, the two contributions are equal and opposite resulting in a calculated 18knuc of unity and at late or “tight” transition states the calculated 18knuc is inverse. The EIE for hydrate formation was calculated in this paper to be 0.93 for hydroxide addition to acetaldehyde, which corresponds to an EIE of 0.97 for water. This inverse calculated EIE is consistent with the values tabulated by Rishavy and Cleland.28 The two important conclusions: (1) that early transition states will lead to significant normal nucleophile KIEs and (2) that late transition states will be dominated by an inverse EIE serve as the basis of a priori estimates of all nucleophile KIEs. 2. Nucleophilic Attack on Electrophilic sp2 Carbon The most extensive class of nucleophilic water 18O KIEs determined is on enzyme catalyzed hydration reactions, where the elements of water are added across a CyC as shown in Scheme 36.3. In all three enzymes for which this 18O KIE has been determined, fumarase,29 enolase30 and enoyl-CoA hydratase,31,32 primary and secondary D KIEs were also determined, permitting conditions that minimized external commitment factors to be identified. Using the experimental conditions which minimized external commitments, significant normal 18(V/KH2O) values were measured using the equilibrium perturbation technique. For fumarase, enoyl-CoA hydratase, and enolase the maximum 18(V/KH2O) values were 1.04, 1.02, and 1.005, respectively. In all three

R

SCHEME 36.3

O

HO

O X

X

R H

902

Isotope Effects in Chemistry and Biology

enzymes pH variation studies have been performed and general base catalysts identified, but in no case has it been proposed that hydroxide is the nucleophile. In all three cases the transition state has been proposed to reflect the addition of water to the electrophilic carbon with general base assistance as shown in Scheme 36.4, with the base being a glutamate residue for enolase33 and enoyl-CoA hydratase (Figure 36.4a). These 18(V/KH2O) results are important because they demonstrate that even when H2O is a nucleophile, and its bond order increased in the transition state, normal KIEs can be observed. This normal effect is attributed to O participation in the reaction coordinate motion. However to date, there have been few high level theoretical studies on hydrolysis reactions where the 18O KIE has been studied to identify the reaction coordinate vibrational motion. The significant differences in the magnitude of the three effects, varying from 1.04 to 1.005, are attributed to the different extents to which the nucleophilic C – O bond forming step is rate determining. In fumarase, where 18(V/KH2O) is the largest, the primary D(V/K) is inverse.29 This inverse D(V/K) is interpreted as indicating that the proton that is added to the adjacent carbon of the CyC, is in a stiffer environment than either bulk solvent or at C3 of the product malate and that C – O bond formation is completely rate determining. For enoyl-CoA hydratase 18(V/KH2O) is 1.02 and the primary D(V/K) is 1.6.31 The normal primary D(V/K) is consistent with either a concerted reaction or a stepwise reaction where the two bond forming reactions are partially rate determining. In either case the secondary a2D(V/K) of 0.85 indicates that the bonding at the electrophilic carbon has been significantly changed. The conversion from sp2 to sp3 and the addition of O will stiffen the environment of the substituted C3 proton predicting an inverse a2D(V/K) if the addition of the nucleophilic water is rate determining. The smallest 18(V/KH2O) of 1.005 for enolase was obtained at low pH and Mg2þ where the primary D(V/K) was 2.9.30 The smaller 18(V/KH2O) combined with the larger D(V/K) suggest that C – H bond formation/cleavage is largely rate determining for this reaction, a conclusion confirmed by the determination of the effect of deuteration on the a2D (V/K).34 The normal 18(V/KH2O) values for all three of these reactions establishes the fact that the nucleophilic addition of water to electrophilic C will be characterized by normal 18(V/KH2O) values whose magnitude will be reduced when other steps in the reaction become rate determining. In addition to changes in chemical mechanism, specific active site interactions can also have a strong influence on the observed nucleophile KIE. An example is the Zn2þ dependent enzyme adenosine deaminase. The addition of water to C6 of adenine is presumed to be an addition– elimination reaction as shown in Scheme 36.5. Because the adenosine product can be quantitatively converted to crystalline hypoxanthine by arsenolysis, the 18O KIE could be determined by conversion of both the H2O and the [6-O ]hypoxanthine to C ; O. The C ; O generated by anaerobic high temperature conversion over glassy carbon can be analyzed by continuous flow isotope ratio mass spectrometry.35 – 37

SCHEME 36.4

Nucleophile Isotope Effects

903 HO

NH2

Zn2+ H

N

O

HO

O

HO

O

O

HO

OH OH

OH

OH

AsO4

N

OH

OH

purine nucleoside phosphorylase

N

N

N

N

N

HN

N

HN

N N

(H)

O

NH2

O 1400°

N

HN N

NH

glassy C (s)

C

O

1400° glassy C

O H

(H)

SCHEME 36.5

Unlike the enzyme catalyzed hydration reactions, the 18(V/KH2O) of 0.986 for this reaction is inverse. Two facets of this reaction could contribute to the observed inverse effect. As shown in Scheme 36.5, the nucleophilic water molecule is coordinated to the active site Zn2þ atom. As noted above divalent metal ion coordination stiffens the environment of the coordinated O atom. Second, 18O EIEs for addition to carbon double bonds are inverse and predicted to be 0.96† for the formation of the proposed tetrahedral intermediate.28 If cleavage of the C – N bond were completely rate determining it would impose a large reverse commitment on the preceding addition of water and 18(V/KH2O) would approach the inverse EIE of 0.96. Thus, the measured value of 0.986 is inconsistent with C – N bond cleavage being completely rate determining. Under similar reaction conditions, Weiss et al. used the primary 15(V/Kadenosine) to determine the reverse commitment factor for C – O bond formation to be 1/1.4. Using the estimated 18O EIE of 0.96, the 18knuc for addition of the Zn2þ bound water (or hydroxide) to C6 would be 1.004. § A larger intrinsic 18knuc, e.g., similar to those measured for the double bond hydration reactions, would require that the observed 18(V/KH2O) be normal. The smaller 18knuc for adenosine deaminase can be attributed to coordination of the nucleophilic water to the divalent metal ion. This suggestion is consistent with the known inverse EIE on water coordination to divalent metal ions.16 This inverse EIE on coordination could potentially be the sole contributor to the observed KIE if the binding of adenosine were to trap the nucleophilic Zn2þ water resulting in a large forward commitment. Such trapping of Zn2þ coordinated water has precedent in the reaction of carbonic anhydrase38 and is suggested by the adenosine deaminase crystal structure.39 However, a large forward commitment for adenosine is not consistent with the observed normal 15(V/Kadenosine).40 3. Attack on Carbon – Oxygen Double Bonds, Addition to Carbonyl Compounds a. Hydrolysis of Formic Acid Derivatives The hydrolyses of esters and amides are the most common biochemical reactions that are characterized by the attack of water on carbon– oxygen double bonds. Marlier and coworkers have †The value of 0.96 comes from the reciprocal of the fractionation factor for malate (1.033) and assuming that replacing a H with N will increase the fractionation factor by 1.008. § This value is calculated using the standard commitment factor equation 18 kobs ¼ ð18 knuc þ cr 18 Keq Þ=ð1 þ cr Þ:

904

Isotope Effects in Chemistry and Biology O

O O

H

R H2O

H

O H

oxidation

O

O

C

O

O NH2

H2O

H

oxidation

O

O

C

O

SCHEME 36.6

taken advantage of the ability to oxidize formate to CO2 without loss of either of the oxygen atoms to use isotope ratio mass spectrometry to determine 18kwater for hydrolysis of methyl formate and formamide. The oxidation of formate directly generates CO2 so the isotope ratio of both the nucleophile water and the hydrolysis products can be determined by isotope ratio mass spectrometric analysis of CO2. One complication of this procedure is that the isotopic composition of the carbonyl O of the substrate must be known, and the reaction must be quantitative so that the isotope effect on the carbonyl O will not result in any fractionation. The hydrolysis and oxidation reactions are shown in Scheme 36.6. Base catalyzed nucleophilic acyl substitution reactions are commonly presumed to proceed through an anionic tetrahedral intermediate as shown in Scheme 36.7. Following formation of the intermediate, any of the three heteroatoms can be expelled resulting in reversal, exchange, or hydrolysis. Following a proton transfer, reversal and exchange are presumed to have identical barriers so a lack of carbonyl isotope exchange implies that hydrolysis is faster. The observed lack of isotope exchange into the formate carbonyl under the reaction conditions studied, indicated that the nucleophilic attack to form a monoanionic tetrahedral intermediate was largely rate determining. For ester hydrolysis the 18kwater is 1.023. The nature of the rate determining transition state was further characterized by the primary carbonyl-13k of 1.034, the secondary Dk of 0.95 and the methoxyl-18k of 1.009, all consistent with the rate determining transition state being nucleophilic addition to the carbonyl. The normal 18kwater represents the 18knuc on the nucleophilic addition if H2O is the nucleophile. If OH2 is the nucleophile, the observed 18 kwater will be 18Kdeprotonation £ 18knuc. This would require the unlikely attribution of an inverse 18 knuc of , 0.98 to the nucleophilic addition step. To account for the first order dependence on hydroxide it was proposed to act as a general base, consistent with the general base catalysis of formate ester hydrolysis observed with succinate.41 Hydrolysis of formamide resulted in a surprisingly similar observed 18kwater of 1.022.42 By running the reaction at high base concentration, the nucleophilic attack at the carbonyl carbon may be made largely rate limiting as rapid deprotonation of the monoanionic tetrahedral intermediate (Scheme 36.7) to the dianion commits the reaction to hydrolysis. As discussed above for the hydrolysis of methyl formate, a relatively low (less than 18Kdeprotonation) but still normal 18 knuc is more consistent with water being the nucleophile than hydroxide. O O R

O− X

+ OH−

reversal R

OH X

anionic tetrahedral intermediate

SCHEME 36.7

exchange

R

hydrolysis

X

O R

O−

Nucleophile Isotope Effects

905 H2N(HO)

O

O NH O

CO2−

O

NH(O) CO − carboxypeptidase 2 H2O

+

NH O

18(V/K 18(V/K

H2O)

= 1.019 amide

O) = 0.998 ester

H2

SCHEME 36.8

b. Carboxypeptidase Catalyzed Amide and Ester Hydrolysis Carboxypeptidase catalyzes the hydrolysis of the C-terminal linkage of peptide amides and esters as shown in Scheme 36.8. This work represents the seminal determination of the 18kwater on any hydrolysis reaction.19 The 18(V/KH2O) values were determined by measuring the isotope ratio of water by equilibration with CO2 and analysis of the CO2 by multiple injections on a GC-quadrupole MS. The isotope ratio of the hydrolysis product was determined with the same GC-quadrupole MS by analyzing the parent ion of the product following methylation with diazomethane. The experiment was undertaken to investigate whether the hydrolysis proceeded by the initial attack of an active site carboxylate to form an anhydride intermediate. Since, the anhydride would be common to both amide and ester hydrolysis the 18(V/KH2O) KIEs should be identical. The measured values of 1.019 ^ 0.002 for amide hydrolysis and 0.998 ^ 0.002 for ester hydrolysis19 were consequently inconsistent with a simple anhydride intermediate mechanism. The difference in 18(V/KH2O) for amide and ester hydrolysis suggests that there is a significant difference in environment of the nucleophilic water molecule in the rate determining steps for each reaction. The simplest interpretation based on Scheme 36.7 is that the normal 18(V/KH2O) for amide hydrolysis implicates the nucleophilic addition as being largely rate determining. The observed value of 1.019 is similar to the values for the hydrolysis of the formate derivatives (Section II.B.3.a). The inverse value for ester hydrolysis is consistent with decomposition of the tetrahedral intermediate contributing to the rate determination. The EIE for forming the tetrahedral intermediate will be significantly inverse due to formation of the stiffer C – O bond. As discussed above for the adenosine deaminase reaction, this EIE is estimated as , 0.96 and will contribute to the observed 18(V/KH2O) This EIE with an assumed intrinsic 18knuc of 1.02 on addition to the carbonyl suggests that there is nearly equal partitioning of the tetrahedral intermediate for ester hydrolysis. Thus, the ratio khydrolysis/kreversal is , unity for ester hydrolysis but must be greater than unity for amide hydrolysis. 4. Hydrolysis of Phosphate Esters Hydrolyses of phosphate esters are crucial reactions in living systems; phosphodiester hydrolysis is a key step in the repair and degradation of DNA and RNA, and phosphomonoester hydrolyses are controlling steps in many signal transduction pathways. Consequently, there has been intense interest in the mechanism of phosphoryl transfer since the elucidation of the structure of DNA and the characterization of ATP as the major source of biochemical free energy. The study of isotope effects on phosphoryl transfer reactions has been intensively pursued by the Cleland and Hengge labs43 – 45 and may be applied to the study of nucleic acid enzymology.46 The hydrolyses of acyclic phosphate esters are believed to proceed through an SN2 type transition state with varying extents of bond formation and cleavage to the nucleophile and leaving group, respectively. The relative amount of bond formation and cleavage will depend, in general, on the character of the nucleophile,

906

Isotope Effects in Chemistry and Biology

leaving group and degree of phosphate esterification. Most isotope effect studies on phosphate ester hydrolysis have employed nitrophenyl esters to enable the use of the remote label method and because their reactivity permits the reactions to be observed at near ambient temperatures. These studies demonstrated that the leaving group effect 18O KIE can vary from between 1.01 to 1.03, while the nonbridging 18O KIEs vary between 0.995 and 1.033.47 Analysis of the combination of KIEs suggests that monoesters hydrolyze via loose transition states and phosphotriesters through tighter transition states.45 Recently several nucleophiles 18O KIEs (18kwater) have been reported for phosphoester hydrolysis which significantly extend the characterization of phosphoryl hydrolysis transition states. The hydroxide catalyzed hydrolysis of the thymidine-50 -p-nitrophenylphosphate was determined to be 1.07. This surprisingly large value arises from hydroxide being the attacking nucleophile.21 The equilibrium deprotonation results in an EIE of 1.04 that is fully expressed in the observed 18kwater. Additionally, an early transition state would contribute a significant contribution from the reaction coordinate motion. The product of these two effects gives rise to an 18kwater that is diagnostically greater than 1.04. This 18kwater was determined using an adaptation of the method introduced by Breslow and coworkers.19 The reaction was run in H18 2 O and the isotope ratio of the water determined after equilibration with CO2 and the isotope ratio of the hydrolysis product determined by whole molecule mass spectral analysis. Two differences were employed; the water was accurately diluted prior to equilibration with CO2 so the analysis could be performed by isotope ratio mass spectrometry and the product thymidine-50 -[18O] phosphate was analyzed by electrospray ionization interfaced with a quadrupole mass spectrometer. Significant instrumental bias could be introduced using selected ion monitoring of the maximum intensity. This bias is minimized by fitting the isotopic envelope to a series of identically shaped peaks23 which yields the relative integrated intensities for each isotopic peak. Figure 36.5 shows a spectrum of the product thymidine-50 -[18O]phosphate and the calculated fit to the series of peaks. The advantages of the electrospray ionization are its sensitivity, which permits low concentrations to be analyzed, and the ability to generate a stable signal by continuous direct infusion, which allows spectra to be acquired for extended times to improve signal to noise. Metal ions, particularly lanthanides and nonexchanging complexes, are known to catalyze the hydrolysis of phosphodiesters.48,49 However, the mechanism of catalysis is not completely understood. Cassano has shown that Mg2þ(H2O)5(OH2) complex accelerates the hydrolysis of thymidine-50 -p-nitrophenylphosphate over 1000-fold.50 That the acceleration came from activation of the nucleophile and not coordination to the phosphate or bridging oxygen was demonstrated by the lack of acceleration of the 3-chloroquinuclidine displacement of nitrophenol. The 18kwater of 1.027 ^ 0.013 for this hydrolysis was determined by the same method as the hydroxide catalyzed hydrolysis, with the isotope ratio of the water being determined by equilibration with CO2 and the product thymidine-50 -phosphate by electrospray Ionization-quadrupole MS. The small normal effect observed for the Mg2þ catalyzed reaction contrasts with that observed for the hydrolysis by hydroxide alone. Three different effects are likely to reduce the magnitude of the observed 18kwater. The observed value is the product of the EIE on formation of the active nucleophile, Mg2þ(H2O)5(OH2), and the intrinsic 18knuc on the displacement reaction. As discussed in Section II.A.3, the EIE for equilibrating water with the Mg2þ(H2O)5(OH2) complex has offsetting contributions from an inverse EIE due to coordination and a normal EIE for deprotonation resulting in a greatly reduced overall EIE for generation of the nucleophilic species. The 18knuc for the attack on phosphorous is then estimated to have a small normal effect. The reduction in the intrinsic 18knuc from that observed for the hydroxide promoted reaction would be attributed to a later transition state, consistent with the Hammond postulate. Hengge has determined the 18knuc for the addition of hydroxide bridging two metal ion centers to a coordinated nitrophenyl phosphate ester as shown in Scheme 36.9 by the remote label method to be 0.937.51 This very large inverse 18knuc coupled with a large 18O KIE on the nitrophenyl leaving

Nucleophile Isotope Effects

907 NO2

CH3O

CH3OO

NO2

O P

O N Co(III) N

H

P

O

O

N

-

-

O H

N Co(III) N

N

18K

~ eq = 0.93

O N Co(III) N

O

H

O

N

N

-

Co(III)

O

N

H

SCHEME 36.9 Formation of phosphorane intermediate has a large inverse

k hyd

N

18

Keq.

group of 1.029 suggests that this hydrolysis reaction proceeds through a phosphorane intermediate. The large inverse 18knuc comes from the complete expression of the EIE for formation of the O – P bond. It is not surprising that the magnitude of this effect is greater than the 0.96 effect on protonation of hydroxide as replacing an O –H bond with a bond to a heavy atom results in an increased fractionation factor.18 For the hydrolysis of the homologous tethered nitrophenyl monoester, the 18knuc and 18kleaving group are 1.013 ^ 0.001 and 1.019 ^ 0.001, respectively. The normal values for both nucleophile and leaving group strongly suggest that the reaction has become concerted (A. Hengge, Personal communication, 2004). The studies described illustrate that determination of the nucleophile isotope effects on phosphate hydrolysis has the potential to characterize the nature of the attacking water molecule. If hydroxide is the direct nucleophile, then a very large 18kwater will be observed arising as the product of the EIE on deprotonation and the KIE that characterizes the reaction coordinate. Coordination of water to divalent metal ions results in a decrease in the observed 18kwater which can nonetheless still is normal due to deprotonation and reaction coordinate motion. In contrast, when there are large reverse commitments arising from slow steps occurring after a reversible addition to the electrophilic center, significant inverse KIEs will be observed. 5.

18

knuc Effect on Hexokinase

The role of ATP as a universal phosphoryl donor in biochemistry makes the determination of the transition states of kinases of general interest. These reactions differ from those discussed previously as the nucleophile is no longer water. In the hexokinase catalyzed reaction the nucleophile is the C6 hydroxyl group of the sugar substrate. The determination of small inverse secondary nonbridging-18(V/KATP) values52 suggests that the phosphoryl transfer transition state was dissociative in character, either an SN1 reaction or one with reduced bond order to both the nucleophile and the leaving group oxygen atoms.44 This conclusion is buttressed by the determination of the primary 18 (V/Kglucose) to be 1.013 (W. W. Cleland, Personal communication, 2003). Double labeled [6-18O,1-13C]glucose was synthesized and mixed with [1-12C]glucose so the KIE could be determined by the remote label method.53 The significantly normal effect obtained under conditions similar to those used for the determination of the nonbridging effect helps to establish the phosphoryl transfer step as at least partially rate determining. Consequently the measured inverse nonbridging-18(V/KATP) values may be more reliably attributed to the chemical phosphoryl transfer step. The modest value of the 18(V/Kglucose) when compared to those observed for hydrolysis reactions suggests that there may be a partial commitment that reduces both the primary and secondary 18O KIEs. The existence of such a commitment would mean that the intrinsic nonbridging-18(V/KATP) was even more inverse than the observed value. However since this is the only determined 18knuc for an alcohol nucleophile, it is

908

Isotope Effects in Chemistry and Biology

uncertain what range of primary 18O KIEs may be expected, but this observation unequivocally establishes that they can have significant normal values.

III.

15

N ISOTOPE EFFECTS

A. EQUILIBRIUM I SOTOPE E FFECTS oN P ROTONATION AND N – C B OND F ORMATION The lone electron pair on amine nitrogens is inherently more nucleophilic than the lone electron pairs of water and alcohol oxygen atoms because of the nitrogen’s reduced electronegativity. This enhanced nucleophilicity, coupled with the increases in pKa (e.g., the pKa of H3Oþ is less than 0, while that for NHþ 4 is 9.5) results in protonated and neutral nitrogen species existing in aqueous solution. Unlike the situation with O where both the neutral and anionic forms of the nucleophile need to be considered, with nucleophilic nitrogen species the nucleophile will always be the unprotonated form and the 15Keq on deprotonation will have to be included in any isotope effect measured below the pKa of the nucleophile. The other significant difference between 15N and 18 O isotope effects is that the 15N EIEs and KIEs are significantly smaller than the analogous 18O isotope effects because the mass ratio of the light and heavy isotopes is decreased by just under twofold. Nitrogen and oxygen nucleophiles, however, share similar mechanisms of activation: desolvation, deprotonation, and general base proton abstraction. Since nitrogen nucleophiles routinely have a single lone pair, coordination to metal ions cannot provide a mechanism of nucleophilic activation. The EIEs on desolvation28 and protonation of NH354 are given in Equation 36.5 and Equation 36.6 NH3 ðaqÞ Y NH3 ðgasÞ NH3 ðaqÞ þ H3 Oþ Y þ NH4 ðaqÞ

15

Keq ¼ 1:007 15

Keq ¼ 0:9811

ð36:5Þ ð36:6Þ

As anticipated, these 15N EIEs indicate that H-bonding to the NH3 provides a slightly stiffer environment than its absence and results in the normal EIE on volatilization. The inverse EIE on protonation of NH3 arises from formation of the new N –H bond. While it is tempting to predict that the same EIE would be observed for all amines, the measured values for several amines varied from 0.978 to 0.988.55 The EIE on deprotonation (which will be a required correction for aqueous reactions run at pH , the pKa of the amine) will be normal and the reciprocal of the values above. All of the 15N nucleophile KIEs (15knuc) discussed below involves the reaction at electrophilic carbon. These reactions form a new N –C bond giving rise to the anticipation that the associated EIEs will be inverse. This is explicitly true for the reactions of tertiary amines, but deprotonation of the product formed from the reaction of hydrogenic amines will only diminish the inverse EIE, e.g., the 15N EIE for the addition of NH3 to form the zwitterion of phenylalanine by addition to cinnamate is 0.984 which is decreased by an additional 1.6% on deprotonation.54

B. KINETIC I SOTOPE E FFECTS ON N UCLEOPHILIC ATTACK AT sp 3 C ARBON The mechanism of SN2 reactions with nitrogen nucleophiles has been a subject of intensive study. Methyl transfer reactions have served as a paradigm for nucleophilic reactions in general.56 In particular, the secondary-D KIEs for transfer of methyl-d3 compounds have been of considerable interest since such effects provide information on whether the transition states are associative or dissociative in character.57 The reaction of p-N,N-dimethyl toluidine with methyl iodide has been characterized both experimentally and computationally.58 – 61 The computational study indicated that normal 15knuc values as great as 1.028 would be observed in early transition states, and the predicted effect decreased monotonically to be inverse when the N –C bond order surpassed 0.5.58 Subsequent experimental investigation confirmed this trend with 15knuc decreasing as the basicity

Nucleophile Isotope Effects

909

of the toluidine derivative decreased.60 Subsequent computational evaluation using a higher level of theory suggested that the 15knuc values on these reactions would be nearly invariant with p-substitution of the dimethyl aniline or with the dielectric constant of the solvent.61 A significant effect on 15knuc was predicted for 2,6-dimethyl-N,N-dimethylaniline. The steric effect of the ortho-methyl groups resulted in an increase in the incipient C – N bond length at the transition state that resulted in an increase in the calculated 15knuc to 1.0032. Kurz and coworkers made similar observations on the SN2 reactions55 of a variety of amines with highly reactive methane sulfonates. The over 20 reported 15knuc values all fell between 0.994 and 1.003, with the normal effects being limited to the most basic amines, and hence the earliest transition states. The consistency of the experimental observations and calculations indicate that only small normal 15knuc values (, 1.005) will be observed for nucleophilic attack of amines at sp3 carbon centers.

C. KINETIC I SOTOPE E FFECTS ON E NZYME C ATALYZED N UCLEOPHILIC ATTACK AT sp 2 C ARBONS 15

N KIEs have been measured for the addition of NH3 to the conjugated carbon –carbon double bond of fumarate catalyzed by aspartase. This reaction is analogous to the hydration reactions catalyzed by fumarase and enolase (Section II.B.2). Aspartase has significant sequence similarity to fumarase and shares fumarate as a common substrate.62 The 15(V/Kaspartate) for elimination of NH3 is 1.0239 ^ 0.0014 under conditions where the primary D(V/Kaspartate) is unity. The combination of these two effects indicate that, as for fumarase, the cleavage or formation of the vicinal C – H bond is fast and the cleavage or formation of the C – N bond is the rate determining step. 15(V/KNH3) for this reaction is 0.99. p p Since C –N bond reaction is rate determining, 15(V/KNH3) will be the intrinsic 15knuc for ammonia adding to fumarate. That an intrinsic KIE is inverse indicates that the e zero point energy effect from bond formation contributes more than the imaginary frequency factor (reaction coordinate motion). Inverse 15N KIEs suggest a moderately late transition state for this nucleophilic addition reaction. The secondary D(V)fumarate-d2 value of 0.81 is consistent with a significant change in hybridization from sp2 to sp3 at the electrophilic carbon as it reflects greater than 60% of the overall EIE. Rehybridization of a sp2 carbon to sp3 has an inverse effect of at least 0.93 and the EIE on conversion of fumarate-d2 to malate-d2 is 0.69.29 The significant change in hybridization is presumed to result from a significant increase in bonding to the incoming ammonia nucleophile, i.e., this secondary effect is also strongly indicative of a late transition state. Heavy atom KIEs on the addition of N-nucleophiles to carbon –oxygen double bonds have been investigated in several systems. The initial determination was on the reaction of aniline with an active ester of propanoate, in acetonitrile.63 If decomposition of the tetrahedral intermediate (Scheme 36.7) is the rate determining step, an inverse 15N KIE reflecting the 15N EIE on formation of the tetrahedral intermediate would be observed. The formation of the new C – N bond is expected to result in a 15N EIE of , 0.97 (Section III.A). Since the observed inverse effect of 0.996 is significantly less inverse than the projected EIE for tetrahedral intermediate formation, the suggestion is forwarded that there is a normal contribution from the formation of the tetrahedral intermediate. Since a partition ratio was not determined, the actual magnitude of the 15knuc on the addition step could not be extracted. However, if the C – N bond shortens on expulsion of the leaving group, there can be a normal contribution from the reaction coordinate motion. This alternative explanation has received support in the study of hydrazinolysis of methyl formate. p p By the Haldane relationship, the 15Keq for this reaction is the ratio of 15(V/K)aspartate/15(V/K)NH3. The 15Keq has been determined to be 1.034 (56. Hermes JD, Weiss PM, Cleland WW). Use of nitrogen-15 and deuterium isotope effects to determine the chemical mechanism of phenylalanine ammonia-lyase. (Biochemistry 1985, 24: 2959–2967.) Making 15 (V/K)NH3 ¼ 0.99.

910

Isotope Effects in Chemistry and Biology

CO2− O

+ NH3 R

knuc

HO H2N

CO2−

CO2− NADPH H2N+

R

R

NADP+

H2N H

CO2−

R

SCHEME 36.10 Chemical steps in the glutamate and alanine dehydrogenase catalyzed reactions.

By varying the pH of the reaction, it has been possible to determine 15N KIEs for the hydrazinolysis of methyl formate in water where either formation of the tetrahedral intermediate, or its breakdown, were cleanly rate determining. At pH 8.0, a very large 18O leaving group KIE of 1.0621 ^ 0.0008 was determined,41 identifying cleavage of this bond during the decomposition of the tetrahedral intermediate as being rate determining. The observed 15N KIE of 0.990 under these conditions64 reflects the product of the EIE for forming the tetrahedral intermediate and the secondary KIE on the decomposition step. The proposed normal secondary effect is ascribed to the participation of the C –N bond in the reaction coordinate: cleavage of the C –O bond is concomitant with a coordinated change in hybridization state of the carbonyl carbon and shortening of the C– N bond. The estimated 15Keq for formation of the intermediate is strongly inverse, , 0.976, requiring a normal 15 N KIE of up to 1.014 on the cleavage of the tetrahedral intermediate. A computational evaluation of the 15N KIE on decomposition of tetrahedral intermediates could provide insight into the origin of this effect. At high pH, a change in rate determining step is reflected in the decrease in the 18O leaving group KIE from 1.062 to 1.004. The increased base concentration results in either formation or deprotonation of the tetrahedral intermediate as being rate determining. The 15N KIE remains inverse at 0.992.64 The hybridization state of the carbonyl carbon in the transition state is revealed as being nearly sp3 hybridized by the strongly inverse secondary formyl-D KIE of 0.72. The formyl-D and 15N KIEs are consistent either with a late transition state for the nucleophilic addition, or as representing the EIEs for tetrahedral intermediate formation if deprotonation is the rate determining step. The modest solvent D2O effect of 1.6 does not unequivocally rule out either of the possibilities.64 The attack of nitrogen on a carbonyl carbon to form a tetrahedral intermediate, knuc constitutes the first step in the alanine and glutamate dehydrogenase catalyzed reductive aminations of a-keto acids65 as shown in Scheme 36.10. This step can be made largely rate determining by using thioNADPH as the reductant. Under those conditions a normal 15(V/KNH3) of 1.0015 was observed.66 As any of the subsequent steps in the reaction mechanism would have large (by comparison) inverse EIEs from the formation of at least one C – N bond, the normal effect necessarily arises from the KIE on carbinolamine formation. However, since it is significantly smaller than the effect on volatilization, the possibility exists that it reflects the desolvation of the N lone pair at the active site. Under all other reaction conditions, significant inverse 15(V/KNH3) effects were observed. Most of these effects were associated with significant primary D KIEs, identifying hydride transfer to the iminium intermediate as being partially rate determining, consistent with the inverse 15(V/KNH3) effects.

IV. PROSPECTUS A sufficient number of 18knuc and 15knuc KIEs have been determined to begin to observe broad similarities and differences. The two isotope effects have in common modest size and significant inverse contributions from the formation of a new heavy atom bond from the electrophilic center to the isotopically labeled nucleophilic atom. This bond formation shifts the position of the experimental window (so that it includes values on both sides of unity), when compared to that of the leaving group, and not its size. Inverse 18O and 15N KIEs will be observed either when there are late transition states or when steps in the reaction after the nucleophilic addition are rate limiting.

Nucleophile Isotope Effects

911

The initial interpretation of both 15N and 18O nucleophile KIEs depends on a characterization of the protonation state of the nucleophile. For 15N KIEs the analysis is straightforward, the nucleophilic N must be neutral, consequently if the reaction conditions are such that it is protonated in the ground state, the appropriate normal EIE on deprotonation must be taken into account. The analysis of 18O KIEs poses greater ambiguity because both the neutral and anionic forms of the nucleophile can participate as the nucleophile. Because the pKa of the nucleophilic O decreases over 10 units during the nucleophilic reaction, these reactions are prone to general base catalysis.67 Both specific base and general base catalysis can lead to significantly more normal observed 18knuc values than the 15knuc values observed for N nucleophiles. 18O KIEs provide even greater variation because of the potential coordination to metal ions and subsequent Ionization. The EIEs on coordination have been characterized, but it is not clear whether the Ionization of the coordinated metal ion will have the same large normal effect observed for solvent water. This is a crucial question that will need to be answered before the experimental results of studies of enzyme catalyzed hydrolysis reactions can be interpreted with confidence. While this chapter is not exhaustive, e.g., the 15N KIEs on amino acid lyases have been omitted because of mechanistic uncertainties,68 it is comprehensive. The number of nucleophile KIEs determined has been limited by real technical difficulties. These limitations are being reduced significantly by advances in whole molecule and continuous flow isotope ratio mass spectrometry. In numerous cases, e.g., in the study of hydrolysis of nitrophenyl phosphate,43,45,49 the nucleophile KIE is the last piece of the puzzle to be determined. The determination of these nucleophile KIEs will help to clarify the bond order to the nucleophile in the transition state and consequently complete the assessment of the dissociative or associative character of the transition state. Using nucleophile KIEs to characterize how enzyme active sites control the activation of water as a nucleophile will help reveal how living systems have contrived to take advantage of kinetically stable, but thermodynamically labile bonds that form the basis of all biological macromolecules.

ACKNOWLEDGMENTS The authors would like to thank Dr. W. W. Cleland and Dr. A. C. Hengge for access to their determination of 18O nucleophile KIEs prior to publication.

REFERENCES 1 Bigeleisen, J. and Wolfsberg, M., Theoretical and experimental aspects of isotope effects in chemical kinetics, Adv. Chem. Phys., 1, 15 – 76, 1958. 2 Streitwieser, A., Jr., Jagow, R. H., Fahey, R. C., and Suzuki, S., Kinetic isotope effects in the acetolyses of deuterated cyclopentyl tosylates, J. Am. Chem. Soc., 80, 2326 –2332, 1958. 3 Hartshorn, S. R. and Shiner, V. J., Calculation of h/d, 12c/13c, and 12c/14c fractionation factors from valence force fields derived for a series of simple organic molecules, J. Am. Chem. Soc., 94, 9002– 9012, 1972. 4 Kirsch, J. F., Secondary kinetic isotope effects, In Isotope Effects on Enzyme-Catalyzed Reactions, Cleland, W. W., O’Leary, M. H., and Northrop, D. B., Eds., University Park Press, Baltimore, MD, pp. 100– 121, 1977. 5 Gorenstein, D. G., Lee, Y.-G., and Kar, D., Kinetic isotope effects in the reactions of aryl-18O-2, 4-dinitrophenyl dibenzyl phosphate and aryl-18O-2,4-dinitrophenyl phosphate. Evidence for monomeric metaphosphate, J. Am. Chem. Soc., 99, 2264– 2267, 1977. 6 Engel, C. K., Mathieu, M., Zeelen, J. P., Hiltunen, J. K., and Wierenga, R. K., Crystal structure of enoyl-coenzyme A (CoA) hydratase at 2.5 angstroms resolution: a spiral fold defines the CoA-binding pocket, Embo J., 15, 5135– 5145, 1996.

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7 Engel, C. K., Kiema, T. R., Hiltunen, J. K., and Wierenga, R. K., The crystal structure of enoyl-CoA hydratase complexed with octanoyl-CoA reveals the structural adaptations required for binding of a long chain fatty acid-CoA molecule, J. Mol. Biol., 275, 847– 859, 1998. 8 Bahnson, B. J., Anderson, V. E., and Petsko, G. A., Structural mechanism of enoyl-CoA hydratase: three atoms from single water are added in either an E1cb stepwise or concerted fashion, Biochemistry, 41, 2621– 2629, 2002. 9 Fauman, E. B., Yuvaniyama, C., Schubert, H. L., Stuckey, J. A., and Saper, M. A., The x-ray crystal structures of Yersinia tyrosine phosphatase with bound tungstate and nitrate. Mechanistic implications, J. Biol. Chem., 271, 18780– 18788, 1996. 10 Zhang, Z. Y., Wang, Y., and Dixon, J. E., Dissecting the catalytic mechanism of protein-tyrosine phosphates, Proc. Natl Acad. Sci. U.S.A., 91, 1624– 1627, 1994. 11 Knofel, T. and Strater, N., Mechanism of hydrolysis of phosphate esters by the dimetal center of 5’-nucleotidase based on crystal structures, J. Mol. Biol., 309, 239– 254, 2001. 12 Knofel, T. and Strater, N., x-ray structure of the Escherichia coli periplasmic 5’-nucleotidase containing a dimetal catalytic site, Nat. Struct. Biol., 6, 448– 453, 1999. 13 Jansco, G. and Van Hook, W. A., Condensed phase isotope effects (especially vapor pressure isotope effects), Chem. Rev., 74, 689– 749, 1974. 14 Green, M. and Taube, H., Isotopic fractionation in the OH2 – H2O exchange reaction, J. Phys. Chem., 67, 1565– 1566, 1963. 15 Baes, C. F. Jr. and Mesmer, R. E., The Hydrolysis of Cations, Wiley, New York, pp. 496, 1976. 16 Feder, H. M. and Taube, H., Ionic hydration: an isotopic fractionation technique, J. Chem. Phys., 20, 1335– 1336, 1952. 16 17 Hunt, H. R. and Taube, H., The relative acidity of H18 2 O and H2 O coordinated to a tripositive ion, J. Phys. Chem., 63, 124– 125, 1959. 18 Cleland, W. W., Measurement of isotope effects by the equilibrium perturbation technique, Methods Enzymol., 64, 104, 1980. 19 Breslow, R., Chin, J., Hilvert, D., and Trainor, G., Evidence for the general base mechanism in carboxypeptidase a-catalyzed reactions: partitioning studies on nucleophiles and H18 2 O kinetic isotope effects, Proc. Natl Acad. Sci. U.S.A., 80, 4585– 4589, 1983. 20 Goshe, M. B. and Anderson, V. E., Determination of amino acid isotope ratios by electrospray ionization- mass spectrometry, Anal. Biochem., 231, 387–392, 1995. 21 Cassano, A. G., Anderson, V. E., and Harris, M. E., Evidence for direct attack by hydroxide in phosphodiester hydrolysis, J. Am. Chem. Soc., 124, 10964, 2002. 22 Begley, I. S. and Sharp, B. L., Characterization and correction of instrumental bias in inductively coupled plasma quadrupole mass spectrometry for accurate measurement of lead isotope ratios, J. Anal. At. Spectrom., 12, 395– 402, 1997. 23 Cassano, A. G., Ph.D. Thesis Case Western Reserve University Molecular Biology, 163, 2004. 24 Marlier, J. F., Multiple isotope effects on the acyl group transfer reactions of amides and esters, Acc. Chem. Res., 34, 283– 290, 2001. 25 Begley, I. S. and Scrimgeour, C. M., Online reduction of H2O for d 2H and d 18O measurement by continuous-flow isotope ratio mass spectrometry, Rapid Commun. Mass Spectrom., 10, 969– 973, 1996. 26 Hogg, J. L., Rodgers, J., Kovach, I., and Schowen, R. L., Kinetic isotope-effect probes of transitionstate structure. Vibrational analysis of model transition states for carbonyl addition, J. Am. Chem. Soc., 102, 79 – 85, 1980. 27 Lewis, D. E., Sims, L. B., Yamataka, H., and McKenna, J., Calculations of kinetic isotope effects in the Hofmann eliminations of substituted (2-phenylethyl)trimethylammonium ions, J. Am. Chem. Soc., 102, 7411– 7419, 1980. 28 Rishavy, M. A. and Cleland, W. W., 13C, 15N, and 18O equilibrium isotope effects and fractionation factors, Can. J. Chem., 77, 967– 977, 1999. 29 Blanchard, J. S. and Cleland, W. W., Use of isotope effects to deduce the chemical mechanism of fumarase, Biochemistry, 19, 4506– 4513, 1980. 30 Anderson, V. E., Ph.D. Thesis University of Wisconsin – Madison Biochemistry, 1981. 31 Bahnson, B. J. and Anderson, V. E., Isotope effects on the crotonase reaction, Biochemistry, 28, 4173– 4181, 1989.

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32 Bahnson, B. J. and Anderson, V. E., Crotonase-catalyzed b-elimination is concerted: a double isotope effect study, Biochemistry, 30, 5894– 5906, 1991. 33 Babbitt, P. C., Hasson, M. S., Wedekind, J. E., Palmer, D. R., Barrett, W. C., Reed, G. H., Rayment, I., Ringe, D., Kenyon, G. L., and Gerlt, J. A., The enolase superfamily: a general strategy for enzyme-catalyzed abstraction of the alpha-protons of carboxylic acids, Biochemistry, 35, 16489– 16501, 1996. 34 Anderson, S. R., Anderson, V. E., and Knowles, J. R., Primary and secondary kinetic isotope effects as probes of the mechanism of yeast enolase, Biochemistry, 33, 10545– 10555, 1994. 35 Koziet, J., Isotope ratio mass spectrometric method for the online determination of oxygen-18 in organic matter, J. Mass Spectrom., 32, 103– 108, 1997. 36 Houerou, G., Kelly, S. D., and Dennis, M. J., Determination of the oxygen-18/oxygen-16 isotope ratios of sugar, citric acid and water from single strength orange juice, Rapid Commun. Mass Spectrom., 13, 1257– 1262, 1999. 37 Kornexl, B. E., Gehre, M., Hofling, R., and Werner, R. A., Online d 18O measurement of organic and inorganic substances, Rapid Commun. Mass Spectrom., 13, 1685– 1693, 1999. 38 Tu, C. K. and Silverman, D. N., Solvent deuterium isotope effects in the catalysis of oxygen-18 exchange by human carbonic anhydrase II, Biochemistry, 21, 6353– 6360, 1982. 39 Wilson, D. K., Rudolph, F. B., and Quiocho, F. A., Atomic structure of adenosine deaminase complexed with a transition-state analog: understanding catalysis and immunodeficiency mutations, Science, 252, 1278– 1284, 1991. 40 Weiss, P. M., Cook, P. F., Hermes, J. D., and Cleland, W. W., Evidence from nitrogen-15 and solvent deuterium isotope effects on the chemical mechanism of adenosine deaminase, Biochemistry, 26, 7378– 7384, 1987. 41 Sawyer, C. B. and Kirsch, J. F., Kinetic isotope effects for reactions of methyl formate-methoxyl-18O, J. Am. Chem. Soc., 95, 7375– 7381, 1973. 42 Marlier, J. F., Dopke, N. C., Johnstone, K. R., and Wirdzig, T. J., A heavy-atom isotope effect study of the hydrolysis of formamide, J. Am. Chem. Soc., 121, 4356– 4363, 1999. 43 Cleland, W. W. and Hengge, A. C., Mechanisms of phosphoryl and acyl transfer, FASEB J., 9, 1585– 1594, 1995. 44 Cleland, W. W., Secondary 18O isotope effects as a tool for studying reactions of phosphate mono-, di-, and triesters, FASEB J., 4, 2899– 2905, 1990. 45 Hengge, A. C., Isotope effects in the study of phosphoryl and sulfuryl transfer reactions, Acc. Chem. Res., 35, 105– 112, 2002. 46 Cassano, A. G., Anderson, V. E., and Harris, M. E., Understanding the transition states of phosphodiester bond cleavage: insights from heavy atom isotope effects, Biopolymers, 73, 110–129, 2004. 47 Anderson, M. A., Shim, H., Raushel, F. M., and Cleland, W. W., Hydrolysis of phosphotriesters: determination of transition states in parallel reactions by heavy-atom isotope effects, J. Am. Chem. Soc., 123, 9246 –9253, 2001. 48 Philip Hendry, A. M. S., Metal ion promoted phosphate ester hydrolysis. Intramolecular attack of coordinated hydroxide ion, J. Am. Chem. Soc., 111, 2521– 2527, 1989. 49 Rawlings, J., Cleland, W. W., and Hengge, A. C., Metal ion catalyzed hydrolysis of ethyl p-nitrophenyl phosphate, J. Inorg. Biochem., 93, 61– 65, 2003. 50 Cassano, A. G., Anderson, V. E., and Harris, M. E., Analysis of solvent nucleophile isotope effects: evidence for concerted mechanisms and nucleophilic activation by metal coordination in nonenzymatic and ribozyme catalyzed phosphodiester hydrolysis, Biochemistry, 43(35), 10547– 10559, 2004. 51 Humphry, T., Forconi, M., Williams, N. H., and Hengge, A. C., An altered mechanism of hydrolysis for a metal-complexed phosphate diester, J. Am. Chem. Soc., 124, 14860 –14861, 2002. 52 Jones, J. P., Weiss, P. M., and Cleland, W. W., Secondary oxygen-18 isotope effects for hexokinasecatalyzed phosphoryl transfer from ATP, Biochemistry, 30, 3634– 3639, 1991. 53 O’Leary, M. H. and Marlier, J. F., Heavy atom isotope effects on alkaline hydrolysis of methyl benzoate, J. Am. Chem. Soc., 101, 3300, 1979. 54 Hermes, J. D., Weiss, P. M., and Cleland, W. W., Use of nitrogen-15 and deuterium isotope effects to determine the chemical mechanism of phenylalanine ammonia-lyase, Biochemistry, 24, 2959– 2967, 1985.

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55 Kurz, J. L., Daniels, M. W., Cook, K. S., and Nasr, M. M., Kinetic nitrogen isotope effects on methyl transfer to amines, J. Phys. Chem., 90, 5357– 5360, 1986. 56 Lowry, T. H. and Richardson, K. S., Mechanism and theory in organic chemistry, Harper & Row, New York, 1976. 57 Ruggiero, G. D. and Williams, I. H., Kinetic isotope effects for gas phase SN2 methyl transfer: a computational study of anionic and cationic identity reactions, J. Chem. Soc., Perkin Trans., 2, 591– 597, 2002. 58 Paneth, P. and O’Leary, M. H., Nitrogen and deuterium isotope effects on quaternization of N, N-dimethyl-p-toluidine, J. Am. Chem. Soc., 113, 1691– 1693, 1991. 59 Gawlita, E., Szylhabel-Godala, A., and Paneth, P., Kinetic effects on the Menshutkin reaction: theory versus experiment, J. Phys. Org. Chem., 9, 41 –49, 1996. 60 Szylhabel-Godala, A., Madhavan, S., Rudzinski, J., O’Leary, M. H., and Paneth, P., Nitrogen and deuterium isotope effects on the Menshutkin reaction, J. Phys. Org. Chem., 9, 35 – 40, 1996. 61 Owczarek, E., Kwiatkowski, W., Lemieszewski, M., Mazur, A., Rostkowski, M., and Paneth, P., Calculations of substituent and solvent effects on the kinetic isotope effects of Menshutkin reactions, J. Org. Chem., 68, 8232– 8235, 2003. 62 Woods, S. A., Miles, J. S., Roberts, R. E., and Guest, J. R., Structural and functional relationships between fumarase and aspartase, Biochem. J., 237, 547, 1986. 63 Kaminski, Z. J., Paneth, P., and O’Leary, M. H., Nitrogen kinetic isotope effects on the acylation of aniline, J. Org. Chem., 56, 5716– 5718, 1991. 64 Marlier, J. F., Haptonstall, B. A., Johnson, A. J., and Sacksteder, K. A., Heavy-atom isotope effects on the hydrazinolysis of methyl formate, J. Am. Chem. Soc., 119, 8838– 8842, 1997. 65 Grimshaw, C. E., Cook, P. F., and Cleland, W. W., Use of isotope effects and pH studies to determine the chemical mechanism of bacillus subtilis L -alanine dehydrogenase, Biochemistry, 20, 5655– 5661, 1981. 66 Weiss, P. M., Chen, C. Y., Cleland, W. W., and Cook, P. F., Use of primary deuterium and 15N isotope effects to deduce the relative rates of steps in the mechanisms of alanine and glutamate dehydrogenases, Biochemistry, 27, 4814– 4822, 1988. 67 Jencks, W. F., Catalysis in Chemistry and Enzymology, McGraw-Hill, New York, p. 644, 1969. 68 Retey, J., Discovery and role of methylidene imidazolone, a highly electrophilic prosthetic group, Biochim. Biophys. Acta, 1647, 179– 184, 2003.

37

Enzyme Mechanisms from Isotope Effects W. Wallace Cleland

CONTENTS I.

Isotope Effect Theory....................................................................................................... 915 A. Notation .................................................................................................................... 916 B. Measurement of Isotope Effects .............................................................................. 916 1. Direct Comparison ............................................................................................ 916 2. Internal Competition ......................................................................................... 917 a. The Remote Label Method — Stable Isotopes .......................................... 918 b. The Remote Label Method — Radioactive Isotopes ................................. 920 3. Equilibrium Perturbation................................................................................... 920 C. Equation for the Isotope Effect................................................................................ 921 D. Determination of Intrinsic Isotope Effects .............................................................. 923 1. Northrop’s Method ............................................................................................ 923 2. Multiple Isotope Effect Method ........................................................................ 924 II. Examples of Mechanistic Analysis.................................................................................. 925 A. Aspartate Transcarbamoylase .................................................................................. 925 B. Malic Enzyme .......................................................................................................... 926 C. Oxalate Decarboxylase ............................................................................................ 927 D. Chorismate Mutase .................................................................................................. 927 E. L-Ribulose-5-P 4-Epimerase.................................................................................... 928 F. Other Enzymes ......................................................................................................... 928 References..................................................................................................................................... 928

I. ISOTOPE EFFECT THEORY Isotope effects are a powerful tool for the study of enzyme mechanisms, since they look at the chemistry while it is taking place and can give information on transition state structure. They are caused by substituting a heavier isotope for a lighter one in a substrate for the enzyme. Isotopic substitution can affect either the rate of a reaction (a kinetic isotope effect) or the equilibrium constant (an equilibrium isotope effect; see Chapter 2). They are called primary isotope effects when a bond is made or broken to the isotopic atom during the reaction, and secondary when the bonding to this atom changes, but no bonds to it are made or broken (see Chapter 39). Secondary isotope effects are called a if bonds are made or broken to an atom attached to the isotopic one, or b if they are made or broken to the second atom from the isotopic one. Thus in the conversion of ethanol to acetaldehyde, deuterium substitution in the CH2 hydrogen atom transferred to NAD during the reaction gives a primary isotope effect, while substitution in the non-transferred CH2 915

916

Isotope Effects in Chemistry and Biology

hydrogen gives an a secondary isotope effect. Substitution in the methyl hydrogens gives a b secondary isotope effect, and running the reaction in D2O so the OH becomes deuterated will give a primary isotope effect. Kinetic isotope effects involve comparison of the stiffness of bonding of the isotopic atom in the transition state relative to the substrate, while equilibrium isotope effects compare the stiffness of bonding of product and substrate. In each case the heavy isotope enriches in the more stiffly bonded position, and the lighter isotope in the less stiffly bonded position. Stretching, bending and torsional vibrations all contribute to the observed isotope effects. Equilibrium isotope effects can be calculated from the fractionation factors of substrate and product, which are equilibrium isotope effects relative to a standard molecule (Refs. 1,2 or table in Chapter 2). The standards used for enzymatic reactions are water for deuterium, tritium or O-18 isotope effects, aqueous CO2 for C-13, and aqueous ammonia for N-15 ones. Equilibrium isotope effects are temperature dependent and extrapolate to unity at infinite temperature. They may be either greater than unity (normal) or less than unity (inverse), depending on whether the isotopic atom is more stiffly bonded in substrate or product. The equilibrium isotope effect on the back reaction is the reciprocal of that in the forward reaction. Primary kinetic isotope effects are always normal, since the vibration corresponding to the reaction coordinate motion has an imaginary frequency in the transition state. The zero-point energy contribution to a primary kinetic isotope effect extrapolates to unity at infinite temperature, but the imaginary frequency factor is temperature independent. Secondary kinetic isotope effects can be either normal or inverse depending on the bonding in the transition state. They are very useful for determining transition state structure, since unless the motion of the isotopic atom is coupled to the reaction coordinate (see Section I.D.2), they do not include any imaginary frequency factor and report on the bonding in the transition state.

A. NOTATION Isotope effects are expressed as the ratio of the parameter of the light isotope to that of the heavy one. The notation used by enzymologists is to indicate the isotope effect by a leading superscript. These are D, T, 13, 14, 15, 18 for deuterium, tritium, C-13, C-14, N-15, and O-18. S-34 or P-33 isotope effects would be expressed with 34 or 33, and Cl-37 ones with 37. Thus Dk is a deuterium isotope effect on a rate constant (kH/kD) and TKeq is a tritium isotope effect on the equilibrium constant (KeqH/KeqT). 13(V/K) is a C-13 isotope effect on V=K; while DV is a deuterium isotope effect on the maximum velocity.

B. MEASUREMENT OF I SOTOPE E FFECTS 1. Direct Comparison For enzymatic reactions studied under steady state conditions, the independent kinetic parameters are V (the maximum velocity) and V=K (apparent first order rate constant at low substrate concentration) for each substrate. For isotope effects that are large enough (in practice, primary and some secondary deuterium ones), one can simply compare reciprocal plots with deuterated and unlabeled substrates. The ratio of the slopes is D(V/K) and the ratio of the vertical intercepts is DV. This is the only way to determine an isotope effect on the maximum velocity. The purity of the substrate is not important for determination of D(V/K), as the effects of inhibitors are diluted out when substrate concentration is extrapolated to zero, but it is essential that the relative concentrations of active substrate be accurately known for both the deuterated and unlabeled substrates. By contrast, the purity of the substrates is critical for determination of DV, but accurate concentrations are not, since the substrate concentrations are extrapolated to infinity. In practice the direct comparison method is the least accurate method for measuring isotope effects, but for large deuterium ones it is the only practical one.

Enzyme Mechanisms from Isotope Effects

917

2. Internal Competition In this method the substrate contains both the heavy and the light atoms and the changes in the ratio of these as the reaction proceeds is used to determine the isotope effect. This is the method that is used for tritium or C-14 isotope effects and would be used for P-33 ones if they are ever measured (P-33 gives twice the mass difference and has twice the halflife of P-32, and thus is the phosphorus isotope of choice). Very accurate determinations of specific activity are required. It also is the method that is used with the natural abundances of C-13 (1.1%), N-15 (0.37%), O-18 (0.2%), and Cl-37 (24%) as labels. In this case, the changes in mass ratio are determined with an isotope ratio mass spectrometer, with N2, CO2, or CH3Cl3 as the molecules containing the labels. For Cl-37 isotope effects, AgCl and a FAB-IRMS method has also been used.4 Use of an isotope ratio mass spectrometer is the most precise method of determining isotope effects, with errors of 0.02% or less. While geochemists determine S-34 isotope effects by putting SO2 into an isotope ratio mass spectrometer, this tends to burn out the filament, and S-34 isotope effects would better be determined on enzymatic reactions by the remote label method outlined below. The internal competition method is the most accurate one of the three that can be used to study an enzymatic reaction, but it only determines isotope effects on V=K for the substrate containing the label. Isotope effects on V can only be determined by direct comparison. That internal competition experiments only determine isotope effects on V=K is readily demonstrated by realizing that the labeled substrate is a competitive inhibitor of the unlabeled one and vice versa. Thus for a tritium label: vH ¼ VH H=½KH ð1 þ T=KT Þ þ H

vT ¼ VT T=½KT ð1 þ H=KH Þ þ T

ð37:1Þ

where H and T are concentrations of unlabeled and tritiated substrates and the subscripts identify the kinetic parameters. Note that for an alternative substrate the competitive inhibition constant is the Km (defined as the apparent dissociation constant in the steady state). These equations can be put in a form with identical denominators by dividing by their respective Km ’s: vH ¼ ðV=KÞH H=½1 þ T=KT þ H=KH ¼ 2dH=dt

ð37:2Þ

vT ¼ ðV=KÞT T=½1 þ H=KH þ T=KT ¼ 2dT=dt

ð37:3Þ

If Equation 37.2 is divided by Equation 37.3: dH=dT ¼ ½ðV=KÞH =ðV=KÞT ½H=T ¼ T ðV=KÞ½H=T

ð37:4Þ

Then, dH/H ¼ T(V/K)(dT/T) which integrates to: T

ðV=KÞ ¼ lnðH=H0 Þ=½lnðT=T0 Þ

ð37:5Þ

When the label is present in small amounts in an internal competition experiment, as is the case with the radioactive isotopes, or with the natural abundances of C-13, N-15, or O-18, the following definitions apply: f ¼ fraction of reaction ¼ ðH0 2 HÞ=H0

ð37:6Þ

R0 ¼ mass ratio of starting substrate ¼ T0 =H0

ð37:7Þ

Rp ¼ mass ratio of product at fraction of reaction f ¼ ðT0 2 TÞ=ðH0 2 HÞ

ð37:8Þ

Rs ¼ mass ratio in residual substrate at f ¼ T=H

ð37:9Þ

918

Isotope Effects in Chemistry and Biology

Then,

H=H0 ¼1 2 f T=T0 ¼1 2 fRp =R0 ¼ ð1 2 f ÞðRs =R0 Þ ¼ ð1 2 f Þ=½1 2 f þ ðfRp =Rs Þ

ð37:10Þ

The equations for calculating the isotope effect are: T

ðV=KÞ ¼ logð1 2 f Þ=½logð1 2 fRp =R0 Þ ¼ logð1 2 f Þ=log½ð1 2 f ÞðRs =R0 Þ ¼ logð1 2 f Þ=log{ð1 2 f Þ=½1 2 f þ ðfRp =Rs Þ }

ð37:11Þ

For measurement from Rp and R0, the f value should be as low as practical (0.2 – 0.4), since Rp rises (for a normal isotope effect) as the reaction proceeds and reaches R0 at an f value of 1.0. For measurement using Rs and R0, one should use an f value of 0.5– 0.7, as Rs/R0 rises gradually during the reaction, reaching the size of the isotope effect at an f value of 0.63– 0.7, and infinity at f ¼ 1.0. The error in f will be more important than that in Rs/R0 for f values above 0.5. While determinations using Rp values can be made for any size isotope effect, one can use Rs values only for ones less than 2. For larger tritium isotope effects, most of the label remains in the substrate, making values determined from Rs very inaccurate. For C-13, N-15, and O-18 isotope effects, it is desirable if possible to determine both Rp and Rs values and thus to have two independent measures of the isotope effect. If they agree, one has confidence that the measured value is valid. If the heavy and light molecules are present in comparable levels, as when using roughly equal levels of deuterated and unlabeled substrate, the equation for the isotope effect, although still derived from Equation 37.5, becomes more complex: D

ðV=KÞ ¼ log½1 2 f =ðXH þ XD Rp =R0 Þ =½log½1 2 f =ðXD þ XH R0 =Rp Þ ¼ log½ð1 2 f Þ=ðXH þ XD Rs =R0 Þ =½log½ð1 2 f Þ=ðXD þ XH R0 =Rs Þ

ð37:12Þ

where XH and XD are initial mole fractions of unlabeled and deuterated substrates. For isotope effects less than 1%, such as Cl-37 ones where the R0 value is , 0.32, Equation 37.11 and Equation 37.12 give the same answer, but for an isotope effect of 2.4 with XH and XD both 0.5 and f ¼ 0:375; the use of Rp/R0 in Equation 37.11 gives 2.26, while the use of Rs/R0 gives 7.28. The isotope ratio mass spectrometer compares the mass ratios in an unknown and in a tank standard of CO2 or N2 and gives an answer expressed as a d value. This is the mass ratio in sample divided by the mass ratio in standard, minus one and multiplied by 1000. A change of 1d thus equals a change of 0.1% in the mass ratio. The R values used in Equation 37.11 and Equation 37.12 above are then equal to the d of the sample divided by 1000, plus one. One must have samples that are close to natural abundance for two reasons. First, the spectrometer is designed to deal with such mass ratios, and if the d value gets too large in the positive or negative direction, the error is magnified and the result may no longer be valid. Second, if one did change the amplifiers so that 50– 50 mixtures of the isotopes could be measured and made up standards that matched this, a contamination of 1% CO2 or N2 from the air would look like a 2% isotope effect. With natural abundance, a 1% contaminant can introduce a maximum error of 1% in the isotope effect minus one, which is negligible. a. The Remote Label Method — Stable Isotopes There is no problem using the isotope ratio mass spectrometer to measure isotope effects on decarboxylations, since the CO2 product is readily isolated on a vacuum line, freed from water by distillation on the line, and put into the spectrometer for analysis. Molecules containing a single nitrogen are readily oxidized to N2 by CuO in a sealed quartz tube at 7758C, followed by separation

Enzyme Mechanisms from Isotope Effects

919

from CO2 and water on the vacuum line. For C-13 isotope effects other than in CO2, a degradation must be devised to isolate each carbon of interest as CO2, while for molecules containing more than one nitrogen, one must degrade the molecule to separate the nitrogens for analysis. For O-18 isotope effects, one usually has no easy way to isolate CO2 without equilibration of the oxygens with water (but see Section II.C). In these cases, the remote label method is the method of choice. To illustrate the method, consider reactions of various nucleophiles with p-nitrophenyl acetate.5 To determine the O-18 isotope effect in the phenolic oxygen, one prepares a molecule with O-18 in this position and N-15 in the nitro group. Triphenyl phosphate is readily nitrated in the para position, and gives p-nitrophenol on hydrolysis. This in turn is exchanged at pH 14 with H2[18O] to give the desired doubly labeled molecule. 0.37% of this is mixed with p-nitrophenyl acetate containing N-14 in the nitro group. Ammonium nitrate depleted of N-15 in both nitrogens is commercially available as a byproduct of isolating N-15. One ends up with remote labeled material containing the natural abundance of N-15 in the nitro group, but with every N-15 accompanied by O-18 in the phenolic oxygen. The isotope effect is determined from mass ratios in the p-nitrophenol product, and the initial and residual p-nitrophenyl acetate. This value is the product of the N-15 and O-18 isotope effects, and division by the N-15 one (determined with natural abundance p-nitrophenyl acetate) gives the O-18 isotope effect. Determination of isotope effects in other positions of p-nitrophenyl acetate only requires synthesis of molecules labeled with N-15 in the nitro group and with O-18 in the carbonyl oxygen, C-13 in the carbonyl carbon, or deuterium in the methyl group. In each case the doubly-labeled molecule is mixed with the one containing N-14 in the nitro group. Note particularly that this allows determination of the b-secondary deuterium isotope effect with a precision of less than 0.1%, which is much more accurate than that which one can do by direct comparison of rates with deuterated and unlabeled molecules. In practice, corrections are needed for incomplete labeling in use of the remote label method. The equation for the isotope effect caused by a single heavy atom in the position of interest is: j

ðV=KÞ per atom ¼ 1 þ ½ðsqÞ 2 1 =½1 2 ðsqÞð1 2 yÞ=i

ð37:13Þ

where: (sq) ¼ the ith root of ðP=RÞ=½1 2 QðP=R 2 1Þ P ¼ q, j(V/K) ¼ observed isotope effect with remote label (i.e., 15,18,18(V/K), where the remote label is N-15, and there are two O-18’s in the position of interest). R ¼ isotope effect in remote labeled position with natural abundance reactant. i ¼ number of discriminating atoms (1, 2, or 3: 2 in the example above). j ¼ nature of discriminating atoms (13, 15, 18, etc.). q ¼ nature of remote label (13, 15). Q ¼ ð1 2 bÞz=ðbxÞ < z=b ¼ degree to which light material in the remote labeled mixture is depleted below natural abundance. b ¼ fraction of double labeled material in the remote labeled mixture. z ¼ fraction of heavy label present in the remote labeled position of the light material used for the remote labeled mixture. x ¼ fraction of heavy label in the remote labeled position of the doubly labeled material used for mixing. y ¼ fraction of heavy discriminating label in the doubly labeled material (or if 2 or 3 discriminating atoms are present, the fraction containing all atoms labeled. For example, 90% labeling with i ¼ 2 gives y ¼ 0.81). It is critical in using the remote label method that the mixture not contain enriched or depleted impurities that will distort the analysis. One should thus mix the two molecules and then repurify the mixture. An enriched contaminant that does not react will lead to isotope effects that are too

920

Isotope Effects in Chemistry and Biology

high. By contrast, a depleted contaminant that is already in the product form will also give too high a result. In most cases, such contaminants will lead to different results when Rp or Rs are used for the analysis, which is a good reason for trying to determine both mass ratios. A number of groups are useful as remote labels. The nitro groups of p-nitrophenol (pK 7) and m-nitrobenzyl alcohol (pK 14) have been useful for reactions where these molecules were the leaving group. The latter is prepared by nitrating benzaldehyde, followed by reduction. The nitrogens of p-amidophenol (pK 8.6) and choline (pK 14) have also been useful remote labels. Another useful remote label is the exocyclic amino group of adenine, which is readily inserted by reaction of chloropurine riboside with ammonia in a bomb. The resulting adenosine can be built into ATP or NAD or similar molecules, and this nitrogen can be later isolated as ammonia by treatment with adenosine deaminase after the residual substrate or product are degraded back to adenosine. Nitrogens are not the only useful remote labels. C-1 of glucose, which can be readily added from cyanide to arabinose, can be converted to CO2 by phosphorylation with hexokinase and treatment with the dehydrogenases that convert glucose-6-P to ribulose-5-P. Any carboxyl group that can be decarboxylated to CO2 is a potential remote label. This method allows one to determine almost any isotope effect in any molecule. b. The Remote Label Method — Radioactive Isotopes Schramm has pioneered the use of C-14 or tritium atoms as remote labels in his work on N-ribosyltransferases.6,7 For example, a molecule containing a C-14 label at C-50 is mixed with a molecule containing tritium at a position of interest (C-10 , C-20 , etc.). The change in the C-14tritium ratio in product or residual substrate is used to calculate the isotope effect. For determination of the N-15 isotope effect at N-9 of AMP, a molecule was synthesized containing N-15 at N-9 as well as C-14 at C-50 , and this was mixed with C-50 tritiated AMP. The C-14-tritium ratio determines the N-15 isotope effect after correction for any tritium isotope effect at C-50 . The reader is referred to Refs. 6 and 7 for further examples of this method. It requires careful counting, and can determine isotope effects only an order of magnitude less accurately than with the isotope ratio mass spectrometer. This is, however, sufficient for all but the most accurate work. 3. Equilibrium Perturbation The third method of measuring isotope effects on enzymatic reactions is equilibrium perturbation, which works only for reactions that are readily reversible. In this method, one makes up reaction mixtures containing all reactants at calculated equilibrium concentrations, but with one substrate labeled with a heavy isotope and the corresponding product unlabeled. These molecules are called the perturbants. One then adds enzyme and observes what happens by following absorbance, CD, or some other parameter that can be monitored continuously. A normal isotope effect will cause the labeled substrate to react more slowly than the unlabeled product, so that the apparent equilibrium will be perturbed towards the labeled substrate. As isotopic equilibrium is approached, however, the apparent equilibrium will return to the starting point, at least if the concentrations are correctly chosen. The apparent isotope effect is determined from the size of the perturbation by the following Equation 37.2: ðAmax 2 A0 Þ=A00 ¼ a21=ða21Þ 2 a2aða21Þ

ð37:14Þ

where Amax 2 A0 is the size of the perturbation and A00 is a concentration factor which limits the size of the perturbation. If there is only one substrate and product, A00 is the reciprocal of the sum of the reciprocal concentrations of the perturbants, but if there are other reactants, the level of any one of these that is less than that of the lowest perturbant concentration will also limit A00 .2 The apparent

Enzyme Mechanisms from Isotope Effects

921

isotope effect, a, is related to the actual isotope effects in forward and reverse reactions by D

D

ðEq:P:Þfor ¼ aðK þ D Keq Þ=ð1 þ KÞ

ð37:15Þ

ðEq:P:Þrev ¼ að1 þ K=D Keq Þ=ð1 þ KÞ

ð37:16Þ

where the leading superscripts indicate a deuterium isotope effect and DKeq is the value in the forward direction. While an isotope effect determined by equilibrium perturbation is similar to one on V=K; they are not usually identical unless there is only one substrate and product, and thus D (Eq.P.) is used in place of D(V/K). Note that D(Eq.P.)rev ¼ D(Eq.P.)for/DKeq. The parameter K in Equation 37.15 and Equation 37.16 is the ratio at equilibrium of the levels of unlabeled perturbants in the system (i.e., [product]/[substrate]), which is not the same as the actual starting levels because of the equilibrium isotope effect. K is thus calculated from DKeq and the actual concentrations of the perturbants used. In practice, a computer program is used to solve these equations.2 The purity of the reactants is not critical as long as one of them is not contaminated by a more slowly or rapidly reacting molecule. Good temperature control is essential, since most enzymatic reactions have an equilibrium constant that varies with temperature. Picking up a cuvet to add enzyme introduces a pulse of heat that is slowly transferred to the contents and results in a spurious perturbation. The best way to control for artifacts is to perturb unlabeled substrate against unlabeled product, and labeled substrate against labeled product, as well as using either labeled substrate or product to cause a perturbation. This method is not as sensitive as internal competition, but can determine isotope effects as small as 1.003, which is considerably better than direct comparison.

C. EQUATION FOR THE I SOTOPE E FFECT If only one step in an enzymatic mechanism is isotope-sensitive, the equation for the isotope effect on V=K is x

ðV=KÞ or x ðEq:P:Þ ¼ ðx k þ cf þ cr x Keq Þ=ð1 þ cf þ cr Þ

ð37:17Þ

where x indicates the nature of the isotopic atom (D, C-13, N-15, etc.) and xk is the intrinsic isotope effect on the isotope-sensitive step. cf and cr are commitments in forward and reverse directions, and each is the ratio of the rate constant for the isotope-sensitive step to the net rate constant for release of a reactant from the enzyme. For cf this is the substrate whose concentration is varied in a direct comparison experiment, the labeled substrate in an internal competition experiment, or the perturbant in an equilibrium perturbation. For cr this is the first product released in direct comparison or internal competition measurements, but the perturbant in an equilibrium perturbation. For mechanism 18, where the isotope sensitive step involves k5 and k6, the values of cf and cr for a direct comparison or internal competition experiment are given by Equation 37.19 and Equation 37.20. k1 A

k3

k5

k7

k2

k4

k6

k8

k9

k11

E O EA O EAp O EPQp O EPQ ! EQ ! E

ð37:18Þ

cf ¼ ðk5 =k4 Þð1 þ k3 =k2 Þ

ð37:19Þ

cr ¼ ðk6 =k7 Þð1 þ k8 =k9 Þ

ð37:20Þ

922

Isotope Effects in Chemistry and Biology

However, for an equilibrium perturbation experiment, where the perturbants are A and Q and the right portion of mechanism 18 becomes mechanism 21, cr is given by Equation 37.22. k7

k9

k11

k8

k10 P

k12 Q

EPQp O EPQ O EQ O E

ð37:21Þ

cr ¼ ðk6 =k7 Þð1 þ k8 =k9 ð1 þ k10 P=k11 ÞÞ

ð37:22Þ

With isocitrate dehydrogenase, no deuterium isotope effect was seen on V=Kisocitrate because cf is so large. When deuterated isocitrate was used in an equilibrium perturbation experiment, however, D (Eq.P.)for was 1.15, since cr, which was for NADPH, was now larger than cf, so that the observed isotope effect was DKeq.8 In the direct comparison experiment, cr was for CO2, which is released rapidly from the enzyme, so the commitment is very low. A similar situation occurs if product P is present in a direct comparison or internal competition experiment. Equation 37.22 represents cr in this case also, and will be a linear function of P concentration if product release is ordered as shown in mechanisms 18 and 21. Both cf and cr have external parts ðk3 k5 =ðk2 k4 Þ and k6 k8 =ðk7 k9 Þ in Equation 37.19 and Equation 37.20 and internal parts (k5 =k4 and k6 =k7 ). The internal commitments are usually pH independent and are unaffected by the levels of other reactants, but this is not true of the external commitments, as noted above in the difference between Equation 37.20 and Equation 37.22. The external forward commitment is often abolished by changes in pH which decrease k3 relative to k2 in Equation 37.19, for example (see Cook chapter). If there are several substrates, the external forward commitment will depend on the levels of substrates other than the labeled or varied one. For example, if the mechanism is ordered k1 A

k3 B

k2

k4

k05

E O EA O EAB ! E þ products

ð37:23Þ

where k05 is the net rate constant for further reaction through the first irreversible step, cf for B is k50 /k4 regardless of the level of A, while that for A is cf ¼ ðk05 =k4 Þð1 þ k3 B=k2 Þ

ð37:24Þ

cf for A thus varies from k05 =k4 when B is very low to infinity when B is saturating. If A is the labeled substrate in an internal competition experiment, one should try to keep the level of B as low as possible (, 10% of Kia Kb =Ka ) so that the external commitment is reduced to that of B ðk05 =k4 Þ: In his chapter, Cook gives a full description of the way external commitments vary with substrate levels for random as well as ordered and equilibrium ordered mechanisms. If Equation 37.17 is divided by xKeq, one obtains the equation for x(V/K) in the reverse reaction, with cf and cr trading places and 1=x Keq being the equilibrium isotope effect in the reverse direction. However, one must be careful about the meanings of cf and cr, and which reactants they are calculated for. For example, in an ordered mechanism if the labeled substrate is A in an internal competition experiment, cf in the forward direction is given by Equation 37.24, while the reverse commitment in the back reaction is figured for the first product released, which is B, and thus has the constant value of k05 =k4 ; unless B is present. The equation for the isotope effect on V in mechanism 18 is: x

V ¼ ðx k þ cVf þ cr x Keq Þ=ð1 þ cVf þ cr Þ

ð37:25Þ

where xk, cr and xKeq have the same meanings as in Equation 37.17. cr is given by Equation 37.20, or Equation 37.22 if P is present. The parameter cVf in Equation 37.25 is not a commitment, but is a special constant found only in the equation for isotope effects on V: It basically is a comparison of

Enzyme Mechanisms from Isotope Effects

923

k5 in mechanism 18, multiplied by the proportion of EA and EA* in the latter form at equilibrium (k3/(k3 þ k4)), and other net rate constants following the catalytic step ðk7 =ð1 þ k8 =k9 Þ; k9 and k11) and those prior to the catalytic step that lead to EA* (k3 in this case). Thus in mechanism 18: cVf ¼ k5 ½k3 =ðk3 þ k4 Þ ½1=k3 þ ð1=k7 Þð1 þ k8 =k9 Þ þ 1=k9 þ 1=k11

ð37:26Þ

If k11, for example, is small, the last term becomes large and one will not see a V isotope effect, even when a large one is seen on V=K: This is a common situation with dehydrogenases where the release of the nucleotide product is slow and rate limiting for V: Division of Equation 37.25 by xKeq will not give the equation for xV in the back reaction. The reverse commitment in the back reaction is cf in the forward one, but cVf in the back reaction is given by an equation similar to Equation 37.26, but involving the net rate constants in the back direction. Equation 37.17 assumes that only one step is isotope sensitive. If each step is isotope sensitive in mechanism 27, for example: k1 A

k3

k2

k4

k 05

E O EA O EAB ! E þ products

ð37:27Þ

the V=K isotope effect is x

ðV=KÞ ¼

x

Keq1 x Keq3 x k05 þ x Keq1 x k3 ðk05 =k4 Þ þ x k1 ðk3 k05 =ðk2 k4 ÞÞ 1 þ ðk05 =k4 Þð1 þ k3 =k2 Þ

ð37:28Þ

In Equation 37.28 the numerator terms include equilibrium isotope effects on steps leading up to the kinetic one. The first term includes xk50 , the second one xk3, and the last one xk1. The partition ratios are the same in corresponding numerator and denominator terms. This pattern is readily expanded for more complex mechanisms, such as 18.

D. DETERMINATION OF I NTRINSIC I SOTOPE E FFECTS From Equation 37.17 and Equation 37.25 it is clear that unless commitments are small and the isotope-sensitive step is rate limiting, one needs a way to determine the intrinsic isotope effect in order for the observed values to be useful. Two methods have been used for doing this. The first involves comparison of deuterium and tritium isotope effects on V=K: The second uses the effects of deuteration on C-13 or other heavy atom isotope effects to solve for intrinsic isotope effects. 1. Northrop’s Method In this method9 one assumes that there is either no reverse commitment, or that the equilibrium isotope effect is unity. Then one can define a single commitment or sum of commitments as “c” and, D

ðV=KÞ 2 1 ¼ ðD k 2 1Þ=ð1 þ cÞ and T ðV=KÞ 2 1 ¼ ðT k 2 1Þ=ð1 þ cÞ so that

½D ðV=KÞ 2 1 =½T ðV=KÞ 2 1 ¼ ðD k 2 1Þ=ðT k 2 1Þ

ð37:29Þ

Since Tk ¼ (Dk)1.44,10 one can rewrite Equation 37.29 as: ½D ðV=KÞ 2 1 =½T ðV=KÞ 2 1 ¼ ðD k 2 1Þ=½ðD kÞ1:44 2 1

ð37:30Þ

One thus has an equation with experimental quantities on the left side and only one unknown on the right side. This equation requires either a computer program for solution, or reference to tables of ½D ðV=KÞ 2 1 =½T ðV=KÞ 2 1 vs: D k:11

924

Isotope Effects in Chemistry and Biology

If there is a finite reverse commitment and DKeq is not unity, Equation 37.30 will not give an exact answer. The procedure then is to divide D(V/K) by DKeq and T(V/K) by TKeq (which is (DKeq)1.44) to give the isotope effects in the reverse direction. These are used in Equation 37.30 to give a value for Dkrev. The true value of Dkfor then lies between the one given by Equation 37.30 using isotope effects measured in the forward direction, and Dkrev DKeq, and the true value of Dkrev lies between the value given by Equation 37.30 and Dkfor/DKeq. 2. Multiple Isotope Effect Method In this method the effect of deuteration on a heavy atom isotope effect is used to determine limits for intrinsic isotope effects, or if there is no reverse commitment, or one uses both primary and secondary deuteration, to determine true values of the intrinsic isotope effects.12 The method requires that all isotope effects be on the same step. Equation 37.17 will apply to all V=K isotope effects, with the superscript x being pri-D, sec-D, or 13 for primary or secondary deuterium, or C-13 isotope effects. The commitments will be the same. However, when 13(V/K) is determined with a deuterated substrate, the forward commitment is divided by Dkfor, and the reverse one by Dkfor/DKeq, since the rate constant for the isotope sensitive step is reduced by these factors, while the net rate constants for reactant release are not changed. One thus has potentially five equations resembling Equation 37.17 for 13(V/K)H, 13(V/K)pri-D, 13(V/K)sec-D, pri-D(V/K), and sec-D(V/K). Since there are only five unknowns (13k, pri-Dk, sec-Dk, cf, and cr; the equilibrium isotope effects are known or determinable separately), the equations can be solved simultaneously for the five unknowns. If cr is negligible, one needs only the effect of primary deuteration on 13(V/K), as there are then only the two intrinsic isotope effects and cf to determine. This was the case for prephenate dehydrogenase, where C-13 isotope effects of 1.0033 and 1.0103 with unlabeled and deuterated substrate and a deuterium isotope effect on V=K of 2.34 gave 13k ¼ 1.0155, Dk ¼ 7.3, and cf ¼ 3.7.13 The increased C-13 isotope effect upon deuteration, but the small intrinsic C-13 isotope effect shows that the reaction is concerted, although with an asynchronous transition state with well advanced C – H cleavage, but much less C –C cleavage. An example of the use of five equations to solve for all of the parameters was a study of glucose-6-P dehydrogenase.14 The resulting values were 13k ¼ 1.041 ^ 0.002, pri-Dk ¼ 5.3 ^ 0.3, sec-D k ¼ 1.054 ^ 0.035, cf ¼ 0.75 ^ 0.26, cr ¼ 0.49 ^ 0.27, and cf þ cr ¼ 1:24 ^ 0:14: Note that 13 k is very well determined, while pri-Dk is somewhat less so. The value of sec-Dk is not significantly different from unity, although since the equilibrium isotope effect is 0.89, any value of unity or above represents a real isotope effect. While the individual values of cf and cr are not well determined, their sum is well determined. When the experiments were repeated in D2O, 13k was not changed (1.044 ^ 0.004), but pri-Dk was reduced to 3.7 ^ 0.3, which is a significant decrease. sec-Dk also appeared to decrease (to 1.00 ^ 0.04), although the difference, while probably real, is not significant. The surprising result was that cf increased to 1.9 ^ 0.6 and cr to 0.65 ^ 0.40, with their sum being 2.5 ^ 0.3, a doubling of the value in water. In this reaction there are three hydrogens in motion in the transition state. The primary hydrogen is being transferred from C-1 of glucose-6-P to NADP. The hydrogen at the 4 position of the nicotinamide ring of NADP is moving from an in plane to an out of plane position in NADPH, and the proton on the 1-OH group of glucose-6-P is being transferred to the aspartate general base on the enzyme. Coupled hydrogen motions of this type involve tunneling and the first deuterium substitution in the system reduces the tunneling and leads to smaller isotope effects on subsequent deuterium substitution. Thus in D2O, where the 1-OH is an OD group, the other deuterium isotope effects are reduced. The striking thing about these results is that the commitments are doubled in D2O. Apparently the effect of the D2O solvent is greater on the rate constants for the conformation changes that precede and follow hydride transfer than on the rate of the latter. Thus one should attempt to interpret the effects of D2O on enzymatic reactions only when the intrinsic isotope effects and commitments are fully determined as in this case.

Enzyme Mechanisms from Isotope Effects

925

A similar coupled motion effect is seen in other dehydrogenases where the motion of an asecondary hydrogen is part of the reaction coordinate motion. With formate dehydrogenase, where there are no commitments since the C-13 isotope effect is the same with unlabeled and deuterated formate, the a-secondary deuterium isotope effects at the 4-position of NAD were reduced half way to the equilibrium isotope effect of 0.89 when deuterated formate was the substrate.15 With NAD, the a-secondary deuterium isotope effect was 1.23, reduced to 1.07 by deuteration of formate, which shows a high degree of coupled motion and a late transition state. As nucleotide substrates with more positive redox potentials replaced NAD and the transition state became earlier, which lessens the coupled motion, the a-secondary deuterium isotope effect decreased to 1.18 (1.03 with formate-d) with thio-NAD, and to 1.06 (0.95 with formate-d) with acetylpyridine-NAD.

II. EXAMPLES OF MECHANISTIC ANALYSIS A. ASPARTATE T RANSCARBAMOYLASE This enzyme catalyzes the reaction of carbamoyl-P with aspartate to give carbamoylaspartate and phosphate. When the C-13 isotope effect in carbamoyl-P was determined as a function of aspartate level, the values varied from 1.0217 at very low aspartate to unity at very high aspartate, with the half-point of the change being at 4.8 mM aspartate.16 This pattern shows that the mechanism is ordered, with carbamoyl-P adding first. The experiments were run by using 12 mM carbamoyl-P and converting half to products, with the mass ratio of the residual substrate (converted to CO2 with acid) being used to determine the isotope effect. To keep the aspartate at the appropriate level, it was added at the same rate as it was consumed, using a pump. This rate was first calculated and an experiment run, at the end of which the aspartate level was determined. The amount of enzyme needed to balance the rate of addition and consumption was then recalculated, and a second reaction was run and analyzed. This enzyme is highly cooperative, with inactive T form or forms, and an active R form. In the absence of modifiers most of the enzyme is in the T form, but aspartate binds only to the R form, so as more is added, the rate increases until all six sites are populated. ATP is an allosteric activator that binds to the R form and restores Michaelis kinetics, while CTP is an allosteric inhibitor that binds to the T form and increases the cooperativity. When the C-13 isotope effects were repeated in the presence of sufficient ATP to restore normal kinetics, or in the presence of sufficient CTP to slow the reaction considerably, the curve of 13(V/KCP) vs. aspartate was identical to the one in the absence of modifiers. This pattern shows that there is only one R form and it always has the same properties, as least as far as the isotope effect goes. Thus the enzyme fits the Monod model of allosteric behavior.17 When the isolated catalytic subunits, which show normal kinetics and no allosteric behavior, were used, the C-13 isotope effect varied from 1.0240 at very low aspartate to 1.0039 at infinite aspartate, showing that the reaction had now become partially random.16 When the slow alternate substrate cysteine sulfinate replaced aspartate, the C-13 isotope effect in carbamoyl-P was 1.039 for both the holoenzyme and the catalytic subunits at all levels of cysteine sulfinate. In this case the kinetic mechanism has become totally random, and 1.039 is probably the intrinsic isotope effect on the chemical step.16 The H124A mutant catalytic subunits gave a C-13 isotope effect of 1.04 at both very low and very high aspartate. Here again the mechanism is random and this is the intrinsic isotope effect on the chemistry.18 The chemical mechanism of this enzyme was addressed by determining the N-15 isotope effects in carbamoyl-P (again by acid degradation of residual substrate to yield ammonia, which was oxidized to N2 for analysis19). In the non-enzymatic breakdown of carbamoyl-P monoanion the isotope effect was 1.0028, while for the dianion it was 1.0114. The monoanion value is a secondary isotope effect, as the product of breakdown is carbamic acid. The dianion value is a primary isotope effect, as the molecule is degraded by internal general base catalysis by the phosphate group to give

926

Isotope Effects in Chemistry and Biology

cyanic acid. In this process an N –H bond is cleaved, which causes the 1.14% isotope effect. When catalytic subunits of wild type or H134A enzyme were used, the N-15 isotope effects were 1.0027 with H134A and aspartate, 1.0024 with wild type and cysteine sulfinate, and 1.0014 with wild type and aspartate. Since the first two combinations give intrinsic C-13 isotope effects, it appears that the values of 1.0027 and 1.0024 represent the intrinsic N-15 value. The value of 1.0014 is reduced by the same degree that the C-13 isotope effect is with wild type enzyme and aspartate. These data suggest that the nitrogen of carbamoyl-P does not undergo a N – H cleavage during the reaction, and thus rule out cyanic acid as an intermediate. The data are consistent with attack of aspartate to give a tetrahedral intermediate which breaks down into product. The small intrinsic N-15 isotope effect is thus a secondary one, presumably resulting from loss of amide resonance in the tetrahedral intermediate.

B. MALIC E NZYME When isotope effects are on different steps, deuteration slows down one step, and thus increases the commitment for the other isotope-sensitive step. In this case the C-13 isotope effect with unlabeled and deuterated substrates and the deuterium isotope effect are not independent, but are related as follows.12 In the direction where the deuterium-sensitive step comes first: ½13 ðV=KÞH 2 1 =½13 ðV=KÞD 2 1 ¼ D ðV=KÞ=D Keq

37:31

while in the reverse direction where the C-13 sensitive step comes first: ½13 ðV=KÞH 2 13 Keq =½13 ðV=KÞD 2 13 Keq ¼ D ðV=KÞ

37:32

These are really the same equation, with the parameters in Equation 37.31 being for the forward reaction, and those in Equation 37.32 for the reverse reaction. However, DKeq is often far enough from unity for experimental data to fit one of these equations and not the other. Then one can deduce which isotope sensitive step comes first, and thus the chemical mechanism. When these equations were applied to chicken liver malic enzyme with NADP as substrate,12 the C-13 isotope effect in the CO2 product was 1.030 with unlabeled malate and 1.025 with 2-deuterated malate. The deuterium isotope effect was 1.47. When these data were inserted into Equation 37.31 with DKeq ¼ 1.18, the result was 1.21 ^ 0.05 ¼ 1.25 ^ 0.03, which is a satisfactory fit. Using Equation 37.32, however with 13Keq ¼ 0.999, the result was 1.20 ^ 0.05 – 1.47 ^ 0.03. Thus dehydrogenation precedes decarboxylation, as one would expect. Later work, however, showed that with nucleotides with higher redox potentials, deuteration increased rather than decreased the C-13 isotope effect.20 This suggested that the reaction had become concerted, and this was confirmed by determining the C-13 isotope effect at C-3 as well as C-4.21 Similar results were reported with enzyme from Ascaris.22,23 Assuming no reverse commitment to decarboxylation, it was possible to calculate the following parameters for the concerted oxidative decarboxylation with acetylpyridine-NADP: cf ¼ 1.1 – 1.7, Dk ¼ 3.1– 3.7, 13 kC-4 ¼ 1.012, 13kC-3 ¼ 1.018.21 While the reaction is concerted, the transition state is asynchronous, with C –C cleavage lagging well behind C – H cleavage. This situation is similar to that seen with prephenate dehydrogenase (Ref. 13; see Section I.D.2). In contrast to the concerted mechanism seen with acetylpyridine-NADP and malate, a stepwise reaction was seen when (2R,3R)-erythro-fluoromalate was used as a substrate with acetylpyridineNADP.24 Thus the C-13 isotope effect in CO2 was reduced from 1.0138 to 1.0087 when malate was deuterated. While the acetylpyridine-NADP reaction has an equilibrium constant two orders of magnitude higher than with NADP, the use of fluoromalate decreases Keq by an order of magnitude. It appears that the conversion of the mechanism from stepwise to concerted results from the more favorable equilibrium for conversion of malate to pyruvate.21

Enzyme Mechanisms from Isotope Effects

927

C. OXALATE D ECARBOXYLASE This enzyme converts oxalate monoanion to CO2 and formate. It contains a Mn ion and requires a catalytic level of O2, which is not used up during the reaction. Since both oxalate and formate as anhydrous salts are oxidized cleanly to CO2 by I2 in dimethyl sulfoxide without oxygen exchange, it is possible to determine both the C-13 and O-18 isotope effects for generating each of the products.25 The C-13 isotope effect for forming CO2 was only 0.8% at pH 5.7 where the chemistry is rate limiting, while the O-18 isotope effect was inverse as expected, since the C – O bond order in CO2 is higher than that in oxalate. The small C-13 isotope effect suggests that C – C cleavage is not the major rate limiting step. By contrast, the C-13 isotope effect for formation of formate was 1.9% and the O-18 one 1.0%, both large. This suggests that the C –O bond order in the carboxyl of oxalate going to formate is reduced in a step prior to the C – C cleavage step. By fitting the isotope effects to the equations for such a model, the mechanism was postulated to involve formation of an oxalate radical in the first step in which an electron was removed from the end that was to become formate and a proton from the end that was to become CO2 (Glu-333 is poised to be the general base). Presumably the catalytic oxygen and the Mn interact in some fashion (perhaps by forming a Mn3þ-superoxide) that allows the metal ion to extract an electron from the bound oxalate. Calculating the equilibrium isotope effects for the radical formation and comparing these with calculated fractionation factors for oxalate with reduced bond order at one end gave 1.15 as the C – O bond order in the end of the radical intermediate that was destined to become formate. The fact that the same number was obtained from analysis of the C-13 and O-18 isotope effects supports this conclusion. A C-O bond order of 1.15 corresponds to 70% of a resonance form with single bonds in this end of the oxalate radical, and this 70% partial charge is sufficient to lead to rapid decarboxylation to give CO2 and a radical anion of formate. The latter will then pick up a proton (probably from Glu-333 via a water molecule) and an electron from Mn to form formate. The equations indicate that the oxalate radical intermediate reacts forward four times faster than it returns to starting material.

D. CHORISMATE M UTASE This enzyme converts chorismate to prephenate in a reaction that involves C – O cleavage at C-5 of the ring, and C – C formation between C-1 of the ring and the CH2 group of the enolpyruvyl side chain. To answer the question whether this reaction is a concerted pericyclic reaction, or stepwise with enolpyruvate and a carbenium ion at C-1 of the ring as intermediates, O-18 isotope effects in the ether oxygen at C-5 and C-13 ones at C-1 of the ring and the CH2 group of the side chain were determined by the remote label method, with the carboxyl at C-1 being the remote label (this carbon in prephenate is acid labile and readily isolated as CO2). With the B. subtilis enzyme, the O-18 isotope effect was 4.5% and the C-13 ones at C-1 and in the CH2 group were 0.43% and 1.3%.26,27 These numbers suggest that the reaction is concerted, with C – O cleavage well advanced in the transition state, but C –C formation lagging behind. This reaction takes place in the absence of the enzyme, but produces both prephenate and p-hydroxybenzoate in about a 10-1 ratio at 608C. The product ratio is not dependent on temperature or pH. The O-18 isotope effects for formation of both products are similar (4.5 – 4.8%), as are the C-13 isotope effects at C-1 (1.2%).27 The reaction to give prephenate also appears concerted, with C –O cleavage well advanced, but with a different degree of C –C formation at C-1. The formation of p-hydroxybenzoate results from a different rotamer of chorismate, with the CH2 of the side chain pulling the proton from C-4 to cause aromatization of the ring. The C-13 isotope effect at C-1 in this case presumably represents a decrease in bond order at this carbon as the result of C – O cleavage leading C– H cleavage.

928

Isotope Effects in Chemistry and Biology

E. L- RIBULOSE- 5-P 4-E PIMERASE This enzyme isomerizes L -Ribulose-5-P to D -xylulose-5-P. Since the change is at C-4, which is two carbons removed from the keto group, the usual enolization mechanism for isomerization cannot apply. The mechanism was demonstrated by determining C-13 isotope effects and deuterium ones.28 At pH 5.5, where the chemistry is largely rate limiting, the C-13 isotope effects were 2.53% at C-3 and 2.05% at C-4, but only 0.7% at C-2 and 0.3% at C-5. The deuterium isotope effects were 3% at C-3 and 14% at C-4. These values show that cleavage occurs between C-3 and C-4 without C – H cleavage at either carbon. The mechanism thus involves a retroaldol cleavage to give the enolate of dihydroxyacetone from carbons 1-3, stabilized by the Zn2þ present in the enzyme, and glycolaldehyde-P from carbons 4 and 5. Rotation of the aldehyde group and recondensation accomplishes the epimerization.

F. OTHER E NZYMES Several other chapters in this volume cover isotope effect studies on enzymatic reactions. In addition, the literature has mechanistic studies on phenylalanime ammonia lyase,29 fumarase,30 aspartate aminotransferase,31,32 biotin carboxylase,33 pyruvate carboxylase,34 adenosine deaminase,35 and OMP decarboxylase,36 that may be of interest to the reader.

REFERENCES 1 Rishavy, M. A. and Cleland, W. W., 13C, 15N, and 18O equilibrium isotope effects and fractionation factors, Can. J. Chem., 77, 967– 977, 1999. 2 Cleland, W. W., Measurement of isotope effects by the equilibrium perturbation technique, Methods Enzymol., 64, 104– 125, 1980. 3 Taylor, J. W. and Grimsrud, E. P., Chlorine isotopic ratios by negative ion mass spectrometry, Anal. Chem., 41, 805– 810, 1969. 4 Lewandowicz, A., Sicinska, D., Rudzinski, J., Ichiyama, S., Kurihara, T., Esaki, N., and Paneth, P., Chlorine kinetic isotope effect on the fluoroacetate dehalogenase reaction, J. Am. Chem. Soc., 123, 9192– 9193, 2001. 5 Hengge, A. C. and Hess, R. A., Concerted or stepwise mechanisms for acyl transfer reactions of p-nitrophenyl acetate? Transition state structures from isotope effects, J. Am. Chem. Soc., 116, 11256– 11263, 1994. 6 Schramm, V. L., Enzymatic transition states and transition state analog design, Annu. Rev. Biochem., 67, 693– 720, 1998. 7 Schramm, V. L., Enzymatic transition-state analysis and transition-state analogs, Methods Enzymol., 308, 301– 355, 1999. 8 Cook, P. F. and Cleland, W. W., pH variation of isotope effects in enzyme-catalyzed reactions. 1. Isotope- and pH-dependent steps the same, Biochemistry, 20, 1797– 1805, 1981. 9 Northrop, D. B., Steady-state analysis of kinetic isotope effects in enzymatic reactions, Biochemistry, 14, 2644– 2651, 1975. 10 Swain, C. G., Stivers, E. C., Reuwer, J. F. Jr, and Schaad, L. J., Use of hydrogen isotope effects to identify the attacking nucleophile in the enolization of ketones catalyzed by acetic acid, J. Am. Chem. Soc., 80, 5885 –5893, 1958. 11 Table for use with Northrop’s method. In Isotope Effects on Enzyme-Catalyzed Reactions, Cleland, W. W., O’Leary, M. H., and Northrop, D. B., Eds., University Park Press, Baltimore, pp. 280– 283, 1977. 12 Hermes, J. D., Roeske, C. A., O’Leary, M. H., and Cleland, W. W., Use of multiple isotope effects to determine enzyme mechanisms and intrinsic isotope effects. Malic enzyme and glucose-6-phosphate dehydrogenase, Biochemistry, 21, 5106– 5114, 1982.

Enzyme Mechanisms from Isotope Effects

929

13 Hermes, J. D., Tipton, P. A., Fisher, M. A., O’Leary, M. H., Morrison, J. F., and Cleland, W. W., Mechanisms of enzymatic and acid-catalyzed decarboxylations of prephenate, Biochemistry, 23, 6263– 6275, 1984. 14 Hermes, J. D. and Cleland, W. W., Evidence from multiple isotope effect determinations for coupled hydrogen motion and tunneling in the reaction catalyzed by glucose-6-phosphate dehydrogenase, J. Am. Chem. Soc., 106, 7263– 7264, 1984. 15 Hermes, J. D., Morrical, S. W., O’Leary, M. H., and Cleland, W. W., Variation of Transition-state structure as a function of the nucleotide in reactions catalyzed by dehydrogenases. 2. Formate dehydrogenase, Biochemistry, 23, 5479– 5488, 1984. 16 Parmentier, L. E., O’Leary, M. H., Schachman, H. K., and Cleland, W. W., 13C isotope effects as a probe of the kinetic mechanism and allosteric properties of Escherichia coli aspartate transcarbamoylase, Biochemistry, 31, 6570– 6576, 1992. 17 Monod, J., Wyman, J., and Changeux, J.-P., On the nature of allosteric transitions: a plausible model, J. Mol. Biol., 12, 88 – 118, 1965. 18 Waldrop, G. L., Turnbull, J. L., Parmentier, L. E., O’Leary, M. H., Cleland, W. W., and Schachman, H. K., Steady-state kinetics and isotope effects on the mutant catalytic trimer of aspartate transcarbamoylase contining the replacement of Histidine 134 by alanine, Biochemistry, 31, 6585– 6591, 1992. 19 Waldrop, G. L., Urbauer, J. L., and Cleland, W. W., 15N isotope effects on nonenzymatic and aspartate transcarbamylase catalyzed reactions of carbamyl phosphate, J. Am. Chem. Soc., 114, 5941– 5945, 1992. 20 Weiss, P. M., Gavva, S. R., Harris, B. G., Urbauer, J. L., Cleland, W. W., and Cook, P. F., Multiple isotope effects with alternative dinucleotide substrates as a probe of the malic enzyme reaction, Biochemistry, 30, 5755– 5763, 1991. 21 Edens, W. A., Urbauer, J. L., and Cleland, W. W., Determination of the chemical mechanism of malic enzyme by isotope effects, Biochemistry, 36, 1141– 1147, 1997. 22 Karsten, W. E. and Cook, P. F., Stepwise versus concerted oxidative decarboxylation catalyzed by malic enzyme: a reinvestigation, Biochemistry, 33, 2096– 2103, 1994. 23 Karsten, W. E., Gavva, S. R., Park, S.-H., and Cook, P. F., Metal ion activator effects on intrinsic isotope effects for hydride transfer and decarboxylation in the reaction catalyzed by the NAD-malic enzyme from Ascaris suum, Biochemistry, 34, 3253– 3260, 1995. 24 Urbauer, J. L., Bradshaw, D. E., and Cleland, W. W., Determination of the kinetic and chemical mechanism of malic enzyme using (2R,3R)-erythro-fluoromalate as a slow alternate substrate, Biochemistry, 37, 18026– 18031, 1998. 25 Reinhardt, L. A., Svedruzic, D., Chang, C. H., Cleland, W. W., and Richards, N. G. J., Heavy atom isotope effects on the reaction catalyzed by the oxalate decarboxylase from Bacillus subtilis, J. Am. Chem. Soc., 125, 1244– 1252, 2003. 26 Gustin, D. J., Mattei, P., Kast, P., Wiest, O., Lee, L., Cleland, W. W., and Hilvert, D., Heavy atom isotope effects reveal a highly polarized transition state for chorismate mutase, J. Am. Chem. Soc., 121, 1756– 1757, 1999. 27 Wright, S. K., DeClue, M. S., Mandal, A., Lee, L., Wiest, O., Cleland, W. W., and Hilvert, D., Isotope effects on the enzymatic and non-enzymatic reactions of chorismate, J. Am. Chem. Soc., in press. 28 Lee, L. V., Vu, M. V., and Cleland, W. W., 13C and deuterium isotope effects suggest an aldol cleavage mechanism for L-ribulose-5-phosphate 4-epimerase, Biochemistry, 39, 4808– 4820, 2000. 29 Hermes, J. D., Weiss, P. M., and Cleland, W. W., Use of nitrogen-15 and deuterium isotope effects to determine the chemical mechanism of phenylalanine ammonia-lyase, Biochemistry, 24, 2959– 2967, 1985. 30 Blanchard, J. S. and Cleland, W. W., Use of isotope effects to deduce the chemical mechanism of fumarase, Biochemistry, 19, 4506– 4513, 1980. 31 Rishavy, M. A. and Cleland, W. W., 13C and 15N kinetic isotope effects on the reaction of aspartate aminotransferase and the tyrosine-225 to phenylalanine mutant, Biochemistry, 39, 7546– 7551, 2000. 32 Wright, S. K., Rishavy, M. A., and Cleland, W. W., 2H, 13C, and 15N kinetic isotope effects on the reaction of the ammonia-rescued K258A mutant of aspartate aminotransferase, Biochemistry, 42, 8369– 8376, 2003.

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33 Tipton, P. A. and Cleland, W. W., Carbon-13 and deuterium isotope effects on the catalytic reactions of biotin carboxylase, Biochemistry, 27, 4325 –4331, 1988. 34 Attwood, P. V., Tipton, P. A., and Cleland, W. W., Carbon-13 and deuterium isotope effects on oxalacetate decarboxylation by pyruvate carboxylase, Biochemistry, 25, 8197– 8205, 1986. 35 Weiss, P. M., Cook, P. F., Hermes, J. D., and Cleland, W. W., Evidence from nitrogen-15 and solvent deuterium isotope effects on the chemical mechanism of adenosine deaminase, Biochemistry, 26, 7378– 7384, 1987. 36 Rishavy, M. A. and Cleland, W. W., Determination of the mechanism of orotidine 50 (-monophosphate decarboxylase by isotope effects, Biochemistry, 39, 4569– 4574, 2000.

38

Catalysis and Regulation in the Soluble Methane Monooxygenase System: Applications of Isotopes and Isotope Effects John D. Lipscomb

CONTENTS I. Introduction ...................................................................................................................... 931 II. sMMO Components ......................................................................................................... 932 III. sMMO Catalytic Cycle .................................................................................................... 933 IV. Kinetic Solvent Isotope Effect used to Probe Proton Donors......................................... 936 V. The Mechanism of Q Reaction with Substrates.............................................................. 938 VI. Arrhenius Plots for the Reactions of Reaction Cycle Intermediates .............................. 942 VII. The Effects of MMOB ..................................................................................................... 943 VIII. A Test of the Molecular-Sieve Model using KIE Studies .............................................. 945 IX. A Tunneling Reaction ...................................................................................................... 946 X. New Insight into the Mechanism of C –H Bond Cleavage............................................. 947 Acknowledgments ........................................................................................................................ 949 References..................................................................................................................................... 949

I. INTRODUCTION Methanotrophic bacteria exist by using methane as a sole source of carbon and energy.1,2 They are typically found in lakes, oceans, and wetlands at the interface between the anaerobic and aerobic environments. At this boundary, they can use O2 from atmospheric mixing or photosynthesis to initiate the oxidation of methane released by anaerobic methanogenic organisms as a consequence of their metabolism. Since methanotrophic bacteria are the only aerobic organisms that efficiently oxidize methane, they play a major role in restricting the egress of methane, a powerful greenhouse gas, into the atmosphere. Recently, the existence of anaerobic organisms that slowly oxidize methane has been reported and these also appear to make a contribution to the savaging of methane.3 Methane has two physical characteristics that are at the heart of understanding how a biological system can cause its oxidization in a controlled reaction. First, it is a small, symmetrical molecule with little hydrogen bonding capability. This severely limits the means by which an enzyme can bind it with specificity. Second, the C – H bond that must be broken during the oxidation reaction has bond dissociation energy (BDE) of 105 kcal per mole, at least 5 kcal per mole stronger than 931

932

Isotope Effects in Chemistry and Biology

the C – H bonds broken in any other biological oxidation reaction. This second point has some significant implications for methanotrophic bacteria because one might argue that any species that could react with methane must also be able to react with virtually any other biomolecules that it contacts. Hence, the key question in methanotrophic metabolism emerges: How can these organisms selectively oxidize methane in a complex background of other hydrocarbons and a sea of other biomolecules? Because no downstream metabolic pathways exist for alternative substrates, the answer to this key question is tantamount to existence for the methanotroph and ultimately for all aerobic species. The essential enzyme in the oxidation of methane is methane monooxygenase or MMO.4,5 MMO is found in two completely different forms in the methanotroph family. All methanotrophs possess a membrane-bound form of MMO (pMMO) that appears to be a copper containing enzyme.6,7 A consensus has not been reached on the precise number and arrangement of coppers and possible role of iron in the enzyme-catalyzed reaction.8 – 11 This is due to the fact that the enzyme activity is rapidly lost when the cell is lysed making it difficult to purify and characterize. However, recent advances have resulted in methods to much more effectively stabilize the enzyme.8,10,12 Characterization of the enzymes from these preparations strongly support the presence of copper clusters in the enzyme, but the structure of the cluster remains a point of controversy. It has been suggested that two types of copper cluster exist, one to carry out the chemistry and one to supply electrons for the reaction.7,12 The second form of MMO (sMMO) is found in the soluble fraction of some species of methanotrophs when grown under low copper concentration conditions.4 Our group and others have developed means to purify sMMO to homogeneity.13,14 In contrast to pMMO, sMMO utilizes a dinuclear non heme iron cluster at the active site.15 – 17 Both forms of MMO exhibit the classic monooxygenase stoichiometry in which one atom of oxygen from O2 is incorporated into the product while the second is reduced to water. The electrons for the latter process formally derive from reduced pyridine nucleotide in both forms of MMO: NADðPÞH þ O2 þ CH4 þ Hþ ! NADðPÞþ þ CH3 OH þ H2 O However, in the case of pMMO it remains controversial whether NAD(P)H is the direct source of electrons or whether membrane bound quinones serve as mediators.8,12 Both enzymes will accept several adventitious substrates, but the range is much greater for sMMO, for which literally hundreds of other substrates are known.18 – 20 The sMMO also catalyzes many more different types of reactions including N-oxidation, epoxidation, dehalogenation, aromatic hydroxylation, and desaturation.18 – 20 The combination of homogeneous, well characterized components and the broad substrate range has allowed a more thorough investigation of the sMMO reaction. This has included extensive use of isotopes and isotope effects. Here we will focus on the application of these techniques in the study of the mechanism of sMMO, but comments on applications to the study of pMMO will be made where appropriate.

II. sMMO COMPONENTS As summarized in Figure 38.1, the sMMO system has three protein components: a 39.7 kDa reductase (MMOR) with both FAD and Fe2S2 redox active centers,14,21 a 15.8 kDa regulatory protein termed component B (MMOB),14,22 and a 245 kDa hydroxylase component (MMOH) containing the active site.13,14 The x-ray crystal structure of MMOH shows that it is a dimeric protomer with an (abg)2 quaternary structure.16,17 Each a-subunit hold a bis-m-hydroxo bridged ˚ below the surface in dinuclear iron cluster required for catalysis. The diiron cluster is buried 12 A the middle of a four a-helix bundle. Importantly for the discussion that follows, the apparent active site cavity located immediately adjacent to the cluster is not directly accessible from the surface. Thus, substrates must either move through the protein matrix or a conformational change must provide access at some point in the catalytic cycle. In the resting enzyme, both irons of the cluster

Catalysis and Regulation in the Soluble Methane Monooxygenase System

933

NADH + CH4 + O2 + H+

MMOR FAD

Fe

H O

O H MMOH

Fe

Fe2S2 Fe

MMOB

H O O H

Fe

NAD+ + CH3 OH + H2O

FIGURE 38.1 Components of the soluble MMO system.

are in the ferric state15,23 and are coordinately saturated. In order to react with O2 by an inner-sphere mechanism, both the redox state and the coordination number must change. MCD/CD24 and X-ray crystallographic studies25 show that when the diiron cluster is reduced by two electrons, the irons each become ferrous and five coordinate. The structural studies show that this occurs by loss of both bridging hydroxides and reorientation of a Glu ligand to form a single atom oxygen bridge. The open coordination sites on the irons each face into the active site cavity, presumably ready to accept a bridging dioxygen.

III. sMMO CATALYTIC CYCLE The catalytic cycle of MMOH has been probed by a combination of spectroscopic and transient kinetic techniques.5,26 Our current view of the steps in the cycle is shown in Figure 38.2a. By stoichiometrically reducing MMOH chemically in the absence of MMOR and NADH, a single starting point is established. Upon rapidly mixing a large excess of O2 with or without MMOB and 2e− HOH H III O Fe ROH

III

Fe

ROH O

HOH II

Fe

III

O H

Fe

Hox

O

T

III

Fe

III

(a)

R

H O O

R

HOH III O III Hred Fe Fe O P* O

R Fe

C O O2 II Fe

Q

IV

IV

Fe

Fe RH

P OH H III O Fe

O

IV

O

Fe

O O Fe

III

O

O

IV

O O

Fe

IV

Fe O

R

H2O

Compound Q cluster

(b)

FIGURE 38.2 (a) Proposed reaction cycle of sMMO. The diiron cluster in the active site of MMOH is depicted in the outer cycle. These structures represent out current view of the reaction cycle intermediates. The central cycle include transient intermediates that have been directly detected or detected by diagnostic chemistry (R). (b) current hypothesis for the structure of Compound Q from spectroscopic studies.

934

Isotope Effects in Chemistry and Biology

substrate, a single turnover cycle ensues, stopping at the resting diferric state. This allows the steps in the cycle to be treated as a series of first-order or pseudo first-order reactions so that the rate constants (actually reciprocal relaxation times) can be determined by multiple exponential fitting. It was discovered early that the presence of stoichiometric MMOB greatly accelerates both the steady-state turnover reaction, and importantly, the reaction of diferrous MMOH with O2.27,28 The latter reaction is accelerated approximately 1000-fold, removing an early roadblock in the cycle and allowing the subsequent intermediates to be detected. Perhaps the most exciting observation was that a yellow transient intermediate formed after O2 addition.27 This species was easily observed because the other forms of MMOH are nearly colorless in the visible spectrum, and it decays rather slowly in the absence of substrate. The latter characteristic allowed the intermediate, termed compound Q, to be trapped and characterized by spectroscopic approaches. Mo¨ssbauer spectroscopy showed that 57Fe-enriched Q exhibits an isomer shift of 0.17 mm/s similar to values observed for heme-containing enzymes and model compounds in the Fe(IV) oxidation state.29,30 Since only a single quadrupole doublet was observed for the two irons in the cluster, both must be in the Fe(IV) state. This was a remarkable finding because it was the first time that a single complex had been shown to contain more than one high valent iron. EXAFS studies of Q showed ˚ apart and that each iron had one relatively short 1.77 A ˚ bond to that the two irons are 2.46 A 31 oxygen. Comparison with model diiron complexes showed that the only structure compatible with these distances is a “diamond core” composed of two single oxygen atoms bridging the irons as illustrated in Figure 38.2b. Each bridge is asymmetric, consisting of a short, strong iron-oxo bond to one iron and a weaker bond to the second. The yellow color of Q allowed the kinetics of its reaction with substrates to be observed directly. It was found that the formation rate constant of Q was not affected by substrates type or concentration, but the decay rate constant increased linearly with substrate concentration as shown in Figure 38.3a.27 A second-order rate constant could be obtained from the slope of the pseudo firstorder rate constant versus concentration plot, and this value changed dramatically with the type of substrate used. Finally, the rate constant for appearance of products in the active site could be determined for some substrates and was found to correlate with the rate constant for Q decay.27 This provided a strong indication that Q is the reaction cycle intermediate that reacts directly with the substrate to form product.

5 Ethane

kQD (s−1)

4

Methane

3

Propene (~Furan)

2

Propene (~Nitrobenzene)

1 0

(a)

MMO Q O IV IV Fe Fe IV

100

200

300

400

[Substrate] mM

500

Fe O

O

Fe O

O III

+

IV

Fe O

III

Fe

(b)

IV

Fe O

III

IV

Fe

600

O IV

Fe

d4-Methane 0

O

O

P450 -cation radical

Fe

III

Fe

O

FIGURE 38.3 Kinetics of reactions of Compound Q with substrates. (a) pseudo first-order rate constants for reaction of Q with the substrates shown on the figure at 48C, pH 7.0. (b) comparison of the reactive intermediates proposed for sMMO and cytochrome P450. There is no evidence that Q assumes a non-diamond core structure, but the open form is shown to facilitate comparison.

Catalysis and Regulation in the Soluble Methane Monooxygenase System

935

The discovery of Q was quite exciting because it has the same overall oxidation state (see Figure 38.3b) as the long sought Fe(IV)-oxo p-cation radical species of cytochrome P-450 thought by many to be responsible for the reaction that breaks the C –H bond of substrates.32 Unlike Q, the reactive P-450 species has proven to be difficult to trap,33 and the nature of its reaction with substrates has thus far not been possible to directly characterize. The ability to directly observe Q has allowed the substrate C – H bond-breaking step of an iron containing monooxygenase to be directly observed for the first time. Once Q was recognized, other intermediates in the reaction cycle were easier to characterize. Those that have been identified are shown as letters in the interior cycle of Figure 38.2a. The diferrous MMOH starting species for the single turnover reaction has an unusual g ¼ 16 EPR signal.15,34 It was shown using rapid-freeze quench techniques that this signal was lost much more rapidly than Q was formed27,28 indicating that there must be at least one intermediate, termed compound P, in the Q formation process. Moreover, the rate constant for the loss of the g ¼ 16 signal was independent of the O2 concentration showing that an irreversible step occurs between O2 binding to the enzyme and the formation of the intermediate species. This oxygen complex was called compound O.28 O retains the g ¼ 16 signal showing that the irons are in the ferrous state, thus the oxygen is bound to the enzyme but not to the irons. A search for P, revealed a rapidly forming and decaying species with a relatively weak chromophore at 700 nm.30 Mo¨ssbauer spectra of P showed that the irons are primarily ferric with some ferrous character, consistent with the formation of a peroxy complex.30,31 Synthesis of the appropriate model complexes showed that the Mo¨ssbauer spectrum of P is characteristic for a bis-m-oxo diferric peroxo complex.35 The decay rate constant for this species matched that for the formation of Q, but the formation rate constant for P was significantly less than that for the decay of O. Consequently, an intervening species Pp was proposed 36 Subsequent transient kinetic studies revealed a lag in the formation of Q consistent with exponential phases associated with intermediates O and Pp.37 Addition of methane or any other substrate to the single turnover reaction had no effect on the rate constants for the cycle through formation of Q, consistent with a true oxygen activation phase in preparation for interaction with substrate.37 The obvious reactivity of Q with substrates led to the concept of a hydrocarbon reaction phase following oxygen activation to form Q.27 Using nitrobenzene as a substrate leads to the generation of a p-nitrophenol product (in addition to the other regioisomers). The p-nitrophenol absorbs at 325 nm in the hydrophobic active site, but this shifts to 404 nm when the product ionizes after release into solution. Consequently, the formation of a terminal product complex, termed T, could be directly observed.27 The rate constant for the breakdown of T was shown to be equal to the turnover number for the enzyme and thus represents the rate limiting step in catalysis. The reaction cycle is conveniently described as occurring in two phases as illustrated in Figure 38.4. In the initial phase, oxygen is activated to form intermediate Q without the participation of substrate. In the second phase, substrate reacts with Q to form products. This CH3OH

0.17 s−1

Hox

Hr

O

T

Hydrocarbon Reaction Phase

Fast

26 s−1

[R] −1 −1

12,500 M s

CH4

O2

Fast

9 s−1

kQD

Q

2.5 s−1

P*

Oxygen Activation Phase

P

FIGURE 38.4 Phases of the sMMO reaction cycle. The rate constants shown are for the reaction at 48C, pH 7.0.

936

Isotope Effects in Chemistry and Biology

organization differs somewhat from the reaction cycle of cytochrome P-450 in which substrate binds before reduction of the heme and the binding and activation of oxygen.38 – 40

IV. KINETIC SOLVENT ISOTOPE EFFECT USED TO PROBE PROTON DONORS A primary regulatory mechanism exercised by enzymes of many types is the control of proton delivery.41 In the MMO reaction, two protons must be ultimately delivered from solution to break the O – O bond of O2 and form water as a product. The step or steps in which this occurs was probed using transient kinetics to explore the deuterium kinetic solvent isotope effects (KSIE) on each step of the catalytic cycle.36 Most steps showed no KSIE, but the Pp to P and P to Q steps showed effects of 1.3 and 1.4, respectively. Moreover, only these steps showed a dependence of the conversion rate constant on solution pH as shown in Figure 38.5a. Each slows with increasing pH, following a simple single ionization titration curve with a pKa value of 7.6. Finally, as shown in Figure 38.5b, the proton inventory plot for each reaction was found to be linear. The proton inventory is described by Equation 38.142,43 where fTi and fRj designate fractionation factors for ith and jth exchangeable proton in the transition state and the reactant state, respectively. The term n is the atom fraction of deuterium in the solvent and kn is the rate constant measured at that n value. The term kH is the rate constant measured in non deuterated solvent. v Y

kn ¼ i¼1 w Y kH j¼1

ð1 2 n þ nfTi Þ ð38:1Þ

ð1 2 n þ nfRj Þ

When fR ¼ 1 for all sites involved in the reactant state, the denominator of Equation 38.1 becomes unity. In the reactant state, the fractionation factors for common proton donors such as hydroxyl, carboxyl, and primary and secondary amine functions are effectively one. In cases where there is an observable isotope effect, the product of the fractionation factors for the transition state and the reactant state cannot be equal. If the overall fractionation factor (i.e., the product of the individual values according to the denominator of Equation 38.1) for the reactant state is known, then that for the transition state can be determined experimentally by fitting the proton inventory plot. Alternatively, if the fractionation factors for the reactant ðfR Þ and product ðfP Þ states as well as 10

P Formation

Q Formation

3

1.0

4

P Decay

1

2 0

(a)

8

7 pH

0.6 1.0

Q Formation

0.8

Q Decay 7

P Formation

0.8 kn /k H

2

6

k obs, s−1

k obs, s−1

8

8

9

0

0.6 0.0

(b)

0.2 0.4 0.6 0.8 1.0 Molar fraction D2O (n)

FIGURE 38.5 Proton dependence of the P and Q reactions. (a) pH dependence of the formation and decay of P and Q. (b) proton inventory plot for the P and Q decay reactions. (From Lee, S.-K. and Lipscomb, J. D., Biochemistry, 38, 4423– 4432, 1999. With permission.)

Catalysis and Regulation in the Soluble Methane Monooxygenase System

937

the extent to which the transition state has progressed toward the product (a) are known, then the fractionation factor for the transition state can be calculated. A single proton transferred gives rise to

fT ¼ ðfR Þ12a ðfP Þa

ð38:2Þ

In cases where the observed KSIE arises from the contribution of a single proton in the transition state, i.e., fR ¼ 1 but fT – 1; the proton inventory plot ðkn =kH vs. nÞ will be linear as described by Equation 38.3, and the inverse of the observed isotope effect ðkD =kH Þ gives the value of fTi : kn ¼ ð1 2 n þ nfT Þ kH

ð38:3Þ

A simple model to account for a linear response invokes a single proton transfer during the reaction. If two or more protons are being transferred in the transition state, then the proton inventory plot is nonlinear because the numerator of Equation 38.1 becomes a quadratic or higher function. Consequently, the most straightforward interpretation of the linear proton inventory plots observe for the isotope sensitive steps in the MMOH catalytic cycle is that a single proton donor with a fraction factor near unity transfers the proton in a single step. There are relatively few potential donors in the vicinity of the diiron cluster as show in Figure 38.6. The most likely donors are Thr213, water in the active site, or water bound as a ligand to one of the irons of the cluster. The fact that the reactions occur with pKa values near neutral make the Thr and active site water less likely as donors, but it supports metal bound water in this function. Transfer of protons within the cluster also preserves the neutrality of the overall cluster, which seems to be a common strategy for metal centers in oxygenase enzymes.44 Another potential proton donor with the characteristics described above is histidine. However, the only His near the cluster is one of its own ligands, which is already deprotonated in forming the

Leu216 Phe236 Leu110

Val112

lle217

Phe188 Thr213

Cys 151

Wat Wat

Val239

Phe192 Wat

Wat

E114

O H147

O

Fe1 O

E209

O

OH

Fe2

HO

O

Asp242

Leu204

E243

Wat

O

O

O

H246

Gln140

E144

Asp143

FIGURE 38.6 Potential sources of protons in the MMOH active site. (Adapted from Lee, S.-K. and Lipscomb, J. D., Biochemistry, 38, 4423–4432, 1999. With permission.)

938

Isotope Effects in Chemistry and Biology

cluster. Another possible source of a His is the MMOB component which binds to the MMOH subunit that holds the diiron cluster. This possibility will be considered further below.

V. THE MECHANISM OF Q REACTION WITH SUBSTRATES The mechanism for C –H bond cleavage is quite controversial.5,26 In our original proposal,14,20 we chose to present a cytochrome P450 like mechanism in which Q breaks the C – H bond of methane homolytically to yield a substrate radical in the active site next to what can be described as a hydroxyl radical bound to the diiron cluster (Figure 38.2a). This radical containing species, termed compound R,27 is too short-lived to trap directly, but evidence supporting its existence is reviewed below. Clearly, other possibilities for the chemistry of this step have also found experimental support. The chemical insight that has derived from the exploration of this chemistry will undoubtedly be one of the most significant legacies of MMO research. The radical nature of the MMO-catalyzed reaction was suggested by several different types of experiments. One of the most compelling involved the use of carrier-free 1-[2H, 3H] R- and S-ethane as illustrated in Figure 38.7.45,46 Oxidation by MMO followed by recovery of the product ethanol and derivatization to yield a diastereomeric molecule allowed facile characterization of the chirality of the product by tritium NMR spectroscopy. It was observed that 35% inversion of stereochemistry occurred during the reaction suggesting that a discrete intermediate, presumably an ethyl radical, was formed. The intramolecular 2H isotope effect was found to be 4.2 based on the product distribution. Similarly, the 3H-isotope effect was estimated to be greater than 12 by quantifying tritium found in the solvent. By contrast to this evidence for an intermediate in the sMMO reaction, the analogous chiral ethane experiment performed using pMMO showed no racemization, favoring a concerted reaction.47 H (R)-(2H1,3H1)-ethane Retention

CH3

T

Inversion

D 53%

12%

OH

26%

H CH3

T

(S)-(2H1,3H1)-ethanol

27%

OH CH3

T

CH3 T

CH3 OH

(R)-(2H1,3H1)-ethanol

(R)-(3H1)-ethanol

7%

H

D

OH

D

9%

52%

H

Inversion

T (S)-(3H1)-ethanol

14%

Retention CH3

D T

(S)-(2H1,3H1)-ethane

FIGURE 38.7 Conversions of carrier free R- and S- 1-[2H, 3H]-ethane catalyzed by sMMO.

Catalysis and Regulation in the Soluble Methane Monooxygenase System

939

Compound Q Decay Rate Constant (mM−1s−1)

Another type of experiment involved the use of the radical clock molecule 1,1-dimethyl cyclopropane which was shown to ring open during the reaction to form low yields of two types of products characteristic of radical and cation intermediates, respectively.48 Another approach involved the use of spin traps which also showed that 13C-labeled substrate radical species can apparently diffuse away from the active site and be trapped to yield long-lived EPR active species.49 Finally, the nature of the C – H bond breaking reaction by Q was probed directly using CD4 and its isotopic isomers.50 It was found that the KIE for the decay rate constant of the Q reaction with methane is at least 50 (see Figure 38.2a). Moreover, the linearity of the plot of apparent rate constant versus 2H content of the methane substrate (Figure 38.8) suggests that this is a primary isotope effect. In the absence of other alternatives, this large KIE is clearly consistent with a C –H bond-breaking step occurring during the Q reaction. However, the exceptionally large value of the KIE suggests that it may have a significant nonclassical component which will be discussed below. For many enzyme systems, the experiments just described might have led to the firm conclusion that the reaction proceeds through a radical, or perhaps cation, intermediate. However, MMO offers a new level of resolution for oxygenases that has facilitated the consideration of other mechanisms. For example, the chiral ethane experiment failed to show complete racemization of the product despite the fact that it involves only simple rotation around an unhindered single C – C bond. Such rotation would normally occur on the 100-femtosecond time scale implying that the resolution at this level is possible for MMOH chemistry. Loss of chirality on this time scale is too fast for traditional radical-rebound chemistry often associated with cytochrome P-450. This initiated proposals for a variety of other mechanisms, some of which are illustrated in Figure 38.9. For example, it was proposed that the reaction may, in fact, be concerted with special characteristics that give it apparent radical character.26 One such proposal is that the difference in vibrational rates of the bonds being broken and formed in the reaction causes the transition state to have a radical component with a lifetime appropriate for the observed degree of racemization of chiral ethane.51,52 Another proposal arising from density functional and quantum mechanical modeling of the reaction based on the active site structure suggests that a radical is formed, but it remains bound in a complex with the OH group of the cluster so that a discrete, freely rotating species is not formed.53,54 A quite different type of proposal, also arising from calculations, is that an iron –carbon bond forms.55 This, in turn, would lead to loss of tetrahedral symmetry of the methane and allow pseudo rotation to account for the loss of chirality.56 Alternatively, the C –H bond may break during formation of the putative Fe –C bond, but the hydrogen may be retained in the cluster on one of the bridging oxygens so that rebound of the OH group occurs internally without complete loss chirality.57

14 12 10 8 6 4 2 0

CH4

CH3D

CH2D2

CHD3

CD4

Substrate

FIGURE 38.8 Rate constant observed for reaction of isotopic isomers of methane shown. (Adapted from Nesheim, J. C. and Lipscomb, J. D., Biochemistry, 35, 10240– 10247, 1996, using data for reactions performed at pH 7.0 and 48C.)

940

Isotope Effects in Chemistry and Biology P

Q H2O

OH III

Fe

O

III

O

IV

Fe

R

IV

Fe

Fe

O H

O

OH III

III

Fe

Fe

O H

R

RH

RH

O

IV

RH

O

IV

RH

III

Fe

Fe

O

O

III

Fe

R

Fe ROH2+

R+ + H2O

III

H O

IV

Fe

Fe

O H

R

FIGURE 38.9 Summary of the general classes of reaction mechanisms proposed for the oxygen-insertion reaction of the MMO catalytic cycle.

The major support for a concerted oxygen-insertion mechanism for MMO in recent years has derived from experiments employing ultrafast radical clock reagents that rearrange on the sub nanosecond time scale.52,54,58 Figure 38.10 compares the products from radical clocks covering a wide range of rearrangement rates. When MMO catalyzes oxidation of some of the ultrafast reagents such as molecule 4 in Figure 38.10, little or no detectable formation of a radical is observed. On the other hand, ultrafast reagents, such as 3, do reveal the formation of radicals, but predict that Substrate

Unrearranged Products

Rearranged Products

OH

OH

1 18%

OH

60, 61 42%

40%

OH

2

HO

OH

81% OH

CH3

6%

OCH3

Ph

4 Ph

Ph

80% OCH3

OCH3 14%

61 Ph 6% 58, 59

OH Ph 55%

O H

Ph

48

13%

OH

3 Ph

Ref

45%

OH

58, 59

5 100%

OH

FIGURE 38.10 Diagnostic substrates used in experiments to detect cation or radical intermediates in the sMMO reaction. The products actually detected are shown. The rearranged products are shown in two columns. Those on the left derive from radical intermediates while those on the right derive from cation intermediates.

Catalysis and Regulation in the Soluble Methane Monooxygenase System

941

the radical lifetime would be too short to allow diffusion of discrete radical species in the active site during the rebound process. Experiments with other, “slower” radical clock and related reagents have shown that the architecture of the enzyme active site may in some way interfere with the radical clock timing.48,59 – 62 All of the clocks that fail to reveal radicals or only show radicals with very short apparent lifetimes are largely unhindered at the site of possible hydrogen atom abstraction. This also includes reagent 5, which has a relatively slow rearrangement rate and does not reveal radical formation. By contrast, many of the slower clocks with rearrangement rates comparable to that of reagent 5, such as reagents 1 and 2, show substantial rearrangements consistent with the formation of relatively long lived intermediates. These reagents all have bulky substituents near the putative site of hydrogen atom abstraction. These observations suggest that the ability of the substrate to closely approach the reactive oxygen species in Q, or interact with amino acids in the active site, may determine whether a radical substrate species lives long enough to be detected. An alternative possibility is the formation of a cation intermediate either together with or instead of the putative radical intermediate.61 This was probed most effectively using the reagents methylcubane,60,61 norcarane,62,63 and ethylbenzene64 that give unique products for radical vs. cation intermediates as illustrated for the norcarane case in Figure 38.11. In each case, when these reagents were used as alternative substrates for MMO, evidence for products from cationic intermediates was observed. For norcarane and ethylbenzene, the characteristic products for radical intermediates were also observed. The source of the cation remains controversial. A reasonable scenario is that the cation derives from further oxidation of a radical intermediate by the Fe(III)Fe(IV)-OH species that results from the initial hydrogen atom abstraction reaction.62,64 Such a species would be quite oxidizing as evidenced by the likely role of such a species in generating the tyrosyl radical of ribonucleotide reductase.65 Another possibility that has been recently been advanced based on observations in the cytochrome P-450 system is that intermediate P is the actual reactive species.66 A protonated peroxy species might attack the substrate, and following loss of water, yield a cationic intermediate. This type of mechanism seems unlikely when considered in the context of methane oxidation, but the low levels of cation-derived products make it difficult to definitively rule out for substrates with weaker bonds. To date, no kinetic evidence for the reaction P with the diagnostic reagents used to detect radicals and cations has been observed. In contrast, many of these reagents have been shown to react rapidly with Q to accelerate the loss of its characteristic yellow chromophore.62

H

H

Radical Product

norcarane (1)

Rebound

sMMO, O2

(9)

Radical Rearrangement (8) Rebound

OH

Cation Product

CH2OH OH

3-cycloheptenol (4) 3-hydroxymethyl-cyclohexene (3) Second e− and H+ abstraction

OH− (10) Cation Rearrangement

(11)

endo-(2a) and exo-2-norcaranol (2b)

FIGURE 38.11 Products derived from norcarane oxidation catalyzed by sMMO. Low yields of hydroxylation at other positions and overoxidation to ketones were also observed.

942

Isotope Effects in Chemistry and Biology

VI. ARRHENIUS PLOTS FOR THE REACTIONS OF REACTION CYCLE INTERMEDIATES The ability to directly observe the interconversion of intermediates in the MMOH catalytic cycle has allowed studies of its dependence on temperature.37,66 As shown in Figure 38.12a, Arrhenius plots for the P formation and decay reactions are linear, independent of the presence or absence of substrates. The linearity of the plots is consistent with single step reactions, and the substrate independence supports the conclusion that Pp and P are not the species that react with substrate.37 By contrast, the Arrhenius plots of the reaction of Q with all substrates show a pronounced dependence on the type and concentration of substrate present. All of these plots appear to be linear with the important exception of when methane is the substrate. As shown in Figure 38.11b, the plots in this case are non linear, exhibiting a break that occurs at a temperature that is methane concentration dependent.37 This concentration dependence suggests that the break is not due to a temperature-dependent phase transition in the solvent or a conformational change in the enzyme. It supports a two-step reaction of Q with methane. Interestingly, the Arrhenius plot of the reaction of Q with CD4 is linear. One explanation for this unusual set of observations invokes a Q reaction that includes a reversible substrate-binding step followed by an effectively irreversible C – H bond-breaking step (Scheme 38.1).37 If a change in rate-limiting step occurs in the temperature range that is observable with stoppedflow spectroscopy (, 4 to 378C), then it would cause a break in the Arrhenius plot if the two reactions have different activation parameters. The point at which the break occurs would depend kQF P

6

3 Q

5

2

3 2 1

In kQD

In kQF

4 None 200 mM Methane 100 mM Methane 400 mM Methane 200 mM Ethane 360 mM Propene 200 mM Propane 400 mM Furan

200 mM Methane 100 mM Methane

1 50 mM Methane 0

0 3.225 3.300 3.375 3.450 3.525 3.600 3.675

(a)

kQD Q T

400 mM Methane

−1 3.33 3.35 3.40 3.45 3.50 3.55 3.60 3.65

(b)

1/T (K−1) × 1000

1/ T (K−1) × 1000

FIGURE 38.12 Arrhenius plots for reaction cycle intermediate interconversion. (a) conversion of P to Q. The reaction was not affected by substrate type or concentration. (b) conversion of Q to T using either methane at the concentration shown (solid lines) or 200 mM ethane (dashed line) as the substrate. (Adapted from Brazeau, B. J. and Lipscomb, J. D., Biochemistry, 39, 13503– 13515, 2000. With permission.) CH4 P

kQ F

Q

kSB

QS

kQD

T

KTD

HOX

CH3OH

SCHEME 38.1 Two-step model for Q reaction with substrates.

Catalysis and Regulation in the Soluble Methane Monooxygenase System

943

on the specific values of the activation parameters and the substrate concentration because only the binding reaction would depend linearly on the latter. If this step is reversible, then the rate constant for the C –H bond-breaking step would also show substrate concentration dependence, but in this case, it would be hyperbolic. Due to the strength of the C – D bond, the bond-breaking reaction would become rate limiting at a much lower temperature for the CD4 reaction compared with that of CH4. Consequently, the break in the Arrhenius plot would occur at temperatures below those observable by stopped flow, and the plot would appear linear in the 4 to 378C range. In this range, the Arrhenius plot for CD4 arises from the C – D bond-breaking reaction, and thus, an isotope effect is observed when a comparison is made to the high temperature region of the Arrhenius plot for the CH4 reaction. Just the opposite occurs in the case of ethane and d6-ethane. The weaker bonds relative to methane cause the binding reaction to remain rate limiting to higher temperatures, and thus in the stopped-flow temperature range, the Arrhenius plot again appears linear. In this case, however, the binding reactions of the two ethanes are being compared in the observable part of the Arrhenius plot, and thus no isotope effect is expected. The fact that the plot of Q decay rate constant versus substrate concentration at any single temperature appears linear raises an interesting question since the two-step binding model just described would predict a hyperbolic plot for a substrate like methane where the binding reaction is relatively fast. The most likely explanation is that the plot is actually hyperbolic, but the KD for substrate binding is relatively high, so the plot appears linear at low substrate concentrations. Evidence in favor of this possibility has been published for the reaction of some alternative substrates.27,67 A second possibility is that the two active sites of the complete MMOH molecule are not independent such that both must bind substrate before either reacts. This would introduce a squared substrate concentration term into the numerator of the hyperbolic expression, and thus give a linear plot under conditions of high substrate concentration relative to the KD value.

VII. THE EFFECTS OF MMOB The effectors protein, MMOB, dramatically influences the MMOH catalytic cycle. In the reconstituted system, the MMOB supplied at nearly the same concentration as the MMOH active sites results in a 150-fold greater turnover number.14 Spectroscopic studies have shown that the complex between MMOB and MMOH results in changes in the environment of the diiron cluster despite the fact that it is buried well below the MMOH surface.68,69 Studies of the regiospecificity of hydroxylation of complex substrates show dramatic changes when MMOB is added to the reaction.70 Transient kinetic studies show that the reduced MMOH reacts with O2 to form intermediate Pp 1000 times faster when MMOB is present.28 These effects are apparently caused by the formation of a high-affinity complex between the MMOH a-subunit and MMOB.68 This complex is energetically coupled to the oxidation state of the diiron cluster such that a 2132 mV change in cluster potential occurs upon MMOH –MMOB complex formation.71 Accordingly, the affinity of the two components decreases by about four orders of magnitude when the diiron cluster is fully reduced. Insight into the detailed interaction of MMOB with MMOH has been derived primarily though NMR investigations. The solution structures of the MMOB components from Methylococcus trichosporium OB3b72 and M. capsulatus Bath73 have been solved and show a tightly folded core region along with extended disordered N- and C-termini (Figure 38.13a). Despite this disorder, at least the N-terminal region plays an important role since proteolysis of the terminal 12 residues of M. capsulatus Bath MMOB results in loss of the MMOB functions.74 NMR was also used to reveal changes in the relaxation properties of individual backbone amide groups in the MMOB structure as MMOH is introduced into the system.73,75 When the MMOH is fully reduced, the affinity for MMOB is sufficiently low that a substantial concentration of protein is in dynamic equilibrium at near stoichiometric concentrations of the components. This experiment shows that one surface

944

Isotope Effects in Chemistry and Biology

ROH Quad mutant MMOB

Hox

T

[R] N107G/S109A/ S110A/T111A

H33A N-terminal cleavage

N-term

H5A

C-term

(a)

RH

Hr

O2

Cleaved MMOB

O

H5A-MMOB

P*

Q

P H33A-MMOB

(b)

FIGURE 38.13 Structure of MMOB and effects of mutagenesis. (a) solution structure of MMOB derived from NMR approaches and sites of mutagenesis. (b) steps in the catalytic cycle of MMOH affected by the MMOB mutants shown on the figure. (Adapted from Wallar, B. J. and Lipscomb, J. D., Biochemistry, 40, 2220– 2233, 2001. With permission.)

of MMOB is involved in the interface and identifies specific regions that appear to form the contact zones.75 These include both regions on the tightly folded core region and the disordered N- and C-terminal regions. The role that the interaction zone residues play in the function of MMOB has been probed by site directed mutagenesis.76 Several single-residue and multiple-residue mutants were generated and their behavior in single and continuous turnover systems assessed. Remarkably, as illustrated in Figure 38.13b, different MMOB mutants affected the rates of different steps throughout the catalytic cycle. This shows that MMOB plays roles in several, perhaps all, of the individual steps of the cycle. In some cases, the step affected is not rate limiting so that no effects are observed in continuous turnover assays. However, using the single-turnover system, the changes induced in the rates of intermediate interconversion are readily detected. Examples of this are found in the MMOB mutants H33A and H5A. These two His residues are found in the disordered N-terminal region. These affect the rate constants for the Pp to P and P to Q steps in the cycle, respectively.76 Thus, removing His residues affects the steps revealed by KSIE studies to require proton transfer.

7

N107G/S109A/S110A/T111A

4 3

wild type

2 1 H33A

0

(a)

0

20

40

60

Methane Turnover

1.6 Q decay rate constant

Q decay rate constant

6 5

1.8

Large Substrate Turnover

80

Furan (mM)

1.0

H5A

0.8 0.6 0.4

N107G/S109A/S110A/T111A

0.0

(b)

H33A

1.2

0.2

H5A

100 120 140 160

wild type

1.4

0

20

40

60

80

100 120 140 160 180

Methane (mM)

FIGURE 38.14 Substrate concentration dependence of Q reactions using wild-type and Quad-mutant forms of MMOB at 48C, pH 7.0. (a) reactions using furan as a substrate. Other substrates larger than methane give similar plots. (b) reactions using methane as the substrate. (Adapted from Wallar, B. J. and Lipscomb, J. D., Biochemistry, 40, 2220– 2233, 2001. With permission.)

Catalysis and Regulation in the Soluble Methane Monooxygenase System

MMOH

FeIV O O

MMOB

945

MMOH

FeIV O

FeIV

O

NO2

Quad MMOB

FeIV

NO2

FIGURE 38.15 Molecular sieve hypothesis for the ability of MMOH to selectively oxidize methane. (Adapted from Wallar, B. J. and Lipscomb, J. D., Biochemistry, 40, 2220– 2233, 2001. With permission.)

One of the most informative mutations is N107G, S109A, S110A, or T111A, or the “Quad” mutant. This mutant affects the rates of the Q reaction with substrates and the dissociation of products from intermediate T.76 Since the rate-limiting product dissociation rate is affected, the continuous turnover rate is altered. Interestingly, the rate increases for assay substrates such as furan and nitrobenzene, so this is a gain of function mutation in many cases. Transient kinetic studies, such as that shown in Figure 38.14a, demonstrate that the Q reaction with large substrates is accelerated about sevenfold. Similarly, the release of the relatively large product, resulting from nitrophenol oxidation, p-nitrophenol, is accelerated about twofold. These rate increases could both be explained on the basis of facilitated transfer of molecules into and out of the active site. This led to a hypothesis for one level of regulation of MMO catalysis in which substrate selectivity is controlled by physical access to the buried active site.76 As noted above, the X-ray crystal structure of MMOH alone shows a closed-active site, consistent with its low reactivity in the absence of MMOB. It is proposed that MMOB opens the MMOH active site when they form a complex such that molecules the size of methane and oxygen gain access, but other larger molecules are excluded. In this way, MMO can catalyze oxidation of methane in the presence of other potential substrates with weaker C –H bonds. MMOH becomes, in effect, a molecular sieve for methane and oxygen as illustrated in Figure 38.15. The hole size of the sieve may be controlled by the MMOB – MMOH interface residues. Thus, in the Quad mutant where the residues are all smaller and more hydrophobic, the hole size might be larger accounting for the increased reactivity of large substrates and release rate of large products. It is unclear at present whether the increase in accessibility is caused by formation of an actual pore or to a more subtle change. One possibility for such a change is an increase in the mobility of the region of MMOH structure over the active site cavity.

VIII. A TEST OF THE MOLECULAR-SIEVE MODEL USING KIE STUDIES The analysis of the Arrhenius plots for Q reaction with substrates led to proposal described above that large substrates bind slowly in the MMOH active site so that the C –H bond-breaking step is not rate limiting and no deuterium KIE is observed. The molecular-sieve model for MMOB function suggests that the hole size into the active site, and thus potentially the binding rate for large substrates, can be increased using the Quad mutant. We might then expect that an isotope effect would be observed for substrates other than methane if the rate of binding is sufficiently increased. Indeed, a KIE of 2 is observed for ethane oxidation when the Quad mutant is used in place of

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TABLE 38.1 Effect of the Quad Mutant MMOB on Observed Q Decay Rates (48C, pH 7.0) MMOB Substrate Methane d4-methane Ethane d6-ethane Propane d8-propane

Rate Constant (M

Quad MMOB

21 21

12,500 250 14,500 14,500 950 950

s

)

KIE

Rate Constant (M 21 s21)

KIE

50

1,500 250 30,000 15,000 2,400 2,400

6

1 1

2 1

MMOB as shown in Table 38.1.77 The maximum isotope effect for this reaction is 4.2 as shown by the chiral ethane experiment describe above.45 The Quad mutant appears to be able to unmask a significant fraction of the isotope effect, strongly supporting both the two-step Q reaction and the molecular-sieve hypothesis. Larger substrates such as propane show accelerated rates for both deuterated and unenriched forms (Table 38.1) consistent with the notion that access is increased, but the large size of the molecule and the comparatively weak C –H bonds do not allow the C – H bond-breaking step to become rate limiting in the stopped-flow observable range, and thus no KIE is observed.

IX. A TUNNELING REACTION The unexpected large deuterium KIE for the Q reaction with methane50 is consistent with a large tunneling channel in the reaction coordinate specifically for this substrate. The small value for the ratio of preexponential factors (, 0.0002) and the large change is the activation enthalpy (7 kcal/mol) are also suggestive of a tunneling reaction.78 If the enzyme is able to use quantummechanical tunneling to specifically accelerate the methane reaction with Q while limiting the reaction of other substrates to largely Arrhenius processes, then a powerful second layer of specificity might be accessed. Unfortunately, the usual approaches to evaluating the extent of tunneling in a chemical reaction are difficult to employ effectively in the case of MMO. The accessible temperature range is limited by aqueous solutions and the potential for conformational changes or changes in rate-limiting step in solvents at low temperature. It is also worth noting that most of the commonly used cryosolvents are also substrates for MMO. Tritiated substrates cannot be used due to the nature of the experiment, which involves direct observation of the single-turnover reaction and thus high enrichments. One way to evaluate tunneling is to attempt to perturb it using the mutant forms of MMOB that affect active site access. As shown in Figure 38.14b and Table 38.1, the reaction of Q with unenriched methane decreases substantially in rate constant when the Quad mutant MMOB is used.77 All other substrates, including CD4, show either unchanged or increased reaction rate constants. The unchanged rate constants presumably reflect rate-limiting C –H or C – D bondcleaving reactions when both MMOB and Quad-mutant MMOB are used. These results suggest that the substantially lower-rate constant for methane cannot be attributed to access problems. The possibility that it is due to perturbation in the extent of tunneling is suggested by the fact that the observed KIE decreases to the classical value of six. Thus, it is possible that the complex between MMOH and MMOB not only facilitates access but also remodels the active site somewhat to promote tunneling specifically in the methane reaction.

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X. NEW INSIGHT INTO THE MECHANISM OF C–H BOND CLEAVAGE The many proposals for the mechanism of C –H bond cleavage discussed above were put forth without knowledge of the profound effects of MMOB specifically on the Q reaction with substrates. As we now view it, the rate constants and activation parameters determined for all substrates other than methane largely pertained to the binding reactions rather than the C – H bond-cleavage reactions. Also, it seems likely that substantial reorganization of the active site occurs as first revealed by the MMOB effects on the regiospecificity of hydroxylation of complex substrates and now by the effects on the tunneling contribution to the reaction. This means that mechanistic conclusions based on DFT and other calculations using the MMOH structure in the absence of MMOB must be evaluated with caution. Another approach to the study of the mechanism of C – H bond cleavage invokes the mutant forms of MMOB to remove at least some of the effects of controlled access to the active site and tunneling in the kinetics of Q reactions with substrates. Examination of the spatial organization of the residues of the Quad mutant based on the solution structure of MMOB shows that these fall into two groups. Residues N107 and S110 extend into the core structure of MMOB while S109 and T111 extend toward the presumed MMOH interface. Mutation of these residues singly and in pairs shows that all affect the kinetics of the MMOH catalytic cycle.79 However, the effects of the Quad mutant are mimicked most closely by the double mutation of S109A, T111A as expected based on their orientation. As shown in Figure 38.16a, the Arrhenius plot of the Q reaction with methane using this double mutant in place of MMOB is linear consistent with the change in rate-limiting step occurring at a somewhat lower temperature due to a faster substrate-binding reaction. The slope of the Arrhenius plot is also changed relative to that of the portion of the normal plot that reflects C –H bond breaking, i.e., the high-temperature region. This is expected, based on the hypothesis that tunneling plays a significant role in the normal reaction while being less significant in the reaction using the mutant MMOBs. As shown in Table 38.2, the mutants discussed here show a range of rates and KIE values for methane oxidation consistent with decreasing the effects of restricted access and tunneling, and this has proven to be useful in mechanistic studies. A method to overcome the restricted access was paradoxically discovered during an attempt to further restrict access by changing residue T111 to a tyrosine (T111Y).79 Instead of blocking the access to the active site as expected, it was observed that the mutation yielded an essentially openactive site. Large substrates reacted very rapidly with Q such that no intermediate was observed. 3.0

10

2.8

WT-MMOB

2.2 DBL2-MMOB

1.8

In kQD(mM−1s−1)

In kQD

2.4 2.0 1.6

d6-ethane

6 4 2

methane

0

1.4

(a)

ethane

8

2.6

d4-methane

−2 0.335 0.340 0.345 0.350 0.355 0.360 0.365 1/T (K−1) × 1000

(b)

98

100 102 104 BDE (kcal mol−1)

106

108

FIGURE 38.16 Q reactions using modified MMOBs that affect access and tunneling. (a) comparison of the Arrhenius plots for the Q reaction with methane using WT or DBL2 mutant MMOB (Brazeau, B. J. and Lipscomb, J. D., unpublished data). (b) modified Polanyi plot for the reaction of Q with substrates using the T111Y mutant of MMOB at 258C, pH 7.0. (Adapted from Brazeau, B. J. and Lipscomb, J. D., Biochemistry, 42, 5618– 5631, 2003. With permission.)

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TABLE 38.2 Effect of Mutation of MMOB on the Observed Deuterium KIE for Reaction of Q with Methane (258C, pH 7.0) MMOB Mutant WT (HTR)b T111A DBL2 T111Y Quad

kHa (M 21 s21)

kD (M 21 s21)

KIE

32,000 28,000 24,000 8,000 4,000

650 650 650 650 650

50 43 37 12 6

a

The error in the kH values is ^5% for WT and ^10% for the other mutants. The error in the kD values is ^15% due to the slow reaction. Data: Brazeau, B. J. and Lipscomb, J. D., unpublished. b The ratio of pre exponential factors is 0.0002 for the reaction with WT-MMOB and about 0.7 and 0.8 for the reactions with DBL2 and Y111Y, respectively. These values represent large extrapolations and thus can be expected to have considerable error.

Approximate rates could be obtained in a double-mix stopped-flow experiment by first making Q without substrate and then rapidly mixing in a substrate. When T111Y MMOB was present, the decay rate of Q was found to be at the limit of resolution of stopped flow at 48C with ethane as the substrate. In fact, the rate constants obtained for this reaction approached diffusion-controlled values consistent with the true reactivity of a species capable of hydrogen atom abstraction from methane. Interestingly, the rate constant for Q decay in the system using T111Y MMOB and methane as the substrate is significantly less than when WT MMOB is used. Under the scenario discussed here, this reflects a significant loss of the tunneling component in the reaction. Thus the T111Y mutant apparently allows the true reaction to be observed without the masking effects of tunneling and restricted access. A modified Polanyi plot of the log of the observed rate constants for methane, ethane, and their deuterated forms versus BDE is shown in Figure 38.16b. The plot is approximately linear as expected if the rate of the reaction is an inverse function of the C– H bond strength. Thus, one of the major anomalies of the MMO reaction discussed above is resolved. The true reaction rate is related to bond strength as expected, but the observed kinetics for the wild type system are dominated by the hydrocarbon size which shows the opposite correlation. Mechanistic insight derives from the linearity of the Polanyi plot. Such plots are linear when the C –H bond-breaking process occurs by hydrogen atom abstraction.80 Recent work has shown that the rate constant of the hydrogen atom abstraction reaction for a given reagent depends upon the strength of the hydrogen bond formed.81 In the case of Q, this would presumably be the strength of the OH bond formed in the intermediate we term R (see Figure 38.2a). The strength of this bond can be estimated based on a standard curve of log-rate constant vs. OH bond strength for a series of characterized H-atom abstracting reagents for a given substrate. Unfortunately, such a standard curve is difficult to construct for ethane or methane because the rates are so slow for most reagents. Nevertheless, the strength of the OH bond in the Q H-atom abstraction reaction can be approximated using substrates with weaker bonds because the curves converge as the C –H bond becomes stronger. A value of roughly 110 kcal/mol is found by this approach, which is consistent with the ability of Q to attack methane by the hydrogen atom abstraction mechanism, we have used in our mechanistic hypothesis. The sMMO system has proven to be a versatile vehicle with which to study oxygenase chemistry. It is also giving us new insight into enzyme catalysis on a much broader front. The ability to rapidly oxidize small, easily labeled molecules and the shear range of reactions catalyzed

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and substrates oxidized allows a nearly unprecedented opportunity to advance our understanding of nature’s catalysts. The use of isotopes has been, and will continue to be, critical in exploiting this opportunity to the fullest.

ACKNOWLEDGMENTS Many talented students have contributed to the work described here. They include: Drs. Brian G. Fox, Wayne A. Froland, K. Kristoffer Andersson, Sang-Kyu Lee, David R. Jollie, Jeremy C. Nesheim (deceased), Yi Liu, Bradley J. Wallar, Yi Jin, Shou-Lin Chang, Brian J. Brazeau, Aimin Liu, Codrina Popescu, John A. Broadwater, Jingyan Zhang, and Hui Zheng. We also gratefully acknowledge the seminal contributions that have come from the laboratories of our collaborators that have allowed the work to expand far beyond the limitations that any single laboratory imposes.

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53 Baik, M.-H., Gherman, B. F., Friesner, R. A., and Lippard, S. J., Hydroxylation of methane by nonheme diiron enzymes: molecular orbital analysis of C– H bond activation by reactive intermediate Q, J. Am. Chem. Soc., 124, 14608– 14615, 2002. 54 Baik, M.-H., Newcomb, M., Friesner, R. A., and Lippard, S. J., Mechanistic studies on the hydroxylation of methane by methane monooxygenase, Chem. Rev., 103, 2385– 2419, 2003. 55 Yoshizawa, K., Ohta, T., Yamabe, T., and Hoffmann, R., Dioxygen cleavage and methane activation on diiron enzyme models: a theoretical study, J. Am. Chem. Soc., 119, 12311– 12321, 1997. 56 Yoshizawa, K., Methane inversion on transition metal ions: a possible mechanism for stereochemical scrambling in metal-catalyzed alkane hydroxylations, J. Organomet. Chem., 635, 100– 109, 2001. 57 Yoshizawa, K. and Yumura, T., A non radical mechanism for methane hydroxylation at the diiron active site of soluble methane monooxygenase, Chem. Eur. J., 9, 2347– 2358, 2003. 58 Liu, K. E., Johnson, C. C., Newcomb, M., and Lippard, S. J., Radical clock substrate probes and kinetic isotope effect studies of the hydroxylation of hydrocarbons by methane monooxygenase, J. Am. Chem. Soc., 115, 939–947, 1993. 59 Jin, Y. and Lipscomb, J. D., Mechanistic insights into C– H activation from radical clock chemistry: oxidation of substituted methylcyclopropanes catalyzed by soluble methane monooxygenase from Methylosinus trichosporium OB3b, Biochim. Biophys. Acta, 1543, 47 – 59, 2000. 60 Jin, Y. and Lipscomb, J. D., Probing the mechanism of C– H activation: oxidation of methylcubane by soluble methane monooxygenase from Methylosinus trichosporium OB3b, Biochemistry, 38, 6178– 6186, 1999. 61 Choi, S.-Y., Eaton, P. E., Kopp, D. A., Lippard, S. J., Newcomb, M., and Shen, R., Cationic species can be produced in soluble methane monooxygenase-catalyzed hydroxylation reactions; radical intermediates are not formed, J. Am. Chem. Soc., 121, 12198– 12199, 1999. 62 Brazeau, B. J., Austin, R. N., Tarr, C., Groves, J. T., and Lipscomb, J. D., Intermediate Q from soluble methane monooxygenase hydroxylates the mechanistic substrate probe norcarane: evidence for a stepwise reaction, J. Am. Chem. Soc., 123, 11831– 11837, 2001. 63 Newcomb, M., Shen, R., Lu, Y., Coon, M. J., Hollenberg, P. F., Kopp, D. A., and Lippard, S. J., Evaluation of norcarane as a probe for radicals in cytochrome P450- and soluble methane monooxygenase-catalyzed hydroxylation reactions, J. Am. Chem. Soc., 124, 6879– 6886, 2002. 64 Jin, Y. and Lipscomb, J. D., Desaturation reactions catalyzed by soluble methane monooxygenase, J. Biol. Inorg. Chem., 6, 717– 725, 2001. 65 Sturgeon, B. E., Burdi, D., Chen, S., Huynh, B.-H., Edmondson, D. E., Stubbe, J., and Hoffman, B. M., Reconsideration of X, the diiron intermediate formed during cofactor assembly in E. coli ribonucleotide reductase, J. Am. Chem. Soc., 118, 7551–7557, 1996. 66 Valentine, A. M., Stahl, S. S., and Lippard, S. J., Mechanistic studies of the reaction of reduced methane monooxygenase hydroxylase with dioxygen and substrates, J. Am. Chem. Soc., 121, 3876– 3887, 1999. 67 Ambundo, E. A., Friesner, R. A., and Lippard, S. J., Reactions of methane monooxygenase intermediate Q with derivitized methanes, J. Am. Chem. Soc., 124, 8770– 8771, 2002. 68 Fox, B. G., Liu, Y., Dege, J. E., and Lipscomb, J. D., Complex formation between the protein components of methane monooxygenase from Methylosinus trichosporium OB3b. Identification of sites of component interaction, J. Biol. Chem., 266, 540– 550, 1991. 69 Pulver, S. C., Froland, W. A., Lipscomb, J. D., and Solomon, E. I., Ligand field circular dichroism and magnetic circular dichroism studies of component B and substrate binding to the hydroxylase component of methane monooxygenase, J. Am. Chem. Soc., 119, 387– 395, 1997. 70 Froland, W. A., Andersson, K. K., Lee, S.-K., Liu, Y., and Lipscomb, J. D., Methane monooxygenase component B and reductase alter the regioselectivity of the hydroxylase component-catalyzed reactions. A novel role for protein –protein interactions in an oxygenase mechanism, J. Biol. Chem., 267, 17588– 17597, 1992. 71 Paulsen, K. E., Liu, Y., Fox, B. G., Lipscomb, J. D., Mu¨nck, E., and Stankovich, M. T., Oxidationreduction potentials of the methane monooxygenase hydroxylase component from Methylosinus trichosporium OB3b, Biochemistry, 33, 713– 722, 1994. 72 Chang, S. L., Wallar, B. J., Lipscomb, J. D., and Mayo, K. H., Solution structure of component B from methane monooxygenase derived through heteronuclear NMR and molecular modeling, Biochemistry, 38, 5799– 5812, 1999.

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73 Walters, K. J., Gassner, G. T., Lippard, S. J., and Wagner, G., Structure of the soluble methane monooxygenase regulatory protein B, Proc. Natl. Acad. Sci. U.S.A., 96, 7877– 7882, 1999. 74 Callaghan, A. J., Smith, T. J., Slade, S. E., and Dalton, H., Residues near the N-terminus of protein B control autocatalytic proteolysis and the activity of soluble methane monooxygenase, Eur. J. Biochem., 269, 1835– 1843, 2002. 75 Chang, S. L., Wallar, B. J., Lipscomb, J. D., and Mayo, K. H., Residues in Methylosinus trichosporium OB3b methane monooxygenase component B involved in molecular interactions with reduced- and oxidized-hydroxylase component: a role for the N-terminus, Biochemistry, 40, 9539– 9551, 2001. 76 Wallar, B. J. and Lipscomb, J. D., Methane monooxygenase component B mutants alter the kinetics of steps throughout the catalytic cycle, Biochemistry, 40, 2220– 2233, 2001. 77 Brazeau, B. J., Wallar, B. J., and Lipscomb, J. D., Unmasking of deuterium kinetic isotope effects on the methane monooxygenase compound Q reaction by site-directed mutagenesis of component B, J. Am. Chem. Soc., 123, 10421– 10422, 2001. 78 Kohen, A. and Klinman, J. P., Enzyme catalysis: beyond classical paradigms, Acc. Chem. Res., 31, 397– 404, 1998. 79 Brazeau, B. J. and Lipscomb, J. D., Key amino acid residues in the regulation of soluble methane monooxygenase catalysis by component B, Biochemistry, 42, 5618– 5631, 2003. 80 Gardner, K. A., Kuehnert, L. L., and Mayer, J. M., Hydrogen atom abstraction by permanganate: oxidations of arylalkanes in organic solvents, Inorg. Chem., 36, 2069– 2078, 1997. 81 Mayer, J. M., Hydrogen atom abstraction by metal-oxo complexes: understanding the analogy with organic radical reactions, Acc. Chem. Res., 31, 441– 450, 1998.

39

Secondary Isotope Effects Alvan C. Hengge

CONTENTS I. II.

Theory and Nomenclature ............................................................................................... 955 Origins of Secondary Isotope Effects .............................................................................. 956 A. Secondary Isotope Effects Resulting from Hybridization Changes........................ 956 B. Secondary Isotope Effects on Acidities................................................................... 959 C. Steric Secondary Isotope Effects ............................................................................. 960 III. Secondary Isotope Effects in Particular Reactions.......................................................... 961 A. Acyl Transfer ........................................................................................................... 961 B. Glycosyl Transfer..................................................................................................... 964 C. N-Ribosyl Hydrolases and Transferases .................................................................. 965 D. Phosphoryl Transfer ................................................................................................. 966 E. Methyl Transfer........................................................................................................ 968 F. Hydride Transfer ...................................................................................................... 969 G. Peptidyl Prolyl Cis –Trans Isomerase...................................................................... 969 References..................................................................................................................................... 975

I. THEORY AND NOMENCLATURE There are a number of extensive discussions of the origins of secondary isotope effects in the literature1 – 5 and in this volume (see Chapter 1). For this reason, only the basic concepts will be presented here, and interested readers are referred to these prior reviews for more in-depth discussions. Isotope effects, as is clear from an examination of the full Bigeleisen equation (see Chapter 1), are affected by changes in any and all vibrational modes; i.e., if a change in environment results in a change in the vibrational spectrum of a molecule, it is a priori evidence that there will be an associated isotope effect. A primary isotope effect results when a bond to an isotopically labeled atom is broken. In this case, the change in the stretching frequency of the scissile bond is the largest term, and, to a first approximation, is often the only one of importance. Since there is no restoring force to this stretching vibration in the transition state, an imaginary frequency results. This contribution to the isotope effect is often called the temperature-independent factor, or the imaginary frequency factor. The other factor in isotope effects is the zero point energy factor, which results from force constant changes to the real vibrational modes of the reactant. When no bond fission to the labeled atom takes place, smaller changes to vibrational modes may still occur and result in a measurable and mechanistically informative isotope effect. These are termed secondary isotope effects, and can be quite sensitive probes of reaction mechanism. Unlike primary isotope effects, which are nearly always normal because the frequency of the vibration that is lost is larger for the lighter isotope, secondary isotope effects may be normal (. 1) or inverse (, 1). A normal isotope effect results when bonding (including bending) to the labeled atom becomes looser, (reduced vibrational frequencies) and an inverse isotope effect results when bonding becomes tighter (increased vibrational frequencies). 955

956

Isotope Effects in Chemistry and Biology

If the isotopic substitution is at an atom directly bonded to the atom undergoing a bonding change, the isotope effect is termed an alpha-secondary (a-28) isotope effect. Secondary isotope effects can often also be observed at positions more distant; b-secondary (b-28) isotope effects, observed when the isotopic label is two bonds removed from the bonding change, are often as large as (a-28) isotope effects. Remote 28 effects at positions even further removed may also be observed. For example, deprotonation of p-nitrophenol is accompanied by an 15N equilibrium isotope effect of 1.0023 ^ 0.0001, determined by isotope ratio mass spectrometry.6 In general, secondary isotope effects are much smaller in magnitude than primary isotope effects. For this reason, the vast majority of secondary isotope effects that have been reported are those resulting from isotopic substitution of hydrogen by deuterium or tritium. However, secondary heavy-atom isotope effects have been reported in a growing number of reactions, and have proven to be of value. Another consideration is that the magnitudes of most secondary kinetic isotope effects are thought to increase as the transition state structure changes from reactant-like to productlike. Primary isotope effects, on the other hand, exhibit a bell-shaped dependence; for example, in deprotonations, the largest primary KIE is observed for a symmetrical transition state, and smaller KIEs are observed for either an early or a late transition state.7 A common notation for isotope effects, which will be used in this chapter, is the use of the leading superscript of the heavier isotope to indicate the isotope effect on the following kinetic quantity; for example, 15k denotes k14/k15, the nitrogen-15 isotope effect on the rate constant k.8 For hydrogen isotope effects, the superscript D or T will be used to denote deuterium or tritium, respectively. Equilibrium isotope effects (EIEs) are designated in a similar manner, but using a capital K, for example, 15K.

II. ORIGINS OF SECONDARY ISOTOPE EFFECTS A. SECONDARY I SOTOPE E FFECTS R ESULTING FROM H YBRIDIZATION C HANGES For hybridization changes of the type sp3 to sp2, or sp2 to sp, normal a-28 isotope effects result. Even though the C – H (D or T) bond stiffens in such hybridization changes, the bending modes loosen, resulting in the normal effect. It is thought that the isotope effect results primarily from changes in the out-of-plane bending vibrations, which become looser (lower in energy), for example, as the tetrahedral reactant is converted into the trigonal planar transition state in an SN1 reaction.9 This exemplifies the importance of bending vibrations that are found in many secondary isotope effects. Kirsch has also presented considerations of the stretching and bending vibrations that contribute to such isotope effects, showing that the out-of-plane bending vibration is mainly responsible.10 The a-DK value for the ionization of benzhydrol is 1.29 (Figure 39.1A).11 For hybridization changes in the reverse direction, inverse secondary isotope effects are seen. For example, the attack of cyanide ion on the aldehyde group of p-methoxybenzaldehyde to form the cyanohydrin (Figure 39.1B) is accompanied by Dk ¼ 0.73.12 a-Secondary KIEs are also used to distinguish between SN1 and SN2 mechanisms.13 In the transition state of an SN2 reaction the central carbon atom resembles an sp2 hybrid, but the close proximity of the nucleophile and leaving group compresses the out-of-plane bending vibrations, thus reducing the normal contribution to the isotope effect that these vibrations normally contribute in an SN1 reaction. As a result, in contrast to the normal values seen in SN1 reactions, a-Dk values for SN2 reactions are typically near unity or slightly inverse. For example, for the solvolysis of p-chloromethyltoluene, which proceeds by an SN1 mechanism, a-Dk ¼ 1.30, while for the SN2 attack of methoxide ion on methyl brosylate (Figure 39.1C), a-Dk ¼ 0.96. Slightly larger (more normal) a-Dk KIEs are found in SN2 reactions of larger substrates. For example, for the SN2 hydrolysis of methyl tosylate, a-Dk ¼ 0.984 per deuterium atom, while in the same reaction with ethyl tosylate, a-Dk ¼ 1.019. Similarly, the hydrolyses of methyl and ethyl bromide exhibit a-Dk values of 0.959 and 0.991 per deuterium, respectively, (data from the compilation in Ref. 13).

Secondary Isotope Effects

957 OH C

(A)

H

L O

C

OH

L

NC C L

(B)

+ HCN Dk=0.73

OCH3 CH3O

(C)

+ H2O

C L

Dk=1.29

H3C

+

CL3 OBs

L C Cl L

Dk=0.96

H2O

OCH3

H3COCL3

+

OBs

L C OH L

H3C

Dk=1.30

+ HCl

L = H or D

FIGURE 39.1 Some representative a-secondary isotope effects in (A) a reaction involving an sp3 to sp2 rehybridization; and (B), an sp2 to sp3 rehybridization. The reactions in (C) are nucleophilic substitutions; the top reaction proceeds by an SN2 mechanism, while the bottom one is an SN1.

A computational examination concluded that these KIEs are the result of an inverse contribution from stretching and a normal bending contribution. For small (i.e., methyl) substrates, the stretching contribution is the larger contributor, and the KIEs are inverse or small and normal. For larger substrates such as ethyl, the bending contribution is the most important, resulting in a trend toward a larger (more normal) a-Dk.9 b-Secondary isotope effects have also long been used as probes of the mechanisms of SN1 solvolysis reactions, following the early work of Lewis14 and Shiner.15 A commonly cited example consists of the b-secondary isotope effects for the solvolysis reactions of the cyclopentyl tosylates shown in Figure 39.2.16 Numerous explanations have been advanced to explain the b-secondary Dk

OTs L

H

OTs

1.15

1.16

L L

OTs

1.22

H L = H or D

FIGURE 39.2 Secondary isotope effects for the solvolysis reactions of cyclopentyl tosylates.

958

Isotope Effects in Chemistry and Biology L Cl

L CH3

Cl

CH3 L

L D

k = 1.14

D

k = 0.986

FIGURE 39.3 Secondary isotope effects for the solvolysis reaction arising from hyperconjugation at the b-position (left) and inductive effects at the bridgehead positions (right).

effects, but the most commonly accepted view is that hyperconjugation resulting from developing carbocation character results in a weakening of the b-C –H(D) bonds. The dependence of the b-secondary isotope effect on the angular orientation of the C – H(D) bond relative to the developing p-orbital on the a carbon atom was clearly demonstrated by Shiner and Humphrey using the bicyclic tertiary chloride shown in Figure 39.3.17 The b secondary KIEs for the solvolysis in 60% ethanol were measured at the bridgehead position and at the b-methylene group. Methylene deuteration results in a normal KIE of 1.14 ^ 0.01, while bridgehead deuteration results in an inverse KIE of 0.986 ^ 0.01.17 Only the methylene hydrogen atoms are situated in a geometry that allows hyperconjugation. The small inverse effect from bridgehead deuteration arises from the inductive effect of deuterium vs. hydrogen (see Section II.B). Hyperconjugation between a C – H(D) bond and the vacant p-orbital of a carbocation is most effective when the dihedral angle is zero. Assuming that this angular dependence obeys the cos2u rule (which is a reasonable assumption; vicinal NMR coupling constants obey the same rule) the angular dependence of the hyperconjugative secondary KIE can be approximated by the equation18: logD kobs ¼ cos2 u log½hyp ðD kÞmax þ logind ðD kÞ This equation is cast in terms of a deuterium isotope effect, but is equally valid for tritium effects. The term hyp(Dk)max refers to the maximum hyperconjugative isotope effect, which occurs when the dihedral angle is 08. The term log ind(Dk) refers to the inductive isotope effect. A review by Hehre and Sunko gives a more thorough discussion of the conformational dependencies of secondary b-deuterium isotope effects on the structures of cationic transition states.3 b-Secondary isotope effects can also result from negative hyperconjugation of C –H bonds. A lone electron pair anti to a C – H bond decreases the C– H stretching frequency by , 60 cm21.19,20 These conformation-dependent changes in stretching frequency result in EIEs on rotamer distribution. These b-secondary EIE have been measured for O- and N-containing heterocycles where the C – D bond preferentially adopts an axial conformation.21,22 Increased negative hyperconjugation was predicted to result in normal EIEs on H-bond formation23 and decreases in C –D stretching frequencies correlate with H-bond strength in alcohols.24 Secondary isotope effects are also used to probe b-elimination reactions (Figure 39.4). The isotope effect at the b-position p results from the change in hybridization of this carbon atom from sp3 to sp2. The a position is affected by a similar hybridization change, as well as by the elemental nature of the leaving group atom. An example of the latter contribution is the fractionation factor of the alpha C – H in secondary alcohols compared to saturated methylenes. The ratio of the fractionation factors of these two species is 1.18, reflecting the fact that replacing a C – H bond with a C– O bond stiffens the bonds to the other hydrogen atoms.4 The equilibrium isotope effect for the sp3 to sp2 conversion is normal, , 1.12, due to loosening of the out-of-plane bending modes. This is the value of the EIE at the b position for the conversion p Some publications reverse the designations of the a and b hydrogens from those shown in Figure 39.4. The convention in organic chemistry is to designate the carbon atom bearing the leaving group as a, and that is the convention used here.

Secondary Isotope Effects

959 Hα

Habs Rα Rβ

Hβ

Lg

Rβ

Rα

Hβ

Hα

FIGURE 39.4 Designation of positions in a b-elimination reaction. Habs is the hydrogen atom abstracted, Lg denotes the leaving group.

shown in Figure 39.4, while the EIE at the a position is typically in the neighborhood of 1.3. The EIEs for specific reactions will vary due to structural effects. In an early transition state, no significant hybridization change has occurred, and a secondary isotope effect close to unity is observed. In a late transition state, the hybridization change is largely complete, and a KIE close to the EIE for a given system will be observed.25 A number of enzymatic b-elimination reactions have been examined using such isotope effects, including enolase, fumarase, a number of ammonia lyases, and serine dehydratase (see Ref. 26 and references therein). More recent studies include that of the O-acetylserine sulfhydrylase-catalyzed b-elimination reaction, in which the secondary kinetic deuterium isotope effect for two deuterium atoms (3,3-d2-O-acetylserine) was 1.11, much smaller than the measured equilibrium value of 1.81, indicating an early transition state if the elimination process is concerted.27

B. SECONDARY I SOTOPE E FFECTS ON ACIDITIES The acidities of carboxylic acids are affected by the substitution of deuterium on the b-carbon atom, and on the a-carbon of formic acid. Though there are discrepancies in the magnitudes of some reported equilibrium isotope effects (DK) on acidities, depending on the methods used to measure them, such isotope effects are normal (Table 39.1) (for a larger compilation of such data, see Ref. 13, page 293). These normal KIEs result from the anharmonicity of C – H vs. C – D bonds, manifested in a reduced C– D bond length. This results in increased electron density at a carbon bearing D relative to H, which is also revealed by the increased shielding observed for 13C NMR resonances. Similar isotope effects have been noted on amine basicity. NMR saturation-transfer measurements on 15N-labeled-ammonium ion show the presence of a a-secondary kinetic isotope effect for dissociation of a proton from ammonium ion, Dk ¼ 1.07 per deuterium atom.28 b-Secondary EIEs have been determined by NMR titrations that compare the pKa values of protonated benzylamine and benzylamine-a-d1, and of N, N-dimethylaniline with its trideuteromethyl analog. In both compounds, deuterium substitution increases the basicity of the TABLE 39.1 Some Representative Secondary Deuterium Isotope Effects on Acidities of Oxygen Acids Acid (L 5 H or D) C6L5OH C6L5COOH C6H5CL2COOH CL3COOH LCOOH CF3CL2OH

D

K

Reference

1.12 ^ 0.02 1.024 ^ 0.006 1.12 ^ 0.02 1.004 to 1.007 1.033 ^ 0.002 1.06 ^ 0.03 1.12 ^ 0.05 1.138 ^ 0.009

94 94 95 96 97 98 99 23

The values shown are for the fully labeled species (not per deuterium).

960

Isotope Effects in Chemistry and Biology

amine since protonation of the amine eliminates the negative hyperconjugation with the b-C –H bonds. For benzylamine, DK ¼ 1.0419 ^ 0.0009 (DpKa ¼ 0.0178); a previous measurement found that dideuteration resulted in DpKa ¼ 0.032.29 For N,N-dimethylaniline, DK ¼ 1.1051 ^ 0.0018 (DpKa ¼ 0.0434).30 The b-DEIE of 1.5 for trimethylamine-d9 in aqueous solution (1.047/deuterium)31 is reduced from the gas phase observation for the protonation [1-D2]ethyl methylamine of 1.09/deuterium.32 Because H-bonding to the N-lone and O-lone pairs in aqueous solution reduces the negative hyperconjugation in the free base, the gas phase experiments result in significantly larger observed EIEs. The b-secondary EIE on the acidity of trifluoroethanol reveals the same effect of negative hyperconjugation applies to alcohols, as the pKa of dideuterotrifluoroethanol is decreased by 0.05 relative to trifluoroethanol, and the C – D stretching frequency decreases 78 cm21, upon ionization in aqueous solution.23 Such effects may also be observed in the ionization of glucose. Measurement of the TK EIEs at all six positions for deprotonation of glucose showed the most pronounced effects at H1 (1.051 ^ 0.0007) and at H4 (1.024 ^ 0.0003).33 The results indicate the presence of unexpected intramolecular sharing of charge in anionic aqueous glucose, with the practical consequence that tritiated glucose elutes ahead of nontritiated compound on anion exchange resin. A previous report noted an analogous isotope effect on the acidity of glucose resulting from deuteration at C1, resulting in a change in retention time in anion exchange chromatography.34 A 15N EIE of 1.0023 ^ 0.0001 accompanies the deprotonation of p-nitrophenol, measured using isotope ratio mass spectrometry.6 This isotope effect results from the delocalization of charge in the phenolate ion, which involves contribution from a quinonoid resonance form in which the bond order between nitrogen and oxygen decreases and the bond order between nitrogen and carbon increases. Since N – O bonds are stiffer than N – C bonds, 15N prefers the neutral phenol, resulting in the normal isotope effect for deprotonation. In addition to their effect on acidities, the different inductive properties of D vs. H are also exhibited in differences in dipole moments. The dipole moments of NL3, CL3NL2, LCNO and (CH3)3CL are all from 0.01 to 0.04 Debye units larger when LyD than for the protio compounds.35

C. STERIC S ECONDARY I SOTOPE E FFECTS The lower zero-point energy of the C – D bond, and the anharmonicity (Chapters 1, 5, and 10, contain further discussion of anharmonicity) of the potential energy surface result in a C –D bond that is slightly shorter than a corresponding C – H bond. This results in steric isotope effects in some systems, and molecular mechanics calculations assign a smaller van der Waals radius to deuterium than to hydrogen.36 As a result of this discrimination, for the racemization of 2,20 -dibromo-4,40 dicarboxybiphenyl Dk ¼ 0.84 (Figure 39.5).37 A number of similar steric isotope effects on bond rotations are described in a review on the topic by Carter and Melander.38 In another study, it was found that the presence of a CL3 group in the 3- or 4-position of methyl pyridine does not result in a significant (H3/D3) isotope effect on the rate of methylation with methyl iodide. However, inverse KIEs result when the CL3 group is in the 1-position in reactions with methyl, ethyl or isopropyl iodide Br L HO2C

CO2H Br L L = H or D

FIGURE 39.5 A compound for which a steric isotope effect on isomerization (rotation around the central bond) arises from the smaller van der Waals radius of deuterium compared to hydrogen.

Secondary Isotope Effects

961

TABLE 39.2 Secondary Isotope Effects for Reactions of Deuteromethyl Pyridines with Alkyl Iodides Pyridine 4-Methyl-d3 3-Methyl-d3 2-Methyl-d3 2-Methyl-d3 2-Methyl-d3

Alkyl Iodide Methyl Methyl Methyl Ethyl Isopropyl

D

k at 258C 0.999 0.991 0.971 0.960 0.932

(Table 39.2).39 The lack of a significant isotope effect in the 3- and 4-CL3-pyridines indicates that the isotope effect is a steric effect, not the result of inductive effects.

III. SECONDARY ISOTOPE EFFECTS IN PARTICULAR REACTIONS Secondary isotope effects have found wide use in the study of the mechanisms and transition states of enzymatic and nonenzymatic reactions, and a comprehensive review of the literature on this topic is beyond the scope of this chapter. In addition to the general reviews cited at the beginning of this chapter, several reviews on the application of secondary isotope effects in the study of enzymatic reactions have been published.10,40 – 42 In the sections below, the aim is to complement these earlier reviews and to focus on reports that are more recent. In this volume, Chapters 30, 34, and 37 among others, contain additional examples of secondary isotope effects on enzymatic reactions.

A. ACYL T RANSFER Secondary b-hydrogen isotope effects, and secondary a-hydrogen isotope effects, have long been used in the study of acyl and formyl transfer reactions, respectively. In the acyl case, the dominant source of the isotope effect arises from changes in hyperconjugation as nucleophilic attack converts the electron-deficient sp2 carbonyl group to a more neutral, tetrahedral carbon atom. Such nucleophilic interaction decreases hyperconjugative effects, increasing the force constants and resulting in an inverse b-secondary effect. One example is the equilibrium addition of methanol to d6-acetone, for which DK ¼ 0.78. Using the method of Sunko et al.18 to calculate the contribution to this observed isotope effect from one deuterium at the optimal dihedral angle of 08 yields a value of 0.89,43 assuming a contribution of 1.013 for the inductive effect.44 Inverse b-28 KIEs have been measured in a number of reactions involving nucleophilic attack at carbonyl in uncatalyzed (see Refs. 40 , 45 and references therein) and enzymatic reactions.46 In contrast, formation of an acylium ion intermediate will increase hyperconjugative effects and result in a normal b-28 effect. For example, for the hydrolysis of acetyl chloride at 2 228C, b-28 Dk for three deuteriums ¼ 1.51, which was interpreted as evidence for a transition state resembling acylium ion.47 The a-28 hydrogen isotope effects in formyl transfer reactions result from the hybridization change at the carbon directly bonded to the labeled atom, and changes to bending vibrations are the dominant contributor. Since the hybridization change is from sp2 to sp3, an inverse effect is expected. The a-28 DK for the equilibrium addition of nucleophiles to a number of formyl and acyl carbon atoms are close to 0.77 (see Ref. 48 and references therein). Kinetic effects for the attack of various nucleophiles on phenyl formates yield KIEs in the range of 0.9 to 0.8.48 The a-28 KIEs for the hydrazinolysis of methyl formate are 0.76 at pH 10, when formation of the tetrahedral intermediate is rate-determining, and near unity (0.98) at pH 8, when breakdown

962

Isotope Effects in Chemistry and Biology

is rate-determining.49 The latter value is indicative of a transition state that has regained the trigonal nature of the reactant. More recently, heavy-atom isotope effects have been measured for a number of acyl transfer reactions from amides and esters. In addition to the formyl-H, the carbonyl-13C, nucleophile-18O, leaving-15N, and carbonyl-18O KIEs have been reported for the alkaline hydrolysis of formamide.50 Of these heavy-atom KIEs, only the last is a secondary effect; this was expected to be normal, as breaking the carbonyl p bond was estimated to result in a maximum 18kcarbonyl from 1.02 to 1.03. However, an inverse value of 0.980 was obtained. Interpretation of the carbonyl KIE is complicated by other factors besides loss of the p bond, including addition of new bending and torsional modes in a tetrahedral transition state. The most interesting result in the alkaline hydrolysis of formamide came from the primary nucleophile-18O KIE of 1.022, which points to a water molecule, presumably one of those solvating the hydroxide ion, as the nucleophile, not hydroxide ion itself.50 For a further discussion of this and other nucleophile isotope effects, see Chapter 36. Isotope effects have been measured for acyl transfers from p-nitrophenyl acetate ( p-NPA) to a variety of anionic O and S nucleophiles.51 The secondary KIEs were measured at the positions of the b-deuterium, the carbonyl 18O, and the 15N in the leaving group. The latter isotope effect measures delocalized charge in the nitrophenolate leaving group, as discussed above in Section II.A. Along with primary isotope effects (the carbonyl 13C and the ester oxygen 18O), the secondary KIEs for reactions of p-NPA with various nucleophiles were used to address the controversial question of whether esters with good leaving groups undergo acyl transfer by a concerted mechanism, instead of via a tetrahedral intermediate. The acyl transfers to anionic nucleophiles with pKa 9.3 and over, all exhibit modest inverse b-Dk of 0.95 to 0.98 (for three deuterium atoms), indicating a transition state with little rehybridization of the carbonyl carbon atom. This would be consistent with either a concerted mechanism, or an early transition state for formation of an intermediate. More telling were the significant 18O KIEs in the scissile oxygen atom (here designated 18kleaving). If a tetrahedral mechanism is followed, since these nucleophiles all have pKa values significantly higher than that of p-nitrophenol, formation of the tetrahedral intermediate should be rate determining, and its collapse to products rapid. Thus, 18kleaving should be small, and would be a secondary effect in this scenario, since this bond is not broken in formation of the tetrahedral intermediate. Very small magnitudes for 18kleaving were observed in the alkaline hydrolysis of methyl formate and methyl benzoate (1.009 and 1.006, respectively), but in the p-NPA reactions, 18kleaving is much larger (1.02 to 1.03), indicative of significant bond fission. This 18 kleaving has a maximum value of about 1.03 with the p-nitrophenol leaving group, while for alkyl leaving groups the maximum is 1.06 to 1.07. The data are more consistent with a concerted mechanism for the tested O and S nucleophiles with p-NPA, and a stepwise mechanism for the hydrolysis of methyl formate and methyl benzoate. The data for the neutral amine nucleophile with p-NPA were similar to those for the hydrazinolysis of methyl formate and methyl benzoate at pH 8, where breakdown of a tetrahedral intermediate is rate-determining.50 The required proton transfer from the nitrogen nucleophile seems to result in a mechanism involving a tetrahedral intermediate even for esters with a very good leaving group. The isotope effects were also measured for the enzymatic acyl transfers from p-NPA with chymotrypsin, carbonic anhydrase, papain, and the acid protease from Aspergillus.52 Because the isotope effects were measured by the remote-label method53 using the nitrogen atom in the leaving group as a reporter, they represent the KIEs for the acyl transfer step from p-NPA to the enzymatic nucleophile. The secondary b-Dk for the enzymatic reactions were consistently smaller than those for the uncatalyzed aqueous reactions with anionic oxygen and sulfur nucleophiles, while the other isotope effects were similar between the enzymatic and nonenzymatic reactions. This was attributed to enzymatic hydrogen bonding or other electrostatic interactions with the ester carbonyl group, serving to maintain in the transition state nearly all of the hyperconjugation with the b-hydrogen atoms present in the ester substrate.

Secondary Isotope Effects

963

The secondary b-deuterium KIEs for the deacylation step were measured for chymotrypsin and for elastase, using p-NPA as the substrate. These isotope effects were obtained from the zero-order 3D rates for the H3 and the D3 esters (V3H 0 and V0 ), under conditions where rapid acylation of enzyme is followed by rate-limiting deacylation of the acyl enzyme.54 The deacylation of acylchymotrypsin exhibits a b-DKIE of 0.94, compared with 0.91 for nonenzymatic protolytically catalyzed hydrolysis model reactions. The b-DKIE for deacylation of acylelastase was even less inverse, 0.98. The results imply the enzymatic deacylation reactions are accompanied by smaller loss of hyperconjugation in the transition state than the nonenzymatic reactions. Similar conclusions result from comparisons of a-deuterium isotope effects for the corresponding deformylation reactions.54 Secondary b-deuterium KIEs have also been measured for the hydrolysis of amides and b-lactams, and for the b-lactamase-catalyzed reaction. For the hydrolysis in alkaline solution of p-nitroacetanilide, the Dk (for three deuterium atoms) varies from 0.967 in 2 mM [– OH], to a maximum value of , 0.98 at 30 mM [– OH], and then decreases to 0.933 in 2 M [– OH].55 These results indicate dependence of the rate-limiting step on hydroxide concentration. In the least basic conditions, decomposition of a monoanionic tetrahedral intermediate is rate limiting, while under the most basic conditions, nucleophilic attack is rate-limiting. These conclusions were subsequently supported by measurements of the 15N KIEs at the amido nitrogen atom. It was found that 15 kamido ¼ 1.035 at 2 mM [–OH], while at 2 M [– OH], 15kamido ¼ 0.995.56 The 15N KIE is a primary one in decomposition of a tetrahedral adduct (2 mM [– OH]), but is an a-secondary isotope effect when the rate-limiting step is formation of the intermediate. The inverse value in the latter case most likely results from restricted bending modes as the carbon rehybridizes from sp2 toward sp3. The b-secondary and solvent deuterium isotope effects have been measured for the hydrolysis of the b-lactam penicillanic acid (Figure 39.6), under acid- and base-catalyzed conditions44 as well as for the hydrolysis of this substrate catalyzed by b-lactamase.43 Direct comparisons of b-Dk values between b-lactams and normal amides cannot be made because the latter involve deuterium substitution in methyl groups that are free to rotate, while in a b-lactam rotation is not possible. Thus, a direct comparison requires the calculation of the contribution to the observed isotope effect for a single deuterium, from the observed effect and the dihedral angle, and the inductive contribution to the observed isotope effect.44 In the alkaline hydrolysis of penicillanic acid, the b-Dk, corrected to represent the effect of a single deuterium at the optimal dihedral angle of 08, is 0.95 ^ 0.01, similar to the value obtained by Stein et al. for p-nitroanilide hydrolysis. This indicates that nucleophilic attack on acyclic amides and b-lactams occurs by similar mechanisms, unaffected by the ring strain in the latter compounds. In contrast, for the acid-catalyzed hydrolysis of penicillanic acid, b-Dk is unity. This was attributed to an acylium ion like transition state, preceded by protonation on nitrogen. Although acyclic amides are well known to protonate on oxygen, it was proposed that for strained, cyclic amides nitrogen protonation might be favored.44 The b-D(V/K) values were measured for the reactions of penicillanic acid, and for the depsipeptide substrate m-[[(phenylacetyl)glycyl]-oxy]benzoic acid, with several class A b-lactamases and one class C b-lactamase.43 The values for b-D(V/K) for the reactions with penicillanic acid were inverse, from 0.88 to 0.93, for all of the enzymes examined, consistent with transition states in which the carbonyl has become tetrahedral, as in the alkaline hydrolysis reaction of penicillanic acid. The b-D(V/K) numbers are also inverse for the reaction of the depsipeptide substrate with the class A

S O

S

−

OH O2C HN

−

N CO2−

FIGURE 39.6 The alkaline hydrolysis of penicillanic acid.

CO2−

964

Isotope Effects in Chemistry and Biology

b-lactamases, but for the class C b-lactamase of Enterobacter cloacae P99, b-D(V/K) is essentially unity, 1.002 ^ 0.012. From this evidence, as well as the absence of a solvent isotope effect, it was concluded that the rate-limiting transition state in the latter reaction does not involve covalent chemistry.

B. GLYCOSYL T RANSFER The equilibrium isotope effects at a number of sites have been reported for the equilibrium between the a- and b-anomeric forms of D -glucopyranose in water. The deuterium isotope effects (DKb/a) were measured by NMR, and computational methods were used to gain insights to their origin and interpretation. DKb/a effects of 1.043 ^ 0.004, 1.027 ^ 0.005, 1.027 ^ 0.004, 1.001 ^ 0.003, 1.036 ^ 0.004, and 0.998 ^ 0.004 on the equilibrium constant were found for the [1-D]-, [2-D]-, [3-D]-, [4-D]-, [5-D]-, and [6,60 -D2]-labeled sugars, respectively. Calculations point to differences in negative hyperconjugation as the sources for the isotope effects at H-1 and H-2. It was concluded that the isotope effect at H-1 is due to np ! sp hyperconjugative transfer from O5 to the axial C-1 – H-1 bond in b-glucose. The isotope effect at H2 is due to rotational restriction of OH-2 to an angle of 1608 in the a form and 608 in the b-sugar, which results in differences in hyperconjugative transfer from O-2 to CH-2. The isotope effects at the H-3 and H-5 positions result primarily from steric repulsion between these positions and the axial anomeric hydroxyl oxygen in a-glucose.57 A similarly impressive set of kinetic isotope effects have been measured for the hydrolysis reactions of methyl a- and b-glucopyranosides,58 and of methyl xylopyranosides in comparison with methyl 5-thioxylopyranosides.59 These isotope effects were measured using the quasiracemate method, in which the optical rotation of a mixture of the labeled D -sugar and the unlabeled L -sugar is followed using a polarimeter. Figure 39.7 shows the positions of the most important isotope effects measured. Normal values for a- and b-Dk along with other data, show that the acidcatalyzed hydrolyses proceed via exocyclic C – O bond fission with no nucleophilic participation in the transition state, supporting the notion of oxo- and thio-carbenium-like transition states. In the glucopyranosides, 18k at the ring oxygen atom are very small (0.991 to 0.996), indicating that charge delocalization involving this oxygen atom lags behind C – O bond fission. In contrast, in reactions of the corresponding a- and b-xylopyranosides, more inverse isotope effects at this position (0.978 to 0.983) indicate that delocalization of positive charge by this atom is coupled with leaving group departure. The difference is ascribed to removal of the C-5-hydroxymethylene group permitting greater overlap between the O-5 and C-1 orbitals in the transition state.59 Oxocarbenium like transition states were also implicated by secondary isotope effects measured in the reactions catalyzed by lysozyme and related enzymes, and for galactosidase (reviewed in Ref. 40). In more recent work, measurements were made of a-28-T(V/K) for a number of exo-aglucanases that catalyze the hydrolysis of a-D -glucopyranosyl fluoride (a-GF) with inversion.60 R HO HO

18

O

18

OCH3

H(D) H(D) OH

13C

FIGURE 39.7 The sites of isotope effect measurement for the hydrolysis of methyl glucopyranosides and methyl xylopyranosides. For glucopyranosides, RyCH2OH, for xylopyranosides, RyH. For the thio analogues, the ring oxygen atom is replaced by sulfur. The a- and b-deuterium isotope effects give information about the degree of rehybridization and hyperconjugative effects in the oxocarbenium-like transition state. The ring-18O effect is sensitive to the degree to which this atom participates in delocalization of positive charge. The two primary isotope effects indicated give information as to the degree of bond fission (methoxyl-18O) and the degree of nucleophilic participation (13C).

Secondary Isotope Effects

965

The three tested glucoamylases and a glucodextranase each exhibit a significant normal T(V/K), ranging from 1.17 to 1.26.60 The results indicate a transition state with significant oxocarbenium ion character, and the possibility of a discrete oxocarbenium ion intermediate was proposed. The KIEs are very similar to those obtained earlier for enzymatic reactions of glycopyranosides catalyzed with retention of configuration, as well as for glucopyranosyl transfers catalyzed with inversion of configuration (see Ref. 60 and references therein).

C. N-RIBOSYL H YDROLASES AND T RANSFERASES A very thorough set of isotope effects, most of them secondary, have been measured by the Schramm group on reactions of N-ribosyl hydrolases and transferases. Figure 39.8 shows the substrate inosine indicating the positions at which kinetic isotope effects were measured, and the transition state information yielded by each. All of these isotope effects have been measured by the remote label method using radioactive labels (3H or 14C), and isotope ratio scintillation counting. The methodology for preparation of the labeled forms of the substrate, and of the measurement of the isotope effects, have been described in a recent review.61 Nucleoside hydrolases catalyze the hydrolysis of inosine. Comparisons of the experimental isotope effects with calculated KIEs for various transition state models showed that the data were most consistent with a loose, oxocarbenium ion-like transition state with low bond order to the leaving group and nucleophile. The limiting mechanisms of a purely dissociative process in which the leaving group departs to form an oxocarbenium ion before the nucleophile adds, or a tight SN2-like transition state, were ruled out.62 The most surprising result was the observation of a 5% (1.05) T(V/K) at H50 , four bonds away from the reaction center. This was attributed to a transition state interaction in which the 50 -hydroxyl is rotated to bring it near the ring oxygen, which participates in oxocarbenium ion formation. The catalytic importance of this interaction is demonstrated by the inactivity of the 50 -deoxy substrate. The presence of such an interaction was confirmed by a subsequent x-ray structure of the enzyme with a bound transition-state analog.63 A related enzyme is purine nucleoside phosphorylase (PNP), which catalyzes the phosphorolysis of inosine. PNP has been shown to be important in T-cell proliferation; since undesirable activation and proliferation of T-cells are associated with a number of human diseases including T-cell leukemias and transplant tissue rejection, its inhibition has been a target for drug design. Measurements of the a-DV as a function of pH suggested a mechanism involving a change in rate-limiting step from one where C –N bond fission is not rate-limiting at pH 7.3 (where D V ¼ 1.01), to a step at higher and lower pH with a transition state exhibiting significant oxocarbenium ion character.64 Subsequently, the transition state for the reaction was probed in more detail using a large set of KIEs (Figure 39.8), and a transition state geometry was obtained by a comparison of the experimental KIEs with those calculated for systematically varied transition state geometries using bond order-vibrational analysis, employing the BEBOVIB-IV program.65 A recent review describes in detail how such calculations are performed.66 This analysis led to the design of several transition-state analogues, among them Immucillin-H67(Figure 39.8). This and related compounds are very potent inhibitors of PNP, and Immucillin-H is currently in clinical trials for the treatment of T-cell leukemia. For the PNP reaction, oxocarbenium ion character was a key transition state feature identified by the KIEs. The positive charge of this species was incorporated in Immucillin by replacing the 40 -oxygen of inosine by an imino group, which has an estimated pKa of 6.5. Carbon replaced the nitrogen atom in the scissile C – N-9 bond, which imparts chemical stability, and increases the pKa of the nitrogen at N-7. As the loose transition state implied by the KIEs is reached, the pKa of this nitrogen atom in the substrate will increase from its value of , 2 in the substrate, to a level such that it is proposed to be protonated (or involved in a strong hydrogen bond). The N-7 position of Immucillin-H has a pKa of . 8.5, resembling the elevated pKa of the corresponding nitrogen in the transition state of the reaction with inosine.

966

Isotope Effects in Chemistry and Biology O N HO

3

15

H O 14 3 H C

3

H

N

NH N

HO

OH N N

H N

3

OH

HO

H N

Inosine

H HO

OH

Immucillin-H

FIGURE 39.8 At left, inosine, showing the positions at which kinetic isotope effects have been measured in reactions of N-ribosyl hydrolases and transferases. The 15N and 14C KIEs measure the degree of bond fission. The tritium effect at H1 is sensitive to rehybridization at C1; those at H2 and H4 give information about geometry changes at these carbon atoms, respectively. The tritium effect at H5 revealed distortion at C5 due to interaction between the C5 hydroxyl and the ring oxygen atom. At right, Immucillin-H, designed as a transition state analog from kinetic isotope effect studies of the purine nucleoside phosphorylase-catalyzed reaction.

The same KIE measurement techniques used for the studies described above have also been used to measure EIEs on binding of glucose to hexokinase; these results are described in detail in Chapter 42.

D. PHOSPHORYL T RANSFER Measurements of 18O isotope effects on reactions of phosphate esters have been useful in examining the mechanisms of phosphoryl transfer reactions. Figure 39.9 shows the position at which isotope effects have been measured in such reactions when the leaving group is p-nitrophenol. Phosphate esters may be monoesters, diesters, or triesters; isotope effects for the reactions of all three types of ester have been measured, with a variety of leaving groups. The primary isotope effect in the bridging position, 18kbridge together with that in the nonbridging atoms, 18knonbridge, give information regarding the extent of bond fission and the hybridization change at phosphorus, respectively. The secondary 18knonbridge reveals whether the phosphoryl group resembles metaphosphate in a loose transition state, or if it has a phosphorane-like structure in an associative mechanism. As is the case for a-hydrogen isotope effects bonded to a carbon atom undergoing a hybridization change, 18knonbridge will be affected by bond order and bending and torsional vibrational modes. In an associative mechanism, the transition state resembles a pentacoordinate phosphorane with five single bonds, and the net bond order to the P – O nonbridge oxygens decreases. In a dissociative mechanism, the phosphoryl group resembles metaphosphate in the transition state, and the bond order to these oxygen atoms increases. The changes in bond order of the nonbridge oxygen atoms in these mechanisms lead to the expectation that 18knonbridge will be inverse for dissociative mechanisms, and normal for associative mechanisms. The contributions from changes in bending

a

−O

Oa

− P Oa

Ob

cNO

2

(a) The nonbridge phosphoryl oxygen atoms (18knonbridge) (b) The bridge oxygen atom, the position of bond cleavage (18kbridge) (c) The nitrogen atom in the leaving group (15k).

FIGURE 39.9 The designations of nonbridging and the bridging oxygen atoms of a phosphate monoester. A diester has two ester groups and two nonbridging oxygens, a triester has only one nonbridging oxygen.

Secondary Isotope Effects

967

modes should be in the opposite direction. The variable multiple bond character of P– O bonds in phosphoryl groups gives these bonds greater possibilities to undergo bond order (stretching force constant) changes than a-C –H bonds. For reactions of dianions of monoesters, where other data support a loose, metaphosphate-like transition state, 18knonbridge, is small and inverse, where the opposite contributions to the isotope effect nearly cancel.68 This isotope effect is normal for reactions of phosphodiesters and triesters, where evidently bond order changes are the dominant contributors to 18knonbridge, reaching a maximum of over 3% (1.033) for diethylphosphorylcholine where a tight, phosphorane-like transition state is implicated.69 This follows the trend predicted from calculations that predict normal 18knonbridge isotope effects for associative transition states and inverse values for dissociative ones.70 More recent ab initio calculations reaffirm these earlier BEBOVIB-IV predictions. Both bridging and nonbridging 18(V/K ) isotope effects have been measured for a number of enzymatic phosphoryl transfer reactions. The KIEs for the phosphotriesterase-catalyzed reaction with two triesters, diethyl p-nitrophenyl phosphate, and diethyl 4-carbamoylphenyl phosphate, showed that the chemical step is not rate-limiting for V/K with the first substrate, but is so with the second one. Significant primary and secondary effects, 18(V/K )bridge ¼ 1.036 and 18 (V/K )nonbridge ¼ 1.018, indicate a transition state with a large degree of P – O(R) bond fission as well as associative character.71 Isotope effects in the analogous positions were measured for the ribonuclease A-catalyzed reaction with the diester substrate uridine-30 -m-nitrobenzylphosphate. This leaving group has a pKa of , 14.9, close to that of the natural substrate. The significant primary isotope effect, 18 (V/K )bridge ¼ 1.036, indicates considerable bond fission in the transition state. The very small 18 (V/K )nonbridge of 1.005 was interpreted as most consistent with a concerted mechanism, rather than one with a phosphorane intermediate or with a mechanism involving proton transfers to or from the phosphoryl group.72 A number of phosphatases have been probed using the substrate p-nitrophenyl phosphate. The protein– tyrosine phosphatases (PTPases) have a conserved Cys nucleophile and an Asp general acid that protonates the leaving group. The secondary 15N isotope effect in the nitro group is useful in these reactions because, as previously mentioned in Section II.B, it is sensitive to charge delocalization. Since the KIEs were measured using the competitive method they are on (V/K ), for the phosphoryl transfer from substrate to the enzymatic nucleophile. The very small 18(V/K )nonbridge results from the PTPase-catalyzed reactions with p-NPP (YOP from Yersinia, PTP1 from mouse, human VHR, and Stp1 from yeast) are indicative of a phosphoryl transfer transition state that is loose, with minimal nucleophilic participation and in which the phosphoryl group resembles metaphosphate. The primary 18(V/K )bridge and secondary 15(V/K ) in the leaving group are consistent with extensive P –O(R) bond cleavage to the leaving group and full neutralization by protonation in the transition state. For all four enzymes, when the general acid Asp is mutated to Asn, the values for 18(V/K )nonbridge become normal (ranging from 1.0019 to 1.0024), revealing that the transition state assumes a somewhat larger degree of nucleophilic participation. The magnitudes of 18(V/K )bridge and 15(V/K ) increase by an amount very close to the corresponding equilibrium isotope effects for the deprotonation of p-nitrophenol, indicating extensive bond cleavage to the leaving group, which now is unprotonated and bears essentially a full negative charge in the transition state.68 Particularly telling in this last regard is 15(V/K ), which goes from near unity in the reactions of the wild type enzymes, to values that range from 1.0024 to 1.0030 in the Asp to Asn mutants, depending on the specific PTPase. These values all indicate the presence of substantial negative charge on the leaving group. A loose, metaphosphate-like transition state for the reaction of alkaline phosphatase with glucose-6-phosphate is implicated by the small, inverse magnitudes of 18(V/K ) (0.9982 at pH 6).73 The same conclusion is drawn from the secondary 18(V/K ) (0.9987 at pH 8.2) in the g-position of ATP, for the phosphoryl transfer from ATP catalyzed by hexokinase with glucose as the substrate. With the slow substrate 1,5-anhydro-D -glucitol, 18(V/K ) ¼ 0.9976 at the same pH.74

968

Isotope Effects in Chemistry and Biology

The primary and secondary 18(V/K ) effects have been measured for the reaction of kanamycin nucleotidyltransferase-I from Staphylococcus aureus, which catalyzes the transfer of the AMP moiety from ATP to the 40 hydroxyl group of kanamycin A. To observe the intrinsic isotope effect in the reaction, the substrate analogue m-nitrobenzyl triphosphate was used, which has a kcat/Km two orders of magnitude lower than the kcat/Km of the KNTase reaction with ATP. The primary 18O isotope effects at pH 5.7 (close to the optimum pH) and at pH 7.7 (away from the optimum pH) are 1.016 ^ 0.003 and 1.014 ^ 0.002, respectively. The secondary 18O isotope effects in the nonbridging oxygen atoms are 1.0033 ^ 0.0004 and 1.0024 ^ 0.0002 at pH 5.7 and 7.7. These isotope effects are consistent with a concerted reaction with a slightly associative transitionstate structure.75

E. METHYL T RANSFER The a-28 Dk has been used to study the transition state of methyl transfer reactions. Recall that a-28 KIEs for SN1 reactions are normal due to loosening of bending modes, while those of SN2 are small or inverse. For the catechol O-methyltransferase-catalyzed methylation of 3,4-dihydroxyacetophenone by S-adenosyl methionine, a significant inverse a-28 D(V/K) ¼ 0.85 ^ 0.05 (for three deuteriums) was measured.76 In contrast, nonenzymatic model reactions (Figure 39.10) exhibited normal Dk values as high as 1.17 for two deuterium atoms, which yields an estimated value of 1.27 for three deuterium atoms.77 These results were interpreted to imply compression along the reaction coordinate of the enzymatic reaction, where the inverse a-28 KIEs result from restricted out-ofplane bending vibrations in a transition state in which the methyl group is sandwiched tightly between the nucleophile and leaving group.76 – 78 In the years since these data were reported, computations have been presented in support of a variety of interpretations. Some computational studies challenged the original interpretation, and concluded that inverse isotope effects in methyl transfers do not implicate a tight (or compressed) transition state; on the contrary, a more inverse a-Dk was asserted to reflect a looser transition state geometry.79,80 However, a later computational reappraisal concluded that inverse values for a-Dk in methyl transfers do result from compressed transition states,81 consistent with the original hypothesis and with other computational work discussed in Section II.A, addressing a-28 contributions to KIEs in SN2 reactions.9 Other calculations of the expected a-Dk and 13k for a model reaction, in which S-adenosyl methionine was modeled by trimethylsulfonium ion, produced values in close agreement with the experimental values for the enzymatic reaction, suggesting that the enzymatic and nonenzymatic reactions have similar transition states.82 In yet another computational study, the intramolecular SN2 methyl transfer was modeled between two oxygen atoms confined within a rigid system.83 Methyl transfer by retention of configuration (same side attack) was predicted when the oxygens are separated by three or four bonds, accompanied by a normal (1.21 to 1.34) a-Dk for three deuterium atoms. When six or more bonds separate the

SCH3 −H HO

CL2 S

CH3

+ O CL2

NO2

NO2

L = H or D

FIGURE 39.10 The Dk values, for two deuterium atoms, for the ring closure reaction are 1.17 for water catalysis, 1.10 for carbonate buffer catalysis, and 0.998 for hydroxide catalysis.

Secondary Isotope Effects

969

oxygens, the reaction proceeds by inversion and is accompanied by an inverse (0.6 to 0.81) a-Dk. Both mechanisms are possible when five bonds separate the oxygen atoms. These authors suggested that the difference in the retention and inversion isotope effects is large enough to constitute an alternative to the chiral methyl group for the determination of the stereochemical course of methyl transfer. If this prediction is correct, it requires that the model system originally studied77 by Mihel et al. (Figure 39.10) must proceed by same side attack instead of back side displacement. It remains to be demonstrated whether this is indeed the case.

F. HYDRIDE T RANSFER Dehydrogenase reactions involving the conversion of the cofactor NADþ to NADH have been probed using two secondary isotope effects, namely, those arising from 15N substitution and from substitution of the 4-H with deuterium or tritium.84 In this reaction, C-4 of the nicotinamide ring (the labeled position) changes its hybridization from sp2 to sp3. The 28-DK isotope effect measured for this equilibrium is inverse, 0.89.85 A secondary 15K isotope effect attributed to the pyramidalization of the nitrogen atom has also been reported. A value of 1.044 in the direction from NADþ to NADH86 in alcohol dehydrogenase was the first such isotope effect in this system reported, but a subsequent measurement for this EIE using lactate dehydrogenase obtained a lower value of 1.010.87 The latter, smaller value is also supported by the 15EIE of 1.004 measured for the equilibrium addition of cyanide addition to NADþ.87 One might reasonably expect to observe experimental kinetic secondary effects at H-4 to lie between 0.89 and unity; a value close to unity would indicate an early, NADþ like transition state, while inverse values would reflect the progress of hydride transfer to C-4 in the transition state. However, following the seminal report by Kurz88 there have been several reports of normal secondary kinetic isotope effects that are outside these limits (see Ref. 89 and references therein). Such larger than expected isotope effects have been attributed to a coupling of the motion of the secondary hydrogen to that of the hydride that is transferred. Further, Huskey and Schowen,90 have modeled these “anomalous” secondary kinetic isotope effects, and concluded that the data require a significant amount of tunneling of the primary hydrogen, accompanied by a coupling of the motions of the primary and secondary hydrogens. This same phenomenon was recently observed in the reaction of NAD-malic enzyme from Ascaris suum, which catalyzes the metal ion-dependent oxidative decarboxylation of L -malate to give pyruvate and CO2.89 a-Secondary tritium kinetic isotope effects were measured with NADþ or APADþ and L -malate-2-H(D) and several different divalent metal ions. The values for a-28-T(V/K ) are slightly higher than unity with NADþ and L -malate as substrates, much larger than the expected inverse isotope effect for a hybridization change from sp2 to sp3. The a-28-T(V/K ) isotope effects are near unity with L -malate-2-D as the substrate. The data were taken as an indication of quantum mechanical tunneling and coupled motion in the malic enzyme reaction when NADþ and malate are used as substrates. With APADþ(3-acetylpyridine adenine dinucleotide) in place of NADþ and Mn2þ as the Lewis acid, inverse values for a-28 T(V/K ) are observed, and a Swain-Schaad exponent of 3.3 is estimated, consistent with rate-limiting hydride transfer and no quantum mechanical tunneling or coupled motion.89

G. PEPTIDYL P ROLYL C IS – T RANS I SOMERASE The cis – trans isomerization about C – N bonds of Xaa-Pro (Figure 39.11) is of interest biochemically and mechanistically. Two such peptidyl prolyl cis – trans iosomerases (PPIs) that have been examined mechanistically are the cyclosporin A binding protein cyclophilin (CyP), and the FK-506 binding protein (FKBP). The isomerization process catalyzed by these enzymes requires rotation around a bond that possesses partial double bond character due to amide

970

Isotope Effects in Chemistry and Biology R C O R' C O

N

R' C

N

O C O R

FIGURE 39.11 The reaction catalyzed by Peptidyl Prolyl cis –trans Isomerase (PPI).

resonance. This could be facilitated in an enzymatic reaction by nucleophilic attack to form a tetrahedral intermediate, with accompanying loss of amide resonance, followed by rotation about the C – N bond, and, finally, collapse of the intermediate. If such a mechanism were involved and rate-limiting, an inverse b-D(V/K ) should be observed, arising from loss of hyperconjugation and the resultant tightening of the bonds to these hydrogens as the carbonyl carbon is converted from sp2 to sp3. However, the observation of a normal D(V/K ) of 1.13 at this position in the PPI-catalyzed isomerization implies the presence of increased hyperconjugation in the transition state.91 – 93 This can be explained if the transition state is characterized by partial rotation about the amide bond. Once the lone pair on nitrogen is twisted out of planarity with the carbonyl group pi bond, resonance delocalization is lost, and the carbonyl bond resembles that of a ketone. Hyperconjugation in such a transition state will be greater than that in the resonance delocalized amide ground state. The uncatalyzed isomerization also exhibits a similar b-secondary deuterium KIE, implying that both reactions have similar transition states. The PPI-catalyzed isomerization exhibits a small enthalpy of activation of 4 kcal mol21, in contrast to the much larger value of 20 kcal mol21 for the uncatalyzed reaction.92 This and other data further support the notion that the rate-limiting step is a physical process, rather than nucleophilic attack at the carbonyl group of the amide. To investigate dependence on enzyme nucleophiles and proton donors in the isomerization catalyzed by FKBP, eight potential catalytic residues were replaced with alanine by site-directed mutagenesis. Each FKBP variant efficiently catalyzed the PPIase reaction, further supporting a chemically unassisted mechanism involving catalysis by distortion, with rate enhancement due in part to desolvation of the peptide bond at the active site.

REFERENCES 1 Collins, C. J. and Bowman, N. S., Isotope Effects in Chemical Reactions, Van Nostrand, New York, 1970. 2 Halevi, E. A., Secondary isotope effects, Prog. Phys. Org. Chem., 1, 109– 221, 1963. 3 Sunko, D. E. and Hehre, W. J., Secondary deuterium isotope effects on reactions proceeding through carbocations, Prog. Phys. Org. Chem., 14, 205– 246, 1983. 4 Hartshorn, S. R. and Shiner, V. J., Calculation of H/D carbon-12/carbon-13, and carbon-12/carbon-14 fractionation factors from valence force fields derived for a series of simple organic molecules, J. Am. Chem. Soc., 94, 9002– 9112, 1972. 5 Melander, L. and Saunders, W. H., Secondary hydrogen isotope effects, In Reaction Rates of Isotopic Molecules, Robert E. Krieger, Malabar, FL, pp. 170– 201, 1987. 6 Hengge, A. C. and Cleland, W. W., Direct measurement of transition-state bond cleavage in hydrolysis of phosphate esters of p-nitrophenol, J. Am. Chem. Soc., 112, 7421– 7422, 1990. 7 Melander, L. and Saunders, W. H., The Range of Normal Isotope Effects, Robert E. Krieger, Malabar, FL, 130– 139, 1987.

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8 Northrop, D. B., Determining the absolute magnitude of hydrogen isotope effects, In Isotope Effects on Enzyme-Catalyzed Reactions, Cleland, W. W., O’Leary, M. H., and Northrop, D. B., Eds., University Park Press, Baltimore, MD, pp. 122– 152, 1977. 9 Poirier, R. A., Wang, Y., and Westaway, K. C., A theoretical study of the relationship between secondary a-deuterium kinetic isotope effects and the structure of SN2 transition states, J. Am. Chem. Soc., 116, 2526 –2533, 1994. 10 Kirsch, J. F., Secondary kinetic isotope effects, In Isotope Effects on Enzyme-Catalyzed Reactions, Cleland, W. W., O’Leary, M. H., and Northrop, D. B., Eds., University Park Press, Baltimore, MD, pp. 100– 121, 1977. 11 Stewart, R., Gatzke, A. L., Mocek, M., and Yates, K., Deuterium isotope effects in organic cations, Chem. Ind. (London), 331–332, 1959. 12 do Amaral, L., Bull, H. G., and Cordes, E. H., Secondary deuterium isotope effects for carbonyl addition reactions, J. Am. Chem. Soc., 94, 7579– 7580, 1979. 13 Westaway, K. C., Secondary hydrogen – deuterium isotope effects and transition state structure in SN2 processes, In Isotopes in Organic Chemistry, Vol. 7, Buncel, E. and Lee, C. C., Eds., Elsevier, 275– 392, New York, 1987. 14 Lewis, E. S. and Boozer, C. E., Isotope effects in the ionization of alkyl chlorosulfites, J. Am. Chem. Soc., 74, 6306– 6307, 1952. 15 Shiner, V. J., Solvolysis rates of some deuterated tertiary amyl chlorides, J. Am. Chem. Soc., 75, 2925– 2929, 1953. 16 Streitweiser, A. Jr., Jagow, R. H., and Suzuki, S., Kinetic isotope effects in the acetolyses of deuterated cyclopentyl tosylates, J. Am. Chem. Soc., 80, 2326– 2332, 1958. 17 Shiner, V. J. and Humphrey, J. S., The effects of deuterium substitution on the rates of organic reactions. IX. Bridgehead b-deuterium in a carbonium ion solvolysis, J. Am. Chem. Soc., 85, 2416– 2419, 1963. 18 Sunko, D. E., Szele, I., and Hehre, W. J., Hyperconjugation and the angular dependence of b-deuterium isotope effects, J. Am. Chem. Soc., 99, 5000– 5005, 1977. 19 Krueger, P. J., Jan, J., and Wieser, H., Infrared spectra: intramolecular trans lone pair interaction with a-CH bonds and the stability of conformers in alcohols and thiols, J. Mol. Struct., 5, 375– 387, 1970. 20 Murphy, W. F., McKean, D. C., Coats, A. M., Kindness, A., and Wilkie, N., Raman and infrared intensities in trimethylamine species, polarizability and dipole derivatives and effective charges, J. Raman Spectrom., 108, 763– 770, 1995. 21 Anet, F. A. L. and Kopelevitch, M., Deuterium isotope and anomeric effects in the conformational equilibria of molecules containing CHD – O groups, J. Am. Chem. Soc., 108, 2109– 2110, 1986. 22 Forsyth, D. A. and Hanley, J. A., Conformationally dependent intrinsic and equilibrium isotope effects in N-methylpiperidine, J. Am. Chem. Soc., 109, 7930– 7932, 1987. 23 Gawlita, E., Lantz, M., Paneth, P., Bell, A., Tonge, P., and Anderson, V. E., H-bonding in alcohols is reflected in the a C– H bond strength: variation of C– D vibrational frequency and fractionation factor, J. Am. Chem. Soc., 122, 11660– 11669, 2000. 24 Maiti, N. C., Carey, P. R., and Anderson, V. E., Correlation of an alcohol’s a C– D stretch with hydrogen bond strength in complexes with amines, J. Phys. Chem. A, 107, 9910– 9917, 2003. 25 Saunders, W. H., Mechanisms of elimination reactions. XXIV. Model calculations of isotope effects in elimination reactions. Effects of the leaving group and of transition-state geometry, Chem. Scr., 8, 27 – 36, 1975. 26 Anderson, V. E., Isotope effects on enzyme-catalyzed b-eliminations, In Enzyme Mechanism from Isotope Effects, Cook, P. F., Ed., CRC Press, Boca Raton, FL, pp. 389– 417, 1991. 27 Hwang, C. C., Woehl, E. U., Minter, D. E., Dunn, M. F., and Cook, P. F., Kinetic isotope effects as a probe of the beta-elimination reaction catalyzed by O-acetylserine sulfhydrylase, Biochemistry, 35, 6358– 6365, 1996. 28 Perrin, C. L. and Engler, R. E., Secondary H/D kinetic isotope effect for dissociation of aqueous ammonium ion, J. Phys. Chem., 95, 8431– 8433, 1991. 29 Bary, Y., Giboa, H., and Halevi, E. A., Secondary hydrogen isotope effects. Part 5. Acid and base strengths: corrigendum and addendum, J. Chem. Soc. Perkin II, 938– 942, 1979. 30 Perrin, C. L. and Ohta, B. K., Stereochemistry of secondary b-deuterium isotope effects on amine basicity, ACS National Meeting, San Francisco. American Chemical Society, 2000.

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55 Stein, R. L., Fujihara, H., Quinn, D. M. et al. Transition-state structural features for anilide hydrolysis from b-deuterium isotope effects, J. Am. Chem. Soc., 106, 1457– 1461, 1984. 56 Hengge, A. C. and Stein, R. L., Role of protein conformational mobility in enzyme catalysis — acylation of a-chymotrypsin by specific peptide substrates, Biochemistry, 43, 742– 747, 2004. 57 Lewis, B. E. and Schramm, V. L., Conformational equilibrium isotope effects in glucose by 13C NMR spectroscopy and computational studies, J. Am. Chem. Soc., 123, 1327– 1336, 2001. 58 Bennet, A. J. and Sinnott, M. L., Complete kinetic isotope effect description of transition states for acid-catalyzed hydrolyses of methyl a- and b-glucopyranosides, J. Am. Chem. Soc., 108, 7287– 7294, 1986. 59 Indurugalla, D. and Bennet, A. J., A kinetic isotope effect study on the hydrolysis reactions of methyl xylopyranosides and methyl 5-thioxylopyranosides: oxygen versus sulfur stabilization of carbenium ions, J. Am. Chem. Soc., 123, 10889– 10898, 2001. 60 Matsui, H., Blanchard, J. S., Brewer, C. F., and Hehre, E. J., Alpha-secondary tritium kinetic isotope effects for the hydrolysis of alpha-D -glucopyranosyl fluoride by exo-alpha-glucanases, J. Biol. Chem., 264, 8714– 8716, 1989. 61 Schramm, V. L., Enzymatic transition-state analysis and transition-state analogs, Methods Enzymol., 308, 301– 355, 1999. 62 Horenstein, B. A., Parkin, D. W., Estupinan, B., and Schramm, V. L., Transition-state analysis of nucleoside hydrolase from Crithidia fasciculata, Biochemistry, 30, 10788– 10795, 1991. 63 Degano, M., Almo, S. C., Sacchettini, J. C., and Schramm, V. L., Trypanosomal nucleoside hydrolase. A novel mechanism from the structure with a transition-state inhibitor, Biochemistry, 37, 6277– 6285, 1998. 64 Stein, R. L. and Cordes, E. H., Kinetic alpha-deuterium isotope effects for Escherichia coli purine nucleoside phosphorylase-catalyzed phosphorolysis of adenosine and inosine, J. Biol. Chem., 256, 767– 772, 1981. 65 Kline, P. C. and Schramm, V. L., Presteady-state transition-state analysis of the hydrolytic reaction catalyzed by purine nucleoside phosphorylase, Biochemistry, 34, 1153– 1162, 1995. 66 Berti, P. J., Determining transition states from kinetic isotope effects, Methods Enzymol., 308, 355– 397, 1999. 67 Miles, R. W., Tyler, P. C., Furneaux, R. H., Bagdassarian, C. K., and Schramm, V. L., One-third-the-sites transition-state inhibitors for purine nucleoside phosphorylase, Biochemistry, 37, 8615– 8621, 1998. 68 Hengge, A. C., Isotope effects in the study of phosphoryl and sulfuryl transfer reactions, Acc. Chem. Res., 35, 105– 112, 2002. 69 Anderson, M. A., Shim, H., Raushel, F. M., and Cleland, W. W., Hydrolysis of phosphotriesters: determination of transition states in parallel reactions by heavy-atom isotope effects, J. Am. Chem. Soc., 123, 9246 –9253, 2001. 70 Weiss, P. M., Knight, W. B., and Cleland, W. W., Secondary 18O isotope effects on the hydrolysis of glucose-6-phosphate, J. Am. Chem. Soc., 108, 2761– 2762, 1986. 71 Caldwell, S. R., Raushel, F. M., Weiss, P. M., and Cleland, W. W., Transition-state structures for enzymatic and alkaline phosphotriester hydrolysis, Biochemistry, 30, 7444– 7450, 1991. 72 Sowa, G. A., Hengge, A. C., and Cleland, W. W., 18O isotope effects support a concerted mechanism for ribonuclease A, J. Am. Chem. Soc., 119, 2319– 2320, 1997. 73 Weiss, P. M. and Cleland, W. W., Alkaline phosphatase catalyzes the hydrolysis of glucose-6phosphate via a dissociative mechanism, J. Am. Chem. Soc., 111, 1928– 1929, 1989. 74 Jones, J. P., Weiss, P. M., and Cleland, W. W., Secondary 18O isotope effects for hexokinase-catalyzed phosphoryl transfer from ATP, Biochemistry, 30, 3634– 3639, 1991. 75 Gerratana, B., Frey, P. A., and Cleland, W. W., Characterization of the transition-state structure of the reaction of kanamycin nucleotidyltransferase by heavy-atom kinetic isotope effects, Biochemistry, 40, 2972– 2977, 2001. 76 Hegazi, M. F., Borchardt, R. T., and Schowen, R. L., a-Deuterium and carbon-13 isotope effects for methyl transfer catalyzed by catechol O-methyltransferase — SN2-like transition state, J. Am. Chem. Soc., 101, 4359 –4365, 1979. 77 Mihel, I., Knipe, J. O., Coward, J. K., and Schowen, R. L., a-Deuterium isotope effects and transitionstate structure in an intramolecular model system for methyl-transfer enzymes, J. Am. Chem. Soc., 101, 4349– 4351, 1979.

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78 Gray, C. H., Coward, J. K., Schowen, K. B., and Schowen, R. L., a-Deuterium and carbon-13 isotope effects for a simple, intermolecular sulfur-to-oxygen methyl-transfer reaction. Transition-state structures and isotope effects in transmethylation and transalkylation, J. Am. Chem. Soc., 101, 4351– 4358, 1979. 79 Boyd, R. J., Kim, C.-K., Shi, Z., Weinberg, N., and Wolfe, S., Secondary H/D isotope effects and transition state looseness in nonidentity methyl transfer reactions. Implications for the concept of enzymatic catalysis via transition state compression, J. Am. Chem. Soc., 115, 10147– 10152, 1993. 80 Wolfe, S. and Kim, C.-K., Secondary H/D effects in methyl-transfer reactions decrease with increasing looseness of the transition structure, J. Am. Chem. Soc., 113, 8056– 8061, 1991. 81 Moliner, V. and Williams, I. H., Influence of compression upon kinetic isotope effects for SN2 methyl transfer. A computational reappraisal, J. Am. Chem. Soc., 122, 10895– 10902, 2000. 82 Zheng, Y.-J. and Bruice, T. C., A theoretical examination of the factors controlling the catalytic efficiency of a transmethylation enzyme: catechol O-methyltransferase, J. Am. Chem. Soc., 119, 8137– 8145, 1997. 83 Wolfe, S., Kim, C.-K., Yang, K., Weinberg, N., and Shi, Z., Transverse compression and the secondary H/D isotope effects in intramolecular SN2 methyl-transfer reactions, Can. J. Chem., 76, 359– 370, 1998. 84 Blanchard, J. S. and Wong, K. K., Isotope effects on enzyme-catalyzed redox reactions, In Enzyme Mechanism from Isotope Effects, Cook, P. F., Ed., CRC Press, Boca Raton, FL, pp. 342– 348, 1991. 85 Cook, P. F., Blanchard, J. S., and Cleland, W. W., Primary and secondary deuterium isotope effects on equilibrium constants for enzyme-catalyzed reactions, Biochemistry, 19, 4853– 4858, 1980. 86 Cook, P. F., Oppenheimer, N. J., and Cleland, W. W., Secondary deuterium and nitrogen-15 isotope effects in enzyme-catalyzed reactions. Chemical mechanism of liver alcohol dehydrogenase, Biochemistry, 20, 1817– 1825, 1981. 87 Burgner, J. W. 2nd, Oppenheimer, N. J., and Ray, W. J. Jr., Remote nitrogen-15 isotope effects on addition of cyanide to NAD, Biochemistry, 26, 91 – 96, 1987. 88 Kurz, L. C. and Frieden, C., Anomalous equilibrium and kinetic a-deuterium secondary isotope effects accompanying hydride transfer from reduced nicotinamide adenine dinucleotide, J. Am. Chem. Soc., 102, 4198– 4203, 1980. 89 Karsten, W. E., Hwang, C.-C., and Cook, P. F., a-Secondary tritium kinetic isotope effects indicate hydrogen tunneling and coupled motion occur in the oxidation of L -malate by NAD-malic enzyme, Biochemistry, 38, 4398– 4402, 1999. 90 Huskey, W. P. and Schowen, R. L., Reaction-coordinate tunneling in hydride-transfer reactions, J. Am. Chem. Soc., 105, 5704 –5706, 1983. 91 Harrison, R. K. and Stein, R. L., Mechanistic studies of enzymic and nonenzymic prolyl cis –trans isomerization, J. Am. Chem. Soc., 114, 3464– 3471, 1992. 92 Harrison, R. K. and Stein, R. L., Mechanistic studies of peptidyl prolyl cis – trans isomerase: evidence for catalysis by distortion, Biochemistry, 29, 1684– 1689, 1990. 93 Harrison, R. K., Caldwell, C. G., Rosegay, A., Melillo, D., and Stein, R. L., Confirmation of the secondary deuterium isotope effect for the peptidyl prolyl cis – trans isomerase activity of cyclophilin by a competitive, double-label technique, J. Am. Chem. Soc., 112, 7063– 7064, 1990. 94 Klein, H. S. and Streitweiser, A. Jr., The inductive effect of deuterium, Chem. Ind. (London), 180, 1961. 95 Halevi, E. A. and Nussim, M., Electron release from carbon– hydrogen and carbon – deuterium bonds, Bull. Res. Council Israel, 5A, 263– 264, 1958. 96 Barnes, D. J. and Scott, J. M. W., Isotope effects on chemical equilibriums. Part 1. Some secondary isotope effects of the second kind, Can. J. Chem., 51, 411–421, 1973. 97 Streitweiser, A. Jr. and Van Sickle, D. E., Acidity of hydrocarbons. IV. Secondary deuterium isotope effects in exchange reactions of toluene and ethylbenzene with lithium cyclohexylamide, J. Am. Chem. Soc., 84, 254– 258, 1962. 98 Ropp, G. A., Isotopically labeled HCO2H and its derivatives. VI. Estimation of the relative ionization constants of HCO2H and formic-D acid at 24 ^ 0.5%, J. Am. Chem. Soc., 82, 4252– 4254, 1960. 99 Bell, R. P. and Jensen, M. B., Dissociation constant of formic-D acid, DCO2H, in aqueous solution, Proc. Chem. Soc., 307– 308, 1960.

40

Isotope Effects in the Characterization of Low Barrier Hydrogen Bonds Perry A. Frey

CONTENTS I. II.

Introduction ...................................................................................................................... 975 Zero-Point Energy Effects and Hydrogen Bonds ............................................................ 977 A. Classification of Hydrogen Bond Types.................................................................. 977 B. NMR Chemical Shifts of Strong Hydrogen Bonds................................................. 978 C. Isotope Effects on Physical Parameters................................................................... 979 1. Isotope Effects on Vibrational Frequencies...................................................... 979 2. Isotope Effects on Chemical Shifts of LBHBs................................................. 980 3. D/H Fractionation Factors................................................................................. 981 D. Strengths of Hydrogen Bonds.................................................................................. 981 III. Low-Barrier Hydrogen Bonds in Enzymes ..................................................................... 982 A. Serine Proteases ....................................................................................................... 982 B. Cholinesterases......................................................................................................... 985 C. D5-3-Ketosteroid Isomerase ..................................................................................... 986 IV. Role of Low-Barrier Hydrogen Bonds in the Actions of Enzymes................................ 987 A. Serine Proteases and Esterases ................................................................................ 987 B. D5-3-Ketosteroid Isomerase ..................................................................................... 990 C. Other Enzymes ......................................................................................................... 990 Acknowledgments ........................................................................................................................ 990 References..................................................................................................................................... 990

I. INTRODUCTION In this article, the focus will be on the importance of zero-point vibrational energy effects in the characterization of low-barrier hydrogen bonds (LBHBs). The most widely studied zero pointphenomena are kinetic-isotope effects. The observation of a kinetic-isotope effect identifies a rate-limiting step in a complex reaction mechanism, and the magnitude of the effect includes information about the structure of the activated complex or transition state. The theory of KIEs is well developed, and the observation of experimental deviations from theory in deuterium or tritium KIEs can give clues to the possible involvement of quantum mechanical tunneling in hydrogen transfer mechanisms. The classical theory of KIEs is centered on the kinetic consequences of zeropoint vibrational energy differences between isotopes of an atom undergoing a chemical change. However, zero point effects also figure prominently in other phenomena, including the isotope effects on the physical properties of hydrogen bonds. 975

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Research on hydrogen bonding during the last 50 years has led to the accumulation of structural and spectroscopic data on exceptionally short hydrogen bonds in many small molecules.1,2 Short hydrogen bonds are defined structurally as follows: (1) the distance between the two heteroatoms (A· · ·B) is very short. (2) the proton lies inline with the heteroatoms. (3) the covalent bond linking hydrogen to one of the heteroatom is elongated, and the hydrogen bond between the proton and the distal heteroatom is short. Due to differences in van der Waals radii for heteroatom, the lengths RA· · ·B of short hydrogen bonds vary with the participating heteroatom. The usual standard for weak hydrogen bonding is a contact closer than the sum of van der Waals radii, which ˚ for hydrogen bonds involving O and N. For strong hydrogen correspond to RA· · ·B # 3.0 to 3.1 A bonding, a reasonable criterion is a contact shorter than twice the sum of covalent radii for the ˚ for O – H· · ·O; participating heteroatoms. By this standard, the values of RA· · ·B are # 2.6 A ˚ for N –H· · ·O; and # 2.80 for N –H· · ·N. # 2.70 A Elongation of the covalent bond in a strong hydrogen bond is central to its characterization and physicochemical properties. Crystallographic data on hydrogen bonds of the type O – H· · ·O reveal correlations among covalent bond lengths (RO – H), distances between heteroatom (RO· · ·O), and hydrogen bond lengths (RH· · ·O). The shorter the hydrogen bond, the closer the heteroatom and the longer the covalent bond. Plots of RO – H against RO· · ·O and of RO – H against RH· · ·O reveal this fact.3,4 One such plot is shown in Figure 40.1 for hydrogen bonds linking two oxygen atoms. Clearly, the shorter the hydrogen bonds the longer the covalent bond. The trade-off indicates increasing covalency for the hydrogen bond with lengthening, and weakening, of the covalent bond. This relationship proves to be the structural basis for other physical properties of hydrogen bonded systems that allow short hydrogen bonds in enzymes to be identified and characterized. Empirical evidence shows that short hydrogen bonds are found between heteroatom that have similar proton affinities and lie close together in the structure.1 The importance of similar proton affinities is intuitively satisfying and compatible with theory. The close proximity of heteroatom in a short hydrogen bond may arise from the strength of the hydrogen bond itself in a vacuum or vapor phase, or it may be brought about by structural constraints within the molecule, or by crystal packing interactions. In an enzyme, close proximity of heteroatoms with similar proton affinities can 1.20

O—H [Å]

1.15 1.10 1.05 1.00 0.95 0.90 1.10

1.20

1.30

1.40

1.50 1.60 1.70 H······O (Å)

1.80

1.90

2.00

2.10

FIGURE 40.1 Correlation of the lengths of the covalent and hydrogen bonds in hydrogen bonded oxygen systems. Shown here is a plot of the lengths of the covalent bonds (RO – H) and hydrogen bonds (RH· · ·O) in hydrogen bonds of the type O– H· · ·O in a large number of molecules. The plot clearly shows that the shorter the hydrogen bond the longer the covalent bond. The covalent bond is taken to be the shorter bond and the ˚ ), and this represents a hydrogen bond the longer. In the limit, the two bonds can be equal in length (,1.2 A symmetrical hydrogen bond. Reproduced from Figure 3.5 of Ref. 4 with permission.

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be brought about by the tertiary structure of the protein. Short hydrogen bonds are stronger, often much stronger, than conventional, weak hydrogen bonds.1

II. ZERO-POINT ENERGY EFFECTS AND HYDROGEN BONDS A. CLASSIFICATION OF H YDROGEN B OND T YPES Zero-point energy effects distinguish the three classes of hydrogen bonds that have been defined as weak, strong, and very strong.1 The drawing in Figure 40.2 illustrates how zero-point energy contributes to the distinction among the three classes. In the weak or conventional hydrogen bond depicted in Figure 40.2A, the electronegative heteroatom A is covalently bonded to the proton H in a polar bond in which the proton is the electropositive end. A weak electrostatic attraction with the electronegative heteroatom B attracts the proton. The two heteroatom exist in different potential energy wells connected through a high barrier by the weak hydrogen bond. The zero-point energies of hydrogen and deuterium lie near the bottom of well A, where the anharmonicity is low, and the two are bonded at about the same distance to heteroatom A. In the strong hydrogen bond in Figure 40.2B, the two heteroatoms lie closer together and the barrier between them is correspondingly lowered to that of the zero-point energy of hydrogen. Then the anharmonicities of hydrogen and deuterium are different, and the proton becomes partially covalently bonded to B as well as A; consequently, its bond to A is longer than a conventional covalent bond, and its hydrogen bond to B is shorter than a conventional hydrogen bond. This type of hydrogen bond may be categorized as a low-barrier hydrogen bond or LBHB.5 In this case, the zero-point difference will lead to isotope effects on several physicochemical parameters. In the very strong hydrogen bond depicted in Figure 40.2C, the heteroatoms are so close and their shared bonding of hydrogen and deuterium is such that the barrier is lowered well below the zero-point energies of both isotopes. In this case, the zero-point effects do not support the same isotope effects but allow crossover and modestly opposite isotope effects.1 In the limit, the barrier can, in theory, vanish in the rare case of

FIGURE 40.2 Zero-point energy relationships among three classes of hydrogen bonds. The potential energy wells for hydrogen-bonded heteroatoms and the zero-point energy levels for hydrogen and deuterium are qualitatively depicted for the three classes of hydrogen bonds defined in Ref. 1. The barrier in the weak hydrogen bond in panel A is lowered if the heteroatoms are merged, as in panels B and C, but only when the proton and heteroatoms are hydrogen bonded inline. In strong hydrogen bonds or LBHBs, panel B, the barrier is near the zero-point energy of hydrogen but above that of deuterium. This leads to isotope effects on such physicochemical parameters as H/D fractionation factor, vibrational frequency of A –H(D), and isotope shift in the 1H(2H) NMR signal. In very strong or single-well hydrogen bonds (SWHBs), panel C, the zero-point energies of hydrogen and deuterium both lie well above the barrier, and the hydron is shared almost equally between the heteroatoms. In the limit of a perfectly symmetrical hydrogen bond, the barrier in theory may be absent. In both LBHBs and SWHBs, there is significant covalency between the proton and both heteroatoms.

978

Isotope Effects in Chemistry and Biology

a perfectly symmetrical hydrogen bond. As shown in Figure 40.1 for hydrogen bonds of the type O – H· · ·O, the full range of bond lengths is found, and the distinction between the two bonds can become the relative lengths of partially covalent hydrogen bonds.

B. NMR C HEMICAL S HIFTS OF S TRONG H YDROGEN B ONDS Elongation of the covalent bond to a hydrogen-bonded proton deshields the proton and moves the NMR signal downfield. This effect arises from decreased shielding by the bonding electrons, which are more attracted to the electronegative heteroatom. The longer the covalent bond, the greater the deshielding effect and the further downfield the 1H NMR chemical shift. Weakly hydrogen-bonded protons display dH-values of 10 to12 ppm, significantly downfield from carbon-bonded protons. In a symmetrical hydrogen bond, a rare species, the proton is equally shared and its average position is equidistant from the heteroatom. The furthest downfield protons in hydrogen-bonded nitrogen and oxygen display chemical shifts of 20 to 22 ppm. A concept of the significance of these values can be acquired by considering the chemical shifts of 30 ppm for a free proton6 and 0 ppm for tetramethylsilane. Considering the protons in tetramethylsilane maximally shielded by bonding electrons, symmetrical hydrogen-bonded protons must be very dramatically deshielded. The unsymmetrical LBHBs between oxygen and nitrogen atoms display 1H NMR chemical shifts of 17 to 19 ppm, representing very significant elongation of the covalent bond and extensive deshielding. The relationship between decreasing heteroatom distances and elongation of covalent bonds to hydrogen on the one hand and proton NMR chemical shift on the other is very well established. The shorter the heteroatom separation, the further downfield the NMR chemical shift of the proton.7, 50 A typical plot of RO· · ·O against chemical shift for hydrogen bonds of the type O – H· · ·O is shown in Figure 40.3. The data show that the chemical shift approaches 22 Ppm with RO· · ·O decreasing to ˚ , the shortest hydrogen bonding distance so far observed in hydrogen bonds of this type. The 2.4 A points in Figure 40.4 fit the equation D ¼ 5:04 2 1:16 lnðdH Þ þ 0:0447ðdH Þ; where D refers to 2.80 2.76

O...H-O Distance

2.72 2.68 2.64 2.60

D=5.04 − 1.16In(δ) + 0.0447(δ)

2.56 2.52 2.48 2.44 2.40 10

12

14

16 Chemical Shift

18

20

FIGURE 40.3 Correlation of heteroatom distances with proton chemical shifts in hydrogen bonded oxygen systems. Shown is a plot of RO· · ·O (O· · ·H – O distance) in Angstrom units against 1H NMR chemical shift in ppm for crystalline amino acids. The distances in the crystals are taken from crystal structures and the chemical shifts from solid state NMR spectra.52 The line is calculated from parameters obtained after fitting to the equation shown. The data are from Ref. 52, and the figure is reprinted from Figure 40.3 of Ref. 53 with permission.

Isotope Effects in the Characterization of Low Barrier Hydrogen Bonds

979

0.4 0.2 0.0 −0.2 −0.4 −0.6 −0.8 −1.0 10

12

14

16

18

20

22

FIGURE 40.4 Trends in the deuterium isotope effect on the proton NMR chemical shift for hydrogen bonds. Values for deuterium isotope shift D[dD– dH] are plotted against the proton chemical shifts in the same molecules. The molecules incorporate intramolecular hydrogen bonds of the types O– H· · ·O and N– H· · ·N and include b-diketones, b-ketoesters, 2-hydroxybenzaldehydes, dicarboxylic acid monoanions, and proton sponges. The weak hydrogen bonds display dH values of 10 to 12 ppm, and the very strong hydrogen bonds display dH values of 20 to 22 ppm. This plot shows that the largest deuterium isotope effects on chemical shift are displayed by hydrogen bonds with dH values of 14 to 19 ppm. These may correspond to the class of strong hydrogen bonds or LBHBs defined in Figure 40.2B. Adapted from data presented in Ref. 1.

RO· · ·O, and this correlation allows a value for RO· · ·O to be calculated from a measured value of dH for a hydrogen bond. It is likely that shielding due to nonbonding electron pairs on oxygen would increase in even shorter hydrogen bonds, with consequent up field movement of the NMR signal. This effect would explain the chemical shift of 16 ppm for the shortest and strongest hydrogen bond ˚. known, that in hydrogen difluoride [F | | | |H| | | |F] − in which the heteroatom separation RF· · ·F is 2.26 A The relationship between chemical shift and hydrogen bonding holds in the absence of additional internal magnetic fields in the molecule, that is, in the absence of a paramagnetic ion or large ring current effects. Such internal magnetic fields can lead to deshielding irrespective of the strength of the hydrogen bond. These effects are absent in the molecules under discussion in this article. However, they are important subjects of study in the structural analysis by NMR of such metal complexes as iron sulfur clusters and of proteins incorporating them within their structures.

C. ISOTOPE E FFECTS ON P HYSICAL PARAMETERS Hydrogen isotope effects on three physicochemical parameters distinguish hydrogen bonds in the three classes. Weak hydrogen bonds display little or no isotope effects on the D/H fractionation factor (f), on the NMR chemical shift, e.g., D½dD – dH ; or on the vibrational frequencies, nAH =nAD : In contrast, owing to the zero-point effects illustrated in Figure 40.2, strong hydrogen bonds or LBHBs display characteristic isotope effects on these parameters. Very strong hydrogen bonds display very small, no isotope effects, or modest reverse effects. 1. Isotope Effects on Vibrational Frequencies The vibrational spectra of strong hydrogen bonds display complex behavior that distinguishes them from weak hydrogen bonds.1 The bands are broadened and red shifted. Interestingly, normal hydrogen bonds display a deuterium isotope effect of nAH =nAD ¼ 1:4 due to the mass difference between hydrogen and deuterium. The deuterium isotope effect is diminished or abolished (nAH =nAD ¼ 1:0) in strong or very strong hydrogen bonds.8 The variation of nAH =nAD with

980

Isotope Effects in Chemistry and Biology

hydrogen bond length follows expectations based on Figure 40.2, where the maximum perturbation on the isotope effect in hydrogen bonded oxygen is expected for LBHBs (Figure 40.2B). The ˚ or less than “normal” value of nAH =nAD ¼ 1:4 is observed whenever RO· · ·O is greater than 2.6 A ˚ . These conditions correspond to weak and very strong hydrogen bonds of oxygen and to the 2.45 A potential energy diagrams in Figure 40.2A and Figure 40.2C. However, nAH =nAD , 1:4 whenever ˚ .9 These values of RO· · ·O correspond to strong hydrogen bonds or LBHBs 2.5 # RO· · ·O , 2.6 A intermediate in length between weak and symmetrical or very strong hydrogen bonds, and to the potential energy diagram in Figure 40.2B. Thus, the zero-point effect in this case diminishes or abolishes the deuterium isotope effect on the vibrational frequency. In a hydrogen bond involving a carboxyl group, the carbonyl stretching frequency can give information about the hydrogen bond. The C – O bonding in a carboxylate or carboxylic acid group is slightly perturbed by a weak hydrogen bond. However, in a strong or very strong hydrogen bond, the electronic charge in a carboxylate group is significantly dispersed, forcing significant changes in the carbonyl stretching frequency. This phenomenon appears in the FTIR spectra of complexes between 1-methylimidazole and 2,2-dichloropropionic acid dissolved in aprotic solvents. The proton affinities of these molecules seem to be well matched in chloroform, and their 1:1 mixture behaves as the LBHB complex 1. O CH3CCl2C

C

O

H N

NCH3

1

It displays the downfield chemical shift of 17.5 ppm. Moreover, its FTIR spectrum displays a carbonyl stretching band at a frequency intermediate between those of 2,2-dichloropropionic acid and its tetrabutylammonium salt.10 Complexes of acids stronger than 2,2-dichloropropionic acid with 1-methylimidazole display typical carboxylate bands and complexes of weaker acids display typical carboxylic acid bands in the FTIR spectra. Confirmation of the LBHB is provided by the effects of isotopic substitution.11 Substitution of deuterium for hydrogen red shifts and weakens the carbonyl stretching band, and substitution of 18O in the carbonyl group induces an isotope shift on the carbonyl frequency. The 18O effect confirms the band assignment, and the deuterium isotope effect confirms the LBHB as of the type represented by the potential energy diagram in Figure 40.2B. 2. Isotope Effects on Chemical Shifts of LBHBs The NMR chemical shift values for hydrogen and deuterium are essentially the same for hydrons engaged in weak and very strong hydrogen bonding. However, the values of dD are up field from those of dH in the cases of strong hydrogen bonds or LBHBs. The difference for a particular hydrogen bond represents a deuterium isotope effect known as the isotope shift. The isotope shift is expressed as D½dD – dH : Figure 40.4 shows a plot of D½dD – dH against dH for hydrogen bonds in small molecules. The molecules include hydrogen bonds of the type O – H· · ·O and N – H· · ·N dissolved in a variety of solvents and at various temperatures. The most conventional hydrogen bonds with chemical shifts of 10 to 12 and the strongest hydrogen bonds with chemical shifts of 20 to 22 display little or no isotope shift, with a tendency toward positive values in the strongest hydrogen bonds. These correspond to the potential energy diagrams in Figure 40.2A and Figure 40.2C, respectively. The negative isotope shifts in Figure 40.4 correspond to intermediate values of chemical shift, with the maximum effect corresponding to dH values of 14 to 19 ppm. As in the case of nAH =nAD ; the maximum isotopic perturbation occurs in LBHBs, hydrogen bonds classed as strong, intermediate in strength between the weak and symmetrical or very strong hydrogen bonds. In molecular complexes incorporating hydrogen bonds of the type N – H· · ·O, the hydrogen bond can be probed by the 1H NMR chemical shift, the 2H isotope shift, the 15N NMR chemical shift, and the 15N isotope effect on dH, that is, the value of JN – H for 15N –H coupling. All three

Isotope Effects in the Characterization of Low Barrier Hydrogen Bonds

981

parameters are sensitive to the length of N –H and give information about the position of the proton in the complex. Based on these parameters for complexes studied under cryogenic conditions to acquire narrow spectral lines, the full range of complexes from N – H· · ·O to the symmetrical 12 N| | | |H| | | |O to N· · ·O –H can be observed. 3. D/H Fractionation Factors The D/H fractionation factor fi for a hydrogen bond is the equilibrium distribution of deuterium to hydrogen (D/H)i relative to the ratio in the solvent (D/H)s, that is fi ¼ ðD=HÞi =ðD=HÞs : Careful measurements of D/H fractionation factors reveal that typical values for weak hydrogen bonds are 1.0 to 1.2. Studies of short hydrogen bonds show values ranging from 0.3 upward, with very low values for LBHBs.51 Symmetrical or very strong hydrogen bonds display low, but not the lowest, fractionation factors. A low fractionation factor is a specific indicator of low-barrier hydrogen bonding or deep-well hydrogen bonding, that is, of strong or very strong hydrogen bonding corresponding to the potential energy diagrams in Figure 40.2B and Figure 40.2C.

D. STRENGTHS OF H YDROGEN B ONDS The strengths of weak, strong (LBHB), and very strong (SWHB) hydrogen bonds are regarded as 2 to 10 kcal mol21, 12 to 24 kcal mol21, and . 24 kcal mol21, respectively. These values refer to the energy difference of the system brought about by the hydrogen bond that is the difference in energy in the presence and absence of the hydrogen bond. This should not be confused with hydrogen-bond strengths defined as equilibria between two hydrogen-bonded states. It is generally accepted that short hydrogen bonds are stronger than long hydrogen bonds. The limited number of measured energies in species such as the hydrated oxonium and hydrated hydroxide ions, in which the lengths of hydrogen bonds are known, supports this.1 In simple molecules, spectroscopic methods (IR and ICR) can give hydrogen-bond energies. Available data show that the energies of hydrogen bonds of the type O – H· · ·O are 5.0 to 7.4 kcal mol21 for values of ˚ to 2.67 A ˚ , respectively. The energies of short hydrogen bonds are 25 to 31 kcal RO· · ·O of 2.98 A 21 ˚ to 2.39 A ˚ , respectively. These energies refer to gas-phase mol for values of RO· · ·O from 2.45 A measurements. Computational methods give similar energies for molecules known to incorporate short hydrogen bonds. The most thoroughly studied case is that of hydrogen maleate shown below as 2a. The calculated values for the difference in energy between 2a and 2b, in which the O –H bond is present but the hydrogen bond is absent, is 29 kcal mol21 in the gas phase.13,14 −

O

O O−

O O

H

O

O

O

2a

2b

H

The strengths of hydrogen bonds depend on the polarity of the medium and are lower in polar than in non polar media. Gas phase calculations generally refer to a dielectric constant of 1.0 for a vacuum. The calculated energy of the hydrogen bond in 2a in a medium with a dielectric constant of 80, as in water, is 2 16 kcal mol21, which is still a strong hydrogen bond.14 This value is compatible with the effect of internal hydrogen bonding on the ionization constants of hydrogen maleate.15 While it is difficult to observe the internal hydrogen bond of hydrogen maleate 2a by 1H NMR in aqueous media, owing to fast chemical exchange with water protons, it is nevertheless present in aqueous acetone. The LBHBs in 2a and in hydrogen 2,2-dimethylmalonate and hydrogen cis-1,2-

982

Isotope Effects in Chemistry and Biology

cyclohexane dicarboxylate are observable by 1H NMR at 2 558C in solutions of 0.31 mole fraction water and 0.69 mole fraction acetone at 2 558C.16 The low temperature suppresses the exchange rate, and the acetone suppresses freezing of the medium. The signals are observable at higher temperatures or higher mole fractions water, with exchange broadening. The strong hydrogen bond in hydrogen cyclopropane 1, 1-dicarboxylate is also observable under the same conditions and even at room temperature.49 The fractionation factor for 2a in water is low, 0.77,17 and the activation enthalpy for exchange with water is high, 8 kcal mol21.18 All evidence confirms the presence of the strong or very strong hydrogen bond in 2a and in analogous hydrogen dicarboxylates in aqueous media and pure water. Spectroscopic data and isotope effects on hydrogen bonds of the type N – H· · ·O indicate that they can be strong, but do not provide a direct measurement of strength. However, the heat of complexation in the formation of complexes such as one above can be measured thermochemically. The heats of complexation for 2,2-dichloropropionic acid with 1-alkylimidazoles are 2 12 to 2 14 kcal mol21, in the range expected for LBHBs.19

III. LOW-BARRIER HYDROGEN BONDS IN ENZYMES The possibility of a role for LBHBs in enzymes was put forward to explain the observation of low D/H fractionation factors and to account for the stabilization of substrate enolates generated at active sites. Low fractionation factors had been reported in enzymatic reactions.20,21 Among the ionizing groups in enzymes, only the thiol group of cysteine would be expected to display low D/ H fractionation. However, cysteine was not found in the relevant active sites; therefore, low D/H fractionation was put forward as evidence for the involvement of LBHBs in transition states in the reactions of certain enzyme substrate complexes. Simultaneously, the question of how an enzyme might stabilize an enolate derived from a substrate such as acetyl CoA with a carbon pKa . 20, or of a carboxylate substrate with a carbon pKa $ 30 came under consideration. The hypothesis that LBHBs might stabilize such high energy intermediates in enzymatic sites was advanced.22 Of the enzymatic systems originally discussed as possibly involving the role of an LBHB, several have since been explained in other ways, or the postulated function of an LBHB has been excluded by experimentation. However, one example of enolate stabilization by an LBHB has been found experimentally in the case of D5-3-ketosteroid isomerase. And the possibility of LBHB involvement in the actions of triose phosphate isomerase and citrate synthase, and related enzymes, remains open.

A. SERINE P ROTEASES An LBHB was assigned to the catalytic diad of His57 and Asp102 within the tetrahedral intermediate in the action of chymotrypsin.5 This assignment was based on spectroscopic and structural evidence. A downfield signal at 18 ppm had been reported in the 1H NMR spectrum of chymotrypsin at low pHs.6 The signal appeared at 15 ppm at pHs higher than 7, the pKa of His57 at the active site, and the signal was assigned to the proton bridging His57-N12 and Asp102-Od1. This signal and its assignment were confirmed in chymotrypsin, trypsin, and a-lytic protease.23,24 The signal appeared further downfield at 18.9 ppm in the tetrahedral complex of chymotrypsin with N-acetyl-L -leucyl-L -phenylalanyl trifluoromethylketone.5,25 In this complex, the hydroxyl group of Ser195 was added to the trifluoromethylketone to form an adduct. The complex was regarded as an analog of the tetrahedral intermediate in catalysis, in which CF3 occupied the position of the leaving group in a tetrahedral intermediate.26 The crystal structure of this adduct showed a very tight contact between His57-N12 and Asp102-Od1.27 The downfield proton appears in tetrahedral adducts of chymotrypsin with peptide trifluoromethylketones at pHs below 12. Under these conditions, His57 is ionized by protonation on Nd1, and this transforms the proton bridging His57-N12 and Asp102-Od1 into an LBHB.

Isotope Effects in the Characterization of Low Barrier Hydrogen Bonds

983

FIGURE 40.5 The structures and electrostatic nature of tetrahedral adducts of chymotrypsin. The chemical transformations in tetrahedral adduct formation at the active site of chymotrypsin are depicted for the reactions of (A) a peptide trifluormethylketone, (B) a peptide boronic acid, (C) a peptide aldehyde.

The LBHB is regarded as a common feature of chymotrypsin at low pH and of tetrahedral adducts of chymotrypsin. The reaction of chymotrypsin with a peptide trifluoromethylketone is illustrated in Figure 40.5A, which shows that the hydroxyl group of Ser195 undergoes nucleophilic addition to the carbonyl group of the trifluoromethylketone. The proton released from Ser195 is abstracted and held by His57, which becomes more tightly bonded to Asp102 through the formation of an LBHB. The adduct is analogous to a tetrahedral intermediate in the action of chymotrypsin, with the CF3 group of the peptide trifluormethylketone in place of the leaving group of a substrate. That the complex is a transition state analogue is supported by the dissociation constant for N-acetyl-L leucyl-L -phenylalanine trifluoromethylketone, which is 0.2 nM.28 The complex is stable because CF3 cannot leave, and this property allows the complex to be studied as an analogue of the tetrahedral intermediate.26 The complex is otherwise similar to the tetrahedral intermediate. The nucleophilic addition of Ser195, with proton transfer to His57, creates a negative charge on the adduct, and this resides with the oxyanion arising from the carbonyl group and held in the oxyanion binding site. This interaction, and the electrostatics overall, are similar to the tetrahedral intermediate in catalysis. Figure 40.5B shows the reaction of chymotrypsin with a peptide boronic acid to form a tetrahedral adduct. The reaction also involves nucleophilic addition by Ser195, in this case to boron, and the proton is transferred to His57. However, nucleophilic addition to a boronic acid does not lead to an oxyanion on the central boron; instead, the negative charge resides on boron itself, and in the active site of chymotrypsin the oxyanion binding site is occupied by a hydroxyl

984

Isotope Effects in Chemistry and Biology

group of the boronic acid adduct. Thus, the electrostatic charge is dislocated in the peptide boronic acid adducts relative to the tetrahedral adduct in catalysis. Other structural differences include an increase in the distance between His57-N12 and Asp102-Od1 relative to the resting enzyme, and the NMR signal for the bridging proton lies up field, at 16.9 ppm29 from its position in chymotrypsin at low pH. The peptide boronic acid adducts display a second downfield proton (18.2 ppm) between His57-Nd1 and the second hydroxyl group of the boronic acid.29 The spectroscopic, chemical, and kinetic facts about a series of peptide trifluoromethylketone adducts with chymotrypsin are set forth in Table 40.1. Included, together with the downfield 1 H NMR chemical shifts ranging from 18.6 to 19 ppm, are the low D/H fractionation factors and the isotope effects on the 3H NMR signals for the triton bridging His57-N12 and Asp102-Od1. The tritium NMR study is of special interest because tritium NMR is done with highly enriched [3H]H2O that retains substantial hydrogen. Thus, both the 3H and 1H NMR signals can be observed in the same samples and directly compared.30 This is shown in Figure 40.6 for three of the peptide trifluoromethylketone adducts with chymotrypsin. The isotope shifts are obvious. The deuterium substitution also leads to a negative isotope shift.31 Included in Table 40.1 are the activation energies for exchange of the LBHB with solvent protons and the rate constants for exchange. High values of the activation energy are expected for strongly bound protons owing to slow dissociation. The values for the complexes of chymotrypsin with peptide trifluoromethylketones are very high and fully compatible with strong hydrogen bonding. High resolution x-ray crystallographic analysis of the complexes with N-acetyl-L-phenylalanine and N-acetyl-L -leucyl-L -phenylalanine trifluoromethylketones show that the contact ˚ , well distance RN· · ·O between His57-e2 and Asp102-Od1 in these complexes are 2.59 to 2.62 A ˚ below the 2.72 A separation for twice the sum of van der Waals radii for nitrogen and oxygen.27,43 Thus, the peptide trifluoromethylketone adducts with chymotrypsin meet three of the four spectral

TABLE 40.1 Physical Properties of the LBHB in Transition State Analogue Complexes of Chymotrypsin Inhibitora Physical Property

AcF–CF3

AcGF– CF3

AcVF–CF3

AcLF–CF3

dH (Ppm)b fc D(dT – dH) ppmd kex (sec21)e DH‡ex (kcal mol21)f KI (mM21)g pKah

18.6 0.32 0.8 282 15 17,30,20,40 10.7

18.7 0.34 0.8 123 16 18,12 11.1

18.9 0.38 — — — 2.8,4.5 11.8

19.0 0.43 0.8 12.4 19 1.2, 2.4 12.1

a

Abbreviations: N-acetyl-L -phenylalanine trifluoromethylketone, AcFCF3; N-acetyl-glycyl-L -phenylalanine trifluoromethylketone, AcGFCF3; N-acetyl-L -valyl-L -phenyl-alanine trifluoromethylketone, AcFCF3, N-acetyl-L -leucyl-L -phenylalanine trifluoromethyl-ketone, AcFCF3. b His57-Hd1. From Ref. 45. c D/H fractionation factor of His57-Hd1. From Ref. 54. d Tritium isotope shift for His57-Hd1. From Ref. 30. e Rate constant for exchange of His57-Hd1 with solvent at 258C. From Ref. 54. f Activation enthalpy for exchange of His57-Hd1. From Ref. 54. g Inhibition constants reported. From Refs. 25, 26, 28. h pKa of His57-H12 reported. From Refs. 44, 45.

Isotope Effects in the Characterization of Low Barrier Hydrogen Bonds

985

3

H

1

H 19.5 19.0 18.5 18.0 17.5

(a) Chemical shift (/ppm)

(b)

19.5 19.0 18.5 18.0 17.5

Chemical shift (/ppm)

(c)

19.5 19.0 18.5 18.0 17.5

Chemical shift (/ppm)

FIGURE 40.6 The proton and triton NMR signals of LBHBs in chymotrypsin-peptide trifluoromethylketone adducts. Shown are the downfield regions of the 3H (top) and 1H NMR spectra of samples of chymotrypsin inhibited by (a) AcF –CF3, (b) AcVF – CF3, and (c) ALF – CF3. Proton and triton spectra were accumulated from the same samples. The figure is adapted from Figure 40.1 of Ref. 30 and used with permission.

and structural criteria for LBHBs between His57 and Asp102. Data on vibrational frequencies are not available. If the LBHB in a chymotrypsin or peptide trifluormethylketone adduct stabilizes adduct formation, it should increase the basicity of His57. In fact, the values of pKa for His57 range from 10.6 to 12 in the four complexes studied, as shown in Table 40.1. Since these complexes are transition state analogs, the high basicities of His57 imply a role for the LBHB in catalysis. N-acetyl-L -leucyl-L -leucine trifluoromethylketone (AcLLCF3) reacts with and inhibits the serine protease subtilisin in the same way that chymotrypsin reacts with the inhibitors in Table 40.1. The downfield proton with a low fractionation factor is also observed in the transition-state analogue complex of subtilisin with AcLLCF3.32 Further, the complex formed with subtilisin incorporating [15N] histidine displays proton-15N coupling. The values of JH – N and dN15 indicate that the LBHB proton is 20 to 30% transferred, that is, that the N – H bond is elongated by ˚. about 0.2 A As originally reported in the case of chymotrypsin,6 serine proteases themselves display the downfield proton NMR signal under weakly acidic conditions, that is, below pH 7 owing to protonation of His57-Nd1. Spectral studies of this LBHB in chymotrysinogen (dH ¼ 18.1 ppm) show that it displays a low D/H fractionation factor (0.4) and a high activation energy for exchange with solvent (10.4 kcal mol21), typical of an LBHB.33 At high pH, where His57-Nd1 is not protonated, the signal appears at 15 Ppm and displays a normal D/H fractionation factor and low activation energy for exchange. These properties are consistent with an LBHB only when His57Nd1 is protonated, as it is in the tetrahedral intermediate. ˚ the imaged hydrogen In an ultra high resolution crystal structure of subtilisin, refined at 0.78 A in the LBHB between His-N12 and Asp-Od1 supports the assignment of an LBHB.34 The structure ˚ and RH – O ¼ 1.5 A ˚ . The N – H bond is clearly elongated by at least is modeled with RN – H ¼ 1.2 A ˚ . Similar contacts are observed in the ultra high resolution structures of other serine 0.2 A proteases.35,36

B. CHOLINESTERASES The active sites of many esterases include serine – histidine – glutamate triads similar to those in serine proteases. Acetylcholinesterase (AchE) and butyrylcholinesterase (BchE) are such enzymes, and they catalyze the hydrolysis of acetylcholine or butyrylcholine to choline and the corresponding acid by a mechanism similar to the action of serine proteases. Available physicochemical evidence of an LBHB between His440 and Glu327 in these enzymes is analogous to that

986

Isotope Effects in Chemistry and Biology

TABLE 40.2 Spectral Properties of the Proton Bridging His440 and Glu327 in Cholinesterases Enzyme/Complex BChE-TMTFAa BChE-paraoxonc HþAchEd AChE-TMTFAe AchE-paraoxonc a b c d e

dH (Ppm)

f (D/H) Fractionation Factor

18.1b 16.1b 17,7e 17.9e 16.6e

0.63b 0.72b — 0.76e —

Tetrahedral adduct of BchE with m-(N,N,N-trimethylammonio) trifluoromethylketone. Data from Ref. 37. Ethylphosphorylserine derivatives of BchE or AChE arising from reaction with paraoxon. AchE at pH 4, with His440 protonated. Data from Ref. 38.

accumulated for serine proteases.37,38 The 1H NMR signals are not as far downfield as for free chymotrypsin, and the D/H fractionation factors are higher than in chymotrypsin. Results for these enzymes are summarized in Table 40.2. The complexes with the trifluoromethylketone TMTFA are analogous to those of chymotrypsin with peptide trifluoromethylketones. The chemical shifts and fractionation factors for these complexes indicate strong hydrogen bonding. However, the LBHBs may not be as strong as in the comparable complexes of chymotrypsin, which display further downfield 1H NMR signals and lower fascination factors (Table 40.1).

C. D5-3-KETOSTEROID I SOMERASE LBHBs were observed in complexes of D5-3-ketosteroid isomerase (KSI) with estradiol or dihydroequilenin.39 The phenolic group is in close OH

HO

OH

HO Dihydroequilenin

Estradiol

˚ ) in the structure of the complex of equilenin with KSI.40 The pKa of the contact with Tyr14 (2.5 A phenolic group is similar to that of the homoenolate intermediate in catalysis, so that dihydroequilenin and estradiol may be regarded as analogues of the intermediate. The assignment of the LBHB is based on the downfield 1H NMR signal and on other properties of the proton associated with this signal. The appearance of this signal in the NMR spectrum depends on the presence of Tyr14 in KSI and on the presence of dihydroequilenin or estradiol in complex with KSI. With dihydroequilenin, the signal appears at 18.2 ppm, and it displays a D/H fractionation factor of 0.34. The rate constant for its exchange with water is 137 sec21, and the activation energy for its exchange is 13 kcal mol21. By all available criteria, an LBHB exists in the complex of KSA with equilenin, and it involves Tyr14 and equilenin. The most obvious assignment is that the LBHB bridges Tyr14 and the phenolic group of equilenin.

Isotope Effects in the Characterization of Low Barrier Hydrogen Bonds

987

IV. ROLE OF LOW-BARRIER HYDROGEN BONDS IN THE ACTIONS OF ENZYMES The role of LBHBs in enzyme action is of considerable interest and significance. Many enzymatic intermediates are high energy species that must be stabilized at enzymatic sites if they are to be kinetically significant. Classical means of stabilizing such intermediates include delocalization of negative charge by iminium groups (CyNHþ) or stabilization by coordination with metal ions. Another means of introducing stabilization on the same order would be the formation of an LBHB in a metastable intermediate. Such intermediates resemble transition states, so that their stabilization would also lower the activation energies for the reactions in which they participate. The LBHBs so far observed in enzyme action are postulated to stabilize metastable intermediates in catalysis. The roles of LBHBs in enzymatic catalysis are considered in the following sections.

A. SERINE P ROTEASES AND E STERASES The mechanisms of the reactions catalyzed by serine proteases and serine esterases include nucleophilic catalysis by serine and covalent acyl – enzyme intermediates. The reactions follow the overall course of Equation 40.1a and Equation 40.1b, where X ¼ N or O. O

O

E-Ser-OH

+

R1

C

XR2

E-Ser-O

C

R1

+

O

E-Ser-O

C

R2XH

(40.1a)

O R1

+ H2O

E-Ser-OH

+

R1

C

OH

(40.1b)

Focusing on Equation 40.1a, serine reacts as a nucleophile in displacing R2XH from the acyl group of the substrate, where X ¼ N or O. Serine is a poor nucleophile, in accordance with its pKa of 13.4.41 To facilitate the reaction, the imidazole group of histidine in the catalytic triad serves as the base to remove the proton from the serine-OH group in the acylation step 1a. Histidine is weak (pKa 7) as a base for serine (pKa 13.4), the ideal base to catalyze the acylation of serine would display a pKa-value between that of serine (13.4) and that of the leaving group, , 9 for a peptide. Such a base would be strong enough to abstract the proton from serine but not too strong to donate a proton to the leaving group. Proton transfer to His57 in a serine protease leads to LBHB-formation with Asp102, a process that should increase the basicity of His57. This occurs in the transition state for the formation of the tetrahedral intermediate, the metastable species in the mechanism of acylati.on. The tetrahedral intermediate resembles the transition state, and should display similar properties. The structurally analogous tetrahedral adducts of peptide trifluoromethylketones with chymotrypsin are transition-state analogues and should give clues to the properties of the tetrahedral intermediate and transition state. These adducts include the LBHB between His57-N12 and Asp102-Od1. Moreover, the values of pKa for His57-Nd1 in these adducts are 10.6 to 12, in between those of serine and the leaving group. Therefore, the LBHB should facilitate the formation and reaction of the tetrahedral intermediate according to the mechanism in Figure 40.7. Note that LBHB formation accompanies tetrahedral intermediate formation in the first step of acylation, and that His57 serves both as the base in the first step and as the acid for proton donation to the leaving group in the second step. The basicity of His57 in the transition state analogue complexes of chymotrypsin is in the optimal range for its function in this mechanism. The “normal” pKa of 7 for His57 in free chymotrypsin must be explained. Protonation of His57-Nd1 in free chymotrypsin leads to the LBHB between His57-N12 and Asp102-Od1, much

988

Isotope Effects in Chemistry and Biology

FIGURE 40.7 A role for the LBHB in the action of chymotrypsin. The mechanism by which serine undergoes acylation in the reactions of serine proteases and esterases is outlined in this figure for the case of chymotrypsin. Formation of the tetrahedral intermediate requires the action of His57 to abstract the proton from Ser195 in the first step. This process is accompanied by LBHB formation between His57-N12 and Asp102-Od1. The strength of this LBHB is postulated to increase the basicity of His57 and its effectiveness as a catalyst for Ser195. Elimination of the leaving group from the tetrahedral adduct is catalyzed by proton transfer from His57.

as in proton transfer to His57 in peptide trifluoromethylketone adducts. The pKa of His57 in the latter adducts is high, but in free chymotrypsin it appears normal.42 This paradox can be resolved by considering the properties of the tetrahedral addition complex of chymotrypsin with N-acetylL -leucyl-L -phenylalaninal (AcLF –CHO). The x-ray crystal structure is nearly the same as that of the complex of chymotrypsin with AcLF –CF3 with H in place of CF3 and one other exception.43 The configuration of the hemiketal carbon in the complex with AcLF –CF3 places the anionic oxygen in the oxyanion binding site. In contrast, the hemiacetal carbon in the complex with AcLF – CHO is epimeric; in one epimer the oxygen is hydrogen bonded to His57, and in the other it is at the oxyanion site. This structural difference between the complexes of AcLF – CF3 and AcLF – CHO arises from a fundamental chemical difference. The reaction of AcLF –CHO with chymotrypsin follows the course in Figure 40.5C.43 The essential chemical difference between complexation by AcLF – CF3 and AcLF – CHO is the destination of the proton derived from Ser195, as shown in Figure 40.5A and Figure 40.5C. This can be explained by the relative basicities of the hemiketal, the hemiacetal, and His57. In the hemiketal complex with AcLF –CF3, His57 is more basic (pKa ¼ 12) than the hemiketal, so that the proton from Ser195 goes to His57 – Nd1. The pKa of the hemiketal is about 9.5 in aqueous solution, and it is lower than five in the active site.25 The tetrahedral intermediate should display a solution-pKa of about 11, also lower than that of His57. In contrast, the hemiacetal

Isotope Effects in the Characterization of Low Barrier Hydrogen Bonds

989

of Ser195 with AcLF – CHO is much more basic (pKa , 13.5) than His57, so that the proton goes to the hemiacetal. With this background, the other ionization properties of the tetrahedral complexes and free chymotrypsin can be explained. In the complex with AcLF – CF3, the pKa-values of both His57 and the hemiketal differ from the solution values for comparable groups by about five pKa-units.25,44,45 Therefore, either protonation of the hemiketal anion or loss of a proton from His57-Nd1 must overcome a barrier DDG8 of about 7 kcal mol21 relative to the comparable groups in solution. The likely barrier is the change in net charge of 2 1 at the active site. The net charge at the active site of the AcLF – CHO-complex, which is structurally similar to the AcLF – CF3-complex is also 2 1 (Figure 40.5C), and the ionization of His57 should face the same barrier of 7 kcal mol21. Yet the pKa of His57 in the AcLF –CHO-complex is 6.5.43 The simplest rationale for the apparently normal pKa is that the barrier to ionization is internally offset; that is, some internal interaction overcomes the natural barrier to an alteration in the net charge of 2 1 in the active site. Such an

FIGURE 40.8 A role for the LBHB in the action of D5-3-KSI. Abstraction of the proton by Asp38 leads to the homoenolate intermediate, and this is postulated to be stabilized by an LBHB between the oxyanion and Tyr14. LBHB formation between Tyr14 and dihydroequilenin or estradiol supports this concept of the mechanism, where the ionized inhibitors are regarded as analogs of the homoenolate ion.

990

Isotope Effects in Chemistry and Biology

internal offset would be the stabilization afforded by the LBHB formed upon protonation of His57. The electrostatics in the ionization of free chymotrypsin are the same as in the ionization of the complex with AcLF –CHO, and the “normal” pKa of 7 for free chymotrypsin can be explained by the same effect. According to the foregoing theory, the internal offset must overcome the 7 kcal mol21 barrier to altering the net charge. This is just the same as the increase in activation energy that occurs whenever the LBHB is disrupted, as pointed out above in the discussion of the mechanism.5,47,48 On this basis, the LBHB contributes about 7 kcal mol21 to lowering the activation energy, and it does so by increasing the basicity of His57 in the transition state and tetrahedral intermediate.

B. D5-3-KETOSTEROID I SOMERASE The interconversion of D5- and D4-3-ketosteroids proceeds by way of enolization and the intermediate homoenolate ion, as illustrated in Figure 40.8. Structural and kinetic results implicate Tyr14, Asp38, and Asp99.40 Asp38, as shown in Figure 40.8, functions as the acid or base catalyst in enolization. The observation of an LBHB between Tyr14 and dihydroequilenin, suggests LBHB formation between Tyr14 and the homoenolate intermediate in the catalytic cycle. The LBHB would stabilize the homoenolate and facilitate its formation.

C. OTHER E NZYMES Enzymes such as triose phosphate isomerase, fumarase, crotonase, and citrate synthase catalyze enolate formation. It is reasonable to expect some means of enolate stabilization to play a significant role in catalysis. In the case of the divalent metal ion independent triose phosphate isomerase, evidence for strong hydrogen bonding has been presented.46 The divalent metal ion dependent triose phosphate isomerase is thought to function by a hydride transfer mechanism that does not involve enolate formation. In the reactions of serine, proteases and D5-3-KSI features of the transition states became spectroscopically and structurally accessible and led to the discovery of LBHBs. If LBHBs are significant in the actions of other enzymes, they may be detected if features the transition states can be unmasked in some way, perhaps by the development of suitable transition state analogues.

ACKNOWLEDGMENTS The author’s research on low-barrier hydrogen bonds was supported by the National Institutes of Health through grants GM30480 and DK28607. Research was carried out in collaboration with John Tobin, Constance Cassidy, Jing Lin, Yaoming Wei, David Neidhart, W. W. Cleland, J. L. Markley, W. M. Westler, and L. A. Reinhardt.

REFERENCES 1 Hibbert, F. and Emsley, J., Hydrogen bonding and chemical reactivity, Adv. Phys. Org. Chem., 26, 255– 379, 1990. 2 Jeffrey, G. A., An Introduction to Hydrogen Bonding, Oxford University Press, New York, 1997. 3 Ichikawa, M., Correlation between two isotope effects in hydrogen bonded crystals: transition temperature and separation of two equilibrium sites, Chem. Phys. Lett., 79, 583–587, 1978. 4 Jeffrey, G. A. and Saenger, W., Hydrogen Bonding in Biological Structures, Verlag, Heidelberg, 1991. 5 Frey, P. A., Whitt, S. A., and Tobin, J. B., A low-barrier hydrogen bond in the catalytic triad of serine proteases, Science, 264, 1927– 1930, 1994. 6 Robillard, G. and Shulman, R. G., High resolution nuclear magnetic resonance study of the histidineaspartate hydrogen bond in chymotrypsin and chymotrypsinogen, J. Mol. Biol., 71, 507– 511, 1972.

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7 Berglund, B. and Vaughn, R. W., Correlations between proton chemical shift tensors, deuterium quadrupole couplings, and bond distances for hydrogen bonds in solids, J. Chem. Phys., 73, 2037– 2043, 1980. 8 Hadzi, F., Infrared spectra of strongly hydrogen bonded systems, Pure Appl. Chem., 11, 435–453, 1965. 9 Novak, A., Hydrogen bonding in solids. Correlation of spectroscopic and crystallograpic data, Struct. Bond., 18, 177– 216, 1974. 10 Tobin, J. B., Whitt, S. A., Cassidy, C. S., and Frey, P. A., Low-barrier hydrogen bonding in molecular complexes analogous to histidine and aspartate in the catalytic triad of serine proteases, Biochemistry, 34, 6919– 6924, 1995. 11 Cassidy, C. S., Reinhardt, L. A., Cleland, W. W., and Frey, P. A., Hydrogen bonding in complexes of carboxylic acids with l-alkylimidazoles: steric and isotope effects on low barrier hydrogen bonding, J. Chem. Soc. Perkin Trans., 2, 635– 641, 1999. 12 Smirnov, S. N., Golubev, N. S., Denisov, G. S., Benedict, H., Schah-Mohammedi, P., and Limbach, H. H., Hydrogen/deuterium isotope effects on the NMR chemical shifts and geometries of intermolecular low-barrier hydrogen bonds, J. Am. Chem. Soc., 118, 4094– 4101, 1996. 13 Murthy, A. J. N., Bhat, S. N., and Rao, C. N. R., Molecular orbital studies of the hydrogen bond, J. Chem. Soc. A, 1251– 1256, 1970. 14 McAllister, M. A., Characterization of low-barrier hydrogen bonds. Three Hydrogen maleate. An ab initio and DFT investigation, Can. J. Chem., 75, 1195 –1202, 1997. 15 Frey, P. A. and Cleland, W. W., Are there strong hydrogen bonds in aqueous solutions?, Bioorg. Chem., 26, 175– 192, 1998. 16 Cassidy, C. S., Lin, J., Tobin, J. B., and Frey, P. A., Low-barrier hydrogen bonding in aqueous and aprotic solutions of dicarboxylic acids: spectroscopic characterization, Bioorg. Chem., 26, 213– 219, 1998. 17 Jarrett, R. M. and Saunders, M., A new method for obtaining isotopic fractionation data at multiple sites in rapidly exchanging systems, J. Am. Chem. Soc., 107, 2648– 2654, 1985. 18 Lin, J. and Frey, P. A., Strong hydrogen bonds in aqueousl and aqueous-acetone solutions of dicarboxylic acids: activation energies for exchange and deuterium fractionation factors, J. Am. Chem. Soc., 122, 11258 –11259, 2000. 19 Reinhardt, L. A., Sacksteder, K. A., and Cleland, W. W., Enthalpic studies of complex formation between carboxylic acids and 1-alkylimidazoles, J. Am. Chem. Soc., 120, 13366– 13369, 1998. 20 Cleland, W. W., Low-barrier hydrogen bonds and low fractionation factor bases in enzymatic reactions, Biochemistry, 31, 317– 319, 1992. 21 Cleland, W. W. and Kreevoy, M. M., Low-barrier hydrogen bonds and enzymic catalysis, Science, 264, 1887– 1890, 1994. 22 Gerlt, J. A. and Gassman, P. G., Understanding the rates of certain enzyme-catalyzed reactions: proton abstraction from carbon acids, acyl-transfer reactions, and displacement reactions of phosphodiesters, Biochemistry, 32, 11943– 11952, 1993. 23 Markley, J. L., Hydrogen bonds in serine proteinases and their complexes with protein proteinase inhibitors. Proton nuclear magnetic resonance studies, Biochemistry, 17, 4648– 4656, 1978. 24 Bachovchin, W. W., Confirmation of the assignment of the low-field proton resonance of serine proteases by using specifically nitrogen 15 labeled enzyme, Proc. Natl Acad. Sci. U.S.A., 82, 7948– 7951, 1985. 25 Liang, T.-C. and Abeles, R. H., Complex of alpha-chymotrypsin and N-acetyl-L -leucyl-L -phenylalanyl trifluoromethyl ketone: structural studies with NMR spectroscopy, Biochemistry, 26, 7603– 7608, 1987. 26 Imperiali, B. and Abeles, R. H., Inhibition of serine proteases by peptidyl fluoromethyl ketones, Biochemistry, 25, 3760– 3767, 1986. 27 Brady, K., Wei, A., Ringe, D., and Abeles, R. A., Structure of chymotrypsin-trifluoromethyl ketone inhibitor complexes: comparison of slowly and rapidly equilibrating inhibitors, Biochemistry, 29, 7600– 7607, 1990. 28 Brady, K. and Abeles, R. H., Inhibition of chymotrypsin by peptidyl trifluoromethyl ketones: determinants of slow-binding kinetics, Biochemistry, 29, 7608– 7617, 1990. 29 Bao, D., Huskey, W. P., Kettner, C., and Jordan, F., Hydrogen bonding to active-site histidine in peptidyl boronic acid inhibitor complexes of chymotrypsin and subtilisin: proton

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31 32 33 34 35 36 37 38 39 40 41 42

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Isotope Effects in Chemistry and Biology magnetic resonance assignments and H/D fractionation, J. Am. Chem. Soc., 121, 4684– 4689, 1999. Westler, W. M., Frey, P. A., Lin, J., Wemmer, D. E., Morimoto, H., Williams, P. G., and Markley, J. L., Evidence for a strong hydrogen bond in the catalytic dyad of transition-state analogue inhibitor complexes of chymotrypsin from proton– triton NMR isotope shifts, J. Am. Chem. Soc., 124, 4196– 4197, 2002. Cassidy, C. S., Lin, J., and Frey, P. A., The deuterium isotope effect on the NMR signal of the lowbarrier hydrogen bond in a transition-state analog complex of chymotrypsin, Biochem. Biophys. Res. Commun., 273, 789– 792, 2000. Halkides, C. J., Wu, Y. Q., and Murray, C. J., A low-barrier hydrogen bond in subtilisin: 1H and 15N NMR studies with peptidyl trifluoromethyl ketones, Biochemistry, 35, 15941– 15948, 1996. Markley, J. L. and Westler, W. M., Protonation-state dependence of hydrogen bond strengths and exchange rates in a serine protease catalytic triad bovine chymotrypsinogen A, Biochemistry, 35, 11092– 11097, 1996. Kuhn, P., Knapp, M., Soltis, S. M., Ganshaw, G., Thoene, M., and Bott, R., The 0.78 A structure of a serine protease: bacillus lentus subtilisin, Biochemistry, 37, 13446– 13452, 1998. Betzel, C., Gourinath, S., Kumar, P., Kaur, P., Perbandt, M., Eschenburg, S., and Singh, T. P., Structure of a serine protease proteinase K from Tritirachium album limber at 0.98 A resolution, Biochemistry, 40, 3080– 3088, 2001. Katona, G., Wilmouth, R. C., Wright, P. A., Berglund, G. I., Hajdu, J., Neutz, R., and Schofield, C. J., X-ray structure of a serine protease acyl – enzyme complex at 0.95-A resolution, J. Biol. Chem., 277, 21962– 21970, 2002. Viragh, C., Harris, T. K., Reddy, P. M., Massiah, M. A., Mildvan, A. S., and Kovach, I. M., NMR evidence for a short, strong hydrogen bond at the active site of a cholinesterase, Biochemistry, 39, 16200– 16205, 2000. Massiah, M. A., Viragh, C., Reddy, P. M., Kovach, I. M., Johnson, J., Rosenberry, T. L., and Mildvan, A. S., Short, strong hydrogen bonds at the active site of human AchE: proton NMR studies, Biochemistry, 40, 5682– 5690, 2001. Zhao, Q., Abeygunawardana, C., Talalay, P., and Mildvan, A. S., NMR evidence for the participation of a low-barrier hydrogen bond in the mechanism of delta 5 to 3 KSI, Proc. Natl Acad. Sci. U.S.A., 93, 8220– 8224, 1996. Choi, C., Ha, N. C., Kim, S. W., Ki, D. H., Park, S., Oh, B. H., and Choi, K. Y., Asp-99 donates a hydrogen bond not to Tyr-14 but to the steroid directly in the catalytic mechanism of D5-3-KSI from Pseudomonas putida biotype B, Biochemistry, 39, 903– 909, 2000. Bruice, T. C., Fife, T. H., Bruno, J. J., and Brandon, N. E., Hydroxyl group catalysis. II. The reactivity of the hydroxyl group of serine. The nucleophilicity of alcohols and the ease of hydrolysis of their acetyl esters as related to their pKa, Biochemistry, 1, 7 – 12, 1962. Zhong, S., Haghjoo, K., Kettner, C., and Jordan, F., Proton magnetic resonance studies of the active center histidine of chymotrypsin complexed to peptideboronic acids: solvent accessibility to the Nd and Ne sites can differentiate slow-binding and rapidly reversible inhibitors, J. Am. Chem. Soc., 117, 7048– 7055, 1995. Neidhart, D., Wei, Y., Cassidy, C. S., Lin, J., Cleland, W. W., and Frey, P. A., Correlation of lowbarrier hydrogen bonding and oxyanion binding in transition state analogue complexes of chymotrypsin, Biochemistry, 40, 2439– 2447, 2001. Cassidy, C. S., Lin, J., and Frey, P. A., A new concept for the mechanism of action of chymotrypsin: the role of the low-barrier hydrogen bond, Biochemistry, 36, 4576– 4584, 1997. Lin, J., Cassidy, C. S., and Frey, P. A., Correlations of the basicity of His 57 with transition state analogue binding, substrate reactivity, and the strength of the low-barrier hydrogen bond in chymotrypsin, Biochemistry, 37, 11940– 11948, 1998. Harris, T. K., Abeygunawardana, S., and Mildvan, A. S., NMR studies of the role of hydrogen bonding in the mechanism of triosephosphate isomerase, Biochemistry, 36, 14661– 14675, 1997. Craik, C. S., Roczniak, S., Largeman, C., and Rutter, W. J., The catalytic role of the active site aspartic acid in serine proteases, Science, 237, 909–913, 1987. Henderson, R., Catalytic activity of chymotrypsin in which histidine-57 has been methylated, Biochem. J., 124, 13 – 18, 1971.

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49 Hess, R. A. and Reinhardt, L. A., Unusual properties of highly charged buffers: large ionization volumes and low barrier hydrogen bonds, J. Am. Chem. Soc., 121, 9867– 9870, 1999. 50 Jeffrey, G. A. and Yeon, Y., The correlation between hydrogen-bond lengths and proton chemical shifts in crystals, Acta Crystallogr. Sect. B, 42, 410– 413, 1986. 51 Kreevoy, M. M. and Liang, T., Structures and isotopic fractionation factors of complexes A1HA21 2 , J. Am. Chem. Soc., 102, 3315– 3322, 1980. 52 McDermott, A. and Ridenour, C. F., Encyclopedia of NMR, Wiley, Sussex England, pp. 3820, 1996. 53 Mildvan, A. S., Harris, T. K., and Abeygunawardana, C., Nuclear magnetic resonance methods for the detection and study of LBHBs on enzymes, Methods Enzymol., 308, 219– 245, 1999. 54 Lin, J., Westler, W. M., Cleland, W. W., Markley, J. L., and Frey, P. A., Fractionation factors and activation energies for exchange of the low barrier hydrogen bonding proton in peptidyl trifluoromethyl ketone complexes of chymotrypsin, Proc. Natl Acad. Sci. U.S.A., 95, 14664– 14668, 1998.

41

Theory and Practice of Solvent Isotope Effects Daniel M. Quinn

CONTENTS I. II.

Introduction ...................................................................................................................... 995 Origins of Solvent Isotope Effects................................................................................... 996 A. Equilibrium Solvent Isotope Effects........................................................................ 996 B. Kinetic Isotope Effects............................................................................................. 997 III. The Kresge – Gross – Butler Equation............................................................................... 998 A. Derivation................................................................................................................. 998 B. The Proton Inventory Technique ........................................................................... 1000 1. Proton Inventories of Elementary Steps ......................................................... 1000 2. Effects of Reactant State Fractionation .......................................................... 1001 3. Proton Inventories of Multistep Enzyme Reactions....................................... 1002 IV. Fractionation Factors...................................................................................................... 1004 A. Reactant State Fractionation Factors of Common Functional Groups ................. 1004 B. Transition State Fractionation Factors................................................................... 1005 V. Practical Considerations................................................................................................. 1006 VI. Examples ........................................................................................................................ 1008 A. Nonenzymic Reactions .......................................................................................... 1008 B. Enzymatic Reactions.............................................................................................. 1009 1. Serine Proteases............................................................................................... 1009 2. Acetylcholinesterase........................................................................................ 1012 3. Carbonic Anhydrase ........................................................................................ 1014 4. Tyrosine Hydroxylase ..................................................................................... 1015 VII. Conclusions .................................................................................................................... 1016 References................................................................................................................................... 1016

I. INTRODUCTION Solvent isotope effects arise in enzymatic reactions when rate or equilibrium constants are measured in H2O, D2O, or mixtures of these isotopic solvents. Since there are multiple isotopically exchangeable sites in an enzyme reaction assembly (i.e., enzyme þ substrate), the origins of solvent isotope effects are conceivably of vexing complexity. The multisite origin of solvent isotope effects is suggested by the wide range of isotopic fractionation factors observed in staphylococcal nuclease1 (fractionation factors are discussed later in this chapter). However, like all isotope effects, when solvent isotope effects and their interpretation are corroborated by additional experimental results, they are of considerable utility in elucidating the elements of structure and function that contribute to enzyme catalytic power. This chapter outlines the conceptual basis of solvent isotope effects, discusses practical considerations when measuring and interpreting them, 995

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Isotope Effects in Chemistry and Biology

and explores the use of solvent isotope effects in several enzyme-catalyzed reactions. The application of solvent isotope effects to enzyme reactions has been extensively reviewed.2 – 8 Consequently, the reader is encouraged to consult these reviews for additional background on solvent isotope effects. Solvent isotope effects are measured as the ratio of equilibrium or rate constants in isotopically various solvents, in particular H2O and D2O for enzyme reactions. The terminology described in this paragraph will be utilized, and expanded, throughout the remainder of this chapter. Consider an D2 O H2 O ; respectively. and Keq equilibrium process whose equilibrium constants in H2O and D2O are Keq The corresponding solvent isotope effect is defined in Equation 41.1: D2 O

H2 O D2 O Keq ¼ Keq =Keq

ð41:1Þ

Kinetic solvent isotope effects are similarly defined in Equation 41.2: D2 O

k ¼ kH2 O =kD2 O

ð41:2Þ

As these equations show, the isotope effect is designated by using a leading superscript that denotes the heavy isotopic species.

II. ORIGINS OF SOLVENT ISOTOPE EFFECTS A. EQUILIBRIUM S OLVENT I SOTOPE E FFECTS The structural features of a reaction assembly that contribute to the magnitude of the solvent isotope effect can be most readily understood by utilizing fractionation factors. Consider the isotopic equilibria in Scheme 41.1. In the scheme both the reactant RH and the product PH possess a single isotopically exchangeable site, and the corresponding isotope exchange equilibrium constants, called fractionation factors, are fR and fP ; respectively:

fR ¼

½RD =½RH ½RD ½RH ¼ ½SOD =½SOH n=ð1 2 nÞ

ð41:3Þ

fP ¼

½PD =½PH ½PD ½PH ¼ ½SOD =½SOH n=ð1 2 nÞ

ð41:4Þ

These equations recognize the fact that the mixed isotopic solvent SOL (L ¼ H or D) is in large excess over the solutes RL and PL, and therefore the ratio in isotopic solvents is determined by n, the atom fraction of deuterium in the solvent. The thermodynamic cycle of Scheme 41.1 in turn relates the magnitude of the equilibrium solvent isotope effect to the values of the fractionation factors: D2 O

D2 O H2 O Keq ¼ Keq =Keq ¼ fR =fP

RH SOD

H2O Keq

SOH

PH SOD

SOH

D2O

RD

Keq

PD

SCHEME 41.1 Thermodynamic cycle for description of equilibrium solvent isotope effects.

ð41:5Þ

Theory and Practice of Solvent Isotope Effects

997

If the equilibrium system possesses w isotopically exchangeable sites whose fractionation factors are different in the reactant and product states, then Equation 41.6 accounts for the solvent isotope effect: D2 O

Keq ¼

w Y i

fRi =

w Y i

fPi

ð41:6Þ

B. KINETIC I SOTOPE E FFECTS A thermodynamic cycle like that in Scheme 41.1 can be constructed from which the relationship between kinetic isotope effects and fractionation factors is derived. This thermodynamic cycle, shown in Scheme 41.2, springs from the transition state theory of Glasstone, Laidler, and Eyring.9 The transition state theory asserts that the rate constant is proportional to the equilibrium constant for conversion of reactants to the transition state: k¼k

kB T ‡ K h

ð41:7Þ

In Equation 41.7, K ‡ is the equilibrium constant, kB, T and h are Boltzmann’s constant, the absolute temperature and Planck’s constant, respectively, and k; the transmission coefficient, accounts for nonequilibrium contributions to the rate constant, such as quantum mechanical tunneling or transition state recrossing. Consequently, Equation 41.8, which relates the kinetic isotope effect to reactant and transition state fractionation factors, arises from the thermodynamic cycle in Scheme 41.2: D2 O

k ¼ kH2 O =kD2 O ¼ kH2 O KH‡ 2 O =kD2 O KD‡ 2 O ¼ fR =fT

ð41:8Þ

Note that the isotope effect on the transmission coefficient, kH2 O =kD2 O ; is a nonthermodynamic source of isotopic fractionation in the transition state and therefore is subsumed into the transition state fractionation factor according to fT ¼ f‡ kD2 O =kH2 O ; where f‡ is the thermodynamic fractionation factor. As for equilibrium isotope effects, an analog of Equation 41.8 can be written that takes into account contributions of x isotopically exchangeable sites whose fractionation factors change on conversion of reactants to the transition state: D2 O

k¼

x Y i

fRi =

x Y i

fTi

ð41:9Þ

Transition state fractionation factors generally cannot be estimated from the value of the solvent isotope effect alone. However, a differential solvent isotope effect technique, the proton inventory

RH

kH 2 O

fR RD

TH

PH fT

kD 2 O

TD

PD

SCHEME 41.2 Thermodynamic cycle for description of kinetic solvent deuterium isotope effects. TH and TD denote isotopic transition states.

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Isotope Effects in Chemistry and Biology

method,2 – 8 is useful for this purpose. The proton inventory method is described in the next section of this chapter.

III. THE KRESGE–GROSS– BUTLER EQUATION A. DERIVATION Solvent isotope effects usually arise from differential isotopic fractionation at multiple exchangeable sites in the reactant and transition states (for kinetic isotope effects) or reactant and product states (for equilibrium isotope effects). A differential solvent isotope effect approach in which rate or equilibrium constants are measured in mixtures of the isotopic solvents H2O and D2O provides, in principle, a means for resolving contributions to the isotope effect from multiple sites. The derivation that follows provides an expression for differential kinetic solvent isotope effect measurements, though an equivalent expression for equilibrium solvent isotope effects can easily be written by analogy. Consider a reaction like that in Scheme 41.2, albeit in which two isotopically exchangeable sites experience changes in fractionation factors as the reactants are converted to the transition state. For the reactions in H2O ðk0 ; kH2 O Þ and in mixtures of H2O and D2O of atom fraction of deuterium n (i.e., kn ), Equation 41.10 and Equation 41.11 are, respectively, derived by using the transition state theory expression in Equation 41.7: k0 ¼ k0 kn ¼

kB T ½TH1 H2 h ½RH1 H2

kB T k0 ½TH1 H2 þ kDH ½TD1 H2 þ kHD ½TH1 D2 þ kDD ½TD1 D2 h ½RH1 H2 þ ½RD1 H2 þ ½RH1 D2 þ ½RD1 D2

ð41:10Þ ð41:11Þ

Equation 41.11 contains concentration terms for the various isotopic versions of reactants and transition states in the mixed isotopic solvent. Dividing Equation 41.10 by Equation 41.11 gives the following expression: k0 ¼ kn

½RD1 H2 ½RH1 D2 ½RD1 D2 þ þ ½RH1 H2 ½RH1 H2 ½RH1 H2 ½TD1 H2 kHD ½TH1 D2 k ½TD1 D2 þ þ DD k0 ½TH1 H2 k0 ½TH1 H2 ½TH1 H2

1þ 1þ

kDH k0

ð41:12Þ

The concentration ratios in Equation 41.12 can be replaced with fractionation factor-containing terms. In analogy to Equation 41.3 and Equation 41.4, Equation 41.13 and Equation 41.14 apply to the singly isotopically exchanged species: ½RD1 H2 fR n ¼ 1 12n ½RH1 H2

kDH ½TD1 H2 fT n ¼ 1 12n k0 ½TH1 H2

½RH1 D2 fR n ¼ 2 12n ½RH1 H2

ð41:13Þ

kHD ½TH1 D2 fT n ¼ 2 12n k0 ½TH1 H2

ð41:14Þ

If one assumes that the rule of the geometric mean operates for doubly isotopically exchanged species,8 then Equation 41.15 and Equation 41.16 result: ½RD1 D2 fR fR n2 ¼ 1 2 2 ½RH1 H2 ð1 2 nÞ

ð41:15Þ

Theory and Practice of Solvent Isotope Effects

999

kDD ½TD1 D2 fT fT n2 ¼ 1 2 2 k0 ½TH1 H2 ð1 2 nÞ

ð41:16Þ

Hence, combining Equations 41.12 to 41.16, one obtains: k0 ð1 2 n þ nfR1 Þð1 2 n þ nfR2 Þ ¼ kn ð1 2 n þ nfT1 Þð1 2 n þ nfT2 Þ

ð41:17Þ

This approach can easily be generalized to the case in which x exchangeable sites experience changes in fractionation factor when the reactants are converted to the transition state: x Y

k0 ¼ i¼1 x Y kn i¼1

ð1 2 n þ nfRi Þ ð1 2 n þ nfTi Þ

ð41:18Þ

A limiting case of Equation 41.18 arises when a very large number of sites each make a small contribution to the observed solvent isotope effect. Consider, for example, a situation in which the solvent isotope effect k0 =k1 arises from fractionation at v sites, each possessing fR ¼ 1:00 and fT ¼ 0:99: In this case Equation 41.18 is rearranged to give Equation 41.19: Zk;n ¼ kn =k0 ¼ ð1 2 n þ nfT Þv

ð41:19Þ

A solvent isotope effect that arises from small contributions at a large number of fractionating sites is referred to as a medium effect, and is denoted herein and in previous publications as a Z term isotope effect, such as in Equation 41.19. This Z term equation can be further simplified by using the following transformations: lnðZk;n Þ ¼ v ln½1 2 nð1 2 fT Þ ¼ vnðfT 2 1Þ

ð41:20Þ

For n ¼ 1, one successively writes Equation 41.21 to Equation 41.23: lnðZk;1 Þ ¼ k1 =k0 ¼ vðfT 2 1Þ

ð41:21Þ

lnðZk;n Þ ¼ n lnðZk;1 Þ

ð41:22Þ

n Zk;n ¼ Zk;1

ð41:23Þ

Equation 41.23 shows that when an observed solvent isotope effect arises from a large number of fractionating sites, the partial isotope effect Zk;n measured in a solvent isotopic mixture is just the full isotope effect Zk;1 raised to the n power. Equation 41.23 was derived for a situation in which isotopic fractionation at very many sites occurs only in the transition state (i.e., fT values are – 1). However, a medium effect could arise as well from many site fractionation in the reactant state (i.e., fR values are – 1) or from a combination of many site fractionations in the reactant and transition states. Equation 41.18 can now be further generalized to a situation wherein an enzymic isotope effect arises from a combination of a medium effect (Equation 41.23) and from pronounced fractionation at a few sites (i.e., these few sites have fR and/or fT values that are quite different than unity).

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Isotope Effects in Chemistry and Biology

This general situation is described by Equation 41.24: x Y

kn ¼ k0 i¼1 x Y

ð1 2 n þ nfTi Þ ð1 2 n þ

i¼1

nfRi Þ

n Zk;1

ð41:24Þ

Equation 41.24, the Kresge – Gross – Butler equation,2 – 8 provides a powerful tool in the measurement of solvent isotope effects. By varying the composition of solvent through a range of atom fractions of deuterium n, one gains additional information that frequently supports, and sometimes gives life to, mechanistic models for enzymic and nonenzymic reactions. The utility of differential solvent isotope effect measurements, analyzed according to particular mathematical reductions of the Kresge – Gross – Butler equation, is outlined in the next section of this chapter.

B. THE P ROTON I NVENTORY T ECHNIQUE 1. Proton Inventories of Elementary Steps Though inspection of Equation 41.24 suggests that there are several origins for solvent isotope effects (transition state fractionation, reactant state fractionation, medium effects), in experimental practice one can usually utilize particular cases of the Kresge – Gross –Butler equation to describe observed differential solvent isotope effects. When medium effects are unimportant, analysis of differential solvent isotope effects according to Equation 41.24 frequently allows one to realize a “proton inventory” of the studied reaction. The proton inventory technique is a particularly useful tool for characterizing the mechanisms of reactions that are accelerated by general acid/base catalysis, and whose transition state structures are therefore stabilized by bridging protonic interactions. Several enzyme reactions that have been so analyzed are discussed later in this chapter. For now, however, it is useful and instructive to discuss various scenarios for analysis of solvent isotope effects by the proton inventory method. Consider, for example, the multistep enzyme reaction outlined in Scheme 41.3. For this mechanism, the relationships among kcat ; kcat =Km and microscopic rate constants are given by Equation 41.25 and Equation 41.26: k3 k5 k3 þ k4 þ k5

ð41:25Þ

kcat k1 k3 k5 ¼ Km k2 ðk4 þ k5 Þ þ k3 k5

ð41:26Þ

kcat ¼ kE ;

If the chemical step is solely rate limiting (i.e., k3 p k5 and k4 p k5 ), then kcat ¼ k3 : Therefore, experimental probes of reaction mechanism and/or transition state structure are probes of the elementary step in which covalent bond rearrangements are occurring. Imagine a reaction in which medium effects and reactant state fractionation do not contribute palpably to the isotope effect

E + A

k1 k2

EA

k3 k4

EP

k5

E + P

SCHEME 41.3 Kinetic mechanism for a three-step enzyme reaction. E, A, and P, respectively, represent the free enzyme, free substrate and the product; EA and EP are, respectively, Michaelis complexes of the substrate and the product. Only the chemical step, denoted by rate constant k3 and its microscopic reverse k4 ; is subject to a solvent isotope effect.

Theory and Practice of Solvent Isotope Effects

1001

1.0

kn /k0

0.8

0.6

0.4 0.0

0.2

0.4

0.6

0.8

1.0

n

FIGURE 41.1 Proton inventories for enzyme-catalyzed reactions are diagnostic of the number and nature of transition state protonic interactions. Each proton inventory is for a reaction that generates a solvent isotope effect D2 O k ¼ 3:0: The linear plot arises from stabilization of the transition state by a single protonic interaction, and is described by kn =k0 ¼ 1 2 n þ 0:33n: Proceeding downward, the next two proton inventories arise, respectively, from reactions that involve equal contributions from two transition-state protons, as described by kn =k0 ¼ ð1 2 n þ 0:58nÞ2 ; and from three transition-state protons, as described by kn =k0 ¼ ð1 2 n þ 0:69nÞ3 : The plot with the most curvature arises from a mechanism in which a large number of transition-state protonic interactions individually make small contributions to the solvent isotope effect, and is described by kn =k0 ¼ 32n : Error bars show that, with a precision of ^ 2%, one-proton catalysis can be distinguished from two- or three-proton catalysis, which can be distinguished from a many-proton isotope effect (a medium effect).

(i.e., Zk;1 , 1; all fRi , 1), thereby reducing Equation 41.24 to the following: kn ¼ k0

x Y i¼1

ð1 2 n þ nfTi Þ

ð41:27Þ

In this situation, the measurement of differential solvent isotope effects and their analysis according to Equation 41.27 allows the experimentalist to enumerate the protonic interactions that contribute to transition state stabilization, and to determine their corresponding fractionation factors. One thereby realizes an inventory of transition state protonic interactions, from which models of reaction mechanism and transition state structure can be gleaned. Figure 41.1 displays proton inventory plots for an elementary reaction step in which one, two, three, and many (a medium effect) transition state protonic interactions make respective equal contributions to a solvent isotope effect D2 O k3 ¼ 3:0: Moreover, the figure shows the resolution of the proton inventory technique when rates or rate constants are determined to a precision of ^ 2%, a statistic that is realizable in many enzyme reaction assays. The point of these comparisons is that, in practice, the proton inventory technique can distinguish cases in which the transition state is stabilized by one, two, three, or many protonic interactions that contribute to the solvent isotope effect. Additional practical considerations to be weighed when utilizing solvent isotope effects and proton inventories to study enzyme reactions are discussed later in this chapter. 2. Effects of Reactant State Fractionation Numerous situations can be imagined in which fractionation in the reactant state contributes to the observed solvent isotope effect. The discussion herein is limited to a few common cases, which are

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Isotope Effects in Chemistry and Biology

2.5

1.0

kn /k 0

2.2

c

0.9

1.9

0.8 a

1.6

0.7 b

1.3

0.6 0.5

1.0 0.0

0.2

0.4

n

0.6

0.8

1.0

FIGURE 41.2 Effects of isotopic fractionation in the reactant state on the shapes of proton inventories. Plot (a) arises from isotopic fractionation at many reactants state sites, and is described by kn =k0 ¼ 0:402n : Plot (b) arises from isotopic fractionation at a single reactant site, and is described by kn =k0 ¼ 1=ð1 2 n þ 0:40nÞ: The scale for plots (a) and (b) is on the left-hand y-axis. Plot (c) shows an upward bulging proton inventory that arises when an isotope effect of 3 generated by a single transition-state proton transfer is partially offset by isotopic fractionation in a single reactant-state site, and is described by kn =k0 ¼ ð1 2 n þ 0:33nÞ=ð1 2 n þ 0:67nÞ: The scale for plot (c) is on the right-hand y-axis.

displayed graphically in Figure 41.2. Case 1, corresponding to curve b in Figure 41.2, involves an inverse solvent isotope effect of D2 O k ¼ 0:4 that arises from a single reactant state protonic site for which fR ¼ 0:4: This case produces maximum observed curvature for solvent isotope effects arising from reactant state fractionation, as the figure illustrates, and is described by Equation 41.28: kn ¼

k0 ð1 2 n þ nfR Þ

ð41:28Þ

This case is readily diagnosed by plotting 1=kn vs. n, which produces a linear plot. Case 2 considers a circumstance where D2 O k ¼ 0:4 arises from small albeit collective contributions to the inverse isotope effect from many sites. In this case kn ¼ k0 Z 2n ; and curve a in Figure 41.2 shows that the curvature of the proton inventory is less than that produced by Case 1. Plotting ln kn vs. n gives a straight line, diagnostic of a medium effect origin for the solvent isotope effect. Therefore, a multiprotonic isotope effect produces a proton inventory that is less curved than a proton inventory arising from a single reactant state site model. This trend contrasts with that which arises from transition state fractionating sites that contribute normal isotope effects, as earlier illuminated in Figure 41.1 and the associated discussion. Case 3 considers a situation in which a solvent isotope effect of 2.0 arises from offsetting contributions from a transition state site with fT ¼ 0:33 and a reactant state site with fR ¼ 0:67: This case gives rise to a proton inventory that bulges upward in shape, as shown in curve c of Figure 41.2. 3. Proton Inventories of Multistep Enzyme Reactions There are two types of kinetic complexity that can affect observations of solvent isotope effects and proton inventories. First is the relatively rare situation in which the reaction proceeds by two or more parallel routes. Consider a situation in which the erstwhile rate limiting k3 process is split into parallel k3a and k3b steps, with the former accompanied by one and the latter two solvent isotope

Theory and Practice of Solvent Isotope Effects

1003

effect producing transition state protonic interactions. Therefore, kcat ¼ k3a þ k3b and Equation 41.29 accounts for the observed proton inventory: kcat;n =kcat;0 ¼ f3a ð1 2 n þ nfT3a Þ þ f3b ð1 2 n þ nfT3b Þ2

ð41:29Þ

In Equation 41.29, f3a ¼

k3a k3a þ k3b

f3b ¼

k3b k3a þ k3b

ð41:30Þ

The fractions of rate determination by the parallel k3a and k3b steps are, respectively, given by f3a and f3b of Equation 41.30. In this situation the shape of the proton inventory will be dictated by the relative values of the rate constants of the parallel steps. If k3a q k3b ðf3a , 1Þ; the proton inventory will be linear, whereas if k3a p k3b ðf3b , 1Þ; the proton inventory will be nonlinear and downward bulging. Other scenarios can be envisioned; e.g., one of two parallel pathways produces an inverse solvent isotope effect and a nonlinear proton inventory that arises from a reactant state fractionation factor fR , 1 (cf., Figure 41.2 and the accompanying discussion). Whatever the detailed mechanistic circumstance, the salient point is that the shape of the proton inventory will be dictated by the more rapid, and hence more rate determining, of the parallel processes. Another and more common origin of kinetic complexity in enzymic and nonenzymic reactions arises when serial microscopic steps contribute comparably to rate limitation. For example, for the mechanism in Scheme 41.3, consider a situation in which k1 and k3 contribute to rate limitation, but product release is rapid (i.e., k5 q k4 ). Equation 41.26 therefore reduces to Equation 41.31: kE ¼

k1 k3 k2 þ k3

ð41:31Þ

If k3 is solvent isotope sensitive but k1 and k2 are not, then the relationship between the observed and intrinsic solvent isotope effects is given by Equation 41.32: D2 O

kE ¼

D2 O

k3 þ C ¼ f1 þ f3 D2 O k3 1þC

ð41:32Þ

In this equation C ¼ k3 =k2 is the commitment to catalysis and is related to the respective fractional rate limitations f1 and f3 of the k1 and k3 steps according to the following expressions: f1 ¼

C k3 ¼ 1þC k2 þ k3

f3 ¼

1 k2 ¼ 1þC k2 þ k3

ð41:33Þ

Equation 41.32 stresses that the intrinsic solvent isotope effect D2 O k3 is expressed in the observed solvent isotope effect D2 O kE just to the extent that k3 is rate limiting. Hence, if the commitment C is large, and therefore f3 is , 0, D2 O kE , 1; whereas when C is , 0 and f3 is , 1 the intrinsic isotope effect is expressed. When isotopically sensitive and insensitive steps contribute comparably to rate limitation, a proton inventory experiment can conceivably provide a resolution of the kinetic complexity of the studied reaction. Again, imagine that k3 alone in Equation 41.31 is subject to a palpable solvent isotope effect, and for the sake of illustration D2 O k3 arises only from isotopic fractionation at a single protonic interaction in the transition state. In this situation, k3;n is a linear function of the isotopic composition of the solvent: k3;n ¼ k3;0 ð1 2 n þ nfT3 Þ

ð41:34Þ

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Isotope Effects in Chemistry and Biology

Substitution of Equation 41.34 into Equation 41.31 gives an expression for kE;n that, on division by kE;0 and subsequent algebraic manipulation, produces the following equation: kE;n ð1 þ CÞð1 2 n þ nfT3 Þ 1 2 n þ nfT3 ¼ ¼ kE;0 1 þ Cð1 2 n þ nfT3 Þ f3 þ f1 ð1 2 n þ nfT3 Þ

ð41:35Þ

When fT3 is , 1, and therefore the observed solvent isotope effect is normal (i.e., . 1), Equation 41.35 predicts a proton inventory that bulges upward in shape. Consequently, fitting proton inventory data to Equation 41.35 allows the intrinsic solvent isotope effect D2 O k3 ¼ 1=fT3 and the commitment to catalysis C (or equivalently, the fractional rate determinations f1 and f3 ; see Equation 41.33) to be determined. Moreover, this information further defines the relative free energies of the serial transition states that contribute to rate limitation, according to DDG‡ ¼ 2RT ln C ¼ 2RT lnðf1 =f3 Þ; which in turn allows one to construct a quantitative free-energy diagram for the reaction. However, because reactant state fractionation can also produce a proton inventory that bulges upward in shape (see Section III.B.2), additional experimental evidence is needed to support a model wherein kinetic complexity arises from serial steps.

IV. FRACTIONATION FACTORS Section II and Section III ascribed the origins of solvent isotope effects to isotopic fractionation at exchangeable sites of reactant and product states (equilibrium isotope effects) or reactant and transition states (kinetic isotope effects). This section reviews typical magnitudes of the fractionation factors of functional groups and other reactive entities that are oft-observed contributors to enzyme-catalyzed reactions.

A. REACTANT S TATE F RACTIONATION FACTORS OF C OMMON F UNCTIONAL G ROUPS Table 41.1 lists fractionation factors of functional groups that are common constituents of the amino acid side chains of enzyme-active sites, as well as those of solvent species that are important in acid and base-catalyzed reactions (for a more extensive list of fractionation factors, see the review by Quinn and Sutton8). Several trends among these values are noteworthy: (a) most amino acid functional groups that serve as nucleophiles (e.g., OH of Ser, NH2 of Lys) or as general acid

TABLE 41.1 Fractionation Factors of Functional Groups Commonly Encountered in Enzyme Reactions Species LO2

f 0.43

L3Oþ RSL

0.69 0.4–0.6

ROL RCOOL RNL2 RNLþ 3 N-Methylacetamide Phenol Hydrogen bonds

1.04 0.92 1.13 1.08 1.12 1.13 ,1.0

Comments

f for aggregate LaO2(LbOLc)3; fa ¼ 1.25, fb ¼ 0.70, fc ¼ 1.00 f per O– L bond Typical range for thiols, including cysteine thiol sidechains of proline racemase12

Reference 10 11 8, and references cited therein 13 13 13 14 13 13 15

Theory and Practice of Solvent Isotope Effects

1005

catalysts (e.g., COOH of Asp and Glu, NþH of the imidazolium group of His) have fR values , 1. Hence these groups are not expected to make sizeable contributions that arise from reactant state fractionation to the solvent isotope effect. (b) Amide NH groups have fR , 1:1: Consequently, when peptide NH groups are involved in catalysis, their isotopic fractionation will make diminutive contributions to the solvent isotope effect. (c) The SH groups of thiols have inverse fractionation factors that generally fall in the range f R ¼ 0.4 to 0.6. Consequently, when a cysteine SH group is involved in catalysis, as a nucleophile for example, one may expect to observe an inverse solvent isotope effect. (d) The fractionation factor for H3Oþ is inverse, fR ¼ 0:69 per exchangeable site. One may reasonably expect similar fractionation factors in reactions wherein lyonium-like species are generated, such as upon equilibrium protonation of the ethereal oxygens of esters or acetals. (e) The fractionation factor for HO2 is fR , 0:4: Note that the predominant contribution to this net fractionation factor is thought to be three fractionation factors of fR ¼ 0:7 each for H-bonding between the solvent and the lone pairs of HO2. Consequently, one may expect a similar fractionation factor in an enzyme-catalyzed reaction that generates an alcoxide species, such as the conjugate base RO2 of an alcohol substrate or the side chains of Ser and Thr. The involvement of species that do not possess exchangeable protons can sometimes generate palpable solvent isotope effects. Such isotope effects arise when transfer of the relevant species from H2O to D2O is accompanied by a nonzero transfer-free energy, DGtr ; and are frequently referred to as medium effects (see Section III.A and Quinn and Sutton8). These effects arise from numerous small contributions from protonic positions in solvating waters. Many biologically important species have transfer-free energies that give rise to modest contributions to the solvent 2 isotope effect, computed as expð2DGtr =RTÞ: e.g., Mg2þ, 0.9; RCO2 2 , , 1; Cl , , 0.8. However, 2 4,16 thiolate anions, RS , have values in the range 0.4 to 0.7, and therefore involvement of the thiolate anion of Cys in an enzyme-catalyzed reaction is expected to generate a prominent medium effect contribution to the observed solvent-isotope effect.

B. TRANSITION S TATE F RACTIONATION FACTORS The chapter by Schowen in this volume provides a thorough exposition of the involvement of H-bonding interactions in enzyme catalysis, and discusses expectations and observations for the magnitudes of the fractionation factors that they generate. Consequently, the discussion herein of H-bond fractionation factors will be brief. Table 41.2 lists expectations for fractionation factors of transition-state protonic interactions, and includes examples of fractionation factors that have been measured in enzyme – ligand complexes that resemble high-energy intermediates and transition states. Most H-bonds (i.e., those with modest H-bond energies of # 5 kcal mol21) have fractionation factors that are near unity, and therefore do not make important contributions to the solvent isotope effect. Generally, such H-bonds involve donor – acceptor pairs for which the DpKa of the donor and the conjugate acid of the acceptor is large. However, H-bond strength increases as the donor/acceptor DpKa decreases,24 and correspondingly the fractionation factor of the bridging proton tends to be more inverse.18 Such stronger H-bonds therefore contribute to the solvent isotope effect. As discussed by Schowen (Chapter 29 in this volume), there are two classes of stronger H-bonds that are frequently primary determinants of the magnitude of solvent isotope effects on enzyme-catalyzed reactions: primary H-bonds arise from proton transfers among enzyme sidechain functions and substrates and are elements of general acid/base catalytic stabilization of the transition state. As Schowen argues, the proton transfers of primary H-bonds likely occur in stable vibrational modes of the transition state, and generate fractionation factors that are typically in the range fT ¼ 0:2 to 0.5. Therefore, even though proton transfers are occurring via primary H-bonds, the isotope effects that they produce are secondary isotope effects, since there is no proton motional amplitude in the reaction coordinate. Secondary H-bonds arise at sites that do not entail proton transfer, but due to structural changes in the reaction assembly when reactants are converted to

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Isotope Effects in Chemistry and Biology

TABLE 41.2 Fractionation Factors of Transition State H-Bonds and Associated Model Complexes f

Comments

Reference

Primary H-bonds

0.2 –0.5

Isotope effects arise from barrierless proton transfers

Secondary H-bonds

0.3 –0.7

Arise from H-bonds that are stronger in the transition state than in the reactant state In MeCN solvent Same f when OR is DMSO and DMF f of His57-Asp102 H-bond in complex of chymotrypsin with N-acetyl-Phe-trifluoroketone f of His57-Asp102 H-bond in complex of chymotrypsin with N-acetyl-Gly-Phe-trifluoroketone f of His57-Asp102 H-bond in complex of chymotrypsin with N-acetyl-Val-Phe-trifluoroketone f of His57-Asp102 H-bond in complex of chymotrypsin with N-acetyl-Leu-Phetrifluoroketone f of His64-Asp32 H-bond in complex of subtilisin with N-carbobenzoxy-Leu-Leu-Phe-trifluoroketone f of His64-Asp32 H-bond in complex of subtilisin with methoxysuccinnyl-Ala-Ala-Pro-2-amino3-phenylethyl boronic acid f of His438-Glu325 H-bond in complex of butyrylcholinesterase with m-(N,N,N-trimethylammonio)trifluoroacetophenone f of His447-Glu334 H-bond in complex of acetylcholinesterase with m-(N,N,N-trimethylammonio)trifluoroacetophenone

Schowen, Chapter 29 in this volume Schowen, Chapter 29 in this volume 17, 18 18 19

Species

(RO–H –OR)2 (RO–H –OR)þ þ RCO2 2 – H–ImH

0.27–0.47 0.36 0.32

þ RCO2 2 – H–ImH

0.34

þ RCO2 2 – H–ImH

0.39

þ RCO2 2 – H–ImH

0.43

þ RCO2 2 – H–ImH

0.85

þ RCO2 2 – H–ImH

0.53

þ RCO2 2 – H–ImH

0.63

þ RCO2 2 – H–ImH

0.76

19 19 19

20 21

22

23

the transition state experience fractionation factor changes (generally from fR , 1 to fT , 1). Secondary H-bonds may give rise to solvent isotope effect contributions that rival in magnitude those of primary H-bonds, as shall be seen later in this chapter when specific enzyme reactions are discussed.

V. PRACTICAL CONSIDERATIONS Precision is a consideration when utilizing the proton inventory technique to tally the contributions of protonic interactions to the solvent isotope effect.3,8 As illustrated in Figure 41.1, when the solvent isotope effect is D2 O k ¼ 3 and the precision of rate or rate constant measurements is ^ 2%, a frequently achievable reproducibility for enzyme kinetics assays, one can distinguish one-proton catalysis from two- or three-proton catalysis, and two- or three-proton catalysis in turn from manyproton catalysis. The pH and pD dependencies of enzyme-catalyzed reactions can, when not properly accounted for, complicate interpretation of the magnitudes of solvent isotope effects and mechanistic implications of proton inventories. This consideration is illustrated in Figure 41.3, which plots simulated pH and pD rate profiles for a reaction whose rate depends on the basic form of a residue that has pKa1 ¼ 5 and the acidic form of a residue that has pKa2 ¼ 9, and for which D2 O k ¼ 2:0: This situation could arise for an enzyme reaction whose rate depends, respectively, on Asp-COO2

Theory and Practice of Solvent Isotope Effects

1007

1.0

Relative Rate

0.8 0.6 0.4 0.2 0.0 4

5

6 7 8 pL (L = H or D)

9

10

FIGURE 41.3 Solvent-isotope effects on the pL-rate profiles of enzyme-catalyzed reactions. The top plot is for the reaction in H2O, and the bottom plot for the reaction is D2O. Each bell-shaped profile is described by R ¼ A=ð1 þ 10pKa1 2pL þ 10pL2pKa2 Þ; where R is the measured relative rate and A is the pL-independent maximum relative rate. For the pH-rate profile the maximum relative rate is set at A ¼ 1; and the respective acid-ionization constants for the acidic and basic limbs of the profile are pKa1 ¼ 5.0 and pKa2 ¼ 9.0, respectively. The pD-rate profile is calculated for a solvent-isotope effect on the maximum rate of 2.0, and therefore A ¼ 0.5, and uses pKa1 ¼ 5.48 and pKa2 ¼ 9.48, as described in the text. Consequently, the inflections on the acidic and basic limbs of the profile are shifted , 0.5 pL units for the reaction in D2O vs. that in H2O, and a similar shift is seen for the profile maxima. R and Lys-NHþ 3 functional groups. Since f , 1 for the species involved in the acid –base equilibria þ (i.e., RCOOH, RNH3 , RNH2), the following version of Equation 41.6 provides the expected solvent isotope effect on the Ka values:

KaD2 O ¼ KaH2 O ðfL3 Oþ Þ3 ¼ KaH2 O ð0:69Þ3

ð41:36Þ

The factor (0.69)3 arises from isotopic fractionation of L3Oþ, the isotopic species of the hydronium ion, which are products of each acid– base equilibrium and possess three equivalent isotopically fractionating sites. Therefore: pKaD2 O ¼ pKaH2 O þ 0:48

ð41:37Þ

The pKa values that dictate the pH dependencies of enzyme-catalyzed reactions frequently shift upward in D2O vs. H2O by , 0.5 pH units,4,8 as Equation 41.37 demonstrates and Figure 41.3 illustrates. Consequently, in order to measure kinetic solvent isotope effects that are not complicated by equilibrium isotope effects on acid dissociation, measurements should be conducted in equivalent buffer of H2O and D2O. Equivalent isotopic buffers are those that contain the same concentrations of all solute species. This strategy insures that the buffer [acid]/[base] ratio is the same in H2O and D2O and, since most buffer pKa values are larger in D2O than H2O by , 0.5 pH units,4 that rates are measured at equivalent positions on the pL-rate profiles (L ¼ H,D). If the respective pH and pD profiles have been determined, then unbiased solvent isotope effects and proton inventories can be realized by measuring rates or rate constant at pL (L ¼ H,D) values where dk=dpL ¼ 0: For the profiles in Figure 41.3, this corresponds to making measurements at the peak of the pL dependencies. Measurement of pL (L ¼ H,D) values for solvent isotopic buffers can be conducted by using pH meters that are equipped with glass combination electrodes. However, one must keep in mind

1008

Isotope Effects in Chemistry and Biology

that the potential that is measured at various pL values is a result of the acid ionization behavior of the glass electrode surface. As for other organic and inorganic acids, the response of a glass electrode is sensitive to the solvent isotope. For buffers in D2O, one gets the pD value by adding 0.4 to the pH meter reading.3,4,8,25

VI. EXAMPLES The mechanistic utility of solvent isotope effects and proton inventories is best illustrated with some exemplary studies from the literature. While not inclusive, the examples that follow are illustrative of the range of mechanistic features that can be addressed by solvent isotope effect measurements.

A. NONENZYMIC R EACTIONS Schowen and coworkers26 used the proton inventory method to characterize transition state proton transfers for the basic methanolysis of aryl-substituted N-methyltrifluoroacetanilides (Scheme 41.4). Proton transfer mediated decomposition of the tetrahedral intermediate T 2 is rate limiting, but the relative timing of proton transfer and leaving group expulsion depends on the aryl substituent Y. When Y ¼ p-MeO, the leaving group is most basic, Dk ¼ 7.4 and fT ¼ 0:14: This large isotope effect and corresponding markedly inverse transition state fractionation factor are consistent with rate-limiting proton transfer from MeOH to the amido N of T 2 to generate a zwitterionic tetrahedral intermediate that rapidly collapses to products. In this case the reaction coordinate motion that transits the rate-limiting transition state is that for O to N proton transfer, which generates a sizeable, primary H/D isotope effect. However, when the leaving group is less basic, smaller isotope effects and less inverse transition state fractionation factors are observed: Y ¼ m-NO2, Dk ¼ 2.6, fT ¼ 0:38; Y ¼ p-Cl, Dk ¼ 3.5, fT ¼ 0:29: For these substrates the mechanism has shifted to one in which proton transfer and leaving group expulsion are synchronous, albeit in which the reaction coordinate is that for the heavy atom rearrangement which effects C – N bond scission. The much smaller isotope effects arise from general acid catalytic primary H-bonds, which give secondary H/D isotope effects, as discussed by Schowen (Chapter 29 in this volume) and summarized above in Section IV.B.

MeO

H

O−

O Y

N C CF3 Me

−OMe

N C

Y

CF3

Me OMe T−

H O− N Y

+

C

CF3

Me OMe +

MeO − + T−

NHMe

Y

+ CF3CO2Me +

MeO −

SCHEME 41.4 Mechanism of basic methanolysis of aryl substituted N-methyltrifluoroacetanilides.

Theory and Practice of Solvent Isotope Effects

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These nonenzymic isotope effects have implications for the interpretation of solvent isotope effects and proton inventories for enzyme-catalyzed reactions. Solvent isotope effects that arise from transition state proton bridges that have fT values in the 0.3 to 0.4 range are secondary isotope effects produced by general acid/base primary H-bonds. This range is included in the fT ¼ 0:2 to 0.5 range cited in Table 41.2 for fractionation factors of primary catalytic H-bonds. Fractionation factors of this magnitude have often been postulated to accord with proton inventory data for enzyme-catalyzed reactions, or have been observed by NMR for enzyme-inhibitor complexes that resemble catalytic transition states (see Table 41.2).

B. ENZYMATIC R EACTIONS 1. Serine Proteases The chemical mechanism of serine protease catalysis involves the operation of an active site AspHis-Ser H-bonded triad that provides nucleophilic and general acid – base functions,27 as outlined in Scheme 41.5. Though some elements of the mechanism remain controversial, consensus appears to be developing that the following protonic interactions contribute to transition state stabilization in serine protease catalysis: (a) the active site His serves, respectively, as a general base catalyst for the formation and a general acid catalyst for the decomposition of tetrahedral intermediates. The associated proton transfers occur in stable modes of the transition state, are referred to as primary H-bonds (see Chapter 29 by Schowen in this volume and Section IV.B of this chapter), and, since they have significantly inverse fractionation factors, will make normal contributions to the solvent isotope effect. (b) As imidazolium cationic character develops in the tetrahedral intermediates and transition states that flank these intermediates, the strength of the Asp-His H-bond increases.28,29 This stronger H-bond stabilizes the tetrahedral intermediates and flanking transition states, albeit modestly, and because of its inverse fractionation factor also makes a normal contribution to the solvent isotope effect. Since this transition state protonic interaction does not result in proton transfer, it is an example of a secondary H-bond (see Chapter 29 by Schowen in this volume). (c) The carbonyl oxyanion of tetrahedral intermediates is stabilized in mammalian serine proteases by H-bonding to two peptide NH groups, which comprise the oxyanion hole.27 Due to a sizeable pKa mismatch between the NH donors and the conjugate acid of the carbonyl oxyanion, these H-bonds are expected to be relatively weak and therefore to have fT , 1: Consequently, these protonic interactions do not make sizeable contributions to the solvent isotope effect. Though the discussion of the preceding paragraph suggests that proton inventories of serine protease reactions will indicate that two protonic interactions contribute to the solvent isotope effect, such is not always the case. Table 41.3 summarizes proton inventory results for various serine protease-catalyzed reactions. As the entries in the table show, for minimal substrates a single transition-state proton contributes to the solvent isotope effect (i.e., the proton inventories are linear), whilst peptide substrates that interact with additional subsites of the extended active sites of the enzymes elicit two-proton catalysis, as signaled by proton inventories that are nonlinear and bulge downward. This is a reasonable evolutionary strategy for maximizing catalytic power and substrate specificity in serine protease reactions. Multiple subsite interactions with extended substrates call more prominently into play the His-Asp H-bond, an interaction that in turn increases the basicity of the His imidazole side chain and consequently provides extra transition-state stabilization.28,29 This idea is supported by the NMR studies reported by Frey and coworkers28 of complexes of chymotrypsin with peptidyl trifluoromethyl ketones. They find a correlation between the basicity of His57 of the chymotrypsin active site and the strength of binding of the inhibitor; for example: N-acetyl-Leu-Val-CF3, Ki ¼ 170 mM, pKa(His57) ¼ 10.3; N-acetyl-Gly-Phe-CF3, Ki ¼ 15 mM, pKa(His57) ¼ 11.1; N-acetyl-Leu-Phe-CF3, Ki ¼ 1.8 mM, pKa ¼ 12.1. Moreover, the potency of inhibition, as measured by pKi, correlated linearly with logðkcat =Km Þ for analogous

1010

Isotope Effects in Chemistry and Biology Deacylation:

Acylation: His

O−

H

N

His

Ser N

H

O

O

R C

R

O

X

R1

C

R2

−

H

N

Ser H

N

C

H

O

O

O O

C

R2

Acylenzyme Intermediate

O Michaelis Complex

His

R C

O−

H

His

Ser

N + N

O

O−

H

O

R1 X

C R2 O_

R2

O_

Tetrahedral Intermediate R2CO2− + H+

His

N

His

Ser N

O−

O

R C O

O

HO C

O

R1XH

H

N + N H

H

R C

Tetrahedral Intermediate

O−

Ser

O

C

R2

H

R C

N

Ser N

H O

O

Acylenzyme Intermediate

SCHEME 41.5 Mechanism of serine hydrolase catalysis. For serine proteases RCO2 2 is Asp, while for cholinesterases RCO2 is Glu. Typical substrates are anilides or peptides (X ¼ NH) for serine proteases, and 2 esters (X ¼ O) or thioesters (X ¼ S) for serine proteases and cholinesterases.

methyl ester substrates. Therefore, P1 and P2 amino acid residues of trifluoroketone inhibitors that confer maximal inhibition also produce the best chymotrypsin substrates. Stein et al.33 characterized transition state structures for chymotrypsin and elastase-catalyzed reactions by measurement of b-deuterium secondary isotope effects and proton inventories. Of particular interest herein are the results for chymotrypsin-catalyzed hydrolysis of N-benzyloxycarbonyl-Gly-p-nitrophenyl ester, a reaction for which kcat is rate limited by deacylation (cf., Scheme 41.5). Kinetic b-deuterium isotope effects for this reaction were used to estimate the fractional tetrahedral character in the transition state as F ¼ 0:84: Imagine that, of the two chemical steps in the deacylation stage of catalysis, formation of the tetrahedral intermediate is rate limiting. One can then use the fractional tetrahedral character in the following equation to estimate the pKa of the attacking water in the transition state: pKa‡ ¼ 15:7 2 ½15:7 2 ð21:7Þ

F

ð41:38Þ

Theory and Practice of Solvent Isotope Effects

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TABLE 41.3 Proton Inventory Results for Serine Protease-Catalyzed Reactions Enzyme Trypsin Chymotrypsin

Porcine pancreatic elastase

Human leukocyte elastase

Substrate N-Benzoyl-L-Arg ethyl ester N-Benzoyl-L-Phe-L-Val-L-Arg p-nitroanilide p-Nitrophenyl acetate N-Benzyloxycarbonyl-L-Phe p-nitrophenyl ester N-Succinyl-L-Ala-L-Ala-L-Pro-L-Phe thiobenzyl ester N-Benzyloxycarbonyl-Gly p-nitrophenyl ester p-Nitrophenyl acetate N-Benzyloxycarbonyl-Gly p-nitrophenyl ester N-Methoxysuccinyl-L-Ala-L-Pro-L-Ala thiobenzyl ester N-Acetyl-L-Ala-L-Pro-L-Ala p-nitroanilide N-Methoxysuccinyl-L-Val thiobenzyl ester N-Methoxysuccinyl-L-Ala-L-Pro-L-Ala p-nitroanilide N-Methoxysuccinyl-L-Ala-L-Ala-L-Pro-L-Val thiobenzyl ester

D2

Ok

xa

Reference

3.03 4.00 2.45 2.45 2.44 3.34 2.45 2.45 3.08 2.15 2.72 3.43 2.87

1 2 1 1 2 2 1 1 2 2 1 2 2

30 30 31 5 32 33 30 33 32 34 32, 35 35 35

In this equation 15.7 is the pKa of the attacking water before the onset of nucleophilic attack on the acylenzyme intermediate, and 15.7 2 (2 1.7) is the pKa change that H2O experiences on equilibrium formation of the tetrahedral intermediate. The pKa ¼ 2 1.7 of H3Oþ is used as an estimate for the pKa of the nucleophilic water in the tetrahedral intermediate. Using F ¼ 0:84 in Equation 41.38 gives pK‡a , 5. One can now estimate the fractional proton transfer to His57 in the transition state:

b¼

10DpKa 1 þ 10DpKa

ð41:39Þ

In Equation 41.39 DpKa is the difference in the pKa values of the imidazole side chain of His57 and the nucleophilic water. If pKa ¼ 7 for His57, then DpKa , 2 and b ¼ 0:99; an indication that proton transfer from the attacking water to His57 is essentially complete in the transition state. This is, of course, a reasonable surmise, since a late transition state is indicated by F ¼ 0.84. The solvent isotope effect for kcat of chymotrypsin-catalyzed hydrolysis of N-benzyloxycarbonyl p-nitrophenyl ester was D2 O kcat ¼ 3:34; and proton inventory analysis indicated that this isotope effect arises from two transition state protonic interactions that each have fT ¼ 0:54: The simplest mechanistic model that accommodates these results is that the primary H-bond between the nucleophilic water and His57 and the secondary H-bond of the Asp-His diad have similar transition state fractionation factors and hence contribute comparably to the solvent isotope effect. One can estimate the fractionation factor of the Asp-His H-bond in the tetrahedral intermediate of the chymotrypsin reaction by using the following equation:

fT ¼ ðfTI Þb

ð41:40Þ

Using fT ¼ 0:54 and b ¼ 0:99 from the preceding paragraph, one calculates fTI ¼ 0:54; a result that is not surprising given the prominent extent of proton transfer from the nucleophilic H2O to His57 in the transition state. Moreover, this estimate of fTI falls in the range of fractionation factors in Table 41.2 for serine protease transition state analog complexes that mimic the deacylation transition state.

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Isotope Effects in Chemistry and Biology

Stein et al.33 reported that for elastase-catalyzed hydrolysis of N-benzyloxycarbonyl-Gly-pnitrophenyl ester the solvent isotope effect was D2 O kcat ¼ 2:45 and the corresponding proton inventory was linear, consistent with a mechanism in which a single transition-state proton contributes to the solvent-isotope effect. Why the marked difference between the chymotrypsin and elastase-catalyzed reactions? The difference can be rationalized by repeating the analysis just applied to the chymotrypsin reaction. For the elastase reaction, Stein et al. estimated F ¼ 0:43 from the b-deuterium secondary isotope effect on kcat : When this measure of fractional tetrahedral character is used in Equation 41.38, one gets pKa‡ , 12 for the nucleophilic water in the transition state. Using this value in Equation 41.39 gives b ¼ 1025 : Consequently, if fTI ¼ 0:54 as in the chymotrypsin reaction, from Equation 41.40 one gets fT ¼ 1:0 for the Asp-His H-bond in the transition state of the elastase-catalyzed reaction. That is, the absence of two-proton catalysis for the elastase reaction is a result of the fact that the transition state is early with respect to development of tetrahedral character, and, pari pasu, is very early with respect to proton transfer from the nucleophilic water to His57. 2. Acetylcholinesterase Acetylcholinesterase (AChE) is a hydrolytic enzyme whose function is to hydrolyze the neurotransmitter acetylcholine at synapses and neuromuscular junctions in the central and peripheral nervous systems, respectively.36 – 38 Since the frequency of cholinergic neurotransmission is several hundred Hz, AChE must clear the synapse of acetylcholine in a few milliseconds, and consequently the enzyme has evolved to very high catalytic efficiency. The chemical mechanism by which AChE achieves this highly efficient catalysis is formally similar to that of the serine proteases in Scheme 41.5. In the numbering system of mammalian AChEs, the constituents of the active site triad are Ser203, Glu334, and His447. Consider the kinetic mechanism for AChE catalysis in Scheme 41.6. The steady-state kinetic parameters that come from this mechanism are given in the following equations: kcat ¼

Vmax k3 k5 ¼ ½E T k3 þ k5

ð41:41Þ

kE ¼

kcat k1 k3 ¼ Km k2 þ k3

ð41:42Þ

Solvent isotope effects on kE are diminutive for various AChEs,39 an observation that is consistent with prominent rate limitation by diffusional encounter of enzyme and substrate. This is the expectation for an enzyme whose catalytic function is highly evolved and that catalyzes an exoergonic reaction. For example, partial rate limitation by diffusional encounter is supported by proton inventory analysis of kE for human AChE-catalyzed hydrolysis of acetylthiocholine,39 as shown in Figure 41.4. The analysis of the proton inventory assigned a medium effect of 1.2 to the diffusional encounter k1 step and to its microscopic reverse k2 step, since D2O is more viscous than H2O by a factor of 1.2. Also, the chemical catalytic step k3 was assumed to involve a single transition state proton that contributes prominently to the solvent isotope effect; this assumption

E + A

k1 k2

EA

k3

F + P

k5

E + P + Q

SCHEME 41.6 Kinetic mechanism of AChE-catalyzed hydrolysis of acetylthiocholine. E and A are the free enzyme and the free substrate, respectively. The Michaelis complex and the acylenzyme intermediate are designated as EA and F, respectively, and the respective products P and Q are thiocholine and acetate.

Theory and Practice of Solvent Isotope Effects

1013

1.0

kn /k0

0.8

0.6

0.4

0.0

0.2

0.4

n

0.6

0.8

1.0

FIGURE 41.4 Proton inventories for human AChE-catalyzed hydrolysis of acetylthiocholine. Filled circles are for the wild-type AChE reaction, and the solid line is a least-squares fit to Equation 41.43 that gives C ¼ 3 ^ 1 and fT3 ¼ 0:33 ^ 0:06; the corresponding intrinsic solvent isotope effect is D2 O k3 ¼ 1=fT3 ¼ 3:0 ^ 0:6: Open circles are for the Glu202Ala mutant reaction, and the solid line is a least-squares fit to Equation 41.43 that gives C ¼ 0:4 ^ 0:3 and fT3 ¼ 0:39 ^ 0:06; the corresponding intrinsic solvent isotope effect is D2 O k3 ¼ 1=fT3 ¼ 2:6 ^ 0:4:

is justified later. When these factors are considered, algebraic manipulation of Equation 41.42 produces Equation 41.43: kE;n ð1 þ CÞð1 2 n þ nfT3 Þ ¼ kE;0 1 þ 1:2n Cð1 2 n þ nfT3 Þ

ð41:43Þ

In this equation, kE,0 and kE;n are values of kE in H2O and in mixtures of H2O and D2O of atom fraction of deuterium n, respectively, C ¼ k3 =k2 is the commitment to catalysis, and fT3 is the fractionation factor generated by the transferring proton in the k3 step. The intrinsic solvent isotope effect is given by D2 O k3 ¼ 1=fT3 : The nonlinear fit to Equation 41.43 of the proton inventory in Figure 41.4 for wild-type human AChE-catalyzed hydrolysis of acetylthiocholine gave C ¼ 3 ^ 1 and D2 O k3 ¼ 3:0 ^ 0:6: Therefore, from Equation 41.33 one calculates that the diffusional encounter step k1 is 75% rate limiting and the chemical step k3 is 25% rate limiting. The effects of viscosity on kE supported this model for the acylation reaction dynamics of human wild-type AChE-catalyzed hydrolysis of acetylthiocholine.39 One might imagine that a judicious choice of active site mutation could result in selective destabilization of the transition state of the chemical step k3 ; and therefore according to Equation 41.43 the observed solvent isotope effect would be increased and the proton inventory would become linear. This is just what happens for the Glu202Ala mutant of human AChE,39 as also shown in Figure 41.4. In the crystal structure of the complex of Torpedo californica AChE and the transition state analog inhibitor m-(N,N,N-trimethylammonio) trifluoroacetophenone (TMTFA), Glu202 makes a close contact not only with the quaternary ammonium function of TMTFA but also with His440 of the catalytic triad.40 Consequently, it is not surprising that the Glu202 to Ala mutation renders the chemical step rate limiting. Moreover, the solvent isotope effect for the Glu202Ala mutant is 2.6 ^ 0.4, in statistical agreement with the intrinsic solvent isotope effect that arose from curve fitting the proton inventory for the wild-type AChE reaction. The linear proton inventory for the Glu202Ala mutant suggests that, at least for the acylation transition state, AChE does not function as a multiproton catalyst. However, one might argue that

1014

Isotope Effects in Chemistry and Biology

mutation has reduced the full panoply of transition state stabilizing interactions that the wild-type enzyme can bring to bear. This does not appear to be the case. For fetal bovine serum, Electrophorus electricus, Torpedo californica, and human AChEs the proton inventories for Vmax (and hence kcat ) of acetylthiocholine, propionylthiocholine, and butyrylthiocholine hydrolyses are linear, indicating that a single proton contributes prominently to the solvent isotope effect.41 According to Equation 41.41 and the acylenzyme trapping experiments of Froede and Wilson,42 both acylation and deacylation contribute to rate limitation of kcat of acetylthiocholine hydrolysis. One therefore might suggest that multiproton catalysis is obscured by kinetic complexity. However, both the k3 and k5 components of kcat are chemical steps that should be similarly affected by protontransfer catalysis involving His440. Moreover, for the alternate aryl substrates phenylacetate43 and o-nitrophenylacetate,44 whose kcat value are similar to that for acetylthiocholine, the proton inventories are again linear. Therefore, the weight of evidence, gleaned from experiments with mutant and wild-type AChEs and with different substrates, supports that AChE is a one-proton catalyst. Simple general acid– base catalysis (i.e., one-proton catalysis) by AChE can be rationalized by considering additional observations. Massiah et al.23 have detected a downfield chemical shift at d ¼ 17:8 in the 1H-NMR spectrum of the complex of TMTFA and human AChE. This resonance, which is attributed to the Glu334-His447 H-bond, has a fractionation factor f ¼ 0:76: Malany et al.39 utilized kinetic b-deuterium isotope effects to characterize the structure of the transition state for the k3 step (acylation) of human AChE-catalyzed hydrolysis of acetylthiocholine. These experiments yielded a value of the fractional tetrahedral character of F ¼ 0:76 in the transition state. Since Ser203 is the nucleophile in the acylation stage of catalysis, the following analog of Equation 41.38 is used: pKa‡ ¼ 16 2 ½16 2 ð22Þ

F

ð41:44Þ

In this equation, 16 and 2 2 are estimates of the pKa values of the g-OH of Ser203 before the onset of nucleophilic attack and in the tetrahedral intermediate, respectively. Therefore, F ¼ 0:76 in Equation 41.44 gives pK‡a , 7 for g-OH of Ser200 in the transition state, which, along with pKa ¼ 6.3 for His440, can be put into Equation 41.39 to yield a fractional extent of proton transfer of b ¼ 0:2: Using f ¼ 0:76 measured by Massiah et al.23 as an estimate of fTI and b ¼ 0:2 in Equation 41.40 gives fT ¼ 0:95 for the Glu334-His447 H-bond in the acylation transition state. Consequently, multiproton catalysis is not observed for AChE-catalyzed hydrolysis of acetylthiocholine because the transition-state fractionation factor of the Glu334-His447 H-bond is only modestly different than unity. The only prominent contribution to the solvent isotope effects of 2 to 3 that are commonly observed for kcat of AChE-catalyzed reactions arises from proton transfer between Ser203 and His447 in the acylation stage of catalysis or between H2O and His447 in the deacylation stage of catalysis. 3. Carbonic Anhydrase The mechanism of carbonic anhydrase (CA) has been presented in considerable detail in the chapter in this volume by Silverman and in a review by Lindskog.45 Accordingly, the discussion herein will focus on the solvent isotope effect and proton inventory analysis for kcat of CA catalysis. CA þ catalyzes the reversible hydration of CO2 to produce HCO2 3 þ H3O . The nucleophilic element for 2 the hydration reaction is a water species, probably HO , bound to an active site Zn2þ. This Zn – OH adds to CO2 to generate the bound HCO2 3 product, whose dissociation is followed by H2O binding to the active site zinc. General-base proton removal from the zinc-bound water by His64 utilizes an intervening chain of two water molecules. Subsequent His64-Hþ deprotonation by buffer returns the enzyme to the correct protonation state for another catalytic cycle.

Theory and Practice of Solvent Isotope Effects

1015

Venkatasubban and Silverman46 characterized the transition state for kcat of CA-catalyzed hydration of CO2 by measuring solvent isotope effects and proton inventories. At high buffer concentration, proton transfer from Zn – OH2 to His64 through the intervening solvent chain is the rate-limiting step for kcat of CO2 hydration. The solvent isotope effect was D2 O kcat ¼ 3:2 and ln kcat;n was a linear function of n, a proton inventory observation that accords with a many-proton origin for the isotope effect (see Equation 41.19 to Equation 41.23 and the accompanying discussion). Therefore, the proton inventory is consistent with concerted proton transfers from Zn– OH2 to His64, a mechanistic event that would produce fractionation factor changes in at least four protons of the water chain on conversion of the reactant to the transition state. 4. Tyrosine Hydroxylase Frantom and Fitzpatrick47 have measured substrate and solvent isotope effects on the tyrosine hydroxylase (TyrH)-catalyzed hydroxylation of tyrosine to 3,4-dihydroxyphenylalanine (DOPA) as the product. This reaction utilizes a mononuclear iron active site that generates a Fe(IV)O electrophile from O2 and tetrahydropterin (PH4) as additional substrates. For the wild-type enzyme, both a-deuterium secondary (measured with 3,5-L2-Tyr, L ¼ H or D) and solvent isotope effects on Vmax are unity,48,49 while for the slower substrate 5-2H-tryptophan an inverse secondary isotope effect Dk ¼ 0.93 is observed.50 Consequently, for the tyrosine hydroxylation reaction, other slow steps must be masking the intrinsic substrate isotope effect that arises from sp2 to sp3 rehybridization in the electrophilic addition step. For the wild-type enzyme, a 1:1 stoichiometry between tyrosine hydroxylation and PH4 oxidation to PH2 is observed. However, for the mutant enzymes F309A, E326A and H336E, these reactions are uncoupled such that a large excess of PH4 is consumed per tyrosine hydroxylated, as outlined in Scheme 41.7. The ratio of tyrosine hydroxylation to PH4 consumption is C ¼ k1 =ðk1 þ k2 Þ; which Frantom and Fitzpatrick call the coupling, and therefore unmasks intrinsic solvent and substrate isotope effects on k1 and k2 ; no matter whether there are other slow steps in the mechanism. For the E326A mutant, respective substrate and solvent isotope effects on the coupling are D C ¼ 0:93 ^ 0:04 and D2 O C ¼ 0:49 ^ 0:02; values that compare favorably to the corresponding isotope effects on Vmax ; D Vmax ¼ 0:88 ^ 0:03 and D2 O Vmax ¼ 0:53 ^ 0:03: Frantom and Fitzpatrick constructed a proton inventory for C by measuring the ratio of DOPA to PH2 products in various mixtures of buffered H2O and D2O. Their experiment provides an example of the utilization of the proton inventory technique to characterize mechanistic feature in an enzymic reaction that involves parallel pathways (see Section III.B.3 of this chapter). The proton inventory was linear and is described by Equation 41.45: Cn =C0 ¼ f2 ð1 2 n þ nfT2 Þ þ f1

ð41:45Þ

In this equation, f1 ¼ k1 =ðk1 þ k2 Þ and f2 ¼ k2 =ðk1 þ k2 Þ are fractions of rate limitation of the parallel k1 and k2 steps. The proton inventory yielded D2 O k2 ¼ 1=fT2 ¼ 2:38 ^ 0:04: This solvent

TyrH + O2 + PH4 + Tyr

k1 TyrH-O2-PH4-Tyr

TyrH-Tyr-X k2

TyrH + DOPA + PH2 + H2O TyrH + Tyr + PH2 + H2O2

SCHEME 41.7 Kinetic mechanism for branching of TyrH-catalyzed hydroxylation of Tyr between productive turnover through k1 vs. unproductive reaction via k2 : The second complex TyH-Tyr-X is probably the complex of Tyr with the Fe(IV)O form of TyrH that is competent for electrophilic attack on the aromatic ring of the bound Tyr substrate.

1016

Isotope Effects in Chemistry and Biology

isotope effect was interpreted as arising from water attack on the Fe(IV)O intermediate, which leads to unproductive turnover of this intermediate and consumption of an equivalent of PH4.

VII. CONCLUSIONS Measurement of solvent isotope effects and proton inventories continues as the premier method for experimental characterization of transition-state protonic interactions, in particular the primary H-bonds that are characteristic of general acid/base catalysis and the secondary H-bonds that accompany rearrangement of the heavy-atom framework of the substrate in the transition state. Recent advances, such as the NMR measurement of fractionation factors of H-bonds in complexes of enzymes with transition state analogs, have extended the mechanistic utility of solvent-isotope effects and proton inventories. Parallel measurements of substrate and solvent-isotope effects provide enhanced mechanistic insights, whilst the measurement of solvent-isotope effects and proton inventories for mutant enzymes can frequently unmask intrinsic behaviors. These advances insure that solvent isotope effects will continue as a useful probe of enzyme reaction mechanisms and transition state structure.

REFERENCES 1 Loh, S. N. and Markley, J. L., Hydrogen bonding in proteins as studied by amide hydrogen H/D fractionation factors: application to staphylococcal nuclease, Biochemistry, 33, 1029– 1036, 1994. 2 Schowen, R. L., Solvent isotope effects on enzymic reactions, In Isotope Effects on Enzyme-Catalyzed Reactions, Cleland, W. W., O’Leary, M. H., and Northrop, D. B., Eds., University Park Press, Baltimore, MD, pp. 64 – 99, 1977. 3 Schowen, K. B. J., Solvent hydrogen isotope effects, In Transition State of Biochemical Processes, Gandour, R. D. and Schowen, R. L., Eds., Plenum, New York, pp. 225– 283, 1978. 4 Schowen, K. B. and Schowen, R. L., Solvent isotope effects on enzyme systems, Methods Enzymol., 87, 551– 606, 1982. 5 Venkatasubban, K. S. and Schowen, R. L., The proton inventory technique, Crit. Rev. Biochem., 17, 1 – 44, 1984. 6 Alvarez, F. J. and Schowen, R. L., Mechanistic deductions from solvent isotope effects, In Isotopes in Organic Chemistry, Vol. 7, Buncel, E. and Lee, C. C., Eds., Elsevier, Amsterdam, pp. 1 – 60, 1987. 7 Kresge, A. J., More O’Ferrall, R. A., and Powell, M. F., Solvent isotope effects, fractionation factors and mechanisms of proton transfer reactions, In Isotopes in Organic Chemistry, Vol. 7, Buncel, E. and Lee, C. C., Eds., Elsevier, Amsterdam, pp. 177– 273, 1987. 8 Quinn, D. M. and Sutton, L. D., Theoretical basis and mechanistic utility of solvent isotope effects, In Enzyme Mechanism from Isotope Effects, Cook, P. F., Ed., CRC Press, Boca Raton, FL, pp. 73 – 126, 1991. 9 Glasstone, S., Laidler, K. J., and Eyring, H., The Theory of Rate Processes, McGraw-Hill, New York, 1941. 10 Gold, V. and Grist, S., Deuterium solvent isotope effect on reaction involving the aqueous hydroxide ion, J. Chem. Soc. Perk. Trans., 2, 89 – 95, 1972. 11 Kresge, A. J. and Allred, A. L., Hydrogen isotope fractionation in acidified solutions of protium and deuterium oxide, J. Am. Chem. Soc., 85, 1541, 1963. 12 Belasco, J. G., Bruice, T. W., Albery, W. J., and Knowles, J. R., Energetics of proline racemase: fractionation factors for the essential catalytic groups in the enzyme– substrate complex, Biochemistry, 25, 2558– 2564, 1986. 13 Jarret, R. M. and Saunders, M., A new method for obtaining isotopic fractionation data at multiple sites in rapidly exchanging systems, J. Am. Chem. Soc., 107, 2648– 2654, 1985. 14 Salomaa, P., Schaleger, L. L., and Long, F. A., Ionization of some weak acids in water – heavy water mixtures, J. Phys. Chem., 68, 410– 411, 1964.

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15 Arnett, E. M. and McKelvey, D. R., Solvent isotope effect on thermodynamics of nonreacting solutes, In Solute – Solvent Interactions, Coetzee, J. F. and Ritchie, C. D., Eds., Marcel Dekker, New York, pp. 343– 398, 1969. 16 Jancso, G. and Van Hook, W. A., Condensed phase isotope effects (especially vapor pressure isotope effects), Chem. Rev., 74, 689– 719, 1974. 17 Kreevoy, M. M., Liang, T. M., and Chang, K.-C., Structures and isotopic fractionation factors of complexes AHA21, J. Am. Chem. Soc., 99, 5207– 5209, 1977. 18 Kreevoy, M. M. and Liang, T. M., Structure and isotopic fractionation factors of complexes A1HA21 2 , J. Am. Chem. Soc., 102, 3315– 3322, 1980. 19 Lin, J., Westler, W. M., Cleland, W. W., Markley, J. L., and Frey, P. A., Fractionation factors and activation energies for exchange of the low barrier hydrogen bonding proton in peptidyl trifluoromethyl ketone complexes of chymotrypsin, Proc. Natl Acad. Sci. USA, 95, 14664– 14668, 1998. 20 Halkides, C. J., Wu, Y. Q., and Murray, C. J., A low-barrier hydrogen bond in subtilisin: 1H and 15N NMR studies with peptidyl trifluoromethyl ketones, Biochemistry, 35, 15941 –15948, 1996. 21 Mildvan, A. S., Massiah, M. A., Harris, T. K., Marks, G. T., Harrison, D. H. T., Viragh, C., Reddy, P. M., and Kovach, I. M., Short, strong hydrogen bonds on enzymes: NMR and mechanistic studies, J. Mol. Struct., 615, 163– 175, 2002. 22 Viragh, C., Harris, T. K., Reddy, P. M., Massiah, M. A., Mildvan, A. S., and Kovach, I. M., NMR evidence for a short, strong hydrogen bond at the active site of a cholinesterase, Biochemistry, 39, 16200– 16205, 2000. 23 Massiah, M. A., Viragh, C., Reddy, P. M., Kovach, I. M., Johnson, J., Rosenberry, T. L., and Mildvan, A. S., Short, strong hydrogen bonds at the active site of human acetylcholinesterase: proton NMR studies, Biochemistry, 40, 5682 –5690, 2001. 24 Shan, S., Loh, S., and Herschlag, D., The energetics of hydrogen bonds in model systems: implications for enzymatic catalysis, Science, 272, 97 – 101, 1996. 25 Salomaa, P., Schaleger, L. L., and Long, F. A., Solvent deuterium (D) isotope effects on acid – base equilibria, J. Am. Chem. Soc., 86, 1 – 7, 1964. 26 Hopper, C. R., Schowen, R. L., Venkatasubban, K. S., and Jayaraman, H., Amide hydrolysis. VII. Proton inventories of transition states for solvation catalysis and proton-transfer catalysis. Decomposition of the tetrahedral intermediate in amide methanolysis, J. Am. Chem. Soc., 95, 3280– 3283, 1973. 27 Hedstrom, L., Serine protease mechanism and specificity, Chem. Rev., 102, 4501– 4523, 2002. 28 Cassidy, C. S., Lin, J., and Frey, P. A., A new concept for the mechanism of action of chymotrypsin: role of the low-barrier hydrogen bond, Biochemistry, 36, 4576 –4584, 1997. 29 Gerlt, J. A., Kreevoy, M. M., Cleland, W. W., and Frey, P. A., Understanding enzymic catalysis: the importance of short, strong hydrogen bonds, Chem. Biol., 4, 259–267, 1997. 30 Elrod, J. P., Hogg, J. L., Quinn, D. M., Venkatasubban, K. S., and Schowen, R. L., Protonic reorganization and substrate structure in catalysis by serine proteases, J. Am. Chem. Soc., 102, 3917– 3922, 1980. 31 Pollack, E., Hogg, J. L., and Schowen, R. L., One-proton catalysis in the deacetylation of acetyl-achymotrypsin, J. Am. Chem. Soc., 95, 968– 969, 1973. 32 Stein, R. L. and Strimpler, A. M., P1 residue determines the operation of the catalytic triad of serine proteases during hydrolysis of acyl enzymes, J. Am. Chem. Soc., 109, 4387– 4390, 1987. 33 Stein, R. L., Elrod, J. P., and Schowen, R. L., Correlative variations in enzyme-derived and substratederived structures of catalytic transition states. Implications for the catalytic strategy of acyl-transfer enzymes, J. Am. Chem. Soc., 105, 2446– 2452, 1983. 34 Hunkapiller, M. W., Forgac, M. D., and Richards, J. H., Mechanism of action of serine proteases. Tetrahedral intermediates and concerted proton transfer, Biochemistry, 15, 5581– 5588, 1976. 35 Stein, R. L., Strimpler, A. M., Hori, H., and Powers, J. C., Catalysis by human leukocyte elastase: proton inventory as a mechanistic probe, Biochemistry, 26, 1305– 1315, 1987. 36 Rosenberry, T. L., Acetylcholinesterase, Adv. Enzymol., 43, 103– 218, 1975. 37 Quinn, D. M., Acetylcholinesterase: enzyme structure, reaction dynamics, and virtual transition states, Chem. Rev., 87, 955– 979, 1987.

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38 Taylor, P. and Radic, Z., The cholinesterases: from genes to proteins, Annu. Rev. Pharmacol. Toxicol., 34, 281– 320, 1994. 39 Malany, S., Sawai, M., Sikorski, R. M., Seravalli, J., Quinn, D. M., Radic, Z., Taylor, P., Kronman, C., Velan, B., and Shafferman, A., Transition state structure and rate determination for the acylation stage of acetylcholinesterase catalyzed hydrolysis of (acetylthio)choline, J. Am. Chem. Soc., 122, 2981– 2987, 2000. 40 Harel, M., Quinn, D. M., Nair, H. K., Silman, I., and Sussman, J. L., The x-ray structure of a transition state analog complex reveals the molecular origins of the catalytic power and substrate specificity of acetylcholinesterase, J. Am. Chem. Soc., 118, 2340– 2346, 1996. 41 Pryor, A. N., Selwood, T., Leu, L.-S., Andracki, M. A., Lee, B. H., Rao, M., Rosenberry, T., Doctor, B. P., Silman, I., and Quinn, D. M., Simple general acid – base catalysis of physiological acetylcholinesterase reactions, J. Am. Chem. Soc., 114, 3896– 3900, 1992. 42 Froede, H. C. and Wilson, I. B., Direct determination of acetyl-enzyme intermediate in the acetylcholinesterase-catalyzed hydrolysis of acetylcholine and acetylthiocholine, J. Biol. Chem., 259, 11010– 11013, 1984. 43 Kovach, I. M., Larson, M., and Schowen, R. L., One-proton catalysis in the deacetylation of acetylcholinesterase, J. Am. Chem. Soc., 108, 3054– 3056, 1986. 44 Acheson, S. A., Dedopoulou, D., and Quinn, D. M., Simple general acid – base catalysis and virtual transition states for acetylcholinesterase-catalyzed hydrolysis of phenyl esters, J. Am. Chem. Soc., 109, 239– 245, 1987. 45 Lindskog, S., Structure and mechanism of carbonic anhydrase, Pharmacol. Ther., 74, 1 – 20, 1997. 46 Venkatasubban, K. S. and Silverman, D. N., Carbon dioxide hydration activity of carbonic anhydrase in mixtures of water and deuterium oxide, Biochemistry, 19, 4984– 4989, 1980. 47 Frantom, P. A. and Fitzpatrick, P. F., Uncoupled forms of tyrosine hydroxylase unmask kinetic isotope effects on chemical steps, J. Am. Chem. Soc., 125, 16190– 16191, 2003. 48 Fitzpatrick, P. F., Steady-state kinetic mechanism of rat tyrosine hydroxylase, Biochemistry, 30, 3658– 3662, 1991. 49 Fitzpatrick, P. F., Studies of the rate-limiting step in the tyrosine hydroxylase reaction: alternate substrates, solvent isotope effects, and transition-state analogs, Biochemistry, 30, 6386– 6391, 1991. 50 Moran, G. R., Derecskei-Kovacs, A., Hillas, P. J., and Fitzpatrick, P. F., On the catalytic mechanism of tyrosine hydroxylase, J. Am. Chem. Soc., 122, 4535– 4541, 2000.

42

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase Brett E. Lewis and Vern L. Schramm

CONTENTS I. II. III.

IV.

V.

Introduction .................................................................................................................... 1020 History of Enzymatic Binding Isotope Effects.............................................................. 1020 Contributions of Binding and Prebinding Steps to Isotope Effects in Enzymology .... 1021 A. BIE and KIE........................................................................................................... 1022 1. Expressions of D(kcat/Km) and T(kcat/Km)........................................................ 1022 2. Expressions of Dkcat ......................................................................................... 1024 B. Prebinding Isomeric Isotope Effects and KIE ....................................................... 1026 1. Effect on Competitive (kcat/Km) KIE Measurements...................................... 1026 a. Regimes I– III ............................................................................................ 1027 b. Regimes IV –VI......................................................................................... 1027 c. Regimes VII – IX ....................................................................................... 1028 2. Effect on Noncompetitive Dkcat Measurements .............................................. 1029 3. Curtin –Hammet Principle............................................................................... 1029 C. Prebinding Isotope Effects and BIE ...................................................................... 1030 D. Conclusions ............................................................................................................ 1032 1. Transition State Studies................................................................................... 1032 2. Determination of Rate-Limiting Steps and Tunneling ................................... 1032 Physical Basis for Binding and Kinetic Isotope Effects ............................................... 1032 A. Frequency Changes due to Reaction and Heavy-Atom Labeling......................... 1033 1. Heavy Atom Labeling ..................................................................................... 1033 2. High-Frequency CH Bond Stretch: Equilibrium Isotope Effects................... 1034 3. Lower-Frequency CN Bond Stretch: Equilibrium Isotope Effects ................ 1035 4. When Does MMI Count?................................................................................ 1036 5. When Does EXC Count? ................................................................................ 1036 B. Alteration in Force Constants ................................................................................ 1037 1. How Many Modes Actually Matter? .............................................................. 1038 2. Isotope Effects from Altering Mode Coupling Partners................................. 1039 3. Sterics and Hyperconjugation ......................................................................... 1042 C. Summary ................................................................................................................ 1045 Example: Glucose and Brain Hexokinase ..................................................................... 1046 A. Methods .................................................................................................................. 1046 B. The Binary Complex.............................................................................................. 1047 C. The Ternary Complex ............................................................................................ 1048

1019

1020

Isotope Effects in Chemistry and Biology

VI. Applications for BIE ...................................................................................................... 1049 VII. Conclusion...................................................................................................................... 1049 Acknowledgments ...................................................................................................................... 1050 References................................................................................................................................... 1050

I. INTRODUCTION Reaction rates in both chemistry and enzymology are subject to the isotopic composition of the reactants.1 – 3 The physical basis for isotope effects are well understood and derive from normalmode force constant changes as a reactant progresses to product. Accurate measurements of the relative reaction rates of isotopically-substituted compounds can provide unique information about the structure of a chemical transition state or even the identity of a rate-limiting step. The most common applications of these kinetic isotope effects in enzymology today are the measure of D(kcat/Km), T(kcat/Km), and Dkcat. Interest in enzymatic transition states is growing, as high affinity inhibitors may be designed from knowledge of their properties. Although binding equilibrium isotope effects have been less studied, they are critical to the goal of understanding enzyme– substrate interactions at each step of catalysis. A transition state may be modeled from kinetic isotope effects (KIE) without knowledge of equilibrium binding isotope effects (BIE), but equilibrium BIE reveal the atomic interactions at the first step — substrate binding — in these molecular machines. This chapter provides a general consideration of KIE including isotopic sensitivity in prebinding, binding, and internal nonchemical steps. We will dissect the equations defining equilibrium and kinetic isotope effects and describe some of the molecular forces responsible for force constant changes. We present some computational analysis of these effects meant to aid the interpretation of both BIE and KIE, and we conclude with a summary of recent findings in the binding of glucose to human brain hexokinase.

II. HISTORY OF ENZYMATIC BINDING ISOTOPE EFFECTS The application of isotope effects in enzymology has blossomed mainly due to the simplifying and descriptive work of Cleland and Northrop in the 1970s and 1980s.4 – 7 Today it is common practice to measure a kinetic isotope effect, either competitively or noncompetitively, and to draw some conclusion either about the nature of a transition state or in identifying a rate-limiting step. Many transition states have been solved8 – 13 (for example Figure 42.1), and countless rate-limiting steps have been identified. These data have led not only to a deeper understanding of enzymatic function but also to the design of transition-state analogs with very tight binding characteristics.14 – 16

O % KIE HN acid PNP IU-NH IAG-NH GI-NH pertussis

< 0.7 3.3 5.1 2.7 2.2 2.0 − 0.8%

2.6% 15.0%

15N

N

3H

30% (solvent 2H2O)

H N

HO O

14C 3H

3H

4.4% OH

3H

OH

OW 16.1%

FIGURE 42.1 Acid-catalyzed hydrolysis compared to enzymatic reaction through kinetic isotope effects. All values not otherwise labeled are for the IU-nucleoside hydrolase reaction. The remote (50 ) kinetic isotope effects are difficult to rationalize without appealing to binding isotope effects. (See Refs. 12,28,90.)

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

1021

In contrast very few binding equilibrium isotope effects (EIE) have been reported. Early isotope studies concentrated on deuterium or tritium effects on the rate of hydride transfer; such effects (typically on the order of 3 to 100) dwarf any contribution from isotopic effects on binding (usually a few percent). In cases where multiple isotope-sensitive steps have been considered,17 – 20 BIE are always omitted. It has become dogmatic that BIE do not matter much to the enzymologist, even though today enzymologists also study much more subtle chemistry. It is clear that isotope effects occur in solution isomeric and conformational equilibria,21 – 24 in enzyme –substrate binding steps,25,26 as well as in transition state formation.8,9,27 – 30 Here we shall revisit BIE and take a comprehensive approach to their role in enzyme –substrate interactions. A limited literature exists on equilibrium BIE in biochemistry. The earliest observation of an isotope effect on a equilibrium binding constant came in 1971, with Bush et al.31 and liver alcohol dehydrogenase (LADH). They determined that the dissociation constant for NADH binding (0.27 mM) was tighter than that for both A-NADD (0.29 mM) and B-NADD (0.46 mM). These normal EIE on association, measured noncompetitively using gel filtration and fluorometry, indicated some stereospecific vibrational loosening of the geminal hydrogen atoms in reduced nicotinamide. Detecting isotopes with either isotope ratio mass spectrometry or scintillation counting, Anderson’s group found a 1.10 ^ 0.03 deuterium isotope effect and 1.085 ^ 0.01 tritium isotope effect, respectively, on the association constant for association of NADþ to the same enzyme.32 Both values represent normal isotope effects and CH bond loosening upon binding. (Within error limits, the effects adhere to the Swain – Schaad relationship.) Several years later, Gawlita et al.33 measured 18O isotope effects for equilibrium binding of [1-18O]oxamate to LADH in the presence of reduced cofactor and found an effect of 0.9840 ^ 0.0027 by isotope competition and isotope ratio mass spectrometry. They modeled oxamate both in solution and docked into the active site of LADH, generating vibrational spectra of each, and found that the inverse effect could be explained by the proximity and orientation of ionic interaction between Arg169 and the oxamate anion. These studies have been motivated by the search for ground-state destabilization in enzymatic reactions. The rate acceleration conferred by enzymes is due to stabilization of the transition state. Lowering the energy of a transition state or finding a different transition state with a lower energy makes it easier for the substrate to achieve the necessary free energy to clear the chemical barrier. It is also possible to decrease the free energy difference needed to achieve the transition state by boosting the energy of a bound substrate. This “ground-state destabilization” is proposed to occur locally in the reactive functionality of the substrate, while powerful active site contacts keep the overall binding energy favorable enough for a reasonable dissociation constant. Anderson found evidence of ground-state destabilization, and work published by our group found evidence as well in hexokinase25,26 (and see Section V in this chapter). Whereas basic questions have been addressed computationally in purely chemical systems (i.e., why axial CH bonds in cyclohexane are more loose than equatorial bonds in that molecule22), there have been only limited computations made for the interactions of small and large molecules such as those encountered in biology.

III. CONTRIBUTIONS OF BINDING AND PREBINDING STEPS TO ISOTOPE EFFECTS IN ENZYMOLOGY The expressions for KIE studies can be expanded to include binding effects, other internal kinetic factors, as well as the usual product release factor, gaining some new insights into the nature of KIE. We also examine the experimental problems associated with conducting the competitive KIE assay in the presence of high- and low-affinity isomeric substrates with different fractionation factors.

1022

Isotope Effects in Chemistry and Biology

A. BIE AND KIE 1. Expressions of D(kcat/Km) and T(kcat/Km) Competitive isotope effect measurements may be made with either hydrogen isotope; we will choose tritium notation for simplicity. We begin with the relatively simple mechanism of Reaction 42.A in Figure 42.2. When the intrinsic KIE, Tk3, is very large as in the case of proton or hydride transfer, one may reasonably assume that Tk3 dominates the observed isotope effect. Northrop5 derives the corresponding expression for the observed value of T(kcat/Km) in terms of the commitment factors Cr and Cf, where Cr ¼ 1 k4 =1 k5 and Cf ¼ 1 k3 =1 k2 : The leading superscript indicates isotopic mass if numeric and isotope effect of D or T: T

kcat Km

¼

T

int Keq Cr þ T k3 þ Cf Cr þ 1 þ Cf

ð42:1Þ

The predictions of this equation are well understood: Tk3 will be observed only if the commitment factors are small. A large reverse commitment, Cr, will weight the observed value to the limit of T int Keq ; whereas a large forward commitment factor will cause the observed effect to move toward int ; unity. Thus, the factors Cr, 1, and Cf serve as weighting terms for each of the isotope effects T Keq T k3 ; or unity, respectively. Now permitting isotope effects on any step yields: T

kcat Km

T

¼

int T k5 Cr þ T KA T k3 þ T k1 Cf KA T Keq Cr þ 1 þ Cf

ð42:2Þ

int where T KA ¼ T k1 =T k2 ; T Keq ¼ T k3 =T k4 and the commitment factors Cr and Cf retain the same definition. Each numerator term consists of (1) an intrinsic KIE, preceded by (2) the product of all

int

K eq

E+S

k1 k2

E.S

k3 k4

E.P

int

k1 k2

K eq

k3 E.S

k4

E'.S

k1 k2

E.S*

k3 k4

E'.S*

int

k1 k2

E.S*

k3 k4

k6

E.P

k7

(42.B)

E+P

K eq k5 k6

E.P*Q

k7

E.P*+Q

E'.S*

k5 k6

FIGURE 42.2 Model reaction mechanisms.

k9

(42.C)

E+P*

int

int

K eq+2

K eq

K eq-1

E+S*

k5

int

int

K eq-1

E+S*

(42.A)

E+P

int

K eq-1

E+S

k5

E.P*Q

k7

E.P*+Q

k9 k10

E'.P*

k11

E+P* (42.D)

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

1023

EIE from substrate binding through its own step and (3) a weighting factor as before. Each numerator term may be thought of as the T(kcat/Km) isotope effect that would be observed if its intrinsic KIE were the exclusively rate-limiting step of the reaction, and these are weighted to determine the observed KIE according to the sharing of rate-limiting status. Adding extra enzyme forms on the EP side in the reaction mechanism expands the Cr weighting factor and, similarly, adding extra steps on the ES side expands the Cf term, until for Reaction 42.C, the isotope effect derivation yields Equation 42.3: 2 6 6 6 6 6 4 T

kcat Km

T

1

k6 1 k8 1k 1k 7 9

T

int T int T int T Keq Keqþ1 k9 KA Keq21

þ

¼

1

k6 1 k8 1k 1k 7 9

T

T

int T k5 KA Keq21

!

1

k6 1k 7

þ

! þ T

T

T

int T int T Keq k7 KA Keq21 1

T

þ KA k3

! þ1þ

1

k5 1k 4

!

k5 1 k4

! þ T k1 1

k3 1 k5 1k 1k 2 4

þ

!

1

k6 1k 7

!3

7 7 !7 7 1 1 k3 k5 7 5 1 1 k2 k4

ð42:3Þ

We have also added a second irreversible step at the end, representing a mechanism with additional requirement for regeneration of the enzyme active form (an isoenzyme mechanism). Each term in the numerator represents total isotope partitioning between reactant and a transition state for a different forward kinetic constant, up to the first irreversible step. Expressed in the traditional way, Equation 42.3 contains terms that resemble the expansion of Northrop’s commitment factors, which would have been Cf ¼ ðk5 =k4 Þð1 þ k3 =k2 Þ and Cr ¼ ðk6 =k7 Þð1 þ k8 =k9 Þ in this mechanism. Of course, since the weighting terms are divided out in the denominator, multiplying the top and bottom by (1k7/1k6) yields the equivalent expression: 2 6 6 6 6 6 4 kcat Km

T

T

int T int T int T Keq Keqþ1 k9 KA Keq21

¼

T

1

k8 1 k9

!

3 þ

T

T

int T int T Keq k7 KA Keq21

7 7 7 !7 ! ! 7 1 1 1 1 1 1 k k k k k k 5 int T k5 1 7 þ T KA T k3 1 5 1 7 þ T k1 1 3 1 5 1 7 þ T KA T Keq21 k6 k4 k6 k2 k4 k6 ! ! ! ! 1 1 1 1 1 1 1 k8 k7 k5 k7 k3 k5 k7 þ 1 1 1 þ1þ 1 þ 1 1 1 k2 k4 k6 k4 k6 k9 k6

ð42:4Þ

and essentially moves the focus of the equation to the right of chemistry. Similarly, multiplying by (1k4/1k5) moves the focus one step to the left. There are three obvious choices for the reference point of the equation: the first step, the chemical step, and the last step. Writing the equation with respect to the binding step is probably the most informative: 2 6 6 6 6 6 4 T

kcat Km

¼

T

T

int T int T int T KA Keq21 Keq Keqþ1 k9

þ

T

T

k2 1 k4 1 k6 1 k8 1 1 1 1 k3 k5 k7 k9 !

! þ

1 k2 1 k4 k2 T T k þ K A 3 1 1k 1k k 3 5 ! ! 3 1 1 1 1 1 1 1 k2 k4 k6 k8 k k k þ 1 21 41 6 þ 1 1 1 1 k3 k5 k7 k9 k3 k5 k7

int T k5 KA Keq21

1

1

T

T

int T int T KA Keq21 Keq k7

!

1

k2 1 k4 1 k6 1 1 1 k3 k5 k7

þ Tk 1

k2 1 k4 1 1 k3 k5

! þ

1

k2 1 k3

!3 7 7 7 7 7 5

! þ1 ð42:5Þ

1024

Isotope Effects in Chemistry and Biology

Each transition state is now weighted by a commitment factor that represents the degree to which its substrate “communicates” with free substrate. This is demonstrated by taking the limit of k5 ! 1: Any quantity of k5’s substrate, E1Sp, produced by the previous step is immediately consumed by k5 ; effectively removing it from equilibrium with the free substrate. The first irreversible step becomes k3 and terms that contain 1k5 in the denominator disappear from the equation. Finally, the product of equilibrium effects at each term then represent the isotope partitioning in the equilibrium between “free” substrate and the substrate for that term’s step. For a transition state to contribute, its substrate may be a fractionated version of substrate, but it must not be a trapped version of substrate. These equations and conclusions are similar to those drawn by Blanchard, Cook, and Cleland,17 – 19 except that the present forms permit the introduction of BIE. It is therefore stressed that free reactant is the true reference state against which isotope partitioning in the transition states must be compared. This is an important practical concept for the modeling phase of transition state analysis. 2. Expressions of Dkcat It does not appear that expanded equations in Dkcat have been recorded. Therefore, we treat our model reactions in the manner of the previous section. It is usually not practical to measure Tkcat with the tritium isotope, so we only treat Dkcat. Starting with Reaction 42.A and the assumption that k3 is the only isotopically sensitive step, the isotope effect on kcat is written with Northrop’s nomenclature:

D

Rf Ef

kcat ¼

int þ D Keq Cr þ D k3

Rf Ef

þ Cr þ 1

;

ð42:6Þ

int where Rf ¼ ð1 k3 =1 k5 Þ; Ef ¼ 1; Cr ¼ ð1 k4 =1 k5 Þ; and D Keq ¼ ðD k3 =D k4 Þ: Including product release D among the sensitive steps adds k5 to two terms for the same reaction:

D D

kcat ¼

Rf Ef

k5

int þ D k5 D Keq C r þ D k3

Rf Ef

þ Cr þ 1

:

ð42:7Þ

The most interesting expansions to Northrop’s expressions come from Reaction 42.B and Reaction 42.C, respectively:

D

k7 þ

D

kcat ¼

1

k3 1k 7

1

k3 1k 7 D

!"

k3 !"

1þ 1

k3 1 k3

1þ

!

1 int Keq

D

int Keq int Keq

!

! 1þ

1þ

D

1 int Keq21

int Keq21 int Keq21

!!#

!!# þ

D

þ k5

1

k3 1k 5

1

k3 1k 5

!" 1þ

!" 1þ

1 int Keq21

D

int Keq21 int Keq21

!# þ

1

k3 1k 3

!#

! ð42:8Þ

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

D

k9

þ D

kcat ¼

1

k3 1 k9 þ

1

k3 1k 9 D

!

k5

! þ 1

k3 1k 5

D

þ k7 1

k3 1k 5 1

1

k3 1k 7

!" 1þ

!"

k3 1 k7

1þ

int Keq int Keq

! 1þ

D

int Keq21 int Keq21

! !# int 1 Keq21 k3 D þ k3 1 int k3 Keq21 ! !!# 1 1 1þ int int Keq Keq21

!!#

D

1þ !"

!"

D

1025

1þ 1

int Keq21

!# þ

1

k3 1k 3

;

ð42:9Þ

!

int int where in both equations Keq21 ¼ 1 k3 =1 k4 and Keq ¼ 1 k5 =1 k6 : Familiar terms appear in these equations. The numerator of Equation 42.8 consists of the intrinsic KIE on each forward step, Dki, multiplied by the measure of whether that step is rate limiting, (1kchemical/1ki), and some nested equilibrium constants which resemble Northrop’s “gating” (or scarcity) measure. The reference point for each “limiticity” factor may be changed by multiplying numerator and denominator by (1ki/1k3). The weighting factors are slightly changed in the denominator: rate-limiting factors remain, but the gating factors contain no isotope effect terms. By rearranging the definition of D int K ¼ ð1 K int =2 K int Þ to ðD K int =1 K int Þ ¼ ð1=2 K int Þ; the parallel between the factors may be seen. The gating terms in the denominator represent scarcity of protio substrate for ki, whereas the same terms in the numerator represent scarcity of deutero substrate for that step. Clearly, each isotope effect depends upon previous isotope effects for substrate. This may exaggerate or dilute an intrinsic effect. The rate-limiting step now carries more sequelae than a simple contribution to Dkcat; all subsequent forward steps whose substrates are in equilibrium with the rate-limiting step now make an additional contribution limited by (1krate-lim/1ki). For instance, a slow k3 (with respect int to k4) gives a large ð1=Keq21 Þ and will cause subsequent equilibrated steps to be more “gating.” Nonequilibrated steps like k9 in Reaction 42.C will not benefit because their substrates cannot recede back along the mechanism (i.e., k8 does not exist in this mechanism). If k5 were rate limiting and slow with respect to k6 ; only Dk7 would benefit in this manner. Therefore we may conclude that equilibrated steps following the rate-limiting step will contribute to the overall D kcat above and beyond what would be expected due to their limiticity alone. Each step gets credit for increasing its substrate’s concentration, and any step whose substrate has a propensity to vanish in the reverse direction (and is subsequently even less abundant) gets even more credit for building concentration. Equation 42.9 demonstrates that added irreversible steps contain no nested equilibrium terms. Therefore, steps or sequences separated by irreversible steps do not interdepend. If k9 were made reversible by k10 as in Reaction 42.D, then the next step (k11) would be represented in Dkcat with no relationship to any terms in the k3 – k7 series except in limiticity. The hypothetical reaction free energy profile illustrated in Figure 42.3 can be used to illustrate these principles. Rate constants k3 and k5 are kinetic barriers to achieving thermodynamically uphill intermediates (these are reflected in rapid k4 and k6), and k7 produces a difficult and irreversible conversion. An isotope effect on (kcat/Km) will only represent partitioning of substrate to the highest transition state, but each step of the energetic barrier gets credit in Dkcat KIE. Every step contributes according to its limiticity; steps in reversible segments get an advantage if previous, reversible steps were uphill energetically. BIE would pose no exception to this rule.

1026

Isotope Effects in Chemistry and Biology

E+S

k1 k2

E.S

k3 k4

E∗.S

k5 k6

E.P

k7

E+P

k9

E+P

FIGURE 42.3 Hypothetical reaction free energy profile. The (kcat/Km) KIE will only represent partitioning of substrate to the highest transition state, but each step of the mountain gets credit in Dkcat KIE.

B. PREBINDING I SOMERIC I SOTOPE E FFECTS AND KIE It is usually assumed that only one isomer of substrate exists in solution, but this assumption is rarely made explicit. Several conformers may compose the substrate equilibrium in solution, and there may also be one or more alternate nonbinding stable isomers (such as a-glucose with glucose oxidase) to the binding substrate. Since equilibrium solution isotope effects in deuterated substrates21,23,24,35 – 39 can rival secondary KIE in magnitude, we will examine several specific cases. 1. Effect on Competitive (kcat/Km) KIE Measurements For the reaction: k1

k3

S1 O S2 ! P k2

the parameters k1 and k2 are catalyst-independent rates that will vary with substrate and solvent, and pre Keq ¼ k1 =k2 : The enzymatic rate k3 depends on total enzyme [E]T and the velocity saturation curve v([S2]), and these parameters can be adjusted relative to k1 and k2 : In general, there will be nine regimes for this reaction (Figure 42.4). The nine regimes are labeled with roman numerals in Figure 42.4. The size of the S1 and S2 labels are meant to reflect the dominant species. Thus, in Regimes I– III, ½S1 p ½S2 ; where S2 is the form binding enzyme, in Regimes IV – VI, [S1] < [S2], and in Regimes VII – IX, ½S1 q ½S2 : The latter represent an unusual case in which the enzyme prefers the least abundant form. Vertical position in the figure represents the relative magnitude of k3 with respect to k1 and k2 ; which are depicted as lines. In approaching a problem like this, we are chiefly concerned with (1) which isotope effect is being observed, (2) which substrate is being observed, and (3) whether it is practical or possible to manipulate k3 to change regime. Since the substrate cannot “know” whether organized solvent or enzyme active site is performing its transformation, we may apply our understanding from enzymatic transition state analysis, except that we must proceed carefully in considering which is the reference state for our analysis. In all cases, it is assumed that substrates S1 and S2 are at equilibrium concentrations before enzyme is added. Critically, it must be remembered that while we shall refer to Tk3 as the enzymatic transition state, it must be independently verified that Tk3 is a fully expressed, intrinsic KIE if one wishes to use the value as reflecting the chemical step.

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

I

IV k1

II S1S2

1027

VII k2

S1S2

S1 S2

VIII

V

k2 III

k1 VI natural substrates

k1 >> k2

k1 ≈ k 2

IX unnatural substrates

k1 << k2

FIGURE 42.4 The nine possible relationships between k1, k2 and k3 for Reaction 42.1 in the text. Kpre eq ¼ k1/k2, determining the relative abundance of S1 and S2 prior to addition of enzyme (k3 ¼ v([S2])[E]T). Constants k1 and k2 are solute and solvent dependent while k3 (reaction progress) may be adjusted by changing enzyme concentration. Roman numerals (I– IX) represent the position of k3 with respect to a system determined by k1 and k2. For example, I represents k3 q k1 q k2 whereas II represents k1 q k3 q k2.

a. Regimes I – III k1 q k2 : Regimes I through III represent the common assumption that one dominant species exists, S2 in our case, and that the enzyme binds this species. Since k1 q k2 and ½S1 p ½S2 ; Tk3 is the only contributor to the observed isotope effect. Any measure of substrate is a measure of S2 only, and measuring substrate or product for isotopic enrichment are equivalent. These results are summarized in Table 42.1. Regime I, for example, is listed in the top left portion of the table, and we find that analyzing the product reveals Tk3, that S1 cannot be measured, analyzing S2 reveals Tk3, and that taking the whole product pool is essentially equivalent to analyzing S2 alone. b. Regimes IV–VI k1 < k2 and ½S1 < ½S2 : These common situations are the most complicated to analyze, particularly Regime V. Fortunately, while the ratio [S1]/[S2] probably cannot be manipulated in general, k3 may be decreased into Regime VI or increased into Regime IV by adjusting enzyme concentration or pH, although the latter is usually undesirable. Presently, we shall not treat Regime V. IV. k3 q k1 ; k2 : Equilibration between S1 and S2 is slow compared to k3 : Table 42.1 shows this regime broken into two phases. At first, enzyme will consume S2 with no significant change in [S1]. During this time S1 will show no isotope enrichment, while measurement of S2 or P will demonstrate the isotope effect on k3 : Measurement of overall substrate during this phase will reveal a mixture of unaltered S1 and altered S2. Once S2 is consumed, any conversion of S1 to S2 will be immediately be followed by rapid conversion to P. The reaction rate in this phase is limited to k1 : Thus formation of S2 may be considered the first irreversible step and measurement of S1 will yield the isotope effect on k1 : S2 may not be measured in this phase because its steady-state concentration

1028

Isotope Effects in Chemistry and Biology

TABLE 42.1 Isotope Effects Measured from Substrate or Product Pools for Each Region in Figure 4 Analyte Product S1 S2 (S1 þ S2)

I. T

k3 a

T

k3 ¼ S2

1st Phase

IV.

T k3 No change T k3 14 14 ð S1 þ D S2 Þ=ð3 S1 þ D3 S2 Þ

2nd Phase T T

VII. T

k1 k1

T

a

a

¼ S1

¼ S1

II. Product S1 S2 (S1 þ S2)

T

VIII.

k3

a

k3 ¼ S2

¼ S1

III.

S1 S2 (S1 þ S2)

T

a

T

T

T

b

pre T ð Keq Þ k3 pre T ðT Keq Þ k3

b

a

Product

k3

a

k3 ¼ S2

k1 k1

VI.

IX.

pre T Weighted mixture of T k3 from [S2] and ðT Keq Þ k3 from [S1]eq

pre T ðT Keq Þ k3

a

a

pre T Weighted mixture of T k3 from [S2] and ðT Keq Þ k3 from [S1]eq

pre T ðT Keq Þ k3 a

¼ S1

Typically cannot be measured independently. Not solved here.

is very small, and care must be taken when analyzing product formation for isotopic enrichment because the product pool will contain isotopes converted during the first phase. VI. k3 p k1 ; k2 : In our case, where only S2 binds but is in “rapid” equilibrium with S1, then the enzymatic transition state would generate an isotope effect of Tk3 from S2 and an isotope effect pre T pre of T Keq may be estimated computak3 from S1. Once the structures of S1 and S2 are known, T Keq tionally or measured experimentally. Identification of the binding conformer typically requires structural data from NMR, crystallography or Raman spectroscopy. Modeling transition states from KIE therefore requires knowledge of this equilibrium. From the perspective of the transition state, there are several starting structures, and their fractionation factors are weighted according to their abundance. At present, the commonly used programs do not handle multiple reactant structures. The most desirable approach, if possible, is to isolate S1 from the substrate pool to evaluate for isotopic enrichment, since the expressed KIE shall reflect the changes in the transition state with respect to S1 as a reference state. c. Regimes VII – IX k1 p k2 : Since ½S1 q ½S2 ; the binding substrate is much less abundant than the alternate forms. Systems of this type are likely to be rare with natural substrates, because enzymes are expected to evolve to match the most abundant species. S2 cannot be measured for isotopic enrichment. VII. k3 q k2 q k1 : Whatever small amount of S2 present is rapidly consumed and k1 becomes the rate-limiting step. As k1 is the first irreversible step, Tk1 is expressed, and the observable substrate is essentially S1. We must be careful not to assign isotope effects here to the enzymatic transition state.

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

1029

VIII and IX. k2 q k3 ; k1 and k2 q k1 q k3 : Since k3 p k2 ; S1 and S2 are in equilibrium compared with reaction progress, and k3 is the first irreversible step. The observed isotope effect pre T is T Keq k3 in both cases, and we may think of isotope partitioning from a k3 transition state as reaching all the way back to S1. 2. Effect on Noncompetitive Dkcat Measurements Prebinding isomeric IE will not affect Dkcat measurement, since common practice is to measure reaction rates at several substrate concentrations and extrapolate to infinite substrate concentration. pre Even in the case of a small Keq or a large CEIE that severely restricts the availability of the binding isomer, it will be clear when saturation occurs. 3. Curtin– Hammet Principle It is worth mentioning this principle at this point, as it deals with rapidly converting forms and competing reactions. In the terms of the previous discussion, the Curtin –Hammet principle applies to a rapid equilibrium between S1 and S2 where an enzyme is available to accept S2 as substrate for reaction, and a different reaction path is available for S1, whether via the same or different enzyme. The amount of each product formed depends on the rate of each reaction path and not on the abundance of one substrate over the other. Owing to the relatively rapid interconversion, neither reaction path starves for substrate, and the faster progresses further. The experimental consequence of the rapid equilibrium is that isotopes are shared by the reactions, and therefore, an isotope effect on even only one reaction will translate into gradual isotopic enrichment of the substrate for the other reaction. This is similar to the problem of two competing reactions run by one enzyme with one substrate (see below). First let us review the method of conversion of observed isotopic enrichment of substrate into isotope effect. For the simple reaction: k

S ! P; we know that the disappearance of S is given by 2 d[S]/dt ¼ k[S]. Isotopically, this is 2 d[1S]/dt ¼ 1 1 k[ S] and 2 d[3S]/dt ¼ 3k[3S]. Dividing to eliminate time dependence gives d[1S]/d[3S] ¼ 1 3 ( k/ k)([1S]/[3S]), and with rearrangement, d[1S]/[1S] ¼ Tkd[3S]/[3S]. Performing the definite integral over both sides with starting conditions 1S0 and 3S0, and the final conditions 1Sf and 3Sf gives ln([1Sf]/[1So]) ¼ Tk ln([3Sf]/[3So]). Since fractional conversion, f , consists of native substrate concentrations, as in (1 2 f ) ¼ [1Sf]/[1So], we may also write that (1 2 f ) ¼ (Rsf/Rso)([3Sf]/[3So]), where Rsf and Rso are the proton:tritium isotope ratios in substrate pool after and before reaction, respectively. We can finally substitute to arrive at the familiar result ln(1 2 f ) ¼ Tk ln[(1 2 f ) (Rso/Rsf)]. However, we are concerned with the problem of two competing reactions: k1

S ! P1 k2

S ! P2 and here we find that 2 d[S]/dt ¼ (k1 þ k2)[S], equivalent in form to our previous case, so we can make the substitutions 1 k1 þ 1 k2 ¼ 1 k and 3 k1 þ 3 k2 ¼ 3 k to yield our starting point. 1

k1 þ 1 k2 3 k1 þ 3 k2

! ¼

lnð1 2 f Þ ¼ T kobs;S : lnð1 2 f ÞðRso =Rsf Þ

1030

Isotope Effects in Chemistry and Biology

This gives Tkobs,S, the isotope effect obtained by observing only the substrate S, and in this case, with so many terms to the left, it is of very little use. We can see, at this point, that

T

kobs;S ¼

1

k1 þ 1 k2 3k þ 3k 1 2

!

1 1 k1 k2 T B 3k þ 3k C k þ T k2 3 RR C B ¼B 1 3 1 C¼ 1 ; @ 1 þ 3 RR k A 1þ 3 2 k1 0

1

3

RR stands for reaction ratio for the tritiated substrate, and it is clear that Tkobs,S is the weighted sum of isotope effects for the individual reactions. We get much further by noticing that the experimentalist can gather more information by examining isotopic ratios in both products. Therefore, we perform similarly to above, with d[P1]/dt ¼ k1[S] and d[P2]/dt ¼ k2[S]. Then [S]dt ¼ d[P1]/k1 ¼ d[P2]/k2, and we can integrate both sides with initial conditions P1 ¼ P2 ¼ 0 and final conditions P1f and P2f, yielding k2[P1f] ¼ k1[P2f]. Writing the last equation twice isotopically and dividing gives Tk2(1P1f/3P1f) ¼ Tk1(1P2f/3P2f), or T k2(RP1f) ¼ Tk1(RP2f), where RP1f and RP2f are protium:tritium isotopic ratios in P1 and P2 after reaction completion, respectively. These last values can usually be measured most easily. Back substituting yields the final relationships: 1 RP2f 3 1 þ ð RRÞ C B RP1f C¼ ¼ T k1 B A @ 1 þ ð3 RRÞ 0

T

kobs;S

0 T

B k1 ¼ T kobs;S B @

T

lnð1 2 f Þ R ln ð1 2 f Þ so Rsf 1

C 1 þ ð3 RRÞ C A RP2f 3 ð RRÞ 1þ RP1f

k2 ¼ T k1

RP2f : RP1f

Therefore, when Tk1 is known, one needs to measure only RP2f/RP1f; otherwise, solving the whole system from scratch requires 3RR, f , and (Rso/Rsf) as well. This analysis is critical if one wants to model the transition states of an enzyme running two reactions on a single substrate, as is the case with both cholera toxin and CD38.40 It is desirable, in order to minimize errors, to measure isotope effects on one reaction in the absence of the other reaction, if possible by leaving out a second substrate, and then measure the other isotope effects in the context of the combined system.

C. PREBINDING I SOTOPE E FFECTS

AND

BIE

The measurement of a BIE is fairly straightforward, as long as there is no substrate turnover. Taking an example ligand-binding scheme as in the upper panel of Figure 42.5, one is interested in isotopic enrichment changes between A and EA. In theory, this would require separating A and EA and measuring the isotopic composition of each. We would have two isotope ratios, that for “free A” and that for “bound A,” and we would divide these to yield the isotope effect. In conducting a BIE experiment, one must beware of multiple substrate isoforms. The lower panel of Figure 42.5 shows a hypothetical two-ligand system. In a competitive assay between light and heavy isotopes, the unbound reactant pool consists of four species, namely AL, AH, BL, and BH, and the bound reactant pool consists of four as well: EAL, EAH, EBL, and EBH. Let us define the following quantities: KA ¼ [EAL]/[E][AL], Kb ¼ [EBL]/[E][BL], and Keq ¼ [BL]/[AL]. For isotope effects we have: TKeq ¼ [BL][AH]/([BH][AL]), TKA ¼ [EAL][AH]/([EAH][AL]),

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

E

+

A

E

+

A

KA KA

1031

EA EA

K eq E

+

KB

B

EB

FIGURE 42.5 (upper) Ligand binds enzyme. (lower) Two ligand isoforms binding the same enzyme. KA and KB are association constants.

and TKB ¼ [EBL][BH]/([EBH][BL]). Dividing the isotope ratio in the bound pool by the isotope ratio in the unbound pool gives our observed isotope effect:

T

Kobs ¼

½EAL ½EAH ½AL ½AH

þ ½EBL þ ½EBH þ ½BL þ ½BH

¼ T KB

T

Keq þ Keq 1 þ Keq

!

0 B B B @

Keq KB KA T Keq T KB Keq KB þ T KA KA 1þ

1 C C C A

We can see that TKobs reduces properly to TKA if Keq ¼ 0, and to TKB if Keq is set to infinity; but if we set the isotope effects on binding to unity, we find something unusual: 0

if

T

KA ¼ T KB ¼ 1;

T

Kobs

Keq KB 1þ T B Keq þ Keq B KA ¼ B Keq KB 1 þ Keq @ T Keq þ KA

1 C C C A

This last expression indicates that in the absence of a BIE, the presence of an isomeric equilibrium isotope effect will appear to the investigator as an observed binding isotope effect. Even then, in the presence of what we will call intrinsic BIE (or true BIE), an isomeric isotope effect can bias the results. If the binding constants are identical, then the expression reduces to unity. If they are very different, however, the isomeric isotope effect can translate into an observed BIE. A 4% isomeric equilibrium isotope effect can translate into a normal or inverse effect of 1.5% if the binding affinities are different by only two orders of magnitude. Even the extreme case, where only one isomer binds but the second still exists demonstrates the significance of the isomeric isotope effect: if KA ¼ 0;

T

T

Kobs ¼ KB

T

! Keq þ Keq ; and if KB ¼ 0; 1 þ Keq

T

Kobs

T

K ¼ T A Keq

T

Keq þ Keq 1 þ Keq

!

Prebinding effects are greatest when Keq ¼ 1, isomer affinities are very disparate, and TKeq is large. They can be minimized when Keq is large or small, affinities are identical, or when TKeq is unity. In the common case of rapidly interconverting free substrate isomers with different fractionation factors, the collective fractionation factor is taken as the product of each factor raised to the exponent of its own mole fraction. On the other hand, if the isomers are not rapidly interconverting, it may be possible to isolate one binding reaction by pretreatment with some agent.

1032

Isotope Effects in Chemistry and Biology

D. CONCLUSIONS There is a growing body of literature on prebinding, binding, and nonchemical KIE and EIE in enzymology. Isotope effects on isomeric equilibrium constants21,23,35 – 37,41,42 and enzyme – ligand binding reactions32,33,43,44 have been measured experimentally and modeled computationally for a number of systems. By contrast, internal equilibrium constants are difficult to determine experimentally, but they may be estimated from knowledge of active site residues and their fractionation factors.45 Extensive progress has been made through calculations on model compounds to describe the molecular forces underlying EIE.21,46,47 Both hyperconjugative and steric differences between isomers can generate EIE, and the same phenomena must underlie internal EIE. Some recent advances in the theory of transient-phase KIE measurement appear promising,48 and there is a growing need to develop techniques to measure these and internal equilibrium IE experimentally. 1. Transition State Studies With one dominant rate-limiting step the common interpretation is that measured isotope effects reflect the cumulative isotope partition between free substrate and the transition state of that ratelimiting step. Although it is usually assumed that chemistry is the only isotopically sensitive step, this assumption is not necessary. The structures for enzymatic transition states can be established without knowledge of binding or other internal equilibrium isotope effects. However, there are two nonexclusive complicating cases. First, where there may be multiple substrate isomers or conformations in solution, these should be taken collectively as a starting point for the transition state modeling. Second, when more than one step is rate limiting, the observed T(kcat/Km) will represent a weighted mixture of each transition state. Therefore, kinetic BIE may contribute. In addition, one may not conclude that expressed KIE imply that chemistry is rate limiting or that commitment factors are low; these parameters should be measured independently. The idea of a “virtual” transition state has appeared before by Schowen20 and Stein,49 in which the contribution of multiple steps to observed KIE is recognized, but Stein has treated the question in terms of stepwise intrinsic KIE weighted by nonisotopic factors, in contrast to both the T(kcat/Km), D(kcat/Km), and the Dkcat equations reported presently. 2. Determination of Rate-Limiting Steps and Tunneling Hydrogen tunneling in enzymatic reactions is typically uncovered by unusually large primary or secondary deuterium and tritium KIE. Expanded forms of Dkcat show that primary isotope effects are multiplicative with equilibrium isotope effects on previous steps. Therefore the accurate deduction of proton-transfer barrier size and shape requires some attention to nonchemical isotope effects. However, binding and pre-BIE do not appear in the equations for Dkcat.

IV. PHYSICAL BASIS FOR BINDING AND KINETIC ISOTOPE EFFECTS The derivation from first principles to our starting point can be found in several places.1,50 We start with the relations describing EIE and KIE, respectively, as a function of the 3N 2 6 normal vibrational frequencies of each molecule of N atoms. For the equilibrium, X –H þ Y – D $ X – D þ Y – H: ! ! 3N26 3N26 Y Y uX – D uY – H 1 D Keq ¼ exp ðuX – H þ uY – D 2 uX – D 2 uY – H Þ 2 uX – H uY – D 3N26 Y

ð1 2 e2uX – H Þð1 2 e2uY – D Þ ð1 2 e2uX – D Þð1 2 e2uY – H Þ

!

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

1033

each transformed frequency u ¼ hv=kB T corresponds to one normal mode of vibration. (To simplify, let X and Y both contain N atoms.) A stable molecule of N atoms requires 3N coordinates to describe its three-dimensional position. To restrict ourselves to an internal frame of reference, we subtract three coordinates for the position of the molecule’s center of mass and another three angles of rotation of the molecule in an outside reference system. In comparing reaction rates of molecules with either H or D isotope, it has been shown that: ! 10 10 0 1 3N26 X 1 3N26 Y uD 3N26 Y 1 2 e2uH !B B C ðu 2 u Þ exp C H D C B CB 2 u– uH C CB HL B D 1 2 e2uD C CB B ! kH ¼ C; B B C – C B – 3N27 2u A B C@ 3N27 X 1 – Y u– uDL @ 3N27 A Y 12e H D A@ – exp ðu 2 u Þ D – 2 H u– 1 2 e2uD H where an additional vibrational mode is missing from the denominator terms since one of the reactant’s stable modes has been converted into an unstable reaction coordinate. In both of these equations, isotope effects on the rates of reaction or equilibrium position depend only upon the values of the vibrational frequencies. These relationships are commonly referred to in shorthand as: D

kH ¼ ðTSDFÞðMMIÞðZPEÞðEXCÞ and

D

Keq ¼ ðMMIÞðZPEÞðEXCÞ;

where TSDF, MMI, ZPE, and EXC refer to transition state decomposition frequency, molecular moment of inertia, zero-point energy, and excited states, respectively. In this section, we walk through a couple of worked examples to illustrate some points about the relative importance of terms in the isotope effect equations.

A. FREQUENCY C HANGES DUE

TO

R EACTION

AND

H EAVY- ATOM L ABELING

In examining the set of frequencies from almost any four species represented in an expression of isotope effect as shown above, we will find four unique sets of frequencies, assuming the reaction is nontrivial. Clearly there will be differences between reactant and transition state or between the two species in chemical equilibrium, but labeling with a heavy atom, whether deuterium, tritium, 14C or 15N, produces its own changes. First we examine the latter, due to heavy-atom labeling. A far more complete theoretical treatment is found in Melander’s Reaction Rates of Isotopic Molecules,3 but we shall attempt some worked examples here. 1. Heavy Atom Labeling Take a C –H stretch in glucose (MW ¼ 180 g/mol). Replacing the proton with a heavier isotope will change the reduced mass, m; of the stretch but not the force constant. At identical temperature, the kinetic energy is unchanged, permitting the relationship between reduced mass and vibrational frequency, v; given by m1 v21 ¼ m2 v22 : v2 ¼ v1

rffiffiffiffiffi m1 m2

1 1 1 ¼ þ m MA MB 1 1 1 ¼ þ m1 MAð1Þ MBð1Þ

1034

Isotope Effects in Chemistry and Biology

1 1 1 þ ¼ MAð2Þ MBð2Þ m2 2 6 6 v2 ¼ v1 6 6 4

1 MAð2Þ

þ

1 MBð2Þ

1 1 þ MAð1Þ MBð1Þ

! 31=2 7 7 !7 7 5

Therefore, beginning with the C – H stretch frequency in wavenumbers v; where v ¼ v·c; vH ¼ 2800 cm21 11=2 1 1 1=2 B 2 þ 179 C C ¼ vH 0:709 ¼ vH B @ 1 1 A þ 1 179 0

vD ¼ vH

mCH mCD

v vD < pHffiffi ¼ 1985 cm21 2 This means that the reactant molecule substituted with deuterium has at least one changed vibrational frequency. 2. High-Frequency CH Bond Stretch: Equilibrium Isotope Effects The first column of Table 42.2 lists the masses of the first simple harmonic oscillator to consider: MA ¼ 1 g/mol and MB ¼ 179 g/mol. This represents a C – H bond in glucose, and we choose an arbitrary stretching frequency of 2800 cm21. The same molecule substituted with one deuterium for the hydrogen atom we have focused upon with then have MA ¼ 2 g/mol and MB ¼ 179 g/mol. The stretching frequency is calculated as above in Section IV.A.1, finding a frequency ratio of 0.709 and a deuterium stretching frequency of 1985 cm21. The transformed frequency, ui, is calculated with u ¼ hv=kB T; and the same is done for the same vibrational mode in our hypothetical product molecule, including a change in frequency by 136 to 2664 cm21 in the nondeuterated form. For the process of conversion of our reactant to product, we would have two frequencies, and the inclusion of deuterated forms increases this to four. The EIE is calculated in Table 42.2 in pieces from the expressions for MMI, ZPE, and EXC given above, and is listed, respectively, as EIE-MMI, EIE-ZPE, and EIE-EXC, and the product of all three is given as “EIE total.” So far we have assumed that the vibrational mode under consideration has merely changed in frequency between reactant and product, and has not disappeared as would occur in a proton-transfer type reaction. We see from the second column of the table that EIE-MMI and EIE-EXC give a contribution of unity, and the real isotope effect comes from EIE-ZPE, with a magnitude of 1.100. We discuss the reasons for the unity contributions below, in Section IV.A.4 and Section IV.A.5. If we are studying the KIE, and our C – H bond becomes broken and not merely strengthened or weakened in the reaction, implying the primary isotope effect, then we may calculate the KIE in pieces as well, treating the frequency as changed from 2800 to 0 cm21 instead of to 2664 cm21. Since our focus is primarily on BIE and not KIE, and the TCDF term is a focus of constant research outside the scope of this article, we include the KIE-MMI, KIE-ZPE, KIE-EXC, and “KIE total” terms for reference in Table 42.2. In a primary KIE one mode has been converted to the unstable reaction coordinate, now possessing an imaginary frequency. A large contribution to the isotope effect (5.063) comes from the fact that the previously stable mode in the numerator of

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

1035

TABLE 42.2 Worked Examples Show Relative Importance of MMI, ZPE, and EXC for Several Cases of Molecule, Substitution, Vibrational Frequency, and Reaction CH Bond in Glucose (MW 5 180 g/mol) Light MA MB Frequency ratio

1 179

CN Bond in Minimal Model (MW 5 12 1 14 g/mol)

Heavy

Light

2 179 0.709

1 181 1.000

3 179 0.581

12 14

Heavy 12 15 0.984

14 14 0.961

vi-reactant (cm21) ui-reactant

2800 13.513

1985.4 9.581

2799.9 13.512

1625.5 7.845

1000 4.826

984.5 4.751

960.8 4.637

vi-ts or -product (cm21) ui-ts or -product

2664 12.856

1889.0 9.116

2663.9 12.856

1546.6 7.464

864 4.170

850.6 4.105

830.1 4.006

EIE-MMI EIE-ZPE EIE-EXC EIE total

1.000 1.100 1.000 1.100

1.000 1.000 1.000 1.000

1.000 1.148 1.000 1.147

1.000 1.005 1.000 1.005

1.000 1.013 0.999 1.012

KIE-TCDF KIE-MMI KIE-ZPE KIE-EXC KIE total

0.709 7.140 1.000 5.063

Not treated herein 1.000 1.000 1.000 1.000

0.581 17.013 1.000 9.881

Not treated herein 0.984 0.961 1.038 1.099 1.001 1.002 1.023 1.058

v should be v-bar.

our KIE expression is now essentially unbalanced by a corresponding frequency in the denominator (formerly with 3N 2 6 modes and now with 3N 2 7 modes), and the rest of the intrinsic isotope effect comes from the leading transition-state decomposition-mode factor. Very large isotope effects are obtained with these measurements, even neglecting the decomposition mode itself. The large percentage mass change in hydrogen isotope substitutions yields excellent sensitivity to changes in force constant. Replacing a deuterium for hydrogen at a site other than the CH bond we have focused upon yields very little change in the frequency of our motion, since the reduced masses in this case can be treated as MA ¼ 1 g/mol and MB ¼ 180 g/mol, as shown in Table 42.2 as well. No ground state change in stretching frequency translates into no overall isotope effect. The tritium EIE is also shown as the last column under the CH bond example in Table 42.2, demonstrating that EIE-MMI and EIE-EXC are still near unity, and the majority of the EIE comes from changes in the zero-point energy term. 3. Lower-Frequency CN Bond Stretch: Equilibrium Isotope Effects Identical calculations for a minimal C – N system with a stable stretch frequency of 1000 cm21 are shown in the CN Bond Example in Table 42.2. One can see that for either a 15N equilibrium isotope effect or a 14C EIE, the change in frequency is much smaller than in the CH bond case, with frequency ratios of 0.984 and 0.979, respectively. The overall EIE is 1.005 for 15N substitution and 1.006 for 14C substitution, while the MMI contribution is unity out to at least 10 decimal places (not entirely shown). The EIE-EXC is 0.9996 for the 15N substitution and 0.9994 14C. Again, the largest contribution to the overall isotope effect

1036

Isotope Effects in Chemistry and Biology

comes from the zero-point term, but the overall effects demonstrated in the table would be within the experimental error. In enzymatic systems, the changes in reduced mass are much smaller, since most substrates are much larger than a simple C – N system. A typical N-glycohydrolase will yield a transition state resembling that in Figure 42.1. Taking the 14C isotope effect for comparison (not shown in Table 42.2): 31=2 2 1 1 þ 6 135 134 7 7 ¼ 0:996 £ 4:826 ¼ 4:808 EIE overall ¼ 1:001: u14 ¼ u12 6 4 1 1 5 þ 133 134 In order to observe 20-heavy atom KIE, the frequency changes in several modes must be quite large. The decomposition mode, resulting in a primary KIE, would result in a vibrational contribution to the isotope effect of only 1.005. However, these 10-heavy atom KIE can be observed, since the decomposition factor is expected to be larger than these numbers. 4. When Does MMI Count? 21 The first pffiffi term21(MMI) is analyzed in the following way. Let vH ¼ 2800 cm ; and vD ¼ ð2800= 2Þ cm : Then let this mode’s frequency shift by 134 wavenumbers. This ought to generate an isotope effect; but examine the effect on MMI: 21 v– H ¼ 2800 2 134 ¼ 2666 cm

[

2666 21 v– D ¼ pffiffi cm 2 1 0 1 uD p ffiffi C B uH ! B 2 C ¼B C ¼ 1: – @ 1 A uD pffiffi 2 u– H

The implicit assumption in the last equality is that between reactant and transition state, there is no change in reduced mass for this vibration. This assumption is approximately valid only in large molecules and for all modes in EIEs and 20-KIE, but only two of three modes in 10-KIE. For 10-KIE, we are left with: ! !1=2 m– u– 1L MMI ¼ : mo u– 2L pffiffi The first part of this term is ð1= 2Þ; for example, in a 10-deuterium KIE. 5. When Does EXC Count? The EXC term yields no contribution for modes when the frequency is greater than about 1500 cm21 for the following reason: v ¼ 1500 cm21 u¼

hc v ¼ ð4:826 £ 1023 cmÞð1500 cm21 Þ kB T

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

1037

uH ¼ 7:2 uD ¼ 5:1 1 2 e2uH ¼ 1 2 0:00072 ¼ 0:99928 1 2 e2uD ¼ 1 2 0:00598 ¼ 0:99402: The ratio between the last two values yields a contribution to KIE of approximately 1.005, at the error limit of experimental observation. Even the total loss of a CH stretching frequency, as in a KIE, and isotopic substitution as drastic as deuteration will contribute only this much to the EXC product. In an EIE, this 1.005 will be mostly offset by mode with a changed frequency, and will contribute even less. The physical reason for this observation is that in a harmonic quantum oscillator, the energy levels are always evenly spaced with values of ðn þ ð1=2ÞÞhn: The distance between each level and the next is simply hn; and therefore the higher the frequency, the larger the distance between energy levels. Above some vibrational frequency, ambient energy in the form of temperature is insufficient to permit significant occupation of excited energy levels for either the CH or CD bond. The isotope effect in this term is derived from the difference in ability of CH or CD to populate these excited levels. The ratio between two numbers very close to unity will also be unity, as seen in the example. This applies to both CH and OH stretches, which oscillate around 2800 and 3800 cm21, respectively.

B. ALTERATION IN F ORCE C ONSTANTS If the isotope effect could be attributed to a change in a single ZPE term only, as in a C –H stretching mode change, an EIE of 10% would correspond to a vibrational shift between reference and observed state of 136 wavenumbers in the case of deuterium label and 94 wavenumbers if tritium label kH hc hc – – ¼ 1:10 ¼ exp ðvH 2 vD 2 v– ððv 2 v– H þ vD Þ ¼ exp H Þ 2 ðvD 2 vD ÞÞ kD 2kB T 2kB T H ¼ exp

hc 1 1 2 pffiffi ðvH 2 v– HÞ 2kB T 2 D 2 KIEðvH 2 v– HÞ ¼

where 2kB T hc

hc ¼ 2:826 £ 1023 cm kB T

lnð1:1Þ 1 1 2 pffiffi 2

¼ 136 cm21

T 2 KIE ¼ 94 cm21 The energy required to change only the zero point of this mode for in the protiated molecule (2800 to 2664 cm21) is given by: DE0 ¼

1 1 hðv1 2 v0 Þ ¼ ð6:626 £ 10234 J secÞð24:01732 £ 1012 sec21 Þ 2 2 ¼ ð1:330938116 £ 1021 JÞ DE0 ¼ 801 J=mol ¼ 191 cal=mol:

We shall see below that the intramolecular forces available to change these force constants through hyperconjugation are well in excess of these requirements. The change in force constant (also called k; but used only in the next equation as force constant) required for a 10% isotope

1038

Isotope Effects in Chemistry and Biology

effect is: v1 ¼ v0 k [ 1 ¼ k0

v1 v0

2

¼

sffiffiffiffi k1 k0

2664 cm21 2800 cm21

!2 ¼ 0:905:

Surprisingly, bond order changes sufficient to cause such large changes can result from a very small increase in the electron population of an antibonding orbital sp. Calculations21 have shown that a 5.5% isotope effect attributed entirely to a CH stretching mode is associated with a 0.55% change in CH bond length and a 0.23% change in bond order calculated from s and sp population estimates by the natural bond orbitals (NBO) package of Gaussian. The consequence here is that isotope effects can be a very sensitive measure of molecular structure. 1. How Many Modes Actually Matter? Take the same CH stretching mode as first treated in Section IV.A.1, but with the deuterium substitution elsewhere: vH ¼ 2800 cm21 11=2 0 1 1 B 1 þ 180 C C vD ¼ vH B @ 1 1 A þ 1 179 vD < vH ! Remote substitutions do not cause perturbation in this mode between reactant and product or transition state. Conversely, modes which deal directly with our substitution will influence MMI, ZPE, and EXC. In a three-dimensional world, each nucleus would participate in three modes, except that due to vibrational coupling in real molecules, this number can be as large as nine. Speaking hypothetically, we can reduce the equations to account for the three modes that should be left to dominate these equations given a single isotopic substitution. For KIE: ! 10 0 10 1 3 X 3 Y 3 1 2uH Y u D 1 2 e !B B C exp ðu 2 u Þ CB D CB C 2 H u– uH C CB HL B D 1 2 e2uD C B CB ! kH ¼ B C; B C – B 2 – CB 2 2 C@ Y X u– 1 – 1 2 e2uH A DL @ Y uD A@ A – exp ðu 2 uD Þ – 2 H u– 1 2 e2uD H and for EIE: D

3 Y uX – D uY – H Keq ¼ uX – H uY – D

!

3 Y

1 exp ðuX – H þ uY – D 2 uX – D 2 uY – H Þ 2 ! 3 Y ð1 2 e2uX – H Þð1 2 e2uY – D Þ : ð1 2 e2uX – D Þð1 2 e2uY – H Þ

!

In the case of hydrogen bonded to carbon, one of these modes will be a C –H stretching mode, as unique in force constant as it is unique in chemical shift, and the other two hypothetical modes

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

1039

would be orthogonal bends. However, it is unreasonable in complicated molecules to expect this to be rigorously true, and very difficult in such molecules to catalogue the vibrational modes affected by heavy atom substitution. The two bending modes just mentioned will actually be spread out over as many as nine molecular normal modes, and will be quite difficult to describe. If the worker has separated out the stretching mode frequency, as it is possible to do, what is left of the overall partition function must necessarily comprise these two orthogonal bends, whether they are present in pure form or not. 2. Isotope Effects from Altering Mode Coupling Partners The frequencies described above correspond to collective motions of atoms in the chemical species, resulting from combinations of individual, more intuitive motions. Coupling motions results in an equal number of normal modes. The frequencies of these modes are not identical to those of the original bond stretches and angle bends that comprise them. This can be illustrated by examining the frequency changes in formaldehyde upon deuteration, see Table 42.3. In comparing the monodeuterated to native, one sees decreases in all frequencies. Both hydrogen atoms participate in all of the vibrational modes, as do the deuterium atoms. However, the monodeuterated molecule has a broken symmetry which has served to uncouple the CH stretch from CD and to isolate a CH in-plane wag. The new CH stretch vibrates at 3199 cm21, halfway between the original modes in which it participated, the symmetric CH stretch (3163 cm21) and the antisymmetric CH stretch (3234 cm21). More general techniques in modal coupling and frequency changes are found in Molecular Vibrations by Wilson et al.51 To root our calculations in reality, we may compare vibrational frequencies calculated using ab initio methods to those observed in experiment. From the literature we find observed values of 1167.3, 1249.1, 1500.2, 1746.0, 2782.5, and 2843.3 cm21 for the ground state vibrations of formaldehyde.52 – 55 It is generally accepted that calculated frequencies may be collectively scaled in order to provide better agreement with experimental values; performing a least-squares fit against the first column of Table 42.3 gives a best-fit scaling factor of 0.884. The standard error of this fit is only 30.79 cm21, small in comparison to the magnitude of the numbers compared. Further, we expect calculations to be more internally consistent than they are with experimental values. In isotope effect calculations, scaling factors are regularly used, bringing even the absolute magnitude of the frequencies within acceptable limits of experimental values. The frequency alterations that occur as a chemical process occurs cause coupling changes that may translate into an observable isotope effect under certain circumstances. TABLE 42.3 Deuteration in Formaldehyde Causes Frequencya (cm21) Shifts and Assignment Changes CH2O 1337 1378 1693 1915 3163 3234 a

CHH oopb wag H, H IPc rock HCH ip scissor HCH ip scissor, CO stretch CH(H) stretch (symm) CH(H) stretch (anti)

CDHO 1138 1212 1572 1875 2361 3199

D,H ip rock CDH oop wag CH ip bend HCD ip scissor, CO stretch CD stretch CH stretch

CD2O 1072 1088 1242 1841 2318 2409

CDD oop wag D, D ip rock DCD ip scissor DCD ip scissor, CO stretch CD(D) stretch (symm) CD(D) stretch (anti)

Calculated with Gaussian (M.J. Frisch et al., Gaussian 94, Revisions C.2, D.4, Gaussian 98, Revision A.6, Gaussian, Inc., Pittsburgh, PA, 1998.) b Out-of-plane. c In-plane.

1040

Isotope Effects in Chemistry and Biology

Let us momentarily study the isotope effect on a hypothetical equilibrium between two species A and B: A – H þ B – D $ A – D þ B –H. The isotope effect, as always, is given by DK ¼ (KH/KD), and in terms of partition functions, Q: 1 3N26 Y sinh 2 uX2i Q¼ uX2i i where ui have been defined above as the transformed normal mode frequencies, and X ¼ A –H, B – D, A – D, and B– H, we have the overall isotope effect: D

K¼

QA – H QB – D : QA – D QB – H

However, returning to the question of coupling, we know that molecules move in complicated fashion, in which the presence of a CH stretching mode resonating at a frequency near the resonating frequency of a second CH stretching mode will cause the combination of the two modes into two resultant motions with two new frequencies. Frequency changes, which are the essence of isotope effects may be caused by the creation, destruction, or change in frequency of other motions in the molecule. For example, in the cleavage reaction of an N-glycosidic bond, the C – N bond stretched at a given frequency, probably close to the frequency of other motions in the molecule. By cleaving this bond, in fact to any extent, in the transition state, we have altered its intrinsic frequency as well as the extent to which it couples with these other motions. Coupled molecular motions of which the C –N stretch was a part will no longer feel its presence and will themselves experience alterations in frequency. To investigate the effect of altering coupling motions, we can look at the isotope effect on an equilibrium between a molecule and a molecule identical to itself except with the coupling motions “turned off,” or more accurately, at least changed. First we choose formaldehyde, in which the C –H bonds are identical and are strongly coupled. In order to uncouple them, we shall transform one hydrogen into a deuterium. In the equilibrium reaction written above, A – H is formaldehyde, B –H is “uncoupled” or monodeuterated formaldehyde, A – D is formaldehyde deuterated at the position where we are studying the isotope effect, and B – D is uncoupled formaldehyde, also deuterated at the position of interest. In order to satisfy the law of conservation of mass and the assumptions of derivation of the partition functions currently written, we can include in our imagination extra protons and deuterons at infinite distance, where they will not affect the results of frequency calculations. We can then compare the partition functions of the four species as an EIE: D

K¼

QHH QDD : QHD QDH

Figure 42.6 illustrates several examples from which we may carefully generalize. Each molecule pictured contains two deuterium atoms, and we are cataloguing the modal coupling between hydrogen atoms at the two positions, and all modes are included in the following calculations, not only stretching modes. The question answered by these calculations is: “Uncoupling the modes of these two atoms by deuterating the first causes frequency changes in the modes of the second corresponding to what isotope effect?” (The order of deuteration does not matter.) The two hydrogen atoms in formaldehyde are chemically equivalent, and we would expect strong coupling. Therefore take the value of 1.022 as nearly maximal. In 2-propanol, CH and OH motions are not coupled at all, whereas CH – CH motions are coupled to the extent of chemical identity and physical proximity between the two hydrogens. In benzene, the first two adjacent deuterations or the last two adjacent deuterations (where the diagrammatic convention is reversed, showing hydrogen atoms where the deuterations will occur instead of deuterium atoms where the deuterations occur in calculation) give a small effect, even though these hydrogens must be

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

1041

O 1.022 H

H

D H

D H3C

1.0005 CH3

O

1.00007

H

D

H H H

HO

H H

D

H3C

D

1.0012

D

H 1.0011

D D

H H3C

HO

D D

H D

1.021

H H

1.00001

D

H H D

C H

HO

D

FIGURE 42.6 Isotope effects due to decoupling normal mode vibrations. These calculations show the deuterium isotope effect due to deuteration; substitution of CD for CH will uncouple its vibrations from others in the molecule, changing all previously-coupled vibrational frequencies. These frequency changes between deuterated and protiated compounds will the produce isotope effects shown. For 2-propanol (left lower column), the values indicate the isotope effect on the second deuteration due to the first. For formaldehyde (top left) and benzene (right column), values represent deuterating the first and second or penultimate and last positions.

chemically equivalent; this implies that the greater the number of coupled equivalent atoms, the smaller the contribution of each to the important modes. Finally, for glucose Table 42.4 lists the effects in D -glucose of backbone hydrogen atoms upon each other; the only experimentally relevant effect appears between the chemically similar methylene hydrogen atoms. This is not the first observation of breakdown of the rule of the geometric mean, which states that the fractionation factor at one site does not depend upon deuteration of another site in the same molecule.56 Deviation from the rule has been attributed to bond –bond interaction in the normal mode molecular vibrations.57 It should be remembered here that these are the greatest possible effects due to coupling changes for the respective molecules. Frequency changes result in coupling changes, and not, in general, in complete uncoupling. In constructing these models, some modes were completely uncoupled through the extreme measure of deuteration. With real equilibrium experiments, it is only possible that conformational changes can possibly lead to such radical uncouplings, and

1042

Isotope Effects in Chemistry and Biology

TABLE 42.4 IE Due to Uncoupling of Normal Vibrational Modes in Glucose. Rows and Columns Represent First and Second Deuterium Substitutions, Respectively

1 2 3 4 5 6 7

1

2

3

4

5

6

1.00073 1.00000 1.00000 1.00002 1.00000 1.00000

1.00070 1.00001 1.00000 1.00000 1.00000

1.00063 1.00001 1.00000 1.00000

1.00075 1.00001 1.00000

1.00017 1.00019

1.01351

7

therefore we would consider the possible contribution of coupling/uncoupling interactions to be insignificant. These coupling changes may play a significant role, however, in KIE if the stable motion being destroyed was coupled with motions of the atoms being probed through heavy-atom substitution. 3. Sterics and Hyperconjugation We now describe the phenomena that contribute to observable BIE. While there are three classes of motions (stretching, bending, and twisting), those equations governing isotope effects are sensitive only to force constant changes. Therefore, we would expect to find isotope effects due to stretching frequency changes and bending frequency changes; they are summarized in Figure 42.7. Closer examination reveals the mechanism of hyperconjugation (see especially Refs. 58 –73). Molecular orbitals are generally classified as bonding, antibonding, or nonbonding. Computationally, these are referred to as natural bond orbitals (NBO), since they correspond to our intuitive understanding. Even in the hydrogen molecule, one observes one bonding orbital and one antibonding orbital. The total bond order, n, is defined by the difference in electron populations Hyperconjugative effects on CH bonds

These effects tend to loosen CH bonds, giving normal IE when they occur.

H

H

H

C

C

C

O

R

Lone pair effect (np--> σ*)

C H(R) Anti-periplanar overlap (σ -- > σ∗)

O− Ionization/ Hydrogen bond effect

Steric effects Steric effects can only tighten a vibrational mode.

H H

bulky

Direct imposition

H Simple angle distortion

FIGURE 42.7 Phenomena contributing to equilibrium isotope effects. Hyperconjugative electron transfer can lower CH bond order and give normal isotope effects (top row), while steric imposition can increase CH vibrational frequencies and give inverse effect (bottom row).

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

antibonding -

σ∗

1043

lone pair – n (nonbonding)

bonding - σ

FIGURE 42.8 Orbitals relevant to CH bond order in 2-propanol. Shown are bonding and antibonding CH-bond orbitals and one of two high energy oxygen – electron lone-pairs. Notice that the higher energy sp antibonding orbital appears to have a node or absence of electron density between nuclei. The antibonding configuration resembles free atoms more than in the bonding orbital.

between these two orbitals, and we have seen above that the force constants relevant to isotope effects depend upon this bond order. It is possible to change the bond order in any molecule by affecting either the population of a bonding orbital or of an antibonding orbital. Total electron count in a conformational equilibrium process is always constant, so in order to move electrons around, there must be two or more participating intramolecular orbitals. Electrons can be seen as molecular residents, occupying bonding, antibonding, and nonbonding orbitals as convenient to best lower their free energy. The relevant orbitals in 2-propanol (and any hydroxyl moiety) are illustrated in Figure 42.8. The bonding and antibonding orbitals comprising the bond order of the central CH bond are shown, as well as the oxygen lone pair responsible for most electron donation into that antibonding orbital. Not shown are the OH antibonding orbital which accepts electrons from the shown CH bonding orbital, and the second oxygen lone pair, which lies orthogonal to the first. The basic requirements for hyperconjugation are that the two orbitals have some nonzero total spatial overlap and that the orbitals possess different energy. Nonzero overlap boils down to symmetry and proximity. The magnitude and angular dependence of some hyperconjugative effects are shown in Figure 42.9. The isotope effects illustrated in Figure 42.9 should be considered from the perspective of the oxygen lone pair electrons in the 2-propanol molecule. The greatest normal isotope effect results from the total deprotonation of the hydroxyl. This deprotonation leaves behind a significantly increased oxygen electron density, which with increased energy and spatial extent results in more electron delocalization into the antibonding orbital of the observed central CH bond. This isotope effect appears in the plot as a straight line because there is no HCOH dihedral angle. Including a hydrogen bond recipient like formate amounts to a partial deprotonation of the hydroxyl group and yields normal isotope effects to a lesser degree, regardless of HCOH angle. Similarly, including the attack by formic acid results in a partial protonation and yields inverse isotope effects, regardless of HCOH angle. The two formate models demonstrate the largest normal isotope effect at 908 of HCOH. This is the point of greatest alignment between the oxygen lone pair electron orbital shown in Figure 42.8 and the antibonding CH orbital also shown there. The formic acid models show a different maximal angle due to orbital rehybridization secondary to the protonative attack. One formate model included an extra constraint to prevent any steric bumping between formate and the central CH bond at zero degrees, resulting in a clear difference in isotope effect between the two models at this

1044

Isotope Effects in Chemistry and Biology

formate & “constrained”

formic acid & “constrained”

(a)

formate “constrained”

formate

1.600

equilibrium tritium isotope effect

1.500

2-propanol (deprotonated) with formate(constrained) with formate 2-propanol alone with formic acid with formic acid (constrained)

1.400 1.300 1.200 1.100 1.000 0.900

(b)

10

30

50

70

90

110 130 150 170

HCOH torsional angle (degrees)

FIGURE 42.9 Hyperconjugation in 2-propanol. (a) The alcohol was computed in hydrogen bonding interaction with formic acid or formate, under constrained or relatively unconstrained conditions. The HCOH dihedral angle of 2-propanol was then taken through 1808 of rotation. In the case of formate, the constrained model was forbidden to collide with the central CH bond of 2-propanol, and in the case of formic acid, the constrained model was forbidden to form two hydrogen bonds, as shown. (b) The calculations were run at b3pw91/6-31gpp, and show that isotope effects at the central CH bond can be generated by manipulating the hydroxyl proton, either in orientation of the HCOH angle or through deprotonation altogether.

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

1045

1.1040

1.1035

0.98 0.95

1.1030

0.93

1.1025

eie - total stretch only

0.90

bending modes

0.88 2.0

1.1020

CH bond length (angstroms)

O-H = 3.5 Å

Equilibrium tritium isotope effect

1.00

CH bond length (angstroms)

2.2

2.4 2.6 2.8 3.0 3.2 OH distance (angstroms)

3.4

1.1015 3.6

FIGURE 42.10 A collision between 2-propanol and formaldehyde. This calculation was performed at b3pw91/6-31gpp and shows that as this model carbonyl is made to collide axially with the central CH bond of 2-propanol, an inverse isotope effect is generated. The CH bond length is shortened, and we find that the isotope effect is due, counterintuitively, to tightening of the stretching and bending modes of the central CH bond.

point. In contrast, the unconstrained formic acid model was permitted to form two hydrogen bonds between the molecules; the deprotonating interaction slightly offsets the inverse influence of the protonating bond, yielding a slightly less inverse overall effect. While hyperconjugation is a relatively indirect approach to altering force constants, it is also possible through direct steric interaction to generate isotope effects, yielding some fascinating results. These are shown in Figure 42.10, with the collision between a CH bond in 2-propanol and the carbonyl oxygen of formaldehyde. This calculation was performed in order to understand a large inverse isotope effect on the association constant for glucose binding to human brain ˚ in an axial attack, we hexokinase.25 As the oxygen –hydrogen distance is decreased from 3.5 A expected to find a tightening of the 2-propanol CH stretching mode, with little effect on the bending modes. Instead, we find that the steric approach causes shortening of the CH bond with concomitant tightening of the bending modes as well. It is clear at this level of theory that large inverse equilibrium isotope effects are possible through steric interactions. It is well known that deuterated and tritiated compounds make better bases and occupy smaller molar volumes than their unlabeled counterparts,23,41,64,73 – 80 and that these phenomena have a solid foundation in chemical physics.2,81 – 84 Any equilibrium interaction that requires basic character will occur more favorably with a deuterated compound. It is also well known that this increased basicity can be modulated by deuterium proximity to the basic group and by orienting the C –D bond parallel to its electron lone pairs.59,61,62,64,73

C. SUMMARY Isotope effects are highly sensitive to molecular structural changes, and this fact makes isotopes extremely useful probes of molecular structure. A careful experimentalist can obtain isotope effect values to high precision, and then it remains to explain them, whether kinetic or equilibrium. This analysis provides an appreciation of the critical nature of estimating the force constants and their changes for internal molecular motions. Much effort has been devoted to this pursuit

1046

Isotope Effects in Chemistry and Biology

in developing force fields derived from model compounds intended for more general use,55,85 but ab initio calculation of force constants and their relative changes is probably the most reliable method. For years, workers were limited to force field models designed for molecular mechanics calculations in order to estimate force constants for stretching and bending modes. These constants would be provided to a program like BEBOVIB86 in reduced, “natural” coordinates and turned into Cartesian internal coordinates for solution. One could use the combination of Gaussian with a program like QUIVER87 only for EIE. Now it is possible to generate transition state models with Gaussian and test them directly with a program like ISOEFF98.88 We feel that this is a far superior method for estimating normal mode frequencies. The computations described here have been made at the highest level possible of theory given available resources, and this should always be the investigator’s rule of thumb.

V. EXAMPLE: GLUCOSE AND BRAIN HEXOKINASE A. METHODS In practice, BIE can be measured by passing a solution containing the binding equilibrium through a membrane with carefully chosen cutoff to restrict the passage of enzyme (Figure 42.11). Isovolumetric samples are taken from each side of the membrane, and analyzed isotopically, yielding the values Rm (ratio of isotopes on the “mixture” side) and Rf (ratio of isotopes in the “free” substrate pool). For example, we consider isotopic enrichment with tritium, and the unenriched substrate is labeled with 14C. Then, since we allow Rm ¼ (14Cf þ 14Cb)/(3Hf þ 3Hb), Rf ¼ (14Cf/3Hf), and Rb ¼ (14Cb/3Hb): Rb ¼

1 ff Rm þ R: 1 2 ff ff 2 1 f

The isotope effect on the binding association constant is then given by Rb/Rf. Radioactive contaminants must be carefully avoided, as they may appear not to bind enzyme and will thus bias the results. The two major sources of errors in experimental determination of BIE are easily minimized, as they derive from separate extrapolations. The first, common to all competitive studies detecting the radioisotopes tritium and 14C, involves the simultaneous measurement of tritium and 14C decays. The spectrum from 5 to 1000 keV in liquid scintillation counting is divided into two windows bordering at approximately 285 keV. Tritium decays are of lower energy and appear only in the first window, but 14C decays spread over both windows. The traditional method for determining each concentration in a sample is to run a 14C standard to determine the ratio of 14C

E+

14C-glu

E•14C-glu

3H-glu

E•3H-glu

14C-glu E +3 H-glu

+

E•14C-glu E• 3H-glu

14C-glu 3H-glu

FIGURE 42.11 Experimental method for measuring binding isotope effects (BIE).

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

1047

counts between the two windows. This ratio and the number of counts in the second window of his sample are used to subtract the appropriate number of counts from the sample’s first window. The end result of such an approach is that one has a direct measure of 14C but not of tritium. We have found that using tritium:14C ratio of no less than 3:1 gives suitable error reduction. The second source of error is in the separation of “free” and “mixed” substrate pools as explained above, one has a direct measure of isotope ratio in the “free” substrate but not the “bound.” The latter must be extrapolated from the “mixed” by subtracting the “free” counts. Maximizing enzyme concentration and minimizing total substrate concentration, while using enough counts for meaningful statistics, addresses this problem by minimizing the magnitude of the subtraction.

B. THE B INARY C OMPLEX Listed in Table 42.5, tritium BIE were measured as described above and in Ref. 25. Statistically significant isotope effects occur at every position measured. Given that hexokinase is known to take either anomer as substrate, it follows that either anomer may bind the enzyme, and it becomes necessary to determine the presence and the contribution of anomeric EIEs. These were measured, as described in Ref. 21, as were the affinity constants for each anomer to human brain hexokinase: Ka ¼ 15 mM and Kb ¼ 24 mM. The anomeric isotope effects, actually measured as deuterium effects, are extrapolated to tritium effects via the Swain – Schaad exponent 1.441, and Table 42.6 explores the contribution of the TKeq to an observed binding experiment for relative TABLE 42.5 Equilibrium Binding IE for Glc to HK Depend upon b – g, CH2-ATP Competitive Labels

Without ATP Analoga

With ATP Analogb

[1-3H]- þ [2- or 6-14C]glucose [2-3H]- þ [2- or 6-14C]glucose [3-3H]- þ [2- or 6-14C]glucose [4-3H]- þ [2- or 6-14C]glucose [5-3H]- þ [2- or 6-14C]glucose [6,6-3H2]- þ [2-14C]glucose

1.027 ^ 0.002 0.927 ^ 0.0003 1.027 ^ 0.004 1.051 ^ 0.001 0.988 ^ 0.001 1.065 ^ 0.003

1.013 ^ 0.001 0.929 ^ 0.002 1.031 ^ 0.0009 1.052 ^ 0.003 0.997 ^ 0.0009 1.034 ^ 0.004

a b

Ref. 25. Ref. 26.

TABLE 42.6 Contribution to BIE by Anomeric Aqueous Equilibrium Ka/Kb Label [1-3H] [2-3H] [3-3H] [4-3H] [5-3H] [6,6-3H2] a b c

T

Keqa

0.01

0.1

0.65b

1

10

100

BIEc

1.063 1.039 1.039 1.001 1.053 0.997

0.964 0.977 0.977 0.999 0.970 1.002

0.971 0.982 0.982 1.000 0.976 1.001

0.994 0.996 0.996 1.000 0.995 1.000

1.000 1.000 1.000 1.000 1.000 1.000

1.020 1.013 1.013 1.000 1.017 1.000

1.024 1.015 1.015 1.000 1.020 1.000

1.027 0.927 1.027 1.051 0.988 1.065

Extrapolated from deuterium isotope effects reported in Lewis and Schramm (ref) by the Swain–Schaad relationship. Determined for this system by fluorescence titration. Experimental from Table 42.1.

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Isotope Effects in Chemistry and Biology

anomeric binding affinities of several orders of magnitude. These can be compared against the last column of that table to show that no single value of affinity ratio can transform the set of anomeric EIE into the observed BIE. Thus, the presence of “intrinsic” BIE is proven. Extensive discussion of the anomeric equilibrium isotope effects and their repercussions may be found in Ref. 21. The experimental binary BIE summarized in Table 42.5 were sufficiently surprising to cause us to measure the [2-3H]glucose effect 60 times under different conditions. This isotope effect is 0.927. We also found large effects at the [4-3H] and [6-3H2] positions of 1.051 and 1.065, respectively. Large normal BIEs are easily explained by familiarity with Figure 42.9b. In a hydroxylated molecule, a significant BIE due to tritiation geminal to the hydroxyl is generated through a deprotonating active site contact. Total deprotonation (ionization) can yield an effect of approximately 1.56. The CH bond geminal to a hydroxyl is exquisitely sensitive to what we may call the “nuclear load” of the nucleophilic oxygen, and we will return to this point below, with the ternary effects. Large inverse isotope effects require the identification of a candidate active site contact with the CH bond itself. In the crystal structure of human brain hexokinase,89 the carbonyl oxygen of Ser603 ˚. makes a close contact to the bound sugar molecule, with O – H distance estimated at 2.67 A (Glucose hydrogen atoms modeled.) The choice of computational model becomes important, as an axial attack by CH4 on the central CH bond of 2-propanol with H – H intermolecular contact was insufficient to generate such a large effect; however, the axial attack by the carbonyl oxygen of formaldehyde, with its larger van der Waals radius, is sufficient.

C. THE T ERNARY C OMPLEX As the isotope effects on the anomeric solution equilibrium were found not to contribute to the binary complex binding data, neither do they matter for glucose binding in the ternary complex of hexokinase; glucose; b,g-CH2-ATP-Mg2þ. Probably the most kinetically important result from the studies with glucose and hexokinase is the change in isotope effect at the [6-3H2] position with addition of the nonhydrolyzable ATP-analog (b,g-CH2-ATP). The value changes from 1.065 to 1.034, as shown in Table 42.5. There are no changes at the [2-3H], [3-3H], or [4-3H] positions. While 1% changes are seen at [1-3H] and [5-3H], we know that chemistry occurs at the nucleophilic 6-hydroxyl. Partial deprotonation of O6 has occurred in the binding of glucose through interaction with Asp657. This is consistent with the idea of a nucleophilic activation in binding. The large normal isotope effect is partially relieved with the binding of ATP analog. While it is possible that the deprotonation by Asp657 is reversed or partial protonation from some other source (i.e., Lys621), it is much more likely that the increased nucleophilicity of O6 has been partially satisfied by attack on the terminal phosphate group itself, as in Figure 42.12. More discussion found in Ref. 26. Clearly, BIE have a role in mechanistic enzymology. It should be recalled that interpretation of any isotope effects is merely comparison of vibrational states, and one must be reasonably confident of the reference state. In the case of glucose

E

+

α-glucose

Kα

E•α-glucose

Keq E

+

β-glucose

Kβ

E•β-glucose

FIGURE 42.12 Human brain hexokinase binds both anomers of glucose and anomerization while bound is very unlikely.

Enzymatic Binding Isotope Effects and the Interaction of Glucose with Hexokinase

1049 ADP O O

Asp657

H

O

O

H

ATP

O H

H

Asp657

O

O

O

H H

P

O

O

O

H

H

O

O

O HO

H H

OH

HO

H

OH

H

H

OH OH

H

FIGURE 42.13 Binding isotope effects depend upon residue ionization state and hydrogen-bond directionality.

binding to human brain hexokinase, this was provided by the fact that we had already modeled anomeric equilibrium isotope effects.

VI. APPLICATIONS FOR BIE In modeling a transition state for a reaction with any amount of forward commitment, the kinetic BIE will always play some role. An equilibrium BIE is most powerful in its own regard when found in combination with a crystal structure or NMR solution structure giving a picture of the binding interactions between ligand and receptor, because it adds an extra dimension or two of understanding. First, and most importantly, structural methods in use today give highly detailed pictures of molecular interaction and conformation, but neither NMR or standard x-ray crystallography report on the presence or location of hydrogens. The enzymologist’s pH studies indicate the pKa’s of important groups in catalysis or binding, but the chronology of their protonation remains a more difficult problem. Groups interacting with substrate but only of secondary kinetic importance go neglected as well. BIE can very sensitively relate this information. Consider Figure 42.13, in which a glucose molecule interacts with a model carboxylic group representing contact to an active site residue. Three different isotope effect patterns are given, whether the carboxylate is deprotonated or if protonated, whether it donates to the 2OH or the 3OH. With good structural information and a number of isotope effects, it is possible to map out an active site hydrogen-bond network. Secondly, the structural methods are limited in their resolution. Hydrogen bonding distances as well as subtle structural distortions and certainly electronic distortions are far better observed with vibrational methods. Properly modeled, BIE can give a piecewise estimation of binding energy, or an excellent energetic estimation of ground state destabilization. Naturally, other methods are available in this area. Infrared spectroscopy will inform upon a specific vibrational mode, but requires a clearly identifiable vibration. Isotope effects report on the total vibrational change of an atom, but will not distinguish between bending, stretching, or twisting modes. Isotope-edited Raman spectroscopy shows each altered mode separately, easing vibrational assignment, but this technique requires high (mM) concentrations of enzyme and isotopic probes distinct from protein complexes which are not always attainable. Thus, the vibrational methods complement one another.

VII. CONCLUSION Chemical species exist on their own in solution equilibria, they bind enzymes, and are transformed by them. The fact that isotopic substitution alters chemical behavior in a meaningful way means that investigators have a microscopic handle on each of these phenomena. While transition state

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modeling represents the holy grail of catalysis, prebinding and BIE forward that quest and yield unique and valuable information in their own right.

ACKNOWLEDGMENTS This work was performed in Bronx, NY, and was supported by research grant GM41916 from the National Institutes of Health.

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Index A AADH see aromatic amine dehydrogenase AB equation, 130 –1 absolute rate theory, 838 absorption chromophore, 323– 9, 334–46 infrared, 177 –83, 365 abundance, 404, 410 acetic acid acetate complexes, 223 acetic acid hydrogen bonds, 223 acetic acid methanol complexes, 535–7, 542 –3 1-acetyl-2-hydroxynaphthalene2, 261 acetylcholine hydrolysis, 1012–14 acetylcholinesterase (AChE), 985–6, 1012–14 N-acetyl-L-leucyl-L-phenylalaninal, 988–90 acids acid-base complexes, 198, 199, 205 –17 acid-base reactions, 451–62, 705–9 acid-dissociated rate constants, 453, 456– 8 acidity, 959– 60 hydrogen-bond symmetrization, 175 –7, 183 –91 activation energy carbon-hydrogen-bond cleavage, 728 –30 concerted transfer, 536 hydrostatic pressures effects, 840 proton transfer, 454–5 stepwise transfer, porphine, 532 –3 activation factors, 754 –7 activation free energy adiabatic proton transfer, 553 barriers, 512 –14, 569 –70 double-well proton potentials, 704 multidimensional tunneling, 597 nonadiabatic proton transfer, 564 proton transfer, 553, 564, 707–8, 710 VTST, 581 –3, 597 activation parameters, 824 –5 acyclic phosphate esters hydrolysis, 905 acyl transfer, 961–4 S-adenosylhomocysteine, 778 S-adenosylhomocysteine hydrolase, 778 ADH see alcohol dehydrogenases; amine dehydrogenases adiabatic corrections, 91–6 adiabatic proton transfer free-energy relationship, 553–8 kinetic isotope effects, 524 –5, 549– 62, 572–3 molecular fragments, 697– 8 polar environments, 549 –62, 572 –3 tunneling, 524–5 adiabatic transitions, 731, 734 aerobic degradation, 879

AIM see Approximate Instanton Method air oxygen, 403 alanine dehydrogenase, 802, 910 alcohol dehydrogenases (ADH) catalysis, 811–30 hydrogen-electron transfer, 491–4 hydrogen transfer dynamics, 822– 6 hydrogen tunneling, 753, 756–7 hydrostatic pressures effects, 842– 4 kinetic isotope effects, 482–7 nuclear quantum mechanics, 632–4 solvent isotope effects, 827–9 alcoholic alkoxide-promoted dehydrohalogenations, 465–72 alcohols anharmonicity constants, 289 to carbanions, 471– 2 equilibrium dissociation constants, 456 frequency shifts, 289 hydron-transfer, 471–2 aliphatic chloroalkanes, 880 aliphatic hydrocarbons, 879, 880 alkoxide-promoted dehydrohalogenations, 465–72 alkyl iodides, 960–1 altitude effects, 393 amide hydrolysis, 903–5 amidogen radicals, 377 amine basicity, 959–60 amine dehydrogenases (ADH), 491–4, 676, 678–85, 737–8 amino acids, 480–2 aminoquinol, 663 ammonia, 156–70 ammonium ions, 265–6 amorphous solids, 591–4 anaerobic degradation, 879 analytic solution, 67 –9 anharmonicity anharmonic corrections, 200– 2 Bigeleisen-Mayer equation, 8–9 CMIE spectroscopy, 134 condensed matter isotope effects, 134, 139–40 constants, 287–9 hydrogen bonds, 154–62, 166– 70, 285, 287–9, 296–7 muonium, 435–6 proton stretching frequencies, 158–62 aniline NH chromophore absorption, 334–8 nitrogen kinetic isotope effects, 909–10 overtone spectra, 318–19 resonantly enhanced two-photon ionization, 330–8 anomalous isotope effects, 285–7 anomeric aqueous equilibrium, 1047–8

1055

1056 anticooperative coupled hydrogen bonds, 222 apo-glucose oxidase reconstitution, 651 Approximate Instanton Method (AIM), 522– 3, 526–9, 533 aqueous solutions, 451– 62, 693, 702, 705–9 archean isotope atmospheric chemistry, 407 argon, 31–2, 127–8, 335 –6, 337 aromatic amine dehydrogenase (AADH), 676, 678–9, 728, 738 aromatic amino acid hydroxylases, 868–70 Arrhenius activation, 582, 754–7, 824–5 Arrhenius behavior, 559, 568–71 Arrhenius plots, 441 –3, 728–9, 942–3 aryl-substituted N-methyltrifluoroacetanilides, 1008–9 Ascaris suum, 799, 802, 805 aspartate transcarboylase, 925–6 atmosphere, 387–411 capital delta constraints, 410–11 carbon cycle, 398–402 carbon dioxide, 375, 402 carbon isotopic effects, 398 –402 carbon monoxide, 372, 374, 400–2 delta constraints, 410–11 geochemical cycle isotopes, 388–90 ice cap archives, 396–8 interstellar media, 407–9 isotope selective spectroscopy, 347–8 lead isotope effects, 407 methane isotopes, 398–400 nitrogen isotopes effects, 402 nitrous oxide, 376 organic molecule deuterium enrichments, 407 –9 oxygen, 381, 403 ozone, 367–8 reference sample constraints, 410 –11 sulfate isotope effects, 403 sulfur, 403 –7 water cycle isotope effects, 390–6 zinc isotope effects, 407 atomic nuclear corrections, 94 –5 atomic properties, 435 atomic vapor laser isotope separation (AVLIS), 53–4 attainment, 67, 68 averaged Landau-Zener probability, 698–9 AVLIS see atomic vapor laser isotope separation 7-azaindole, 538 –9, 542– 3

B B3LYP, 113 Bacillus stearothermophilus, 482, 753, 756–7, 823 Baker Campbell Hausdorf theorem, 478 band shape, 296 –7 bare isotopic hydrogen nuclei properties, 434 barriers barrierless association reactions, 587 free energy, 512–14, 556–8, 569–70 heights, 581 recrossings, 512–14 see also low-barrier hydrogen bonds base-promoted elimination, 422– 3 basis sets, 111, 716

Index bath mode distribution, 477 ß-dicarbonyl enol tautomers, 241–2 ß-diketones, 272–3 BEBO see Bond-Energy–Bond-Order ß-elimination, 958–9 Bell tunneling correction, 683 bending frequency changes, 1042–5 benzaldehyde, 817–21 benzene carbon isotope effects, 338–46 condensed matter isotope effects, 127–8, 132–3, 141–2 muonium reactions, 442–3 overtone spectra, 318–19 reduced partition functions, 16, 17 sulfonates, 426 vapor pressure isotope effect, 127– 8, 132–3 benzenesulfonates, 426 benzhydrol, 956 benzoic acid dimers, 537 benzyl alcohol, 817–21, 824–5, 828–9, 842–4 benzyl chlorides, 425–6 benzylic anions, 471–2 benzylic hydroxylation, 866–7 BIE see binding isotope effects Bigeleisen–Mayer equation, 6–12, 955 bi-imidazoline complexes, 506 bimolecular reactions, 584–90 binary complexes, 1047–8 binding equilibrium isotope effects, 1019–51 binding force constants, 22– 3 binding isotope effects (BIE), 1019–51 biogenic methane isotopes, 399 1,8-bis(dimethylamino)naphthalene, 242–3, 262–3, 271 ß-lactams, 963–4 BO see Born–Oppenheimer Bond-Energy–Bond-Order (BEBO) 454–6, 460–1 bonds angles, 436–7 breakage, 685 dissociation energy, 931– 2 lengths, 254, 275, 436–7, 439–41, 976– 7 stretch, 1034–6, 1038–9 Born–Oppenheimer (BO), approximation adiabatic corrections, 91 –6 Bigeleisen–Mayer equation, 8–12 corrections, 9–12, 18 isotope theoretical chemistry, 90–6 muonium, 439–41 proton transfer, 695–6 vibrational isotope effects, 297–8 Born–Oppenheimer proton energies, 716 boron, 46, 78– 9 boronic acid, 983–4 bovine serum copper amine oxidase (BSAO), 664 brain hexokinase, glucose, 1046–9 bridged ring compounds, 534–5, 542–3 bridge energy, 285–6 bridging isotope effects, 967 bromine, 156–70, 299–301 2-bromopropane, 466 Bronsted coefficients, 556–7

Index Bronsted relation, 454 BSAO see bovine serum copper amine oxidase ß-secondary isotope effects, 957–8, 961, 963–4 ß-thioxoketones, 273 –4 t-butyl radicals, 438–9 butyrylcholinesterase, 985 –6

C Caldariomyces fumago, 886–7 calix[]arenes, 537 Calutron, 56 canonical variational theory (CVT), 580, 600–1 capital delta constraints, 410–11 carbamoylaspartate, 925 –6 carbamoyl-P, 925–6 carbanions, 471– 2 carbohydrates, 263–4 carbon acid, 466–7 carbonates, 377–8 carbon chemical shifts, 267, 271–3 carbon cycle, 398–402 carbon dioxide, 374–6, 847–8, 850 carbon disulfide, 134 –5 carbon–hydrogen bond cleavage, 725 –39, 865, 938–40, 947–9 carbon–hydrogen bond stretch, 1034– 6, 1038–9 carbon–hydrogen chromophore absorption, 323 –9, 338–46 carbonic anhydrase carbon dioxide hydration, 847–8, 850 catalytic mechanisms, 847 –57 concerted multiple proton transfer, 539– 41 nuclear quantum mechanical effect, 631 solvent isotope effects, 847 –57, 1014–15 carbon isotope effects, 338–46, 417–19, 423–4, 427–8 carbon isotope enrichment, 46 carbon kinetic isotope effects, 417 –19, 423– 4, 427–8 carbon labelling, 419 carbon monoxide, 372– 4 carbon-oxygen double bonds, 903–5 carbon substitution, 16, 17 carbonyl compounds, 903 –5 carboxylic acids, 245–6, 537, 959– 60 carboxypeptidase, 905 Cartesian coordinates, 98– 9 Cartesian displacements, 282 catalysis alcohol dehydrogenase, 811 –30 carbonic anhydrase, 847 –57 enzymatic carbon–hydrogen-bond cleavage, 725–39 nuclear quantum mechanics, 635, 637–8 solvent hydrogen isotope effects, 847 –57 see also enzyme-catalyzed reactions catalytic acceleration, 766 catalytic cycles, 933 –6 catalytic efficiency, 796–807, 861–2, 864–5, 921 –3, 967–8 catalytic-hydrogen bonds, 772, 776–7, 781–7 catalytic mechanisms, 847 –57 catalytic proficiency, 766, 768

1057 catalytic triads, 779–87 cation intermediates, 941 CCSD calculations, 113– 14 cell models, 129–30 centrifuge, 47, 53 centroid probabilities, 629 characteristics hydrogen bonds, 773– 5, 975–90 low-barrier hydrogen bonds, 975–90 methane, 931–2 muons, 434–5 charge-assisted hydrogen bonds, 233 CHARMM parameters, 484 chemical dynamics in complex systems, 476–9 chemical exchange, 45–7, 50–2, 60 –1 chemically bound muon states, 435–41 chemical modification, 816 chemical process simulations, 623–6 chemical shifts FHN hydrogen bonds, 217–22 hydrogen bonds in solution, 238–9 intramolecular hydrogen bond compounds, 253, 254, 258–9 low-barrier hydrogen bonds, 980–1 nitrosyl hydride, 215–17 strong hydrogen bonds, 978–9 chemical steps, 744–53, 794–5 chemical waste disposal, 70 chloranil, 841–2 chlorinate aliphatic hydrocarbon reduction, 879 chlorine anharmonicity, 156 hydrogen-bonded complexes, 156–70 hydrogen-bond symmetrization, 175–7, 183–91 internal-return mechanisms, 470 isotopic ratio measurements, 877–8 kinetic isotope effects, 601–2, 875–88 4-chlorobenzoil-CoA dehalogenase, 885–6 4-chlorobenzoyl-coenzyme-A, 885–6 m-chlorobenzyl p-substituted benzenesulfonates, 426 chloroform, 317, 323–9 Fe(III)-heme-chloroperoxidase, 886–7 cholinesterases, 985–6 chondrites, 377–8 chorismate mutase, 927 chromatography, 74–7, 132–3, 420–1 chromophore absorption, 323–9, 334–46 chymotrypsin, 982– 5, 987–90, 1009–12 a-chymotrypsin, 786 classical calculations, force fields, 282–4 classical molecular dynamics simulations, 500, 505, 508 classical systems, equilibrium in ideal gases, 4–5 classification, hydrogen bonds, 154, 255, 977–8 close separation separative capacity, 64–5 Clostridium oroticum, 801 CMIE see condensed matter isotope effects cobalamine complexes, 735–8 coenzyme B12 complexes, 735–8 coenzyme binding, 815 coherent tunneling, 446 coherent two-proton transfer, 714–15

1058 collidine complexes, 210–21 combustion, 372, 401 commitments to catalysis, 795, 839, 1022 competitive isotope effect measurements, 1022, 1026– 9 composition, atmospheric sulfur, 405–7 compressibility, 120 computational issues, 234–6, 826 computer simulations, 621 –40 concentration, 401, 405 –6 concertedness acetic-aced methanol complex, 535–7, 542 –3 enzymatic reactions, 638, 802–7 kinetic isotope effects, 522, 535–43, 638 multiple proton transfer, 522, 535–43 pH dependence, 802–7 proton transfer, 802– 7 rate-limiting steps, 422–3 condensation, 125–7, 134–5 condensed matter environments electron-coupled proton transfer, 710 –20 proton conductivity, 691–722 proton transfer, 691– 722 hydrogen-bonded systems, 702– 10 molecular fragments, 693–702 condensed matter isotope effects (CMIE), 120 –48 anharmonicity, 134, 139–40 benzene, 127–8, 132 –3, 141– 2 ethylene, 140–1, 142 examples, 140–5 excess free energy, 138–9 molar volume isotope effects, 120, 136–8 molecular properties, 124– 34 overview, 25–32 spectroscopy, 134– 5 vapor pressure isotope effects, 120–33 water, 132–3, 142 –5 condensed phase molecular properties, 124– 34 nuclear quantum mechanical effect, 626–30 separated isotope measurements, 122 –3 VTST, 594–9 conformation change signaling, 787–8 cooperative coupled hydrogen bonds, 222 copper amine oxidases, 662– 5 copper complexes, 738 copper dependent enzymes, 870 –1 copper proteins, 660–2 corner cutting, 480–2 coupling constants, 165–71, 239–40, 253 electron-proton transfer, 499–515, 710–20 hydrogen bonds, 222–6, 296–8 hydrogen-electron transfer, 491 –4 intramolecular dynamics, 315 –17 partners, 1039–42 proton electron transfer, 499 –515, 710 –20 symmetry, 479 –80 covalent bond elongation, 976–7 p-cresol methylhydroxylase flavocytochrome, 801 critical property isotope effects, 137 –8

Index crossover effects, 443–4 cryostats, 30, 31 crystalline surfaces, 590–1 crystallography acids, 183–4 carbonic anhydrase isozymes, 849–50 chloride, 183–4 halides, 183–4 hydrogen bonds, 177–8, 183–4, 194–6 hydrogen halides, 183–4 ice, 177–8 muonium diffusion, 446 strong acids, 183–4 symmetrization, 177–8, 183–4 crystals, 446, 590–1 Curtin–Hammet principle, 1029– 30 CVT see canonical variational theory cyanide ions, 425–6 cytochrome P-450, 658–60, 665, 866– 8, 941

D DßM see dopamine ß-monooxygenase dead end protonation, 804– 6 dehalogenase catalytic reactions, 879–86 dehalogenation, 465–72, 875–6 dehydrohalogenations, 465–72 delta constraints, 410–11 demixing, 145–8 density functional theory (DFT), 110, 112 desolvation, 895– 8 deuterated phenols, 289 deuteration, formaldehyde, 1039–41 deuterium chloride, 187–90 exchange reactions, 51 –2 fractionation, 204–5, 217, 218 hailstorms, 393– 5 ice, 181–3 ice cores, 398 isotopic fractionation, 204–5, 217, 218 kinetic isotope effects, 23–5, 426–7, 936, 947–8 soluble methane monooxygenase, 936, 947–8 solvent isotope effects, 936, 947–8 water in precipitation, 393–6 deuterium isotope effects adiabatic corrections, 94 –5 Born–Oppenheimer approximation, 94–5 collidine-acid complexes, 210–18 coupled hydrogen bonds, 222–6 dinitrogen monohydride, 205– 10 hydrostatic pressures effects, 841 intramolecular hydrogen bonds, 269–73 nitrosyl hydride, 210–18 NMR chemical shifts, 217–22 NMR parameters, 202–18 pyridine-acid complexes, 210–18 strong dinitrogen monohydride, 205–10 deuterium substitution hydrogen-bond symmetrization, 181– 3, 187–90

Index hydrogen motion vibrations, 284 ideal gas phase molecules, 113–15 proton stretching frequencies, 162– 5 reduced partition functions, 16–17 vibrational spectroscopy, 284 deuterocarbons effects, 127–8, 131–3 deuteroethylenes, 136, 140– 1, 142 deuteromethyl pyridines, 960 –1 deuteron transfer, 454–5, 855–6 DFT see density functional theory DHFR see dihydrofolate reductase diabatic basis sets, 716 diabatic free energy, 716–19 diabatic proton transfer, 696– 7 diabatic states, 716–18 2,7-dibromo-1,8-bis(dimethylamino)naphthalene, 299–301 ß-dicarbonyl enol tautomers, 241 –2 dicarboxylic acids, 243 –4 1,2-dichloroethane, 880–3 dicopper complexes, 738 trans-dideuteroethylene, 30–1 dielectric constants, 217, 221 –2 dielectric continuums, 235 dielectric effect, 134, 135 differential binding, 621–2 diffraction, 194, 206–7, 240 –1 diffusion, 46–7, 54– 5, 446–7 diffusion coefficients, 708–9 dihydroequilenin, 986 dihydrofolate, 511–12 dihydrofolate reductase (DHFR), 500, 508, 511–14, 748 dihydrogen triacetate hydrogen bonds, 223 dihydroorotate dehydrogenase, 801 2,6-dihydroxy acylaromatics, 262–3 dihydroxyphenylalanine (DOPA), 653 –5, 1015– 16 ß-diketones, 272–3 1,1-dimethylcyclopropane, 939 dimethylsulfide, 405 dinitrogen monohydride, 205–10, 298–9 dioxygen, 443–4 dip spectroscopy, 310 direct comparison techniques, 916, 922 dispersed polarons, 626–7, 630–1 displacement chromatography, 74–5 dissociation, 320–9, 366, 451– 62 distillation, 44– 7, 50, 60 DMANH, 262 –3, 265, 268–9, 271 DOIT code, 522, 529, 533 donor-acceptor distances, 693– 5 donor-acceptor modes, 564 donor-acceptor vibrations, 510–11 DOPA see dihydroxyphenylalanine dopamine ß-monooxygenase (DßM), 660 –2, 870– 1 double proton transfer, 522, 529 –37, 542 –3 double-resonance, 310, 331– 3, 346 double-well potentials, 197, 232– 47, 703–5 dual-temperature principle, 52 dynamical barrier recrossings, 512–14 dynamic coupling, 711, 712–14, 719 dynamics

1059 complex systems, 476–9 enzyme catalysis, 638–9, 744–5 hydrogen transfer, 479– 82, 822–6 intramolecular, 306–48 molecular dynamics, 307– 8 quantum dynamics, 311–13, 500, 505, 508, 583–4 ultrafast intramolecular, 311–17 vapor pressure isotope effect, 124–34 dynamic solvent effects, 445

E EA-VTST/MT, 594, 596–9 ECPT see electron-coupled proton transfer effective upper limits, 752–3 effusion, 54 –5 EIE see equilibrium isotope effects eigenvalues, 16, 33, 90 Eigen–Weller model, 452–3 elastase-catalyzed reactions, 1010–11 electrical anharmonicity, 296–7 electric charge, 374–5 electric fields, 160–2, 264–5 electrolysis, 45–7, 55 –6 electro-magnetic method, 47, 56 electron count, 1043 electron-coupled proton transfer (ECPT), 499–515, 710–20 electron-hydrogen transfer, 491–4 electronic coupling, 298 electronic energies, 440 electronic Schro¨dinger equation, 90 electronic structure, 110–12 electronic wave functions, 8 electron transfer (ET), 500– 1, 626, 651–5 electrophilic fluorination, 420 electrophilic sp carbon, 901–3 electroweak charges, 307–8 elementary proton transfer, 693–702 elution chromatography, 74–5 embedded-atom method, 590 emissions, 399 empirical determinations, 67 –9 empirical valence bonds (EVB), 511, 625–6 enaminones, 260–1 energetics, 234–5 energy adiabatic corrections, 92 –3 Born–Oppenheimer approximation, 92–3, 94 intramolecular vibrational, 325– 9, 334–46 minimum energy paths, 583, 586–7 muonium, 439–41 vibrational, 96–9, 325–9, 334–46 see also activation energy; activation free energy; free energy; potential energy; zero-point energy engineering gated motion, 682–5 enol tautomers, 241– 2 enoyl-CoA hydratase, 896– 7 enrichment profiles, 75–7

1060 ensemble-averaged variational transition-state theory with multidimensional tunneling (EA-VTST/MT), 594, 596–9 enthalpy, 824–5 entropy, 824–5 environmental issues, 875– 6 environment effects, 197–8 enzymatic activation, 645 –66 enzymatic acyl transfers, 962 enzymatic ß-elimination, 959 enzymatic binding isotope effects equilibrium isotope effects, 1020, 1021– 47 glucose, 1019–51 hexokinase interaction, 1019–51 historical overviews, 1020–1 kinetic isotope effects, 1020, 1021– 47 enzymatic carbon–hydrogen-bond cleavage, 725–39 applications, 735–8 kinetic isotope effects, 726, 728–9, 730–4 proteinless systems, 729– 30 rate constants, 727–8 semiclassical Instanton theory, 731, 734 temperature dependence, 726, 728 –9, 730– 4 theoretical models, 730–5 enzymatic hydrogen transfer, 482–91 enzymatic hydrogen tunneling, 744–60 amine dehydrogenases, 676, 678–85 flavoprotein catalyzed substrate oxidations, 671–85 kinetic isotope effects, 671 –3 quinoprotein catalyzed substrate oxidations, 671–85 temperature dependence, 673–7 enzymatic reactions concertedness, 638 halogenation, 886 –7 hydrostatic pressures effects, 839–44 kinetic isotope effects, 475 –95, 638 nuclear quantum mechanics, 621–40 proton coupled electron transfer, 508–14 solvent isotope effects, 995 –6, 1002–4, 1006– 16 vibrationally enhanced tunneling, 475 –95 enzyme-catalyzed reactions carbon–hydrogen-bond cleavage, 725–39 carbon isotope effects, 428 computer simulations, 621– 40 hydration, 901–3 hydrogen bonds, 766, 777 –88 hydrogen tunneling, 744–60 intermediate partitioning, 861–71 nonenzyme catalyzed reactions, 794–5 nucleophilic attack at sp carbon, 909–10 pH dependence, 794, 802–8 solvent isotope effects, 995 –6, 1006–16 substrate dependence, 794, 796–802, 807–8 transition state stabilization, 766–8 enzyme mechanisms from isotope effects, 915– 28 examples, 925–8 intrinsic isotope effects, 923–5 enzymes catalytic acceleration magnitude, 766 concerted proton transfer, 539–41 dead end protonation, 804–6

Index EA-VTST/MT, 603–5 enzyme-reactant complexes, 805–6 enzyme-substrate interactions, 1020 hydrogen bonding site mutations, 777–80 hydrogen bond symmetry and isotope effects, 246–7 low-barrier hydrogen bonds, 982–90 motion, 512–14 soluble methane monooxygenase, 939 VTST/MT, 599–600, 603–5 equalities, 99 equation for the isotope effect, 921–3 equilibrium conformation isotope effects, 437–8 expectation values, 161–2, 169–70 fluctuations, 744–5 general overview, 3– 4 ideal gases, 4–12 lines, 62 perturbation, 899, 920–1 ratio, 996, 1022–9 secondary zone approximation, 599 solvation paths, 594–5 solvent isotope effects, 996–7 equilibrium constants adiabatic corrections, 93 –6 gas phase equilibrium isotope effects, 100–2 hydrogen bonds in solution, 237 isotope enrichment, 44 photoacids, 453, 456–8 photo-dissociation, 453 solvent isotope effects, 996–7 statistical mechanics, 100– 2 equilibrium isotope effects (EIE) enzymatic binding isotope effects, 1020– 47 hydrogen tunneling, 745 hydroxide formation, 898 intramolecular hydrogen bonds, 269–75 molecular oxygen enzymatic activation, 647– 50 nucleophilic isotope effects, 895–9, 908 solvent isotope effects, 996–7 statistical mechanics, 100– 9 equine liver alcohol dehydrogenase, 807 Escherichia coli nucleotidase, 896–7 esterases, 987– 90 ester hydrolysis, 903–7 estradiol, 986 ET see electron transfer ethane, 938 ethanolamine, 681–2, 737 ethanolic sodium ethoxide, 471 ethanol oxidation, 827– 8 ethoxide-promoted dehydrochlorination, 470 ethylene, 29– 31, 140–2 evaporation, 390– 1, 393 EVB see empirical valence bonds evolution infrared spectra, 292–4 excess free energy, 138–9 excess proton conductivity, 693, 702, 705–9 exchange distillation, 46, 51–2, 60 excitation, 321– 2, 330, 332–3 excited states

Index acids in aqueous solutions, 451–62 enzymatic binding isotope effects, 1033–7 molecular oxygen enzymatic activation, 648 –9 photo-dissociation reactions, 451–62 exotic nuclei, 417–28 carbon kinetic isotope effects, 417– 19, 423–4, 427–8 enzyme-catalysed reactions, 428 fluorine kinetic isotopic effects, 417–19, 421 –3 kinetic methods, 420 –1 nucleophilic aliphatic substitutions, 423–6 rate-limiting steps, 421–3 secondary deuterium kinetic isotope effects, 426– 7 transition-state structure, 423–6 expectation values, 161 –4, 166, 169–70 experimental conditions, 258–9 explicit-bath models, 594–9 explosion risks, 71–2 extraterrestrial solids, 377–9 Eyring activation parameter, 824– 5 Eyring equation, 767, 770

F FAB-IRMS, 879 Fajan, K., 1 fast atom bombardment–isotope ratio mass spectroscopy (FAB– IRMS), 879 FC see Fermi-contact feed supply considerations, 72 Fe(III)-heme-chloroperoxidase, 886–7 Fermi-contact (FC) 165, 168– 70 Fermi resonances, 297, 315–16, 336–7 field effects, 166–70 field shifts, 11–12 field strengths, 158–62, 169–70 finite nuclear mass corrections, 93–4 first order rules, 12–13 fixed donor-accepter distances, 693 –5 flavin adenine dinucleotide activation, 650 –3 flavin-dependent putidaredoxin reductase, 658–60 flavoenzyme half-reactions, 672–3 flavoproteins, 671 –85 fluorine chemical shifts, 267 FHF hydrogen bonds, 223 –4 FHN hydrogen bonds, 217–22 FLF hydrogen bonds, 224–6 hydrogen-bonded complexes, 156–70 kinetic isotopic effects, 417–19, 421–3 labelling, 419–20 NMR chemical shifts, 217– 22 fluoroacetate dehalogenase, 885 fluoroketone, 422–3 flux autocorrelation functions, 478–9 force constants, 22, 97–9, 1037–46 force fields, 282 –4 formaldehyde, 1039–41, 1045 formate dehydrogenase, 797–8 formate models, 1043–4 formic acid, 234–5, 522, 537, 903 –5 formyl transfer, 961

1061 Fo¨rster cycle, 456–7 fossil methane isotopes, 399 fractionation constants, 400–1 molecular forces, 11 nitrosyl hydride, 217, 218 ozone isotopologues, 364, 367 solvent isotope effects, 1001–3 water cycle isotope effects, 391–2 fractionation factors Bigeleisen–Mayer equation correction, 9–12 hydrogen bonds, 204– 5, 217, 218, 237–8, 981 liquid vapor, 25–9, 120, 138–9 low-barrier hydrogen bonds, 981 nitrosyl hydride, 217, 218 soluble methane monooxygenase, 936–7 solvent isotope effects, 996–7, 1004–6 vapor pressure isotope effects, 123–4 fragment mass spectra, 322– 3 Frank –Condon nuclear overlap, 674–5 free energy adiabatic proton transfer, 553–8 barriers, 512–14, 556–8, 563, 566, 569–70 carbonic anhydrase catalysis, 854–5 condensed matter isotope effects, 138–9 deuteron transfer rates, 454–5 double-well proton potentials, 704–5 electron-coupled proton transfer, 716–19 enzyme catalysis computer simulations, 630 hydron transfer, 198, 199 molecular fragments, 695–6, 701–2 nonadiabatic proton transfer, 563–6, 569–70 photo-dissociation, 454–6 proton coupled electron transfer, 502–5, 509 proton transfer, 454–5, 551–2, 695– 6, 701–2, 854–5 surfaces, 502–5, 509, 695–6 vapor pressure isotope effect, 124 VTST, 582 see also activation free energy free radicals, 438–9, 738 freon mixtures, 210–11 frequency isotopic ratio, 284–5, 289, 294–6 frequency shifts, 287–9 fumarate bonds, 909 Fusarium oxysporum, 804

G gas centrifuge, 47, 53 gaseous diffusion, 47, 54–5 gas-liquid chromatography, 132–3 gas phase, 100–9, 584–90, 600– 3 gas solubility, 132–3 gating hydrogen tunneling, 675, 757–8 molecular fragments, 700–1 photon transfer, 513–14, 700–1 proton coupled electron transfer, 513–14 trimethylamine dehydrogenase, 682–5 GAUSSIAN, 111–13 Gaussian basis sets, 111

1062 generalized Langevin equation, 476–7 geochemical cycle isotopes, 388–90 geometry coupled hydrogen bonds, 222 –6 geometric isotope effects (GIE) 195 –6, 200–2, 211– 16, 222–6 geometric mean, 26, 865, 1041 hydrogen bonds, 195–6, 200–5, 211 –16, 222– 6, 234– 5 GF matrix techniques, 15, 18, 33, 97, 282–3 Gibb– Helmholtz equation, 582 GIE see geometric isotope effects global spin-spin coupling constants, 168–70 glucose with hexokinase, 1019–51 glucose oxidase, 650–3, 665 glutamate, 910 glycine, 538–9 glycosyl transfer, 964–5 Goeppert-Mayer, Maria, 3– 4, 6–7 Golden Rule, 522, 524, 528 –9, 731– 4 graphical solution, material balance equation, 69 greenhouse gas concentrations, 398 Greenland, 395, 396 Greiff, L.J., 2–3 ground-state acids, 456 –7 expectation values, 161 –2, 169– 70 free-energy curves, 554– 5 nuclear tunneling, 744 –5 vibrationally adiabatic potential curves, 589 Gueron, Jules, 28

H hailstorms, 393–5 haloacetates, 885 DL-2-haloacid dehalogenase, 883 –5 haloalkane dehalogenase LinB, 881–2 haloalkane dehalogenases, 880–3 haloalkanoic acids, 883 –5 Hamiltonians, 477– 8 Hansenulapolymorpha, 664 harmonic approximation, 282–4 force constants, 97– 9 frequency equalities, 99, 112– 15 oscillators/oscillation, 477, 730 –1 Hartree–Fock method (HF) 110–11, 113 heats of solution, condensed matter isotope effects, 120 heats of vaporization, 120 heavy atom kinetic isotope effects, 19–20 heavy atom labeling, 1033–4 heavy water production, 25–6 height equivalent of theoretical plates (HETP), 65, 67 –9, 77–8, 79–81 Helmholtz free energy difference, 124 Henry’s law, 122–3 heterogeneous environments, 721 heterotetrameric sarcosine oxidase, 676, 682 –5 HETP see height equivalent of theoretical plates

Index hexokinase, 907–8, 1019–51 HF see Hartree–Fock method high-frequency CH bond stretch, 1034–5 high hydrostatic pressures effects, 837–44 high pressures acids, 175–7, 183–91 hydrogen-bond symmetrization, 175– 91 ice, 175–83 strong acids, 175–7, 183– 91 high-resolution spectroscopy, 311–17 high target enrichment failures, 70–1 historical overviews, 1–3, 1020–1 HLADH see horse liver alcohol dehydrogenase homoconjugate nitrosyl hydride, 216– 17 homogeneous isotropic media, 721 horse liver alcohol dehydrogenase (HLADH), 483–7, 753, 812–30 hot-band transitions, 326–7 human acetylcholinesterase, 1013– 14 human carbonic anhydrase, 848–9 human leukocyte elastase, 787–8 hybridization, 956–9 hydride transfers, 511–12, 630, 969 hydrocarbon effects, 127–8, 131–3 hydrochloric acid, 22 –4, 175– 7, 183–91, 601–2 hydrogen abstraction, 441–2 atomic properties, 435 electron transfer, 491–4 exchange reactions, 51 –2 fractionation, 204–5, 217, 218, 1005–6 interstellar media, 388, 407–9 isotope enrichment, 45 kinetic isotope effects, 21–5 liquid vapor fractionation factors, 26–8 substitutions, 113–15 transfer, 479–82, 822–6 tunneling enzyme catalysis, 744–60 hydrostatic pressures effects, 838– 42 kinetic isotope effects, 744–60 reaction rates, 753–4 Swain–Schaad exponential relationship, 746–53 temperature dependence, 753–7 ultra-light isotopes, 433–48 vapor pressure isotope effect, 127– 8, 131–3 VTST/MT, 601–2 hydrogen bonds carbon–hydrogen cleavage, 725– 39, 865, 938–40, 947–9 carbon–hydrogen stretch, 1034–6, 1038–9 categorization, 772 characteristics, 773–5, 975–90 charge-assisted, 233 classification, 154, 255, 977– 8 complexes concerted proton transfer, 537 infrared spectra, 154–65 proton stretching frequencies, 158–62 spin-spin coupling constants, 165–71

Index two-bond spin-spin coupling constants, 165–71 X –H stretching bands, 154–65 condensed matter environments, 702– 10 correlations, 194 –6, 198 –202, 205– 16 dimers, 537 double-well proton potentials, 232– 47, 703– 5 enzyme catalysis, 766, 777–88 fractionation factors, 1005–6 geometry, 195–6, 200 –5, 211– 16, 222–6, 234–5 nucleophilic isotope effects, 895–8 physical parameter isotope effects, 979–81 potential shape, 292–4 single-well potentials, 232–47, 709–10 site mutations, 777–80 in solution current work, 241 –7 double-well potentials, 232 –47 equilibrium constants, 237 fractionation factors, 237–8 infrared spectroscopy, 240 neutron diffraction, 240 –1 NMR chemical shifts, 238–9 NMR coupling constants, 239 –40 observation methods, 236–41 single-well potentials, 232–47 x-ray diffraction, 240–1 spectroscopy, 193– 227, 240, 774–5 stabilization energy, 769–70 strength, 771–5 structure, 771– 5 symmetrization acids, 175– 7, 183–91 deuterium substitution, 181–3, 187 –90 high pressures, 175– 91 ice, 175–83, 190–1 strong acids, 175–7, 183 –91 symmetry in single and double-well potentials, 232–47 thermochemistry, 774–5 transition states, 766, 768–70, 1005–6 types, 154 vibrational isotope effects, 281–301 zero-point energy, 977–82 see also low-barrier hydrogen bonds hydrogen fluoride, 217–22, 422–3 hydrogen halides, 154– 62, 166–70, 175–7, 183–91 hydrogen isotope effects adiabatic corrections, 94–5 Born–Oppenheimer approximation, 94–5 collidine-acid complexes, 210 –21 coupled hydrogen bonds, 222 –6 dinitrogen monohydride, 205–10 double-well potentials, 232 –47, 703 –5 muonium, 94–5 nitrosyl hydride, 210–18 NMR studies, 193 –227 applications, 205 –26 chemical shifts, 217 –22 theory, 194– 204 origins, 196–8 pyridine-acid complexes, 210 –18

1063 single-well potentials, 232–47, 709– 10 hydrogen peroxide, 380–1, 655 hydrogen phthalate anions, 235–6 hydrogen sulfide, 405 hydrolysis, 895–8, 903–7, 1012–14 hydrons potential influence, 196–7 tautomerism, 204 transfer, 197–9, 466–8, 471–2 see also protons hydroperoxides, 380–1, 655 hydrostatic pressures effects, 837–44 theory, 837–41 hydroxide formation, 898 hydroxonium ions, 705 hydroxyacyl aromatics, 260–2 hydroxyarenes, 451 hydroxybenzaldehydes, 256–7 hydroxyl, 372 hydroxylase components, 932– 4, 937, 943–5 hydroxylation, 861– 71 9-hydroxyphenalen-1-one, 234–5 hyperconjugation, 958, 1042–5 hyperfine interactions, 438–9

I ice, 142–4, 175–83, 190–1 icebergs, 395 ice caps/cores, 396–8 ideal gas phase molecules, 113–15 bi-imidazoline complexes, 506 imidogen chromophore absorption, 334–8 implicit bath methods, 594–5 incoherent tunneling, muonium, 446 inert gases, 127–8 infrared (IR) absorption, 177–83, 365 double resonance, 331–3, 346 infrared resonance enhanced multiphoton ionization, 309 intensity, 283–4 spectra, 154– 65, 281– 301 spectroscopy hydrogen bonds in solution, 240 hydrogen-bond symmetrization, 177–83 infrared spectroscopy with mass and isotope selection, 317–20, 347–8 isotope selective infrared spectroscopy, 306–48 ozone isotopologues, 365 principles, 306–8 inner-sphere electron transfer, 652–5 Instanton theory, 522–3, 526–9, 533, 731, 734 instrumentation, oxygen-18 isotope effects, 646–7 integrated intensity, 296–7 interaction distance fluctuations, 700– 1 interaction zone residues, 944–5 intermediate compounds, 934–49 intermediate partitioning, 861–71 intermolecular forces, 125– 7 intermolecular hydrogen bonds, 245–7

1064 intermolecular interactions, 197–8 intermolecular proton transfer, 856 –7 internal competition, measuring isotope effects, 917 –20, 922 internal-return mechanisms, 465–72 interstellar media, 388, 407–9 intervening water molecules, 847– 57 intramolecular dynamics, 306 –48 overview, 308–11 spectroscopic states, 313–17 intramolecular hydrogen bonds classification, 255 NMR studies, 253 –75 definitions, 255 equilibrium isotope effects, 269 –75 experimental conditions, 258–9 static systems, 259–69 theory, 255 –8 resonance-assisted hydrogen bonds, 254 –5, 259 –62, 268– 9 symmetry and isotope effects, 241–5 intramolecular isotope effects, 20, 21, 241 –5, 862– 4 intramolecular proton transfer, 850 –6 intramolecular quantum dynamics, 311–13 intramolecular vibrational energy redistribution, 334– 46 chloroform CH chromophore absorption, 325– 9 time-scales, 325–9 vibrational mode specificity, 334–8 intrinsic free-energy barriers, 556–8 intrinsic isotope effects, 268–9, 275, 923 –5 intrinsic kinetic isotope effects, 746– 8 inventory atmospheric sulfur, 405 see also proton inventory inverse vibrational frequencies, 283 ion exchange, 53, 72–82 ion-migration, 45 ion-pair hydrogen bonds, 154 IR see infrared IRMS see isotopic ratio mass spectrometry iron, 506, 886– 7 irreversible processes, 53–6 iso-mechanisms, 839–40, 848 ISOS see isotopomer selective overtone spectroscopy isotope chemistry, 12– 18, 89–115 isotope enrichment, 42–83 cascade theory, 56–8 chemical waste disposal, 70 electrolysis, 45– 7, 55–6 electro-magnetic method, 47, 56 explosion risks, 71–2 gaseous diffusion, 47, 54–5 high target failures, 70 –1 irreversible processes, 53–6 laser isotope separation, 47, 53–4 molecular effusion, 54–5 nonsteady state phenomena, 53, 72– 82 overview, 42–3 process, 44–56 reversible processes, 44–53, 72 –82

Index separation stage cascading, 42– 3, 56–66 startup, 66– 8 steady state phenomena, 44–53, 56–66 thermal diffusion, 46–7, 55 working material explosions, 71– 2 isotope ratio determinations, 899 isotope ratio mass spectrometry (IRMS), 877–8, 900 isotope selective infrared spectroscopy, 306– 48 chloroform, 317, 323–9 intramolecular dynamics, 308–11 isotope selective overtone spectroscopy, 320, 323–5, 329–48 isotope solution excess free energy, 138– 9 isotope theoretical chemistry, 12–18, 89 –115 Born–Oppenheimer approximation, 90–6 isotope effects numerical calculations, 109–15 molecular vibrations, 96–9 numerical calculations, 109–15 potential energy surfaces, 96–9 statistical mechanics, 100– 9 symmetry numbers, 104–9 vibration energy levels, 96– 9 isotopic balance, 42–3 isotopic substitution, 169–71, 284–5 isotopic waters, 144–5 isotopomer harmonic frequency, 99 isotopomer selective overtone spectroscopy (ISOS), 320, 323–5, 347–8

J Jameson theory, 255–6 jet-cooled aniline bands, 335–7 jet-cooled benzene spectra, 341–6

K ketopantoate reductase, 799, 805–6 ketosteroid isomerases, 986, 990 KIE see kinetic isotope effects kinetic complexity effects, 751–2 kinetic constants, 817–18 kinetic energy, 370–1 kinetic isotope effects (KIE) alcohol dehydrogenase catalysis, 812–13 alcoholic alkoxide-promoted dehydrohalogenations, 465–72 carbon monoxide isotopologues, 372–3 chlorine, 875–88 computer simulations, 622–40 enzymatic binding isotope effects, 1020, 1021–47 carbon–hydrogen-bond cleavage, 726, 728–34 catalysis, 622–40, 671– 85, 744–60 hydrogen transfer, 482–91 hydrogen tunneling, 671–3, 744–60 reactions, 475–95, 622–40, 671–85, 744–60 flavoprotein catalyzed substrate oxidations, 671– 85 hydride transfers in solutions, 630

Index hydrogen tunneling, 671–3, 744–60 hydrostatic pressures effects, 837 lactate dehydrogenase, 487 –91 low-barrier hydrogen bonds, 975 molecular fragments, 701– 2 molecular oxygen enzymatic activation, 646 –66 multiple proton transfer, 521 –44 concerted transfer, 522, 535 –43 stepwise transfer, 522, 529 –35 muonium, 441–5 nucleophilic reactions, 899–910 overview, 18–25 photo-dissociation reactions, 451–62 primary, 465 –72, 549 –73 proton coupled electron transfer, 499–515 proton transfer, 521–44, 701–2 quinoprotein catalyzed substrate oxidations, 671–85 short-lived radionuclides, 417–19 temperature dependence, 753–7 vibrationally enhanced tunneling, 475 –95 VTST/MT, 600 –5 kinetics alcohol dehydrogenase catalysis, 827 –9 enzyme catalyzed reactions, 622–40, 671 –85, 744–60, 796–9, 800–2 excited-state acids, 456–7 exotic nuclei, 420 –1 kinetic solvent isotope effects, 936–8, 947 –8, 996– 8 Kirschenbaum, Isidor, 3 Kohn–Sham theorems, 111 Kramers methodology, 476, 478–9 Kresge–Gross–Butler equation, 998 –1004

L labelling, 419–20, 424–6, 918 –20, 1033–4 lactate dehydrogenase (LDH), 487–91, 630 –1 LADH see liver alcohol dehydrogenase Landau–Teller formulation, 370 –1 Landau–Zener probability, 698–9 large-curvature tunneling (LCT) 583, 588 laser excitation, 332 –3 laser-induced photoacidity, 451 laser isotope separation, 47, 53–4 lattice vibrations, 181–2 LBHB see low-barrier hydrogen bonds LCT see large-curvature tunneling LDH see lactate dehydrogenase lead isotope effects, 407 leaving groups, 426 Le Chatelier’s principle, 837–8 lengths, bonds, 254, 275, 436–7, 439 –41, 976 –7 leuco-crystal violet oxidation, 841 –2 leukocyte elastase, 787–8 LFER see linear free energy relationships Libby, Willard F., 4 liberation modes, 289 ligands, 777– 9 Lindemann, F.A., 1–2

1065 linear free energy relationships (LFER), 630 lipoxygenase enzymatic carbon–hydrogen-bond cleavage, 736– 7 hydrogen-electron transfer, 491–4 nuclear quantum mechanics, 634–5 oxygen-18 isotope effects, 655–7, 665 proton coupled electron transfer, 508–11 Lippincott– Schroeder potential, 291– 2 liquefied freon mixtures, 210–11 liquid chromatography, 420–1 liquid hydrogen low temperature distillation, 25 –6 liquid phase, 603 liquids, 120–33, 591– 9 liquid scintillation, 420–1 liquid vapor fractionation factors (LVFF), 25 –9, 120, 138–9 liquid–vapor interfaces, 391 liquid water, 447 lithium, 45–6, 73, 206 liver alcohol dehydrogenase (LADH) nuclear quantum mechanics, 633–4 pH dependence, 807 proton coupled electron transfer, 500, 508, 511–12 liver NADP-malic enzyme, 804 lone pair electrons, 1043 long-range isotope effects, 264, 273–4 long-range proton transfer, 707–9 Lorentzian functions, 343–5 low-barrier hydrogen bonds (LBHB) alcohol dehydrogenase catalysis, 827–9 characterization, 975–90 cholinesterases, 985–6 double-well potentials, 232–3 enzymes, 982–90 ketosteroid isomerases, 986, 990 serine hydrolases, 783–5 serine proteases, 982–5, 987– 90 single-well potentials, 232–3 vibrational isotope effects, 298– 301 zero-point energy, 975, 977–82 low-frequency CH bond stretch, 1035–6 low temperature spectroscopy, 210–11 L-ribulose-5-P, 4-epimerase, 928 LVFF see liquid vapor fractionation factors

M McCabe –Thiele diagrams, 61– 4 MADH see methylamine dehydrogenase maleate anions, 234 malic enzymes, 799, 802, 804, 805, 926 malonic acid decarbonoxylation, 20, 21 mannitol dehydrogenase, 798 Marcus-like models, 757–8 Marcus theory, 853–7 Martian meteorites, 379 masking kinetic isotope effects, 480– 2 mass differences, 306–7 effect, 446–7 mass-dependent isotopic fractionation, 876

1066 mass-independent fractionation, 362 mass-independent isotope effects, 362 molecular oxygen enzymatic activation, 648–9 scaling, 557–8, 586 selective IR spectroscopy, 317 –29 selective overtone spectroscopy, 320– 9 weighting, 586 material balance, 42–3, 67–9 matrix effects, 158 –62 matrix elements, 15, 18, 33, 97, 282–3 measurements internal-return mechanisms, 465–72 isotope effects, 465 –72, 916– 21, 995, 996 solvent isotope effects, 995, 996 mechanisms alcohol dehydrogenase catalysis, 812 –16 carbonic anhydrase catalysis, 847–57 electron-coupled proton transfer, 715 –19 enzyme mechanisms from isotope effects, 915–28 internal-return, 465 –72 iso-mechanisms, 839 –40, 848 nucleophilic isotope effects, 893–5 oxygen-insertion, 940– 1 ping pong, 800 –2, 848 resonantly enhanced two-photon ionization, 330 –3 soluble methane monooxygenase, 938–43 substrate dependence, 798– 9, 801–2 unmasking chemistry, 816 –21 vibrationally assisted dissociation and photofragment ionization, 320–3 medium-range isotope effects, 264 MEP see minimum energy paths metal ions, 898–9, 906–7 metal-mediated oxygen activation, 653–4 meteorites, 379 methane, 127–8, 398–400 methane monooxygenases (MMO), 657 –8, 665, 931– 49 methanol to benzylic anions, 471 –2 methanolic sodium methoxide-promoted dehydrochlorination, 468, 469 methanolic sodium methoxide-promoted hydron exchange, 467– 8 methanotrophic bacteria, 931–2 methoxide-catalyzed hydron-exchange reaction, 466– 7 methoxide-promoted dehydrohalogenation, 466 –70 methylamine dehydrogenase (MADH), 676, 678 –82 methylamines, 676, 678–82, 737, 868 methyl-chymotrypsin, 786 methyl glucopyranosides, 964 methyl group hydroxylation reactions, 865 methylmalonyl A mutase (MMCoAM), 726, 728, 730, 735– 6, 738 Methylococcus capsulatus, 657, 943 Methylococcus trichosporium OB3b, 943 methyl transfer, 968– 9 N-methyltrifluoroacetanilides, 1008–9 microbial dehalogenase catalysis, 879–86 microcanonical optimized multidimensional tunneling (mOMT), 583

Index microscopic simulations, 621–40 migration effects, 392–3 MINIMAX coefficients, 14 minimum energy paths (MEP), 583, 586–7 minimum reflux ratio, 63– 4 mixed labeling experiments, 749 MMCoAM see methylmalonyl-coenzyme A mutase MMI see molecular moment of inertia MMO see methane monooxygenase (MMO); methane monooxygenases Moeller Plesset (MP) perturbation, 111, 113 molar density, 136–8 molar volume isotope effects (MVIE), 120, 136–8 molecular balance, 42 –3 molecular dynamics electroweak charges, 307–8 mass differences, 306–7 Pauli identity, 307 simulations, 500, 505, 508 spins, 307 symmetry selection rules, 307 molecular effusion, 54– 5 molecular forces, 11 molecular fragments, 693–702 molecular hydrogen-muonium reactions, 441–2 molecular mechanics see quantum mechanical and molecular mechanics molecular moment of inertia (MMI), 648–9, 1033–6 molecular oxygen, 645–66 molecular properties, 124–34 molecular sieve model, 945–6 molecular spectroscopy, 311–13 molecular structure, 12–18 molecular theory, 29 molecular vibrations, 96–9, 181–2, 730–1 mole fractions, 70–1 moments of inertia, 648–9, 1033, 1034–6 mOMT see microcanonical optimized multidimensional tunneling monatomic systems, 127–9 morphinone reductase (MR), 675–7 Morse function, 290–1 Morse parameters, 436, 438 Morse potentials, 731–4 MP see Moeller Plesset MR see morphinone reductase multidimensional tunneling, 580– 607, 680–2 multiphasic pressure functions, 841 multiple charge transfer, 500, 505 multiple isotope effect method, 924–5 multiple proton transfer, 521– 44 concerted transfer, 522, 535– 43 stepwise transfer, 522, 529–35 multistate continuum theory, 500, 505, 508 multistep enzyme reactions, 1002–4 muonium anharmonicity, 435–6 chemically bound states, 435–41 diffusion, 446–7 equilibrium conformation, 437–8 hydrogen isotope effects, 94–5

Index kinetic isotope effects, 441 –5, 602– 3 mass effect, 446–7 structural isotope effects, 435–41 ultra-light isotopes, 433–48 vibrating species, 435 –41 VTST/MT, 602 –3 zero-point energy, 435–8, 441–2 muscle isoform, 487–91 mutagenesis, 816 –18 mutations, 683– 4, 777–80 MVIE see molar volume isotope effects Myers, Lawrence S., 8

N NAD analogues in solution, 630 NAD cofactors, 482– 3, 488–9 NADH, 933 NAD-malic enzyme, 799, 802, 805 NADP enzymatic hydrogen tunneling, 675–7 NADP-malic enzyme, 804 Nafion membranes, 721 1-naphthol 5-sulfonate, 459–60 negative isotope effects, 266 negative muons, 434 NES see nonequilibrium solvation neutron diffraction, 194, 206– 7, 240–1 nitrate aerosols, 380 nitric oxide isotopomers, 70– 1 nitroalkane oxidase, 804 nitrogen chemical shifts, 265–7 chromophore absorption, 334– 8 dinitrogen monohydride, 205–10, 298 –9 expansion, 337 hydrogen-bonded complexes, 156–70 interstellar media, 388, 407 –9 isotope enrichment, 46, 50, 58, 63–4, 70–2, 79–81 nitrogen-carbon bonds, 908 nucleophilic reactions, 908–10 separation, 79 –81 vapor pressure isotope effect, 127–8 p-nitrophenyl acetate, 962 nitrosyl hydride, 210–17, 218 nitrous oxide, 376–7, 403 Nitrox process, 58, 63–4 NMD see nonmass-dependent fractionation NMR see Nuclear Magnetic Resonance no-mixing cascades, 56–7 nonadiabatic proton transfer, 549 –53, 562– 73 nonadiabatic transitions, 731–4 noncompetitive isotope effects, 1029 nonenzymatic reactions, 759, 794–5, 1008–9 nonequilibrium solvation (NES), 592– 6 nonmass-dependent fractionation (NMD), 362, 368–71, 375–8 nonmass-dependent isotope effects, 361– 82 atmospheric oxygen, 381 atmospheric sulfur, 407

1067 carbon dioxide, 374–6 carbon monoxide, 372–4 fractionation, 377–9 hydrogen peroxide, 380–1 nitrate aerosols, 380 nitrous oxide, 376– 7 oxygen, 381 ozone, 364–71 sulfur, 378–9 troposphere, 388 nonresonance-assisted hydrogen bonds, 262–4, 269 nonrigid molecules, 106 nonsteady state phenomena, 53, 72–82 nontautomeric Schiff bases, 266 Northrop’s method, 923–4 NQM see nuclear quantum mechanical effects N-ribosyl hydrolases, 965–6 nuclear coupling, 298 nuclear field shift, 10–11 Nuclear Magnetic Resonance (NMR) chemical shifts, 215– 22, 238–9, 978–9 coupling constants, 165–71, 239–40 deuterium bond isotope effects, 202–5 FHN hydrogen bonds, 217–22 hydrogen bonds, 165– 71, 193–227, 238–40, 253–75, 773 intramolecular hydrogen bonds, 253–75 definitions, 255 equilibrium isotope effects, 269–75 experimental conditions, 258– 9 static systems, 259–69 theory, 255–8 spectroscopy, 193–227 fractionation, 204–5 hydrogen bonds applications, 205–26 geometries, 202–5 theory, 194–204 two-bond spin-spin coupling constants, 165–71 nuclear mass corrections, 93–4 nuclear quantum mechanical (NQM) effects catalytic contributions, 635, 637–8 enzyme catalysis, 622–40 hydride transfers in solutions, 630 microscopic simulations, 621–40 simulations, 621–40 nuclear Schro¨dinger equation, 90–1 nuclear shielding, 254, 256, 275 nuclear spin, 10 nuclear wave functions, 8 nucleophilic isotope effects, 893–911 nucleophilic activation, 893–5 oxygen isotope effects, 895–908 reaction mechanisms, 893–5 nucleophilic reactions activation, 893–5 aliphatic substitutions, 423–6 aromatic substitution, 421–2 attack at sp carbon, 908–10 fluorination, 420 kinetic isotope effects, 899–910

1068 nitrogen isotope effects, 908– 10 oxygen isotope effects, 895–908 numerical calculations, 7–8, 109–15

Index oxygenated proteins, 648 ozone, 364–71, 403

P O octanol hydroxylation, 867 OMP decarboxylase, 766, 768–9 OMT see optimized multidimensional tunneling one-photon IR absorption, 310 optimized multidimensional tunneling (OMT), 583, 588, 600– 1 ordered water, 850 organic molecule deuterium enrichments, 407 –9 osmium–benzoquinone complexes, 506 –7 outer-sphere electron transfer, 652 –4 outer-sphere reorganization energy, 504–5 over-barrier trajectories, 527 overtone spectroscopy aniline, 318– 19 benzene, 318–19 isotope selective, 320, 323 –5, 329– 48 isotopomer selective, 320, 323–5, 347 –8 mass selective, 320 –9 resonantly enhanced ionization, 329– 46 overtone spectroscopy by vibrationally assisted dissociation and photofragment ionization (OSVADPI), 320– 9, 347 oxalate decarboxylase, 927 oxalate monoanion, 927 oxyanion holes, 785–7 oxygen-18 isotope effects copper amine oxidases, 662–5 copper proteins, 660–2 cytochrome P-450, 658– 60, 665 dopamine ß-monooxygenase, 660–2 enzymatic activation, 645 –66 glucose oxidase, 650–3, 665 lipoxygenase, 655–7, 665 methane monooxygenases, 657 –8, 665 molecular oxygen enzymatic activation, 645–66 P-450, 658 –60, 665 peptidylglycine a-hydroxylating monooxygenase, 660– 2 soybean lipoxygenase, 655– 7, 665 tyrosine hydroxylase, 653 –5, 665 oxygen acids, 959 activation, 652 –5 chemical shifts, 266–7 exchange, 852–3 fractionation, 377–8, 394–5 ice cores, 398 insertion mechanisms, 940 –1 isotope enrichment, 47 nonmass-dependent isotope effects, 362–4, 381 nucleophilic reactions, 895–908 rate-determining trapping, 656–7 substitution, 16 water in precipitation, 395– 6

P-450, 658– 60, 665, 866–8, 941 partially adiabatic proton transfer, 697–8 partition function ratios, 103–4 partition functions, 102– 9 partitioning intermediates, 861–71 Pauli identity, 307 PCET see proton coupled electron transfer pD dependency, 1006–8 penicillanic acid, 963–4 1,4(Z,Z)pentadiene-containing fatty acids, 655 pentaerythritol tetranitrate, 675– 7 peptide boronic acid, 983– 4 peptide trifluoromethylketones, 982–5 peptidylglycine a-hydroxylating monooxygenase, 660–2, 748 peptidyl prolyl cis-trans isomerases, 969–70 peptidyl trifluoromethyl ketones, 1009–12 perturbation equilibrium, 899, 920– 1 isotope selective spectroscopy, 328–9 Moeller Plesset, 111, 113 reduced partition functions, 13–17 statistical mechanical, 13– 14 time-dependent, 731 phase diagrams, 182– 3 phase studies, 178–83 pH dependency, 794, 802–8, 1006–8 phenols, 287–9, 456– 62 phenomenological Marcus-like models, 757–8 phenylalanine hydroxylase, 868– 70 phosphate esters, 905– 7, 966 phosphodiester hydrolysis, 905 phosphomonoester hydrolysis, 905 phosphoryl transfer, 966–8 photoacids, 366, 451– 62 photodissociation, 366, 451–62 photofragment spectroscopy, 310 photolysis, 366–7, 376–7, 378 physical characteristics, methane, 931–2 physical parameter isotope effects, 979–81 physical properties, muonium, 434–5 physical steps, 794–5 Pimentel proton affinity, 295 ping-pong mechanisms, 800–2, 848 point approximation, 202–4 Polanyi plots, 947–8 polar environments, 549– 73 polar ice, 396 polarizability, 298 pollutants, 888 polyatomic systems, 125, 127–30 polycrystallines, 205–10 polymer-polymer mixtures, 145– 8 polymer-solvent mixtures, 145–8 polynomial expansions, 14 –18, 33 porphine, 522, 529–35, 542–3

Index positive muons, 434 potential barrier heights, 20, 21 potential energy distributions, 282, 284–5 functions, 290–2 surfaces, 96–9, 160–2, 371, 695–6 potential evolution, 292 –4 potential free-energy surfaces, 695 –6 potentials, 254 potential shape, 292–4 PPV generating sequences, 487 prebinding isotope effects, 1026–31 precipitation, 392–6 pressure alcohol dehydrogenase catalysis, 825 –6 demixing, 148 hydrostatic pressures, 837 –44 ozone isotopologues, 366 primary amine dehydrogenases, 737– 8 primary geometric hydrogen bond isotope effects, 200–2 primary hydrogen bonds, 1005 primary isotope effects, 267–8 primary kinetic isotope effects, 465–72, 549–73 product end refluxer, 58 –61 promoting modes, 422 –3, 465– 72, 638 promoting vibrations, 479– 82, 483–7, 491–4, 513 –14 2-propanol, 1043–5 proteinless systems, 729– 30 proteins motion, 826 promoting vibrations, 491–4 soluble methane monooxygenase, 932 –3 protonation, 299 –301, 908 proton coupled electron transfer (PCET), 499–515, 710–20 proton inventory carbonic anhydrase catalysis, 850 –1 concerted multiple proton transfer, 542– 3 conformation change signaling, 787 –8 Kresge–Gross–Butler equation, 1000–4 multistep enzyme reactions, 1002–4 serine hydrolase oxyanion holes, 786 soluble methane monooxygenase, 936 solvent isotope effects, 1000–4, 1006, 1008–11, 1013– 14 protons conducting Nafion membranes, 721 conductivity, 691–722 donor-accepter modes, 564 donor-acceptor vibrations, 510–11 donors, 510– 11, 564, 936–8 hops, 705, 706 ordering, 184 shared hydrogen bonds, 154 stretching frequencies, 158 –65 stretching vibrations, 180 –1, 184, 189–91, 290 –2 vibrations, 2, 90–2, 180 –1, 184, 189 –91, 287 –9, 510–11 see also hydrons proton sponges, 242– 3, 262–3, 271

1069 proton transfer acid-base reactions, 452–3 alcohol dehydrogenase catalysis, 827–9 carbonic anhydrase catalysis, 850–7 condensed matter environments, 691–722 diabatic proton transfer, 696–7 double, 522, 529–37, 542–3 double-well proton potentials, 703–5 elementary, 693–702 free energy correlations, 454–5 homogeneous isotropic media, 721 hydrogen-bonded systems, 702–10 intervening water molecules, 847–57 intramolecular hydrogen bonds, 254–5 molecular fragments, 693–702 pH dependence, 802–7 secondary carbon kinetic isotope effects, 427–8 single-well potentials, 709–10 theory, 501 Zundel complexes, 706–9 see also adiabatic proton transfer; multiple proton transfer; nonadiabatic proton transfer Pseudomonas sp., 798, 883 pterin cofactors, 653–4 purpurogallin studies, 262 putidaredoxin, 658–60 pyridine-acid complexes, 210–17, 218, 245–6 pyridine complexes, 287– 8 2-pyridone 2-hydroxypyridine, 537, 538

Q Q-branch regions, 324– 9 QCP see quantized classical paths QM/MM see quantum mechanical and molecular mechanics quantized classical paths (QCP), 627–31, 634–5 quantum chemical kinetics, 347–8 quantum corrections, 198–202 quantum dynamics, 311–13, 583–4 Quantum Kramers methodology, 476, 478–9 quantum mechanical and molecular mechanics (QM/MM) enzymatic hydrogen tunneling, 680– 2, 758–9 enzyme catalysis simulations, 624– 6 hydrogen tunneling, 680–2, 758– 9 proton coupled electron transfer, 500, 505, 508 serine hydrolase oxyanion holes, 786–7 quantum mechanics classical calculations, 282–4 dynamics, 583–4 equilibrium in ideal gases, 4–5 tunneling, 104, 680–2, 744, 758–9 quantum nontunneling proton transfer, 549–62, 572–3 quantum statistics, 712 quantum tunneling, 104, 680–2, 744, 758–9 quasiharmonic approximation, 630–1 quasiparticles, 628–9 quasithermodynamics, 581–2 quinoline N-oxides, 262–3

1070 quinoproteins, 671–85 Quiver program, 112 Q vibrational modes, 564 –6

R radioactive carbon, 426 –7 radioactive isotopes, 920 RAHB see resonance-assisted hydrogen bonds rain, 392 Raman active vibrational modes, 184 Raman scattering, 185–90 random mechanisms, 798–9, 801–2 Raoult’s law, 122 –3 rare gases, 127– 9 rate coefficient ratios, 372–3 rate constants alcohol dehydrogenase catalysis, 814 –21 concerted multiple proton transfer, 542 –3 electron-coupled proton transfer, 718 –19 enzymatic binding isotope effects, 1022–9 enzymatic carbon– hydrogen-bond cleavage, 727–8 equilibrium isotope effects, 102 –4 gas phase equilibrium isotope effects, 102–4 kinetic isotope effects, 18 –19, 20– 5 muonium reactions, 441 –2 nonadiabatic proton transfer, 562– 7 solvent isotope effects, 996 substrate dependence, 797– 9 rate-determining trapping, 656– 7 rate enhancements, 766, 768 rate-limiting steps, 421– 3, 975, 1032 rate promoting vibrations, 479– 87, 491–4, 513– 14 rate theory, 853 –7 ratio of equilibrium, 996, 1022–9 reactant state fractionation, 1001–5 reaction asymmetry, 559, 567 reaction coordinates, 589, 592 –4, 597 reaction cycle intermediate conversion, 942 –3 reaction energy, 440 reaction free energy, 553, 563, 566 reaction rates, 363, 753–4 reaction symmetry, 455 reactive nuclear modes, 704 reduced-dimensional-bath models, 594 –6 reduced dioxygen species, 648–9 reduced hyperfine coupling, 438– 9 reduced mass considerations, 750 –1 reduced partition function ratio (RPFR), 5–8, 13–18, 33, 128– 9, 138–9 reduced partition functions (RPF), 49, 363 –4 reductase components, 932, 933 –4 reevaporation effects, 393 reference sample constraints, 410 –11 reflux, 58 –61, 63–4 regulatory protein components, 932, 943–8 relative carbon kinetic isotopic effects, 423 –4 relative rate coefficients, 365, 370 remote labelling, 918 –20 removal effects, 407

Index removal processes, 399 reorganization free-energy, 563 reorientation of the dividing surfaces (RODS), 587 reservoirs, 390, 391–2 residence times, 405–6 residue identification, 489– 91 resonance-assisted hydrogen bonds (RAHB), 233, 254–5, 259–62, 268–9 resonance splitting, 328–9 resonantly enhanced two-photon ionization (RE2PI), 329–46 reversible processes, 44– 53, 56–66, 72–82 RFPR see reduced partition function ratio N-ribosyl hydrolases, 965–6 L-ribulose-5-P4-epimerase, 928 Rice –Ramsperger –Kassel–Marcus (RRKM) theory, 369–70 Rittenberg, David, 2 RODS see reorientation of the dividing surfaces root-mean-square displacements, 437 rotational partition functions, 104–9 Roth, Etienne, 28–9 rovibrational energy, 439– 41 ground states, 440 theory, 255–6, 439–41 RPF see reduced partition functions RPFR see reduced partition function ratio RRKM see Rice –Ramsperger –Kassel–Marcus rubisco oxygen reactions, 646, 665 Rule of the Geometric Mean, 26, 865, 1041 ruthenium polypyridyl complexes, 506–7 Rydberg correction, 93– 4

S salicylic acid, 244–5 SAM Laboratory, Columbia University, 3 scattering resonances, 371 Schiff bases, 244–5, 266, 270–1 Schro¨dinger equation, 90–1 scintillation, 420–1 SCT see small-curvature tunneling seasonal effects, 393 secondary carbon kinetic isotope effects, 427–8 secondary catalytic-hydrogen bonds, 772, 777, 781–7 secondary deuterium kinetic isotope effects, 426–7 secondary geometric hydrogen bond isotope effects, 200–2 secondary hydrogen bonds, 200–2, 1004–6 secondary isotope effects, 955–70 a-secondary isotope effects, 956–7, 961 secondary isotope effects acyl transfer, 961– 4 glycosyl transfer, 964– 5 hydride transfer, 969 methyl transfer, 968–9 origins, 956–61 peptidyl prolyl cis-trans isomerases, 969–70 phosphoryl transfer, 966–8 N-ribosyl hydrolases, 965–6

Index theory, 955 –6 transferases, 965–6 secondary Swain–Schaad relationship, 748–53 semiclassical Instanton theory, 731, 734 semiclassical limits, 750–3 separable equilibrium solvation (SES), 594– 5 separation capacity, 64– 5 cascading, 42–3, 56–66 exchange reactions, 51–2, 72–82 factors, 11–12, 42 –8, 51–2, 67–9, 73 –7 ion exchange, 72 –82 isotope enrichment, 42–8 uranium, 11–12 vapor pressure isotope effects, 120–3 sequential mechanisms, 796 –800 serine hydrolases, 779 –87 serine proteases, 982–5, 987–90, 1009–12 SES see separable equilibrium solvation syn-sesquinorbornene disulfones, 534 SHAKE algorithm, 484 SHIE see solvent hydrogen isotope effects short hydrogen bonds, 232 –3, 976– 7 short-lived radionuclides, 417– 28 short-range proton transfer, 706–7 short, strong H-bonds (SSHB), 232–3 silver chloride, 879 simple-hydrogen bonds, 772, 775 simulations alcohol dehydrogenase catalysis, 814 –16 chemical process simulations, 623–6 computer simulations, 621– 40 microscopic, 621–40 molecular dynamics, 500, 505, 508 Singh–Wolfsberg correction, 13 –14 single-well potentials, 232–47, 709–10 sink rates, 400–1 site-directed mutagenesis, 816–18 SLO see soybean lipoxygenase-1 small-curvature tunneling (SCT), 583, 588 sMMO see soluble methane monooxygenase SMOC see standard mean ocean chloride Soddy, F., 1 soft-mode analysis, 180 –1 solids, vapor pressure, 120–33 solid state reactions, 590 –1 solid surface reactions, 590–1 soluble methane monooxygenase (sMMO), 931 –49 Arrhenius plots, 942–3 carbon-hydrogen bond cleavage, 938–40, 947–9 catalytic cycles, 933– 6 components, 932– 3 kinetic solvent isotope effects, 936 –8 molecular sieve model, 945– 6 Polanyi plots, 947–8 proton donors, 936–8 reaction cycle intermediate conversion, 942– 3 reaction mechanisms, 938 –43 substrate reactions, 938–41 tunneling, 946 solute isotope effects, 145 –8

1071 solute-solvent coupling, 592–4 solute-solvent separation, 591–2 solvation, 235– 6, 592–4 solvent effects, 422 solvent hydrogen isotope effects (SHIE), 847–57 solvent induced barriers, 445 solvent isotope effects acetylcholinesterase, 1012–14 alcohol dehydrogenase catalysis, 827–9 carbonic anhydrase, 1014– 15 3,4-dihydroxyphenylalanine, 1015–16 enzyme reactions, 995–6, 1002–4, 1006–16 examples, 1008– 16 fractionation factors, 996– 7, 1004–6 Kresge –Gross– Butler equation, 998–1004 NMR chemical shifts, 217–22 non enzymatic reactions, 1008– 9 origins, 996–8 polymer–polymer mixtures, 145–8 polymer-solvent mixtures, 145–8 reactant state fractionation factors, 1001–5 theory and practice, 995–1016 transition state fractionation factors, 1004–6 tyrosine hydroxylase, 1015–16 solvent reorganization free-energy, 563 soybean lipoxygenase-1 (SLO), 508–11, 726–8, 736–7, 755–6 soybean lipoxygenase, 655–7, 665 spatial isotope effects, 264–5 spectral density, 296–7 spectroscopy condensed matter isotope effects, 134–5 dip, 310 fast atom bombardment-isotope ratio mass spectroscopy, 879 high-resolution, 311– 17 hydrogen bonds, 193– 227, 240, 774–5 isotope selective spectroscopy, 320, 323–5, 329–48 low temperature, 210– 11 molecular, 311– 13 Nuclear Magnetic Resonance, 193–227 photofragment, 310 vapor pressure isotope effects, 130 vibrational, 177, 282–4 see also infrared spectroscopy; overtone spectroscopy spin, 307 spin bosons, 626–7 spin-spin coupling constants, 165–71 square cascades, 61 –4 SSHB see short, strong H-bonds SSZ see static-secondary-zones stable isotopes, 918–20 stack plumes, 406–7 standard free-energy, 454–5 standard mean ocean chloride (SMOC), 877–8 standard-state free energy of activation, 582 standard state Helmholtz free energy, 124 startup, cascades, 66–8 static-secondary-zones (SSZ), 597–9 static systems, 259–69, 597–9 statistical coupling, 718

1072 statistical mechanics, 13–14, 100 –9 steady-state cascade theory, 56–66 kinetic constants, 817– 18 kinetics, 817– 18, 827 oxidation, 827–9 phenomena, 44 –53, 56– 66, 817–18, 827–9 stepwise reactions, 422–3, 522, 529– 35, 711– 14 sterics binding isotope effects, 1042–5 compression, 260–1 effects, 260– 1, 422, 424 –5, 960– 1 enforced hydrogen bonds, 233 hindrance, 422 secondary isotope effects, 960–1 STO3G, 111, 113–14 stopped-flow methods, 672–3, 675 stratospheric carbon dioxide, 375 stratospheric ozone, 367–8 strengths, hydrogen bonds, 771–5, 981–2 stretching frequency changes, 1042–5 stretching vibrations, 180 –1, 189 –91, 289– 92 strong acids, 175–7, 183–91 strong dinitrogen monohydride, 205 –10 strong hydrogen bonds, 693, 721, 978 –9 see also low-barrier hydrogen bonds structural isotope effects, 435–41 structures alcohol dehydrogenase catalysis, 812 –14 carbonic anhydrase isozymes, 849 –50 hydrogen bonds, 771–5 substitution hydrogen bonds, 284–5 leaving groups, 426 substrate effects, 425–6 thermal vibrational averaging, 169 –71 zero-point motion, 169– 71 substrate capture, 839–40, 843 –4 substrate isotope effects, 827–9 substrates binding isotope effects, 1027–30 bond breakage, 685 dependence, 794, 796–802, 807–8 effects, 425– 6 labelled nucleophiles, 425 –6 oxidation, 679–82 soluble methane monooxygenase, 938–41 stickiness, 795 unmasking chemistry, 816 –17 subtilism, 786 sulfates, 378–9, 397, 403 sulfur, 47, 378–9, 396 –7, 405 –7 sulfur dioxide, 406 –7 sum rules, 99, 226 Swain–Schaad relationship, 550, 560, 571 –2, 746– 53 symmetry of coupling, 479 –80 factors, 701 ice phases, 178– 83 numbers, 104 –9 polar environment proton transfer, 550

Index selection rules, 307 see also hydrogen-bond symmetrization synchronous electron-proton transfer, 720 syn-sesquinorbornene disulfones, 534 synthesis, radionuclides, 419–20

T tapered cascades, 57 tautomeric equilibrium, 232 tautomerization, 538 Teller –Redlich product rule, 99, 112–15 temperature dependence adiabatic proton transfer, 559 alcohol dehydrogenase catalysis, 822–5 chlorine kinetic isotope effects, 876 concerted transfer, 536 enzymatic carbon–hydrogen-bond cleavage, 726, 728–34 hydrogen tunneling, 673–7 reactions, 509–10, 673–7, 753–7 free energy profiles, 198, 199 hydrogen transfer, 822– 5 hydrogen tunneling, 673–5, 753– 7 kinetic isotope effects, 673–5, 753–7 molar volume isotope effects, 136 nonadiabatic proton transfer, 568–71 nuclear quantum mechanics, 629– 30 ozone isotopologues, 366 reaction rates, 753–4 secondary isotope effects, 955 stepwise transfer, 529–30, 532–4 temperature-independent factor (TIF), 876, 955 temperature-induced solvent isotope effects, 217–22 ternary complexes, 813– 14, 1048– 9 terreactant mechanisms, 801–2 terrestrial solids, 377–9 tetrachlorop-benzoquinone, 841–2 tetrahydrofolate, 511–12 theory chemical dynamics in complex systems, 476–9 electron transfer, 500–1 enzymatic carbon–hydrogen-bond cleavage, 730– 5 enzyme mechanisms from isotope effects, 915–25 hydrogen bond characteristics, 773–4 hydrogen bond isotope effects, 194–204 hydrogen tunneling, 757–9 hydrostatic pressures effects, 837–41 intermediate partitioning, 862– 6 intramolecular hydrogen bonds, 255–8 multiple proton transfer, 523–9 proton coupled electron transfer, 502–5 proton transfer, 501, 523– 9 secondary isotope effects, 955–6 solvent isotope effects, 995–1016 transition-states, 19 –25, 102– 4, 522–4, 580–607, 753–4 VTST/MT, 584–600 thermal diffusion, 46–7, 55 thermal rate constants, 525, 527 thermal vibrational averaging, 169– 71

Index THERMISTP program, 112–15 thermochemistry, 774–5 thermodynamics condensed matter isotope effects, 125–7, 130, 144–5 efficiency, 56–7 equilibrium constants, 100– 2 isotopic waters, 144– 5 precision, 130 thermophilic ADH, 632 –3, 753, 756–7, 823 Thermus thermophilus, 483 ß-thioxoketones, 273 –4 thymidylate synthase, 748 TIF see temperature-independent factor time-dependence intramolecular dynamics, 311– 17 isotope enrichment, 66–8 material balance equation, 69 perturbation theory, 731 time of flight (TOF) mass spectrometers, 320, 322–3 time-independent states, 311 –17 time-scales, 325–9 TMADH see trimethylamine dehydrogenase TOF see time of flight topa quinone (TPQ) oxidases, 662–5 Torpedo californica AChE, 1013–14 totally adiabatic proton transfer, 698 totally diabatic proton transfer, 696–7 total reflux, 63 TPQ see topa quinone; trihydroxyphenylalanine quinone traditional hydrogen bonds, 154 transfer electron, 500– 1, 626, 651 –5 electron-hydrogen, 491–4 electron-proton, 499 –515, 710 –20 enzymatic acyl, 962 glycosyl, 964– 5 hydrides, 511 –12, 630, 969 hydrogen, 479– 91, 822–6 hydrons, 197 –9, 466–8, 471–2 multiple charge, 500, 505 muonium, 444–5 proton electron, 499–515, 710–20 see also proton transfer transferases, 965– 6 transient conditions, 67–9 transient intermediates, 934 –49 transient kinetics, 814 –16, 827– 9 transient oxidation, 827–9 transient reactions, 814 –16, 819– 20, 827–9 transition mechanisms, 718–19 transition probability, 698–9 transition-states enzymatic binding isotope effects, 1032, 1033 fractionation, 1004–6, 1008–9 stabilization, 766 –70 structure, 423– 6 theory, 19– 25, 102–4, 522–4, 580–607, 753–4 translational partition functions, 104 –9 transmission, 261–2 coefficients, 256– 7, 582, 587 –9, 597– 8, 698–9, 719 1,3,5-triacetyl-2,4,6-trihydroxybenzene, 257 –8

1073 trifluoromethylketones, 982–5 trihydroxyphenylalanine quinone (TPQ), 662–5 trimethylamine dehydrogenase (TMADH), 676, 682–5 triphenylmethane, 467–8 triple-minimum potentials, 530–1 tropospheric budget, 400–1 tropospheric isotopic effects, 388 tropospheric nitrous oxide, 376 tropospheric sulfur, 405–6 TRTEST, 114– 15 trypsin, 779–80 tryptophan hydroxylase, 868–70 tryptophan tryptophylquinone (TTQ), 678–82 tunneling acids, 190 enzymatic binding isotope effects, 1032 hydrogen bonds, 286– 7 hydrogen transfer, 822– 3 limits, 634–5 magnitude, 567 multiple proton transfer, 524–5 muonium reactions, 442–3, 446 mutations, 683–4 proton transfers, 524–5, 549–53, 562–73 rate constants, 524–5, 527 soluble methane monooxygenase, 946 transmission coefficients, 698–9 vibrational mode specificity, 334–8 turnover, 405, 796 two-bond spin-spin coupling constants, 165–71 two-dimensional tunneling, 731 two-level intramolecular dynamics, 314–16 two-oscillator models, 730–1 two-state intramolecular dynamics, 314–16 two-state proton transfer, 452–3 tyrosine hydroxylase, 653–5, 665, 868– 70, 1015–16 tyrosine oxidation, 506–8

U Ubbelohde effect, 200–2, 256–7 ultrafast intramolecular dynamics, 311–17 ultra-light isotopes, 433–48 ultraviolet (UV) double resonance, 331–3, 346 ultraviolet (UV) photolysis, 376–8 unimolecular rate expressions, 504 unimolecular reactions, 590 unmasking chemistry, 816–21 unusual-hydrogen bonds, 772, 776 uranium fractionation factors, 9–12 isotope enrichment, 47, 53, 54, 56, 81–2 Urey, H.C., 2–3, 8 UV see ultraviolet

V vacuum ultraviolet (VUV) lasers, 309–10 valence coordinates, 97–8 valine, 485–7 van’t Hoff activation parameter, 824–5

1074 vapor, water cycle isotope effects, 390– 2 vapor-liquid isotope fractionation, 391 –2 vapor phase, 121–3 vapor pressure isotope effects (VPIE) argon, 31–2, 127–8 condensed matter, 25–6, 30 –2, 120, 124–34 excess free energy, 138–9 liquids, 120–33 molecular properties, 124– 34 solids, 120 –33 vibrational dynamics, 124 –34 variational reaction paths (VRP), 587 variational transition-state theory with multidimensional tunneling (VTST/MT), 580–607 enzymatic hydrogen tunneling, 680–2 kinetic isotope effects, 600 –5 previous reviews, 583 software, 605 theory, 584 –600 vehicular exhaust, 379 velocity dependence, 80–1 vibrating species structural isotope effects, 435 –41 vibration dynamics, 124–34 energy levels, 96 –9 frequency, 9– 12, 979–80, 1039–41 hydrogen tunneling, 751 promotion, 479–87, 491–4, 513 –14 uranium separation factors, 11–12 VPIEs, 127– 8 vibrational energy, 325 –9, 334– 46, 648–9 vibrational isotope effects, 281–301, 979–80 vibrationally adiabatic potentials, 589 vibrationally assisted dissociation and photofragment ionization, 320–9 vibrationally enhanced tunneling, 475 –95, 513 –14 vibrationally excited benzene, 338 –46 vibrationally excited molecules, 329–46 vibrational mode specificity, 334–8 vibrational partition functions, 102 –4 vibrational spectroscopy, 177, 282 –4 vibronic models, 636– 7 volcanic sulfates, 397 VPIE see vapor pressure isotope effects VRP see variational reaction paths VTST/MT see variational transition-state theory with multidimensional tunneling

W water activation, 895 –9 condensed matter isotope effects, 132– 3, 142–5 coordination, 898 –9 cycle isotope effects, 390 –6 excess proton conductivity, 705 –6 H/D exchange reactions, separation factors, 51 –2 ice, 179– 83 kinetic isotope effects, 899 molecules, 437, 706, 847– 57 muonium diffusion, 447

Index phase exchange, 390– 1 proton hops, 706 proton transfer, 847–57 reduced partition functions, 16 thermodynamic properties, 144–5 trimers, 542–3 vapor pressure isotope effects, 132– 3 wires, 537–41, 742–3 wave functions, 8, 90 –6 Wentzel–Kramers–Brillouin (WKB) like approximations, 588 Westheimer–Melander (W–M) picture, 550, 561 Wigner tunnelling correction, 104 wild alcohol dehydrogenase catalysis, 817–18 WIMPER method, 14–16 WINIMAX coefficients, 14 WKB see Wentzel –Kramers–Brillouin W– M see Westheimer –Melander Wolfsberg, Max, 18, 20– 1 working material explosions, 71 –2

X Xanthobacter autotrophicus, 880 X–H stretching bands, 154–65 X–H–Y hydrogen bonds, 154–65 x-ray diffraction, 194, 240–1

Y Yankwich–Calvin experiments, 20, 21 yeast alcohol dehydrogenase, 482–7, 812–30, 842– 4 yeast formate dehydrogenase, 844 yellow transient intermediates, 934–49

Z Zassenhaus expansion, 478 zero-order approximation, 16–17 zero-point energy (ZPE) adiabatic proton transfer, 553–5, 559– 62 Bigeleisen–Mayer equation correction, 10 enzymatic binding isotope effects, 1033– 5, 1037 hydrogen bonds, 977–82 hydrogen tunneling, 746, 750–1 intramolecular hydrogen bonds, 254, 269 isotope enrichment, 50 low-barrier hydrogen bonds, 975, 977– 82 molecular oxygen enzymatic activation, 648– 9 muonium, 435–8, 441–2 ozone isotopologues, 370–1 proton transfer, 550, 552– 5, 559–62 secondary isotope effects, 955 zero-point-inclusive barrier heights, 581 zero-point motion, 169–71 zero-point stretching vibrations, 198–202 zero-time dependence, 69 zinc isotope effects, 407 ZPE see zero-point energy Zundel complexes, 705– 9, 714–15 Zundel polarizability, 298