ADVANCES IN CHEMICAL PHYSICS VOLUME 142
EDITORIAL BOARD BRUCE J. BERNE, Department of Chemistry, Columbia University,...
144 downloads
1037 Views
6MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
ADVANCES IN CHEMICAL PHYSICS VOLUME 142
EDITORIAL BOARD BRUCE J. BERNE, Department of Chemistry, Columbia University, New York, New York, U.S.A. KURT BINDER, Institut fu¨r Physik, Johannes Gutenberg-Universita¨t Mainz, Mainz, Germany A. WELFORD CASTLEMAN, JR., Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania, U.S.A. DAVID CHANDLER, Department of Chemistry, University of California, Berkeley, California, U.S.A. M.S. CHILD, Department of Theoretical Chemistry, University of Oxford, Oxford, U.K. WILLIAM T. COFFEY, Department of Microelectronics and Electrical Engineering, Trinity College, University of Dublin, Dublin, Ireland F. FLEMING CRIM, Department of Chemistry, University of Wisconsin, Madison, Wisconsin, U.S.A. ERNEST R. DAVIDSON, Department of Chemistry, Indiana Univeristy, Bloomington, Indiana, U.S.A. GRAHAM R. FLEMING, Department of Chemistry, The University of California, Berkeley, California, U.S.A. KARL F. FREED, The James Franck Institute, The University of Chicago, Chicago, Illinois, U.S.A. PIERRE GASPARD, Center for Nonlinear Phenomena and Complex Systems, Universite´ Libre de Bruxelles, Brussels, Belgium ERIC J. HELLER, Department of Chemistry, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, U.S.A. ROBIN M. HOCHSTRASSER, Department of Chemistry, The University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A. R. KOSLOFF, The Fritz Haber Research Center for Molecular Dynamics and Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel RUDOLPH A. MARCUS, Department of Chemistry, California Institute of Technology, Pasadena, California, U.S.A. G. NICOLIS, Center for Nonlinear Phenomena and Complex Systems, Universite´ Libre de Bruxelles, Brussels, Belgium THOMAS P. RUSSELL, Department of Polymer Science, University of Massachusetts, Amherst, Massachusetts, U.S.A. DONALD G. TRUHLAR, Department of Chemistry, University of Minnesota, Minneapolis, Minnesota, U.S.A. JOHN D. WEEKS, Institute for Physical Science and Technology and Department of Chemistry, University of Maryland, College Park, Maryland, U.S.A. PETER G. WOLYNES, Department of Chemistry, University of California, San Diego, California, U.S.A.
ADVANCES IN CHEMICAL PHYSICS VOLUME 142
Series Editor STUART A. RICE Department of Chemistry and The James Franck Institute The University of Chicago Chicago, Illinois
Copyright # 2009 by John Wiley & Sons, Inc. All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Catalog Number: 58-9935 ISBN: 978-0-470-46499-1 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
CONTRIBUTORS TO VOLUME 142 BENJAMIN AUER, Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, USA BJO¨RN H. JUNKER, Leibniz Institute of Plant Genetics and Crop Plant Research, 06466 Gatersleben, Germany YU-SHAN LIN, Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, USA JAMES L. SKINNER, Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, USA RALF STEUER, Manchester Interdisciplinary Biocentre, School of Chemical Engineering and Analytical Science, University of Manchester, Manchester M17DN, United Kingdom MARTIN A. SUHM, Institut fu¨r Physikalische Chemie, Universita¨t Go¨ttingen, Go¨ttingen, 37077 Deutschland, Germany
v
INTRODUCTION Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields. Any attempt to do more and be broadly educated with respect to a large domain of science has the appearance of tilting at windmills. Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists. This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field. STUART A. RICE
vii
CONTENTS HYDROGEN BOND DYNAMICS By Martin A. Suhm
IN
ALCOHOL CLUSTERS
VIBRATIONAL LINE SHAPES, SPECTRAL DIFFUSION, HYDROGEN BONDING IN LIQUID WATER By James L. Skinner, Benjamin M. Auer, and Yu-Shan Lin
AND
1
59
AND
COMPUTATIONAL MODELS OF METABOLISM: STABILITY REGULATION IN METABOLIC NETWORKS By Ralf Steuer and Bjo¨rn H. Junker
105
AUTHOR INDEX
253
SUBJECT INDEX
277
ix
5
4
3 2 1 0
NH = 0 NH = 1
P(ω)
0.002
Total
0.001
0 NT = 1 NT = 2
0.002 P(ω)
NT = 3 NT = 4
0.001
0
Total
3000
3200
3400
3600
3800
ω (cm-1) Figure 4, Chapter 2. Top panel: Average number of H bonds to the H atom (in HOD/D2O), hnH i, and total number of H bonds to the HOD molecule, hnT i, versus OH stretch frequency. Middle panel: Frequency distributions with NH ¼ 0 and 1. Bottom panel: Frequency distributions with NT ¼ 1; 2; 3; 4.
0.003 1N 2N 2SH 3SH 2SD 3SD 3D 4D Total
P(ω)
0.002
0.001
0
3000
3200
3400 ω (cm-1)
3800
3600
Figure 5, Chapter 2. Frequency distributions for the eight H-bond classes of HOD molecules in D2O. Labels are as described in the text and in Table I.
1 -1
ω = 2500 cm -1 ω = 2530 cm -1 ω = 2560 cm -1 ω = 2590 cm -1 ω = 2620 cm
0.9
C2(t)
0.8
0.7
0.6
0.5
0
100
200
300
400
500
t (fs) Figure 8, Chapter 2. Frequency-dependent orientation TCFs for HOD/H2O at room temperature. Sub-ensembles are defined according to the value of the OD stretch frequency at t ¼ 0, and the curves correspond to five sub-ensembles as labeled in the graph.
C2(100 fs)
0.9
Theory
0.8
0.7 1°C 25°C 65°C
C2(100 fs)
0.9
Experiment
0.8
0.7
2500
2540
2580
2620
ω (cm-1) Figure 9, Chapter 2. Experimental [59] and theoretical values of the polarization anisotropy time correlation function at 100 fs, as a function of OD stretch frequency, for three different temperatures.
Figure 2, Chapter 3. Current mathematical representations of metabolism utilize a hierarchy of descriptions, involving different levels of detail and complexity. Current approaches to metabolic modeling exhibit a dichotomy between large and mostly qualitative models versus smaller, but more quantitative models. See text for details. The figure is redrawn from Ref. 23.
Figure 6, Chapter 3. Enzymes act as recycling catalysts in biochemical reactions. A substrate molecule binds (reversible) to the active site of an enzyme, forming an enzyme–substrate complex. Upon binding a series of conformational changes is induced that strengthens the binding (corresponding to the induced-fit model of Koshland [48]) and leads to the formation of an enzymeproduct complex. To complete the cycle, the product is released, allowing the enzyme to bind further substrate molecules. (Adapted from Ref.1).
(A)
(B)
Figure 11, Chapter 3. Allosteric regulation: A conformational change of the active site of an enzyme induced by reversible binding of an effector molecule (A). The model of Monod, Wyman, and Changeux (B): Cooperativity in the MWC is induced by a shift of the equilibrium between the T and R state upon binding of the receptor. Note that the sequential dissociation constants KT and KR do not change. The T and R states of the enzyme differ in their catalytic properties for substrates. Both plots are adapted from Ref. 140.
and λ
max ℑ
Figure 12, Chapter 3. Nonaqueous fractionation (NAF), enables the determination of metabolite concentrations and enzyme activities at the subcellular level. The figure is adapted from [203].
λmax
5
max ℜ
eigenvalues λ
10
ℑ
λmax ℜ
0 HO
SN −5 (i) −10 1
(ii)
(iii)
0 −1 −2 −3 Influence ξ of ATP on ν ∈ [1,−∞)
(iv) −4
1
Figure 28, Chapter 3. The eigenvalues of the Jacobian of minimal glycolysis as a function of the influence x of ATP on the first reaction n1 ðATPÞ (feedback strength). Shown is the largest real part of the eigenvalues (solid line), along with the corresponding imaginary part (dashed line). Different dynamic regimes are separated by vertical dashed lines, for lmax < > 0 the state is unstable. Transitions occur via a saddle-node (SN) and a Hopf (HO) bifurcation. Parameters are n0 ¼ 1, TP0 ¼ 1, ATP0 ¼ 0:5, AT ¼ 1, and y ¼ 0:8.
max
20
and λI
max
λR
(d)
10
(a)
0
(c)
−10
(b)
−20 1
0.5
0
−0.5
−1
−1.5
v1 ATP
negative feedback θ
ATP [mM]
(A)
v1
θATP=0.8
(B)
(C)
θv1 =−0.06 ATP
(D)
v1 θATP =−0.7
3
3
3
2
2
2
2
1
1
1
1
3
0 0
10 20 time [min]
30
0 0
10 20 time [min]
30
0 0
5 time [min]
v1
θATP=−1.0
0 0
5 time [min]
Figure 31, Chapter 3. Dynamics of glycolysis. Upper panel: The eigenvalue with the largest r1 real part lmax < as a function of the feedback strength yATP of ATP on the combined PFK-HK reaction. All other saturation parameter are unity ymx ¼ 1. Shown is lmax < (solid line) together with the imaginary max part lmax (dashed line). At the Hopf bifurcation a complex conjugate pair of eigenvalues lmax I < & il= crosses the imaginary axis. Note the similarity to Fig. 28 (Minimal glycolysis). Lower panel: Upon variation of yr1 ATP four dynamic regimes can be distinguished. Shown are the corresponding time courses of ATP using an explicit kinetic model at the points (a, b, c, d) indicated in the plot. (a) A small negative v1 real part lmax < , corresponding to slow relaxation to the stable steady state (yATP ¼ 0:8). (b) An optimal response to perturbations, as determined by a minimal largest eigenvalue lmax (yv1 R ATP ¼ (0:06). (c) Oscillatory return to the stable steady state. The metabolic state is stable, but with nonzero v1 imaginary eigenvalues (yv1 ATP ¼ (0:7). (d) Sustained oscillations yATP ¼ (1:0. All different regimes can be deduced solely from the Jacobian and are only exemplified using the explicit kinetic model.
Figure 38, Chapter 3. A bifurcation diagram for the model of the Calvin cycle with product and substrate saturation as global parameters. Left panel: Upon variation of substrate and product saturation (as global parameter, set equalfor all irreversible reactions), the stable steady state is confined to a limited region in parameter space. All other parameters fixed to specific values (chosen randomly). Right panel: Same as left panel, but with all other parameters sampled from their respective intervals. Shown is the percentage Z of unstable models, with darker colors corresponding to a higher percentage of unstable models (see colorbar for numeric values).
Figure 39, Chapter 3. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO), or via saddle-node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details.
HYDROGEN BOND DYNAMICS IN ALCOHOL CLUSTERS MARTIN A. SUHM Institut fu¨r Physikalische Chemie, Universita¨t Go¨ttingen, 37077 Go¨ttingen, Germany CONTENTS I. Introduction II. Issues A. Structures and Topologies B. Energetics C. Cooperativity D. Hydrogen Bond Isomerism and Conformational Isomerism E. O H Stretching Dynamics F. Isotope and Overtone Effects G. C O and C H Stretching Dynamics H. Torsional Dynamics I. Tunneling Dynamics J. Chirality Recognition K. Cation Solvation L. Anion Solvation III. Experimental Methods A. Microwave Spectroscopy B. Infrared Absorption C. Raman Scattering D. Crossed Beam Techniques E. UV–IR Coupling F. VUV–IR Coupling IV. Computational Methods A. Empirical Force Fields B. Electronic Structure Calculations C. Internuclear Dynamics V. Systems A. Methanol B. Ethanol
Advances in Chemical Physics, Volume 142, edited by Stuart A. Rice Copyright # 2009 John Wiley & Sons, Inc.
1
2
martin a. suhm
C. Linear Alcohols D. Bulky Alcohols E. Unsaturated and Aromatic Alcohols F. Fluoroalcohols G. Trifluoroalcohols H. Chlorinated Alcohols I. Ester Alcohols J. Polyols and Sugars VI. Conclusions Acknowledgments References
I.
INTRODUCTION
Alcohols are the conceptually simplest organic molecules that undergo classical hydrogen bonding. Given the enormous importance of hydrogen bonding in complex organic and biological matter [1, 2], it is imperative to understand its dynamics for such simple, yet realistic, model systems. By adding one molecule at a time, the evolution from single molecules to condensed phases can be mapped out in a molecular cluster approach [3]. In contrast to the more elementary, more abundant, but completely singular water system [4], alcohols can be tailored by modifying their molecular backbone [5]. This ‘‘chemical’’ dimension renders them particularly valuable for supramolecular design [6]. In terms of hydrogen bond topology, it is the reduced dimensionality which makes alcohols attractive. Compared to the complex three-dimensional network present in water, the propensity for ring and chain aggregation in alcohol clusters [7] provides an elementary starting point for the investigation of energy flow along a sequence of intermolecular interactions [8], with important applications in solution and neat liquid phases [9–11]. The coexistence of hydrophobic and hydrophilic domains also leads to interesting surface effects [12] and microstructure in liquid alcohols [13], quite in contrast to water. Alcohols are clearly among the most elementary and longest known [14] protagonists in gasphase supramolecular chemistry [15]. A dedicated review on hydrogen bonding in isolated alcohol clusters bridging methanol on one side [16] and sugars on the other [17] appears timely. In 1996, there were about 12 citations to publications including the keywords jet*, hydrogen*, and alcohol*, according to the Web of Science [18]. In 2006, there were more than 200. In view of several available reviews on aromatic systems [19–21], the focus will be on the less-studied aliphatic alcohols, which come closer to being amphiphilic models. For solutions, where the first studies using nuclear magnetic resonance (NMR) and infrared (IR) spectroscopy date back more than half a century, a recent review concentrating on sterical
hydrogen bond dynamics in alcohol clusters
3
hindrance effects is available [22]. In the solid [23], packing effects always compete with the intrinsic properties of the isolated or cooperative hydrogen bonds. While clusters are also postulated as highly fluxional units in the supercritical state [24, 25] and in solution [26], their detailed understanding rests on a proper characterization at lower temperatures. At room temperature, the cluster concentration in the vapor phase of alcohols is fairly low and thermal excitation still makes an interpretation of the spectra difficult [27]. Therefore, the present review concentrates on cold molecular aggregates, which are most conveniently produced and studied in adiabatic gas expansions or jets [28]. The ultimate goal is to use the detailed insights gained in such low-temperature gasphase studies to better understand the hydrogen bond and conformational behavior in the liquid state [29, 30]. After raising a selection of topical issues in this field and briefly introducing some spectroscopic and numerical techniques to probe the hydrogen bond dynamics, recent results for alcohol clusters are presented in order of increasing complexity. They are followed by some general conclusions and an outlook on future research goals. II.
ISSUES
Among the wealth of issues relevant to hydrogen bonding in alcohol clusters, this review will focus on aspects related to hydrogen bond patterns and on the dynamical implications over a wide range of time scales. Some key questions connected to these aspects will be formulated. A.
Structures and Topologies
O H ! ! ! O hydrogen bonds have a strong preference for a nearly linear arrangement. Furthermore, electrostatic forces or lone electron pair considerations direct the hydrogen that is attached to the accepting oxygen into an approximately tetrahedral angle with respect to the hydrogen bond. The tetrahedral lone-pair picture has recently been debated [31], based on electron density maps and earlier structural evidence on poly-alcohols [32]. While there is certainly significant acceptor potential in the region between the two lone pairs, the two studies [31, 32] may be biased in overestimating it slightly. Pauli repulsion will have to be included in the recent study [31] and distortions due to the optimization of multiple hydrogen bonds and steric constraints in the solid state have to be considered in the earlier analysis [32]. When taken into account, both effects are likely to recover a certain tetrahedral preference in isolated hydrogen bonds. This may or may not be cast into a lone-pair picture. At least it is an extremely useful ordering principle for alcohol cluster structures, which does not rule out exceptions.
4
martin a. suhm
In line with this, alcohols form unsymmetric dimers with well-separated donor and acceptor roles [33] and a more or less pronounced preference for one of the acceptor lone pairs, depending on secondary interactions (see Fig. 1). For trimers, the option to form a ring with three hydrogen bonds usually wins over the hydrogen bond strain and over steric repulsion between the alkyl groups which this induces. However, the open-chain structure with two unstrained hydrogen bonds is not too far in energy and always should be considered for vibrationally excited clusters [34] and whenever secondary interactions come into play [35, 36]. Furthermore, trimer formation can be suppressed at least at elevated temperatures [37], if the alkyl chain becomes too bulky. For the tetramer, a cyclic structure involves less strain and less sterical hindrance than in the trimer and is thus particularly attractive (Fig. 1). The alkyl groups can alternate between positions above and below the hydrogen bond plane and better avoid each other, if they are too big. Again, this alternation can also be interpreted as being due to a lone-pair preference. In terms of pure repulsion, a planar arrangement of all heavy atoms is indeed competitive, if the alkyl group
O H
:
donor
R
O
repulsion
:
R
R H
anticooperative
H
O donor
R down
R
R down
:
H O
O H
O R down
R
O H :
:
OH
:
up
R O H :
cooperative
up
up
:
:
acceptor
:
:
H
HO R
up
Figure 1. Illustration of lone-electron-pair preferences in alcohol dimers, cooperative and anticooperative binding sites for a third monomer, ring strain and steric repulsion in alcohol trimers, alternation of residues in alcohol tetramers, and chain, branch, and cyclic hydrogen bond topologies in larger clusters.
hydrogen bond dynamics in alcohol clusters
5
is not too big. Rings with homodromic hydrogen bond patterns remain energetically attractive for larger clusters [38], but the entropic advantage of chain structures, where the terminal alcohol molecules only form single strong hydrogen bonds, and branched topologies, where alcohol molecules serve as double acceptors, tends to grow. Chain topologies can reduce steric congestion by forming helical structures, whereas branching tends to be more sterically demanding. Isomerism is therefore an important issue beyond a cluster size n ¼ 4 [39] and possibly even before. In the solid, infinite chains and helices are often realized for simple alcohols [40], but cyclic structures are also conceivable [41] and quite abundant for bulky species [7, 42–44]. The structure of liquid alcohols is heavily debated [45]. Entropy arguments would predict winding chains of variable length to be quite important. The missing hydrogen bond compared to cyclic clusters can be partly compensated by the polar environment and by branching points. A finite cluster model of liquid alcohols [45] will necessarily be biased toward small clusters, closed rings, and compact structures, because it cannot reproduce the dramatic increase of conformational and topological entropy in extended flexible chains and dynamical network structures. Nevertheless, a detailed characterization of small clusters can bring us closer to a structural understanding of liquid alcohols. The basic aggregation pattern in alcohol clusters can of course be influenced in any desired direction by the design of the alkyl group, a feature that makes alcohols attractive in molecular recognition studies. Molecular additives can further modify the topological preferences. By offering a pure hydrogen bond acceptor group such as an ether, terminated chain structures can be favored over rings [46]. By adding a local or global charge, major disruptions of the ring topology are possible, because charge coordination competes with the hydrogen bond network [47, 48]. However, the fundamental preference of alcohol clusters to form hydrogen-bonded ring patterns is never lost completely and reappears whenever other constraints start to relax. B.
Energetics
Unfortunately, not many techniques allow us to probe the binding energy of a hydrogen-bonded complex directly [20, 49–51]. With very few exceptions [52, 53], they are restricted to aromatic (p-) systems, where the intrinsic strength of a single alcoholic hydrogen bond is typically superimposed [54] and may even be overwhelmed [55] by p-interactions. Therefore, one often has to rely on quantum chemical sources for energy information [56, 57]. It is essential to calibrate these techniques against the few available experimental benchmark data, such as for methanol dimer [52], phenol–methanol [20], or 1-naphthol complexes [51]. Relative energy orders of isomers are even more important [50] and can sometimes be obtained by jet relaxation studies [58]. The strength of a hydrogen bond can be influenced by introducing electron-donating and electron-withdrawing alkyl
6
martin a. suhm
groups. Quite naturally, the hydrogen bond donor quality increases for electronegative substituents, whereas the acceptor quality decreases. For homodimers—that is, complexes built from identical subunits—the two influences compete with each other. Furthermore, the organic substituents can undergo their own intermolecular interactions, either among themselves or with the functional units of the O H! ! ! O hydrogen bond. Therefore, cluster binding energies are measures of the total interaction between the interacting molecules, which may or may not be dominated by a single classical hydrogen bond interaction. Donor–acceptor roles can become quite intricate in multifunctional systems [59]. This is another important motivation for studying the simplest prototype systems, where any secondary interactions are minimized. C.
Cooperativity
Another factor that influences the aggregation pattern and energetics of alcohol clusters is cooperativity [60]. Once a molecule engages as a hydrogen bond donor, it automatically becomes a better acceptor and vice versa due to the polarization of the O H bond. This favors chain-like and even more cyclic topologies over branched networks of hydrogen bonds [61]. The latter are less stable, because two or more molecules must compete for the electron density at the acceptor oxygen (Fig. 1). It is more favorable for the third molecule to extend the polarization chain of the other two, rather than to interrupt it. The prototype system for this is hydrogen fluoride [62], which, more so than the reactive OH radical [63], may serve as the topological parent compound for alcohol aggregation. Its pronounced hierarchy of interactions (strong 1-D aggregation via cooperative hydrogen bonds, weak 3-D aggregation via dispersive forces) can be systematically attenuated by increasing the size of the alkyl group. This hierarchy is responsible for cluster formation in alcohol vapor even under thermodynamic equilibrium conditions [14, 64, 65]. Although cooperative effects are sometimes invoked whenever a property (such as a hydrogen bond length) changes from the dimer to larger aggregates [66], a many-body decomposition approach can uncover non-pairwise additive effects more rigorously [67]. The natural cluster size to study cooperativity is a trimer. The total energy of a trimer EABC can be decomposed into monomer energies EA , EB , EC , pair interaction terms VAB ¼ EAB EA EB , VAC ¼ EAC EA EC , VBC ¼ EBC EB EC , and a three-body interaction VABC ¼ EABC
EAB
EAC
EBC þ EA þ EB þ EC
such that EABC ¼ EA þ EB þ EC þ VAB þ VAC þ VBC þ VABC
hydrogen bond dynamics in alcohol clusters
7
Only the effects of the three-body interaction term VABC are truly cooperative effects in a trimer, although properties may of course also change with cluster size in a strictly pairwise additive model, where VABC ¼ 0. The formalism may easily be extended to larger clusters and indeed three-body effects tend to be more important in larger clusters than in trimers [68]. For chain-like or cyclic hydrogen bond patterns between three alcohol molecules A, B, and C, VABC is usually negative (attractive). If molecule B acts as an acceptor for both A and C, VABC is typically repulsive (positive), because A and C compete for the electron density at B [61]. This anti-cooperativity provides the main explanation why branching of hydrogen-bonded chains is discouraged in alcohols. D. Hydrogen Bond Isomerism and Conformational Isomerism For the reasons outlined above, hydrogen bond isomerism in alcohols is less pronounced than it might be on statistical grounds, considering that every acceptor oxygen offers a choice between two lone electron pairs. For ring topologies, there are of course different ways of arranging the alkyl groups already in the trimer and different ways of puckering the ( OH)n ring, starting with the tetramer or pentamer. Like isomerism within the alkyl chain [69], these are conformational choices that leave the classical hydrogen bond pattern intact. Hydrogen bond isomerism is less abundant. Lasso structures [39], in which double acceptor alcohol units come into play, only become competitive when the ring strain has leveled off—that is, for n $ 4. The simple reason is that any molecule that is taken out of the ring makes the cycle smaller and increases ring strain. This penalty adds to the anti-cooperative effect present in double-acceptor centers. Open-chain structures are possibly competitive in small, highly strained clusters and become asymptotically equivalent to rings for n ! 1. The best way to stabilize them for intermediate cluster sizes appears to be the introduction of secondary interactions in the alkyl group. Such a secondary stabilization can be an aromatic substituent [35]. When mixed clusters of alcohols are formed, the issue of donor–acceptor isomerism comes into play [58]. Both alcohols can act as donors and acceptors, but the difference between their donor (QD ) and acceptor (QA ) qualities (QD QA ) will not be the same. The molecule that features the smaller difference will preferentially act as an acceptor. The molecule that has the larger difference will prefer the donor position. If the roles are interchanged, the hydrogen bond strength of the complex decreases, but the structure may still represent a local minimum on the potential energy hypersurface. The determination of donor and acceptor qualities in hydrogen-bonded clusters is not straightforward. Energetic quantities such as binding energies are difficult to attribute to single interaction sites. Vibrational red shifts of the O H stretching fundamental may be more suitable parameters to analyze the donor–acceptor
8
martin a. suhm
preference [58], because they closely correlate to hydrogen bond length and strength [70]. Furthermore, they are experimentally more easily accessible. E.
O H Stretching Dynamics
The infrared (IR) spectrum provides some of the most clear-cut observables for hydrogen bonding phenomena. Alcohol clusters have been studied in most detail in the O H stretching fundamental range. The reasons for this are both technical and scientific. Tunable IR lasers have traditionally been versatile and powerful in the 3-mm window. The effects of hydrogen bonding are also particularly pronounced in this range, as was recognized long ago [71]. Cooperativity and decreasing ring strain induce progressive bathochromic shifts with cluster size [16, 35]. The square of the O H stretching transition dipole moment, responsible for IR activity, can be orders of magnitude larger in hydrogen-bonded clusters than in the alcohol monomer. Therefore, even in the absence of size-selectivity and sensitive laser sources, alcohol clusters can be detected and characterized by their O H stretching signature [65]. The bathochromic shift or red shift of the O H oscillator is a sensitive measure of hydrogen bond strength. Its accurate modeling is quite sophisticated, but simple approaches often profit from favorable error compensation [72]. Furthermore, the frequency of an O H oscillator correlates more or less linearly with the length of the O H bond, with a red shift of about 14 cm 1 for a bond length extension by ˚ [63]. In larger clusters, the intrinsic red shift of the individual oscillators 0.001 A is superimposed by coupling effects among the originally degenerate oscillators of the individual alcohol monomers, the so-called Davydov couplings [16]. The strongest red shift is observed for concerted in-phase O H stretching motion of all members of the hydrogen bond cycle. This is an early indicator for concerted hydrogen transfer between the molecules [73], in which the hydrogen-bonded protons switch their chemical bond partners in a cyclic way. The result is an equivalent hydrogen bond pattern running in the opposite direction. The pronounced red shift also reflects cooperativity, because a stretched O H bond has an increased dipole moment, which enhances the intermolecular interaction. Infrared enhancement in the O H stretching fundamental upon hydrogen bond formation can be large, but may be smaller than predicted by traditional quantum chemistry methods [74]. There are few ways to experimentally determine absolute infrared enhancements by hydrogen bond formation [74], because the experimental number density of the clusters usually remains unknown. Instead, theoretical band strengths are often used to estimate the cluster number density [75]. The situation is more favorable if a cluster contains two or more nonequivalent O H groups. In such a case, the intensity ratio between these groups can be determined by direct absorption methods [30, 76]. As a rule, Raman scattering cross sections are less sensitive to hydrogen bonding [16] but show similar qualitative trends.
hydrogen bond dynamics in alcohol clusters
9
The splitting patterns of the degenerate O H oscillators upon cluster formation [77] can be described by a simple model, which is inspired by Hu¨ckel molecular orbital theory [16, 78]. These Davydov splittings reflect a periodic flow of energy among the coupled oscillators. For trimers, its period T is roughly related to the coupling constant W (in cm 1 ) involved, according to T'
1 3cW
where c is the speed of light. Note that the dissipative formula for the lifetime t t'
1 2pcW
has also been invoked in this context [78]. It agrees quite closely with the halfperiod of the oscillation. There are usually further, slower dissipative processes, by which the energy deposited in the O H stretching manifold is redistributed within the alcohol molecules and into the hydrogen bond [21]. If these IVR processes are sufficiently fast and the density of coupling states is high enough, they can be detected as a contribution to the linewidth of the cluster O H stretching band [16]. In any case, the two time–wavenumber relationships listed above are useful qualitative and semiquantitative concepts to translate spectral features into temporal evolutions of a localized wavepacket. For monomers and dimers, where the energy dissipation out of a locally excited O H oscillator is relatively slow, IVR usually has to be detected by timeresolved experiments [21], which can provide further insights into the sequential mechanism. The Davydov coupling constants W may be studied as a function of cluster geometry, cluster size, isotope composition, and alkyl group substitution [16]. They contain valuable information about the nature of the hydrogen bond interaction in alcohol clusters. The mechanism by which the Davydov couplings between initially degenerate O H oscillators arises may be described in different ways. One may interpret it as a through-hydrogen-bond process, similar to classical oscillator coupling through chemical bonds. At the other end, one may interpret it as a purely through-space long-range coupling of the oscillating dipoles. Considering that hydrogen bonds between alcohols are dominated by dipole–dipole interactions, an excitonic dipole–dipole model appears to be adequate [78]. The coupling constant can then be estimated from the geometry and transition dipole moment of the cluster [78]. F.
Isotope and Overtone Effects
A characteristic feature of hydride stretches in general [79] and the O H oscillator in alcohols in particular [80] is its frequency isolation from other
10
martin a. suhm
degrees of freedom. This is an important cause for the relatively slow energy flow out of the O H stretching state, and it invites reduced dimensionality treatments. Even at the harmonic level, this feature can be exploited to predict hydrogenbond-induced red shifts in dimers, where Davydov couplings are small due to the mismatch of zeroth-order frequencies [80]. Beyond the harmonic approximation, the localization of the O H stretching manifold invites experimental overtone studies [81–83] to extract anharmonicity constants and effective harmonic frequencies, which can be directly compared to theoretical predictions [16]. This works well for alcohol monomers, whereas for clusters the dramatic hydrogenbond-induced intensity enhancement is lost in the overtone range due to a cancellation of electrical and mechanical anharmonicity contributions [84, 85] (see Fig. 2). Therefore, overtone vibrations of isolated hydrogen-bonded clusters
Figure 2. Extraction of anharmonicity constants oe xe from the comparison of fundamental OH stretching spectra (center) with overtone spectra (top) and OD spectra (bottom) for the case of jetcooled trifluoroethanol (M) and its most stable dimer conformation, which features a hydrogen bond donor stretching band (Dd ) and an acceptor stretching band (Da ). The deuteration analysis yields slightly different constants than the overtone approach and underestimates the hydrogen bond effect on donor stretching modes [89].
hydrogen bond dynamics in alcohol clusters
11
are rarely observed [86], whereas they have been discussed in condensed phases [87] and have been important in the early days of hydrogen bond spectroscopy [71]. Recently, an alternative approach to O H anharmonicity constants based on deuteration effects on the spectrum has been proposed [16, 88]. It is not as accurate as the overtone approach, because mode mixing is more likely for OD stretching modes and the constants are quite sensitive to this mixing [16]. However, it has provided first insights into the evolution of anharmonicity constants with cluster size. For methanol, it was shown in this way that dimerization leaves anharmonicity fairly unaffected, whereas anharmonic effects increase in the cyclic clusters [16]. This explains in part why harmonic predictions of bathochromic dimer shifts have been so successful in the past. However, very recent overtone measurements in supersonic jets [89] (see Fig. 2) indicate that the deuteration approach may indeed be of limited accuracy for the donor vibration in alcohol dimers. Deuteration can also strengthen hydrogen bonds [90–92]. This is a zeropoint energy effect. The high-frequency libration and torsion modes of the O H group [93] decrease in energy, when the hydrogen atom is replaced by deuterium. In the monomers, the corresponding modes either have no (in the case of external rotations) or only little (in the case of internal rotations) zeropoint energy. Therefore, the torsional isotope effect is much smaller for monomers. The net effect is an increase in dimer binding energy upon deuteration. There is a counteracting effect from the O H stretching mode itself. Due to the red shift in the complex, its zero point energy is smaller in the hydrogen-bonded form than in the isolated molecule. Therefore, the effect of deuteration is larger in the monomer than in the cluster. For hydrogen bonds between alcohol molecules, this counteracting effect is usually smaller than the librational contribution. As a net effect, deuteration strengthens the hydrogen bonds in alcohol clusters. G.
C O and C H Stretching Dynamics
Although it is only a secondary effect, the influence of hydrogen bonding on the C O stretching dynamics has received a lot of attention [94, 95]. This also has technical reasons, because the C O stretching modes fall into the CO2 laser range for several alcohols. Size selectivity is usually needed, because the contributions from different small clusters tend to overlap [96]. If available, structural details can be extracted from the coupling of the oscillators located on the individual monomers [94, 97, 98]. However, the C O stretching mode is less isolated from other normal vibrations than the O H stretching mode. Unusual evolutions of its frequency with cluster size or aggregation state may thus be due to mode mixings [65]. Even the dynamics of C H stretching modes can reflect the hydrogen bond status of alcohol molecules [99, 100], in particular the local hydrogen bond
12
martin a. suhm
topology involved. Because the effects are considerably smaller than the O H shifts and the monomer tends to dominate supersonic jet expansions, sizeselective techniques are typically required to detect such spectroscopic cluster signatures. In favorable cases [30], cluster C H absorptions can also be detected by direct absorption techniques on the slope of the dominant monomer bands. H.
Torsional Dynamics
A primary effect of hydrogen bonding is observed for the torsional dynamics of the O H group. Torsion is orthogonal to the C O H bending mode and can be described as either (a) a hindered rotation of the entire alcohol molecule out of the hydrogen bond constraint or (b) a twisting motion around the C O bond within the monomer. In the first case, the alkyl group moves in a conrotatory way with respect to the O H group, whereas its motion is disrotatory in the second case. As a third possibility, the torsional motion in the cluster may be decoupled from the alkyl group motion, corresponding to a mixture of the monomer rotation and monomer torsion limits (see Fig. 3). Torsion around C C and C O bonds connects different alcohol isomers. The analysis of interactions between torsional states which are concentrated in different torsional wells can provide important information on energy differences between conformations [101, 102]. Conformational isomerism in alcohols is so subtle that it cannot be easily separated from intermolecular influences in
Torsion
Libration
Rotation Figure 3. Librational OH modes in hydrogen-bonded alcohol clusters may be correlated with overall rotation (bottom left) and torsion (top left) of the monomer (illustrated for methanol), but methyl rotation is actually decoupled from OH torsion by hydrogen bonding. Note that the wavenumbers of monomer rotation (' 4 cm 1 ) and torsion (' 280 cm 1 ) are much lower than that of the cluster libration (' 600 cm 1 ) [93].
hydrogen bond dynamics in alcohol clusters
13
condensed phases. Even the delicate interactions in rare gas matrices can overturn the intrinsic preference of the isolated molecule or molecular pair [80, 103]. Therefore low-temperature vacuum-isolated molecules are imperative in this field, if a reliable characterization of the unperturbed energy sequence of torsional isomers is sought. On the other hand, laser-induced torsional isomerization processes are much easier to study in cryogenic matrices [103, 104]. A deeper understanding of the dynamics among such torsional states can also contribute to the design of Brownian molecular machines [105]. I.
Tunneling Dynamics
Several motions in alcohol clusters involve barriers that may be overcome by tunneling rather than by classical over-the-barrier motion. The concerted proton exchange mode between different alcohol molecules has already been mentioned and still remains to be detected in the O H stretching spectrum of methanol tetramer, where it should be accelerated compared to the vibrational ground state [106]. Methyl torsions in the alkyl groups [107] also belong to this category and may be accelerated or decelerated by the hydrogen bond interaction. Heavy atom tunneling of entire alkyl groups between the different sides of the hydrogen bonded ring plane is much less likely even for methanol, in contrast to the analogous but lighter water case [108]. These motions correspond to hindered rotation of the monomer and are slowed down considerably in the complex due to the librational constraints. Finally, there can be rather large torsional tunneling splittings due to O H torsion in the monomers (see the previous section), which are likely to be quenched almost completely by the intermolecular hydrogen bond. These tunneling processes in the monomers can themselves be affected by weaker intramolecular hydrogen bond interactions, such as C H ! ! !O contacts. J. Chirality Recognition Chirality or handedness is an important aspect for most organic molecules. In the case of alcohols, chirality is usually introduced by chemically or isotopically [109] different substituents at the a-C or a more distant site in the alkyl backbone. When two chiral molecules interact via alcoholic hydrogen bonds, their relative chirality will influence the interaction energy and the vibrational spectrum. This is an intermolecular variant of diastereoisomerism [110]. Dimers formed by two monomers of the same handedness are distinguishable from those of opposite handedness in the case of two constitutionally identical monomers. Beyond this special case, which has been denoted as a ‘‘molecular handshake’’ [111], chirality recognition may of course also occur between two different chiral alcohols [112, 113], where some convention [114] has to be used to define like and unlike partners. Depending on how close the chirality centers are to the O H groups and how well the organic rests can accommodate the hydrogen bond
14
martin a. suhm
constraint, the observed differences between the two diastereomers may be more or less pronounced. For dimers of monofunctional alcohols, which are held together by nothing but one strong contact, only small differences are expected and are indeed found in the vibrational spectrum [115], whereas the differences in the UV spectrum are larger [112]. In the room temperature liquid, the differences are usually negligible in such a case [116]. Microwave spectroscopy is particularly well-suited to detect structural and spectral discrepancies between the two kinds of molecular pairs [117]. It is sensitive to the more subtle dispersion- and Pauli repulsion-like secondary interactions that convey the chirality information in the case of a single hydrogen bond contact [54]. In addition to optical spectroscopy, mass-spectrometric techniques are also useful to unravel chirality recognition effects in the gas phase [118]. One may use the stronger term chirality discrimination when a substantial suppression of one intermolecular diastereomer with respect to the other occurs. This requires multiple strong interactions between the two molecular units and therefore more than simple monofunctional alcohols. Some examples where one of the molecules involved is a chiral alkanol are reported in Refs. 112 and 119–121. Pronounced cases of higher-order chirality discrimination have been observed in clusters of hydroxyesters such as methyl lactate tetramers [122] and in protonated serine octamers [15, 123, 124]. The presence of an alcohol functionality appears to be favorable for accentuated chirality discrimination phenomena even in these complex systems [113, 123, 125, 126]. Because the border between chirality recognition and discrimination is quite undefined, it is suggested that the two may be used synonymously whenever both molecular partners are permanently chiral [127]. Even achiral alcohols may be involved in chirality recognition events by switching between labile enantiomeric conformations, depending on the permanent handedness of the binding partner. An example of such a chirality induction event involves the interaction of ethanol with a permanently chiral ether [128] and is illustrated in Fig. 4. Again, the chirality recognition is so weak that microwave spectroscopy is typically required to disentangle the different variants. Chirality induction is very important in organic synthesis, where chiral catalysts are used to favor the formation of one enantiomer over the other. An intramolecular variant, where the permanently chiral center of a chlorinated alcohol leads to a helicity preference in the intramolecular hydrogen bond conformation, was also studied by microwave spectroscopy [129]. If both alcohol monomers forming a dimer are on average achiral, one may still have chirality synchronization events, where the two monomers match their transient chiral conformations when they bind to each other. A particularly simple example is that of ethanol dimer, where the lowest-energy conformer involves two gauche monomers of the same helicity [80, 91]. However, the energy difference to other conformers is so small that efficient isomerizing collisions in a supersonic jet expansion are required to favor the lowest-energy form over the others. A more
hydrogen bond dynamics in alcohol clusters
15
Figure 4. Symmetry breaking of the ethanol torsion potential (top, two gauche and one trans conformation) by interaction with a chiral acceptor molecule (dimethyl oxirane, bottom), in this case RR trans-2,3-dimethyloxirane [128]. Note that trans ethanol is less stable in the complex and that the two gauche (g) forms differ in energy.
pronounced example for chirality synchronization was found in trifluoroethanol [30, 76]. Here, the energy difference between a homoconformational and a heteroconformational dimer is predicted to be small, but they have a different hydrogen bond topology and only one of them is formed in significant amounts in a jet expansion [30, 76]. The intermediate case of fluoroethanol shows evidence for chirality recognition among the four dimer conformations that are observed in the spectrum, but no substantial chirality synchronization [130]. Chirality recognition phenomena in clusters involving no, one, two, or more permanently chiral constituents have recently been summarized [127]. K.
Cation Solvation
Alcohols are important solvents for ions, and the study of solute–solvent clusters promises to provide insights into the solvation process. The solvation of protons by methanol [47], ethanol [131], and higher alcohols [132] has been studied in detail and leads to interesting hydrogen bond topologies [133]. Solvation of larger cations by alcohol molecules has been investigated for many years and may help in understanding the essentials of ion–water interactions [134].
16
martin a. suhm L.
Anion Solvation
Anion solvation in alcohol clusters has been studied extensively (see Refs. 135 and 136 and references cited therein). Among the anions that can be solvated by alcohols, the free electron is certainly the most exotic one. It can be attached to neutral alcohol clusters [137], or a sodium atom picked up by the cluster may dissociate into a sodium cation and a more or less solvated electron [48]. Solvation of the electron by alcohols may help in understanding the classical solvent ammonia and the more related and reactive solvent water [138]. By studying molecules with amine and alcohol functionalities [139] one may hope to unravel the essential differences between O- and N-solvents. One should note that dissociative electron attachment processes become more facile with an increasing number of O H groups in the molecule [140]. III.
EXPERIMENTAL METHODS
As became obvious in the preceding section, progress in understanding alcohol clusters very much depends on the ability to generate these clusters in supersonic jet expansions or in other variants of low temperature isolation and to detect their dynamics via spectroscopic methods. Therefore, some important spectroscopic tools employed in this field shall be summarized, with focus on the alcoholic systems that have been addressed by them. Solution [22, 26, 141, 142] and supercritical [24–26] state techniques will not be covered systematically. A.
Microwave Spectroscopy
Microwave spectroscopy is probably the ultimate tool to study small alcohol clusters in vacuum isolation. With the help of isotope substitution and auxiliary quantum chemical calculations, it provides structural insights and quantitative bond parameters for alcohol clusters [117, 143]. The methyl rotors that are omnipresent in organic alcohols complicate the analysis, so that not many alcohol clusters have been studied with this technique and its higher-frequency variants. The studied systems include methanol dimer [143], ethanol dimer [91], butan-2-ol dimer [117], and mixed dimers such as propylene oxide with ethanol [144]. The study of alcohol monomers with intramolecular hydrogen-bond-like interactions [102, 110, 129, 145–147] must be mentioned in this context. In a broader sense, this also applies to isolated n-alkanols, where a weak Cg H ! ! ! O hydrogen bond stabilizes certain conformations [69, 102]. Microwave techniques can also be used to unravel the information contained in the IR spectrum of clusters with high sensitivity [148]. Furthermore, high-resolution UV spectroscopy can provide accurate structural information in suitable systems [149, 150] and thus complement microwave spectroscopy.
hydrogen bond dynamics in alcohol clusters B.
17
Infrared Absorption
Infrared spectroscopy is the workhorse in this field, because it can quickly provide dynamical details, discriminate between different cluster sizes and phases [40], and sample a wide spectral range. It often yields valuable feedback for quantum chemical calculations. In contrast to some action spectroscopy techniques, IR absorption spectroscopy is not intrinsically size-selective. All cluster sizes generated in the expansion are observed together, and indirect methods of size assignment are needed. For the study of alcohol clusters, direct absorption in supersonic jets [151] is particularly powerful. The best compromise between sensitivity, spectral resolution, spectral coverage, reproducibility, and simplicity is probably achieved by the synchronization of pulsed slit jet expansions with FTIR scans [65, 152–154]. One realization is illustrated in Fig. 5. Because this technique uses incoherent light sources, it works best for the large cross sections of hydrogen-bonded O H stretching fundamentals, but it has also been applied to overtones [89], framework vibrations [65, 155], and even intermolecular modes [93, 156]. Where spectral resolution is more important than spectral coverage, cavity-ring-down laser absorption spectroscopy is a competitive technique [75, 157]. While high spectral resolution can be achieved by both FTIR [158] and laser absorption methods [75, 157], the sensitivity drops significantly with increasing resolution in the FTIR case. Anyway, high spectral resolution is most useful for small monomers and possibly its dimers [158, 159], but not for larger organic clusters.
Figure 5. Schematic drawing of a high-throughput pulsed slit jet FTIR setup involving a 600mm nozzle that is synchronized to the interferometer scans [154].
18
martin a. suhm
Infrared absorption spectroscopy is also a powerful tool for matrix isolation studies, which have been carried out extensively for alcohol clusters [34, 88, 103]. Recently, the gap between vacuum and matrix isolation techniques for direct absorption spectroscopy has been closed by the study of nano-matrices— that is, Ar-coated clusters of alcohols [80]. Furthermore, alcohol clusters can be isolated in liquid He nanodroplets, where metastable conformations may be trapped [160]. C.
Raman Scattering
Whenever symmetry or quasi-symmetry plays a role, particularly in cyclic alcohol clusters, spontaneous Raman scattering off supersonic jet expansions provides valuable complementary information on the cluster dynamics [16, 77]. Even for nonsymmetric clusters and monomer conformations, it may be used to complement IR spectroscopy, because the Raman rovibrational selection rules often ensure a more narrow band profile [69, 161]. The applicability of spontaneous Raman scattering to jet-generated hydrogen-bonded clusters is a very recent advance [77], whereas it is more established for the characterization of jet expansions in general [162] and complexes of simple molecules [163, 164]. For molecules with a suitable UV chromophore, more sensitive stimulated techniques coupled with ionization or fluorescence [165, 166] can be applied. By using VUV radiation, these techniques may also be useful for aliphatic alcohols [167], although fragmentation issues have to be addressed in detail. The nonlinear CARS technique is more widely applicable [168, 169], but it suffers from the dependence on the square of the molecule density in jet applications. In condensed phases, the noncoincidence effect between IR and Raman spectra provides insights into the intermolecular coupling [170, 171]. The combination of IR and Raman spectroscopy is also useful in the study of alcohol clusters in the supercritical state [25]. D.
Crossed Beam Techniques
Cluster size assignment is often a challenge for direct absorption or light scattering techniques that do not provide reliable mass information. For dimers, a combination of pressure or concentration dependence and spectral survey is usually unambiguous. In favorable cases, this also applies to trimers, tetramers, and maybe pentamers. Beyond that size, some kind of mass information is required. Aliphatic alcohols and their clusters suffer from heavy fragmentation upon ionization. This is also the case for close-to-threshold ionization [172]. If there is no suitable aromatic chromophore for soft ionization, the most important technique involves deflection of the clusters by a crossed rare gas beam [96, 98, 173, 174]. Using this technique, one can bracket the cluster size from above based on the deflection angle and from below based on the largest fragment that is observed.
hydrogen bond dynamics in alcohol clusters E.
19
UV–IR Coupling
Once the alcohol or at least the cluster contains a soft ionization or fluorescence chromophore, a wide range of experimental tools opens up. Experimental methods for hydrogen-bonded aromatic clusters have been reviewed before [3, 19, 175]. Fluorescence can sometimes behave erratically with cluster size [176], and short lifetimes may require ultrafast detection techniques [177]. However, the techniques are very powerful and versatile in the study of alcohol clusters. Aromatic homologs of ethanol and propanol have been studied in this way [35, 120, 121, 178, 179]. By comparison to the corresponding nonaromatic systems [69], the O H ! ! ! p interaction can be unraveled and contrasted to that of O H ! ! !F contacts [30]. Attachment of nonfunctional aromatic molecules to nonaromatic alcohols and their clusters can induce characteristic switches in hydrogen bond topology [180], like aromatic side chains [36]. Nevertheless, it is a powerful tool for the sizeselected study of alcohol clusters. In addition, there is a large number of studies involving aromatic alcohols such as phenol [166] or naphthol, which have in part been reviewed before [21]. These include time-resolved studies [21], proton transfer models [181], and intermolecular vibrations via dispersed fluorescence [182]. Such doubleresonance and more recently even triple-resonance studies [183] provide important frequency- and time-domain insights into the dynamics of aromatic alcohols, which are not yet possible for aliphatic alcohols. F.
VUV–IR Coupling
In principle, UV–IR coupling becomes more generally applicable at short UV wavelengths, where even aliphatic alcohols and their clusters absorb photons. In a series of recent papers [172, 184, 185], this approach has been explored. For alcohol monomers, the VUV laser photon alone is either not or just barely able to ionize the molecule. Vibrational excitation of the molecule by a preceding IR laser opens either the ionization channel or additional ion fragmentation channels. For hydrogen-bonded alcohol dimers, the employed VUV excitation is typically above the ionization threshold, but the preceding IR laser can induce dissociation of the dimer. This process depletes the ion signal, and the wavenumber dependence of the depletion is interpreted as the vibrational spectrum. As will be discussed later on, severe spectral distortions including spectral broadening, bidirectional signals, and band shifts arise and tend to become worse for larger clusters. Comparison to direct absorption results can shed light onto the underlying mechanisms of the complex high-energy processes. Even the spectroscopy of strongly bound core electrons reveals some sensitivity to the hydrogen bond interaction in alcohols [186], but the sensitivity currently restricts the study to fairly large clusters.
20
martin a. suhm IV.
COMPUTATIONAL METHODS
Computational methods that assist the characterization of alcohol cluster dynamics are essential, numerous, and diverse. Here, we can only briefly mention some of them that turn out to be particularly useful for the analysis presented below. Where available, we refer to authoritative reviews on these subjects. A.
Empirical Force Fields
If the investigated systems are large, the accuracy requirements low, and the computational speed demands high, there is no alternative to empirical force fields. The spectroscopic study of small alcohol clusters appears quite orthogonal to these constraints, because the systems are comparatively small, accurate spectra are required, and relatively few dimensions are expected to contribute to the dynamics. Nevertheless, simple pairwise additive empirical force fields [187] can be re-parameterized to become useful for spectroscopic purposes [72]. To avoid individual solutions for every alcohol system, transferable force fields are to be favored [188]. Given a good-quality force field, a range of classical and quantum nuclear dynamics techniques can be applied to extract the spectra (vide infra). However, the limits of such approaches are obvious, and they are most useful when interpolating between experimental data. This remains partially true for more sophisticated force fields, even if intra- and intermolecular degrees of freedom are coupled [189]. Therefore, empirical force fields are more frequently applied to the simulation of liquid structure and dynamics and to biomolecule–solvent systems, where they have become invaluable [190]. B.
Electronic Structure Calculations
Potential energy hypersurfaces can be generated pointwise by resorting to a plethora of approximations to the solution of the electronic Schro¨dinger equation. In the early days of hydrogen-bonded cluster investigations, the Hartree–Fock (HF) method was the only useful ab initio approach available. Its deficiencies due to the neglect of electron correlation are now well known. In particular, the HF level does not recover the full hydrogen bond energy and vibrational shift. However, fortunate error compensation by basis set superposition errors has often been exploited to provide a reasonable and computationally economic description of some aspects of the hydrogen bond dynamics in alcohol clusters [39, 191]. Considering the current progress in local correlation methods [192], this may not be needed in the future. An inexpensive access to electron correlation is provided by density functional and hybrid functional techniques. A range of these techniques is implemented in quantum chemistry packages such as Gaussian [193]. They
hydrogen bond dynamics in alcohol clusters
21
describe classical hydrogen bond interactions including cooperativity effects reasonably well, in particular in their hybrid variants which include some HF exchange [194]. Therefore, their application to alcohol clusters is very popular [195–197]. Spectroscopic data can be used to explore their limits. For larger alcohols, layered approaches such as ONIOM extend the applicability [38]. However, the fundamental inability of current functionals to describe dispersion forces becomes more and more critical with system size. In this situation, (semi) empirically dispersion-augmented approaches can be of some use [198]. A more rigorous, but also more expensive, approach is Møller–Plesset perturbation theory. Usually, the second-order (MP2) level is employed, but it requires significantly larger basis sets for convergence than Hartree–Fock or density functional approaches. In the prediction of frequency shifts upon hydrogen bond formation, there can be large changes from second to fourth order, with the MP4 results in better agreement with experiment [199]. If affordable, there is a range of very accurate coupled-cluster and symmetryadapted perturbation theories available which can approach spectroscopic accuracy [57, 200, 201]. However, these are only applicable to the smallest alcohol cluster systems using currently available computational resources. Nearlinear scaling algorithms [192] and explicit correlation methods [57] promise to extend the applicability range considerably. Furthermore, benchmark results for small systems can guide both experimentalists and theoreticians in the characterization of larger molecular assemblies. Spectroscopic applications usually require us to go beyond single-point electronic energy calculations or structure optimizations. Scans of the potential energy hypersurface or at least Taylor expansions around stationary points are needed to extract nuclear dynamics information. If spectral intensity information is required, dipole moment or polarizability hypersurfaces [202] have to be developed as well. If multiple relevant minima exist on the potential energy hypersurface, efficient methods to explore them are needed [203, 204]. C.
Internuclear Dynamics
Within the Born–Oppenhimer approximation, the electronic structure is permanently optimized while the dynamics of the nuclei evolves. The latter thus happens on multidimensional potential energy hypersurfaces which are either defined empirically or else derived from solutions of the electronic Schro¨dinger equation, as outlined above. In the latter case, they may be calculated on the fly or represented on low-dimensional grids or else they have to be approximated by analytical expressions, not unlike the approach employed for empirical force fields. All these strategies are either laborious or computationally demanding and far from routine usage. Except for on-the-fly and to some extent grid methods, they are also highly specific to a given system and cannot be automated easily.
22
martin a. suhm
Therefore, a drastic simplification, the so-called double harmonic approximation, is very popular in theoretical cluster spectroscopy. The dependence of the restoring force and of the electric dipole moment on the vibrational displacement from a minimum structure is assumed to be strictly linear, and the resulting linear system of equations is diagonalized to yield harmonic fundamental vibrations (normal modes) and intensities. Analytical derivative techniques render this approximation very efficient from a quantum chemical point of view. Deviations from real spectra are dealt with by scaling approaches, by looking at differences between monomer and cluster fundamentals, and by other error compensation tools. Obviously, interesting dynamical phenomena such as overtone transitions, combination bands, Fermi resonances, vibrational Franck–Condon patterns, torsional modes, tunneling splittings, and other anharmonic effects are not captured by such an approach. Nevertheless, it often provides a useful zero-order picture of the dynamics, unless multiple minima separated by low potential barriers are involved. Often, one does not need the entire set of normal modes but rather only a small subset. In this case, selective algorithms such as mode-tracking can be helpful [122, 205]. For the highly localized O H stretching vibrations in alcohols, one can even restrict the normal mode analysis to one or a few local modes [80]. Starting from the normal mode approximation, one can introduce anharmonicity in different ways. Anharmonic perturbation theory [206] and local mode models [204] may be useful in some cases, where anharmonic effects are small or mostly diagonal. Vibrational self-consistent-field and configuration-interaction treatments [207, 208] can also be powerful and offer a hierarchy of approximation levels. Even more rigorous multidimensional treatments include variational calculations [209], diffusion quantum Monte Carlo, and time-dependent Hartree approaches [210]. Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born–Oppenheimer or fictitious Car–Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. Sometimes, it is claimed that the finite temperature in classical simulations accounts for anharmonicity. This may be coincidentally true for nearly
hydrogen bond dynamics in alcohol clusters
23
harmonic low-frequency vibrations at environmental temperatures. However, classical molecular dynamics cannot account for the anharmonicity of a highfrequency oscillator like the O H stretching mode in an alcohol, at least not using reasonable temperatures. To recover the fundamental frequency of such an oscillator, a simulation temperature of more than 5000 K would indeed be needed. V.
SYSTEMS
The alcohols and their clusters will be discussed in order of increasing chemical complexity. With growing complexity, more and more of the issues discussed in Section II come into play and can be addressed by the experimental methods outlined in Section III in combination with computational approaches such as those mentioned in Section IV. A.
Methanol
For organic hydrogen bonds, methanol takes the role that HF has for inorganic hydrogen bonds—it is the simplest conceivable prototype. Its cluster spectroscopy has been reviewed together with that of water clusters [98]. While the monomer vibrational dynamics is in general well-studied [214–217], different values for the fundamental O H stretching band center are in use [63, 64, 75, 173, 189, 218]. Based on combined Raman and IR evidence, a value of 3684– 3686 cm 1 appears well-justified [16, 65, 77, 82, 216]. It serves as an important reference for vibrational red shifts in methanol clusters. The methanol dimer is structurally well-characterized [143]. It features a clear distinction between hydrogen bond donor and acceptor O H groups. The acceptor band is only slightly shifted to lower wavenumber, relative to the free monomer [75, 77]. It coincides with a monomer transition from the excited methyl rotor tunneling state. The donor band is shifted by 111 cm 1 to the red. Harmonic predictions of this red shift are much larger, even at fairly high levels of electronic structure treatment [16]. This may be due to higher-order electron correlation and anharmonic effects [199]. Upon deuteration of the bridging proton, the shift reduces to 80 cm 1 , again much less than the best harmonic predictions. Even deuteration of the free O H group affects the red shift of the hydrogen-bonded O H noticeably, although the two oscillators are welldecoupled [58]. This confirms the high sensitivity of vibrational frequency shifts to details of the hydrogen bond environment. The methanol trimer is arguably one of the most interesting clusters. The unexpected structure in its O H stretching spectrum [65, 75, 77, 173] has only recently found a consistent explanation [16]. It is not related to structural isomers [64, 75, 195, 219, 220] but rather to simultaneous excitation and de-excitation of low-frequency methyl umbrella modes [16, 65], that is, a
24
martin a. suhm
Figure 6. The complex OH stretching spectrum of methanol trimer (bottom) can be explained by sum (nS ), difference (nD ), and hot bands (nH ) involving the OH fundamental (nF ) and two umbrella modes of the methyl groups, which are nearly degenerate in the ground state but soften and split after OH stretching excitation. nR is the predominantly Raman active concerted stretching mode [16].
vibrational Franck–Condon effect (see Fig. 6). It is now clear that methanol trimers in free jets only occur in a cyclic, chiral [5] structure, in which two methyl groups point above and one below the hydrogen-bonded plane. This results in two nearly degenerate, strongly IR-active O H stretching modes and
hydrogen bond dynamics in alcohol clusters
25
a very weak but strongly Raman-active in-phase stretching mode [77], similar to the phenol case [166]. Its assignment permits to quantify the average O H oscillator coupling matrix element around 20 cm 1 [16]. In contrast to the unsymmetric trimer, the cyclic tetramer allows for an alternating arrangement of the methyl groups, as one would expect it in a simple lone-pair orbital picture (Fig. 1). The S4 -symmetric structure leads to a symmetric double minimum potential for concerted fourfold proton transfer. This proton transfer has been predicted to be accelerated substantially by symmetric O H stretching excitation. The predicted tunneling splitting in a reduced dimensionality treatment [106] is on the order of 1 cm 1 . While this prediction is still uncertain, it presents a challenge for Raman spectroscopic measurements. Currently [16], the upper experimental limit is 7 cm 1 , in good agreement. Better cooling of the clusters in the Raman jet experiment may help in tightening this experimental bound, but IVR processes in which the energy flows out of the O H stretching manifold could prevent the detection of a splitting. Energy redistribution within the O H stretching manifold is characterized by a nearest-neighbor coupling constant of 30 cm 1 and a second-nearest neighbor coupling of 10 cm 1 , that is, it happens on a picosecond time scale [16]. The preparation of single isomers for methanol dimer, trimer, and presumably tetramer [16] in a supersonic jet expansion contrasts the structural diversity that can be prepared and manipulated in cryogenic matrices [34]. It underscores the ability of supersonic jet expansions to funnel all intermolecular isomers down to the global minimum, if there are no major barriers to overcome on the way. O H stretching bands of larger methanol clusters start to overlap. Investigations on their dynamics and isomerism [160, 196, 197] typically require size-resolved studies [174]. The recently proposed size-specific VUV-IR technique is not practical for this purpose, because it produces strongly broadened, spectrally shifted and most likely fragmentation-affected bands for methanol trimer and larger clusters [172]. The anharmonicity of the O H stretching oscillators changes with cluster size. For the monomer, the anharmonicity constant is on the order of 90 cm 1 . A coarse deuteration analysis [16, 88] suggests that it increases by more than 20% upon trimer and tetramer formation [16]. More accurate overtone analyses are possible in a rare gas matrix [88], but the matrix shift complicates a direct comparison to theory. As an example, the overtone-deduced anharmonicity of methanol monomer in a nitrogen matrix [88] is 85 cm 1 , whereas in vacuum [16] it is 92 cm 1 . The deuteration-estimated anharmonicity is 91 cm 1 for the monomer and 97 cm 1 for the dimer donor in the nitrogen matrix, whereas it is 87 cm 1 for the monomer and 89 cm 1 for the dimer donor in vacuum. Clearly, only a vacuum overtone measurement would be fully conclusive, but as the matrix study [88]
26
martin a. suhm
shows, this is very challenging for methanol dimer and completely out of reach for trimers. However, calculations also support the idea of a moderate increase of anharmonicity upon deuteration in the case of methanol [199]. Instead of embedding methanol clusters in a bulk Ar matrix, one can decorate the preformed clusters with Ar atoms. This nanocoating is observed for Ar as a carrier gas, if the expansion is lean enough in methanol to reach low temperatures. The sign and size of the nanomatrix shifts reflects the cluster– matrix interaction. In the case of the open dimer, a red shift is observed because the Ar atoms can serve as secondary weak hydrogen bond acceptors. The trimer is actually blue-shifted, supporting its closed cycle structure and suggesting that packing effects dominate the matrix interaction [158]. Instead of nanocoating the preformed methanol clusters with Ar, one can also attach more than one methanol unit to preformed Ar clusters using a molecular beam pickup technique [97]. This can result in different cluster structures, although the available results in the C O stretching region [97] are less straightforward to interpret than O H stretching data. Methanol clusters have indeed been studied using several other intramolecular excitations, such as the O H bending fundamental [88] and the C O stretching mode [65, 98]. Here, the different cluster sizes are not so wellseparated and size resolved methods are helpful. A strong cluster size dependence and mode coupling is observed for intermolecular librational bands, which assisted the assignment of the IR-active libration modes of methanol tetramer in a free jet expansion [93]. They are found to be quite anharmonic. Although these modes correlate with overall rotation and methyl torsion of the monomer, they do not involve significant methyl group motion in the cluster (see also Fig. 3). These individual cluster librational bands are two orders of magnitude more narrow than the overall librational band profile. This shows that the broad librational bands in liquid alcohols are largely due to different hydrogen bond environments, rather than to rapid vibrational energy flow. The librational pattern of the cyclic tetramer spectrum has similarities to that of the extended crystalline solid [40], and the IR intensity of the highfrequency libration band serves as a puckering indicator for the methyl groups (see Fig. 7, band C). The effect of librational and torsional excitation on the O H overtone dynamics in methanol clusters would be of interest, given the influence detected in the methanol monomer [83]. Furthermore, torsional states offer important doorways for vibrational energy flow in alcohols [142]. At the low-frequency end of methanol cluster dynamics, hindered monomer rotations and translations appear in the IR spectrum [156, 221–223]. When methanol interacts with other molecules, there are some characteristic differences to water [36]. For example, methanol can symmetrically bind two HCl molecules to its lone pairs without significant energy penalty compared to a cooperative ring arrangement [61]. This is not the case for water, because its
hydrogen bond dynamics in alcohol clusters
27
Figure 7. Librational infrared spectra of methanol clusters [93] (bands B and C due to the tetramer, broad profile due to large clusters, cluster size increases from bottom to top) compared to the absorptions in amorphous and crystalline (zig-zag) solid methanol [40]. The large clusters compare well to the amorphous solid, whereas the ring tetramer may be viewed as a small model of the zig-zag chains in the crystal. Note that the high-frequency band C acquires IR intensity through puckering of the methyl groups above (u) and below (d) the hydrogen bond plane.
lone pairs are less electron-rich. Some other complexes of methanol are discussed in the following chapters. B.
Ethanol
Ethanol introduces the issue of conformational isomerism around the C O bond (Fig. 4, top). The transiently chiral gauche conformations (gþ and g ) are 0.5 kJ/mol higher in energy than the anti or trans form (t) [101]. While the trans form thus dominates at low temperatures in the isolated molecule, crystalline ethanol consists of alternating gauche and trans conformations arranged in infinite hydrogen-bonded chains [224, 225]. To investigate the influence of
28
martin a. suhm
aggregation on conformational isomerism, hydrogen-bonded ethanol dimer was studied repeatedly. Out of the nine distinguishable dimer conformers in a simple counting scheme involving acceptor lone pairs [65], about six may be expected to be relatively stable [33]. Earlier theoretical studies were not very conclusive concerning the subtle energy sequence of these isomers. For an accurate theoretical description of dimers, it is essential that the differences in monomer conformation are well-described. Direct absorption measurements in He supersonic jet expansions [65, 157] and Raman jet spectroscopy [77] reveal at least three, more likely four, competing dimer conformers even at low temperatures. They can be converted into a single most stable conformer by adding Ar to the expansion as a relaxation promoter [80]. This global minimum conformation is among the isomers with the strongest red shift. Extensive ab initio calculations [80] indicate that it is a homoconfigurational gauche dimer; that is, both ethanol units occur in the less stable gauche conformation and match their helicities. Rewardingly, this conformation also exhibits a strongly red-shifted O H stretching band in the calculations, in line with the experimental correlation between shift and stability [80]. Hence, the conformational preference of the monomer is reversed in the dimer, although the driving forces behind this isomerization only involve weak hydrogen bonds and dispersion interactions, whereas the classical O H ! ! ! O hydrogen bond does not discriminate significantly between the conformations. The subtlety of this conformational isomerism is underscored by the fact that matrix embedding of the dimer recovers the trans conformation for both alcohol units [103]. However, spectra recorded immediately after deposition can resemble the jet spectra quite closely under certain conditions [226]. The three most stable dimer conformations according to the correlated ab initio calculations [80] are also the ones that are observed in a high-resolution microwave study [91]. The ethanol dimer donor band obtained by the nonresonant ion dip IR technique (i.e., VUV–IR coupling [184]) is broad and unstructured. It falls on the slope of an even broader feature that has the opposite sign; that is, the IR excitation enhances ionization instead of depleting it. The fact that the depletion signal is blue-shifted by more than 30 cm 1 from the narrow and well-structured, true direct absorption or Raman band [65, 77, 80, 157], is probably related to this complex phenomenon. The O H stretching spectra of ethanol trimers and larger clusters cannot be conformationally resolved in a slit jet expansion [65, 77, 157]. VUV-IR spectra [184] are even broader, sometimes by an order of magnitude, and band maxima deviate systematically by up to þ50 cm 1 from the direct absorption spectra. We note that ethanol dimers and clusters have also been postulated in dilute aqueous solution and discussed in the context of the density anomaly of water– ethanol mixtures [227]. Recently, we have succeeded in assigning Raman OH stretching band transitions in ethanol!water, ethanol2 !water, and ethanol!water2 near 3550, 3410, and 3430 cm 1 , respectively [228].
hydrogen bond dynamics in alcohol clusters
29
Isolated ethanol clusters have earlier been studied using CO2 lasers and sizeselective action spectroscopy in the C O stretching range [94]. Attaching or surrounding the ethanol dimer with Ar atoms gives rise to vibrational shifts in both spectral ranges [80, 94]. Mixed dimers of ethanol and methanol have been studied as well [58] and show that methanol prefers to act as a hydrogen bond donor. This is in line with the improved acceptor character of ethanol, caused by the inductive effect of the Ca methyl group. In contrast to ethanol dimer, the trans preference of the acceptor molecule is preserved in this mixed complex. The same is true for phenol–ethanol dimer [229], but in both cases the preference is reduced compared to the monomer. In summary, in a simple but subtle case of conformational control, the preference of ethanol for a stretched trans conformation can be attenuated and inverted by offering it a range of donor alcohol molecules. This control will be lost completely at elevated temperatures, where dynamically assembled ethanol dimers are important for the properties of the supercritical state [24]. C.
Linear Alcohols
The dynamical features observed for methanol and ethanol invite an extension to longer alkyl chains, to see whether any new aspects related to the increased conformational freedom come into play. The O H group is a sensitive probe for at least the nearest-neighbor torsional states [230]. This is already true at room temperature and even more so in supersonic jets [65, 69, 157]. While isolated linear alkanes prefer an all-trans conformation if they are not too long, the presence of a terminal O H group induces a gauche conformation along the Ca Cb bond (Gt in propanol; see Fig. 8). The preferred O H orientation relative to the alkyl backbone remains trans, like in ethanol. This is consistent with the relaxation behavior of n-propanol and longer-chain alkanols in supersonic jet expansions [69]. Van der Waals interactions of the oxygen atom with Cg H have been postulated as a reason [69, 231]. However, the alltrans conformation is also quite low in energy [102]. These robust experimental findings encourage accurate quantum chemical studies of the conformational landscape of n-alcohols, including higher order electron correlation [69, 232]. Once an accurate description of the potential energy hypersurface is achieved, zero-point energy contributions must be addressed. It remains to be seen whether a consistent picture can be achieved at the harmonic level or whether anharmonic contributions are important in the torsional subspace. In order to obtain robust conformational assignments from vibrational spectra without rotational resolution, it is important to predict reliable monomer frequency shifts between conformations. Harmonic B3LYP predictions were shown to correlate reasonably well with experiment [69], and simple rules based on repulsive and attractive intra-monomer interactions were developed. However, the predicting power of the B3LYP method for the energy sequence
30
martin a. suhm
Figure 8. IR and Raman OH stretching spectra of n-propanol monomers and dimers reflecting conformational diversity. The Raman spectrum reveals the dominance of the internally hydrogenbonded Gt monomer most clearly, whereas the IR spectrum indicates more than five different dimer conformations in the red-shifted dimer spectrum [69].
is much inferior [69]. For a spectral separation of the different conformations, Raman spectroscopy proves to be powerful, because the spectra are dominated by narrow Q-branches. Future improvements of the Raman jet setup [77] will also allow for Ar relaxation experiments. Currently, a combination of IR and Raman experiments is most conclusive in the monomer regime. Beyond ethanol, the number of n-alkanol dimer conformations becomes too large to be vibrationally resolved, even in supersonic jets. For n-propanol, more than five isomers are discernible in the donor O H stretching spectrum (see Fig. 8). For longer chains, there is a smaller number of dominant conformations [69]. Ar relaxation shows that the most stable n-propanol and n-butanol dimers are those with the largest observed red shifts. For longer chains, the situation is more complex. However, the window of observed O H stretching bands is quite independent of chain length beyond propanol. A subtle chain-length alternation effect is observed upon coating the alcohol molecules and dimers with Ar. For monomers, the O H stretching frequency of even-membered chains is perturbed more strongly than that of odd-membered chains. For dimers, the opposite pattern is observed [69]. The modeling of such
hydrogen bond dynamics in alcohol clusters
31
subtle embedding effects is challenging [233]. However, it is important, because isomerization studies among the different conformers are most elegantly carried out in cryogenic matrices [103]. Chain length and chain conformation can also influence the energy flow out of the O H oscillator into the rest of the molecule or molecular cluster [21]. D. Bulky Alcohols Branching of the alkyl chain attached to the O H group opens up a variety of perspectives. It increases the bulkiness of the alcohol with consequences for aggregation [234, 235], it allows for the introduction of permanent chirality [117], and it can help to introduce larger energy differences among conformations. With respect to bulkiness, t-butyl alcohol is the simplest monoconformational example. It is therefore investigated intensely and used as a model system for amphiphilic behavior [236]. t-Butyl alcohol is thought to form micellar structures or microscopic aggregates in the liquid [13], in aqueous solution [237] and also in the supercritical state [25]. Its crystal structures are diverse and complex [44, 238]. Therefore, the investigation of isolated t-butyl alcohol clusters is of some interest. Their O H stretching infrared spectrum has been characterized [39] and is found to exhibit unusual structure even for larger clusters. The origin of this structured spectrum, which falls in a region of C H stretch–torsion combinations, remains to be understood in detail. Large cluster formation still appears to be feasible, although the bulky t-butyl groups certainly interfere with each other and the tetramer may be viewed as a kind of magic number cluster [213]. This is confirmed by a study of mixed clusters of naphthol and t-butyl alcohol [235]. Clusters of even more bulky alcohols such as adamantanols and multiply branched alcohols have been studied [65, 72]. In contrast to earlier evidence around room temperature [37], the low-temperature environment appears to be able to stabilize aggregates beyond the dimer [65]. Crystal structure determination even reveals tetrahedral arrangements of the O H groups in extremely bulky alcohol tetramers, although the proton positions are disordered [239]. The hydrogen-bond-induced red shift in branched alcohol dimers can be predicted using a fairly simple molecular modeling approach, as long as conformational isomerism does not complicate the picture [72]. With respect to chirality, 2-butanol is the simplest alcoholic model system. Due to a single strong recognition center (the hydrogen-bonded O H group), the energy differences between a dimer built from homoconfigured (e.g., lefthanded) monomers and a dimer between heterocofigured (mixed left- and righthanded) molecules are not very large. Spectral differences in the IR are therefore quite small, whereas conformations of the two diastereomeric dimers can be easily told apart in microwave studies of chirality recognition [117].
32
martin a. suhm E.
Unsaturated and Aromatic Alcohols
The simplest unsaturated alcohol with sp3 -carbon O H is allyl alcohol (propenol). The monomer occurs in two energetically similar conformations in the gas phase [145, 240], which are both stabilized by intramolecular O H ! ! !p interactions. The dimer has only been studied in matrix isolation [241]. Spectroscopic evidence for an intermolecular O H ! ! ! p hydrogen bond was found. A vibrational (IR and Raman) supersonic jet measurement would be able to unravel the different monomer and dimer conformations involved. Among the unsaturated alcohols involving an O H group attached to a sp2 carbon, only the resonance stabilized case of malonaldehyde shall be briefly discussed. It features a strong intramolecular hydrogen bond between the enol O H and the remaining aldehydic C O group. The hydrogen atom is bound in a double-minimum potential, and the resulting tunneling splitting [210, 242] has recently been studied as a function of vibrational excitation [155, 243]. Remarkable decreases of the tunneling splitting, equivalent to increases in the tunneling period, have been found for some vibrations [155]. Due to the strong intramolecular hydrogen bond, the clustering tendency of malonaldehyde is small, unless one studies excited conformations [104]. The same applies to tropolone [244]. Aromatic alcohol clusters have been well-studied, also for methodical reasons. The UV chromophore can be exploited for sensitive detection of the IR spectrum [35, 36, 120, 179]. Time-domain experiments become possible [21], which show that the initial energy flow out of the O H stretching mode occurs primarily via C H stretching and bending doorway states. Like in the case of carboxylic acid dimers [245], the role of the hydrogen bond is to shift the O H stretching mode closer to these doorway states and thus to accelerate the initial energy flow. In terms of binding energy, classical hydrogen bonding may be enhanced by the increased acidity of the phenolic O H, if the acceptor alcohol is nonphenolic [149, 182]. On the other hand, p interactions compete with classical hydrogen bonds [19, 54] and may even dominate the interaction, such as in the dimer of 1-naphthol [55]. This is not yet the case for phenol dimer, the prototype compound in this class [246, 247]. F.
Fluoroalcohols
The hydrogen bonds in aliphatic alcohol clusters can be modified in a systematic, yet subtle, way by replacing hydrogen atoms of the alkyl group by fluorine atoms [248, 249]. This leads to only modest changes in spatial extension, but it introduces polarity into the hydrophobic alkyl chains. Despite their polarity, the fluorine atoms are not considered to be attractive hydrogen bond acceptors [250]. Fluorinated alkanes have quite remarkable properties that can be related to this combination of polarity and weak hydrogen bond propensity. Alcohols with
hydrogen bond dynamics in alcohol clusters
33
fluorine atoms at the a-carbon are usually not stable with respect to HF elimination [251, 252]. Therefore, fluorinated ethanols are the simplest stable model systems. 2-Fluoroethanol has been studied in detail. The unsymmetric methyl group substitution increases the number of spectroscopically distinguishable isomers from 2 to 5. However, the intramolecular interaction of the O H group with the F atom stabilizes one out of these conformations by more than 6 kJ/mol. This is enough to form it almost exclusively under supersonic jet expansion conditions [130]. Even in the room-temperature gas phase, there is no sound evidence for other conformations. There has been a lot of debate on whether this intramolecular O H ! ! !F interaction should be considered a hydrogen bond, given that the arrangement of the three atoms is far from linear. Because analogous intermolecular interactions with more favorable geometry show important features of weak hydrogen bonds [130], we will also discuss the bent intramolecular contact in these terms, being aware of alternative viewpoints [253]. What makes the global minimum structure of fluoroethanol particularly interesting is its chirality, although this chirality is of course not relevant for the conventional spectroscopy of the monomer. Due to the intramolecular O H ! ! !F contact, hydrogen-bonded 2-fluoroethanol dimers may be expected to be structurally less diverse than ethanol dimers. Indeed, there is no indication for dimers built from metastable monomer conformations in the supersonic jet expansion [130]. However, the fluorine atom of the hydrogen bond donor alcohol can be engaged in a secondary interaction with the O H group of the acceptor, which would be ‘‘dangling’’ in the case of regular ethanol dimers. Depending on whether or not this interaction is established, the dimers are classified as insertion (i) or addition (a) complexes [154]. Insertion means that the acceptor O H opens and bridges the weak intramolecular O H ! ! !F contact, resulting in a much more favorable and even slightly cooperative O H ! ! !O H ! ! !F arrangement. Addition complexes do not involve such an insertion, but rather conserve the intramolecular contacts of the acceptor and to some extent also of the donor molecule. Obviously, the donor geometry is somewhat distorted by the competition of the acceptor molecule, which is why these complexes have also been called ‘‘associated’’ [130]. Considering the chiral nature of the fluoroethanol monomer, one may therefore expect at least four distinguishable dimer conformations. Heterocofigurational (het) or homoconfigurational (hom) pairings may be combined with inserted or associated hydrogen bond topologies. The choice between the two acceptor oxygen lone electron pairs may double this number, but calculations show that one of the lone pairs is usually more attractive than the other, because it allows for a compact (c) rather than open (o) dimer structure, with increased interaction energy.
34
martin a. suhm
A supersonic jet spectrum indeed shows evidence for four dimer isomers based on the red-shifted donor O H stretching vibrations [130]. In the acceptor region, only two bands are observed, the other two most likely overlapping with the monomer band. The fact that they overlap already indicates that they are of the associated type, where the O H group remains largely unaffected. Stronger evidence comes from an Ar relaxation study. Already traces of the more efficient relaxation promoter are enough to deplete the two least red-shifted donor bands, whereas the two acceptor bands persist. Therefore, a picture analogous to that in ethanol dimer emerges. The most stable dimers exhibit the strongest red shifts and are thus of the inserted type. Calculations suggest that the energy difference is quite subtle, making this a suitable reference system for quantum chemical calculations of weak secondary interactions. Chirality recognition is only weak, but the experimental evidence indicates that the heteroconfigurational dimer may be slightly more stable, opposite to the case of ethanol. We note that 2,2-difluoroethanol behaves quite similar to fluoroethanol. Again, there is only one dominant monomer conformation and up to four dimer isomers are observed. However, in this case the relaxation behavior is completely opposite. The least red-shifted isomers are now most stable and correlate well with associated complexes. This illustrates the subtle interplay between weak O H ! ! !F hydrogen bonds and similarly weak C H ! ! !F interactions, which can also form, even in associated complexes. Fluorinated alcohols are not only interesting as model systems for weak hydrogen bonds with implications in the life sciences [254] and as chemical sensor materials [255], but also provide excellent reaction media [256, 257] and peptide solvents [258–260] with conformation-modulating properties. In both cases, molecular aggregates are thought to play an important role. One of the most widely used fluorinated alcohols is 2,2,2-trifluoroethanol, which will be in the focus of the following section. G.
Trifluoroalcohols
Trifluoromethanol is only metastable with respect to decomposition into F2 C O and HF [261]. The simplest stable alcohol with a CF3 group is 2,2,2trifluoroethanol. Like ethanol, it can occur in a gauche and a trans conformation, but the transiently chiral gauche conformation [146, 262] is strongly favored over the trans form due to intramolecular interactions [263]. It is actually questionable whether the trans form represents a stable local minimum in the isolated molecule [30], whereas it has been found to be abundant in the liquid [264]. It is therefore of interest to investigate for which cluster size the trans conformation starts to become important. The interconversion between the two gauche forms of trifluoroethanol is an order of magnitude slower than in ethanol [146], a consequence of the stabilization of the gauche forms due to the
hydrogen bond dynamics in alcohol clusters
35
intramolecular contact. On the other hand, the anharmonicity of the O H oscillator in gauche trifluoroethanol [30] (oe xe ¼ 85 cm 1 ) is comparable to that of ethanol monomer [81, 265] (oe xe ¼ 88 cm 1 ), based on fundamental and overtone data [89] (see Fig. 2). In analogy to fluoroethanol, which is also locked in a chiral gauche conformation, one would expect to observe up to four trifluoroethanol dimer conformations, differing in their hydrogen bond topology (inserted versus associated) and in their relative monomer chirality or helicity (hom versus het). Calculations suggest that two of these may not be present in large abundance, because they are higher in energy by a few kJ/mol, whereas the other two are predicted to be energetically nearly degenerate (see Fig. 9). However, the experimental spectra [30, 76] (see Fig. 2) strongly support a single dimer conformation, even under the mildest expansion conditions, where relaxation over barriers of more than a few kJ/mol should be inhibited. The spectral details are consistent with a homoconfigurational inserted dimer; that is, both monomer units have the same helicity sense and the acceptor O H inserts into the intramolecular contact of the donor, forming a weak O H ! ! !F hydrogen bond. The energetically almost equivalent heteroconfigurational isomer without insertion is not observed. O H torsional tunneling between the enantiomeric forms of the dimer should be efficiently quenched. Interconversion between the two nearly isoenergetic, but topologically different, isomers involves a barrier of close to 10 kJ/mol (see Fig. 9). Therefore, it is not easy to explain the complete
Figure 9. The torsional potential of trifluoroethanol (dashed line, gþ/t/g , in analogy to the ethanol case shown in Fig. 4) is only distorted slightly if it acts as a hydrogen bond acceptor toward a gþ trifluoroethanol unit [30] (full line). Nevertheless, only the compact inserted homochiral dimer (left, ijcjhom) is observed in the jet experiment, not the compact associated heterochiral dimer (right, ajcjhet).
36
martin a. suhm
absence of the heteroconfigurational isomer in a helium supersonic jet expansion, if it is nearly isoenergetic to the homoconfigurational dimer. It remains to be seen which mechanism is responsible for this extreme form of chirality synchronization [30, 76]. In ethanol [80], the addition of an efficient collision partner (Ar) was required to induce relaxation over barriers on the order of only 3 kJ/mol [33] into the global homoconfigurational minimum structure. A transient mechanism proposed recently for isotope-labeled benzene dimer isomerization [266] does not apply to the high-barrier situation in trifluoroethanol dimer. Molecular dynamics simulations of the collisional cooling process using model potentials [72, 267] or preferably quantum chemical calculations [30, 268] might shed some light onto this interesting dynamical process, as would microwave structural information [91]. Trifluoroethanol dimer is also an ideal model system to investigate the influence of hydrogen bonding on O H stretching anharmonicity. While an isotope study indicated only moderate changes in anharmonicity [30], the recent direct observation of overtone transitions [89] shows that classical hydrogen bonding to oxygen increases the monomer anharmonicity by about 15%, whereas hydrogen bonding to fluorine indeed preserves it (see Fig. 2). In this study, it was possible to quantify the intensity drop from the fundamental O H transition to the overtone for the first time in an alcohol dimer. The O H group bound to fluorine experiences a drop by a factor of 30 in band strength, still comparable to that of the monomer [13]. However, the O H stretching mode facing the oxygen atom of the neighboring molecule drops by a factor of 400, when going from the fundamental to the overtone [89]. This explains on a quantitative level why it is so difficult to observe hydrogen-bonded overtone vibrations and thus to extract reliable anharmonicities. Moving to larger clusters, there is some indirect spectroscopic evidence that monomer trans conformations start to play a role [30] in the few trimer structures which are stable in a supersonic jet expansion. In contrast to methanol trimer [65, 77], the two most strongly IR-active O H stretching modes are split significantly and the in-phase stretching mode gathers substantial IR intensity. The analysis of the coupling pattern [30] confirms the lack of quasi-symmetry and suggests the involvement of different monomer conformations in the trimer. Analysis of the deuterated trimer indicates an increased anharmonicity of the O H stretching mode and pronounced cooperativity effects. According to model simulations and liquid state studies [264], the trans fraction increases with cluster size, until it is comparable to the gauche fraction. A detailed characterization of the solution dynamics of trifluoroethanol is essential for a deeper understanding of its protein solvation and conformational modulation aspects [260]. While the O H stretching mode is a sensitive indicator of weak hydrogen bonds to fluorine and the hydrogen bond topology, there are other dynamical
hydrogen bond dynamics in alcohol clusters
37
consequences of such interactions. C H stretching modes that are in direct contact with F atoms shift characteristically [30]. This is also evident in a comparison of gas- and liquid-state spectra of such alcohols. Similar effects of fluorination have been studied in clusters between alcohols and aromatic compounds [112, 175, 219]. Beyond trifluoroethanol, new aspects come into play. By fluorinating a methyl group in 2-propanol, permanent chirality can be introduced into the molecule [269, 270]. Substitution of both methyl groups by CF3 simplifies the isomer pattern, because the resulting hexafluoroisopropanol favors the nonchiral trans conformation [269]. Currently, we are investigating the influence of aliphatic chain length on the interaction of the CF3 group with the alcohol function in linear trifluoroalkanols and the resulting competition between intramolecular folding and intermolecular aggregation. H.
Chlorinated Alcohols
Chlorinated methanols are only metastable [271], like their fluorinated counterparts. Chlorinated ethanols are stable. Like fluorinated alcohols, they are popular as solvents for peptides [272] because of their characteristic hydrogen bonding properties. Trichloroethanol shows a weaker tendency to dimerize than trifluoroethanol, whereas further aggregation is more favorable, according to solution studies [141]. Matrix isolation studies are also available [262, 273]. We are currently investigating the jet spectra of 2,2,2-trichloroethanol and its clusters. In contrast to trifluoroethanol, more than one dimer conformation is found, thus underscoring the singularity of the previously described chirality synchronization phenomenon. Some chloropropanols are permanently chiral variants of 2-chloroethanol and therefore show a preference between the two gauche conformations that are stabilized by an intramolecular hydrogen bond. This intramolecular case of chirality induction has been investigated by microwave spectroscopy [129]. I.
Ester Alcohols
Like O H groups, carbonyl groups are of major importance in biomolecules and their function. Therefore, an interesting class of model compounds is represented by hydroxyesters. Glycolates are the simplest representatives. They form clusters in which O H ! ! !O H hydrogen bonds compete with O H ! ! !C O hydrogen bonds. The latter are intrinsically stronger, but the former offer cooperative enhancement, if more than two O H groups are involved [274]. When a methyl group is added to the O H-carrying carbon, a chiral center is created in the immediate neighborhood of the O H group. This leads to fascinating chirality recognition phenomena in lactates, in which the relative handedness of neighboring lactate units decides about the preferred hydrogen bond topology in a tetrameric cluster [122, 125, 275]. This tetramer features one of the most
38
martin a. suhm
complex gas-phase chirality discrimination phenomena that have been structurally characterized to date. An even more complex alcoholic system involving chirality discrimination is the protonated serine octamer, which has been largely studied via mass spectrometry [15, 123, 124]. Hydroxyesters have also been combined with simple aliphatic and aromatic alcohols [126, 154] to investigate chirality recognition phenomena, the question of kinetic versus thermodynamic control, and more generally the preference for insertion or addition complexes. J. Polyols and Sugars While the hydrogen-bond-driven aggregation of isolated O H groups is now understood fairly well, nature usually employs sugars [59] for molecular recognition tasks [17]. From a hydrogen bond perspective, sugars can be mimicked most easily by polyols, molecules with multiple O H groups attached to an aliphatic backbone. The strength of intramolecular hydrogen bonds in diols can be tuned via the distance of the two O H groups at the carbon backbone [276]. Glycol as the simplest representative of 1,2-diols has been studied extensively in a range of temperatures [277], including supersonic jet and other rotational spectroscopy techniques [278]. By replacing two C H bonds by C CH3 groups, chirality is introduced and diastereoisomerism as an intramolecular variant of chirality recognition occurs [110]. 1,2-Diols can be used to observe hydrogen-bonded O H overtones [276, 277], because the intramolecular constraints prevent an optimum hydrogen bond geometry. This may reduce the cancellation effect between electrical and mechanical anharmonicity. The liquid state of glycol has been investigated by a range of methods including the development of model potentials [279]. Therefore, isolated cluster studies of this elementary bifunctional prototype would be desirable. The prototype for molecules involving three alcohol groups is glycerol. Its hydrogen bond topology [280, 281] is strongly temperature-dependent, and conformer relaxation in the jet expansion [282] appears to be efficient. In order to model sugars more closely, further functionalities have to be included. Glycidol may be considered as a simplified model that includes an O-heterocycle. Its dimer has been studied successfully with respect to chirality recognition phenomena [111]. More realistic models for open sugars involve keto or aldehyde groups. Glycolaldehyde may thus be viewed as the simplest realistic sugar model, a diose. Its transition from the internally hydrogenbonded gas-phase structure to the intermolecular C O/O H hydrogen bond network in aggregates has been studied by vibrational spectroscopy [283]. In the thermodynamically stable solid, the carbonyl groups are absent due to chemical dimerization. Higher homologs such as dihydroxyacetone also show an interesting interplay between chemically and hydrogen-bond-directed aggregation [284]. Larger sugars [17, 207, 285], their adducts [286], and their
hydrogen bond dynamics in alcohol clusters
39
aggregates up to amorphous sugar nanoparticles [284, 287, 288] as well as sugars in solution [29] represent systems of increasing complexity in which the essential binding motifs of the smaller models can be identified. While experimental data on sugars may not always be conclusive for the properties of isolated hydrogen bonds [32], the opposite is certainly true. By studying clustering in alcohols of increasing complexity and poly-functionality, one will ultimately arrive at an even more detailed understanding of the dynamics and aggregation of naturally occurring mono- [59] and oligo-saccharides [17], which are thought to play a key role in cellular recognition. VI.
CONCLUSIONS
This review has discussed alcohol clusters from methanol [98] to sugars [17]. It has tried to show that a reductionist gas-phase spectroscopy approach to the complexity of organic and biomolecular matter can be fruitful. By studying the most elementary model systems, one can extract the structural, energetic, and dynamic essentials from the enormous hyperspace spanned by the myriads of degrees of freedom involved in biological systems, hydrogen-bonded liquids, and polymers. These essentials include conformational dynamics, molecular recognition, cooperativity, and solvation, among other things. More often than not, one finds that weak hydrogen bonds are decisive for the detailed conformation of alcohols and their complexes. Infrared, Raman, microwave, and double resonance techniques turn out to offer nicely complementary tools, which usually can and have to be complemented by quantum chemical calculations. In both experiment and theory, progress over the last 10 years has been enormous. The relationship between theory and experiment is symbiotic, as the elementary systems represent benchmarks for rigorous quantum treatments of clear-cut observables. Even the simplest cases such as methanol dimer still present challenges, which can only be met by high-level electron correlation and nuclear motion approaches in many dimensions. On the experimental side, infrared spectroscopy is most powerful for the O H stretching dynamics, whereas double resonance techniques offer selectivity and Raman scattering profits from other selection rules. A few challenges for accurate theoretical treatments in this field are listed in Table I. Although most of the reported gas-phase experiments do not investigate the temporal evolution of alcohol clusters explicitly, the frequency-domain spectral information can nevertheless be translated into the time domain, making use of some elementary and robust relationships between spectral and dynamical features [289]. According to this, the 10-fs period of the hydrogen-bonded O H oscillator is modulated and damped by a series of other phenomena. Energy flow into doorway states is certainly slower than for aliphatic C H bonds [290]; but on a time scale of a few picoseconds, energy will nevertheless have
hydrogen bond dynamics in alcohol clusters
41
Acknowledgments I owe many thanks to my current and former co-workers, whose essential contributions on the dynamics of alcohol clusters are reflected in the cited references. Like me, quite a few of them coincidentally consume more alcohol in research than at social occasions. I also wish to thank my colleagues and collaborators in this field, listed in the cited references. I apologize for any important omissions, which seem unavoidable in such a wide field. Furthermore, I wish to thank the Go¨ttingen mechanical workshops for their competent technical realization of the high-throughput supersonic jet experiments. I acknowledge the generous and flexible financial support by the Fonds der Chemischen Industrie, by the collaborative research centers SFB 357 and SFB 602, by the DFG research training group 782 (www.pcgg.de), and more recently also by the Raman DFG grant Su 121/2.
References 1. L. P. Kuhn, The hydrogen bond. I. Intra- and intermolecular hydrogen bonds in alcohols. J. Am. Chem. Soc. 74, 2492–2499 (1952). 2. D. S. Dwyer and R. J. Bradley, Chemical properties of alcohols and their protein binding sites. Cell. Mol. Life Sci. 57, 265–275 (2000). 3. B. Brutschy and P. Hobza, Van der Waals molecules III: Introduction. Chem. Rev. 100, 3861– 3862 (2000). 4. F. N. Keutsch and R. J. Saykally, Water clusters: Untangling the mysteries of the liquid, one molecule at a time. Proc. Natl. Acad. Sci. USA 98, 10533–10540 (2001). 5. W. O. George, B. F. Jones, and R. Lewis, Water and its homologues: A comparison of hydrogenbonding phenomena. Philos. Trans. R. Soc. A 359, 1611–1629 (2001). 6. J.-M. Lehn, Supramolecular Chemistry, Wiley-VCH, New York, 1995. 7. C. Pratt Brock and L. L. Duncan. Anomalous space-group frequencies for monoalcohols Cn Hm OH. Chem. Mater. 6, 1307–1312 (1994). 8. K. J. Gaffney, I. R. Piletic, and M. D. Fayer, Hydrogen bond breaking and reformation in alcohol oligomers following vibrational relaxation of a non-hydrogen-bond donating hydroxyl stretch. J. Phys. Chem. A 106, 9428–9435 (2002). 9. K. J. Gaffney, Paul H. Davis, I. R. Piletic, N. E. Levinger, and M. D. Fayer, Hydrogen bond dissociation and reformation in methanol oligomers following hydroxyl stretch relaxation. J. Phys. Chem. A 106, 12012–12023 (2002). 10. E. T. J. Nibbering and T. Elsaesser, Ultrafast vibrational dynamics of hydrogen bonds in the condensed phase. Chem. Rev. 104, 1887–1914 (2004). 11. J. B. Asbury, T. Steinel, and M. D. Fayer, Hydrogen bond networks: Structure and evolution after hydrogen bond breaking. J. Phys. Chem. B 108, 6544–6554 (2004). 12. K. R. Wilson, R. D. Schaller, D. T. Co, R. J. Saykally, B. S. Rude, T. Catalano, and J. D. Bozek, Surface relaxation in liquid water and methanol studied by x-ray absorption spectroscopy. J. Chem. Phys. 117, 7738–7744 (2002). 13. A. Perera, F. Sokolic´, and L. Zoranic´, Microstructure of neat alcohols. Phys. Rev. E 75, 060502(R) (2007). 14. W. Weltner, Jr., and K. S. Pitzer, Methyl alcohol: The entropy, heat capacity and polymerization equilibria in the vapor, and potential barrier to internal rotation. J. Am. Chem. Soc. 73, 2606– 2610 (1951).
42
martin a. suhm
15. B. Baytekin, H. T. Baytekin, and C. A. Schalley, Mass spectrometric studies of non-covalent compounds: Why supramolecular chemistry in the gas phase. Org. Biomol. Chem. 4, 2825– 2841 (2006). 16. R. Wugt Larsen, P. Zielke, and M. A. Suhm. Hydrogen-bonded OH stretching modes of methanol clusters: A combined IR and Raman isotopomer study. J. Chem. Phys. 126, 194307 (2007). 17. J. P. Simons, R. A. Jockusch, P. C ¸ arc¸abal, I. Hu¨nig, R. T. Kroemer, N. A. Macleod, and L. C. Snoek, Sugars in the gas phase. Spectroscopy, conformation, hydration, co-operativity and selectivity. Int. Rev. Phys. Chem. 24, 489–531 (2005). 18. The Thomson Corporation, ISI Web of Science, 2007. 19. T. S. Zwier, The spectroscopy of solvation in hydrogen-bonded aromatic clusters. Annu. Rev. Phys. Chem. 47, 205–241 (1996). 20. M. Mons, I. Dimicoli, and F. Piuzzi, Gas phase hydrogen-bonded complexes of aromatic molecules: Photoionization and energetics. Int. Rev. Phys. Chem. 21, 101–135 (2002). 21. Y. Yamada, Y. Katsumoto, and T. Ebata, Picosecond IR-UV pump-probe spectroscopic study on the vibrational energy flow in isolated molecules and clusters. Phys. Chem. Chem. Phys. 10, 1170–1185 (2007). 22. J. S. Lomas, Self- and hetero-association of sterically hindered tertiary alcohols. J. Phys. Org. Chem. 18, 1001–1012 (2005). 23. T. Steiner, The hydrogen bond in the solid state. Angew. Chem. Int. Ed. 41, 48–76 (2002). 24. S. J. Barlow, G. V. Bondarenko, Y. E. Gorbaty, T. Yamaguchi, and M. Poliakoff, An IR study of hydrogen bonding in liquid and supercritical alcohols. J. Phys. Chem. A 106, 10452–10460 (2002). 25. J.-M. Andanson, J.-C. Soetens, T. Tassaing, and M. Besnard, Hydrogen bonding in supercritical tert-butanol assessed by vibrational spectroscopies and molecular-dynamics simulations. J. Chem. Phys. 122, 174512 (2005). 26. T. Yamaguchi, New horizons in hydrogen bonded clusters in solution. Pure Appl. Chem. 71, 1741–1751 (1999). 27. A. J. Barnes, H. E. Hallam, and D. Jones, Vapour phase infrared studies of alcohols: I. Intramolecular interactions and self-association. Proc. R. Soc. London A 335, 97–111 (1973). 28. D. R. Miller, Free jet sources, in Atomic and Molecular Beam Methods, G. Scoles, ed., University Press, Oxford, 1988, Chapter 2, pp. 14–53. 29. N. A. Macleod, C. Johannessen, L. Hecht, L. D. Barron, and J. P. Simons, From the gas phase to aqueous solution: Vibrational spectroscopy, Raman optical activity and conformational structure of carbohydrates. Int. J. Mass Spectrom. 253, 193–200 (2006). 30. T. Scharge, C. Ce´zard, P. Zielke, A. Schu¨tz, C. Emmeluth, and M. A. Suhm, A peptide cosolvent under scrutiny: Self-aggregation of 2,2,2-trifluoroethanol. Phys. Chem. Chem. Phys. 9, 4472–4490 (2007). 31. I. Olovsson, The role of the lone pairs in hydrogen bonding. Z. Phys. Chem. 220, 963–978 (2006). 32. J. Kroon, J. A. Kanters, J. G. C. M. van Duijneveldt–van de Rijdt, F. B. van Duijneveldt, and J. A. Vliegenthart, O H ! ! ! O hydrogen bonds in molecular crystals. A statistical and quantummechanical analysis. J. Mol. Struct. 24, 109–129 (1975). 33. V. Dyczmons, Dimers of ethanol. J. Phys. Chem. A 108, 2080–2086 (2004).
hydrogen bond dynamics in alcohol clusters
43
34. S. Coussan, A. Loutellier, J. P. Perchard, S. Racine, A. Peremans, A. Tadjeddine, and W. Q. Zheng, Infrared laser induced isomerization of methanol polymers trapped in nitrogen matrix. I. Trimers. J. Chem. Phys. 107, 6526–6540 (1997). 35. K. Le Barbu-Debus, N. Seurre, F. Lahmani, and A. Zehnacker-Rentien, Formation of hydrogenbonded bridges in jet-cooled complexes of a chiral chromophore as studied by IR/UV double resonance spectroscopy. )2-Naphthyl-1-ethanol/ðmethanolÞn¼1;2 complexes. Phys. Chem. Chem. Phys. 4, 4866–4876 (2002). 36. N. Seurre, J. Sepiol, F. Lahmani, A. Zehnacker-Rentien, and K. Le Barbu-Debus, Vibrational study of the S0 and S1 states of 2-naphthyl-1-ethanol/ (water)2 and 2-naphthyl-1-ethanol/ (methanol)2 complexes by IR/UV double resonance spectroscopy. Phys. Chem. Chem. Phys. 6, 4658–4664 (2004). 37. F. A. Smith and E. C. Creitz, Infrared studies of association in eleven alcohols. J. Res. Natl. Bur. Standards 46(2), 145–164 (1951). 38. M. M. Pires and V. F. DeTuri, Structural, energetic, and infrared spectra insights into methanol clusters ðCH3 OHÞn , for n ¼ 2–12, 16, 20. ONIOM as an efficient method of modeling large methanol clusters. J. Chem. Theory Comput. 3, 1073–1082 (2007). 39. D. Zimmermann, T. Ha¨ber, H. Schaal, and M. A. Suhm, Hydrogen-bonded rings, chains and lassos: The case of tert-butyl alcohol clusters. Mol. Phys. 99, 413–426 (2001). 40. M. Falk and E. Whalley, Infrared spectra of methanol and deuterated methanols in gas, liquid, and solid phases. J. Chem. Phys. 34, 1554 (1961). 41. W. Saenger and K. Lindner, OH clusters with homodromic circular arrangement of hydrogen bonds. Angew. Chem., Int. Ed. 19, 398–399 (1980). 42. R. Taylor and C. F. Macrae, Rules governing the crystal packing of mono- and dialcohols. Acta Cryst. B (Struct. Sci.) 57, 815–827 (2001). 43. A. Garcı´a Fraile, D. G. Morris, A. Garcı´a Martı´nez, S. de la Moya Cerero, K. W. Muir, K. S. Ryder, and E. Teso Vilar. Self-recognition and hydrogen bonding by polycyclic bridgehead monoalcohols. Org. Biomol. Chem. 1, 700–704 (2003). 44. P. A. McGregor, D. R. Allan, S. Parsons, and S. J. Clark, Hexamer formation in tertiary butyl alcohol (2-methyl-2-propanol, C4 H10 O). Acta Cryst. B (Struct. Sci.) 62, 599–605 (2006). 45. R. Ludwig, The structure of liquid methanol. ChemPhysChem 6, 1369–1375 (2005). 46. P. Raveendran, D. Zimmermann, T. Ha¨ber, and M. A. Suhm, Exploring a hydrogen-bond terminus: Spectroscopy of eucalyptol–alcohol clusters. Phys. Chem. Chem. Phys. 2, 3555–3563 (2000). 47. A. Fujii, S. Enomoto, M. Miyazaki, and N. Mikami, Morphology of protonated methanol clusters: An infrared spectroscopic study of hydrogen bond networks of Hþ ðCH3 OHÞn (n ¼ 4–15). J. Phys. Chem. A 109, 138–141 (2005). 48. I. Dauster, M. A. Suhm, U. Buck, and T. Zeuch, Experimental and theoretical study of the microsolvation of sodium atoms in methanol clusters: Differences and similarities to sodiumwater and sodium-ammonia. Phys. Chem. Chem. Phys. 10, 83–95 (2008). 49. A. Latini, D. Toja, A. Giardini-Guidoni, S. Piccirillo, and M. Speranza, Energetics of molecular complexes in a supersonic beam: A novel spectroscopic tool for enantiomeric discrimination. Angew. Chem. Int. Ed. 38, 815–817 (1999). 50. M. Mons, F. Piuzzi, I. Dimicoli, A. Zehnacker, and F. Lahmani, Binding energy of hydrogenbonded complexes of the chiral molecule 1-phenylethanol, as studied by 2C-R2PI: Comparison between diastereoisomeric complexes with butan-2-ol and the singly hydrated complex. Phys. Chem. Chem. Phys. 2, 5065–5070 (2000).
44
martin a. suhm
51. C. Wickleder, D. Henseler, and S. Leutwyler, Accurate dissociation energies of O H ! ! ! O hydrogen-bonded 1-naphthol ! solvent complexes. J. Chem. Phys. 116, 1850–1857 (2002). 52. A. Bizzarri, S. Stolte, J. Reuss, J. G. C. M. van Duijneveldt–van de Rijdt, and F. B. van Duijneveldt, Infrared excitation and dissociation of methanol dimers and trimers. Chem. Phys. 143, 423–435 (1990). 53. L. Belau, K. R. Wilson, S. R. Leone, and M. Ahmed, Vacuum ultraviolet (VUV) photoionization of small water clusters: Correction. J. Phys. Chem. A 111, 10885–10886 (2007). 54. K. Le Barbu, V. Brenner, Ph. Millie´, F. Lahmani, and A. Zehnacker-Rentien, An experimental and theoretical study of jet-cooled complexes of chiral molecules: The role of dispersive forces in chiral discrimination. J. Phys. Chem. A 102, 128–137 (1998). 55. M. Saeki, S.-i. Ishiuchi, M. Sakai, and M. Fujii, Structure of the jet-cooled 1-naphthol dimer studied by IR dip spectroscopy: Cooperation between the p–p interaction and the hydrogen bonding. J. Phys. Chem. A 111, 1001–1005 (2007). 56. K. E. Riley and P. Hobza, Assessment of the MP2 method, along with several basis sets, for the computation of interaction energies of biologically relevant hydrogen bonded and dispersion bound complexes. J. Phys. Chem. A 111, 8257–8263 (2007). 57. A. D. Boese, J. M. L. Martin, and W. Klopper, Basis set limit coupled cluster study of H-bonded systems and assessment of more approximate methods. J. Phys. Chem. A 111, 11122–11133 (2007). 58. C. Emmeluth, V. Dyczmons, and M. A. Suhm, Tuning the hydrogen bond donor/acceptor isomerism in jet-cooled mixed dimers of aliphatic alcohols. J. Phys. Chem. A 110, 2906–2915 (2006). 59. T. Suzuki, The hydration of glucose: The local configurations in sugar-water hydrogen bonds. Phys. Chem. Chem. Phys. 10, 96–105 (2008). 60. A. Karpfen, Cooperativity effects in hydrogen bonding. Adv. Chem. Phys. 123, 469–510 (2002). 61. M. Weimann, M. Fa´rnı´k, and M. A. Suhm, Cooperative and anticooperative mixed trimers of HCl and methanol. J. Mol. Struct. 790, 18–26 (2006). 62. M. A. Suhm and D. J. Nesbitt, Potential surfaces and dynamics of weakly bound trimers: Perspectives from high resolution IR spectroscopy. Chem. Soc. Rev. 24, 45–54 (1995). 63. J. Demaison, M. Herman, and J. Lievin, The equilibrium OH bond length. Mol. Phys. 26, 391– 420 (2007). 64. J. R. Dixon, W. O. George, F. Hossain, R. Lewis, and J. M. Price, Hydrogen-bonded forms of methanol: IR spectra and ab initio calculations. J. Chem. Soc., Faraday Trans. 93, 3611–3618 (1997). 65. T. Ha¨ber, U. Schmitt, and M. A. Suhm, FTIR-spectroscopy of molecular clusters in pulsed supersonic slit-jet expansions. Phys. Chem. Chem. Phys. 1, 5573–5582 (1999). 66. Bretta F. King and F. Weinhold, Structure and spectroscopy of (HCN)n clusters: Cooperative and electronic delocalization effects in C H ! ! ! N hydrogen bonding. J. Chem. Phys. 103, 333– 347 (1995). 67. M. Quack and M. A. Suhm, Potential energy hypersurfaces for hydrogen bonded clusters (HF)n, in Conceptual Perspectives in Quantum Chemistry, Conceptual Trends in Quantum Chemistry, Vol. III, J.-L. Calais and E. S. Kryachko, ed., Kluwer, Dordrecht, 1997, pp. 415– 463. 68. M. Quack, J. Stohner, and M. A. Suhm, Analytical three-body interaction potentials and hydrogen bond dynamics of hydrogen fluoride aggregates (HF)n, n $ 3. J. Mol. Struct. 599, 381–425 (2001).
hydrogen bond dynamics in alcohol clusters
45
69. T. N. Wassermann, P. Zielke, J. J. Lee, C. Ce´zard, and M. A. Suhm, Structural preferences, argon nanocoating, and dimerization of n-alkanols as revealed by OH stretching spectroscopy in supersonic jets. J. Phys. Chem. A 111, 7437–7448 (2007). 70. M. Rozenberg, A. Loewenschuss, and Y. Marcus, An empirical correlation between stretching vibration redshift and hydrogen bond length. Phys. Chem. Chem. Phys. 2, 2699–2702 (2000). 71. R. M. Badger and S. H. Bauer, Spectroscopic study of the hydrogen bond. II. The shift of the O H vibrational frequency in the formation of the hydrogen bond. J. Chem. Phys. 5, 839–851 (1937). 72. C. Ce´zard, C. A. Rice, and M. A. Suhm, OH-stretching redshifts in bulky hydrogen-bonded alcohols: Jet spectroscopy and modeling. J. Phys. Chem. A 110, 9839–9848 (2006). 73. J. M. Lopez, F. Ma¨nnle, I. Wawer, G. Buntkowsky, and H.-H. Limbach, NMR studies of double proton transfer in hydrogen bonded cyclic N; N 0 -diarylformamidine dimers: Conformational control, kinetic HH/HD/DD isotope effects and tunneling. Phys. Chem. Chem. Phys. 9, 4498– 4513 (2007). 74. M. N. Slipchenko, K. E. Kuyanov, B. G. Sartakov, and A. F. Vilesov, Infrared intensity in small ammonia and water clusters. J. Chem. Phys. 124, 241101 (2006). 75. R. A. Provencal, J. B. Paul, K. Roth, C. Chapo, R. N. Casaes, R. J. Saykally, G. S. Tschumper, and H. F. Schaefer III, Infrared cavity ringdown spectroscopy of methanol clusters: Single donor hydrogen bonding. J. Chem. Phys. 110, 4258–4267 (1999). 76. T. Scharge, T. Ha¨ber, and M. A. Suhm, Quantitative chirality synchronization in trifluoroethanol dimers. Phys. Chem. Chem. Phys. 8, 4664–4667 (2006). 77. P. Zielke and M. A. Suhm, Concerted proton motion in hydrogen-bonded trimers: A spontaneous Raman scattering perspective. Phys. Chem. Chem. Phys. 8, 2826–2830 (2006). 78. M. Fa´rnı´k and D. J. Nesbitt, Intramolecular energy transfer between oriented chromophores: High-resolution infrared spectroscopy of HCl trimer. J. Chem. Phys. 121, 12386–12395 (2004). 79. M. Quack, U. Schmitt, and M. A. Suhm, FTIR spectroscopy of hydrogen fluoride clusters in synchronously pulsed supersonic jets: Isotopic isolation, substitution and 3-D condensation. Chem. Phys. Lett. 269, 29–38 (1997). 80. C. Emmeluth, V. Dyczmons, T. Kinzel, P. Botschwina, M. A. Suhm, and M. Ya´n˜ez, Combined jet relaxation and quantum chemical study of the pairing preferences of ethanol. Phys. Chem. Chem. Phys. 7, 991–997 (2005). 81. H. L. Fang and D. A. C. Compton, Vibrational overtones of gaseous alcohols. J. Phys. Chem. 92, 6518–6527 (1988). 82. D. Rueda, O. V. Boyarkin, T. R. Rizzo, I. Mukhopadhyay, and D. S. Perry, Torsion–rotation analysis of OH stretch overtone-torsion combination bands in methanol. J. Chem. Phys. 116, 91–100 (2002). 83. P. Maksyutenko, O. V. Boyarkin, T. R. Rizzo, and D. S. Perry, Conformational dependence of intramolecular vibrational redistribution in methanol. J. Chem. Phys. 126, 044311, (2007). 84. W. A. P. Luck and W. Ditter, Die Assoziation der Alkohole bis in u¨berkritische Bereiche. Ber. Bunsenges. Phys. Chem. 72, 365–492 (1968). 85. T. Di Paolo, C. Bourde´ron, and C. Sandorfy, Model calculations on the influence of mechanical and electrical anharmonicity on infrared intensities: Relation to hydrogen bonding. Can. J. Chem. 50, 3161–3166 (1972). 86. S. A. Nizkorodov, M. Ziemkiewicz, D. J. Nesbitt, and A. E. W. Knight, Overtone spectroscopy of H2O clusters in the vOH ¼ 2 manifold: Infrared–ultraviolet vibrationally mediated dissociation studies. J. Chem. Phys. 122, 194316 (2005).
46
martin a. suhm
87. L. England-Kretzer and W. A. P. Luck, Band analysis of CH3OH and CH3OD H-bond complexes in the first overtone region. J. Mol. Struct. 348, 373–376 (1995). 88. J. P. Perchard and Z. Mielke, Anharmonicity and hydrogen bonding I. A near-infrared study of methanol trapped in nitrogen and argon matrices. Chem. Phys. 264, 221–234 (2001). 89. T. Scharge, D. Luckhaus, and M. A. Suhm, Observation and quantification of the hydrogen bond effect on O H overtone intensities in an alcohol dimer. Chem. Phys. 346, 167–175 (2008). 90. J. T. Farrell, Jr., M. A. Suhm, and D. J. Nesbitt, Breaking symmetry with hydrogen bonds: Vibrational predissociation and isomerization dynamics in HF–DF and DF–HF isotopomers. J. Chem. Phys. 104, 9313–9331 (1996). 91. J. P. I. Hearn, R. V. Cobley, and B. J. Howard, High-resolution spectroscopy of induced chiral dimers: A study of the dimers of ethanol by Fourier transform microwave spectroscopy. J. Chem. Phys. 123, 134324 (2005). 92. P. Zielke and M. A. Suhm, Raman jet spectroscopy of formic acid dimers: Low frequency vibrational dynamics and beyond. Phys. Chem. Chem. Phys. 9, 4528–4534 (2007). 93. R. Wugt Larsen and M. A. Suhm, Cooperative organic hydrogen bonds: The librational modes of cyclic methanol clusters. J. Chem. Phys. 125, 154314 (2006). 94. M. Ehbrecht and F. Huisken, Vibrational spectroscopy of ethanol molecules and complexes selectively prepared in the gas phase and adsorbed on large argon clusters. J. Phys. Chem. A 101, 7768–7777 (1997). 95. S. Coussan, Y. Boutellier, J. P. Perchard, and W. Q. Zheng, Rotational isomerism of ethanol and matrix isolation infrared spectroscopy. J. Phys. Chem. A 102, 5789–5793 (1998). 96. U. Buck and I. Ettischer, Vibrational predissociation spectra of size-selected methanol clusters: New experimental results. J. Chem. Phys. 108, 33–38 (1998). 97. F. Huisken and M. Staemmler, Infrared spectroscopy of methanol clusters adsorbed on large Arx host clusters. J. Chem. Phys. 98, 7680–7691 (1993). 98. U. Buck and F. Huisken, Infrared spectroscopy of size-selected water and methanol clusters. Chem. Rev. 100, 3863–3890 (2000). 99. C. J. Gruenloh, G. M. Florio, J. R. Carney, F. C. Hagemeister, and T. S. Zwier, C H stretch modes as a probe of H-bonding in methanol-containing clusters. J. Phys. Chem. A 103, 496–502 (1999). 100. S. Jarmelo, N. Maiti, V. Anderson, P. R. Carey, and R. Fausto, Ca H bond-stretching frequency in alcohols as a probe of hydrogen-bonding strength: A combined vibrational spectroscopic and theoretical study of n-[1-D]propanol. J. Phys. Chem. A 109, 2069–2077 (2005). 101. C. R. Quade, A note on internal rotation–rotation interactions in ethyl alcohol. J. Mol. Spectrosc. 203, 200–202 (2000). 102. A. Maeda, F. C. De Lucia, E. Herbst, J. C. Pearson, J. Riccobono, E. Trosell, and R. K. Bohn, The millimeter- and submillimeter-wave spectrum of the Gt conformer of n-propanol (nCH3CH2CH2OH). Astrophys. J. Suppl. Ser. 162, 428–435 (2006). See also abstract RI14, 61. Mol. Spectr. Symp. 103. S. Coussan, M. E. Alikhani, J. P. Perchard, and W. Q. Zheng, Infrared-induced isomerization of ethanol dimers trapped in argon and nitrogen matrices: Monochromatic irradiation experiments and DFT calculations. J. Phys. Chem. A 104, 5475–5483 (2000). 104. A. Trivella, S. Coussan, T. Chiavassa, P. Theule´, P. Roubin, and C. Manca, Comparative study of structure and photo-induced reactivity of malonaldehyde and acetylacetone isolated in nitrogen matrices. Low Temp. Phys. 32, 1042–1049 (2006). 105. R. D. Astumian, Design principles for Brownian molecular machines: How to swim in molasses and walk in a hurricane. Phys. Chem. Chem. Phys. 9, 5067–5083 (2007).
hydrogen bond dynamics in alcohol clusters
47
106. M. V. Vener and J. Sauer, Vibrational spectra of the methanol tetramer in the OH stretch region. Two cyclic isomers and concerted proton tunneling. J. Chem. Phys. 114, 2623–2628 (2001). 107. B. Fehrensen, D. Luckhaus, M. Quack, M. Willeke, and T. R. Rizzo, Ab initio calculations of mode selective tunneling dynamics in 12CH3OH and 13CH3OH. J. Chem. Phys. 119, 5534–5544 (2003). 108. D. Sabo, Z. Bacˇic´, S. Graf, and S. Leutwyler, Four-dimensional model calculation of torsional levels of cyclic water tetramer. J. Chem. Phys. 109, 5404–5419 (1998). 109. R. Berger, M. Quack, A. Sieben, and M. Willeke, Parity-violating potentials for the torsional motion of methanol (CH3OH) and its isotopomers (CD3OH, (CH3OH) C-13, CH3OD, CH3OT, CHD2OH, and CHDTOH). Helv. Chim. Acta 86, 4048–4060 (2003). 110. J. Paul, I. Hearn, and B. J. Howard, Chiral recognition in a single molecule: A study of homo and heterochiral butan-2,3,-diol by Fourier transform microwave spectroscopy. Mol. Phys. 105, 825–839 (2007). 111. N. Borho and M. A. Suhm, Glycidol dimer: Anatomy of a molecular handshake. Phys. Chem. Chem. Phys. 4, 2721–2732 (2002). 112. A. Al Rabaa, K. LeBarbu, F. Lahmani, and A. Zehnacker-Rentien, Van der Waals complexes between chiral molecules in a supersonic jet: A new spectroscopic method for enantiomeric discrimination. J. Phys. Chem. A 101, 3273–3278 (1997). 113. K. Le Barbu-Debus, F. Lahmani, A. Zehnacker-Rentien, N. Guchhait, S. S. Panja, and T. Chakraborty, Fluorescence spectroscopy of jet-cooled chiral (þ= )-indan-1-ol and its cluster with (þ= )-methyl- and ethyl-lactate. J. Chem. Phys. 125, 174305 (2006). 114. R. S. Cahn, Sir Christopher Ingold, and V. Prelog, Spezifikation der molekularen Chiralita¨t. Angew. Chem. 78, 413–447 (1966). 115. N. Borho, Diploma Thesis, Go¨ttingen, 2001. 116. M. A. Czarnecki, Near-infrared spectroscopic study of hydrogen bonding in chiral and racemic octan-2-ol. J. Phys. Chem. A 107, 1941–1944 (2003). 117. A. K. King and B. J. Howard, A microwave study of the hetero-chiral dimer of butan-2-ol. Chem. Phys. Lett. 348, 343–349 (2001). 118. M. Speranza, Chiral clusters in the gas phase. Adv. Phys. Org. Chem. 39, 147–281 (2004). 119. A. Latini, M. Satta, A. Giardini Guidoni, S. Piccirillo, and M. Speranza, Short-range interactions within molecular complexes formed in supersonic beams: Structural effects and chiral discrimination. Chem. Eur. J. 6, 1042–1049 (2000). 120. K. Le Barbu, F. Lahmani, and A. Zehnacker-Rentien, Formation of hydrogen-bonded structures in jet-cooled complexes of a chiral chromophore studied by IR/UV double resonance spoctroscopy: Diastereomeric complexes of )-2-naphthyl-1-ethanol with )-2-amino-1-propanol. J. Phys. Chem. A 106, 6271–6278 (2002). 121. N. Seurre, J. Sepiol, Katia Le Barbu-Debus, F. Lahmani, and A. Zehnacker-Rentien, The role of chirality in the competition between inter- and intramolecular hydrogen bonds: jet-cooled van der Waals complexes of ())2-naphthyl-1-ethanol with ())1-amino-2-propanol and ())2amino-1-butanol. Phys. Chem. Chem. Phys. 6, 2867–2877 (2004). 122. T. B. Adler, N. Borho, M. Reiher, and M. A. Suhm, Chirality-induced switch in hydrogen-bond topology: Tetrameric methyl lactate clusters in the gas phase. Angew. Chem. Int. Ed. 45, 3440– 3445 (2006). 123. R. Graham Cooks, D. Zhang, K. J. Koch, F. C. Gozzo, and M. N. Eberlin, Chiroselective selfdirected octamerization of serine: Implications for homochirogenesis. Anal. Chem. 73, 3646– 3655 (2001).
48
martin a. suhm
124. U. Mazurek, Some more aspects of formation and stability of the protonated serine octamer cluster. Eur. J. Mass Spectrom. 12, 63–70 (2006). 125. N. Borho and M. A. Suhm, Self-organization of lactates in the gas phase. Org. Biomol. Chem. 1, 4351–4358 (2003). 126. N. Seurre, K. Le Barbu-Debus, F. Lahmani, A. Zehnacker-Rentien, N. Borho, and M. A. Suhm, Chiral recognition between lactic acid derivatives and an aromatic alcohol in a supersonic expansion: Electronic and vibrational spectroscopy. Phys. Chem. Chem. Phys. 8, 1007–1016 (2006). 127. A. Zehnacker and M. A. Suhm, Chirality recognition between neutral molecules in the gas phase. Angew. Chem. Int. Ed. 47, 6970–6992 (2008). 128. N. Borho and Y. Xu, Molecular recognition in 1:1 hydrogen-bonded complexes of oxirane and trans-2,3-dimethyloxirane with ethanol: A rotational spectroscopic and ab initio study. Phys. Chem. Chem. Phys. 9, 4514–4520 (2007). 129. T. Goldstein, M. S. Snow, and B. J. Howard, Intramolecular hydrogen bonding in chiral alcohols: The microwave spectrum of the chloropropanols. J. Mol. Spectrosc. 236, 1–10 (2006). 130. T. Scharge, C. Emmeluth, Th. Ha¨ber, and M. A. Suhm, Competing hydrogen bond topologies in 2-fluoroethanol dimer. J. Mol. Struct. 786, 86–95 (2006). 131. N. Solca` and O. Dopfer, Hydrogen-bonded networks in ethanol proton wires: IR spectra of ðEtOHÞq Hþ -Ln clusters (L ¼ Ar=N2 ; q - 4; n - 5). J. Phys. Chem. A 109, 6174–6186 (2005). 132. T. D. Frigden, L. MacAleese, T. B. McMahon, J. Lemaire, and P. Maitre, Gas phase infrared multiple-photon dissociation spectra of methanol, ethanol and propanol proton-bound dimers, protonated propanol and the propanol/water proton-bound dimer. Phys. Chem. Chem. Phys. 8, 955–966 (2006). 133. N. Uras-Aytemiz, J. P. Devlin, J. Sadlej, and V. Buch, HCl solvation at the surface and within methanol clusters/nanoparticles II: Evidence for molecular wires. J. Phys. Chem. B 110, 21751–21763 (2006). 134. J. M. Lisy, Spectroscopy and structure of solvates alkali-metal ions. Int. Rev. Phys. Chem. 16, 267–289 (1997). 135. C. A. Corbett, T. J. Martinez, and J. M. Lisy, Solvation of the fluoride anion by methanol. J. Phys. Chem. A 106, 10015–10021 (2002). 136. W. H. Robertson, K. Karapetian, P. Ayotte, K. D. Jordan, and M. A. Johnson, Infrared predissociation spectroscopy of I ! ! ! ðCH3 OHÞn ; n ¼ 1; 2: Cooperativity in asymmetric solvation. J. Chem. Phys. 116, 4853–4857 (2002). 137. A. Kammrath, G. B. Griffin, J. R. R. Verlet, R. M. Young, and D. M. Neumark, Time-resolved photoelectron imaging of large anionic methanol clusters: ðMethanolÞn (n . 145–535). J. Chem. Phys. 126, 244306 (2007). 138. Y. Ferro, A. Allouche, and V. Kempter, Electron solvation by highly polar molecules: Density functional theory study of atomic sodium interaction with water, ammonia, and methanol. J. Chem. Phys. 120, 8683–8691 (2004). 139. Y. Liu, Corey A. Rice, and M. A. Suhm, Torsional isomers in methylated aminoethanols: A jetFTIR study. Can. J. Chem. 82, 1006–1012 (2004). 140. B. C. Iba˘nescu, O. May, A. Monney, and M. Allan, Electron-induced chemistry of alcohols. Phys. Chem. Chem. Phys. 9, 3163–3173 (2007). 141. P. C. Blainey and P. J. Reid, FTIR studies of intermolecular hydrogen bonding in halogenated ethanols. Spectrochim. Acta Part A 57, 2763–2774 (2001).
hydrogen bond dynamics in alcohol clusters
49
142. M. A. F. H. van den Broek, H.-K. Nienhuys, and H. J. Bakker, Vibrational dynamics of the C O stretch vibration in alcohols. J. Chem. Phys. 114, 3182–3186 (2001). 143. C. L. Lugez, F. J. Lovas, J. T. Hougen, and N. Ohashi, Global analysis of a-, b-, and c-type transitions involving tunneling components of K ¼ 0 and 1 states of the methanol dimer. J. Mol. Spectrosc. 194, 95–112 (1999). 144. N. Borho and Y. Xu, Lock-and-key principle on a microscopic scale: The case of the propylene oxide–ethanol complex. Angew. Chem. Int. Ed. 46, 2276–2279 (2007). 145. S. Melandri, P. G. Favero, and W. Caminati, Detection of the syn conformer of allyl alcohol by free jet microwave spectroscopy. Chem. Phys. Lett. 223, 541–545 (1994). 146. L. H. Xu, G. T. Fraser, F. J. Lovas, R. D. Suenram, C. W. Gillies, H. E. Warner, and J. Z. Gillies, The microwave-spectrum and OH internal-rotation dynamics of gauche-2,2,2-trifluoroethanol. J. Chem. Phys. 103, 9541–9548 (1995). 147. H. Møllendal, Structural and conformational properties of 1,1,1-trifluoro-2-propanol investigated by microwave spectroscopy and quantum chemical calculations. J. Phys. Chem. A 109, 9488–9493 (2005). 148. K. O. Douglass, J. E. Johns, P. M. Nair, G. G. Brown, F. S. Rees, and B. H. Pate, Applications of Fourier transform microwave (FTMW) detected infrared-microwave double-resonance spectroscopy to problems in vibrational dynamics. J. Mol. Spectrosc. 239, 29–40 (2006). 149. A. Westphal, C. Jacoby, C. Ratzer, A. Reichelt, and M. Schmitt, Determination of the intermolecular geometry of the phenol-methanol cluster. Phys. Chem. Chem. Phys. 5, 4114–4122 (2003). 150. J. T. Yi and D. W. Pratt, Rotationally resolved electronic spectroscopy of tryptophol in the gas phase. Phys. Chem. Chem. Phys. 7, 3680–3684 (2005). 151. D. J. Nesbitt, High-resolution, direct infrared laser absorption spectroscopy in slit supersonic jets: Intermolecular forces and unimolecular vibrational dynamics in clusters. Annu. Rev. Phys. Chem. 45, 367–399 (1994). 152. D. Luckhaus, M. Quack, U. Schmitt, and M. A. Suhm, On FTIR spectroscopy in asynchronously pulsed supersonic jet expansions and on the interpretation of stretching spectra of HF clusters. Ber. Bunsenges. Phys. Chem. 99, 457–468 (1995). 153. C. A. Rice, N. Borho, and M. A. Suhm, Dimerization of pyrazole in slit jet expansions. Z. Phys. Chem. 219, 379–388 (2005). 154. N. Borho, M. A. Suhm, K. Le Barbu-Debus, and A. Zehnacker, Intra- vs. intermolecular hydrogen bonding: Dimers of alpha-hydroxyesters with methanol. Phys. Chem. Chem. Phys. 8, 4449–4460 (2006). 155. T. N. Wassermann, D. Luckhaus, S. Coussan, and M. A. Suhm, Proton tunneling estimates for malonaldehyde vibrations from supersonic jet and matrix quenching experiments. Phys. Chem. Chem. Phys. 8, 2344–2348 (2006). 156. Y. Liu, M. Weimann, and M. A. Suhm, Extension of panoramic cluster jet spectroscopy into the far infrared: Low frequency modes of methanol and water clusters. Phys. Chem. Chem. Phys. 6, 3315–3319 (2004). 157. R. A. Provencal, R. N. Casaes, K. Roth, J. B. Paul, C. N. Chapo, R. J. Saykally, G. S. Tschumper, and H. F. Schaefer, Hydrogen bonding in alcohol clusters: A comparative study by infrared cavity ringdown laser absorption spectroscopy. J. Phys. Chem. A 104, 1423–1429 (2000). 158. T. Ha¨ber, U. Schmitt, C. Emmeluth, and M. A. Suhm, Ragout-jet FTIR spectroscopy of cluster isomerism and cluster dynamics: From carboxylic acid dimers to N2O nanoparticles. Faraday
50
martin a. suhm Discuss. 118, 331–359 (2001) þ contributions to the discussion on pp. 53, 119, 174–175, 179– 180, 304–309, 361–363, 367–370.
159. M. Herman, R. Georges, M. Hepp, and D. Hurtmans, High resolution Fourier transform spectroscopy of jet-cooled molecules. Int. Rev. Phys. Chem. 19, 277–325 (2000). 160. M. Behrens, R. Fro¨chtenicht, M. Hartmann, J.-G. Siebers, U. Buck, and F. C. Hagemeister, Vibrational spectroscopy of methanol and acetonitrile clusters in cold helium droplets. J. Chem. Phys. 111, 2436–2443 (1999). 161. W. Richter and D. Schiel, Raman spectrometric investigation of the molecular structure of alcohols; Conformational dynamics of ethanol. Ber. Bunsenges. Phys. Chem. 85, 548–552 (1981). 162. B. Mate´, I. A. Graur, T. Elizarova, I. Chirokov, G. Tejeda, J. M. Ferna´ndez, and S. Montero, Experimental and numerical investigation of an axisymmetric supersonic jet. J. Fluid Mech. 426, 177–197 (2001). 163. J. M. Ferna´ndez, G. Tejeda, A. Ramos, B. J. Howard, and S. Montero, Gas-phase Raman spectrum of NO dimer, J. Mol. Spec. 194, 278–280 (1999). 164. G. Tejeda, J. M. Ferna´ndez, S. Montero, D. Blume, and J. P. Toennies, Raman spectroscopy of small para-H2 clusters formed in cryogenic free jets. Phys. Rev. Lett. 92, 223401–1 (2004). 165. P. M. Felker, P. M. Maxton, and M. W. Schaeffer, Nonlinear Raman studies of weakly bound complexes and clusters in molecular beams. Chem. Rev. 94, 1787–1805 (1994). 166. T. Ebata, T. Watanabe, and N. Mikami, Evidence for the cyclic form of phenol trimer: Vibrational spectroscopy of the OH stretching vibrations of jet-cooled phenol dimer and trimer, J. Phys. Chem. 99, 5761–5764 (1995). 167. Y. Matsuda, M. Hachiya, A. Fujii, and N. Mikami, Stimulated Raman spectroscopy combined with vacuum ultraviolet photoionization: Application to jet-cooled methanol clusters as a new vibrational spectroscopic method for size-selected species in the gas phase. Chem. Phys. Lett. 442, 217–219 (2007). 168. M. Maroncelli, G. A. Hopkins, J. W. Nibler, and T. R. Dyke, Coherent Raman and infrared spectroscopy of HCN complexes in free jet expansions and in equilibrium samples. J. Phys. Chem. 83, 2129–2146 (1985). 169. A. Furlan, S. Wu¨lfert, and S. Leutwyler, CARS spectra of the HCl dimer in supersonic jets. Chem. Phys. Lett. 153, 291–295 (1988). 170. M. G. Giorgini, Raman noncoincidence effect: A spectroscopic manifestation of the intermolecular vibrational coupling in dipolar molecular liquids. Pure Appl. Chem. 76, 157–169 (2004). 171. M. Paolantoni, P. Sassi, A. Morresi, and R. S. Cataliotti, Raman noncoincidence effect on OH stretching profiles in liquid alcohols. J. Raman Spectr. 37, 528–537 (2006). 172. Y. J. Hu, H. B. Fu, and E. R. Bernstein, Infrared plus vacuum ultraviolet spectroscopy of neutral and ionic methanol monomers and clusters: New experimental results. J. Chem. Phys. 125, 154306 (2006). 173. F. Huisken, M. Kaloudis, M. Koch, and O. Werhahn, Experimental study of the O H ring vibrations of the methanol trimer. J. Chem. Phys. 105, 8965–8968 (1996). 174. C. Steinbach, M. Fa´rnı´k, I. Ettischer, J. Siebers, and U. Buck, Isomeric transitions in sizeselected methanol hexamers probed by OH-stretch spectroscopy. Phys. Chem. Chem. Phys. 8, 2752–2756 (2006). 175. B. Brutschy, The structure of microsolvated benzene derivatives and the role of aromatic substituents. Chem. Rev. 100, 3891–3920 (2000).
hydrogen bond dynamics in alcohol clusters
51
176. Y. Nosenko, A. Kyrychenko, R. P. Thummel, J. Waluk, B. Brutschy, and J. Herbich, Fluorescence quenching in cyclic hydrogen-bonded complexes of 1H-pyrrolo[3,2-h]quinoline with methanol: Cluster size effect. Phys. Chem. Chem. Phys. 9, 3276–3285 (2007). 177. Y. Nosenko, M. Kunitski, R. P. Thummel, A. Kyrychenko, J. Herbich, J. Waluk, C. Riehn, and B. Brutschy, Detection and structural characterization of clusters with ultrashort-lived electronically excited states: IR absorption detected by femtosecond multiphoton ionization. J. Am. Chem. Soc. 128, 10000 (2006). 178. N. Guchhait, T. Ebata, and N. Mikami, Discrimination of rotamers of aryl alcohol homologues by infrared–ultraviolet double-resonance spectroscopy in a supersonic jet. J. Am. Chem. Soc. 121, 5705–5711 (1999). 179. M. Mons, E. G. Robertson, and J. P. Simons, Intra- and intermolecular p-type hydrogen bonding in aryl alcohols: UV and IR–UV ion dip spectroscopy. J. Phys. Chem. A 104, 1430– 1437 (2000). 180. R. N. Pribble, F. C. Hagemeister, and T. S. Zwier, Resonant ion-dip infrared spectroscopy of benzene-ðmethanolÞm clusters with m ¼ 1–6. J. Chem. Phys. 106, 2145–2157 (1997). 181. Y. Matsumoto, T. Ebata, and N. Mikami, Structure and photoinduced excited state keto-enol tautomerization of 7-hydroxyquinoline-ðCH3 OHÞn clusters. J. Phys. Chem. A 106, 5591–5599 (2002). 182. C. Plu¨tzer, C. Jacoby, and M. Schmitt, Internal rotation and intermolecular vibrations of the phenol–methanol cluster: A comparison of spectroscopic results and ab initio theory. J. Phys. Chem. A 106, 3998–4004 (2002). 183. V. A. Shubert and T. S. Zwier, IR–IR–UV hole-burning: Conformation specific IR spectra in the face of UV spectral overlap. J. Phys. Chem. A 111, 13283–13286 (2007). 184. Y. J. Hu, H. B. Fu, and E. R. Bernstein, Infrared plus vacuum ultraviolet spectroscopy of neutral and ionic ethanol monomers and clusters. J. Chem. Phys. 125, 154305 (2006). 185. Y. J. Hu, H. B. Fu, and E. R. Bernstein, IR plus vacuum ultraviolet spectroscopy of neutral and ionic organic acid monomers and clusters: Propanoic acid. J. Chem. Phys. 125, 184309 (2006). 186. M. Abu-samha, K. J. Borve, J. Harnes, and H. Bergersen, What can C1s photoelectron spectroscopy tell about structure and bonding in clusters of methanol and methyl chloride. J. Phys. Chem. A 111, 8903–8909 (2007). 187. J. M. Wang, P. Cieplak, and P. A. Kollmann, How well does a restrained electrostatic potential (RESP) model perform in calculating conformational energies of organic and biological molecules? J. Comput. Chem. 21, 1049–1074 (2000). 188. W. T. M. Mooij, F. B. van Duijneveldt, J. G. C. M. van-Duijneveldt–van de Rijdt, and Bouke P. van Eijck, Transferable ab initio intermolecular potentials. 1. derivation from methanol dimer and trimer calculations. J. Phys. Chem. A 103, 9872–9882 (1999). 189. U. Buck, J.-G. Siebers, and R. J. Wheatley, Structure and vibrational spectra of methanol clusters from a new potential model. J. Chem. Phys. 108, 20–32 (1998). 190. M. Karplus, Molecular dynamics of biological macromolecules: A brief history and perspective. Biopolymers 68, 350–358 (2003). 191. M. Gerhards, K. Beckmann, and K. Kleinermanns, Vibrational analysis of phenol(methanol)1. Z. Phys. D 29, 223–229 (1994). 192. H.-J. Werner, F. R. Manby, and P. J. Knowles, Fast linear scaling second-order Møller–Plesset perturbation theory (MP2) using local and density fitting approximations. J. Chem. Phys. 118, 8149–8160 (2003). 193. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, J. A. Montgomery, Jr., T. Vreven, K. N. Kudin, J. C. Burant, J. M. Millam, S. S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G. A. Petersson, H. Nakatsuji,
52
martin a. suhm
M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J. E. Knox, H. P. Hratchian, J. B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, P. Y. Ayala, K. Morokuma, G. A. Voth, P. Salvador, J. J. Dannenberg, V. G. Zakrzewski, S. Dapprich, A. D. Daniels, M. C. Strain, O. Farkas, D. K. Malick, A. D. Rabuck, K. Raghavachari, J. B. Foresman, J. V. Ortiz, Q. Cui, A. G. Baboul, S. Clifford, J. Cioslowski, B. B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. L. Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, M. Challacombe, P. M. W. Gill, B. Johnson, W. Chen, M. W. Wong, C. Gonzalez, and J. A. Pople, Gaussian 03, Revisions B.04 and C.02., Gaussian Inc., Pittsburgh PA, 2003. 194. C. Maerker, P. von Rague´ Schleyer, R. Liedl, T.-K. Ha, M. Quack, and M. A. Suhm, A critical analysis of electronic density functionals for structural, energetic, dynamic and magnetic properties of hydrogen fluoride clusters. J. Comput. Chem. 18, 1695–1719 (1997). 195. O. Mo´, M. Ya´n˜ez, and J. Elguero, Study of the methanol trimer potential energy surface. J. Chem. Phys. 107, 3592–3601 (1997). 196. F. C. Hagemeister, C. J. Gruenloh, and T. S. Zwier, Density functional theory calculations of the structures, binding energies, and infrared spectra of methanol clusters. J. Phys. Chem. A 102, 82–94 (1998). 197. S. L. Boyd and R. J. Boyd, A density functional study of methanol clusters. J. Chem. Theory Comput. 3, 54–61 (2007). 198. T. Schwabe and S. Grimme, Double-hybrid density functionals with long-range dispersion corrections: Higher accuracy and extended applicability. Phys. Chem. Chem. Phys. 9, 3397– 3406 (2007). 199. A. Bleiber and J. Sauer, The vibrational frequency of the donor OH group in the H-bonded dimers of water, methanol and silanol. Ab initio calculations including anharmonicities. Chem. Phys. Lett. 238, 243–252 (1995). 200. T. H. Dunning, Jr, A road map for the calculation of molecular binding energies. J. Phys. Chem. A 104, 9062–9080 (2000). 201. E. M. Mas, R. Bukowski, and K. Szalewicz, Ab initio three-body interactions for water. I. Potential and structure of water trimer. J. Chem. Phys. 118, 4386–4403 (2003). 202. W. Klopper, M. Quack, and M. A. Suhm, HF dimer: Empirically refined analytical potential energy and dipole hypersurfaces from ab initio calculations. J. Chem. Phys. 108, 10096–10115 (1998). 203. D. J. Wales, J. P. K. Doye, M. A. Miller, P. N. Mortenson, and T. R. Walsh, Energy landscapes: From clusters to biomolecules. Adv. Chem. Phys. 115, 1–111 (2000). 204. F. Schulz and B. Hartke, Structural information on alkali cation microhydration clusters from infrared spectra. Phys. Chem. Chem. Phys. 5, 5021–5030 (2003). 205. C. Herrmann, J. Neugebauer, and M. Reiher, Finding a needle in a haystack: direct determination of vibrational signatures in complex systems. New J. Chem. 31, 818–831 (2007). 206. V. Barone, Anharmonic vibrational properties by a fully automated second-order perturbative approach. J. Chem. Phys. 122, 014108 (2005). 207. S. K. Gregurick, J. H.-Y. Liu, D. A. Brant, and R. B. Gerber, Anharmonic vibrational selfconsistent field calculations as an approach to improving force fields for monosaccharides. J. Phys. Chem. B 103, 3476–3488 (1999). 208. J. M. Bowman, S. Carter, and X. C. Huang, MULTIMODE: a code to calculate rovibrational energies of polyatomic molecules. Int. Rev. Phys. Chem. 22, 533–549 (2003). 209. X.-G. Wang and T. Carrington, Jr., Six-dimensional variational calculation of the bending energy levels of HF trimer and DF trimer. J. Chem. Phys. 115, 9781–9796 (2001).
hydrogen bond dynamics in alcohol clusters
53
210. M. D. Coutinho-Neto, A. Viel, and U. Manthe, The ground state tunneling splitting of malonaldehyde: Accurate full dimensional quantum dynamics calculations. J. Chem. Phys. 121, 9207–9210 (2004). 211. G. V. Mil’nikov, O. Ku¨hn, and H. Nakamura, Ground-state and vibrationally assisted tunneling in the formic acid dimer. J. Chem. Phys. 123, 074308 (2005). 212. J.-W. Handgraaf, E. J. Meijer, and M.-P. Gaigeot, Density-functional theory-based molecular simulation study of liquid methanol. J. Chem. Phys. 121, 10111–10119 (2004). 213. G. S. Fanourgakis, Y. J. Shi, S. Consta, and R. H. Lipson, A spectroscopic and computer simulation study of butanol vapors. J. Chem. Phys. 119, 6597–6608 (2003). 214. A. Serrallach, R. Meyer, and H. H. Gu¨nthard, Methanol and deuterated species: Infrared data, valence force field, rotamers, and conformation. J. Mol. Spectrosc. 52, 94–129 (1974). 215. J. Florian, J. Leszczynski, B. G. Johnson, and L. Goodman, Coupled-cluster and density functional calculations of the molecular structure, infrared spectra, Raman spectra, and harmonic force constants for methanol. Mol. Phys. 91, 439–447 (1977). 216. R. H. Hunt, W. N. Shelton, F. A. Flaherty, and W. B. Cook, Torsion–rotation energy levels and the hindering potential barrier for the excited vibrational state of the OH-stretch fundamental band n1 of methanol. J. Mol. Spectrosc. 192, 277–293 (1998). 217. O. V. Boyarkin, T. R. Rizzo, and D. S. Perry, Intramolecular energy transfer in highly vibrationally excited methanol. II. Multiple time scales of energy redistribution. J. Chem. Phys. 110, 11346–11358 (1999). 218. V. Venkatesan, A. Fujii, T. Ebata, and N. Mikami, Infrared and ab initio studies on 1,2,4,5tetrafluorobenzene clusters with methanol and 2,2,2-trifluoroethanol: Presence and absence of an aromatic C H ! ! ! O hydrogen bond. J. Phys. Chem. A 109, 915–921 (2005). 219. G. S. Tschumper, J. M. Gonzales, and H. F. Schaefer III, Assignment of the infrared spectra of the methanol trimer. J. Chem. Phys. 111, 3027–3034 (1999). 220. M. Mandado, A. M. Grana, and R. A. Mosquera, On the structures of the methanol trimer and their cooperative effects. Chem. Phys. Lett. 381, 22–29 (2003). 221. W. F. Passchier, E. R. Klompmaker, and M. Mandel, Absorption spectra of liquid and solid methanol from 1000 to 100 cm 1 . Chem. Phys. Lett. 4, 485–488 (1970). 222. J. R. Durig, C. B. Pate, Y. S. Li, and D. J. Antion, Far-infrared and Raman spectra of solid methanol and methanol-d4 . J. Chem. Phys. 54, 4863–4870 (1971). 223. K. N. Woods and H. Wiedemann, The influence of chain dynamics on the far-infrared spectrum of liquid methanol. J. Chem. Phys. 123, 134506 (2005). 224. P.-G. Jo¨nsson, Hydrogen bond studies. CXIII. The crystal structure of ethanol at 87 K. Acta Crystallogr. B32, 232–235 (1976). 225. M. Rozenberg, A. Loewenschuss, and Y. Marcus, IR spectra and hydrogen bonding in ethanol crystals at 18–150 K. Spectrochim. Acta Part A 53, 1969–1974 (1997). 226. A. J. Barnes and H. E. Hallam, Infrared cryogenic studies, part 5—Ethanol and ethanol-d in argon matrices. Trans. Faraday Soc. 66, 1932–1940 (1970). 227. N. Nishi, S. Takahashi, M. Matsumoto, A. Tanaka, K. Muraya, T. Takamuku, and T. Yamaguchi, Hydrogen bonding cluster formation and hydrophobic solute association in aqueous solution of ethanol. J. Phys. Chem. 99, 462–468 (1995). 228. M. Nedic´, T. N. Wassermann, Z. Xue, P. Zielke, M. A. Suhm, Raman spectroscopic evidence for the most stable water/ethanol dimer and for the negative mixing energy in cold water/ethanol trimers. Phys. Chem. Chem. Phys. 10, 5953–5956 (2008).
54
martin a. suhm
229. D. Spangenberg, P. Imhof, W. Roth, C. Janzen, and K. Kleinermanns, Phenol-(ethanol)1 isomers studied by double-resonance spectroscopy and ab initio calculations. J. Phys. Chem. A 103, 5918–5924 (1999). 230. B. T. G. Lutz and J. H. van der Maas, The sensorial potentials of the OH stretching mode. J. Mol. Struct. 436–437, 213–231 (1997). 231. K. Ohno, H. Yoshida, H. Watanabe, T. Fujita, and H. Matsuura, Conformational study of 1-butanol by the combined use of vibrational spectroscopy and ab initio molecular orbital calculations. J. Phys. Chem. 98, 6924–6930 (1994). 232. K. Kahn and T. C. Bruice, Focal-point conformational analysis of ethanol, propanol, and isopropanol. ChemPhysChem 6, 487–495 (2005). 233. A. V. Bochenkova, M. A. Suhm, A. A. Granovsky, and A. V. Nemukhin, Hybrid diatomics-inmolecules-based quantum mechanical/molecular mechanical approach applied to the modeling of structures and spectra of mixed molecular clusters Arn ðHClÞm and Arn ðHFÞm . J. Chem. Phys. 120, 3732–3743 (2004). 234. B. Lutz, R. Scherrenberg, and J. van der Maas, The associative behaviour of saturated alcohols with reference to the molecular structure in the OH proximity. An FT–IR temperature study. J. Mol. Struct. 175, 233–238 (1988). 235. M. Saeki, S.-i. Ishiuchi, M. Sakai, and M. Fujii, Structure of 1-naphthol/alcohol clusters studied by IR dip spectroscopy and ab initio molecular orbital calculations. J. Phys. Chem. A 105, 10045–10053 (2001). 236. D. T. Bowron, J. L. Finney, and A. K. Soper, The structure of pure tertiary butanol. Mol. Phys. 93, 531–543 (1998). 237. D. T. Bowron and J. L. Finney, Association and dissociation of an aqueous amphiphile at elevated temperatures. J. Phys. Chem. B 111, 9838–9852 (2007). 238. A. V. Iogansen and M. S. Rozenberg, Infrared bands of the OH (OD) group, hydrogen bonding, and special features of the structure of crystals of tert-butanol at 15–300 K. J. Struct. Chem. 30, 76–83 (1989). 239. H. Serrano-Gonza´les, K. D. M. Harris, C. C. Wilson, A. E. Aliev, S. J. Kitchin, B. M. Kariuki, M. Bach-Verge´s, C. Glidewell, E. J. MacLean, and W. W. Kagunya, Unravelling the disordered hydrogen bonding arrangement in solid triphenylmethanol. J. Phys. Chem. B 103, 6215–6223 (1999). 240. F. Vanhouteghem, W. Pyckhout, C. Van Alsenoy, L. Van den Enden, and H. J. Geise, The molecular structure of gaseous allyl alcohol determined from electron diffraction, microwave, infrared and geometry-relaxed ab-initio data. J. Mol. Struct. 140, 33–48 (1986). 241. A. Aspiala, T. Lotta, J. Murto, and M. Ra¨sa¨nen, IR-induced processes for allyl alcohol dimers involving an intermolecular OH . . . p bond in low-temperature matrices. Chem. Phys. Lett. 112, 469–472 (1984). 242. T. Baba, T. Tanaka, I. Morino, K. M. T. Yamada, and K. Tanaka, Detection of the tunnelingrotation transitions of malonaldehyde in the submillimeter-wave region. J. Chem. Phys. 110, 4131–4133 (1999). 243. C. Duan and D. Luckhaus, High resolution IR-diode laser jet spectroscopy of malonaldehyde. Chem. Phys. Lett. 391, 129–133 (2004). 244. R. L. Redington and T. E. Redington, Implications of comparative spectral doublets observed for neon-isolated and gaseous tropolone(OH) and tropolone(OD). J. Chem. Phys. 122, 124304, (2005). 245. C. Emmeluth, M. A. Suhm, and D. Luckhaus, A monomers-in-dimers model for carboxylic acid dimers. J. Chem. Phys. 118, 2242–2255 (2003).
hydrogen bond dynamics in alcohol clusters
55
246. M. Schmitt, U. Henrichs, H. Mu¨ller, and K. Kleinermanns, Intermolecular vibrations of the phenol dimer revealed by spectral hole-burning and dispersed fluorescence spectroscopy. J. Chem. Phys. 103, 9918–9928 (1995). 247. M. Schmitt, M. Bo¨hm, C. Ratzer, D. Kru¨gler, K. Kleinermanns, I. Kalkman, G. Berden, and W. Leo Meerts, Determining the intermolecular structure in the S0 and S1 states of the phenol dimer by rotationally resolved electronic spectroscopy. ChemPhysChem 7, 1241–1249 (2006). 248. W. J. Middleton and R. V. Lindsey, Jr., Hydrogen bonding in fluoro alcohols. J. Am. Chem. Soc. 86, 4948–4952 (1964). 249. A. Kova´cs and Z. Varga, Halogen acceptors in hydrogen bonding. Coord. Chem. Rev. 250, 710– 727 (2006). 250. J. D. Dunitz and R. Taylor, Organic fluorine hardly ever accepts hydrogen bonds. Chemistry Eur. J. 3, 89–98 (1997). 251. R. D. Suenram, F. J. Lovas, and H. M. Pickett, Fluoromethanol: Synthesis, microwave spectrum, and dipole moment. J. Mol. Spectrosc. 119, 446–455 (1986). 252. S. Andreades and D. C. England, a-haloalcohols. J. Am. Chem. Soc. 83, 4670–4671 (1961). 253. J. M. Bakke, L. H. Bjerkeseth, T. E. C. L. Rønnow, and K. Steinsvoll, The conformation of 2fluoroethanol—Is intramolecular hydrogen bonding important? J. Mol. Struct. 321, 205–214 (1994). 254. P. Maienfisch and R. G. Hall, The importance of fluorine in the life science industry. Chimia 58, 93–99 (2004). 255. J. W. Grate, Hydrogen-bond acidic polymers for chemical vapor sensing. Chem. Rev. 108, 726– 745 (2008). 256. J.-P. Be´gue´, D. Bonnet-Delpon, and B. Crousse, Fluorinated alcohols: A new medium for selective and clean reaction. Synlett, 18–29 (2004). 257. A. Berkessel and J. A. Adrio, Dramatic acceleration of olefin epoxidation in fluorinated alcohols: Activation of hydrogen peroxide by multiple H-bond networks. J. Am. Chem. Soc. 128, 13412–13420 (2006). 258. M. Buck, Trifluoroethanol and colleagues: cosolvents come of age. recent studies with peptides and proteins. Q. Rev. Biophys. 31, 297–355 (1998). 259. D. Roccatano, G. Colombo, M. Fioroni, and A. E. Mark, Mechanism by which 2,2,2trifluoroethanol/water mixtures stabilize secondary-structure formation in peptides: A molecular dynamics study. Proc. Natl. Acad. Sci. USA 99, 12179–12184 (2002). 260. A. Starzyk, W. Barber-Armstrong, M. Sridharan, and S. M. Decatur, Spectroscopic evidence for backbone desolvation of helical peptides by 2,2,2-trifluoroethanol: An isotope-edited FTIR study. Biochemistry 44, 369–376 (2005). 261. K. O. Christe, J. Hegge, B. Hoge, and R. Haiges, Convenient access to trifluoromethanol. Angew. Chem. Int. Ed. 46, 6155–6158 (2007). 262. M. Perttila, Vibrational-spectra and normal coordinate analysis of 2,2,2-trichloroethanol and 2,2,2-trifluoroethanol. Spectrochim. Acta Part A 35, 585–592 (1978). 263. M. L. Senent, A. Perez-Ortega, A. Arroyo, and R. Domı´nguez-Go´mez, Theoretical investigation of the torsional spectra of 2,2,2-trifluoroethanol. Chem. Phys. 266, 19–32 (2001). 264. I. Bako, T. Radnai, and M. C. Bellissent-Funel, Investigation of structure of liquid 2,2,2trifluoroethanol: Neutron diffraction, molecular dynamics, and ab initio quantum chemical study. J. Chem. Phys. 121, 12472–12480 (2004). 265. H. L. Fang and R. L. Swofford, Molecular conformers in gas-phase ethanol: A temperature study of vibrational overtones. Chem. Phys. Lett. 105, 5–11 (1984).
56
martin a. suhm
266. U. Erlekam, M. Frankowski, G. von Helden, and G. Meijer, Cold collisions catalyse conformational conversion. Phys. Chem. Chem. Phys. 9, 3786–3789 (2007). 267. M. Fioroni, K. Burger, A. E. Mark, and D. Roccatano, A new 2,2,2-trifluoroethanol model for molecular dynamics simulations. J. Phys. Chem. B 104, 12347–12354 (2000). 268. M. L. Senent, A. Nino, C. Munoz-Caro, Y. G. Smeyers, R. Dominguez-Gomez, and J. M. Orza, Theoretical study of the effect of hydrogen-bonding on the stability and vibrational spectrum of isolated 2,2,2-trifluoroethanol and its molecular complexes. J. Phys. Chem. A 106, 10673– 10680 (2002). 269. H. Schaal, T. Ha¨ber, and M. A. Suhm, Hydrogen bonding in 2-propanol: The effect of fluorination. J. Phys. Chem. A 104, 265–274 (2000). 270. M. Fioroni, K. Burger, and D. Roccatano, Chiral discrimination in liquid 1,1,1-trifluoropropan2-ol: A molecular dynamics study. J. Chem. Phys. 119, 7289–7296 (2003). 271. T. J. Wallington, W. F. Schneider, I. Barnes, K. H. Becker, J. Sehested, and O. J. Nielsen, Stability and infrared spectra of mono-, di-, and trichloromethanol. Chem. Phys. Lett. 322, 97– 102 (2000). 272. M. Jackson and H. H. Mantsch, Halogenated alcohols as solvents for proteins: FTIR spectroscopic studies. Biochim. Biophys. Acta 1118, 139–143 (1992). 273. V. Moulin, A. Schriver, L. Schriver-Mazzuoli, and P. Chaquin, Infrared spectrum of 2,2,2trichloroethanol isolated in gas matrices. Ab initio optimization of conformers and potential energy calculations. Chem. Phys. Lett. 263, 423–428 (1996). 274. N. Borho and M. A. Suhm, Tailor-made aggregates of a-hydroxy esters in supersonic jets. Phys. Chem. Chem. Phys. 6, 2885–2890 (2004). 275. M. Fa´rnı´k, M. Weimann, C. Steinbach, U. Buck, N. Borho, T. B. Adler, and M. A. Suhm, Sizeselected methyl lactate clusters: Fragmentation and spectroscopic fingerprints of chiral recognition. Phys. Chem. Chem. Phys. 8, 1148–1158 (2006). 276. R. Iwamoto, T. Matsuda, and H. Kusanagi, Contrast effect of hydrogen bonding on the acceptor and donor OH groups of intramolecularly hydrogen-bonded OH pairs in diols. Spectrochim. Acta Part A 62, 97–104 (2005). 277. D. L. Howard and H. G. Kjaergaard, Influence of intramolecular hydrogen bond strength on OH-stretching overtones. J. Phys. Chem. A 110, 10245–10250 (2006). 278. D. Christen and H. S. P. Mu¨ller, The millimeter wave spectrum of aGg0 ethylene glycol: The quest for higher precision. Phys. Chem. Chem. Phys. 5, 3600–3605 (2003). 279. D. P. Geerke and W. F. Van Gunsteren, The performance of non-polarizable and polarizable force-field parameter sets for ethylene glycol in molecular dynamics simulations of the pure liquid and its aqueous mixtures. Mol. Phys. 105, 1861–1881 (2007). 280. R. Chelli, F. L. Gervasio, C. Gellini, P. Procacci, G. Cardini, and V. Schettino, Density functional calculation of structural and vibrational properties of glycerol. J. Phys. Chem. A 104, 5351–5357 (2000). 281. C. S. Callam, S. J. Singer, T. L. Lowary, and C. M. Hadad, Computational analysis of the potential energy surfaces of glycerol in the gas and aqueous phases: Effects of level of theory, basis set, and solvation on strongly intramolecularly hydrogen-bonded systems. J. Am. Chem. Soc. 123, 11743–11754 (2001). 282. G. Maccaferri, W. Caminati, and P. G. Favero, Free jet investigation of the rotational spectrum of glycerol. J. Chem. Soc. Faraday Trans. 93, 4115–4117 (1997). 283. M. Jetzki, D. Luckhaus, and R. Signorell, Fermi resonance and conformation in glycolaldehyde particles. Can. J. Chem. 82, 915–924 (2004).
hydrogen bond dynamics in alcohol clusters
57
284. M. Jetzki and R. Signorell, The competition between hydrogen bonding and chemical change in carbohydrate nanoparticles. J. Chem. Phys. 117, 8063–8073 (2002). 285. R. A. Jokusch, F. O. Talbot, and J. P. Simons, Sugars in the gas phase—Part 2: The spectroscopy and structure of jet-cooled phenyl b-D-galactopyranoside. Phys. Chem. Chem. Phys. 5, 1502– 1507 (2003). 286. E. C. Stanca-Kaposta, D. P. Gamblin, J. Screen, B. Liu, L. C. Snoek, B. G. Davis, and J. P. Simons, Carbohydrate molecular recognition: A spectroscopic investigation of carbohydratearomatic interactions. Phys. Chem. Chem. Phys. 9, 4444–4451 (2007). 287. R. Signorell, M. K. Kunzmann, and M. A. Suhm, FTIR investigation of non-volatile molecular nanoparticles. Chem. Phys. Lett. 329, 52–60 (2000). 288. G. Firanescu, D. Hermsdorf, R. Ueberschaer, and R. Signorell, Large molecular aggregates: From atmospheric aerosols to drug nanoparticles. Phys. Chem. Chem. Phys. 8, 4149–4165 (2006). 289. M. Quack and M. A. Suhm, Spectroscopy and quantum dynamics of hydrogen fluoride clusters, in Molecular Clusters, Advances in Molecular Vibrations and Collision Dynamics, Vol. III, J. M. Bowman and Z. Bacˇic´, eds., JAI Press, London, 1998, pp. 205–248. 290. M. Quack, Spectra and dynamics of coupled vibrations in polyatomic molecules. Annu. Rev. Phys. Chem. 41, 839–874 (1990). 291. T. Scharge, T. N. Wassermann, M. A. Suhm, Weak hydrogen bonds make a difference: Dimers of jet-cooled halogenated ethanols. Z. Phys. Chem. 222, 1407–1452 (2008).
VIBRATIONAL LINE SHAPES, SPECTRAL DIFFUSION, AND HYDROGEN BONDING IN LIQUID WATER JAMES L. SKINNER, BENJAMIN M. AUER, AND YU-SHAN LIN Theoretical Chemistry Institute and Department of Chemistry, University of Wisconsin, Madison, Wisconsin 53706, USA CONTENTS I. Introduction II. Theoretical Formalisms A. Line Shapes 1. Classical Approach 2. Mixed Quantum/Classical Approach: Single Chromophore 3. Mixed Quantum/Classical Approach: Coupled Chromophores and the Time-Averaging Approximation 4. Raman Line Shapes B. Echoes and Other Nonlinear Experiments III. Theoretical Implementation A. Approaches Based on the Simulation Potential B. Approaches Based on Ab Initio Calculations on a Single Water Molecule in an Inhomogeneous Electric Field C. Approaches Based on Ab Initio Calculations on Water Clusters D. Calculation of Transition Dipoles E. Calculation of Transition Polarizabilities F. Calculation of Coupling Frequencies G. Hydrogen Bonding IV. HOD/D2O A. Line Shapes B. Hydrogen Bonding C. Two-Pulse Echoes D. Three-Pulse Echoes and 2DIR Spectra E. Frequency-Dependent Anisotropy Decay V. HOD/H2O and HOT/H2O A. Line Shapes B. Three-Pulse Echoes and 2DIR Spectra C. Frequency-Dependent Anisotropy Decay
Advances in Chemical Physics, Volume 142, edited by Stuart A. Rice Copyright # 2009 John Wiley & Sons, Inc.
59
60
james l. skinner, benjamin m. auer, and yu-shan lin
VI. H2O A. Line Shapes B. Ultrafast Experiments VII. Conclusions Acknowledgments References
I.
INTRODUCTION
Water is endlessly fascinating and important. From Empedocles in ancient Greece to Tsou Yen in ancient China, water was thought to be one of the four or five elements of the universe. Today it plays central roles in diverse fields from earth and atmospheric sciences to biology. The pure substance has a complicated and anomalous phase diagram, due in large part to the important and somewhat elusive concept of hydrogen bonding. All of this and much more is discussed in the delightful, informative, and engaging book by Philip Ball [1]. Even a single phase of water, the liquid, has yet to give up all its secrets. In his provocative recent essay entitled ‘‘Water—An Enduring Mystery,’’ Ball summarizes the current state of affairs in his first sentence: ‘‘No one really understands water’’ [2]. Vibrational spectroscopy can help us escape from this predicament due to the exquisite sensitivity of vibrational frequencies, particularly of the OH stretch, to local molecular environments. Thus, very roughly, one can think of the infrared or Raman spectrum of liquid water as reflecting the distribution of vibrational frequencies sampled by the ensemble of molecules, which reflects the distribution of local molecular environments. This picture is oversimplified, in part as a result of the phenomenon of motional narrowing: The vibrational frequencies fluctuate in time (as local molecular environments rearrange), which causes the line shape to be narrower than the distribution of frequencies [3]. Thus in principle, in addition to information about liquid structure, one can obtain information about molecular dynamics from vibrational line shapes. In practice, however, it is often hard to extract this information. Recent and important advances in ultrafast vibrational spectroscopy provide much more useful methods for probing dynamic frequency fluctuations, a process often referred to as spectral diffusion. Ultrafast vibrational spectroscopy of water has also been used to probe molecular rotation and vibrational energy relaxation. The latter process, while fundamental and important, will not be discussed in this chapter, but instead will be covered in a separate review [4]. In addition to the effects of motional narrowing, vibrational line shapes for the OH stretch region of water are complicated by intramolecular and intermolecular vibrational coupling. This is because (in a zeroth-order local-mode picture) all OH stretch transition frequencies in the liquid are degenerate, and so the effects of any
vibrational line shapes, spectral diffusion
61
vibrational coupling might be expected to be substantial. The intramolecular interaction between OH stretches on the same molecule arises from potential and momentum coupling [5], and for the isolated molecule it is about 50 cm 1 . Intermolecular coupling can be approximated by transition dipole interactions, whose typical magnitude for nearby pairs of OH stretches is on the order of 30 cm 1 [6]. The effects of these couplings are mitigated to some extent by the substantial transition frequency fluctuations of the local OH stretch modes. These fluctuations also serve to break the local symmetry for an individual water molecule, meaning that symmetric and antisymmetric stretch vibrations are not a useful basis set. Thus the situation can best be thought of as a vibrational exciton problem in the local mode basis, with substantial dynamic diagonal fluctuations, and complicated and dynamic off-diagonal couplings [7]. As described above, it is probably adequately clear that the vibrational spectroscopy of water is complicated indeed! One can simplify the situation considerably by considering dilute isotopic mixtures. Thus one common system is dilute HOD in D2 O. The large frequency mismatch between OH and OD stretches now effectively decouples the OH stretch from all other vibrations in the problem, meaning that the OH stretch functions as an isolated chromophore. Of course the liquid is now primarily D2 O instead of H2 O, which has slightly different structural and dynamical properties, but that is a small price to pay for the substantial simplification this modification brings to the problem. Polarized (VV) and depolarized (VH) Raman and IR line shapes for the OH stretch region of HOD/D2 O were measured some time ago [8–12]. Under ambient conditions the line shapes peak near 3400 cm 1, which represents a very significant red shift of some 300 cm 1 from the gas-phase stretch, and have full-width-half-maximum (FWHM) line widths of between 250 and 300cm 1 . The IR line shape is slightly red-shifted from the Raman line shapes, and the latter have shoulders at about 3625 cm 1 . This red shift is thought to be due to non-Condon effects [13, 14], which means that the OH stretch transition dipole depends on molecular environment, and in general increases as the OH stretch frequency decreases. The shoulder in the Raman spectra has been attributed to HOD molecules lacking a hydrogen (H) bond to the H atom [15–19]. Two- and three-pulse echo [10, 20–26] and other ultrafast [16, 17, 27–37] experiments have also been performed on this system. One such integrated three-pulse echo peak shift experiment [10] shows an interesting oscillation at short times (with a period of about 180 fs) that has been attributed to underdamped H-bond stretching [15, 23, 38–42], and a long-time decay on a 1.4-ps time scale that is thought to be due to dynamics of H-bond rearrangement [15, 16, 24, 38, 40, 42, 43]. Other ultrafast pump-probe experiments measure rotational relaxation of the OH-bond unit vector, which was found to be frequency dependent [44–48]. A similar situation occurs for the OD stretch region of dilute HOD in H2 O, where now the OD stretch functions as an isolated chromophore. In this case, IR
62
james l. skinner, benjamin m. auer, and yu-shan lin
and Raman spectra [9, 11, 12, 49–52] peak near 2400 cm 1, with widths of 160–180 cm 1 . Ultrafast echo [53–55] and pump-probe [56, 57] experiments have been performed on this system as well, the former showing a similar spectral diffusion time of about 1.4 ps. Pump-probe rotational relaxation experiments [48, 58, 59] again show a frequency dependence in the amplitude of the anisotropy decay at short times [59]. The IR spectrum has also been measured for the OT stretch of dilute HOT in H2 O [60], which shows a maximum near 2100 cm 1 and a width of about 130 cm 1 . Efforts were made to understand the scaling of the peak frequency and width as the relevant isotope varies from H to D to T [60]. Vibrational spectra for H2 O itself are quite different [49, 61–70]. The IR line shape is peaked at about 3400 cm 1 and has a weak shoulder at about 3250 cm 1 and a FWHM of about 375 cm 1 . The VV Raman line shape is bimodal, with strong peaks at about 3250 and 3400 cm 1 and a FWHM of about 425 cm 1 , while the VH line shape peaks at about 3460 cm 1 , is quite asymmetric, and has a FWHM of about 300 cm 1 . Interpretation of these spectra is controversial. Some suggest assignments based on the symmetric and antisymmetric normal modes of isolated water molecules and/or different classes of H-bonding environments [49, 63–68], while others emphasize the fundamental excitonic nature of the problem [7, 69–72], and hence the collective origin of spectral peaks. After overcoming substantial technical obstacles, primarily due to the very high absorptivity of water in the OH stretch region, ultrafast echo experiments on H2 O have recently been performed [73, 74]. Spectral evolution has been shown to evolve on a very fast (50 fs) time scale. Innovative IR pump–Raman probe experiments have also been performed on neat water in an effort to unravel the dynamics of the vibrational substructure [18, 19, 75–77]. Over the last 30 years there have appeared many calculations of these various line shapes and ultrafast observables, which start with different formalisms, emphasize different features, use different techniques, and arrive at different results. These results are in varying degrees of agreement with experiment and carry different molecular interpretations. Better simulation models, better methods for estimating frequencies, and new theoretical formulations have recently led to improved agreement between theory and experiment, presumably providing more definitive interpretations. In this chapter we review these theoretical approaches to calculating and interpreting the various line shapes and ultrafast observables. II.
THEORETICAL FORMALISMS A.
Line Shapes
We begin by considering the absorption line shape for a liquid. If the light is polarized along lab-fixed axis p, the line shape is given by the Fourier transform
vibrational line shapes, spectral diffusion
63
of the quantum dipole time-correlation function (TCF) [78]: IðoÞ # Re
Z
1
dt e
iot
Tr½rmp ð0Þmp ðtÞ&
ð1Þ
0
where mp is the component in direction p of the dipole operator for the liquid, mp ðtÞ is the Heisenberg time-dependent operator, r is the equilibrium density operator, and the trace is over the Hilbert space of the liquid. Theoretical issues for even the simplest liquid-state system preclude an exact evaluation of the line shape using this approach. 1.
Classical Approach
If one is interested in spectroscopy involving only the ground Born–Oppenheimer surface of the liquid (which would correspond to IR and far-IR spectra), the simplest approximation involves replacing the quantum TCF by its classical counterpart. Thus mp becomes a classical variable, the trace becomes a phase-space integral, and the density operator becomes the phasespace distribution function. For light frequency o with ho > kT, this classical approximation will lead to substantial errors, and so it is important to multiply the result by a quantum correction factor; the usual choice for this application is the harmonic quantum correction factor [79–84]. Thus we have IðoÞ # QH ðoÞ Re
Z
1
dt e
iot
hmp ð0Þmp ðtÞi
ð2Þ
0
where QH ðoÞ is the harmonic quantum correction factor, and the brackets denote a classical statistical mechanical average. The success of this approach depends on the accuracy of the potential surface for intermolecular and intramolecular motion, the accuracy of the form chosen for the classical dipole, and the adequacy of the classical approximation. Of course there are many models for the classical simulation of the intermolecular motion of water [85, 86], and for this problem many of these are probably quite adequate. The issue of the intramolecular motion is more problematic. Models for flexible water often add an intramolecular potential to an existing rigid water model. However, it is not clear that the assumed intermolecular interaction between rigid water molecules leads to the correct vibrational coupling in a flexible model. Furthermore, flexible water simulation models are often parameterized from vibrational spectroscopy, sometimes making meaningful comparison with (vibrational) experiments not possible. Finally there is the issue of the classical approximation, even when modified with a quantum correction factor. One problem is that for OH stretch frequencies, hoOH ) kT. That means that regions of the potential relevant for the 0 ! 1 fundamental
64
james l. skinner, benjamin m. auer, and yu-shan lin
transition are significantly higher in energy than those sampled in a classical simulation. Secondly, the OH stretch potential is very anharmonic, which means that the harmonic or nearly harmonic frequencies associated with a classical simulation will be a poor approximation to the actual transition frequencies. Fully classical approaches do, however, have three significant advantages. First, they are simple to implement; second, it is no more difficult to model H2 O, where OH stretches are coupled, than it is to model HOD/D2 O (whereas in the approaches to follow this is most certainly not the case!); and third, vibrational energy relaxation is automatically included (see below). Results from several of these classical approaches have appeared in the literature [79, 87–91]. 2.
Mixed Quantum/Classical Approach: Single Chromophore
In the mixed quantum/classical approach, one assumes that the degrees of freedom in the problem can be neatly divided into several classes. First there are the vibrations of interest, which are those being probed spectroscopically. These typically have frequencies high compared to kT, and so these are treated quantum mechanically. Secondly, there are low-frequency degrees of freedom that are associated with translations, rotations, and possibly low-frequency vibrations, which are treated classically and which modulate the high-frequency vibrations (see below). These degrees of freedom, which ideally (but not necessarily in practice) have frequencies less than kT, are often called the ‘‘bath.’’ Finally there are other degrees of freedom, typically high-frequency vibrations that are thought not to contribute to the spectroscopy of interest, which are not treated at all. Thus, for example, for the case of HOD/D2 O, for spectroscopy in the OH stretch region the single OH stretch is treated quantum mechanically. The OD stretches and all the intramolecular bends are sufficiently off-resonant from the OH stretch that they do not couple effectively; and furthermore, these modes have frequencies that are sufficiently high that they are not populated in thermal equilibrium (that is, they are all in their ground quantum states). These modes, then, are typically simply neglected. Finally, all of the translational and rotational degrees of freedom constitute the bath. In the case of neat H2 O (for spectroscopy in the OH stretch region) all of the OH stretches are treated quantum mechanically, the bends are neglected (although this is not necessarily a good approximation because of the possible 2:1 bend–stretch Fermi resonance), and the translations and rotations constitute the bath. It is easiest to formulate this problem in the case of a single high-frequency vibrational mode, or chromophore, so let us consider this situation first. For the absorption line shape, which involves only the ground and excited state of the chromophore, a crucial element is the 0 ! 1 transition frequency and its dependence on the classical bath coordinates. Second, one needs (in the case of IR spectroscopy) the projection of the transition dipole in the direction p of the electric field axis. This projection can depend on bath coordinates in two ways.
65
vibrational line shapes, spectral diffusion
First, as the molecule on which the chromophore sits rotates, this projection will change. Second, the magnitude of the transition dipole may depend on bath coordinates, which in analogy with gas-phase spectroscopy is called a nonCondon effect. For water, as we will see, this latter dependence is very important [13, 14]. In principle there are off-diagonal terms in the Hamiltonian in this truncated two-state Hilbert space, which depend on the bath coordinates and which lead to vibrational energy relaxation [4]. In practice it is usually too difficult to treat both the spectral diffusion and vibrational relaxation problems at the same time, and so one usually adds the effects of this relaxation phenomenologically, and the lifetime T1 can either be calculated separately or determined from experiment. Within this approach the line shape can be written as [92–94] IðoÞ # Re
Z
1
dt e 0
iot
D
" Z t #E mp ð0Þmp ðtÞ exp i dt0 oðt0 Þ e
t=2T1
ð3Þ
0
where mp ðtÞ is the fluctuating (since it depends on the classical bath coordinates) component of the transition dipole, oðtÞ is the fluctuating transition frequency, and the brackets indicate a classical average. To evaluate this average, the trajectories for mp ðtÞ and oðtÞ are obtained from a classical molecular dynamics simulation of the bath variables only (the high-frequency modes treated quantum mechanically and those neglected are both constrained to be rigid in this classical simulation). If the dynamics of the system are sufficiently slow, then in the above oðt0 Þ can be replaced by oð0Þ, and mp ðtÞ can be replaced by mp ð0Þ, in which case the line shape becomes IðoÞ # hmp ð0Þ2 Lðo
oð0ÞÞi
ð4Þ
where LðoÞ ¼
1=2pT1 o2
þ ð1=2T1 Þ2
ð5Þ
This is known as the inhomogeneous limit (convoluted with Lorentzian lifetime broadening). 3.
Mixed Quantum/Classical Approach: Coupled Chromophores and the Time-Averaging Approximation
Now suppose that the system of interest has N vibrational chromophores whose frequencies are in the same region, all of which need to be treated quantum mechanically. Such, for example, is the situation for the OH stretch region of
66
james l. skinner, benjamin m. auer, and yu-shan lin
H2 O, where N=2 molecules have N OH stretch chromophores. In this case we need to label the transition dipoles and frequencies by an index i that runs from 1 to N. In addition, in general these chromophores interact, with couplings (in frequency units) oij . In this case the above mixed quantum/classical formula can be generalized to [95–98] IðoÞ # Re
Z
0
1
dt e
iot
X
hmpi ð0ÞFij ðtÞmpj ðtÞie
t=2T1
ð6Þ
ij
where Fij ðtÞ are the elements of the matrix FðtÞ, which satisfies the equation _ FðtÞ ¼ iFðtÞkðtÞ
ð7Þ
subject to the initial condition that Fij ð0Þ ¼ dij and with kij ðtÞ ¼ oi ðtÞdij þ oij ðtÞð1
dij Þ
ð8Þ
Thus kðtÞ is a matrix whose diagonal elements are the fluctuating transition frequencies, oi ðtÞ, and whose off-diagonal elements are the fluctuating couplings oij ðtÞ. Note that with a single chromophore, Eq. (7) can be integrated analytically to obtain Eq. (3). For coupled chromophores the substantial complication is that since the matrix FðtÞ does not commute at different times, Eq. (7) has to be integrated numerically. Moreover, the ensemble average in Eq. (6) is usually implemented as a time average over a single matrix trajectory, which means that Eq. (7) needs to be integrated repeatedly from different starting points. For a few coupled chromophores, this presents no great problems [95, 96, 99, 100]; but given the poor scaling of matrix operations, this process can be very timeconsuming for several hundred or more chromophores [97]. This has prompted a number of theorists to explore alternative approximate methods to tackle this problem [71, 95, 96, 99, 101–111]. In the inhomogeneous limit discussed earlier for a single chromophore, kðtÞ in Eq. (7) can be replaced with kð0Þ, and so now FðtÞ ¼ expðikð0ÞtÞ. kð0Þ can be diagonalized by an orthogonal transformation: M T kð0ÞM ¼ l and then the line shape can be written as X IðoÞ # hc2k Lðo k
ð9Þ
lk Þi
ð10Þ
vibrational line shapes, spectral diffusion where lk are the eigenvalues of kð0Þ and X ck ¼ mpi ð0ÞMik
67
ð11Þ
i
Unlike the general situation described above, this inhomogeneous limit (sometimes called the static averaging approximation [105, 108, 112]) is numerically tractable, because it involves many fewer matrix operations. The above suggested to us an approximate way to deal with dynamical effects for many coupled chromophores [99]. It is based on the idea that motional narrowing can be thought of as arising from a distribution of timeaveraged frequencies. For an isolated chromophore, motional narrowing occurs when the dynamics of the frequency fluctuations are sufficiently fast that they ‘‘self-average’’ (to some extent). The idea then is that one can average a frequency trajectory over time windows of duration T, to obtain a distribution of these time-averaged frequencies, which is of course narrower than the actual distribution of frequencies. This idea has antecedents in earlier papers by Belch and Rice [7], Hermansson and co-workers [113], and Buch [110] but was formalized only recently [99]. For the case of coupled chromophores the approximation is to average the matrix kðtÞ [98]. Thus in the time-averaging approximation (TAA) we replace kðtÞ in Eq. (7) with kT ¼
1 T
Z
T
dt0 kðt0 Þ
ð12Þ
0
which depends parametrically on the averaging time T, and in the case of the IR line shape we replace mjp ðtÞ by mjp ðTÞ. For any one realization of the trajectory kðtÞ, kT is diagonalized by the orthogonal transformation N T kT N, and then the line shape can be written approximately as [99] IðoÞ #
X
hdk ð0Þdk ðTÞLðo
gk Þi
ð13Þ
k
where dk ðtÞ ¼
X
mip ðtÞNik
ð14Þ
i
and gk are the eigenvalues of kT . The ensemble average in Eq. (13) is implemented as a time average over different starting points in the trajectory. Thus in this approach, computational requirements are similar to those in the static averaging approximation, but now dynamical effects are included (albeit approximately).
68
james l. skinner, benjamin m. auer, and yu-shan lin
From an analysis of the isolated-chromophore Kubo model [3] we showed [99] that the appropriate averaging time is given approximately by the very simple formula T ’ 5=", where " is the FWHM of the actual line shape. Of course for an isolated chromophore there is no problem obtaining the actual line shape and, consequently, no need to use the TAA. But for many interacting chromophores, the proposed strategy is to calculate first the line shape for noninteracting chromophores (without non-Condon effects or rotational and lifetime broadening), identify the appropriate averaging time T, and then apply the TAA to the coupled system. We have tested the TAA on simple models with two coupled chromophores (the two OH stretch local modes in water) by comparing to exact calculations [99]. We expect that the TAA will work well for many coupled chromophores as long as the dynamics are not too fast, and the coupling between chromophores is not too large, as in the case of water discussed herein. 4.
Raman Line Shapes
In the usual Raman experiment [78] a polarized laser propagating in, say, the direction of the lab-fixed Y axis excites the system, and scattered light is collected at 90/ , say along the lab-fixed X axis. The excitation light is typically polarized along the Z axis (perpendicular to the scattering plane). The scattered light can be collected with a polarizer in either the Z direction or the Y direction. In the former case, often denoted as the YðZZÞX geometry, this is called the VV spectrum, since the polarization of both excitation and scattered light is ‘‘vertical’’ (compared to the ‘‘horizontal’’ scattering plane). It is also called the polarized spectrum. The latter or YðZYÞX case is called the VH spectrum, since the polarization of the scattered light is ‘‘horizontal.’’ This is also called the depolarized spectrum. Sometimes all the scattered light of any polarization is collected (this is called the unpolarized spectrum), which is related to the VV and VH spectra by Iunp ðoÞ ¼ IVV ðoÞ þ IVH ðoÞ, where o is the frequency difference between the excitation and scattered light. Often one is interested in the ‘‘isotropic’’ spectrum, which can be obtained from Iiso ðoÞ ¼ IVV ðoÞ ð4=3ÞIVH ðoÞ. From a theoretical perspective, since the designation of the lab-fixed axes is arbitrary, what is relevant is the relative orientation of the polarizations of the excitation and scattered light. Thus the line shape for excitation light polarized along axis p, and scattered light polarized along axis q (p or q denote X, Y, or Z axes in the lab frame) is called Ipq ðoÞ. When p ¼ q this is IVV , and when p 6¼ q this is IVH . Mixed quantum/classical formulae for Ipq ðoÞ are identical to those for the IR spectrum, except mpi is replaced by apqi, which is the pq tensor element of the transition polarizability for chromophore i. Thus we have, for example [6], Z 1 X ð15Þ dt e iot hapqi ð0ÞFij ðtÞapqj ðtÞie t=2T1 Ipq ðoÞ # Re 0
ij
vibrational line shapes, spectral diffusion B.
69
Echoes and Other Nonlinear Experiments
Developments in ultrafast light sources have revolutionized vibrational spectroscopy over the last 15 or so years, beginning with the first (two-pulse) photon echo experiments performed by the Fayer group using the Stanford free-electron laser [114], and the first three-pulse photon echo performed by the Hochstrasser group [115]. The basic idea is to subject the sample to a sequence of three ultrafast pulses, typically all polarized in the same direction and with the same center frequency but with different wavevectors, separated by delay times t and T, and then collect the emitted light by various means as a function of time t after the third pulse [92]. This gives a signal as a function of three time variables (that is, it is a three-dimensional spectroscopy). Often the intensity is integrated over the last time t, producing an ‘‘integrated’’ photon echo. The peak in the intensity along the t axis as a function of T is called the three-pulse echo peak shift (3PEPS) experiment. Often the signal amplitude is Fourier transformed along the t and t axes, leading to ‘‘2DIR’’ spectra that depend parametrically upon the ‘‘waiting’’ time T. Interested readers can consult the many reviews on this subject [116–119]. Ultrafast spectroscopy is so important because it provides dynamical information that is very hard or impossible to access from IR and Raman spectra. For systems with a single chromophore, this dynamical information is often characterized by the frequency TCF, CðtÞ ¼ hdoð0ÞdoðtÞi
ð16Þ
where doðtÞ ¼ oðtÞ hoi. CðtÞ, also sometimes called the spectral diffusion TCF, decays (to zero) on the time scale that frequency fluctuations become uncorrelated due to molecular motions in the chromophore’s local environment. This TCF also can have significant structure, corresponding (for example) to inertial motion, underdamped intermolecular vibrations, and so on. If the frequency fluctuations were described by a Gaussian process, then in fact all information about spectral diffusion is contained in this TCF [92]. In favorable circumstances (fluctuations are Gaussian, Condon approximation is adequate), unfortunately not very well met for water (see below) [120], this spectral diffusion TCF can be extracted from ultrafast experiments, providing a very useful way of probing molecular dynamics in liquids. In other cases, one really needs to calculate the actual nonlinear experimental observables to make a direct connection to experiment, meaning that the extraction of relevant dynamical information from experiment is not as straightforward. Certainly, one reasonable approach is to (a) refine theoretical models until calculated nonlinear observables are in agreement with experiment and (b) analyze properties of the model (for example the frequency TCF) to provide a molecular interpretation. For systems
70
james l. skinner, benjamin m. auer, and yu-shan lin
of coupled chromophores, molecular interpretation of these nonlinear experiments is more difficult. Three-pulse echo experiments on isolated chromophores probe the 1 ! 2 transition in addition to the fundamental. Therefore, in order to model nonlinear spectra, in addition to the trajectories for mp ðtÞ and oðtÞ, one needs the trajectory for this 1 ! 2 transition frequency; and in addition to the excited lifetime T1 , one needs the lifetime of the second excited state. From these trajectories, one can calculate the required nonlinear response functions and then the various nonlinear spectra [92, 120, 121]. Another very informative nonlinear experiment involves a typical pumpprobe technique, but with varying laser polarization. These experiments, again for isolated chromophores, measure the rotational anisotropy TCF [122] C2 ðtÞ ¼ hP2 ð^ uð0Þ 1 ^ uðtÞÞi
ð17Þ
where ^ u is the unit vector of the chromophore’s transition dipole, and P2 ðxÞ is the second Legendre polynomial. These anisotropy decay experiments can be performed with pump and probe beams of different frequency, and it has been particularly interesting to try to interpret these results. III.
THEORETICAL IMPLEMENTATION
Within the mixed quantum/classical approach, at each time step in a classical molecular dynamics simulation (that is, for each configuration of the bath coordinates), for each chromophore one needs the transition frequency and the transition dipole or polarizability, and if there are multiple chromophores, one needs the coupling frequencies between each pair. For water a number of different possible approaches have been used to obtain these quantities; in this section we begin with brief discussions of each approach to determine transition frequencies. For definiteness we consider the case of a single OH stretch chromophore on an HOD molecule in liquid D2 O. Each of the approaches is based on the premise that it makes sense to focus on the Born–Oppenheimer potential for the OH stretch for fixed bath variables. Such a potential has vibrational eigenvalues, and for example h times the transition frequency of the fundamental is simply the difference between the first excited and ground state eigenvalues. Thus in essence this is an adiabatic approximation—the assumption is that the vibrational chromophore is sufficiently ‘‘fast’’ compared to the bath coordinates. To the extent that the h times frequency of the chromophore is large compared to kT, and those of the bath are small compared to kT, this separation of time scales exists and so this should be a reasonable approximation. For water, as discussed earlier, some of the bath variables (librations) have frequencies somewhat larger than kT=h, and
vibrational line shapes, spectral diffusion
71
so it is not completely clear that this is a good approximation. Nonetheless, all of the approaches described below can be thought of as different methods for obtaining this Born–Oppenheimer potential. A.
Approaches Based on the Simulation Potential
As mentioned earlier, in the classical simulation of the liquid the OH bond is rigid. To obtain the Born–Oppenheimer potential for OH motion, then, one usually proceeds by assuming a reference potential for the OH stretch of the isolated HOD molecule, which is anharmonic and which is taken from either ab initio calculations or experiment. Then for a given bath configuration, one can compute a perturbation to the reference potential by scanning the OH stretch coordinate. Empirical simulation potentials are usually based on site–site interactions, and as the OH coordinate is scanned (keeping the HOD center of mass and orientation fixed) the site positions on the HOD molecule are modified, thus changing the potential energy. Typically, most of this perturbation comes from site–site electrostatic interactions. For a given bath configuration, one can calculate the eigenvalues numerically for the reference potential plus the bathinduced perturbation to determine the transition frequency. Alternatively, one can use perturbation theory to calculate the bath-induced change in the reference eigenvalues. These two methods typically lead to very similar results. This method has been applied to water and many other problems with significant success [15, 38–40, 43, 106, 113, 123–127]. One worry is that the form of the simulation potential may not be up to the task of producing accurate enough vibrational frequencies. That is, the site parameters of a simulation potential are usually adjusted to give bulk structural or thermodynamic properties of the liquid. In some cases there is a competition, for example, between Lennard Jones and Coulomb interactions such that these liquid properties are given correctly, but the parameters themselves are not completely physical. Thus it is not always clear, for the delicate problem of vibrational frequencies, that this approach will be sufficiently accurate. B.
Approaches Based on Ab Initio Calculations on a Single Water Molecule in an Inhomogeneous Electric Field
A second approach [128] uses the idea that most of the bath-induced perturbation comes from electrostatics. In principle, at every time step in the simulation, one could imagine performing ab initio calculations for an HOD molecule in the field of the point charges on the surrounding water molecules to determine the Born– Oppenheimer curve, but in practice these repeated ab initio calculations make this approach too slow. Mukamel and co-workers solved this problem by performing ab initio calculations for a water molecule in an inhomogeneous electric field and then parameterizing the Born–Oppenheimer potential in terms of the field and field gradients. In this manner all electronic structure
72
james l. skinner, benjamin m. auer, and yu-shan lin
calculations are done in advance, and then during the course of the simulation the electric fields and field gradients at every time step lead to the OH transition frequency [128]. This powerful approach would be expected to improve upon the simulation potential-based approach described above. One minor concern is that the inhomogeneous electric field from the empirical simulation potential is not necessarily a good representation of the actual inhomogeneous electric field in the liquid. A potentially more serious concern is that perhaps electrostatics alone misses important quantum chemical contributions such as exchange and charge transfer [129]. C.
Approaches Based on Ab Initio Calculations on Water Clusters
One way to include these local quantum chemical effects is to perform ab initio calculations on an HOD molecule in a cluster of water molecules, possibly in the field of the point charges of the water molecules surrounding the cluster. In 1991 Hermansson generated such clusters from a Monte Carlo simulation of the liquid, and for each one she determined the relevant Born–Oppenheimer potential and the vibrational frequencies. The transition-dipole-weighted histogram of frequencies was in rough agreement with the experimental IR spectrum for HOD/D2 O [130]. We wanted to extend this approach to include dynamical effects on line shapes. As discussed earlier, for this approach one needs a trajectory oðtÞ for the transition frequency for a single chromophore. One could extract a water cluster around the HOD molecule at every time step in an MD simulation and then perform an ab initio calculation, but this would entail millions of such calculations, which is not feasible. Within the Born–Oppenheimer approximation the OH stretch potential is a functional of the nuclear coordinates of all the bath atoms, as is the OH transition frequency. Of course we do not know the functional. Suppose that the transition frequency is (approximately) a function of a one or more collective coordinates of these nuclear positions. A priori we do not know which collective coordinates to choose, or what the function is. We explored several such possibilities, and one collective coordinate that worked reasonably well was simply the electric field from all the bath atoms (assuming the point charges as assigned in the simulation potential) on the H atom of the HOD molecule, in the direction of the OH bond. In our first implementation [13, 120, 121, 131] of this idea, we took the transition frequency to be a linear function of this electric field. We determined the coefficients of this linear function by fitting to the ab initio frequencies from water clusters (and in this case the clusters were not surrounded by point charges from the other molecules in the simulation). In the liquid simulation we simply calculate this electric field at every time step and then use this linear map (in this case the electric field was the full Ewald field from the simulation) to determine the frequency. In our later implementation [6, 98] we took the
vibrational line shapes, spectral diffusion
73
transition frequency to be a quadratic function of this field, and we again determined the coefficients by fitting to ab initio frequencies from water clusters, this time in the presence of the point charges of all the surrounding solvent molecules within half the box length. In the liquid simulation we then calculate exactly this field (meaning that there is no extrapolation in going from the clusters to the liquid). It is important to make three points about this approach. First, because the electric field used to determine the fit is due to the point charges in a simulation model, different simulation models would in principle have different maps. Second, although this approach may sound like Stark shifting, that is not the case. The actual frequencies come from ab initio calculations on collections of water molecules (not a single molecule), and these frequencies are then fit to a form involving a collective coordinate, which in our case happens to be the electric field, but it could have been something else (and indeed, in other similar implementations the collective coordinate was chosen to be the electrostatic potential [132]). Thus it is not surprising that the linear coefficient of the electric field is not the same as the linear Stark coefficient [133]. Third, our work was preceded by Buch’s innovative work on ice and clusters [111, 134]. She also used an electric field map, different in detail but similar in spirit to what we have been doing, but in her case the map comes from comparing to IR experiments on specific gas-phase clusters. We will now illustrate this approach with specific results for the HOD/D2 O system [98]. An SPC/E water simulation [135] with 128 D2 O molecules at the experimental density for heavy water and 300 K was run, and water clusters surrounded by point charges were extracted as described earlier. DFT electronic structure calculations were performed to obtain the Born–Oppenheimer surface for a tagged bond representing the OH stretch, and a discrete variable representation approach was used to obtain the lowest two eigenvalues numerically. The resulting OH transition frequency (slightly scaled to reproduce the gas-phase frequency) was fit to a quadratic function of the electric field due to the point charges of all other water molecules within half the simulation box length. The frequencies for 200 such clusters together with the quadratic fit are shown in Fig. 1 (bottom panel). One sees that the correlation between the fit and the points is reasonably good, but there is still considerable scatter. This means that for any one given configuration, using the electric field map to determine the frequency will result in some error. It is also interesting to consider the ensemble of frequencies produced by the map [98]. To this end we rerun the simulation and calculate the electric field at every putative H atom from all the point charges out to half the box length. This generates a distribution of fields, which through the map leads to a distribution of frequencies. This distribution can be compared to the histogrammed distribution of actual ab initio frequencies, now from 999 clusters. This
74
james l. skinner, benjamin m. auer, and yu-shan lin 7 6
µi / µg
5 4 3 2 1 0 3800
ωi (cm-1)
3600 3400 3200 3000 2800
0
0.02
0.04
0.06
0.08
Ei Figure 1. Bottom panel: OH stretch frequencies, oi , for water clusters and the surrounding point charges, versus electric field Ei (in atomic units). The solid line is the best quadratic fit. Top panel: Dipole derivative, m0i , (relative to the gas-phase value) for water clusters and the surrounding point charges, versus electric field Ei . The solid line is the best linear fit.
comparison is shown in Fig. 2. One sees that the agreement is remarkably good. Thus, although the map may be somewhat inaccurate for any one particular configuration, for the ensemble it is quite accurate. And since we will be considering only ensemble experiments, such a map proves to be adequate. D. Calculation of Transition Dipoles It has been known for some time that the magnitude of the transition dipole of a local OH stretch in water depends very strongly on the degree of hydrogen bonding [136–139]. This is a non-Condon effect, meaning that this magnitude depends on the bath coordinates. To generate the required trajectory of mp ðtÞ, the
vibrational line shapes, spectral diffusion
75
0.003
P(ω)
0.002
0.001
0
3000
3200
3400
3600
3800
ω (cm-1) Figure 2. The histogram is the distribution of OH stretch frequencies for the water clusters and surrounding point charges, and the solid line is the distribution of frequencies from the quadratic electric field map.
projection of the transition dipole on lab-fixed axis p, one needs to consider both rotations and non-Condon effects. In the gas-phase molecule it is known that the transition dipole does not actually point in the direction of the bond [139]. In ice, however, it does point nearly along the bond [139]. For simplicity, we and others assume that in the liquid it also points along the OH bond. Therefore the orientational component of mp ðtÞ is simply ^ uðtÞ 1 ^ p, where ^ u is the bond vector. The non-Condon effect is taken into account with two factors: (a) the magnitude of the dipole derivative and (b) the 0–1 position matrix element [13]. The former has been parameterized in terms of bath variables in several ways [134, 140]. We find [6, 13, 98] that there is a reasonable linear correlation between this dipole derivative and the electric field discussed above, as shown in Fig. 1 (top panel). The matrix element depends weakly on the transition frequency, and we have included this dependence through a simple linear map [6, 13, 98]. E.
Calculation of Transition Polarizabilities
To obtain Raman spectra one needs the trajectories of the pq tensor elements of the chromophore’s transition polarizability. Actually, for the isotropic Raman spectrum one needs only the average transition polarizability. This depends weakly on bath coordinates; and this, together with the weak frequency dependence of the position matrix element, was included in our previous calculations [13, 98, 121]. For the VV and VH spectra, others have implemented
76
james l. skinner, benjamin m. auer, and yu-shan lin
a bond polarizability model [7, 71, 97, 110], where the transition polarizability for each OH bond has parallel and perpendicular components that have a fixed ratio. In this model, non-Condon effects are not included. In our later calculations [6] we have adopted this model, and following others we take this ratio of components to be 5.6 [141, 142]. Others have chosen different but similar values [71, 97]. To be slightly more accurate, in the Raman spectra shown herein we include these weak non-Condon effects, keeping the ratio of the parallel and perpendicular transition polarizabilities fixed at 5.6, but allowing each of them to depend on bath coordinates through the linear electric field map [98], and we also include the weak frequency dependence of the position matrix element [98]. F.
Calculation of Coupling Frequencies
For the spectroscopy of H2 O, one needs to calculate coupling frequencies between different OH stretch chromophores. The intramolecular coupling has potential and momentum contributions, and for the isolated molecule it is about 50 cm 1 [6]. In condensed phases this coupling is smaller [71]. Some workers in this field therefore neglect this coupling [97], and others take it to be constant [140]. From our ab initio calculations on clusters we have found that the intramolecular coupling frequency varies considerably. We have found it convenient to parameterize this coupling in terms of the electric fields at the two H atoms, by comparing to these ab initio calculations [6, 99]. From our molecular dynamics (MD) simulations we find that the average coupling in the liquid is about 27 cm 1 and that the distribution of couplings is quite broad [6]. Therefore to us it seems important to include this coupling and its dependence on bath coordinates. For the intermolecular coupling one usually assumes a transition dipole interaction [7, 71, 97, 110, 111, 134]. While this would not be expected to be accurate at very short distances, and indeed this is not a particularly good approximation [6], there is no other convenient way to proceed. We parameterized the position of the point dipole along the OH bond by comparing to ab initio calculations [6]. G.
Hydrogen Bonding
One of the goals of vibrational spectroscopy is to say something about H-bonding structure and dynamics in liquid water. For example, one often talks about the number of H bonds per molecule in liquid water, about the time scale for making and breaking H bonds, and about spectral assignments in terms of local H-bonding environments. In order to discuss these topics one needs a definition of an H bond in the liquid. This is a controversial subject, because there is no unique definition and not even agreement on whether such a definition is meaningful.
vibrational line shapes, spectral diffusion
77
In a recent paper we have reviewed this situation [143], commenting on and extending several energetic and geometric definitions and proposing a new electronic-structure-based definition. The definition is based on the concept that H-bonding arises from electron donation from lone-pair orbitals on one molecule into the empty OH antibonding orbital on another [144, 145]. We found that this definition is compatible with a geometric definition, but not one of the usual ones involving donor–acceptor OH 1 1 1O or HO 1 1 1 O angles [143]. Rather, it involves the angle that the acceptor–donor O 1 1 1 H ray makes with the out-of-plane axis on the acceptor molecule. Thus for a given H atom and a given O atom on another molecule, define this O 1 1 1 H distance to be r and define the angle this O 1 1 1 H ray makes with the (closer) out-of-plane axis to be c; then if r < ½2:307 þ 0:343 lnð1
˚ 0:404c þ 0:0971c2 Þ&A
ð18Þ
there is an H bond between the O and H atoms. Note that c is in radians and is defined from 0 to p=2. From a simulation of SPC/E water at ambient conditions, this definition gives 3.36 H bonds per molecule [143]. IV.
HOD/D2O
As mentioned in the introduction, this is a convenient system for theoretical and experimental study because the OH stretch absorbs in the same region as in liquid water, but in this case, this stretch is decoupled from the other stretches in the system, and even from the bend overtone. A.
Line Shapes
IR line shapes for this system have been measured at room and other temperatures. Isotropic, depolarized, and unpolarized Raman line shapes have also been measured. For this system all Raman line shapes are similar, peaking at about 3430 cm 1 , with a shoulder at about 3625 cm 1 (at room temperature). The IR line shape is red-shifted by about 30 cm 1 and does not show the blue shoulder. Experimental IR [10] and unpolarized Raman [12] line shapes are shown in Fig. 3. A number of researchers [15, 38–40, 43, 113, 124–126, 128, 146] have used mixed quantum/classical models, mostly as described in Section III.A, to calculate vibrational line shapes for this system, and several are in fair agreement with experiment. Here we describe our latest work involving approaches discussed in Section III.C. Our theoretical line shapes are calculated as briefly described in previous sections and in published work [98]. From an MD simulation of SPC/E heavy water, we determine the electric field on each putative H atom. We then use electric field maps to determine the transition frequency and dipole derivative. The orientational contribution to mp ðtÞ we
78
james l. skinner, benjamin m. auer, and yu-shan lin 1 IR - Experiment IR - Theory
I(ω)
0.8 0.6 0.4 0.2
1 Raman - Experiment Raman - Theory
I(ω)
0.8 0.6 0.4 0.2 0
3000
3200
3400 ω (cm-1)
3600
3800
Figure 3. Experimental [10, 12] and theoretical IR and unpolarized Raman line shapes for HOD/D2O at room temperature.
simply obtain from the projection of the OH-bond unit vector on an arbitrary lab-fixed axis p. From the frequency and transition dipole projection trajectories we then calculate the IR line shape from Eq. (3). We use the experimental value of T1 ¼ 700 fs for this system [10]. Experimental line shapes often have slightly overlapping resonances, leading to issues having to do with the baseline and to where the tails of the spectra should be truncated (for normalization purposes). In previous papers [6, 13, 98, 120, 121, 131] we have dealt with these issues as best we could, and we compared area-normalized theoretical and experimental line shapes. Here, however, to avoid these issues we compare line shapes by setting the peak intensity of the theoretical line shape to match experiment. Our theoretical IR line shape is shown in Fig. 3. The agreement with experiment is reasonable but certainly not perfect. In particular, although the theoretical peak frequency is excellent, the theoretical width is a bit too large. The Raman line shape is calculated with the bond polarizability model as described above. The unpolarized Raman line shape computed from the sum of the VV and VH line shapes is shown in Fig. 3. One again sees fair agreement between theory and experiment, with excellent peak position and evidence of a
vibrational line shapes, spectral diffusion
79
shoulder on the blue side, but a width that is somewhat too large. It is clear that the red shift and lack of shoulder in the IR line shape relative to the Raman is due to non-Condon effects; the increasing transition dipole strength with decreasing frequency (increasing electric field) emphasizes the red side of the frequency distribution. In a recent experimental study involving the temperature dependences of the IR and Raman line shapes, Loparo et al. [14] confirmed that non-Condon effects are important in experimental (and theoretical!) line shapes, and they found a frequency dependence to the dipole derivative that is qualitatively similar to the form used in our work. B.
Hydrogen Bonding
One of the main points of studying vibrational spectroscopy in water is to provide insight into H-bonding [15, 38]. To this end, one can try to interpret these spectra in terms of H-bonding. For a given simulation snapshot, for each pair of D and O atoms (on different molecules) we determine if they are H-bonded or not by Eq. (18). Each putative H atom is involved in nH H bonds (nH ¼ 0, 1, or 2; see below) and through the electric field map has an OH transition frequency assigned to it [98]. For all such atoms with frequencies in a narrow window, we can average the number of H bonds and then plot the result, hnH i, as a function of frequency, as shown in Fig. 4 (top panel). The results are quite striking: If the OH frequency is less than 3500 cm 1 , the H atom is most definitely hydrogenbonded, since the average number of H bonds is 1 (H atoms with 2 H bonds are very rare). Above 3500 cm 1 the number gradually decreases to near 0, indicating that most of these atoms have broken H bonds. We can also prepare a similar plot, now not for the number of H bonds to the H atom, but for the total number of H bonds formed with the HOD molecule, nT , as shown in Fig. 4 (top panel). In this case for low frequency the number starts above 4 (our definition does not preclude 5 or more H bonds) and then slowly decreases as frequency increases, finally decreasing more rapidly above 3500 cm 1 and reaching a value of less than 2 for the highest frequency. Note that hnT i hnH i is not constant, but rather decreases from about 3 to less than 2 as frequency increases. Thus the H-bonding state of the D and O atoms on the HOD molecule depend quite strongly on the frequency of the OH stretch! One sees that as this frequency increases, even before H bonds to the H break, the average environment around the HOD molecule is such that some of the other H bonds have broken. And at the high-frequency end, when the H bonds to H are completely broken, there are fewer than two remaining H bonds to the HOD molecule. These interesting results can be understood more fully by considering different classes of H-bonded molecules [98]. We can describe the H-bonding configuration of the HOD molecule by the number of H bonds from neighboring molecules to the H atom, nH (as above), the number of H bonds to the D atom, nD , and the
80
james l. skinner, benjamin m. auer, and yu-shan lin 5
4
3 2 1 0
NH = 0 NH = 1
P(ω)
0.002
Total
0.001
0 NT = 1 NT = 2
0.002 P(ω)
NT = 3 NT = 4
0.001
0
Total
3000
3200
3400
3600
3800
ω (cm-1) Figure 4. Top panel: Average number of H bonds to the H atom (in HOD/D2O), hnH i, and total number of H bonds to the HOD molecule, hnT i, versus OH stretch frequency. Middle panel: Frequency distributions with NH ¼ 0 and 1. Bottom panel: Frequency distributions with NT ¼ 1; 2; 3; 4. See color insert.
number of H bonds, to the O atom, nO . The total number of H bonds discussed above is then nT ¼ nO þ nH þ nD . Note that for configurations generated from an SPC/E simulation and for any reasonable (including our new) H-bond definition, nO ¼ 0; 1; 2; 3, and nH or nD ¼ 0, 1, 2. Thus every HOD molecule can be assigned to one of 36 classes. This large number of classes makes it difficult to visualize and discuss results. However, since the probability of obtaining either nO ¼ 0 or 3 is small, we can profitably define a new number NO, which is 1 if nO is 0 or 1, and 2 if nO is 2 or 3. Thus molecules with NO ¼ 1 correspond for the most part to those
81
vibrational line shapes, spectral diffusion
with 1 H bond to the O, while molecules with NO ¼ 2 correspond for the most part to those with 2 H bonds to the O. Likewise, the probability of obtaining nH ¼ 2 is small, and so we can define a new number NH , which is 0 if nH ¼ 0, and 1 if nH ¼ 1 or 2, and similarly for D. This then gives a more manageable number of 8 H-bond classes, each of which is described by the triplet of numbers NO , NH , and ND [98]. Alternatively, we can describe each class by the total number of H bonds NT ¼ NO þ NH þ ND , the number of H-bond donors NH þ ND , and if there is a single donor (in which case the latter number is 1), whether it is the H or the D. Thus, for example, the triplet NO ¼ 2, NH ¼ 1, and ND ¼ 0 can be labeled 3SH : ‘‘Three H bonds with the single donor being the H.’’ (Note, however, that because of the way we have combined the original 36 classes, a member of the class 3SH will occasionally have 4 H bonds.) The translation between these two different ways of labeling environments is given in Table I, where D (as the first letter) means double donor, S means single donor, and N means non-donor [98]. For each class of HOD molecule, we can consider the distribution of OH stretch frequencies. Each of these eight distributions has a certain average frequency hoi, and the probability of being in each of the these eight classes, f , is shown in Table I [98]. Clearly, these eight distributions sum to give the overall frequency distribution. Before we consider each of these distributions separately, it is instructive to combine the classes is two different ways. First we can combine all classes with no H bond to the H (1N , 2SD , 2N , 3SD , or those with NH ¼ 0) and combine all classes with one hydrogen bond to the H (2SH , 3DD , 3SH , 4DD , or those with NH ¼ 1). The frequency distributions for these two combinations are shown in Fig. 4 (middle panel). Now it is clear why hnH i behaves the way it does. Below 3400 cm 1 , only one distribution—that with one H bond—has nonzero weight, and so hnH i ¼ 1. As the frequency increases, one samples both distributions and TABLE I Statistics of the Frequency Sub-distributionsa NO 1 1 1 1 2 2 2 2
NH
ND
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
Label
hoi
f
1N 2SD 2SH 3D 2N 3SD 3SH 4D
3654 3665 3458 3473 3610 3624 3381 3400
0.014 0.056 0.056 0.227 0.013 0.080 0.080 0.475
a NO , NH , and ND are described in the text. Each sub-distribution is labeled as described in the text. hoi is the average frequency (in cm 1) of each sub-distribution, and f is the fraction of molecules in each sub-distribution.
82
james l. skinner, benjamin m. auer, and yu-shan lin
hnH i decreases. Finally, above 3730 cm 1 only the distribution with no H bond has nonzero weight, and so hnH i ’ 0. Thus in this frequency range all H atoms have broken H bonds. Comparing these two distributions to the Raman spectrum in Fig. 3 makes it clear that the shoulder in this spectrum is due to H atoms with broken H bonds. Similarly, we can combine all classes with a given NT —that is, those with NT ¼ 1 (1N ), with NT ¼ 2 (2N , 2SD and 2SH ), with NT ¼ 3 (3SD , 3SH , and 3D ), and with NT ¼ 4 (4D ). These four distributions are also shown in Fig. 4 (lower panel), which facilitates the understanding of the frequency dependence of hnT i. That is, as frequency increases, one passes through the four distributions (in the order NT ¼ 4; 3; 2; 1) and in doing so hnT i decreases. Finally, in Fig. 5 we show all eight frequency distributions for the eight H-bonding classes [98]. What is particularly striking is the breadth of the four distributions corresponding to NH ¼ 1. Thus, for example, a chromophore with an OH stretch frequency anywhere from 3200 to 3600 cm 1 could be in any of these four H-bonding classes. This should put to rest the appealing but apparently incorrect idea that peaks or shoulders in vibrational spectra in this frequency range (especially for neat H2 O; see below) can be attributed to molecules in particular molecular environments (ice-like, bifurcated H-bond, etc.). On the other hand, the four distributions with NH ¼ 0 are relatively narrow. While these four distributions overlap considerably, one can still say with confidence that molecules absorbing near the blue edge of the line are unlikely to have an H bond to the H atom. 0.003 1N 2N 2SH 3SH 2SD 3SD 3D 4D Total
P(ω)
0.002
0.001
0
3000
3200
3400 ω (cm-1)
3600
3800
Figure 5. Frequency distributions for the eight H-bond classes of HOD molecules in D2O. Labels are as described in the text and in Table I. See color insert.
vibrational line shapes, spectral diffusion C.
83
Two-Pulse Echoes
As evidenced from the above discussion, vibrational line shapes provide information mostly about intermolecular structure. Transient hole burning and more recently echo experiments, on the other hand, can provide information about the dynamics of spectral diffusion. The first echo experiments on the HOD/ D2 O system involved two excitation pulses, and the signal was detected either by integrating the intensity [20] or by heterodyning [22]. The experiments were analyzed with the standard model assuming Gaussian frequency fluctuations. The data were consistent with a spectral diffusion TCF that was bi-exponential, involving fast and slow times of about 100 fs and 1 ps, respectively. We calculated the two-pulse echo signal and spectral diffusion TCF from our earlier theoretical model [147], obtaining qualitative agreement with experiment. We also raised cautionary flags about using the cumulant truncation approximation and about invoking models of combined homogeneous and inhomogeneous broadening. The latter issue was illustrated by a study of line shapes and echoes with a generalized Kubo model [148]. D. Three-Pulse Echoes and 2DIR Spectra Shortly thereafter came reports of integrated three-pulse photon echoes, especially using the echo peak shift to provide information about spectral diffusion [21, 23]. In one experiment [10, 23] the peak shift shows an intriguing oscillation at short times with a period of about 180 fs, followed by a slower relaxation with a decay time of 1.4 ps. The three-pulse echo amplitude can also be heterodyned, leading to 2DIR experiments [24–26]. The latter experiments provide a wealth of information, and there are several ways to extract the desired spectral diffusion dynamics [149]. On the theoretical side, a number of researchers used models of the sort described in Section III.A to calculate three-pulse echo intensities and 2DIR spectra [41, 43, 98, 120, 121], and several of these are in qualitative agreement with experiment. Our 2003 paper [41] showed an oscillation in the peak shift, as observed experimentally, but our longer-time decay was too fast, presumably as a result of our oversimplified (TIP4P [150]) water simulation model. We also considered [151] the polarizable fluctuating charge model of Rick, Stuart, and Berne [152], finding that it leads to a slower long-time decay, in better agreement with experiment. In addition, we came to appreciate the importance of including non-Condon effects in our spectroscopic calculations of nonlinear observables [120]. We also tried to assess whether a combination of linear and nonlinear spectroscopy on this system could discriminate among popular water simulation models [121]. The results were not definitive, but we did suggest that the SPC/E model seemed to be the best compromise, performing reasonably well for several different observables. In our latest work involving our quadratic
84
james l. skinner, benjamin m. auer, and yu-shan lin 30
20 τ*(T) / fs
Theory Experiment
10
0 0
0.2
0.4
0.6 T (ps)
0.8
1
Figure 6. Theoretical [98] and experimental [10] three-pulse echo peak shifts as a function of waiting time t2 , for HOD/D2O at room temperature.
electric field frequency map, peak shift and 2DIR observables were in reasonable agreement with experiment [98]. A comparison between theoretical and experimental peak shifts is shown in Fig. 6. The interpretation of the features of the echo peak shift are that the short-time oscillation is due to underdamped H-bond stretching [15, 23, 38–42] and that the longer decay time on the order of 1.4 ps is due to structural relaxation involving making and breaking H bonds [15, 16, 24, 38, 40, 42, 43]. E.
Frequency-Dependent Anisotropy Decay
In a pump-probe experiment, if one probes both parallel and perpendicular to the pump, one can obtain the second-rank orientation TCF in Eq. (17) [122]. Pumping and/or probing at different frequencies [44–48] allows one to measure the reorientation of different sub-ensembles of molecules. Early such experiments on HOD/D2 O showed that molecules on the red side of the line exhibited a slower decay than did those on the blue side [44], although later experiments showed that this difference was only for times less that 1 ps [46]. We simulated [38] the orientation TCF for sub-ensembles of molecules that have different OH stretch frequencies at t ¼ 0. We found that within 100 fs there was an initial drop that was frequency-dependent, with a larger amplitude of this drop for molecules on the blue side of the line. For times longer than about 1 ps the decay times for all frequencies were the same. We argued that since molecules on the red side of the line have stronger H bonds, they are less free to rotate than molecules on the blue side, leading to a smaller initial decay. For times
vibrational line shapes, spectral diffusion
85
longer than the spectral diffusion time, memory of the initial frequency is lost, and so the sub-ensembles must become equivalent, producing identical long-time decays. Similar calculations (although in this case the molecules were required to be in a certain frequency bin at times 0 and t, to mimic the situation where pump and probe beams are both tuned to the same frequency) were carried out by Laage and Hynes [153], with similar but more detailed conclusions. Our calculations are only in qualitative agreement with experiment; the most salient points of discrepancy are that our overall decays are significantly faster than experiment [46], and the theory shows that most of the frequency dependence comes in the initial (less than 100 fs) short-time decay, whereas the experiment shows significant frequency dependence out to 1 ps. V.
HOD/H2O AND HOT/H2O
The system of dilute HOD in H2O is equally good for probing the structure and dynamics of water with an isolated chromophore (in this case the OD stretch), and it may be even better for two reasons. First, in this case the solvent is water, not heavy water; and second, the excited state vibrational lifetime of the OD stretch is somewhat longer (1.45 ps [55]) than that of the OH stretch in HOD/ D2O, providing a wider dynamic window before effects of local heating due to energy deposition from population relaxation occur. A.
Line Shapes
IR and Raman line shapes have been measured for HOD/H2O. They peak near 2500 cm 1 and have line widths in the 160 to 180 cm 1 range. Corcelli et al. [151] calculated these line shapes using the approaches described in Section III.C, for the SPC/FQ model, for temperatures of 10–90/ C, finding quite good agreement with experiment. More recently, we have extended the method involving the quadratic electric field map for HOD/D2O [98] to HOD/H2O [52] and have calculated IR and unpolarized Raman line shapes. These line shapes, in comparison with experimental line shapes [12, 52], are shown in Fig. 7. Agreement between theory and experiment is excellent for both the IR and Raman. Nicodemus and Tokmakoff [60] have measured the IR line shape in the OT stretch region of dilute HOT in H2O. The peak is at 2121 cm 1 , and the width is 127 cm 1 . They are particularly interested in the scaling as one goes from OH to OD to OT, in terms of the peak frequencies and line widths, and its implications for the appropriateness of using electric field fluctuations to describe line broadening. B.
Three-Pulse Echoes and 2DIR Spectra
2DIR spectra have been obtained by the Fayer group on HOD/H2O [53–55]. They characterize the extent of spectral diffusion by the waiting time dependence
86
james l. skinner, benjamin m. auer, and yu-shan lin 1 IR - Experiment IR - Theory
I(ω)
0.8 0.6 0.4 0.2 0 1
Raman - Experiment Raman - Theory
I(ω)
0.8 0.6 0.4 0.2 0 2200
2400
2600
2800
ω (cm-1) Figure 7. Experimental [12, 52] and theoretical IR and unpolarized Raman line shapes for HOD/H2O at room temperature.
of the ‘‘dynamic line width,’’ and from their analysis they find that the long-time decay of the frequency TCF has a time constant of 1.4 ps [55]. We have calculated [121] the dynamic line width for this system, for several water simulation models, and the results are in qualitative agreement with experiment. We also calculated [121] the ‘‘nodal slopes’’ [24, 149] of the 2DIR spectra as a function of waiting time, and again our results are in fair agreement with experiment [55]. The important conclusion from these echo studies on this system and on HOD/D2O is that the decay time for the long-time spectral diffusion in water is about 1.4 ps. This important time, presumably (as discussed earlier) related to the making and breaking of H bonds, had not been obtained by previous experiments. C.
Frequency-Dependent Anisotropy Decay
Orientational relaxation measurements have also been performed on this system. Earlier papers showed no frequency dependence to the decay rate [48, 57, 58], but a recent study by the Fayer group [59] showed that the very short-time decay has a frequency dependence, in agreement with our theoretical calculations for
87
vibrational line shapes, spectral diffusion
the HOD/D2O system [38]. Thus Moilanen et al. [59] showed that by 100 fs the orientational TCF has lost between 10% and 20% of its amplitude, with the shorter drop corresponding to the middle of the line and the longer drop corresponding to the blue side of the line (they did not perform experiments on the red side of the line). The amplitude of the initial drop was analyzed by considering inertial motion in a harmonic cone. We note that this experimental study used a broad-band pump, so all frequencies were excited, and the signal is frequency analyzed with an array detector, and so the resolution is excellent. These authors also reported theoretical calculations of this frequencydependent rotational relaxation. The theory of Auer et al. [98] using the quadratic electric field map, originally developed for HOD/D2O, was extended to the HOD/H2O system [52]. As before [38], the orientation TCF was calculated for those molecules within specified narrow-frequency windows (those selected in the experiment) at t ¼ 0. TCFs for selected frequency windows, up to 500 fs, are shown in Fig. 8. One sees that in all cases there is a very rapid decay, in well under 50 fs, followed by a pronounced oscillation. The period of this oscillation appears to be between about 50 and 80 fs, which corresponds most likely to underdamped librational motion [154]. Indeed, the period is clearly longer on the blue side, consistent with the idea of a weaker H bond and hence weaker restraining potential. At 100 fs the values of the TCFs show the same trend as in experiment, although the theoretical TCF loses
1 -1
ω = 2500 cm -1 ω = 2530 cm -1 ω = 2560 cm -1 ω = 2590 cm -1 ω = 2620 cm
0.9
C2(t)
0.8
0.7
0.6
0.5
0
100
200
300
400
500
t (fs) Figure 8. Frequency-dependent orientation TCFs for HOD/H2O at room temperature. Subensembles are defined according to the value of the OD stretch frequency at t ¼ 0, and the curves correspond to five sub-ensembles as labeled in the graph. See color insert.
88
james l. skinner, benjamin m. auer, and yu-shan lin
between 15% and 25% of its amplitude (compared to between 10% and 20% in experiment). Moilanen et al. [59] also performed experiments at two other temperatures, both higher (65/ C) and lower (1/ C). At the higher temperature they found a steeper frequency dependence to the value of the TCF at 100 fs, with larger values (than at 25/ C) at the lower-frequency end, and smaller values at high frequencies. At the lower temperature, remarkably, they found that the 100-fs drop was roughly frequency-independent! We have repeated the theoretical calculations at these two other temperatures. In disagreement with experiment we find that at each frequency the drop is larger the higher the temperature, and that all temperatures show a similar frequency dependence. These experimental and theoretical values of the TCFs at 100 fs are summarized in Fig. 9. Presumably the discrepancy between theory and experiment is due to inadequacies of the SPC/E simulation model, particularly at the lower and
C2(100 fs)
0.9
Theory
0.8
0.7 1°C 25°C 65°C
C2(100 fs)
0.9
Experiment
0.8
0.7
2500
2540
2580 ω
2620
(cm-1)
Figure 9. Experimental [59] and theoretical values of the polarization anisotropy time correlation function at 100 fs, as a function of OD stretch frequency, for three different temperatures. See color insert.
vibrational line shapes, spectral diffusion
89
higher temperatures (at which it was not parameterized). For example, SPC/E water is known to melt at 215 K [155]. Thus while the 1/ C experiment shows interesting and anomalous behavior 1/ above the melting point, presumably due to large clusters and/or spatial correlations, the theoretical model is some 60/ above its melting point and shows nothing unusual. VI.
H 2O
As discussed earlier, vibrational spectroscopy in neat water is much more complicated, due to the effects of both intramolecular and intermolecular coupling between OH stretch chromophores. A.
Line Shapes
The IR spectrum of water at room temperature and one atmosphere pressure [61–63] is peaked at about 3400 cm 1 and has a weak shoulder at about 3250 cm 1 and a FWHM of about 375cm 1 . Raman spectra are quite different [49, 64–70]: The VV spectrum is bimodal, with strong peaks at about 3400 and 3250 cm 1 , and an FWHM of about 425 cm 1 , while the VH spectrum peaks at about 3460 cm 1 , is quite asymmetric, and has a FWHM of about 300 cm 1 . Note that the gas-phase water molecule has symmetric and antisymmetric stretch fundamentals (both of which are IR and Raman active) at 3657 and 3756 cm 1 , respectively, and so the liquid-state spectra are significantly red-shifted from these values; furthermore, the breadths of the liquid-state spectra are substantially larger than this gas-phase splitting. A common interpretation of these spectra comes from fitting them to several Gaussians, centered at different positions, each with a different width. One then often attributes each Gaussian to a particular molecular environment and/or normal mode. Thus, for example, both IR and Raman spectra of liquid water have been deconstructed in this way, and the various Gaussians are attributed to molecules in different classes of H-bonded environments, sometimes with further designations of symmetric or antisymmetric stretch normal modes [49, 63–68, 156]. The results from such a procedure, together with the observation of approximate isosbestic points in temperature-dependent spectra, have been interpreted as supporting mixture models of water [49, 63–68, 156]. However, Geissler has recently shown that isosbestic points can occur quite generally and therefore do not necessarily imply multiple species [157]. Others have argued that the complications of intramolecular and intermolecular vibrational coupling make simple interpretations of these spectra difficult [7–9, 69, 70, 72]. In fact, many believe that the lower-frequency peak in the VV Raman spectrum arises from a collective mode [7, 69, 70, 72]. Theoretical work on vibrational spectroscopy in H2O has been limited. Rice and co-workers emphasized the role of intramolecular and intermolecular
90
james l. skinner, benjamin m. auer, and yu-shan lin
vibrational coupling, diagonalizing the vibrational Hamiltonian for configurations (albeit time-averaged for 66 fs) generated from an MD simulation [7, 72]. A similar approach has been implemented more recently by Bourˇ [158]. A related approach was also presented by Reimers and Watts [159]. Classical line shape calculations using an MD simulation of flexible molecules [79, 87–90], or from a Car–Parrinello simulation [91], have also been performed. Very recently mixed quantum/classical approaches that include dynamical effects have been presented by Buch et al. [71, 110] and by Torii [97]. In the former calculation the configurations were generated by MD simulation, the local mode anharmonic OH stretch frequencies were determined by an electric field map briefly described earlier, and were then time-averaged for 1 ps to approximate the effects of motional narrowing. Intermolecular couplings were taken to arise from interacting transition dipoles for the OH stretch chromophores. Similar calculations by Torii [97] treat the effects of motional narrowing exactly, but in doing so require extensive numerical computations. Herein we present calculations [6] for liquid H2O that are similar in spirit but different in detail from those of Buch [71, 110] and Torii [97]. The MD simulations are of the SPC/E model [135]. Local-mode anharmonic frequencies are generated from our most recent map developed for the HOD/D2O system [98], as are our transition dipoles. The relatively small intramolecular coupling fluctuates with molecular environment, and is determined by a separate map parameterized from ab initio calculations on clusters. The form of the intermolecular couplings is transition dipole, which is tested and parameterized from additional ab initio calculations. The effects of motional narrowing are taken into account approximately with the TAA [99]. In order to calculate IR and Raman line shapes within the TAA we need to specify the averaging time T. To determine this time we follow our earlier suggestion [99], calculating the line shape for the case of uncoupled chromophores, neglecting rotations, non-Condon effects, and lifetime broadening. The resulting line shape has a FWHM of " ¼ 349 cm 1 . As discussed previously [99], the averaging time is determined using the simple relationship T ¼ 5=". This results in an averaging time of T ¼ 76 fs, which gave an excellent approximation to the motionally-narrowed line shape (for the uncoupled situation) [99], and which will be used in all subsequent calculations. For water the experimental lifetime is T1 ¼ 260 fs [160]. We first focus on the IR line shape [6]. In Fig. 10 we plot the calculated (298 K) and experimental [62] (298 K) line shapes. The calculations agree well with the experiment, in terms of both peak position and FWHM, although the theory misses the weak shoulder at about 3250 cm 1 . The calculated (298 K) and the experimental [67] (295 K) Raman VV and VH line shapes are plotted in Fig. 11. The theoretical VV line shape captures the double-peaked nature of the experiment, although the relative intensity of the two peaks is not quite right. Moreover, the theory captures (but overemphasizes) the weak shoulder at
vibrational line shapes, spectral diffusion 1
IR
91
Theory Experiment
I(ω)
0.8 0.6 0.4 0.2 0
3000
3200
3400
3600
3800
ω (cm-1) Figure 10. Theoretical and experimental [62] IR line shapes for H2O at room temperature.
3650 cm 1 . The theoretical VH line shape is in good agreement with experiment, in terms of both its peak position and its markedly asymmetric shape. Given that the theoretical line shapes provide reasonable approximations to experiment, we can now attempt to understand the differences between the 1
Raman VV Theory Experiment
I(ω)
0.8 0.6 0.4 0.2 0 1
Raman VH
Theory Experiment
I(ω)
0.8 0.6 0.4 0.2 0 3000
3200
3400 ω
3600
3800
(cm-1)
Figure 11. Bottom panel: Theoretical and experimental [67] Raman VH line shapes for H2O at room temperature. Top panel: Theoretical and experimental [67] Raman VV line shapes for H2O at room temperature.
92
james l. skinner, benjamin m. auer, and yu-shan lin 15 R(ω) Rm(ω)
R(ω)
10
5
0
0.002 P(ω)
P(ω) Pc(ω)
0.001
0
3000
3200
3400 ω (cm-1)
3600
3800
Figure 12. Bottom panel: Theoretical distributions of instantaneous frequencies for the uncoupled ðPðoÞÞ and coupled (Pc ðoÞ) chromophores. Top panel: Inverse participation ratios RðoÞ and Rm ðoÞ. Both panels are for H2O at room temperature.
different line shapes and to interpret the various spectral features [6]. To examine the issue of coupling between OH stretch chromophores further, in Fig. 12 we compare the distribution of frequencies of the uncoupled chromophores to that for the fully coupled system. Thus the former is simply the distribution of local-mode frequencies oi , formally given by PðoÞ ¼ hdðo
oi Þi
ð19Þ
and is the same as shown in Figs. 2, 4, and 5. To obtain the distribution for the coupled system, at every time step we diagonalize kðtÞ and then the distribution is Pc ðoÞ ¼
1 X hdðo 2N k
ok Þi
ð20Þ
where ok are the 2N eigenvalues (N ¼ 128, the number of molecules). As shown in the figure, these two distributions are quite similar! Perhaps this is not too
vibrational line shapes, spectral diffusion
93
surprising, since the typical coupling matrix elements ð 25 cm 1 Þ are quite a bit smaller in magnitude than the fluctuations in the diagonal frequencies (say the FWHM, roughly 400 cm 1, of PðoÞ) [6]. Next we can consider the instantaneous eigenstates themselves. At each time step, kðtÞ is diagonalized by the orthogonal transformation BT kB. One can get a rough idea about over how many local-mode chromophores the eigenstates extend by considering the inverse participation ratio:
Rk ¼
2N X
jBik j
i¼1
4
!
1
ð21Þ
For example, if eigenstate k is equally pffiffiffispread among n chromophores, then for n of the terms in the sum jBik j ¼ 1= n, and is zero for the other terms, and so Rk ¼ n. Thus the value of the inverse participation ratio is roughly the number of chromophores involved in any given eigenstate. One can average Rk over eigenstates within a narrow frequency window to obtain RðoÞ, formally given by RðoÞ ¼
1 X hRk dðo 2N k
ok Þi=Pc ðoÞ
ð22Þ
This is also shown in Fig. 12. Remarkably, RðoÞ is as large as 12 near the center of the band! And even for 100 cm 1 on either side, it is still about eight. Thus the eigenstates are surprisingly delocalized. However, one can recall from studies of Anderson localization [161] that to produce localization (in the exponential sense) the width of the diagonal disorder must be significantly larger than the magnitude of the off-diagonal interactions; and even when the states are localized, they can still extend over many chromophores. Thus in terms of delocalization, a little coupling goes a long way! It is also of interest to determine, on the average, if the eigenstates extend primarily over OH groups on different molecules, or over both OH groups on each involved molecule. That is, if an eigenstate extends over n OH chromophores, are these chromophores on as many as n molecules, or are they on closer to n=2 molecules? To this end, let us replace the chromophore label i, by an equivalent set of two indices: a, which runs over the N molecules, and b, which goes from 1 to 2, indicating the two OH chromophores on each molecule. We can then define a molecular inverse participation ratio for each eigenstate k, by 0 ( )2 1 1 N 2 X X Rmk ¼ @ jBabk j2 A ð23Þ a¼1
b¼1
94
james l. skinner, benjamin m. auer, and yu-shan lin
Consider again the situation where an eigenstate is equallyp spread over n of the ffiffiffi chromophores, such that for n ab pairs we have jBabk j ¼ 1= n (while the others are zero). If the eigenstate involves OH groups on different molecules, then for each involved molecule only one term (with b ¼ 1 or 2) contributes, and so Rmk ¼ n. On the other hand, if the eigenstate is on both OH groups of each involved molecule, then for those molecules terms with b ¼ 1 and 2 both contribute, leading to Rmk ¼ n=2. Thus in either case, Rmk is roughly the number of molecules over which the eigenstate extends. One can average over eigenstates as in Eq. (22) to obtain Rm ðoÞ. This is also shown in Fig. 12. One sees that Rm ðoÞ is closer to RðoÞ than to RðoÞ=2, showing that for the most part the eigenstates extend over OH groups on different molecules. Thus, for example, at the center of the band, the eigenstates extend over approximately 10 molecules! Similar conclusions were reached by Buch et al. [71] in their analysis of the eigenstates in the top five layers of water at the liquid–vapor interface. In order to understand the effects of delocalization on spectroscopy, we have performed a detailed analysis of the (relevant for spectroscopy) weighting of the excitonic states [6]. We find that the seemingly modest coupling between the local modes, which indeed produces a modest change in the frequency distribution, produces a weighted frequency distribution that is dramatically different from the frequency distribution itself, including the introduction of a new frequency at about 3250 cm 1 . It is clear that the latter is a direct result of the significant delocalization of the eigenstates. This intensity enhancement at low frequencies is reminiscent of, and indeed has the same origin as, the enhanced optical activity of the eigenstate at the bottom of the exciton band for linear arrays with negative coupling among local-mode states [162]. This discussion corroborates earlier arguments in favor of collective excitations in liquid water and their role in vibrational spectra [7, 69–72]. In summary, then, we see that the relatively modest coupling between localmode chromophores produces a relatively small difference between the distributions of local-mode frequencies and of frequencies for the coupled chromophores. On the other hand, the coupling is large enough to delocalize the instantaneous eigenstates over up to 12 chromophores. In terms of spectroscopy, this delocalization has profound consequences, because it significantly affects the weighting of the different eigenstates. In particular, this delocalization is responsible for the appearance of a new collective characteristic frequency at about 3250 cm 1 . The IR and Raman VV and VH spectra are all different because of the different angular factors in the relevant linear combinations of the eigenstates [6]. Circumstances are such that in the VV spectrum the collective mode is prominent, whereas in the VH and IR spectra it is not. The IR spectrum is red-shifted significantly from the VH spectrum due to non-Condon effects.
vibrational line shapes, spectral diffusion B.
95
Ultrafast Experiments
It is more difficult to perform ultrafast spectroscopy on neat H2O (than it is on HOD/D2O or HOD/H2O) since the neat fluid is so absorptive in the OH stretch region. One innovative and very informative technique, developed by Dlott, involves IR pumping and Raman probing. This technique has a number of advantages over traditional IR pump-probe experiments: The scattered light is Stokes-shifted, which is less attenuated by the sample, and one can simultaneously monitor the populations of all Raman-active vibrations of the system at the same time. These experimental have been brought to bear on the spectral diffusion problem in neat water [18, 19, 75–77]. Photon echo and IR pump-probe experiments can also be performed on neat water, but one needs a very small sample. Fabrication of nano-fluidic Si3Ni4 sample cells have opened up this new and exciting field, and data from these experiments, performed by the Elsaesser and Miller groups, have recently been reported [73, 74]. At room temperature, spectral evolution occurs within 50 fs, and polarization anisotropy decays within 75 fs. At temperatures just about the freezing point, spectral evolution slows down dramatically [74]. Theoretical calculations for ultrafast neat water spectroscopy are difficult to perform and difficult to interpret (because of the near-resonant OH stretch coupling). One classical calculation of the 2DIR spectrum even preceded the experiments [163]! Torii has calculated the anisotropy decay [97], finding reasonable agreement with the experimental time scale. Mixed quantum/ classical calculations of nonlinear spectroscopy for many coupled chromophores is a daunting task. We developed the TAA for linear spectroscopy, and Jansen has very recently extended it to nonlinear spectroscopy [164]. We hope that this will allow for mixed quantum/classical calculations of the 2DIR spectrum for neat water and that this will provide the context for a molecularlevel interpretation of these complex but fascinating experiments. VII.
CONCLUSIONS
We have described our most recent efforts to calculate vibrational line shapes for liquid water and its isotopic variants under ambient conditions, as well as to calculate ultrafast observables capable of shedding light on spectral diffusion dynamics, and we have endeavored to interpret line shapes and spectral diffusion in terms of hydrogen bonding in the liquid. Our approach uses conventional classical effective two-body simulation potentials, coupled with more sophisticated quantum chemistry-based techniques for obtaining transition frequencies, transition dipoles and polarizabilities, and intramolecular and intermolecular couplings. In addition, we have used the recently developed time-averaging approximation to calculate Raman and IR line shapes for H2O (which involves
96
james l. skinner, benjamin m. auer, and yu-shan lin
many coupled chromophores). In most cases our agreement with experiment is reasonably good, and we believe we have reached a reasonable molecular-level understanding of most of the features in the line shapes and ultrafast observables. Of course there remains room for improvement in the theoretical calculations. One could do a better job on the ab initio calculations of clusters (higher-level theory, bigger basis sets or even extrapolation to the complete basis set limit [165, 166], larger clusters, larger number of molecules in the simulation, and hence more surrounding point charges in the electronic structure calculations). There surely exist better maps from nuclear coordinates of the liquid molecules to the OH stretch frequency of a tagged bond, as well as better maps for the transition dipoles and polarizabilities and the intramolecular coupling. One can surely improve upon the transition dipole approximation for the intermolecular coupling [167]. One should probably include the bend vibrational degrees of freedom, because they may couple to the stretches through a 2:1 Fermi resonance. Certainly, it would be better to use a simulation model with three-body interactions [86], or at least one with polarizability [85]. And as mentioned earlier, using a classical simulation model, especially with regard to the librations, may be problematic, suggesting the advisability of pursuing a model with quantum dynamics [168–174]. One problem yet to be solved theoretically involves ultrafast echo and pumpprobe experiments on H2O. Jansen has extended the time-averaging approximation to nonlinear ultrafast spectroscopy [164], meaning that one is now in the position of calculating 2DIR spectra for liquid water, which would allow for direct comparison with results from the exciting new experiments [73, 74]. Another theoretical frontier involves the study of the vibrational spectroscopy of water at other conditions, or in other phases. Here it will be crucially important to use more robust water models, since many effective two-body simulation models were parameterized to give agreement with experiment at one state point: room temperature and one atmosphere pressure. We have already seen that using these models at higher or lower temperatures even for liquid water leads to discrepancies. We note that a significant amount of important theoretical work on ice has already been published by Buch and others [71, 72, 111, 175, 176]. A final theoretical frontier involves water in heterogeneous environments. One such important problem is the water liquid/vapor interface. Terrific and informative vibrational sum-frequency generation experiments [177–186] on this problem have been performed over the last 10 or 15 years, leading to a good understanding of how dynamics and hydrogen bonding differ at this interface from in the bulk [187–192]. A significant amount of theoretical work has been completed on this system as well [71, 110, 140, 193–201]. Very recently, ultrafast surface-selective vibrational spectroscopy experiments have also appeared [202–204], and are in need of a good theoretical description. Other
vibrational line shapes, spectral diffusion
97
examples of important heterogeneous environments amenable to vibrational spectroscopy experiments include water in salt solutions [205, 206] (and in particular it would be great to reach an understanding of the century-old Hoffmeister series ordering [207–209]), confined water (for example, in nanotubes and reverse micelles) and biological water (in and around biomacromolecules) [187, 210–213], and all of these problems await further theoretical (and in many cases experimental) study. We close by hoping that in the next book or article that Philip Ball writes about water, he can honestly say that we now understand at least a little about this fascinating liquid! After this manuscript was submitted, a theoretical paper on 2DIR line shapes for neat H2O appeared [214]. Acknowledgments The authors thank former group members Chris Lawrence, Andrei Piryatinski, Steve Corcelli, and J. R. Schmidt for laying the groundwork for much of the research discussed herein. We also thank the National Science Foundation for support of this work through grants CHE-0446666 and CHE0750307.
References 1. P. Ball, Life’s Matrix: A Biography of Water, Farrar, Straus, and Giroux, New York, 1999. 2. P. Ball, Nature 452, 291 (2008). 3. R. Kubo, Adv. Chem. Phys. 15, 101 (1969). 4. J. L. Skinner, (to be submitted). 5. E. B. Wilson, J. C. Decius, and P. C. Cross, Molecular Vibrations, Dover, New York 1980. 6. B. Auer and J. L. Skinner, J. Chem. Phys. 128, 224511 (2008). 7. A. Belch and S. Rice, J. Chem. Phys. 78, 4817 (1983). 8. T. T. Wall and D. F. Hornig, J. Chem. Phys. 43, 2079 (1965). 9. M. Falk and T. A. Ford, Can. J. Chem. 44, 1699 (1966). 10. C. J. Fecko, J. J. Loparo, S. T. Roberts, and A. Tokmakoff, J. Chem. Phys. 122, 054506 (2005). 11. G. E. Walrafen, J. Chem. Phys. 48, 244 (1968). 12. J. D. Smith, C. D. Cappa, K. R. Wilson, R. C. Cohen, P. L. Geissler, and R. J. Saykally, Proc. Natl. Acad. Sci. USA 102, 14171 (2005). 13. S. A. Corcelli and J. L. Skinner, J. Phys. Chem. A 109, 6154 (2005). 14. J. J. Loparo, S. T. Roberts, R. A. Nicodemus, and A. Tokmakoff, Chem. Phys. 341, 218 (2007). 15. C. P. Lawrence and J. L. Skinner, Chem. Phys. Lett. 369, 472 (2003). 16. R. Laenen, C. Rauscher, and A. Laubereau, Phys. Rev. Lett. 80, 2622 (1998). 17. R. Laenen, C. Rauscher, and A. Laubereau, J. Phys. Chem. B 102, 9304 (1998). 18. Z. Wang, A. Pakoulev, Y. Pang, and D. D. Dlott, J. Phys. Chem. A 108, 9054 (2004). 19. Z. Wang, A. Pakoulev, Y. Pang, and D. D. Dlott, Chem. Phys. Lett. 378, 281 (2003). 20. J. Stenger, D. Madsen, P. Hamm, E. T. J. Nibbering, and T. Elsaesser, Phys. Rev. Lett. 87, 027401 (2001).
98
james l. skinner, benjamin m. auer, and yu-shan lin
21. J. Stenger, D. Madsen, P. Hamm, E. T. J. Nibbering, and T. Elsaesser, J. Phys. Chem. A 106, 2341 (2002). 22. S. Yeremenko, M. S. Pshenichnikov, and D. A. Wiersma, Chem. Phys. Lett. 369, 107 (2003). 23. C. J. Fecko, J. D. Eaves, J. J. Loparo, A. Tokmakoff, and P. L. Geissler, Science 301, 1698 (2003). 24. J. D. Eaves, J. J. Loparo, C. J. Fecko, S. T. Roberts, A. Tokmakoff, and P. L. Geissler, Proc. Natl. Acad. Sci. USA 102, 13019 (2005). 25. J. J. Loparo, S. T. Roberts, and A. Tokmakoff, J. Chem. Phys. 125, 194521 (2006). 26. J. J. Loparo, S. T. Roberts, and A. Tokmakoff, J. Chem. Phys. 125, 194522 (2006). 27. H. Graener, G. Seifert, and A. Laubereau, Phys. Rev. Lett. 66, 2092 (1991). 28. R. Laenen, K. Simeonidis, and A. Laubereau, J. Phys. Chem. B 106, 408 (2002). 29. G. M. Gale, G. Gallot, F. Hache, N. Lascoux, S. Bratos, and J.-Cl. Leicknam, Phys. Rev. Lett. 82, 1068 (1999). 30. G. M. Gale, G. Gallot, and N. Lascoux, Chem. Phys. Lett. 311, 123 (1999). 31. S. Bratos, G. M. Gale, G. Gallot, F. Hache, N. Lascoux, and J.-Cl. Leicknam, Phys. Rev. E 61, 5211 (2000). 32. G. Gallot, N. Lascoux, G. M. Gale, J.-Cl. Leicknam, S. Bratos, and S. Pommeret, Chem. Phys. Lett. 341, 535 (2001). 33. G. Gallot, S. Bratos, S. Pommeret, N. Lascoux, J-Cl. Leicknam, M. Kozin¨ski, W. Amir, and G. M. Gale, J. Chem. Phys. 117, 11301 (2002). 34. Z. Wang, Y. Pang, and D. D. Dlott, J. Chem. Phys. 120, 8345 (2004). 35. Z. Wang, Y. Pang, and D. D. Dlott, Chem. Phys. Lett. 397, 40 (2004). 36. S. Woutersen and H. J. Bakker, Phys. Rev. Lett. 83, 2077 (1999). 37. D. Schwarzer, J. Lindner, and P. Vo¨hringer, J. Chem. Phys. 123, 161105 (2005). 38. C. P. Lawrence and J. L. Skinner, J. Chem. Phys. 118, 264 (2003). 39. C. P. Lawrence and J. L. Skinner, J. Chem. Phys. 117, 8847 (2002). 40. R. Rey, K. B. Møller, and J. T. Hynes, J. Phys. Chem. A 106, 11993 (2002). 41. A. Piryatinski, C. P. Lawrence, and J. L. Skinner, J. Chem. Phys. 118, 9672 (2003). 42. R. Rey, K. B. Møller, and J. T. Hynes, Chem. Rev. 104, 1915 (2004). 43. J. D. Eaves, A. Tokmakoff, and P. L. Geissler, J. Phys. Chem. A 109, 9424 (2005). 44. S. Woutersen, U. Emmerichs, and H. J. Bakker, Science 278, 658 (1997). 45. H.-K. Nienhuis, R. A. van Santen, and H. J. Bakker, J. Chem. Phys. 112, 8487 (2000). 46. H. J. Bakker, S. Woutersen, and H.-K. Nienhuys, Chem. Phys. 258, 233 (2000). 47. J. J. Loparo, C. J. Fecko, J. D. Eaves, S. T. Roberts, and A. Tokmakoff, Phys. Rev. B 70, 180201 (2004). 48. Y. L. A. Rezus and H. J. Bakker, J. Chem. Phys. 125, 144512 (2006). 49. J. R. Scherer, M. K. Go, and S. Kint, J. Phys. Chem. 78, 1304 (1974). 50. H. R. Wyss and M. Falk, Can. J. Chem. 48, 607 (1970). 51. W. A. Senior and R. E. Verrall, J. Phys. Chem. 73, 4242 (1969). 52. Y.-S. Lin and J. L. Skinner, (to be submitted). 53. T. Steinel, J. B. Asbury, S. A. Corcelli, C. P. Lawrence, J. L. Skinner, and M. D. Fayer, Chem. Phys. Lett. 386, 295 (2004). 54. J. B. Asbury, T. Steinel, C. Stromberg, S. A. Corcelli, C. P. Lawrence, J. L. Skinner, and M. D. Fayer, J. Phys. Chem. A 108, 1107 (2004).
vibrational line shapes, spectral diffusion
99
55. J. B. Asbury, T. Steinel, K. Kwak, S. A. Corcelli, C. P. Lawrence, J. L. Skinner, and M. D. Fayer, J. Chem. Phys. 121, 12431 (2004). 56. M. F. Kropman, H.-K. Nienhuys, S. Woutersen, and H. J. Bakker, J. Phys. Chem. A 105, 4622 (2001). 57. T. Steinel, J. B. Asbury, J. Zheng, and M. D. Fayer, J. Phys. Chem. A 108, 10958 (2004). 58. Y. L. A. Rezus and H. J. Bakker, J. Chem. Phys. 123, 114502 (2005). 59. D. E. Moilanen, E. E. Fenn, Y.-S. Lin, J. L. Skinner, B. Bagchi, and M. D. Fayer, Proc. Natl. Acad. Sci. USA 105, 5295 (2008). 60. R. A. Nicodemus and A. Tokmakoff, Chem. Phys. Lett. 449, 130 (2007). 61. J. E. Bertie, M. K. Ahmed, and H. H. Eysel, J. Phys. Chem. 93, 2210 (1989). 62. J. E. Bertie and Z. Lan, Appl. Spec. 50, 1047 (1996). 63. J. B. Brubach, A. Mermet, A. Filabozzi, A. Gerschel, and R. Roy, J. Chem. Phys. 122, 184509 (2005). 64. G. E. Walrafen, J. Chem. Phys. 47, 114 (1967). 65. W. F. Murphy and H. J. Bernstein, J. Phys. Chem. 76, 1147 (1972). 66. G. D’Arrigo, G. Maisano, F. Mallamace, P. Migliardo, and F. Wanderlingh, J. Chem. Phys. 75, 4264 (1981). 67. G. E. Walrafen, M. R. Fischer, M. S. Hokmabadi, and W. H. Yang, J. Chem. Phys. 85, 6970 (1986). 68. D. M. Carey and G. M. Korenowski, J. Chem. Phys. 108, 2669 (1998). 69. J. L. Green, A. R. Lacey, and M. G. Sceats, J. Phys. Chem. 90, 3958 (1986). 70. D. E. Hare and C. M. Sorensen, J. Chem. Phys. 96, 13 (1992). 71. V. Buch, T. Tarbuck, G. L. Richmond, H. Groenzin, I. Li, and M. J. Schultz, J. Chem. Phys. 127, 204710 (2007). 72. S. Rice, M. Bergren, A. Belch, and N. Nielson, J. Phys. Chem. 87, 4295 (1983). 73. M. L. Cowan, B. D. Bruner, N. Huse, J. R. Dwyer, B. Chugh, E. T. J. Nibbering, T. Elsaesser, and R. D. J. Miller, Nature 434, 199 (2005). 74. D. Kraemer, M. L. Cowan, A. Paarmann, N. Huse, E. T. J. Nibbering, T. Elsaesser, and R. J. D. Miller, Proc. Natl. Acad. Sci. USA 105, 437 (2008). 75. J. C. Dea`k, S. T. Rhea, L. K. Iwaki, and D. D. Dlott, J. Phys. Chem. A 104, 4866 (2000). 76. A. Pakoulev, Z. Wang, and D. D. Dlott, Chem. Phys. Lett. 371, 594 (2003). 77. Z. Wang, Y. Pang, and D. D. Dlott, J. Phys. Chem. A 111, 3196 (2007). 78. D. A. McQuarrie, Statistical Mechanics, Harper and Row, New York, 1976. 79. H. Ahlborn, X. Ji, B. Space, and P. B. Moore, J. Chem. Phys. 111, 10622 (1999). 80. L. Frommhold, Collision-Induced Absorption in Gases, 1st edition, Cambridge Monographs on Atomic, Molecular, and Chemical Physics, Vol. 2 Cambridge University Press, England, 1993. 81. P. H. Berens, S. R. White, and K. R. Wilson, J. Chem. Phys. 75, 515 (1981). 82. J. S. Bader and B. J. Berne, J. Chem. Phys. 100, 8359 (1994). 83. S. A. Egorov, K. F. Everitt, and J. L. Skinner, J. Phys. Chem. A 103, 9494 (1999). 84. J. L. Skinner and K. Park, J. Phys. Chem. B 105, 6716 (2001). 85. B. Guillot, J. Mol. Liquids 101, 219 (2002). 86. R. Kumar and J. L. Skinner, J. Phys. Chem. B 112, 8311 (2008). 87. R. Bansil, T. Berger, K. Toukan, M. A. Ricci, and S. H. Chen, Chem. Phys. Lett. 132, 165 (1986).
100
james l. skinner, benjamin m. auer, and yu-shan lin
88. J. Martı´, J. A. Padro´, and E. Gua`rdia, J. Mol. Liq. 62, 17 (1994). 89. J. Martı´, E. Gua`rdia, and J. A. Padro´, J. Chem. Phys. 101, 1083 (1994). 90. H. Ahlborn, B. Space, and P. B. Moore, J. Chem. Phys. 112, 8083 (2000). 91. P. L. Silvestrelli, M. Bernasconi, and M. Parrinello, Chem. Phys. Lett. 277, 478 (1997). 92. S. Mukamel, Principles of Nonlinear Optical Spectroscopy, Oxford, New York, 1995. 93. J. G. Saven and J. L. Skinner, J. Chem. Phys. 99, 4391 (1993). 94. M. D. Stephens, J. G. Saven, and J. L. Skinner, J. Chem. Phys. 106, 2129 (1997). 95. T. l. C. Jansen and J. Knoester, J. Phys. Chem. B 110, 22910 (2006). 96. R. D. Gorbunov, P. H. Nguyen, M. Kobus, and G. Stock, J. Chem. Phys. 126, 054509 (2007). 97. H. Torii, J. Phys. Chem. A 110, 9469 (2006). 98. B. Auer, R. Kumar, J. R. Schmidt, and J. L. Skinner, Proc. Natl. Acad. Sci. USA 104, 14215 (2007). 99. B. Auer and J. L. Skinner, J. Chem. Phys. 127, 104105 (2007). 100. T. l. C. Jansen, D. Cringus, and M. S. Pshenichnikov, preprint (2009). 101. T. l. C. Jansen, W. Zhuang, and S. Mukamel, J. Chem. Phys. 121, 10577 (2004). 102. T. l. C. Jansen, A. G. Dijkstra, T. M. Watson, J. D. Hirst, and J. Knoester, J. Chem. Phys. 125, 044312 (2006). 103. S. Mukamel and D. Abramavicius, Chem. Rev. 104, 2073 (2004). 104. W. Zhuang, D. Abramavicius, T. Hayashi, and S. Mukamel, J. Phys. Chem. B 110, 3362 (2006). 105. J. Choi, S. Hahn, and M. Cho, Int. J. Quant. Chem. 104, 616 (2005). 106. D. W. Oxtoby, D. Levesque, and J.-J. Weis, J. Chem. Phys. 68, 5528 (1978). 107. K. F. Everitt, C. P. Lawrence, and J. L. Skinner, J. Phys. Chem. B 108, 10440 (2004). 108. S. Hahn, S. Ham, and M. Cho, J. Phys. Chem. B 109, 11789 (2005). 109. J.-H. Choi, H. Lee, K.-K. Lee, S. Hahn, and M. Cho, J. Chem. Phys. 126, 045102 (2007). 110. V. Buch, J. Phys. Chem. B 109, 17771 (2005). 111. V. Buch, S. Bauerecker, J. P. Devlin, U. Buck, and J. K. Kazimirski, Int. Rev. Phys. Chem. 23, 375 (2004). 112. Z. Ganim and A. Tokmakoff, Biophys. J. 91, 2636 (2006). 113. L. Ojama¨e, J. Tegenfeldt, J. Lindgren, and K. Hermansson, Chem. Phys. Lett. 195, 97 (1992). 114. D. Zimdars, A. Tokmakoff, S. Chen, S. R. Greenfield, M. D. Fayer, T. I. Smith, and H. A. Schwettman, Phys. Rev. Lett. 70, 2718 (1993). 115. P. Hamm, M. Lim, and R. M. Hochstrasser, Phys. Rev. Lett. 81, 5326 (1998). 116. P. Hamm and R. M. Hochstrasser, Structure and dynamics of proteins and peptides: Femtosecond two-dimensional infrared spectroscopy, in Ultrafast Infrared and Raman Spectroscopy, Markel Dekker, New York, 2001, p. 273. 117. E. T. J. Nibbering and T. Elsaesser, Chem. Rev. 104, 1887 (2004). 118. J. Zheng, K. Kwak, and M. D. Fayer, Acc. Chem. Res. 40, 75 (2007). 119. M. Cho, Chem. Rev. 108, 1331 (2008). 120. J. R. Schmidt, S. A. Corcelli, and J. L. Skinner, J. Chem. Phys. 123, 044513 (2005). 121. J. R. Schmidt, S. T. Roberts, J. J. Loparo, A. Tokmakoff, M. D. Fayer, and J. L. Skinner, Chem. Phys. 341, 143 (2007). 122. G. Lipari and A. Szabo, Biophys. J. 30, 489 (1980). 123. D. W. Oxtoby, Adv. Chem. Phys. 47 (Part 2), 487 (1981).
vibrational line shapes, spectral diffusion
101
124. L. Ojama¨e, K. Hermansson, and M. Probst, Chem. Phys. Lett. 191, 500 (1992). 125. K. B. Møller, R. Rey, and J. T. Hynes, J. Phys. Chem. A 108, 1275 (2004). 126. E. Harder, J. D. Eaves, A. Tokmakoff, and B. J. Berne, Proc. Natl. Acad. Sci. USA 102, 11611 (2005). 127. M. Diraison, Y. Guissani, J.-Cl. Leicknam, and S. Bratos, Chem. Phys. Lett. 258, 348 (1996). 128. T. Hayashi, T. l. C. Jansen, W. Zhuang, and S. Mukamel, J. Phys. Chem. A 109, 64 (2005). 129. S. Li, J. R. Schmidt, S. A. Corcelli, C. P. Lawrence, and J. L. Skinner, J. Chem. Phys. 124, 204110 (2006). 130. K. Hermansson, S. Knuts, and J. Lindgren, J. Chem. Phys. 95, 7486 (1991). 131. S. A. Corcelli, C. P. Lawrence, and J. L. Skinner, J. Chem. Phys. 120, 8107 (2004). 132. K. Kwac and M. Cho, J. Chem. Phys. 119, 2247 (2003). 133. K. Hermansson, J. Chem. Phys. 99, 861 (1993). 134. J. Sadlej, V. Buch, J. K. Kazimirski, and U. Buck, J. Phys. Chem. A 103, 4933 (1999). 135. H. J. C. Berendsen, J. R. Grigera, and T. P. Straatsma, J. Phys. Chem. 91, 6269 (1987). 136. C. M. Huggins and G. C. Pimentel, J. Phys. Chem. 60, 1615 (1956). 137. G. C. Pimentel and A. L. McClellan, The Hydrogen Bond, W. H. Freeman and Company, San Francisco, 1960. 138. D. N. Glew and N. S. Rath, Can. J. Chem. 49, 837 (1971). 139. E. Whalley and D. D. Klug, J. Chem. Phys. 84, 78 (1986). 140. A. Morita and J. T. Hynes, Chem. Phys. 258, 371 (2000). 141. J. G. Scherer and R. G. Snyder, J. Chem. Phys. 67, 4794 (1977). 142. J. E. Bertie, B. F. Francis, and J. R. Scherer, J. Chem. Phys. 73, 6352 (1980). 143. R. Kumar, J. R. Schmidt, and J. L. Skinner, J. Chem. Phys. 126, 204107 (2007). 144. A. E. Reed and F. Weinhold, J. Chem. Phys. 78, 4066 (1983). 145. F. Weinhold and C. Landis, Valency and Bonding: A Natural Bond Orbital Donor–Acceptor Perspective, Cambridge University Press: Cambridge 2005. 146. T. l. C. Jansen, T. Hayashi, W. Zhuang, and S. Mukamel, J. Chem. Phys. 123, 114504 (2005). 147. A. Piryatinski, C. P. Lawrence, and J. L. Skinner, J. Chem. Phys. 118, 9664 (2003). 148. J. R. Schmidt, N. Sundlass, and J. L. Skinner, Chem. Phys. Lett. 378, 559 (2003). 149. S. T. Roberts, J. J. Loparo, and A. Tokmakoff, J. Chem. Phys. 125, 084502 (2006). 150. W. L. Jorgensen, J. Chandrasekhar, J. D. Madura, R. W. Impey, and M. L. Klein, J. Chem. Phys. 79, 926 (1983). 151. S. A. Corcelli, C. P. Lawrence, J. B. Asbury, T. Steinel, M. D. Fayer, and J. L. Skinner, J. Chem. Phys. 121, 8897 (2004). 152. S. W. Rick, S. J. Stuart, and B. J. Berne, J. Chem. Phys. 101, 6141 (1994). 153. D. Laage and J. T. Hynes, Chem. Phys. Lett. 433, 80 (2006). 154. C. P. Lawrence and J. L. Skinner, J. Chem. Phys. 117, 5827 (2002). 155. C. Vega, E. Sanz, and J. L. F. Abascal, J. Chem. Phys. 122, 114507 (2005). 156. D. A. Schmidt and K. Miki, J. Phys. Chem. A 111, 10119 (2007). 157. P. L. Geissler, J. Am. Chem. Soc. 127, 14930 (2005). 158. P. Bourˇ, Chem. Phys. Lett. 365, 82 (2002). 159. J. R. Reimers and R. O. Watts, Chem. Phys. 91, 201 (1984). 160. A. J. Lock and H. J. Bakker, J. Chem. Phys. 117, 1708 (2002).
102
james l. skinner, benjamin m. auer, and yu-shan lin
161. J. L. Skinner, J. Phys. Chem. 98, 2503 (1994). 162. D.-J. Heijs, A. G. Dijkstra, and J. Knoester, Chem. Phys. 341, 230 (2007). 163. R. DeVane, B. Space, A. Perry, C. Neipert, C. Ridley, and T. Keyes, J. Chem. Phys. 121, 3688 (2004). 164. T. l. C. Jansen and W. M. Ruszel, J. Chem. Phys. 128, 214501 (2008). 165. A. Halkier, H. Koch, P. Jørgensen, O. Christiansen, I. M. B. Nielsen, and T. Helgaker, Theor. Chem. Acc. 97, 150 (1997). 166. S. S. Xantheas, C. J. Burnham, and R. J. Harrison, J. Chem. Phys. 116, 1493 (2002). 167. P. Hamm and S. Woutersen, Bull. Chem. Soc. Jpn. 75, 985 (2002). 168. R. A. Kuharski and P. J. Rossky, J. Chem. Phys. 82, 5164 (1985). 169. J. A. Poulsen, G. Nyman, and P. J. Rossky, Proc. Natl. Acad. Sci. USA 102, 6709 (2005). 170. J. A. Poulsen, G. Nyman, and P. J. Rossky, J. Chem. Th. Comp. 2, 1482 (2006). 171. H. A. Stern and B. J. Berne, J. Chem. Phys. 115, 7622 (2001). 172. G. S. Fanourgakis, G. K. Schenter, and S. S. Xantheas, J. Chem. Phys. 125, 141102 (2006). 173. F. Paesani, W. Zhang, D. A. Case, T. E. Cheatum III, and G. A. Voth, J. Chem. Phys. 125, 184507 (2006). 174. F. Paesani, S. Iuchi, and G. A. Voth, J. Chem. Phys. 127, 074506 (2007). 175. V. Buch and J. P. Devlin, J. Chem. Phys. 110, 3437 (1999). 176. M. J. Wo´jcik, K. Szczeponek, and S. Ikeda, J. Chem. Phys. 117, 9850 (2002). 177. Q. Du, R. Superfine, E. Freysz, and Y. R. Shen, Phys. Rev. Lett. 70, 2313 (1993). 178. Q. Du, E. Freysz, and Y. R. Shen, Science 264, 826 (1994). 179. X. Wei and Y. R. Shen, Phys. Rev. Lett. 86, 4799 (2001). 180. V. Ostroverkhov, G. A. Waychunas, and Y. R. Shen, Phys. Rev. Lett. 94, 046102 (2005). 181. Y. R. Shen and V. Ostroverkhov, Chem. Rev. 106, 1140 (2006). 182. M. G. Brown, E. A. Raymond, H. C. Allen, L. F. Scatena, and G. L. Richmond, J. Phys. Chem. A 104, 10220 (2000). 183. E. A. Raymond, T. L. Tarbuck, and G. L. Richmond, J. Phys. Chem. B 106, 2817 (2002). 184. E. A. Raymond, T. L. Tarbuck, M. G. Brown, and G. L. Richmond, J. Phys. Chem. B 107, 546 (2003). 185. M. J. Shultz, S. Baldelli, C. Schnitzer, and D. Simonelli, J. Phys. Chem. B 106, 5313 (2002). 186. D. Liu, G. Ma, L. M. Levering, and H. C. Allen, J. Phys. Chem. B 108, 2252 (2004). 187. J. Faeder and B. M. Ladanyi, J. Phys. Chem. B 104, 1033 (2000). 188. I.-F. W. Kuo and C. J. Mundy, Science 303, 658 (2004). 189. P. Liu, E. Harder, and B. J. Berne, J. Phys. Chem. B 108, 6595 (2004). 190. P. Liu, E. Harder, and B. J. Berne, J. Phys. Chem. B 109, 2949 (2005). 191. V. Buch, A. Milet, R. Va´cha, P. Jungwirth, and J. P. Devlin, Proc. Natl. Acad. Sci. USA 104, 7342 (2007). 192. J. Chowdhary and B. M. Ladanyi, J. Phys. Chem. B 110, 15442 (2006). 193. I. Benjamin, Phys. Rev. Lett. 73, 2083 (1994). 194. D. S. Walker, D. K. Hore, and G. L. Richmond, J. Phys. Chem. B 110, 20451 (2006). 195. A. Morita and J. T. Hynes, J. Phys. Chem. B 106, 673 (2002). 196. A. Morita, J. Phys. Chem. B 110, 3158 (2006).
vibrational line shapes, spectral diffusion
103
197. A. Perry, H. Ahlborn, B. Space, and P. B. Moore, J. Chem. Phys. 118, 8411 (2003). 198. A. Perry, C. Neipert, C. R. Kasprzyk, T. Green, B. Space, and P. B. Moore, J. Chem. Phys. 123, 144705 (2005). 199. A. Perry, C. Neipert, C. Ridley, B. Space, and P. B. Moore, Phys. Rev. E 71, 050601 (2005). 200. A. Perry, C. Neipert, B. Space, and P. B. Moore, Chem. Rev. 106, 1234 (2006). 201. E. Brown, M. Mucha, P. Jungwirth, and D. J. Tobias, J. Phys. Chem. B 109, 7934 (2005). 202. A. N. Bordenyuk and A. V. Benderskii, J. Chem. Phys. 122, 134713 (2005). 203. J. A. McGuire and Y. R. Shen, Science 313, 1945 (2006). 204. M. Smits, A. Ghosh, M. Sterrer, M. Mu¨ller, and M. Bonn, Phys. Rev. Lett. 98, 098302 (2007). 205. S. Park and M. D. Fayer, Proc. Natl. Acad. Sci. USA 104, 16731 (2007). 206. J. D. Smith, R. J. Saykally, and P. L. Geissler, J. Am. Chem. Soc. 129, 13847 (2007). 207. L. M. Pegram and M. T. Record, Jr., J. Phys. Chem. B 111, 5411 (2007). 208. Y. J. Zhang and P. S. Cremer, Curr. Opin. Chem. Biol. 10, 658 (2006). 209. P. B. Petersen and R. J. Saykally, Annu. Rev. Phys. Chem. 57, 333 (2006). 210. M. R. Harpham, B. M. Ladanyi, N. E. Levinger, and K. W. Herwig, J. Chem. Phys. 121, 7855 (2004). 211. M. R. Harpham, B. M. Ladanyi, and N. E. Levinger, J. Phys. Chem. B 109, 16891 (2005). 212. I. R. Piletic, D. E. Moilanen, D. B. Spry, N. E. Levinger, and M. D. Fayer, J. Phys. Chem. A 110, 4985 (2006). 213. D. B. Spry, A. Goun, K. Glusac, D. E. Moilanen, and M. D. Fayer, J. Am. Chem. Soc. 129, 8122 (2007). 214. A Paarmann, T. Hayashi, S. Mukamel, and R. J. D. Miller, J. Chem. Phys. 128, 191103 (2008).
COMPUTATIONAL MODELS OF METABOLISM: STABILITY AND REGULATION IN METABOLIC NETWORKS RALF STEUER Manchester Interdisciplinary Biocentre, School of Chemical Engineering and Analytical Science, University of Manchester, Manchester M1 7DN, United Kingdom ¨ RN H. JUNKER BJO Leibniz Institute of Plant Genetics and Crop Plant Research (IPK), 06466 Gatersleben, Germany All exact science is dominated by the idea of approximation —Betrand Russell
CONTENTS I. Introduction II. Cellular Metabolism and the Art of Modeling A. From Topology to Kinetics: A Hierarchy of Models 1. Detailed Kinetic Models 2. Topological Network Analysis 3. Stoichiometric Analysis 4. Intermediate Approaches B. The Rationale of Mathematical Modeling III. The Basic Concepts of Metabolic Modeling A. Computational Models of Metabolism B. The Properties of the Stoichiometric Matrix 1. The Left Nullspace 2. The Right Nullspace 3. An Example
Advances in Chemical Physics, Volume 142, edited by Stuart A. Rice Copyright # 2009 John Wiley & Sons, Inc.
105
106
ralf steuer and bjo¨rn h. junker C.
IV.
V.
VI.
VII.
VIII.
Enzymes and Chemical Reaction Rates 1. Chemical Equilibrium and Thermodynamics 2. Michaelis–Menten Kinetics 3. Reversible Michaelis–Menten Kinetics 4. The Equilibrium Constants and Haldane Relationship 5. Steady-State Kinetics of Multisubstrate Reactions 6. Inhibition and Allosteric Control of Enzyme Activity D. Putting the Parts Together: A Short Guide From Measuring Metabolites to Metabolomics A. The Problem of Organizational Complexity B. Targeted Analysis of Metabolites C. High-Throughput Measurements: Metabolomics Topological and Stoichiometric Analysis A. Topological Network Analysis B. Flux Balance Analysis and Elementary Flux Modes 1. Elementary Flux Modes (EFMs) 2. Flux Balance Analysis (FBA) C. The Limits of Flux Balance Analysis Measuring the Fluxome: 13C-Based Flux Analysis A. Steady-State Metabolic Flux Analysis 1. The Experimental Concepts 2. Evaluation of a Carbon Labeling Experiment B. Dynamic Flux Analysis Formal Approaches to Metabolism A. The Dynamics of Complex Systems 1. The Stability of Simple Pathways 2. Bistability and Hysteresis 3. The Jacobian Matrix and Linear Stability Analysis 4. Dynamics of Metabolism: A Minimal Model of Glycolysis B. Metabolic Control Analysis 1. The Summation and Connectivity Theorems 2. Scaled Coefficients and Their Interpretation 3. A Brief Criticism of Metabolic Control Analysis C. Biochemical Systems Theory and Related Approaches 1. Biochemical Systems Theory 2. Linear-Logarithmic Kinetics 3. Convenience Kinetics and Related Approaches Structural Kinetic Modeling A. Definitions: The Jacobian Matrix Revisited 1. The Matrix L 2. The Saturation Matrix hlx 3. An Alternative Derivation and the Relationship with MCA B. Rewriting the System: A Simple Example C. Detecting Dynamics and Bifurcations: Glycolysis Revisited 1. Evaluating the Dynamics D. Yeast Glycolysis: A Monte Carlo Approach 1. Defining the Structural Kinetic Model 2. An Analysis of the Parameter Space 3. Sampling the Parameters 4. The Possible Function of Glycolytic Oscillations
computational models of metabolism
107
E.
Thermodynamics and Reversible Rate Equations 1. The Contribution from Allosteric Regulation 2. The Contribution from Kinetics 3. The Contribution from Thermodynamics 4. Parameterizing the Jacobian 5. Examples and Pitfalls in the Sampling of the Parameter F. Complex Dynamics: A Model of the Calvin Cycle IX. Stability and Regulation in Metabolism A. Identifying Stabilizing Sites in Metabolic Networks B. The Robustness of Metabolic States 1. The Role of Feedback Mechanisms 2. The Robustness of Metabolic States X. Epilogue: Toward Genome-Scale Kinetic Models Acknowledgments References
I.
INTRODUCTION
Despite their often overwhelming complexity, all living organisms are essentially chemical systems, predicated by a set of chemical reactions taking place in an aqueous solution within membrane-bounded compartments. Yet this entangled set of reactions gives rise to behavior that seems to be vastly different from any other known chemical system, enabling cells and organisms to engage in seemingly purposeful behavior, to grow, and to reproduce [1]. Deciphering the architecture underlying these interconnected physicochemical processes remains one of the greatest challenges of our time. As one of the integrant characteristics of life, the organization and functioning of metabolic processes has been a focus of research for more than a century and constitutes the traditional subject of biochemistry. More recently, starting in the 1990s, the advent of high-throughput technologies and other methodological innovations has created unprecedented new opportunities to study the mechanisms and interactions that govern metabolic processes. Complementing the massively parallel monitoring of cellular components on the transcriptional (transcriptomics) and translational (proteomics) level, rapid developments in mass spectrometry (MS)-based methods do now allow a quantification of metabolic compounds (metabolomics) on a scale that was impossible a mere decade ago [2–6]. This rather recently acquired ability to generate transcriptomic, proteomic, and metabolomic data in almost intimidating quantities has far-reaching impact on all biological sciences and already shapes the face of most of the current molecular biology. However, paraphrasing Poincare´, a mere accumulation of data and disconnected facts is no more knowledge than a heap of stones is a house. Concomitant to the development of novel experimental methods,
108
ralf steuer and bjo¨rn h. junker
molecular biology has undergone a profound change in the necessity to employ mathematical and computational methods to facilitate the analysis—and eventually understanding—of the intertwined processes taking place in living cells. These new and more quantitative forms of analysis, usually attributed to the emergent field of Systems Biology [7–9], are based on rigorous integration of high-throughput data generation and new computational tools, to eventually discern the principles that govern cellular behavior. As a characteristic paradigm, Systems Biology involves the development of computational cellular models, at multiple levels of abstraction, to achieve a quantitative and predictive understanding of cellular functions [10, 11]. Within this contribution, we seek to describe and discuss the ways and means of constructing a computational representation of cellular metabolic processes. Our main focus is the development of explicit kinetic models of cellular metabolic networks, integrating data from heterogeneous sources, quantitative experiments, and computer modeling. The contribution seeks to cover three important aspects of current computational approaches to metabolism: (i) classic enzyme-kinetics that describe the interactions between the building blocks of large metabolic models, (ii) experimental advances in the accessibility of cellular variables, in particular 13 C-based flux measurements, and (iii) formal frameworks to build and evaluate large-scale kinetic models of metabolism. As a matter of course, the review presented here is far from comprehensive. The construction of kinetic models has a long history in the chemical and biochemical sciences, and a number of authoritative monographs and articles covering the field have appeared in the past decades. Among those sources that have been particularly helpful in preparing this book, we would like to acknowledge several works specifically dealing with models of metabolism [12–16], as well as some monographs on nonlinear phenomena and emergent properties of complex systems [17, 18]. This review: In Section II, we highlight the most common approaches to construct computational models of metabolic systems. Subsequently, and prior to giving a more detailed description of the computational methods involved, we briefly consider the rationales of mathematical modeling, thus providing a foundation for all subsequent sections. Though often eclipsed by more imminent pragmatic issues, fundamental questions, such as what constitutes a ‘‘good’’ model, still set the standards and requirements for all further experimental and computational efforts. Continuing with Section III, we then provide a detailed description of the classic methods to construct mechanistic kinetic models of metabolic processes, ranging from properties of the stoichiometric matrix (Section III.B) to enzyme-kinetic mechanisms (Section III.C). The section concludes with a brief summary of resources and databases to construct kinetic models of metabolic processes.
computational models of metabolism
109
Any attempt to describe a natural systems in computational terms is futile in the absence of an experimental accessibility of the state of the system. In Section IV we thus examine recent improvements in analytical technologies for metabolite analysis, summoned mainly under the term metabolomics, which aims at a comprehensive quantitation of all metabolites (the metabolome) within a biological organism. As one of the most versatile and successful computational approaches to metabolism to date, Section V then describes the computational evaluation of the stoichiometric matrix. The main topics within Section V are topological analysis (Section V.A) and the related concepts of elementary flux modes (EFM) and flux balance analysis (FBA) (Section V.B). The limits of flux balance analysis are briefly discussed in Section V.C. Likewise, building on knowledge of the stoichiometry, but again taking the experimental perspective, metabolic flux measurements are described in Section VI. The experimental determination of metabolic fluxes using 13 C-based flux analysis is a pivotal technology to obtain an understanding of metabolic systems, and it differs substantially from concentration measurements. In subsequent sections, we then resume the efforts to obtain a computational kinetic description of metabolic systems. Section VII addresses formal mathematical approaches to elucidate the functioning of metabolic systems, including basic concepts from dynamic systems theory (Section VII.A), Metabolic Control Analysis (Section VII.B), as well as alternative and heuristic approaches to model large metabolic systems (Section VII.C). Continuing with an alternative approach, Section VIII describes a recent computational framework to deal with incomplete and uncertain knowledge of the kinetics of metabolic networks. Instead of constructing a metabolic model at a particular point in parameter space, an ensemble of models is evaluated that is consistent with the available experimental information. Section VIII also serves to outline a general approach to bridge the gap between topology and dynamics of metabolic pathways. In Section IX we focus on some properties of large-scale metabolic networks, specifically their dynamic stability. Following the arguments of R. M. May, we argue that large metabolic networks are prone to instability—with a network of regulatory interaction to ensure the functioning of the metabolic state. The section describes strategies to detect stabilizing sites in metabolic networks (Section IX.A) and discusses the dynamic robustness of metabolic states (Section IX.B). Finally, Section X provides a summary of the results and outlines the path toward large-scale kinetic models of metabolism. II.
CELLULAR METABOLISM AND THE ART OF MODELING
Living cells are open self-sustained systems that continuously exchange energy and matter with their outside world, allowing them to maintain internal order and to synthesize the building blocks that are necessary for survival and growth [1].
110
ralf steuer and bjo¨rn h. junker
Within each cell, matter and energy taken up from the environment undergo a series of transformations, collectively denoted as cellular metabolism. On a first approximation, metabolism may be subdivided into two opposing streams of interconversions: catabolism, the breakdown of energy-rich nutrients and macromolecules into smaller units, producing precursor metabolites and activated carrier molecules (ATP, NADH, NADPH) that serve as energy currency or electron donors for other cellular processes; and anabolism, the production of new macromolecules and cell components through processes that require energy and reducing power obtained from catabolism. The overall organization is frequently described as a bow-tie architecture [11, 19] with a relatively small number of common metabolic intermediates acting as a hinge between the ramified pathways of catabolism and anabolism. See Fig. 1 for a schematic overview. Metabolism is a highly dynamic process. Almost all organisms are exposed to a constantly changing environment and often exhibit sophisticatedly evolved mechanisms to react to changes and to adapt to environmental and intracellular conditions. All living cells regulate and control their metabolic activities by an intricate network of regulatory feedback mechanisms, usually involving multiple
Figure 1. The organization cellular metabolism. Left panel: Cellular metabolism is tightly embedded into the hierarchies of cellular organization. Right panel: The metabolic network is characterized by two opposing streams of reactions, connected by a relatively small number of common metabolic precursors and intermediates (a bow-tie architecture [11, 24]). Three stages may be distinguished [1, 24]: first, the breakdown of large macromolecules to simple units (catabolism), consisting of linear, convergent, and mostly organism-specific pathways; second, central metabolism, characterized by redundant and interlooped pathways that are ubiquitous and conserved across many species; finally, macromolecule biosynthesis (anabolism), characterized again by divergent linear pathways [1, 24].
computational models of metabolism
111
levels of cellular organization: On the metabolic level, the activity of enzymes can be actively influenced by allosteric effectors or covalent modification. On the transcriptional and translational level, the amount of enzymes present in a certain condition is determined by the activity of transcription factors, as well as by other proteins that influence the synthesis and decay of enzymes. The extent to which each of these different mechanisms contributes to overall metabolic regulation in any given situation is subject to active debate [20, 21]. Noteworthy, and discussed in more detail in Section IX, even within seemingly constant conditions, the dynamic properties of the diverse regulatory interactions and feedback mechanisms play a crucial role to ensure stable intracellular conditions and to prevent the depletion of metabolic intermediates [22, 23]. To elucidate the strategies that the evolution of cellular metabolism has developed to maintain the function and stability of metabolism is one of the constitutional questions of Systems Biology, and it will be repeatedly addressed in this chapter. In particular, properties like metabolic homeostasis, along with the closely related concepts of robustness and stability, are genuine system properties, that is, properties that emerge as a result of interactions between components and are only intelligible in terms of these interactions. This emphasis on the emergent properties of cellular regulation also highlights the twofold provenance of Systems Biology [9]. Besides its origin in molecular biology, Systems Biology is deeply rooted in nonequilibrium thermodynamics and self-organization of complex systems—a field in which mathematical modeling and formal analysis have a considerably longer history than within mainstream molecular biology. In addition to the contribution to discern the organizing principles of life, mathematical modeling and formal analysis may also serve more modest and pragmatic roles. In this study, our interest in the ways and means of good model making is thus also motivated by the various ways in which an understanding of metabolism has direct impact on current applications of molecular biology. The discussion of metabolic models is founded upon the most relevant applications of metabolic modeling: (i) A link from genotype to phenotype: As depicted in Fig. 1, cellular metabolism is tightly embedded into the hierarchies of cellular organization, thus mediating between genotype and phenotype. Knowledge of the metabolic state of a mutant enables us to elucidate the functions of genes that produce no overt phenotype when inactivated [2, 25]. The computational analysis of metabolic networks is thus part of a functional genomic strategy to uncover identifying novel functions through the integration of metabolomic and transcriptomic data [2, 25, 26]. (ii) Metabolism-related diseases and medical applications: Enzyme abnormalities and other disturbances of enzyme function, often having genetic causes, account for a large number of human diseases [27, 28]. In silico models of complex cellular processes aid in defining and understanding abnormal metabolic states, such as the high glucose concentration in blood of diabetes patients [29, 30]. Enzymopathies in the context of the
112
ralf steuer and bjo¨rn h. junker
red blood cell are also briefly discussed in Section IX. (iii) Biotechnology and metabolic engineering: Metabolic engineering, defined as the targeted and purposeful alteration of metabolic pathways in order to produce a set of desired products, is one of the main driving forces behind the efforts to construct largescale metabolic models [31–35]. A number of challenges of outstanding relevance, ranging from global crop supply to the synthesis of biofuels, directly relate to our ability to utilize microbial or plant metabolic pathways in a purposeful way. A.
From Topology to Kinetics: A Hierarchy of Models
Given the inherent complexity of cellular processes, a comprehensive mathematical description of metabolism cannot necessarily—nor should it— be given in terms of a single model. Rather, mathematical representations of cellular metabolism have many facets, ranging from purely topological or stoichiometric descriptions to mechanistic kinetic models of metabolic pathways. The variety of different representations of metabolism, each corresponding to a different level of detail and a different level of available information, is not easily classified into a simple scheme. A possible categorization, though neither comprehensive nor complete, is shown in Fig. 2. 1.
Detailed Kinetic Models
Probably the most straightforward and well-known approach to metabolic modeling is to represent metabolic processes in terms of ordinary differential
Figure 2. Current mathematical representations of metabolism utilize a hierarchy of descriptions, involving different levels of detail and complexity. Current approaches to metabolic modeling exhibit a dichotomy between large and mostly qualitative models versus smaller, but more quantitative models. See text for details. The figure is redrawn from Ref. 23. See color insert.
computational models of metabolism
113
equations (ODEs). Similar to other chemical processes, changes in metabolite concentrations are described by a mass-balance equation that incorporates kinetic details of reaction mechanisms and their associated kinetic parameters. Tracing back to the beginning of the last century, detailed kinetic models have contributed significantly to our understanding of the principles of metabolic regulation [12–16, 36–39]. However, despite their general applicability, the construction of large kinetic models faces a number of substantial difficulties: In contrast to the situation in many chemical systems, kinetic parameters in biological systems are often context specific. For example, the catalytic activity of enzymes may depend on temperature and other conditions in a complicated and nonlinear way. The difficulty to obtain reliable estimates of kinetic parameters is certainly one of the main hindrances to construct kinetic models on a cellular or compartmental scale. Despite recent attempts to construct ‘‘genome-scale’’ kinetic models of cellular metabolism [40–44], explicit kinetic modeling is currently often limited to smaller (sub)systems or individual pathways. In addition, the intelligibility of kinetic models that involve several hundreds of equations may be scrutinized, as is discussed is Section II.B. Nonetheless, the construction of explicit kinetic models allows a detailed and quantitative interrogation of the alleged properties of a metabolic network, making their construction an indispensable tool of Systems Biology. The translation of metabolic networks into ordinary differential equations, including the experimental accessibility of kinetic parameters, is one of the main aspects of this contribution and is described in Section III. 2.
Topological Network Analysis
In the face of the inherent limitations that hamper the construction of large-scale kinetic models, topological and graph-theoretic approaches have attracted considerable interest recently [45–48]. In particular, recent advances in genome sequencing and annotation, and thus the possibility to reconstruct large ‘‘genome-scale’’ metabolic networks for several organisms [49–51], have triggered an extensive interest in the topological characteristics of metabolic networks. Indeed, topological network analysis has a number of considerable advantages as compared to the construction of explicit kinetic models. Topological Network Analysis does not presuppose any knowledge of kinetic parameters, thus allowing an analysis of less well-characterized organisms. It is applicable to extensively large systems, consisting of several thousands of nodes, far beyond the realm of current kinetic models. It allows us to investigate a wide variety of topological properties without undue computational effort, such as the degree distribution [19], average pathlength [52], hierarchies and modularity [53–55], as well as topological robustness [56, 57], thus contributing to better understanding of metabolic network architecture. The topological analysis of metabolic networks is briefly considered in Section V.A.
114
ralf steuer and bjo¨rn h. junker
Nonetheless, an interpretation of metabolic networks entirely in topological terms also gives rise to several profound objections. Most importantly, topological network analysis fails to incorporate the specific distinctive properties of a metabolic systems as a network of biochemical interconversions. Despite the superficial similarities between large classes of biological networks, the structure and function of metabolic systems is fundamentally different from many other networks of cellular interactions [58]. To allow an investigation of the structure and function of metabolic systems, we have to go beyond merely topological arguments. 3.
Stoichiometric Analysis
A considerable improvement over purely graph-based approaches is the analysis of metabolic networks in terms of their stoichiometric matrix. Stoichiometric analysis has a long history in chemical and biochemical sciences [59–62], considerably pre-dating the recent interest in the topology of large-scale cellular networks. In particular, the stoichiometry of a metabolic network is often available, even when detailed information about kinetic parameters or rate equations is lacking. Exploiting the flux balance equation, stoichiometric analysis makes explicit use of the specific structural properties of metabolic networks and allows us to put constraints on the functional capabilities of metabolic networks [61,63–69]. Considering a trade-off between knowledge that is required prior to the analysis and predictive power, stoichiometric network analysis must be regarded as the most successful computational approach to large-scale metabolic networks to date. It is computationally feasible even for large-scale networks, and it is nonetheless far more predictive that a simple graph-based analysis. Stoichiometric analysis has resulted in a vast number of applications [35,67,70–74], including quantitative predictions of metabolic network function [50, 64]. The two most well-known variants of stoichiometric analysis, namely, flux balance analysis and elementary flux modes, constitute the topic of Section V. Despite its predictive power and successful application on a variety of largescale metabolic networks, stoichiometric analysis also encompasses a few inadequacies. In particular, stoichiometric analysis largely relies on the steadystate assumption and is not straightforwardly applicable to analyze complex time-dependent dynamics in metabolic systems. Similarly, stoichiometric analysis does not allow us to account for allosteric regulation, considerably delimiting its capabilities to predict dynamic properties. See also Section V.C for a discussion of the limits of stoichiometric analysis. 4.
Intermediate Approaches
For our purposes, of particular interest are methods that aim to bridge the gap between stoichiometric analysis and explicit kinetic modeling [75]. A variety of
computational models of metabolism
115
results can be obtained that require no information about the quantitative values of kinetic parameters but nonetheless allow to access the possible dynamic behavior of a set of chemical or biochemical reactions. In this respect, an early body of theory was developed by Horn, Jackson, and Feinberg [76–78], providing a methodology for systems governed by mass-action kinetics. Their Chemical Reaction Network Theory aims to connect aspects of network structure to various kinds of unstable dynamics in a systematic way [79, 80]. Analogously, Stoichiometric Network Analysis (SNA), developed by B. L. Clarke, allows us to draw conclusions about possible instabilities in chemical reaction networks, mainly based on knowledge of network topology [79, 80]. In the face of lacking and incomplete enzyme-kinetic data, there is renewed interest in such semiquantitative approaches to the analysis of biochemical reaction networks: methods that do not require extensive kinetic information, but still allow precise predictions on the dynamics of biochemical networks [23,75,84]. In this respect, a commonly employed ansatz is based on replacing the actual (and unknown) rate equations with heuristic counterparts, thus requiring only minimal biological data to make quantitative assertions about network behavior [85–89]. A closely related method, denoted as Structural Kinetic Modeling (SKM), is described in Section VIII. In fact, a large variety of dynamic properties is readily accessible using only a local linear approximations of the system. SKM aims to provide a parametric linear representation of a metabolic network, such that each parameter has a well-defined and straightforward interpretation in biochemical terms. Instead of focusing on a particular set of differential equations, this parametric representation allows us to evaluate large ensembles of possible models, each restricted to comply with the available biochemical knowledge. In this way, it is possible to evaluate the stability with respect to perturbations, the existence of bifurcations, and oscillatory regions as well as several other characteristic dynamic features of the system. The analysis given in Section VIII seeks to provide a general example of how to elucidate the transition from the structure to the dynamics of metabolic pathways. B.
The Rationale of Mathematical Modeling
Prior to immersing into more specific computational or experimental details, we need to define our rationale of metabolic modeling. Undoubtedly, all computerbased simulations of metabolic pathways involve a certain level of mathematical abstraction that should be guided by biochemical knowledge, by experimental accessibility of system parameters and variables, and, importantly, by the questions we are seeking to address [33,90,91]. Nonetheless, and despite the fundamental role of mathematical modeling in what is now termed Systems Biology, there is a surprising level of dissent on what the features and the scope of a ‘‘good’’ model should be. Rather, the need for mathematical models is often taken to be self-evident, requiring no further justification.
116
ralf steuer and bjo¨rn h. junker
Consequently, we have to touch upon at least some operational issues to define our approach to the ways and means of constructing models of metabolism. At the most basic level, surveying the current literature, we face a strong dichotomy between a quest for elaborate large-scale models of cellular pathways and minimal (skeleton) models, tailored to explain specific dynamic phenomena only. Certainly a naive pursuit of any of these two options must fail: While ‘‘whole cell models,’’ as recently advocated in the literature [7,40–42,92,93], are certainly a desirable goal, the inherent limitations of a computer-based replicate reality are obvious. Apart from the inevitable computational complexity of such models in combination with a notorious lack of reliable quantitative information about the kinetic parameters, the possible benefits of such detailed large-scale models can be scrutinized. In the past decades, mathematical modeling often deliberately sacrificed aspects of biological realism for the sake of interpretability. Thus, at the very least, the quest for genome-scale dynamic models of metabolism also entails a shift in paradigm toward comprehensive, and possibly predictive, but not necessarily intelligible, in silico models [23]. On the other hand, minimal models have been a trademark of theoretical biology during the past decades. While minimal models are extensively utilized to explain, for example, oscillatory phenomena in yeast glycolysis or the Calvin cycle [94–96], these models often face difficulties when required to go beyond the specific phenomena they are constructed to explain. The apparent dichotomy between the large-scale and minimal models reflects a more profound paradox of mathematical modeling, encapsulated in the two basic requirements for a good model: On one hand, a good model must go beyond the data. In particular, the purpose of a good model is not exhausted by calculating numbers that conform reasonably to experimental data—which, using the words of J. E. Bailey, ‘‘is in itself, not a distinguished endeavor; it is not particularly difficult, and it teaches little’’ [33]. On the other hand, no model may escape its conceptual provenance: Extrapolation of model properties has conceptual limits. Again following the words of Bailey, ‘‘modeling is relatively meaningless without explicit definition, at the outset, of its purpose’’ [33]. In particular, as argued by Wiechert and Takors, ‘‘the assumption that there is a ‘‘true’’ model for a complex biological system is a misconception’’ [97]. Rather, each model has (i) a scope for which it is postulated to be valid, such as a certain temperature range, certain physiological conditions, or time ranges; (ii) an experimental frame, dictated by the experimental accessibility and precision of model observables; and (iii) is accompanied by a measure of precision, specifying the extent to which the natural system has to be reproduced. That is, whether only qualitative features, such as bistability or oscillations, or quantitative features should to be accounted for [97].
computational models of metabolism
117
As a guideline for the present work, we mainly adopt the stance expressed by J. L. Casti [90]: A model is a mathematical representation of the modelers’s reality, a way of capturing some aspects of a particular reality within the framework of a mathematical apparatus that provides us with a means for exploring the properties of the reality mirrored in the model.
Our assumption is that the working of the living cell may be simulated on the basis of known physicochemical laws. Consequently, the translation (or encoding) of a metabolic systems into mathematical terms allows a formal interrogation of the systems behavior—with the inferred properties becoming predictions about the natural system. Figure 3 depicts the relationship between the real world and mathematical models. Mathematical modeling can be understood as a systematic and ordered way to describe our current knowledge of metabolic processes [91, 98, 99]. In this respect, two important roles of mathematical modeling in current biology should be highlighted: First, a model serves to formalize and communicate beliefs and assumptions about cellular pathways and interactions. Already the simplest model of a possible reaction mechanism, such as the basic Michaelis–Menten scheme discussed subsequently, greatly simplifies the exchange and interpretation of information. Rather than having to communicate an entire set of measurements and graphs, the (alleged) kinetic properties of an enzyme may be summarized and communicated in the form of a Michaelis constant. Second, a model serves to organize
Figure 3. The modeling relation, as adapted from J. L. Casti [90]: The encoding operation provides the link between a natural system (real world) and its formal representation (mathematical world). A set of rules and computational methods allows to infer properties (theorems) of the formal system. Using a decoding relation, we can interpret those theorems in terms of the behavior of the natural system. In this sense, the inferred properties of the formal system become predictions about the natural system, allowing us to verify the consistency of the encoding. The modeling process needs to provide the appropriate encoding/decoding relations that translate back and forth between thereal world and the mathematical world.
118
ralf steuer and bjo¨rn h. junker
parts into a coherent whole [33]. Usually facts mean little in isolation—only when interpreted within the wider framework of a model or a theory, discrepancies and inconsistencies can be identified. In this respect, mathematical modeling provides a systematic framework for the formulation of theories on the functioning of cellular processes. We note that our approach to modeling is necessarily interdisciplinary: In the following, our focus are on the encoding and decoding relations depicted in Fig. 3, rather than on a minute account of the (important) peculiarities and methodologies that apply in either the mathematical or the real world. Finally, we need to emphasize a conceptual difference between models of cellular processes and other models of complex self-organized systems—a difference that might rationalize some of the doubts raised with respect to mathematical modeling as a credible research tool in biological science [33]. Across many scientific disciplines, mathematical reasoning has proven to be exceptionally successful as a tool to understand the properties and behavior of complex systems. However, the complexity of most of these natural and technological processes arises from the interaction of a large number of rather simple units that obey a set of common and rather simple rules. One of the hallmarks of complex systems theory is to explain how apparently complex behavior, such as pattern formation or other instances of self-organization, emerges from simple interactions. In contrast, almost all cellular networks are highly heterogeneous systems. Enzymes, for example, are individual entities, shaped by evolution for a specific purpose in a specific intracellular environment. Sometimes referred to as ‘‘dual causality’’ [50], the functional properties of a given enzyme are determined not only by physicochemical principles, but also—and crucially—by an (ongoing) evolutionary history that results in a functional individuality of cellular components. Consequently, modeling of cellular processes has to deal with considerable heterogeneity in the functions, the kinetic parameters, and the interactions of proteins and enzymes, making a computational representation difficult. A similar divergence to classic analysis of complex systems can be recognized with respect to the properties that we seek to explain. Neglecting some sophistication, complex systems theory is often concerned with how complexity, usually identified with nontrivial temporal or spatial behavior, arises from simplicity. In contrast, the questions that are addressed by modeling of cellular processes are often the opposite: Rather than seeking to explain complex phenomena in terms of simple interactions, we often seek to explain how seemingly simple behavior arises from the interactions of a large set of rather complicated heterogeneous units. For example, the optimal function of many cellular regulatory systems is, at least for most of the time, to prevent any ‘‘complex’’ time-dependent changes. In this sense, and as will be discussed in Section IX, the simple dynamics exhibited by most metabolic
computational models of metabolism
119
networks—often just stable steady states—are usually a greater challenge to explain than would be the occurrence of complex or chaotic dynamics. In other words, the complexity of networks of regulatory interactions manifests itself in the absence of complicated dynamics—a situation very unlike conventional chemical modeling. These differences probably contribute to the fact that mathematical modeling is, as yet, not seen as a mainstream research tool in many areas of molecular biology. However, as will be described in the remainder of this chapter, many obstacles in the construction of kinetic models of cellular metabolism can be addressed using a combination of novel and established experimental and computational techniques, enabling the construction of metabolic models of increasing complexity and size. III.
THE BASIC CONCEPTS OF METABOLIC MODELING
As outlined in the previous section, there is a hierarchy of possible representations of metabolism and no unique definition what constitutes a ‘‘true’’ model of metabolism exists. Nonetheless, mathematical modeling of metabolism is usually closely associated with changes in compound concentrations that are described in terms of rates of biochemical reactions. In this section, we outline the nomenclature and the essential steps in constructing explicit kinetic models of metabolic networks. Modeling biochemical reactions by enzyme-kinetic rate equations, organized into a system of ordinary differential equations, has a long history. As detailed in Ref. [36], the first numerical simulation of a biochemical system was published by B. Chance in 1943, solving the equations for the behavior of a simple enzymatic system using a mechanical differential analyzer [100]. Not long thereafter, the construction of explicit kinetic models was pioneered in the 1950s and 1960s by people like D. Garfinkel, L. Garfinkel, B. Chance, and J. Higgins, see Ref. [36] for an early review. These early models still set the blueprint for most current modeling efforts, aiming at the construction of models of increasing size and complexity. This section mainly builds upon classic biochemistry to define the essential building blocks of metabolic networks and to describe their interactions in terms of enzyme-kinetic rate equations. Following the rationale described in the previous section, the construction of a model is the organization of the individual rate equations into a coherent whole: the dynamic system that describes the timedependent behavior of each metabolite. We proceed according to the scheme suggested by Wiechert and Takors [97], namely, (i) to define the elementary units of the system (Section III.A); (ii) to characterize the connectivity and interactions between the units, as given by the stoichiometry and regulatory interactions (Sections III.B and III.C); and (iii) to express each interaction quantitatively by
120
ralf steuer and bjo¨rn h. junker
Figure 4. Following the scheme described by Wiechert and Takors [97], a mathematical model of metabolism is easily constructed. However, in practice, a number of obstacles hamper the construction of large-scale kinetic models.
specifying the biochemical rate equations and associated parameter values (Section III.C). A schematic overview is given in Fig. 4. A.
Computational Models of Metabolism
We seek to describe the time-dependent behavior of a metabolic network that consists of m metabolic reactants (metabolites) interacting via a set of r biochemical reactions or interconversions. Each metabolite Si is characterized by its concentration Si ðtÞ " 0, usually measured in moles/volume. We distinguish between internal metabolites, whose concentrations are affected by interconversions and may change as a function of time, and external metabolites, whose concentrations are assumed to be constant. The latter are usually omitted from the m-dimensional time-dependent vector of concentrations SðtÞ and are treated as additional parameters. If multiple compartments are considered, metabolites that occur in more than one compartments are assigned to different subscripts within each compartment. The concentrations of metabolites are affected by enzyme-catalyzed reactions, by transport between compartments, and by import and export processes, among other possible processes, such as dilution by cell growth. Each such interconversion or reaction is characterized by two quantities: (i) the stoichiometric
computational models of metabolism
121
coefficients Nij that specify the amounts (up to an arbitrary scaling) in which the participating metabolites are produced or consumed in the reaction; and (ii) a (often nonlinear) function n(S, k) that specifies the rate of the reaction as a function of the concentration vector S and a set of kinetic parameters k. As an example, consider the first step in glycolysis D-Glucose
þ ATP ! D-Glucose-6-phosphate þ ADP
ð1Þ
catalyzed by the enzyme hexokinase (HK, EC 2.7.1.1). The respective stoichiometric coefficients for the reactants glucose (Glc), ATP, glucose-6phosphate (G6P), and ADP are Glc ATP %1 % 1
G6P þ1
ADP þ1
ð2Þ
For biochemical reactions, the stoichiometric coefficients are usually integer and correspond to the (relative) molecularities in which the reactants enter the reaction. Note that for transport processes, and unless metabolites are measured in absolute quantities, the stoichiometric coefficient also reflect differences in compartmental volumes. The rate function nHK of the hexokinase reaction can, for example, be described by an irreversible random-order bireactant mechanism [101], ½Glc( ½ATP( Km;Glc Km;ATP nHK ðS; kÞ ¼ ½Glc( ½ATP( ½Glc( ½ATP( 1þ þ þ Km;Glc Km;ATP Km;Glc Km;ATP Vm
ð3Þ
specified by three kinetic parameters Vmax , Km;Glc , and Km; ATP . Once the stoichiometric coefficients Nij and rate functions nj for all reactions and transport processes are assembled, the time-dependent concentration of a metabolic reactant Si ðtÞ is described by the dynamic mass balance equation r dSi ðtÞ X ¼ Nij nj ðS; kÞ dt j¼1
ð4Þ
or, equivalently, in matrix notation dSðtÞ ¼ NmðS; kÞ dt
ð5Þ
where S denotes m-dimensional time-dependent vector of concentrations, N the m ) r-dimensional stoichiometric matrix, and mðS; kÞ the r-dimensional vector of rate equations. Note that the vector of rate equations may also contain terms
122
ralf steuer and bjo¨rn h. junker
Figure 5. A minimal model of glycolysis: One unit of glucose (G) is converted into two units of pyruvate (P), generating a net yield of 2 units of ATP for each unit of glucose. Gx , Px , and Glx are considered external and are not included into the stoichiometric matrix. A: A graphical depiction of the network. B: The stoichiometric matrix. Rows correspond to metabolites, columns correspond to reactions. C: A list of individual reactions. D: The corresponding system of differential equations. Abbreviations: G, glucose (Glc); TP, triosephosphate, P, pyruvate.
for dilution (growth) and biomass formation. An example of a simple metabolic network is shown in Fig. 5. Given the functional form of the rate equations, the values of the kinetic parameters, and an initial condition Sð0Þ, the timedependent behavior of the metabolic system is fully specified. Equation (5) describes the variation of a metabolite concentration over time as proportional to the rates in which a metabolite is synthesized minus the rates at which it is consumed. A stationary and time-invariant state of metabolite concentrations S0 (steady state) is characterized by the steady-state condition dSðtÞ ¼0 dt
)
NmðS0 ; kÞ ¼ 0:
ð6Þ
Note that Eq. (6) includes thermodynamic equilibrium (m0 ¼ 0) as a special case. However, usually the steady-state condition refers to a stationary nonequilibrium state, with nonzero net flux and positive entropy production. We emphasize the distinction between network stoichiometry and reaction kinetics that is implicit in Eqs. (5) and (6). While kinetic rate functions and the associated parameter values are often not accessible, the stoichiometric matrix is usually (and excluding evolutionary time scales) an invariant property of metabolic reaction networks, that is, its entries are independent of temperature, pH values, and other physiological conditions.
computational models of metabolism
123
Aiming to construct explicit dynamic models, Eqs. (5) and (6) provide the basic relationships of all metabolic modeling. All current efforts to construct large-scale kinetic models are based on an specification of the elements of Eq. (5), usually involving several rounds of iterative refinement. For a schematic workflow, see again Fig. 4. In the following sections, we provide a brief summary of the properties of the stoichiometric matrix (Section III.B) and discuss the most common functional form of enzyme-kinetic rate equations (Section III.C). A selection of explicit kinetic models is provided in Table I. TABLE I Selected Examples of Explicit Kinetic Models of Metabolisma Pathway b
Horseradish peroxidase Glycolysis and respirationc Glycolysis Carbohydrate metabolism Glycolysis (energy) Glutamate metabolism Glycolysis (erythrocytes) Energy metabolism Leaf carbon metabolism Erythrocytes Calvin cycle Calvin cycled Calvin cycle Erythrocytes Glycolysis and fermentation TCA cycle Carbohydrate metabolisme TCA cycle Erythrocytes Central metabolism Yeast glycolysis Pentose phosphate pathway Erythrocytes Calvin cycle Glycolysis Sucrose accumulation Threonine pathway Glycolysis Central carbon metabolism Glycolysis (L. lactis) Urea cycle Mitochondrial metabolism Sucrose breakdown Yeast glycolysisf Sucrose-to-starch pathway
Author and References
Year
Chance [100] Chance et al. [104] Garfinkel and Hess [105] Wright et al. [106] Sel’kov [94] van den Berg and Garfinkel [107] Rapoport et al. [108] Achs et al. [109] Hahn [110] Holzhu¨tter et al. [111] Hahn [112] Giersch [95] Petterson and Ryde-Petterson [113] Joshi and Palsson [114] Galazzo and Bailey [115] Wright et al. [116] Wright and Albe [117] El-Mansi et al. [118] Schuster and Holzhu¨tter [119] Rizzi et al. [120] Teusink et al. [121] Vaseghi et al. [122] Mulquiney and Kuchel [38,123] Poolman et al. [124,125] Wolf et al. [126] Rohwer et al. [127] Chassagnole et al. [128] Hynne et al. [101] Chassagnole et al. [129] Hoefnagel et al. [130] Maher et al. [131] Yugi and Tomita [43] Junker [132] Klipp et al. [133] Assmus [134]
1943 1960 1964 1968 1968 1971 1976 1977 1984 1985 1986 1986 1988 1989 1990 1992 1994 1994 1995 1997 1999 1999 1999 2000 2000 2001 2001 2001 2002 2002 2003 2004 2004 2005 2005
(continued)
124
ralf steuer and bjo¨rn h. junker TABLE I (Continued)
Pathway Leaf carbon metabolism TCA cycle Hepatocyte metabolism
Author and References
Year
Zhu et al. [135] Wu et al. [136] Maria et al. [137]
2007 2007 2008
a The list is not comprehensive and focuses one explicit kinetic models of metabolic pathways— minimal models, stoichiometric descriptions, and signaling are not considered. For additional examples, see also the model repositories listed in Table IV. bAccording to Ref. 36, the first simulation of a biochemical system, using a mechanical differential analyzer. cThe earliest computer-based simulation of glycolysis, involving 22 reactions. dA minimal generic model of oscillatory transients in photosynthesis. eNote that several early models are based on simplified mass-action kinetics. fAn integrative model of yeast osmotic shock, also including signaling pathways.
We emphasize that any utilization of Eq. (5) already rest upon a number of (often reasonable) assumptions. Equation (5) represents an ordinary deterministic differential equation, based on assumption of homogeneity, free diffusion, and random collision, and neglecting spatial [102] or stochastic effects [103]. While such assumptions are often vindicated for microorganisms, the application of Eq. (5) to other cell types, such as human or plant cells, sometimes mandates careful verification. B.
The Properties of the Stoichiometric Matrix
The stoichiometric matrix N is one of the most important predictors of network function [50,61,63,64,68] and encodes the connectivity and interactions between the metabolites. The stoichiometric matrix plays a fundamental role in the genome-scale analysis of metabolic networks, briefly described in Section V. Here we summarize some formal properties of N only. The stoichiometric matrix N consists of m rows, corresponding to m metabolic reactants, and r columns, corresponding to r biochemical reactions or transport processes (see Fig. 5 for an example). Within a metabolic network, the number of reactions (columns) is usually of the same order of magnitude as the number of metabolites (rows), typically with slightly more reactions than metabolites [138]. Due to conservation relationships, giving rise to linearly dependent rows in N, the stoichiometric matrix is usually not of full rank, but rankðNÞ + m + r
ð7Þ
The stoichiometric matrix is characterized by its four fundamental subspaces, two of which are described in more detail below. An examples of each subspace is given in Section III.B.3.
computational models of metabolism 1.
125
The Left Nullspace
The left nullspace E of the stoichiometric matrix N is defined by a set of linearly independent vectors ej that are arranged into a matrix E that fulfills [50, 96] EN ¼ 0
ð8Þ
Note that the E is not unique, each nonsingular linear transformation is again a valid representation of the left nullspace. The matrix E consists of m % rankðNÞ rows, corresponding to mass-conservation relationships (and a linearly dependent rows) in N. In particular, E
dS d ¼ ES ¼ 0 dt dt
)
ES ¼ const:
ð9Þ
To explicitly account for mass conservation, we distinguish between rankðNÞ independent and m % rankðNÞ dependent concentrations, Sind and Sdep , respectively. Choosing E ¼ ½%L0 1(, and rearranging N and S accordingly, we obtain dSdep dSind ¼ L0 dt dt
with
!
Sind S¼ Sdep
"
ð10Þ
Denoting with N 0 the matrix that consists only of the first rankðNÞ linearly independent rows of N (corresponding to the independent species Sind ), the full set of differential equations is given as ! " ! " dS d Sind 1 0 ¼ ¼ 0 N mðS; pÞ S L dt dt dep
ð11Þ
Since Sdep ¼ L0 Sind þ const:, it is sufficient to consider the time evolution of the independent species Sind dSind ¼ N 0 mðS; pÞ dt
ð12Þ
Note that the reduced stoichiometry N 0 is related to the full stoichiometry N by a transformation using the link matrix L ! " 1 ð13Þ N ¼ LN 0 with L :¼ L0 In Sections VII.A and VII.B, we make use of the fact that the link matrix L allows to account for linear dependencies within the partial derivatives. In particular, the dependence of a (vector) function f ðSÞ ¼ f ðSind ; Sdep Þ is expressed
126
ralf steuer and bjo¨rn h. junker
entirely in terms of the independent variables. For the partial derivative, using Sdep ¼ L0 Sind þ const:, we obtain df ðSÞ qf ðSÞ qf ðSÞ dSdep qf L ¼ þ ¼ dSind qSind qSdep dSind qS 2.
ð14Þ
The Right Nullspace
The right nullspace or kernel of N is defined by r % rankðNÞ linearly independent columns ki , arranged into a matrix K that fulfills NK ¼ 0
ð15Þ
Again, the matrix K is not unique and only defined up to a (nonsingular) transformation. An analysis of the right nullspace K provides the conceptual basis of flux balance analysis and has led to a plethora of highly successful applications in metabolic network analysis. In particular, all steady-state flux vectors m 0 ¼ mðS0 ; pÞ can be written as a linear combination of columns ki of K, such that m0 ¼
r%rankðNÞ X
ki ai ¼ Ka
ð16Þ
i¼1
with ai 2 R arranged into a column vector a. Note that nontrivial solutions of Eq. (15) only exist if the steady-state equation is under determined, that is, rankðNÞ < r. As for most metabolic networks, the number of fluxes (columns of N) is larger than the number of (independent) metabolites (rows of N), Eq. (15) puts constraints on the feasible steady-state flux distributions. A thorough analysis of calculability of flux vectors in underdetermined metabolic networks is given in Ref. [138]. 3.
An Example
As a simple example, consider the minimal glycolytic pathway shown in Fig. 5. The stoichiometric matrix N has m ¼ 5 rows (metabolites) and m ¼ 6 columns (reactions and transport processes). The rank of the matrix is rankðNÞ ¼ 4, corresponding to m % rankðNÞ ¼ 1 linearly dependent row in N. The left nullspace E can be written as E ¼ ½0
0 0
1
1(
)
L0 ¼ ½0
0
0
% 1(
ð17Þ
computational models of metabolism
127
reflecting the mass conservation relationship ES ¼ const:, thus dADP dATP þ ¼0 dt dt
and
ATP þ ADP ¼ const:
ð18Þ
Choosing the concentration of ADP as the dependent variable, the network is described by a set of four linearly independent differential equations. The link matrix L is 3 2 1 0 0 0 ! " 60 1 0 0 7 6 7 1 7 L¼ ð19Þ ¼6 60 0 1 0 7 L0 40 0 0 1 5 0 0 0 %1
with N ¼ LN 0 . The right nullspace K is spanned by r % rankðNÞ ¼ 2 column vectors, 2
1 61 6 62 K¼6 62 6 40 2
3 1 17 7 17 7 with 17 7 15 0
2 3 2 3 1 1 617 617 6 7 6 7 627 617 0 7 7 6 m ¼ 6 7a1 þ 6 6 1 7a2 2 6 7 6 7 405 415 2 0
ð20Þ
It can be straightforwardly verified that indeed NK ¼ 0. Each feasible steadystate flux m0 can thus be decomposed into the contributions of two linearly independent column vectors, corresponding to either net ATP production (k1 ) or a branching flux at the level of triosephosphates (k2 ). See Fig. 5 for a comparison. An additional analysis of the nullspace in the context of large-scale reaction networks is given in Section V. C.
Enzymes and Chemical Reaction Rates
The next step in formulating a kinetic model is to express the stoichiometric and regulatory interactions in quantitative terms. The dynamics of metabolic networks are predominated by the activity of enzymes—proteins that have evolved to catalyze specific biochemical transformations. The activity and specificity of all enzymes determine the specific paths in which metabolites are broken down and utilized within a cell or compartment. Note that enzymes do not affect the position of equilibrium between substrates and products, rather they operate by lowering the activation energy that would otherwise prevent the reaction to proceed at a reasonable rate. A detailed kinetic description of enzyme-catalyzed reactions is paramount to kinetic modeling of metabolic networks—and one of the most challenging steps
128
ralf steuer and bjo¨rn h. junker
in the construction of large-scale models of metabolism. Elaborate descriptions of the fundamentals of enzyme kinetics are found in a variety of monographs, most notably the book of E. Segel [139], among many other works on the subject [96,140]. In the following, we summarize some essential properties of typical rate equations. Enzyme-catalyzed reactions can be described at least at two distinct levels. At the basic level, the interconversion of substrates by enzymes is governed by a set of elementary steps, including enzyme–substrate binding, isomerization and dissociation steps, see Fig. 6 for a schematic depiction. Assuming the intracellular medium is an ideal solution, each elementary step is governed by mass-action kinetics, that is, the reaction rates are proportional to the probability of collision of the reactants. For a reaction of the type aA þ bB $ sS þ pP
ð21Þ
the forward and reaction rates are given as nþ ¼ kþ ½A(a ½B(b
and
n% ¼ k% ½S(s ½P(p
ð22Þ
respectively, where k/ are the elementary rate constants and the exponents correspond to the molecularities (stoichiometric coefficients) in which the reactants enter the reaction. In principle, once the stoichiometry and rate constants of all elementary steps are specified, the dynamic behavior of the entire metabolic network can be evaluated using the dynamic mass-balance Eq. (5). However, such an approach is only rarely employed in practice. The numerical simulation of enzymatic
Figure 6. Enzymes act as recycling catalysts in biochemical reactions. A substrate molecule binds (reversible) to the active site of an enzyme, forming an enzyme–substrate complex. Upon binding, a series of conformational changes is induced that strengthens the binding (corresponding to the induced-fit model of Koshland [148]) and leads to the formation of an enzyme–product complex. To complete the cycle, the product is released, allowing the enzyme to bind further substrate molecules. (Adapted from Ref. 1). See color insert.
computational models of metabolism
129
systems is considerably simplified by replacing the elementary steps with overall enzymatic reactions. After a brief digression to the elementary concepts of thermodynamics, the most common forms of enzyme-kinetic rate laws are discussed in Sections III.C.2–III.C.6. It should be emphasized that Eq. (22) is already based on a number of preconditions. In particular, the intracellular medium may significantly deviate from a ‘‘well-stirred’’ ideal solution [141–143]. While the use of Eq. (22) is often justified, several authors have suggested to allow noninteger exponents in the expression of elementary rate equations [96,142,144]—corresponding to a more general form of mass-action kinetics. A related concept, the power-law formalism, developed by M. Savageau and others [145–147], is addressed in Section VII.C. 1.
Chemical Equilibrium and Thermodynamics
In chemical equilibrium, the forward and reverse reaction rates are equal and there is no net production of intermediates. The equilibrium constant Keq is given as the ratio of reactants in equilibrium. For the elementary reaction shown in Eq. (21), we obtain Keq ¼
½S(seq ½P(peq ½A(aeq ½B(beq
¼
kþ k%
ð23Þ
In general, for any chemical reaction to proceed, it must be energetically favorable, as specified by the associated change in Gibbs free energy !G. The Gibbs free energy !G is a function of the displacement of the reaction from equilibrium: Processes with !G > 0 are endergonic and may not proceed spontaneously, whereas processes with !G < 0 are exergonic, that is, they are energy releasing and may proceed spontaneously. The change in Gibbs free energy of a reaction is specified by two addends: The first term is given by the standard free energy !G0, corresponding to the intrinsic characters of the involved molecules, measured under standard conditions. The second term relates to the concentrations of the involved molecules at the time of reaction, !G ¼ !G0 þ RT ln
½S(s ½P(p ½A(a ½B(b
ð24Þ
with R ¼ 8:314472 JK%1 mol%1 and T denoting the universal gas constant and absolute temperature, respectively. The first addend !G0 is closely related
130
ralf steuer and bjo¨rn h. junker
to thermodynamic equilibrium constant Keq . With !G ¼ 0 in equilibrium and Eq. (23), we obtain ) * !G0 !G0 ¼ %RT ln Keq and Keq ¼ exp % ð25Þ RT Note that care must be taken to obtain the value for !G0, as the definition of standard conditions sometimes differ. The relationship of !G to the displacement from thermodynamic equilibrium becomes obvious if Eq. (24) is rewritten in terms of the massaction ratio " of the participating molecules " :¼
½S(s ½P(p a
½A( ½B(
b
)
!G ¼ RT ln
" Keq
ð26Þ
As long as the mass-action ratio of the involved molecules is below its equilibrium value, the reaction proceeds towards chemical equilibrium. It should be emphasized that the value of !G entails no predication about the time scale in which chemical equilibrium is attained. In the context of large-scale models of metabolic networks, thermodynamic considerations have attracted renewed interest recently. In particular, changes in free energy are entirely dictated by chemical properties and concentrations of metabolites, and they do not hinge on specific knowledge of enzyme-kinetic mechanisms. Likewise, thermodynamic constraints are not organism-specific and often accessible even for parts of the metabolic map for which detailed kinetic information is lacking. Incorporating thermodynamic properties provides a link between concentrations and flux and allows the inclusion of thermodynamic realizability as an additional constraint in the large-scale analysis of metabolic networks [74,149–151]. 2.
Michaelis–Menten Kinetics
A kinetic description of large reaction networks entirely in terms of elementary reactionsteps is often not suitable in practice. Rather, enzyme-catalyzed reactions are described by simplified overall reactions, invoking several reasonable approximations. Consider an enzyme-catalyzed reaction with a single substrate: The substrate S binds reversibly to the enzyme E, thereby forming an enzyme–substrate complex ½ES(. Subsequently, the product P is irreversibly dissociated from the enzyme. The resulting scheme, named after L. Michaelis and M. L. Menten [152], can be depicted as kþ1
k2 E þ S %k! % ½ES( %! P þ E %1
computational models of metabolism
131
Using mass-action kinetics for the elementary steps, the rates of change for the substrate and enzyme–substrate complex concentrations are d½S( ¼ %kþ1 ½E(½S( þ k%1 ½ES( dt d½ES( ¼ þkþ1 ½E(½S( % k%1 ½ES( % k2 ½ES( dt
ð27Þ ð28Þ
To derive the well-known overall rate equation for the process, two different simplifying assumptions may be invoked: (i) Rapid Equilibrium. The reversible binding of the substrate is considered to be much faster than the slow irreversible dissociation of the product. In this case, the dissociation reaction only has a minor effect on the equilibrium of the enzyme–substrate complex. With k2 1 k%1 , the concentration of the complex is approximated by Kd ¼
k%1 ½E(½S( ¼ ½ES( kþ1
thus
½ES( ¼
½E(½S( Kd
ð29Þ
with Kd denoting the equilibrium constant. (ii) Quasi-Steady-State Assumption (QSSA). For a large number of enzymes, the assumption k2 1 k%1 is not valid and is also not needed to derive an algebraic form of the overall rate equation. Briggs and Haldane [153] provided an alternative derivation, based on the steady-state approximation. We assume that the enzyme–substrate complex achieves a steady state shortly after the beginning of reaction and rapidly adjusts to a slowly varying substrate concentration. We obtain d½ES( ¼0 dt
)
½ES( ¼
kþ1 ½E(½S( ½E(½S( ¼ KM k%1 þ k2
ð30Þ
with KM :¼
k%1 þ k2 kþ1
ð31Þ
denoting the Michaelis–Menten constant. The steady-state assumption has become a widely accepted dogma underlying the derivation of most rate equations. Nonetheless, note that the functional form of the derived concentration of the complex is identical under both assumptions, although with a different numerical value for the proportionality constant. In the limit k2 1 k%1 , the relationship KM 2 Kd holds. To obtain an expression for the Michaelis–Menten rate equation, the dissociation of the product from the complex needs to be evaluated. Using a
132
ralf steuer and bjo¨rn h. junker
convenient trick, the reaction rate is multiplied with unity to introduce the total enzyme concentration ET ¼ ½E( þ ½ES( 1¼
ET ET ¼ ET ½ES( þ ½E(
ð32Þ
Considering the rate equation n2 ¼ k2 ½ES( and making use of the steady-state approximation for ½ES(, we obtain n2 ¼ k2 ½ES( ¼ k2 ET
½ES( k2 ET ½S( ¼ ½E( þ ½ES( KM þ ½S(
ð33Þ
To recognize that Eq. (33) indeed is the overall reaction of the Michaelis–Menten scheme, an additional requirement is that the concentration of enzyme-bound substrate is negligible compared to the total substrate concentration. The corresponding differential equations of the irreversible Michaelis–Menten scheme can then be simplified to dð½S( þ ½ES(Þ d½S( 2 ¼ %n2 ðSÞ ddt ddt
and
d½P( ¼ þn2 ðSÞ ddt
ð34Þ
Using the abbreviation Vm :¼ k2 ET to denote the maximal velocity of reaction, the resulting rate equation nðSÞ ¼
Vm ½S( KM þ ½S(
ð35Þ
constitutes the fundamental equation of enzyme kinetics. First derived by V. Henri [154, 155] and later evaluated and named after L. Michaelis and M. L. Menten [152], the Michaelis–Menten rate equation is depicted in Fig. 7. (B) 1 ν(S) = VM
0.5
0 0
ν(S) = 1/2 VM KM=1.0 2 4 6 8 substrate concentration S [au]
10
Michaelis−Menten rate ν(S)
Michaelis−Menten rate ν(S)
(A)
1 ν(S) = VM
0.5 ν(S) = 1/2 V
M
K =1.0 0 −2 10
M
0
10 substrate concentration S [au]
2
10
Figure 7. The Michaelis–Menten rate equation as a function of substrate concentration S (in arbitrary units). Parameters are KM ¼ 1 and Vm ¼ 1. A: A linear plot. B: A semilogarithmic plot. At a concentration S ¼ KM , the rate attains half its maximal value Vm .
computational models of metabolism
133
Different from conventional chemical kinetics, the rates in biochemical reactions networks are usually saturable hyperbolic functions. For an increasing substrate concentration, the rate increases only up to a maximal rate Vm , determined by the turnover number kcat ¼ k2 and the total amount of enzyme ET . The turnover number kcat measures the number of catalytic events per seconds per enzyme, which can be more than 1000 substrate molecules per second for a large number of enzymes. The constant Km is a measure of the affinity of the enzyme for the substrate, and corresponds to the concentration of S at which the reaction rate equals half the maximal rate. For S 1 KM , most of the enzymes are free, that is, most active sites are not occupied. For S 3 KM , there is an excess of substrate, that is, the active sites of the enzymes are saturated with substrate. The ratio kcat =Km is a measure for the efficiency of an enzyme. In the extreme case, almost every collision between substrate and enzyme leads to product formation (low Km, high kcat ). In this case the enzyme is limited by diffusion only, with an upper limit of kcat =Km 4 108 % 109 M %1 s%1 . The ratio kcat =Km can be used to test the rapid equilibrium assumption. For a recent discussion of the limits of the QSSA see Ref. [156]. 3.
Reversible Michaelis–Menten Kinetics
Though the assumption of an irreversible dissociation of the product considerably simplifies the mathematical analysis, all enzymatic reactions are inherently reversible. To account for the presence of a significant amount of product within the intracellular medium, we must allow the reverse reaction [96,140,157]. In this case, using an augmented scheme, kþ1
kþ2
E þ S %k! % ½ES( %k! % PþE %1 %2 a modified differential equation for the enzyme–substrate complex is obtained d½ES( ¼ þkþ1 ½E(½S( % k%1 ½ES( % kþ2 ½ES( þ k%2 ½E(½P( dt
ð36Þ
Using the quasi-steady-state approximation and the conservation of total enzyme ET ¼ ½E( þ ½ES(, the concentration of the complex is given as ½ES( ¼
kþ1 ET ½S( þ k%2 ET ½P( kþ1 ½S( þ k%1 þ kþ2 þ k%1 ½P(
ð37Þ
Together with the net reaction for product formation n ¼ kþ2 ½ES( % k%2 ½E(½P(, the reversible Michaelis–Menten equation can be derived:
134
ralf steuer and bjo¨rn h. junker
nðS; PÞ ¼
Vmþ K½S(mS % Vm% K½P( mP 1 þ K½S(mS þ K½P( mP
ð38Þ
In Eq. (38), the elementary rate constants are replaced by the abbreviations [96,157] Vmþ :¼ kþ2 ET
and
Vm% :¼ k%1 ET
ð39Þ
for maximal reaction velocities, as well as KmA :¼
k%1 þ kþ2 kþ1
and KmP :¼
k%1 þ kþ2 k%2
ð40Þ
for Michaelis–Menten constants. Note that in the limit k%2 ! 0 ( KmP ! 1), the irreversible equation is recovered. Instead of the particular irreversible scheme shown above, also two intermediate complexes [ES] and [EP] can be considered. Interestingly, in this case, an identical algebraic form for the rate–equation is obtained, although the definition of the rate constants in terms of elementary constants is slightly more complex. 4.
The Equilibrium Constants and Haldane Relationship
For reversible enzymatic reactions, the Haldane relationship relates the equilibrium constant Keq with the kinetic parameters of a reaction. The equilibrium constant Keq for the reversible Michaelis–Menten scheme shown above is given as Keq ¼
½P(eq ½S(eq
¼
kþ1 kþ2 k%1 k%2
ð41Þ
In thermodynamic equilibrium, the overall reaction is characterized by nðSeq ; Peq Þ ¼ 0. Evaluating Eq. (38), the following relationship must hold Keq ¼
½P(eq ½S(eq
¼
KmP Vmþ KmS Vm%
ð42Þ
The importance of the Haldane relationship Eq. (42) relates to the fact that the kinetic parameters of a reversible enzymatic reaction are not independent but are constraint by the equilibrium constant of the overall reaction [157]. Rewriting the rate equation Eq. (38), + , + , Vmþ ½P( Vmþ ½S( % K½P(eq KmS ½S( % Keq + , nðS; PÞ ¼ ð43Þ ¼ 1 þ K½S(mS þ K½P( KmS 1 þ K½P( þ ½S( mP mP
computational models of metabolism
135
emphasizes the inhibitory character of increasing product concentration upon the reaction rate. 5.
Steady-State Kinetics of Multisubstrate Reactions
Similar to irreversible reactions, biochemical interconversions with only one substrate and product are mathematically simple to evaluate; however, the majority of enzymes correspond to bi- or multisubstrate reactions. In this case, the overall rate equations can be derived using similar techniques as described above. However, there is a large variety of ways to bind and dissociate multiple substrates and products from an enzyme, resulting in a combinatorial number of possible rate equations, additionally complicated by a rather diverse notation employed within the literature. We also note that the derivation of explicit overall rate equation for multisubstrate reactions by means of the steady-state approximation is a tedious procedure, involving lengthy (and sometimes unintelligible) expressions in terms of elementary rate constants. See Ref. [139] for a more detailed discussion. Nonetheless, as the functional form of typical rate equations will be of importance for the parameterization of metabolic networks in Section VIII, we briefly touch upon the most common mechanisms. Restricting ourselves to the rapid equilibrium approximation (as opposed to the steady-state approximation) and adopting the notation of Cleland [158–160], the most common enzyme-kinetic mechanisms are shown in Fig. 8. In multisubstrate reactions, the number of participating reactants in either direction is designated by the prefixes Uni, Bi, or Ter. As an example, consider the Random Bi–Bi Mechanism, depicted in Fig. 8a. Following the derivation in Ref. [161], we assume that the overall reaction is described by nrbb ¼ kþ ½EAB( % k% ½EPQ(. Using the conservation of total enzyme ET ¼ ½E( þ ½EA( þ ½EB( þ ½EAB( þ ½EPQ( þ ½EP( þ ½EQ(
ð44Þ
and expressing each complex in terms of its dissociation constant, a similar reasoning as in Eq. (32) can be utilized. We obtain 1¼
ET ET 1 ¼ 6 ET ½E( 1 þ ½A( þ ½B( þ ½A(½B( þ ½P( þ ½Q( þ ½P(½Q( Ka Kb Ka Kb Kq Kq Kp Kq
ð45Þ
The resulting overall rate equation, in analogy to Eq. (38), is nRandomBiBi ¼
½P(½Q( Vmþ ½A(½B( Ka Kb % Vm% Kp Kq ½B( ½A(½B( ½P( ½Q( ½P(½Q( 1 þ ½A( K a þ Kb þ K a Kb þ Kq þ Kq þ K p K q
ð46Þ
with Vm/ :¼ k/ ET . Note that Eq. (46) is based on a number of simplifying assumptions. In addition to the rapid equilibrium approximation, we assume that
136
ralf steuer and bjo¨rn h. junker
Figure 8. The most common enzyme mechanisms, represented by their corresponding Cleland plots: The order in which substrates and products bind and dissociate from the enzyme is indicated by arrows. (a) The Random Bi Bi Mechanism: Both substrates bind in random order. (b) The Ordered Sequential Bi Bi Mechanism: The substrates bind sequentially. (c) The Ping-Pong Mechanism: The enzyme exists in different states E and E7 . A substrate may transfer a chemical group to the enzyme. Only upon release of the first substrate, the chemical group is transferred to the second substrate.
the binding and release of substrates and products does not depend on whether the other substrate or product is already bound, that is, the corresponding dissociation constants are identical. Despite its limitations, the reversible Random Bi-Bi Mechanism Eq. (46) will serve as a proxy for more complex rate equations in the following. In particular, we assume that most rate functions of complex enzyme-kinetic mechanisms can be expressed by a generalized mass-action rate law of the form Q Q Vmþ i ½S(i % Vm% j ½P(j ð47Þ ngeneric ðS; PÞ ¼ FðS; P; kÞ
computational models of metabolism
137
As proposed by Heinrich and Schauer [96], Eq. (47) provides a generic functional form for most common rate equations, with FðS; P; kÞ denoting a polynomial with positive coefficients and S and P the substrates and products of the reactions, respectively. See also Section VII.C.3 for a more detailed discussion. For an explicit derivations of more complicated rate equations, we refer to the extensive literature on the subject (see, for example, Refs. [139,140,157]). In particular, a thorough recent discussion to determine and parameterize a kinetic scheme, also pointing out several shortcomings and possible pitfalls in the interpretation of kinetic constants found in the literature, is given in Ref. [162]. Nonetheless, and despite the importance of explicit kinetic schemes for metabolic modeling, one should heed the warning denoted as Cleland’s Caution in Ref. [163]: ‘‘The traditional method for introducing this subject is to lead the reader through a tortuous maze of algebra to try to convince him that the resulting rate equations have some basis in reality and can be applied to experimental data.’’ Indeed, one may scrutinize the applicability of some of the subtleties found in more complicated rate functions. As vigorously argued by Savageau [142, 144], . . . the postulates of the Michaelis–Menten Formalism and the canons of good enzymological practice in vitro, which serve so well for the elucidation of isolated reaction mechanisms, are not appropriate for characterizing the behavior of integrated biochemical systems.
Given the uncertainties in the detailed functional form of the rate equations, an increasing number of authors opt for using heuristic rate laws that capture the generic dependencies of typical reactions [89,161,164,165]—at least when dealing with large-scale reaction networks. Indeed, as is described in Section VIII, detailed knowledge of the specific functional form is not always necessary. A variety of dynamic properties is entirely specified by knowledge of the (local) derivative of the rate equation, and any rate equation consistent with a given derivative may account for an observed or alleged behavior. Nonetheless, the rate equations given in Eqs. (46) and (47) set the benchmark to which heuristic approaches must be compared. 6.
Inhibition and Allosteric Control of Enzyme Activity
One of the most distinguishing features of metabolic networks is that the flux through a biochemical reaction is controlled and regulated by a number of effectors other than its substrates and products. For example, as already discovered in the mid-1950s, the first enzyme in the pathway of isoleucine biosynthesis (threonine dehydratase) in E. coli is strongly inhibited by its end product, despite isoleucine having little structural resemblance to the substrate or product of the reaction [140,166,167]. Since then, a vast number of related
138
ralf steuer and bjo¨rn h. junker
examples have been described in the literature, and several theoretical models have been proposed to account for the various mechanisms of regulatory enzymes [140,168–170]. Inhibitors may compete with substrates for the binding site of the enzyme (competitive inhibition), thereby blocking or reducing the rate with which the reaction proceeds. Likewise, many enzymes have more than one binding site, and the binding of a molecule to a site other than the active site may alter the catalytic properties of the enzyme, causing a decreased or increased activity of the enzyme (allosteric regulation). Ligand-induced conformational changes may also occur for effectors that are itself substrates of the reaction. In particular, cofactors such as ATP frequently act as allosteric effectors while simultaneously being cosubstrates of the reaction. Related to the induced fit model of Koshland [148], the binding of a substrate molecule may also cause a modification of the enzyme, without an additional allosteric site involved. Allosteric regulation often results in sigmoidal rate equations, as opposed to the hyperbolic Michaelis–Menten kinetics. Unfortunately, and unlike the stoichiometric properties and some conventional enzyme mechanism, the regulatory modifiers of enzymes are often not well characterized. While the kinetic properties of substrates, products, and known effectors can be straightforwardly measured, the combinatorial number of potential modifiers, sometimes acting only in conjunction with other modifiers, make even their identification a difficult task. Likewise, while the stoichiometry, in particular in central metabolism, is well conserved across many species, the underlying network of regulatory interactions can differ substantially even among related species. Although we know of no dedicated study, it must be expected that the evolution of regulatory interactions occurs on a significantly faster time scale than the evolution of catalytic properties, analogous to the findings obtained for transcription factors and transcriptional regulation [171]. Despite the difficulties in their identification, feedback inhibition and allosteric regulation are defining characteristics of metabolic networks. An important rationale for the development of detailed kinetic models is to elucidate and understand the mechanisms of (allosteric) regulation in largescale metabolic networks [31]. Unlike the catalytic properties of enzymes, whose evolution can often be rationalized based on the requirement to synthesize certain cellular building blocks, the function of allosteric interactions is usually less straightforwardly accessible—and difficult to rationalize using intuitive reasoning alone. Again constituting a genuine system property, the network of allosteric regulation has evolved to ensure the functionality of cellular metabolism to maintain metabolic homoeostasis and to adapt tochanging environmental or intracellular conditions, as will be discussed in Section IX.
computational models of metabolism
139
Figure 9. Various types of inhibition that occur for Michaelis–Menten kinetics. Shown is competitive (A), uncompetitive (B), and noncompetitive inhibition (C). The corresponding rate laws are listed in Table II, (see text for details).
Aiming at a computer-based description of cellular metabolism, we briefly summarize some characteristic rate equations associated with competitive and allosteric regulation. Starting with irreversible Michaelis–Menten kinetics, the most common types of feedback inhibition are depicted in Fig. 9. Allowing all possible associations between the enzyme and the inhibitor shown in Fig. 9, the total enzyme concentration ET can be expressed as ET ¼ ½E( þ ½ES( þ ½EI( þ ½ESI(
ð48Þ
Using the rapid equilibrium assumption between the inhibitor and the enzyme, the expressions for the complexes are given as ½EI( ¼
½E(½I( KI
and
½ESI( ¼
½ES(½I( KI7
ð49Þ
with KI ¼ k%i =kþi and KI7 ¼ k%i2 =kþi2 . Proceeding as in Eq. (32), with ½ES( ninhib ¼ k2 ½ES( ¼ k2 ET |ffl{zffl} ½E( þ ½ES( þ ½EI( þ ½ESI(
ð50Þ
Vm
and using the expression ½ES( ¼ ½E(½S(=KM , we obtain the functional form of the rate equation ninhib ðS; IÞ ¼ +
Vm ½S( 1þ
½I( KI
,
+ , KM þ ½S( 1 þ K½I(7
ð51Þ
I
In the case of competitive inhibition (Fig. 9A), the inhibitor I competes with the substrate S for the active site of the enzyme. Setting KI7 ! 1, the corresponding rate equation is Vm ½S( , nci ðS; IÞ ¼ + ½I( 1 þ KI KM þ ½S(
ð52Þ
140
ralf steuer and bjo¨rn h. junker
The inhibition can be interpreted as an increase of the Michaelis constant KM . In the case of uncompetitive inhibition (Fig. 9B), the binding of the substrate to the enzyme is not affected. However, the [ES] complex becomes inactive upon binding of the inhibitor. Using KI ! 1, the corresponding rate equation is nuci ðS; IÞ ¼
Vm ½S( + , KM þ ½S( 1 þ K½I(7
ð53Þ
I
In the case of noncompetitive inhibition (Fig. 9C), the inhibitor may bind to the [ES] complex, as well as to the free enzyme. In the simplest case, with KI ¼ KI7 , we obtain Vm ½S( , nnci ðS; IÞ ¼ + ½I( 1 þ KI ðKM þ ½S(Þ
ð54Þ
corresponding to a decrease of the apparent maximal reaction rate VM . The corresponding rate equations for reversible reactions are listed in Table II. Additional scenarios that are not considered here include mixed inhibition with KI 6¼ KI7 and a possible residual catalytic activity of the ESI complex. Note that within each equation, the concentration of free inhibitor was used, a derivation in terms of total inhibitor IT involves additional terms. To account for positive cooperativity and sigmoidal rate equations, a number of theoretical models for allosteric regulation have been developed. Common to most models is the assumption (and requirement) that enzymes act as multimers and exhibit interactions between the units. We briefly mention the most TABLE II Various Types of Inhibition that Occur for Michaelis–Menten Kineticsa Type
Competitive
Uncompetitive
Noncompetitive a
Irreversible
n¼
V ½S( ) M * ½I( þ ½S( KM 1 þ KI
VM ½S( ) * n¼ ½I( KM þ ½S( 1 þ ‘ KI n¼+
VM ½S( , 1 þ K½I(I ðKM þ ½S(Þ
Reversible ½S( ½P( Vmþ % Vm% KS KP n¼ ½S( ½P( ½I( þ þ 1þ KM KP KI ½S( ½P( % Vm% Vmþ K K S ) *) P * n¼ ½S( ½P( ½I( 1þ þ 1þ ‘ KM KP KI n¼+
Vmþ K½S(S % Vm% K½P(P ,+ , 1 þ K½S(M þ K½P(P 1 þ K½I(I
Shown is competitive (A), uncompetitive (B), and noncompetitive inhibition (C), corresponding to the cases shown in Fig. 9. The noncompetitive case assumes KI ¼ KI7 . The reversible equations are adapted from [172].
141
computational models of metabolism
well-known cases, namely, the models of (i) Hill, (ii) Monod, Wyman, and Changeux (MWC), and (iii) Koshland, N%emethy, and Filmer (KNF). (i) The Hill Equation. Probably the most straightforward way to account for sigmoidal kinetics in multimeric enzymes is to assume that an already bound substrate to one site enhances the binding to other sites, giving rise to positive cooperativity. In the extreme case, a single substrate may induce the occupation of all n remaining binding sites, resulting in the transition ½E( þ n½S( $ ½ESn (
ð55Þ
Correspondingly, the fractional occupation of binding sites is of the form ½ESn ( ½S(n ½S(n ¼ n ¼ n ET Kd þ ½S( KS þ ½S(n
ð56Þ
with Kd and KS denoting dissociation and half-saturation constants, respectively. The functional form of the Hill equation (56) was originallysuggested in 1910 by A. V. Hill to describe the binding of oxygen to hemoglobin [173]. Note that, apart from the rather informal motivation above, the Hill equation is not based on a mechanistic interpretation. Rather, the Hill coefficient n is often utilized as a heuristic and not necessarily integer measure of cooperativity [140]. Figure 10 shows a corresponding sigmoidal rate law of the form nhill ðSÞ ¼
Vm ½S(n KSn þ ½S(n
ð57Þ
in comparison to the hyperbolic Michaelis–Menten equation. 1
rate ν
hill
(n=1,2,4)
n=1 0.8 0.6
n=2 n=4
0.4 0.2 0 −2 10
−1
0
1
10 10 10 substrate concentration S [au]
2
10
Figure 10. A sigmoidal rate equation as a function of the substrate concentration S. Shown is the rate for n ¼ 1 (Michaelis–Menten, dotted line), n ¼ 2 (dashed line), and n ¼ 4 (solid line). For increasing n, the rate equation is increasingly switch-like. Parameters are Vm ¼ 1 and KS ¼ 1.
142 (A)
ralf steuer and bjo¨rn h. junker (B)
Figure 11. Allosteric regulation: A conformational change of the active site of an enzyme induced by reversible binding of an effector molecule (A). The model of Monod, Wyman, and Changeux (B): Cooperativity in the MWC is induced by a shift of the equilibrium between the T and R state upon binding of the receptor. Note that the sequential dissociation constants KT and KR do not change. The T and R states of the enzyme differ in their catalytic properties for substrates. Both plots are adapted from Ref. 140. See color insert.
(ii) The concerted model of Monod, Wyman, and Changeux (MWC). Proposed in 1965, the MWC or concerted model takes into account the molecular details of allosteric regulation [140,174]. It is assumed that the enzyme consists of two or more subunits with each subunit having two different states, the relaxed (R) and tense (T) state. The transitions between the two conformations may only occur in a concerted fashion, that is, the transition of one subunit into the other state requires the transition of all other subunits into the other state. Enzymes composed of heterogeneous subunits are not allowed. Cooperativity arises as the enzyme is predominantly in one state (T), while the effector binds easier to the other state (R). As more effector is added, the enzyme gradually swings over to the tighter-binding state R. See Fig. 11 for an illustration. Several modification of this scheme were subsequently proposed in the literature [140,175]. (iii) The sequential model by Koshland, N%emethy, and Filmer. The sequential model, proposed in 1966 [176], is an extension of the induced-fit model of Koshland [148]. The binding of an effector to one subunit changes the confirmation of that subunit and thereby alters the interaction of the subunit with its neighbors. The conformational changes in the enzyme are thus sequential, with positive or negative interactions between subunits; that is, the binding of a second effector molecule can be enhanced or suppressed compared to the binding of the first molecule. It should be noted that the concerted and sequential models can be interpreted as limiting cases of more general models involving all possible conformational changes of a multimeric enzyme. However, as for multisubstrate reactions discussed above, the resulting equations are usually hardly intelligible, difficult to distinguish experimentally, and often of little practical use [140].
computational models of metabolism
143
In the following, we mainly use variants of the functional form given in Eq. (56) to describe cooperativity and allosteric regulation in metabolic systems. In particular, within Section VII.C, we discuss a general functional form of rate equations, including allosteric interaction. D. Putting the Parts Together: A Short Guide Following the scheme presented in Fig. 4, we now have assembled the building blocks that are necessary for the formulation of an explicit kinetic model. Although currently often restricted to medium-scale representations of metabolic pathwaysor (sub)networks, the parts may now be organized into a coherent whole, building the kinetic model of a metabolic network. However, obviously, there are several additional constraints to consider. First, an important, and unfortunately often neglected, step in constructing a kinetic model, is to define its purpose and scope. As emphasized in Section II.B, a model might be little more than an arbitrary collection of rate equations and parameters without explicit definition of its purpose [33]. Similarly, the construction of a model requires a scope; that is, we need to define those properties that we do not want to model [97]. Obviously, special care should taken about these steps, and the decisions have to be thoroughly justified, because almost no other part in the process will have a similar influence on the outcome. Second, we need to keep in mind that a model is not the answer to all questions. A kinetic model does not generate an explanation for all observations. In particular, and a platitude maybe, but nonetheless often ignored, a model cannot explain phenomena that depend on interactions or properties that are not included within the model description. This is of particular relevance when aiming to predict the flux distributions following a mutation or perturbation. As mutations and other perturbations of the original network usually entail significant changes on the transcriptional level, a metabolic model will fail to predict the actual response of the system. On the positive side, however, if validly constructed, a metabolic model is the key to a vast variety of properties that cannot be accessed by intuitive reasoning alone. Keeping these words of caution in mind, we provide some brief guidelines for constructing an explicit kinetic model in the following. In proceeding with the model construction process, we follow the overview given in Fig. 4. First, a list of all participating reactions is collected, most conveniently making use of the pathways of interest in one of the online databases summarized in Table IV. Once the set of reactions is compiled, most of the modeling tools collected in Table III will automatically extract the stoichiometric matrix. Note that metabolites that are only either produced or consumed must be defined as external. In biological terms, an external metabolite can be an actual extracellular substrate or a metabolic pool that is large enough that it can be assumed constant for the modeling purposes (e. g., storage starch). Having
144
ralf steuer and bjo¨rn h. junker TABLE III Selected Software Packages for Kinetic Simulations [39, 177]
Name Copasi CellDesigner E-CELL Cellware JDesigner/Jarnac SB Toolbox SBRT
URL http://www.copasi.org/ http://www.celldesigner.org/ http://www.e-cell.org/ http://www.cellware.org/ http://sbw.kgi.edu/sbwWiki/sysbio/jdesigner/ http://www.sbtoolbox2.org/ http://www.bioc.uzh.ch/wagner/software/SBRT/
Reference [178] [179] [40] [180] [181] [182] [183]
established the stoichiometry and thereby the structure of the network, we have to test if the metabolic network is consistent. For example, elementary mode analysis or flux balance analysis (see Section V.C) will locate reactions that are not used, or metabolites that are only be depleted or accumulated. Subsequent to eliminating the structural problems, we assign a detailed rate equation to each reaction. The simplest choice is mass action kinetics, independent of the concentration of enzymes. Usually, however, saturation kinetics as described in Section III.C is used. We need to have an idea about the mechanism of the enzyme, and decide if the enzyme should be modeled as reversible or not. Relevant information on kinetic properties is obtained from databases such as BRENDA (see also Table IV). Also, in many modeling tools, the rate laws corresponding to the most common mechanisms are preimplemented. Nonetheless, sometimes, and especially when multiple inhibitors are involved, the rate law needs to be implemented manually, requiring a thorough knowledge of enzyme kinetics. Avoiding this level of complexity and sacrificing some aspects of reality, the use of heuristic rate laws in this step is discussed in Section VII.C. As the next step, the rate laws have to be parameterized. First, we need the maximal catalytic activity of the enzyme, Vm , which will limit the maximal possible flux mediated by this reaction. We note that Vm is not the specific activity that is measured using purified enzyme and expressed per milligram purified enzyme, but rather reflects the absolute activity that can be observed in a cell or tissue extract, expressed per cell volume or gram fresh weight. This absolute activity is usually measured in vitro in extracts using spectrophotometric methods. These assays recently have been the basis for developing substrate cycling assays, which are currently the most sensitive assays for enzyme activity, and the ones with the highest throughput [184]. The enzyme activity depends on the amount of the enzyme, which is regulated by gene expression and other regulatory mechanisms, and thus must be measured under each environmental condition and in each genotype separately. In contrast to the condition specificity of the Vm values, the binding constants of metabolite to enzymes, the Michaelis constants Km , are not dependent on enzyme concentration but rather result from the structure and amino acid
computational models of metabolism
145
TABLE IV Useful Resources and Webpages to Assemble Information on Kinetic Models, Including Pathway Databases and Model Repositories Name
Description
KEGG [186–188]
Kyoto Encyclopedia of Genes and Genomes http://www.genome.jp/kegg/ Database of metabolic pathways and enzymes http://MetaCyc.org/ A curated resource of core pathways in human biology http://www.reactome.org/ The BRaunschweig ENzyme DAtabase http://www.brenda-enzymes.info/ System for the Analysis of Biochemical Pathways - Reaction Kinetics http://sabio.villa-bosch.de/SABIORK/ Thermodynamics of Enzyme-Catalyzed Reactions http://xpdb.nist.gov/enzyme\_thermodynamics/ A database of annotated published models http://www.ebi.ac.uk/biomodels/ A tool for simulation of kinetic models from a curated database http://www.jjj.bio.vu.nl/ XML-based language to store and exchange models http://www.CellML.org/ See also the CellML Repository: http://www.CellML.org/models The Systems Biology Markup Language http://www.sbml.org/ Standardized minimal information in the annotation of models http://www.ebi.ac.uk/compneur-srv/miriam/ A portal site for Systems Biology http://systems-biology.org
METACYC [189] Reactome [190] BRENDA [191,192] SABIO RK [193,194] NIST [195] BioModels [196] JWS Online [197] CellML [198, 199]
SBML [200, 201] MIRIAM [185] Systems Biology
composition of the enzyme. Thus, under certain circumstances it is legitimate to postulate that the Km values will not change in different environmental conditions. Furthermore, often the Km values for the same enzymes in related species will be similar. Consequently, the Km values are often recruited from databases, such as BRENDA—but for the interpretation of the model, this approximation should be kept in mind. Approximation by in vitro data from the literature or databasesis often the only possibility, because measurement of Km values is very time consuming: The enzyme has to be purified and assayed with different substrate concentrations, usually not a feasible procedure to conduct for all parameters in a model. Similar restrictions apply for inhibitory constants—that is, Ki values. With a complete set of parameters assembled the process of model construction is finished—and the model may be interrogated using the methods described in the subsequent sections. In particular, for the exchange of kinetic
146
ralf steuer and bjo¨rn h. junker
models within the scientific community, some data exchange formats have been developed, the most well-known of which is the Systems Biology Markup Language (SBML). Obviously, in order to be useful for other researcher, a model should come with a certain amount of metainformation, as defined in data standards, such as MIRIAM (minimum information requested in the annotation of biochemical models [185]). Once the model is completed, it can be deposited in model repositories, a list of which is given in Table IV. However, in most cases, there is no complete set of parameters available and we have to instead rely on alternative and additional sources to describe the kinetics of metabolic processes. To this end, and complementing the bottom-up approach described above, the focus of the next section is on direct measurements of metabolite concentrations. IV
FROM MEASURING METABOLITES TO METABOLOMICS
For the quantitative description of the metabolic state of a cell, and likewise— which is of particular interest within this review—as input for metabolic models, experimental information about the level of metabolites is pivotal. Over the last decades, a variety of experimental methods for metabolite quantification have been developed, each with specific scopes and limits. While some methods aim at an exact quantification of single metabolites, other methods aim to capture relative levels of as many metabolites as possible. However, before providing an overview about the different methods for metabolite measurements, it is essential to recall that the time scales of metabolism are very fast: Accordingly, for invasive methods samples have to be taken quickly and metabolism has to be stopped, usually by quick-freezing, for example, in liquid nitrogen. Subsequently, all further processing has to be performed in a way that prevents enzymatic reactions to proceed, either by separating enzymes and metabolites or by suspension in a nonpolar solvent. Metabolite measurements can be quantitative (absolute) or semiquantitative (relative). While the former approach imposes a greater work load, because it requires analysis of purified standards in addition to the biological samples, the latter only compares the normalized signal intensities between two samples. However, although large data sets with relative metabolite concentrations are very helpful for functional genomics, they are only of limited use as a data source for kinetic models. Even though relative measurements enable comparison between different samples, they do not enable comparison between two metabolites within a single sample. The values for metabolite concentrations are needed at different occasions for modeling metabolism: (i) as an additional data source to validate kinetic models that are constructed in a bottom-up approach, (ii) as starting point for steady-state search algorithms, (iii) as additional experimental data for
computational models of metabolism
147
parameter estimation, and (iv) for the definition of a metabolic state in the context of structural kinetic modeling, discussed in Section VIII. The various approaches for measuring metabolites can be categorized into different groups [2]: Target(ed) analysis, which has been performed since several decades, describes the determination and quantification of a small set of known metabolites using one particular analytical technique. Metabolite profiling refers to the analysis of a larger set of identified and unknown metabolites in a more unbiased manner, often with the aim of monitoring the impact of genetic or environmental perturbations on metabolism [202]. Not easily distinguished from metabolite profiling, metabolomics aims at determining as many metabolites as possible using complementary analytical methodologies to ensure maximal comprehensiveness. A.
The Problem of Organizational Complexity
Using the words of an engineer, multicellular organisms are composed of large numbers of ‘‘containers’’. At all times, each such container has its own timedependent state, involving different amounts and concentrations of all components. Furthermore, the organization of biological material is hierarchical: an organism is divided into organs, which are divided into tissues, which consist of different types of cells, with each cell containing different compartments. Unfortunately, it is currently not possible to sample at the smallest level of the hierarchy: the single compartment. Even worse, conventional sampling will usually result in a mixture of different cell types, thus resulting in an averaging effect—and making the results hard to interpret in the context of kinetic models. Nevertheless, there are several techniques that overcome these problems at least partially. One such technique, the nonaqueous fractionation (NAF) [203], enables the determination of metabolite concentrations and enzyme activities at the subcellular level (see Fig. 12). The technique is based on the fact that most cellular components are polar and that metabolism is unable to operate in a nonpolar medium. Biological material is first snap-frozen to stop metabolism, then homogenized, and subsequently freeze-dried. The cellular debris, which is as small as fractions of compartments, is suspended into a gradient of two nonpolar solvents, a relatively light one (e. g., hexane) and a relatively heavy one (e. g., tetrachloroethylene). In a centrifugation step, the cell fragments tend to migrate into the area of the gradient that represent their own density. The gradient is then divided into fractions and the activity of marker enzymes, known to be located exclusively in a certain compartment, is measured. Subsequently, the remaining metabolites and enzymes of interest are measured in the fractions. Through a comparison of the distribution of the marker enzymes with the metabolites and enzymes of interest, the subcellular distribution can be approximated. NAF is currently the only technique that is
148
ralf steuer and bjo¨rn h. junker
Figure 12. Nonaqueous fractionation (NAF), enables the determination of metabolite concentrations and enzyme activities at the subcellular level. The figure is adapted from [203]. See color insert.
able to yield quantitative information of subcellular metabolite concentrations and enzyme activities. However, disadvantages are the very tedious procedure, and the requirement of large amounts of material, usually several grams. The consequence of the latter is that a variety of tissues, rather than a single cell type, is usually analyzed together. Another group of techniques to acquire more specific metabolic information is single cell sampling [204]. One possibility to analyze the content of a single cell is by using microcapillaries. As the resulting sample is in the picoliter range, the analysis is limited to a few metabolites and proteins. Other methods that are often assigned to single cell sampling are actually dissecting and collecting very similar cells, after chemical embedding or cryofixation, either manually or by laser-dissection. The latter has gained particular attention as it is simple, reproducible, precise, and fast, and it allows the collection of large amounts of cell-specific material. Importantly, when interpreting the results obtained from a metabolic model, we always have to take into account the source and thus the reliability of the data that were used to parameterize these models. In the usual case that tissues and compartments are mixed in the sampling procedures, interpretation has to be limited to phenomena that are not influenced by the averaging effect. B.
Targeted Analysis of Metabolites
Prior to the genomics era, most metabolite measurements were performed in targeted approaches such as spectrophotometric assays or HPLC with UV detection. Today, these methods are still the best choice for the analysis of small numbers of metabolites. In spectrophotometric assays, metabolites are first
computational models of metabolism
149
chemically extracted from the biological material and then assayed in vitro by coupling them via enzymes and/or other reactants to the generation of a colored or UV-absorbing substance. The concentration of this substance can be determined by a photometer. All coupling agents are present in excess, so that the metabolite of interest is transformed quantitatively. Care has to be taken in the setup of the assay: Other reacting metabolites might be present in the metabolite extracts, possibly resulting in artifacts. Since the 1990s, major technical breakthroughs have been achieved in highthroughput metabolite quantifications. Conventional metabolite assays are usually parallelized using standardized microtiter-plates, the most common ones contain 96 or 384 wells on an area of 128 by 85 mm. A large number of laboratory consumables and devices have been created for these standardized formats: plastic ware, pipettes, centrifuges, incubators, spectrophotometers, and even liquid handling robots. Using these techniques, it is possible for a single person to perform hundreds of metabolite measurements per day, by the help of robots even thousands. Another important novelty is cycling assays, by means of which it was possible to significantly decrease the so far relatively high detection limits in spectrophotometric metabolite quantifications [205]. In these assays, the metabolite of interest is coupled via one or more enzymes to the production of a certain intermediate metabolite, which in turn is going into a cycle of consumption and regeneration. Thereby, the readout of the spectrophotometer is not a single value, but a continuously increasing optical density, from the slope of which the concentration of the metabolite of interest can be calculated. C.
High-throughput Measurements: Metabolomics
Another strategy to increase the throughput of metabolite measurements is to determine a large number of metabolites in one machine run. In this respect, new technologies, usually based on mass spectrometry, had to be developed or at least existing techniques had to be combined—giving rise to the new area of research denoted as metabolomics (sometimes also called metabonomics in the medical field), in an analogy with other high-throughput ‘‘-omics’’ fields [2–6,206–210]. Metabolomics aims at determining the metabolome, defined as ‘‘the quantitative complement of all the low molecular weight molecules present in cells in a particular physiological or developmental state’’ [206]. However, unlike genomics and transcriptomics, which have already reached their goal of sequencing all genes and quantifying all transcripts, respectively, metabolomics will most probably never reach full coverage. The number of metabolites in a metabolome is estimated to be around 500, 700, and 3000 for bacteria, yeast, and human beings, respectively [211], while for plants this number was estimated to be up to 25,000. It is most probably due to this large number of metabolites that in the late 1990s metabolomics was largely advanced within the plant sciences.
150
ralf steuer and bjo¨rn h. junker
Usually, metabolomics approaches are performed by coupling chromatography with mass spectrometry. For these techniques, metabolites are first extracted from biological material and then, depending on the chromatography step, some functional groups of the metabolites are derivatized to increase volatility and thermostability. In the chromatography step, the compounds are separated by various chemical properties, while in the mass spectrometry step the masses of the compounds (or of their fragments) are analyzed. While a mass spectrometer is not able to differentiate between two stereoisomers, the chromatography step usually is. The fact that most biomolecules are chiral explains to some extent the success story of this analytical combination for metabolite measurements. Besides variations in the principles of the chromatograph and the mass spectrometer, different ionization techniques can be used and different mass spectrometers can be arranged in series. Furthermore, in the preprocessing steps different derivatization procedures can be applied, depending on the group of metabolites that should be analyzed. The first machines for metabolomics, and still the most commonly used ones, are GC-EI-Q-MS, that is, a gas chromatograph coupled to a quadrupole mass spectrometer with electron impact ionization, usually just referred to as GC-MS [202]. The success of the GC-MS for metabolomics can be explained by its numerous advantages: Compared to other systems, a GC-MS is relatively inexpensive, easy to use, and reproducible, and the separation of the compounds is very good. Using a time-of-flight (TOF) mass spectrometer instead of a quadrupole significantly increases the sensitivity and decreases the time of a machine run by about a factor three [212]. Although it has been reported occasionally that more than a 1000 ‘‘metabolites’’ can be detected with this method, this number must be taken with caution. Although in a plant extract there may be a high number of GC peaks (corresponding to derivatives) and up to 5000 MS fragments [213], the number of known and unknown metabolites responsible for these peaks and fragments is somewhat lower, and the number of metabolites of known structure that can be reliably measured is usually around 100. For some more GC peaks it is possible to recognize functional groups based on the MS fragmentation pattern. Nevertheless, using GC-based technologies, the quantification of several important intermediates of central metabolism, especially phosphorylated intermediates, is not very reliable, presumably because these compounds and their derivatives are not thermostable. For an analysis of these groups of metabolites, an LC-MS (liquid chromatography or HPLC coupled to MS) is more suitable, because it eliminates the need for volatility and thermostability and thereby eliminates the need for derivatization. Using a triple quadrupole MS, most of the intermediates in glycolysis, in the pentose phosphate pathway, and in the tricarboxylic acid cycle were measured in E. coli [214].
computational models of metabolism
151
Capillary electrophoresis (CE) either coupled to MS or to laser-induced fluorescence (LIF) is less often used in metabolomics approaches. This method is faster than the others and needs a smaller sample size, thereby making it especially interesting for single cell analysis [215]. The most sensitive mass spectrometers are the Orbitrap and Fourier transform ion cyclotron resonance (FT-ICR) MS [213]. These machines determine the mass-to-charge ratio of a metabolite so accurate that its empirical formula can be predicted, making them the techniques of choice for the identification of unknown peaks. All MS technologies require the establishment of method-specific mass libraries so that compounds in the spectra can be identified [212], a tedious task that has been restricted to large laboratories. Nevertheless, some of these efforts are driven by the metabolomics community, thereby requiring some sort of standardization to conduct comparable experiments, as has been proposed with the ArMet standard [216]. Last but not the least, metabolomics experiments generate large amounts of data that need sophisticated analysis methods to extract biological information, usually based on multidimensional statistics [3, 5, 58, 209, 217, 218]. Metabolomics experiments as the basis for an analysis of the possible dynamics of metabolic networks are discussed in Section VIII. V.
TOPOLOGICAL AND STOICHIOMETRIC ANALYSIS
Parallel to the advances in high-throughput metabolite measurements to characterize cellular states, the most significant progress in the computational analysis of cellular metabolism has been on the topological and stoichiometric level. Propelled by the possibility to reconstruct microbial metabolic networks on a genome-scale, constraint-based stoichiometric analysis has become a key aspect of Systems Biology—up to the point that the term metabolic modeling has become almost synonymous with constraint-based modeling. Stoichiometric network analysis builds upon a computational interrogation of the stoichiometric matrix N, whose properties were already described in Section III.B. The reconstruction of the stoichiometric matrix is a laborious multistage process, based on (i) biochemical data, (ii) genome annotation, (iii) indirect evidence by synthesizing capacities of organisms, and (iv) expert and bibliographic knowledge [50, 219]. A number of useful resources for network reconstruction have been compiled in Section III.D (see also the monograph of B. Ø. Palsson for a more extensive account [50]). Beginning in the late 1990s with simpler microbial organisms, such as H. influenzae [220], a large number of genome-scale reconstructions are available today, including first steps toward the reconstruction of the human metabolic map [51, 221, 222]. It must be emphasized, though, that metabolic reconstructions are not yet completed. No organism is fully characterized on the metabolic level, and, unlike genome sequencing, the metabolic reconstruction of an organism has no
152
ralf steuer and bjo¨rn h. junker
clear end—similar to the corresponding difference between metabolomics and genome sequencing discussed in the previous section. Nonetheless, the topological and stoichiometric analysis of metabolic networks is probably the most powerful computational approach to large-scale metabolic networks that is currently available. Stoichiometric analysis draws upon extensive work on the structure of complex reaction systems in physical chemistry in the 1970s and 1980s [59], and can be considered as one of the few theoretically mature areas of Systems Biology. While the variety and amount of applications of stoichiometric analysis prohibit any comprehensive summary, we briefly address some essential aspects in the following. A.
Topological Network Analysis
From a purely topological point of view, a metabolic network can be interpreted as a bipartite graph, consisting of two sets of nodes that represent metabolites and biochemical interconversions. The two disjoint sets of nodes are connected by a set of (directed or undirected) edges, specifying which metabolites participate in a reaction or biochemical interconversion. As proposed by a number of authors, the bipartite graph may either be collapsed into a substrate graph [45], with edges indicating that two metabolites participate in a common reaction, or into a reaction graph [52], with edges indicating that two reactions share a common metabolic intermediate. Following the recent out burst of complex network analysis [47,48,223], the collapsed substrate and reaction graphs have become the objects of interest for many applications of complex network theory. ^ is defined as Formally, a binary connectivity matrix N 3 ^ ij ¼ 1 if Nij 6¼ 0 N ð58Þ 0 if Nij ¼ 0 ^ encoding the topological properties of a metabolic network [50]. The with N number rm i of reactions that metabolite i participates in is then given as the sum ^ over the ith row of N X ^ ij rm ð59Þ N i ¼ j
whereas the sum over a column specifies the number rrj of metabolites that participate in the jth reaction X ^ ij rrj ¼ N ð60Þ i
Of particular interest are the adjacency matrices with respect to reactions ^ ^TN Ar ¼ N
ð61Þ
computational models of metabolism
153
^N ^T As ¼ N
ð62Þ
and substrates
defining the reaction and substrate graphs, respectively. Note that the substrate adjacency matrix has a strong structural similarity to the Jacobian matrix (see also Sections VII.A and VIII). In the past decade, a large number of studies emphasized the heterogeneous ‘‘scale-free’’ degree distribution of metabolic networks: Most substrates participate in only a few reactions, whereas a small number of metabolites (‘‘hubs’’) participate in a very large number of reactions [19,45,52]. Not surprisingly, the list of highly connected metabolites is headed by the ubiquitous cofactors, such as adenosine triphosphate (ATP), adenosine diphosphate (ADP), and nicotinamide adenine dinucleotide (NAD) in its various forms, as well as by intermediates of glycolysis and the tricarboxylic acid (TCA) cycle. While this heterogeneous degree distribution itself is no news to most biochemists [22], several other topological properties have also been used to characterize metabolic networks. Examples include the remarkably short average pathlength between metabolites (‘‘small-world property’’)–an indicator of the time required to spread information or perturbations within the network [52], the topological damage generated by the deletion of enzymes [57], the comparison of topological structure between various organisms [19], as well as hierarchies and modularity in metabolic networks [53–55]. However, and notwithstanding its merits, we emphasize that the graph-based analysis of metabolic networks has several significant drawbacks. In particular, due to being a network of biochemical interconversions, many aspects of metabolic networks differ fundamentally from those of other networks of cellular interactions [23,58,217,233,234]. Moreover, metabolic networks are actually hypergraphs, that is, networks in which edges (reactions) connect to several nodes (metabolites), necessitating the use of more advanced methods than graph theory for their analysis. B.
Flux Balance Analysis and Elementary Flux Modes
Stoichiometric analysis goes beyond topological arguments and takes the specific physicochemical properties of metabolic networks into account. As noted above, based on the analysis of the nullspace of complex reaction networks, stoichiometric analysis has a long history in the chemical and biochemical sciences [59–62]. At the core of all stoichiometric approaches is the assumption of a stationary and time-invariant state of the metabolite concentrations S0 . As already specified in Eq. (6), the steady-state condition dSðtÞ ¼0 dt
)
NmðS0 ; kÞ ¼ 0
ð63Þ
154
ralf steuer and bjo¨rn h. junker
puts constraints on the feasible flux distributions that can be utilized to predict and explore the functional capabilities of metabolic networks [50, 61, 63, 64, 68,69]. We emphasize that the steady-state condition, as utilized within stoichiometric analysis, does not necessarily presuppose a time-invariant state. All results can be formulated in terms of time-dependent concentration changes !SðtÞ. If, after a time T no net change !SðTÞ ¼ 0 has occurred, we obtain N
Z
T
mðSÞ dt ¼ 0
ð64Þ
0
Equation (64) justifies the application of flux-balance analysis even in the face of (i) fast short-term fluctuations and (ii) periodic long term—for example, circadian—variability. The steady state balance condition restricts the feasible steady-state flux distributions to the flux cone P ¼ fm0 2 Rr : Nm0 ¼ 0g. The reduction of the admissible flux space, with some of its algebraic properties already summarized in Section III.B, is exploited by several computational approaches, most notably Flux Balance Analysis (FBA) [61, 71, 235] and elementary flux modes (EFMs) [96, 236–238]. 1.
Elementary Flux Modes (EFMs)
Aiming at a network-based pathway analysis, the elementary flux modes provide a handle on the set of possible pathways through a metabolic network. In particular, each feasible steady-state flux distribution can be represented by a nonnegative combination of generating vectors that span the flux cone defined above. The EFMs, as proposed by S. Schuster [236, 238], are a set of generating vectors that are defined as a minimal set of reactions capable of working together in a steady state. The metabolic network is decomposed into distinct, but possibly overlapping, pathways—allowing an exhaustive enumeration of all feasible flux vectors. The set of EFMs is unique for a given metabolic network and all feasible flux vectors can be described as linear combinations of EFMs. An example is given in Fig. 13. Note that EFMs are often found to correspond to distinct modes of behavior of the system: Although an observed flux distribution can be an arbitrary combination of all possible flux modes, many biologically realized flux distributions closely relate to one (or few) single flux modes only [239, 240]. The concept of elementary flux modes has resulted in a vast number of applications to analyze and predict the functionality of metabolic networks [64, 65, 138, 241–243]. Software resources that allow for the computation of elementary flux modes are listed in Table V [178, 224]. It should be noted that, due to their definition as an exhaustive enumeration of possible flux distributions,
155
computational models of metabolism
Figure 13. The elementary flux modes: Shown is a minimal representation of the TCA cycle, consisting of m ¼ 5 metabolites and r ¼ 9 reactions. The system gives rise to six elementary modes, depicted on the right: The classic textbook TCA cycle (EFM1). The cycle using a bypass of the pyruvate kinase (EFM2). Withdrawal of cycle intermediates (EFM3). Same as above, but using anaplerotic reactions (EFM4). Withdrawal of pyruvate (EFM5). Same as above, but using anaplerotic reactions (EFM6).
an analysis in terms of elementary flux modes is currently limited to mediumsized metabolic networks. Closely related to elementary flux modes are extreme pathways [50], extreme currents defined by B. L. Clarke [59], and the utilization of Petri nets in the analysis of metabolic networks [244, 245]. TABLE V Several Programs and Toolboxes for Stoichiometric Analysis Name
Ref a
CellNetAnalyzer COBRAb Metatool 5.1c rYANAd SNAe EXPA
[224] [225] [226, 227] [228,229] [230] [231]
Pathway Analyser SBRT Copasif
[232] [183] [178]
a
URL http://www.mpi-magdeburg.mpg.de/projects/cna/cna.html http://systemsbiology.ucsd.edu/downloads/COBRAToolbox/ http://pinguin.biologie.uni-jena.de/bioinformatik/networks/ http://yana.bioapps.biozentrum.uni-wuerzburg.de/ http://www.bioinformatics.org/project/?group\_id=546 http://systemsbiology.ucsd.edu/Downloads/Extreme\_Pathway\_Analysis http://sourceforge.net/projects/pathwayanalyser http://www.bioc.uzh.ch/wagner/software/SBRT/ http://www.copasi.org/
Requires MATLAB. b Requires MATLAB. c Requires MATLAB or GNU octave (www. octave.
org). d Written in Java and distributed under the GNU General Public License (GPL). MATHEMATICA. f Basic evaluation of elementary flux modes only.
e
Requires
156
ralf steuer and bjo¨rn h. junker 2.
Flux Balance Analysis (FBA)
Probably the most prominent approach to large-scale metabolic networks is constraint-based flux balance analysis. The steady-state condition Eq. (63) defines a linear equation with respect to the feasible flux distributions m 0 . Formulating a set of constraints and a linear objective function, the properties of the solution space P can be exploredusing standard techniques of linear programming (LP). In this case, the flux balance approach takes the form: Constraint-Based Stoichiometric Modeling maximize Z ¼ wT 6 m 0 subject to: Nm0 ¼ 0 and nmin + n0i + nmax i i with i ¼ 1; . . . ; r In addition to constraining the flux distribution to lie in the nullspace of N, additional upper and lower bounds may be defined—reflecting, for example, measured maximal velocities Vm or other known capacity constraints. To obtain a solution to the optimization problem, the choice of the objective function is critical. Frequently used objective functions include maximal biomass yield, maximal energy (ATP) production, among various other possible choices. While in many scenarios optimization for biomass, and hence growth, was demonstrated to be a suitable choice, other conditions warrant a different objective function—making the solution to the optimization problem to some extent arbitrary. Not surprisingly, the choice of the objective function is considered a bottle neck of flux balance analysis and any particular choice is often not without controversy [246]. In particular, maybe apart from metabolic maps of entire microbial organisms, most metabolic pathways do not allow to define an optimal ‘‘function’’ entirely in terms of metabolic flux. Furthermore, even when a suitable objective function is available, the solution is usually not unique. To this end, secondary auxiliary objectives might be defined, for example, a maximal ATP yield using a minimal number of enzymatic steps. In this case, similar restrictions as already noted above apply. Despite its widely recognized limitations, flux balance analysis has resulted in a large number of successful applications [35, 67, 72–74], including several extensions and refinements. See Ref. [247] for a recent review. Of particular interest are recent efforts to augment the stoichiometric balance equations with thermodynamic constraints—providing a link between concentration and flux in the constraint-based analysis of metabolic networks [74, 149, 150]. For a more comprehensive review, we refer to the very readable monograph of Palsson [50].
computational models of metabolism C.
157
The Limits of Flux Balance Analysis
From a theoretical perspective, and provided that the network structure and some information about input and output fluxes are available, the intracellular steadystate fluxes can be estimated utilizing flux balance analysis. In conjunction with large-scale concentrations measurements, as described in Section IV, this allows, at least in principle, to specify the metabolic state of the system. However, FBA in itself is not sufficient to uniquely determine intracellular fluxes. In addition to the ambiguities with respect to the choice of the objective function, flux balance analysis is not able to deal with the following rather common scenarios [248]: (i) Parallel metabolic routes cannot be resovled. For example, in the simplest case of two enzymes mediating the same reaction, the optimization procedure can only assign the sum of a flux of both routes, but not the flux of each route. (ii) Reversible reaction steps can not be resolved, only the sum of both directions, that is, the net flux. (iii) Cyclic fluxes cannot be resolved as they have no impact on the overall network flux. (iv) Futile cycles, which are common in many organisms, are not present in the FBA solution, because they are usually not ‘‘optimal’’ with respect to any optimization criterion. These shortcomings necessitate a direct experimental approach to metabolic fluxes, as detailed in the next section. However, prior to proceeding with the experimental approaches, we point out one additional shortcoming of FBA that is only rarely mentioned in the literature. Formally, the flux balance condition is nothing but the zeroth term in a Taylor expansion of Eq. (5). As will be made explicit in Section VIII, the steady-state assumption does not take into account any dynamic property of the metabolic state: Neither can we ascertain if a given flux distribution actually corresponds to a stable steady state nor is it possible to account for allosteric regulation. Given the rich dynamics displayed even by simple metabolic systems, as will be discussed in Section VII.A, the neglect of dynamic properties considerably delimits the capabilities of FBA as a predictor of network function. In particular for biotechnological applications, FBA runs the risk of selecting solutions that might be optimal with respect to an objective function but are highly undesirable based on their dynamic properties. In Section VIII, we thus propose to augment the description of the system, taking the next terms in the expansion into account. VI.
MEASURING THE FLUXOME: ANALYSIS
13
C-BASED FLUX
Given the inherent limits of a purely computational approach to obtain an estimate of the flux distribution of a metabolic system, an experimental determination of metabolic fluxes is paramount to the construction and validation
158
ralf steuer and bjo¨rn h. junker
of metabolic models. Conclusions about fluxes are often inferred from gene expression data (e.g., the pathway is upregulated)—a common way of reasoning that might be valid in special cases but cannot be generalized: As we progress from gene via transcript, protein, and enzyme activity to in vivo metabolic conversion rates (metabolic flux), the assumption of a simple linear causality does not hold [249]. In many cases, it has been demonstrated that there is no strict correlation between transcript and protein abundance [250]. Consequently, at the level of central metabolism in vivo flux cannot be predicted from such measurements [251–253]. Furthermore, it was recently shown that metabolic fluxes in canola embryos can change dramatically as a result of differences in supplied nutrients without considerable changes in enzyme activities [254]. These facts again emphasize the need for a direct experimental determination of metabolic fluxes. In addition, for modeling of metabolism, quantitative flux measurements are crucial to determine one set of the variables of a metabolic system, namely, the steady-state flux m0 . Isotope labeling has long been used to elucidate pathway structures—for example, the path of carbon in photosynthesis [255]. For this technique, an isotopically labeled metabolic intermediate is supplied to the biological material and the label is followed through metabolism. Besides carbon, nitrogen and other atoms can be labeled, but we will not consider these here. Sometimes semiquantitative information is derived by measuring the total isotope enrichment in a metabolite. Nevertheless, the information obtained when performing a carbon labeling experiment (CLE) is far more than just the total enrichment, as we will see below. We have to distinguish between methods based on radioactive (e.g., 14 C) and stable (e.g. 13C) isotopes. Due to their radiation, radioactive isotopes can be measured very easily in metabolic intermediates or end products. They have been widely used to estimate unidirectional fluxes, usually by chemically fractionating the labeled biological material and measuring the radioactivity in the fractions [e.g., 256, 257]. However, the resolution of this approach is limited because due to the problems arising with the radioactivity, usually only the label in some pools of metabolites (e.g., organic acids, starch) is analyzed. Stable isotopes open the door for more detailed analysis. The stable label in metabolic intermediates and end products is in the majority of the cases determined by GCMS (see also Section IV.C). There are several applications to determine dynamic information on fluxes (which will be mentioned in brief in Section VI.B); however, the most fine-grained flux information can be obtained by steady-state metabolic flux analysis using 13C (13C-MFA). Therefore, this method is the main focus of this section. A.
Steady-State Metabolic Flux Analysis
As discussed in Section V.B, flux balance analysis allows to estimate intracellular steady-state fluxes provided the network structure and some information about
computational models of metabolism
159
Figure 14. Principle for measuring bidirectional fluxes by 13 C metabolic flux analysis. In a carbon labeling experiment, 1-13 C-glucose is provided in the medium, and the culture is grown until a steady state is reached. Glucose can either go directly via the hexose phosphate pool (Glu-6P and Fru-6P) into starch, resulting in labeling hexose units of starch only at the C1 position, or it can be cleaved to triose phosphates (DHAP and GAP), from which hexose phosphates can be resynthesized, which will result in 50% labeling at both the C1 and the C6 position (assuming equilibration of label by scrambling at the level of triose phosphates). From the label in the hexose units of starch, the steady-state fluxes at the hexose phosphate branchpoint can be calculated; for example, if we observe 75% label at the C1 and 25% at the C6 position, the ratio of v5 to v7 must have been 1 to 1. All other fluxes can be derived if two of the fluxes of v1, v6, and v7 are known (e.g., v2 = v1 ; v3 = v5 + v6 ).
input and output fluxes are available, albeit without yielding unique solutions for reversible reaction steps, cyclic fluxes, and parallel routes. Steady-state metabolic flux analysis, sometimes also called network flux analysis, aims to overcome these problems by taking isotope data into account—further constraining the flux in the metabolic network. The principle of metabolic flux analysis, and how bidirectional fluxes can be resolved, is depicted in Fig. 14. However, despite the conceptual simplicity of the approach, the quantitative evaluation of metabolic flux analysis requires a mathematical framework that is
160
ralf steuer and bjo¨rn h. junker
highly intimidating to many biologists [258–262]. Even though we do not describe the mathematical framework in detail, we outline the most important concepts and ideas in Section VI.A.2. 1.
The Experimental Concepts
Let us assume that we perform a CLE: We supply a cell culture with a medium containing labeled carbon substrate and harvest the cells after a while. Now, the labeled material contains not only the information about how much label has been taken up but also which metabolites are labeled, in which position with which probability, and even which combinations of labeled atoms within one molecule occur with which probability. Nevertheless, to be able to calculate bidirectional steady-state fluxes from this CLE later, we need to obey several rules when performing the experiment [248]: At first, the biological material needs to be in stationary physiological and metabolic state for the entire duration of the experiment (from some hours to some days). While for bacterial cultures this can be guaranteed by using a Chemostat system, in higher organisms this causes severe constraints. For steady-state metabolic flux analysis of plants, usually some plant organs (embryos, roots) or cells are cultivated in a defined media under constant conditions [252, 263]. These approaches are always the best possible compromise between applicability of the method on the one hand and the closeness to the natural situation (e.g., in planta) on the other hand. Furthermore, the substrate must contain a carbon chain (for reasons that become clear from Fig. 14); that is, CO2 is not a suitable substrate. This requirement has the consequence that the system must grow hetero- or at least mixotrophically, and photosynthetic organisms cannot be studied straightforward. Additionally, the information yield of a CLE is directly dependent of the metabolite that is labeled and the position at which this metabolite is labeled. A rational design procedure was developed that allows the composition of an optimal mixture of labeled substrates for maximal information yield [248]. Similar to the required metabolic steady state, time scales must be sufficiently large to ensure also the isotopic steady state at the end of the experiment. This time can be especially long if storage compounds such as starch or protein are concerned. An important fact to keep in mind is also that labeled and unlabeled substrates do not contain 100% and 0% of label, respectively. On the one hand, commercially available labeled substrates always have residual unlabeled atoms. On the other hand, each stable isotope has a certain percentage of natural abundance—for example, 1.13% in the case of 13C. This means that an unlabeled glucose molecular ion, for example, will cause an M þ 1 peak in a mass spectrometer that is approximately 7% of all mass peaks for the glucose molecular ion, a number that has to be taken into account (compare to Section IV.C). This effect is even more important when GC-MS is used to measure the
computational models of metabolism
161
isotope distribution, as the necessary derivatizing agents usually contain silicon, which has a much higher natural abundance of 4.7% for 29Si and 3.1% for 30Si. After the labeling experiment, the biomass is harvested, chemically fractionated, derivatized, and, finally, in most cases analyzed by GC-MS (see below and Section IV.C). Electron impact ionization is used, which leads to fragmentation of the metabolite, so that not only the molecular ion is measured but also several fragments, a circumstance that makes the measured information more valuable for subsequent data evaluation. 2.
Evaluation of a Carbon Labeling Experiment
To understand the evaluation of a CLE, we need to introduce some terms: The word isotopomer is a combination of the terms isotope and isomer. An isotopomer is one of the different labeling states in which a particular metabolite can be encountered [248]; that is, a molecule with n carbon atoms has 2n isotopomers. These are usually either depicted using outlined and filled circles for unlabeled and labeled atoms, respectively (see Fig. 14), or are described in text format; for example, C#010 would be the isotopomer of a three-carbon molecule labeled at the second position. An isotopomer fraction is the percentage of molecules in this specific labeling state. The positional enrichment is the sum of all isotopomer fractions in which a specific carbon atom in a specific metabolite is labeled [248]. Consequently, the usage of isotopomers enables to account for more information: While a molecule with n carbon atoms will yield n positional enrichments, there are 2n % 1 isotopomer fractions (the 2n th measurement is redundant as, by definition, isotopomer fractions must sum up to unity) [260]. The measurements of the labeled metabolites may be performed with GC- or LC-MS, or by NMR. Because it is the most commonly used method, we will only consider GC-MS based approaches here. Obviously and unfortunately, it is not possible to directly measure the isotopomer enrichments by GC-MS, because the apparatus only yields total masses of molecules or fractions thereof, but not directly the position of a label. Each MS peak is produced by all isotopomers with the same molecular weight—that is, the same number of labeled carbon positions. Sometimes this concept is also called mass isotopomers [264]. In a so-called retrobiosynthetic approach, it has been shown that the labeling state of many intracellular pools can be determined indirectly by measuring the labels in macromolecular biomass components at steady state; for example, the labeling state of alanine from hydrolyzed protein reflects the label of pyruvate [265]. Using this approach, it is possible to quantify fluxes into storage components. Once the labeling data are available, they have to be corrected for initial biomass and natural abundance of the isotopes (see above). Next, for the part of metabolism of interest, we need to model a carbon transition network, in which
162
ralf steuer and bjo¨rn h. junker
Figure 15. An example network. Left: The structure of the network, given by the stoichiometry. Right: The carbon transition network, which is the basis for isotopomer balances. The fate of each carbon atom in each reaction step is shown.
the fate of each carbon atom in each reaction step needs to be known (see Fig. 15 for an example). Using this network, the CLE can be simulated by assuming arbitrary fluxes, and the resulting isotopomer labeling pattern can be calculated [258,259]. This step is the computationally most expensive one. Isotopomer balance equations are nonlinear and iterative numerical approaches were used in the past. However, severe instabilities were observed when large exchange fluxes were present. This means that an analytical procedure was needed, but solving a nonlinear system consisting of thousands of equations (one for each isotopomer of each metabolite) is not a trivial task. In an elegant move to reduce this complexity, the concept of cumomers was introduced [260]. This word again is a combination of two terms. Cumomer fraction means cumulated isotopomer fraction [260], which means the percentage of all isotopomers labeled in at least one certain position (e.g., cumomer 1xx is the sum of isotopomers 100 þ 101 þ 110 þ 111). The complete set of cumomer fractions can be calculated from the complete set of isotopomer fractions. Using this trick, the simulation can be run in less than a second [248]. In the next step, using the isotopomer pattern obtained from the simulation, the (putative) outcome of the measurement can be predicted on the basis of the assumed fluxes. Then, prediction and measurement are compared. Using optimization algorithms, the guessed fluxes can be varied systematically and iteratively, until the closest match is found [248]. We now have determined the steady-state flux values that yielded the observed labeling pattern. Nevertheless, without statistical analysis these values
computational models of metabolism
163
are seemingly precise but statistically worthless [248]. Thus, a sensitivity matrix is generated that contains the sensitivities of the measurements with respect to the estimated fluxes. From this matrix, the covariance matrix can be computed and, in turn, a confidence region for the fluxes can be derived [261]. Using a w2 test on the differences between experiment and simulation remaining after the optimization procedure, the goodness of fit can be judged, and outliers in the data resulting from measurement errors can be eliminated. To simplify the procedures of data evaluation, several computer programs have been published (e.g., [262, 266]). FiatFlux [266] is a software package for 13 C-MFA that is, compared with other tools with similar purpose, relatively easy to use. Nevertheless, it is restricted to some specific substrates, and requires a MATLAB license and at least basic knowledge of the program. Probably the most advanced and most commonly used software for 13C MFA is 13C-FLUX [262]. This stand-alone software runs on UNIX-based systems and is very general, but at the price that it cannot be used without a rough understanding of the underlying mathematical principles [248]. B.
Dynamic Flux Analysis
A number of approaches exist that are not as restricted as 13C-MFA, especially concerning the requirement of a physiological, metabolic, and isotopic steady state. However, it has to be said in advance that these methods usually do not come close to the power of 13C-MFA: Either they cannot resolve bidirectional fluxes or the resulting flux maps are not very detailed. Furthermore, as some approaches for dynamic flux analysis rely on kinetic rate laws to simulate enzymatic reactions, the border between dynamic flux analysis and kinetic metabolic modeling becomes blurred. This ambiguity is also emphasized by the fact that the term kinetic model is occasionally used for the evaluation of dynamic carbon labeling experiments [267, 268]. However, in this review, we use the term kinetic model only for enzyme-kinetic computational models that are used to simulate dynamic behavior of metabolite concentrations and enzymatic fluxes, but not to simulate dynamic labeling patterns. Approaches for dynamic flux analysis vary in their techniques; some examples are given below, others are discussed in Ref. [267]. The first example of a dynamic flux analysis was a study performed in the 1960s [269]. In the yeast Candida utilis, the authors determined metabolic fluxes via the amino acid synthesis network by applying a pulse with 15Nlabeled ammonia and chasing the label with unlabeled ammonia. Differential equations were then used to calculate the isotope abundance of intermediates in these pathways, with unknown rate values fitted to experimental data. In this way, the authors could show that only glutamic acid and glutamine-amide receive their nitrogen atoms directly from ammonia, to then pass it on to the other amino acids.
164
ralf steuer and bjo¨rn h. junker
More recently, Roessner-Tunali et al. [270] incubated discs isolated from potato tubers in media containing unilabeled 13C-glucose and measured the total label in several metabolites at different time points by GC-MS. From this data, they were able to estimate a set of 27 unidirectional carbon exchange rates. Even though this approach is relatively straightforward if compared to steady-state MFA, the gained information is less complex: Estimated fluxes are unidirectional, and parallel routes and cycles (e.g., anaplerotic fluxes), cannot be resolved. Huege et al. [271] cultivated Arabidopsis plants in 13CO2 atmosphere, transferred the plants to normal atmosphere, and monitored the dilution of isotopes in several metabolite pools. Through evaluation of the mass isotopomer distribution, metabolite partitioning processes could be monitored. However, due to the lack of absolute metabolite concentrations, no absolute fluxes could be calculated. Nevertheless, building upon this method, suitable approaches for flux analysis in autotrophic tissue might be derived in the future. VII.
FORMAL APPROACHES TO METABOLISM
Based on the accessibility of high-quality experimental information, we now focus on aspects of model interrogation and analysis. The question how cells actually control and distribute their flux under different conditions requires a mathematical and formal approach to metabolic regulation. The knowledge obtained by quantitative experiments must be, in the sense of Section II.B, encoded into a mathematical system, scrutinized utilizing the tools of formal analysis, and eventually decoded back into predictions about the natural system. Several formal frameworks and methodologies have been proposed that allow us to infer the emergent properties of model of metabolism. For our purposes, these formal frameworks act as toolboxes, enabling the systematic interrogation of metabolic models and thus enabling the transformation of information encoded in the mathematical abstractions into knowledge about metabolic systems. In the following sections, we briefly discuss elements of dynamic systems theory, allowing us to categorize and predict dynamic properties from the knowledge of the interactions (Section VII.A). We then summarize the key definitions of Metabolic Control Analysis (MCA), a forebear of systems biology that defines a quantitative link between the (local) properties of enzymes and the (global) response of concentrations on the network level (Section VII.B). In the last part of this section, we outline several recent approaches that aim to circumvent the inevitable lack of information about detailed rate laws and kinetic parameters (Section VII.C). A.
The Dynamics of Complex Systems
There is no shortage of dynamic phenomena observed in cellular systems [272]. Quite on the contrary, cellular metabolism is a highly dynamic system, and
computational models of metabolism
165
numerous examples of complex dynamic behavior have been reported in the literature. Among the most well-known instances of complex dynamics are temporal variations in the concentrations of metabolic intermediates of the yeast glycolytic pathway [273, 274] and the photosynthetic Calvin cycle [95, 275, 276], along with other descriptions of complex behavior in biochemical systems [37, 96, 98, 277–279]. Though the physiological significance of dynamic phenomena is sometimes unknown, most dynamic properties are closely associated with cellular function [280]. An example is the circadian rhythmicity of metabolism, intimately linked to cellular redox state, energy balance, and mitochondrial respiration [281, 282]. From a more general point of view, dynamic properties of cellular regulatory systems, such as multistability, sustained oscillations, or irreversible switching, constitute the conceptual basis for many, if not most, physiological properties of living cells. Examples include time-keeping by circadian clocks [283], the regulation of cell division [284, 285], cellular signaling [98, 286, 287], or cell differentiation [288, 289]. The established tools of nonlinear dynamics provide an elaborate and versatile mathematical framework to examine the dynamic properties of metabolic systems. In this context, the metabolic balance equation (Eq. 5) constitutes a deterministic nonlinear dynamic system, amenable to systematic formal analysis. We are interested in the asymptotic, the linear stability of metabolic states, and transitions between different dynamic regimes (bifurcations). For a more detailed account, see also the monographs of Strogatz [290], Kaplan and Glass [18], as well as several related works on the topic [291–293]. 1.
The Stability of Simple Pathways
Consider the simplest case of a metabolic network, namely, a single metabolite S that is synthesized and consumed by the reactions nsyn and ncon , respectively. The time dependence of the concentrations SðtÞ is then described by the corresponding mass balance equation 2 dSðtÞ X ¼ nj ðS; kÞ ¼ nsyn % ncon dt j¼1
ð65Þ
where both fluxes may depend on the substrate concentration SðtÞ. A timeinvariant steady-state value S0 is attained if the rate of synthesis equals the rate of consumption dSðtÞ ¼0 dt
,
nsyn ¼ ncon
ð66Þ
Taking into account the typical functional form of Michaelis–Menten kinetics (see Section III.C), the rate of consumption will usually increase with increasing
166
ralf steuer and bjo¨rn h. junker
Figure 16. The stability of a steady state, as determined by a rate balance plot [287]. Left panel: The rate of synthesis nsyn and consumption ncon of a substrate S. Right panel: The (net) flux difference nnet ¼ nsyn % ncon . The steady-state value S0 is locally stable: After transient perturbation the concentration SðtÞ will return to its nominal value S0 . See text for details.
concentration S, whereas the rate of synthesis will decrease (product inhibition). A schematic depiction is shown in Fig. 16. Thus, if the actual concentration SðtÞ < S0 is below the steady-state value S0 , the rate of synthesis exceeds the rate of consumption, and the concentration increases. Vice versa, if the actual concentration SðtÞ > S0 is larger than the steady-state value S0 , the rate of consumption exceeds the rate of synthesis, and the concentration decreases. Given the functional form of the rate equations shown in Fig. 16, the steady-state value S0 is locally stable: After transient perturbation, the concentration SðtÞ will return to its steady-state value S0 . A slightly different scenario is shown in Fig 17. Again the net flux, the difference between the rate of synthesis and therate of consumption, is shown as a function of the substrate concentration. However, in this case, the flux balance
Figure 17. The flux balance equations gives rise to three possible solutions: While the outer solutions are stable, the middle state is unstable: If the actual concentration SðtÞ is below the nominal value S0 , the net flux is negative. The concentration SðtÞ will decrease even more. Vice versa, if the actual concentration SðtÞ is above the nominal value S0 , the net flux is positive, leading to a further increase in SðtÞ.
167
computational models of metabolism
equation does not give rise to a single solution S0 , but allows three different values that fulfill the steady-state equation (multistability). Using a similar reasoning as above, we can conclude that the two outer states are again stable, whereas the intermediate state is unstable: Any perturbation from the intermediate steady state will be amplified further. Note that if a perturbation of any of the stable states drives the concentration SðtÞ across the unstable state, the system will not return to its original state. The unstable state constitutes a separatrix; that is, it forms a boundary between two basins of attraction. Crucial for the later analysis, the decision whether a state is locally stable or not is entirely determined by the slope of the zero-crossing at the steady state (the partial derivative). If the net flux of Eq. (65) has a positive slope, any infinitesimal perturbation will be amplified. 2.
Bistability and Hysteresis
The concept of multistability is further exemplified using the hypothetical minimal metabolic pathway shown in Fig 18: A metabolite A is synthesized with a constant rate n1 and consumed by two reactions n2 ðAÞ and n3 ðAÞ. The rate equations are adapted from [96], n1 ¼ const
n2 ðAÞ ¼ k2 ½A(
n3 ðAÞ ¼
fluxes νsyn and νcon
2
k3 ½A( + ,n 1 þ ½A( KI
ð67Þ
1.5
1
0.5 ν
con
0 0
1
2
3
4
=ν +ν 2
5
ν
3
syn
6
7
8
=ν
1
9
10
concentration A [a.u.]
Figure 18. A simple bistable pathway [96]. Left panel: The metabolite A is synthesized with a constant rate n1 and consumed with a rate ncon ¼ n2 ðAÞ þ n3 ðAÞ, with the substrate A inhibiting the rate n3 at high concentrations (allosteric regulation). Right panel: The rates of nsyn ¼ n1 ¼ const: and ncon ¼ n2 ðAÞ þ n3 ðAÞ as a function of the concentration A. See text for explicit equations. The steady state is defined by the intersection of synthesizing and consuming reactions. For low and high influx n1 , corresponding to the dashed lines, a unique steady state A0 exists. For intermediate influx (solid line), the pathway gives rise to three possible solutions of A0 . The rate equations are specified in Eq. 67, with parameters k2 ¼ 0:2, k3 ¼ 2:0, KI ¼ 1:0, and n ¼ 4 (in arbitrary units).
168
ralf steuer and bjo¨rn h. junker
0
steady state concentration A [a.u.]
10
8 0
stable state A 6
4
2
unstable state A0 SN SN
0 0
0.5
1 1.5 constant influx ν1 [a.u.]
2
Figure 19. The steady-state solutions A0 of the pathway shown in Fig. 18 as a function of the influx n1 . For an intermediate influx, two pathways exist in two possible stable steady states (black lines), separated by an unstable state (gray line). The stable and the unstable state annihilate in a saddlenode bifurcation. The parameters are k2 ¼ 0:2, k3 ¼ 2:0, KI ¼ 1:0, and n ¼ 4 (in arbitrary units).
The rate equation n3 ðAÞ is assumed to follow a Hill-type substrate inhibition, reminiscent of the inhibition of phosphofructokinase by ATP in minimal models of glycolysis [96, 126]. As depicted in Fig. 18 (right plot), the intersections of synthesizing and consuming reactions allow more than one steady-state value A0 . The possible steady-state solutions A0 are depicted in Fig. 19 as a function of the constant influx n1 . For low influx, the system exhibits a unique steady state with low A0. Increasing the influx, an additional steady state arises and three possible states exist (two stable and one unstable). For still increasing influx, the lower steady state collides withthe unstable state and the system again exhibits a single steady state (see also the dashed lines in Fig. 18). The pathway acts as a bistable switch and exhibits hysteresis. Bistability and switching are crucial concepts of cellular regulation [80, 98] and can often be detected using the graphical methods outlined above. See also [98,287] for several illustrative examples. For a number of pathways, transitions between different states were observed, either in silico, in vivo, or both. Examples include the glycolytic pathway [273, 294], the Calvin cycle [113, 125], and models of the human erythrocytes [295, 296]. 3.
The Jacobian Matrix and Linear Stability Analysis
Aiming at a more formal analysis, the asymptotic stability of a steady-state value S0 of a metabolic system upon an infinitesimal perturbation is determined by linear stability analysis. Given a metabolic system at a positive steady-state value
computational models of metabolism
169
S0 , the system of differential equations specified in Eq. (5) can be approximated by a Taylor series expansion 5 dS qm 5 ¼ NmðS0 Þ þ N 55 ðS % S0 Þ þ 6 6 6 ð68Þ dt |fflfflffl{zfflfflffl} qS S0 |fflfflffl{zfflfflffl} ¼0 ¼:M
The first term in Eq. (68) describes the steady-state properties of the system, as exploited by flux balance analysis to constrain the stoichiometrically feasible flux distributions. Since we consider infinitesimal perturbations only, quadratic terms in the expansion are neglected. In this case, the time-dependent behavior of an infinitesimal perturbation !SðtÞ ¼ S % S0 in the vicinity of S0 is described by a linear differential equation d !SðtÞ ¼ M!SðtÞ dt
ð69Þ
with M denoting the Jacobian matrix. Comparing with Eq. (68), the Jacobian matrix is defined as the stoichiometric matrix multiplied by partial derivatives of the rate equations at the state S0 5 qm 55 M :¼ N 5 ð70Þ qS S0
The solution of Eq. (69), and thus the asymptotic (t ! 1) behavior of the perturbation, is entirely specified by the eigenvectors and eigenvalues of the Jacobian matrix M. Assume all eigenvalues li of M are ordered such that l1 corresponds to the eigenvalue with maximal real part <ðl1 Þ " <ðl2 Þ " . . . " <ðlm Þ. The dominant time scales in response to the perturbation are then described by !SðtÞ ¼ a1 E1 exp½l1 t( þ a2 E2 exp½l2 t( þ 6 6 6
ð71Þ
with Ei denoting the eigenvector corresponding to the eigenvalue li and ai denoting proportionality constants depending on the initial conditions. The state S0 is asymptotically stable if, and only if, the largest real part within the spectrum of eigenvalues <ðl1 Þ ¼ lmax Re < 0 is negative. If there exists a single li with <ðli Þ > 0, the steady state is unstable. Imaginary parts within the spectrum of eigenvalues correspond to oscillatory dynamics. Specifically, consider a two-dimensional system at a steady-state value S0 that, upon linearization around S0 , gives rise to the Jacobian ! " a b ð72Þ M¼ c d
170
ralf steuer and bjo¨rn h. junker
Figure 20. The stability diagram. Depending on the trace Tr and determinant ! ¼ ad % bc of the Jacobian matrix M, the steady state can be classified intodistinct dynamic regimes. The parabola indicates the line Tr2 ¼ 4!, corresponding to the occurrence of imaginary eigenvalues. For an interpretation of the different dynamic regimes, see text.
The eigenvalues are determined by the solution of the characteristic equation det½M % 1l( ¼ 0
)
l2 % lTr þ ! ¼ 0
ð73Þ
with Tr ¼ a þ d and ! ¼ ad % bc denoting the trace and determinant of M, respectively. The dynamic properties of the steady-state S0 can be classified according to the different regimes shown in Fig. 20: (i) stable node, the system will return to the state S0 after an infinitesimal perturbation; (ii) stable focus, corresponding to a pair of complex conjugate eigenvalues with negative real part, resulting in damped transient oscillations; (iii) saddle point, at least one direction is unstable: an infinitesimal perturbation will be amplified; (iv) unstable focus, corresponding to a pair of complex conjugate eigenvalues and undamped transient oscillations; (v) unstable node, both eigenvalues have positive real part. The distinct categories shown in Fig. 20 also hold in more than two dimensions. In this case, the (possibly complex conjugate) eigenvalue with the maximal real part is used for the classification.
computational models of metabolism
171
Note that in systems with mass conservation, the Jacobian is rank deficient and not invertible. In this case, a reduced Jacobian is constructed, taking into account the independent variables only. The reduced Jacobian M 0 is defined as 5 5 0 0 qm 5 M :¼ N 5 L ð74Þ qS S0
with L denoting the link matrix (see Section III.B for details). The reduced Jacobian M 0 is usually nonsingular [96]. Of particular interest are transitions between the qualitatively different regimes (bifurcations). The transition from a stable node to a saddle, with one real eigenvalue crossing the imaginary axis, is denoted as saddle-node bifurcation. The transition from a stable focus to an unstable focus, with two complexconjugate eigenvalues crossing the imaginary axis, is termed Hopf bifurcation. The former is of relevance in systems with bi- or multistability, whereas the latter indicates a small amplitude limit cycle emerging (at least transiently) from the stable node. Besides the two most well-known cases, the local bifurcations of the saddlenode and Hopf type, biochemical systems may show a variety of transitions between qualitatively different dynamic behavior [13, 17, 293, 294, 297–301]. Transitions between different regimes, induced by variation of kinetic parameters, are usually depicted in a bifurcation diagram. Within the chemical literature, a substantial number of articles seek to identify the possible bifurcation of a chemical system. Two prominent frameworks are Chemical Reaction Network Theory (CRNT), developed mainly by M. Feinberg [79, 80], and Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81– 83]. An analysis of the (local) bifurcations of metabolic networks, as determinants of the dynamic behavior of metabolic states, constitutes the main topic of Section VIII. In addition to the scenarios discussed above, more complicated quasiperiodic or chaotic dynamics is sometimes reported for models of metabolic pathways [302–304]. However, apart from few special cases, the possible relevance of such complicated dynamics is, at best, unclear. Quite on the contrary, at least for central metabolism, we observe a striking absence of complicated dynamic phenomena. To what extent this might be an inherent feature of (bio)chemical systems, or brought about by evolutionary adaption, will be briefly discussed in Section IX. 4.
Dynamics of Metabolism: A Minimal Model of Glycolysis
Probably the most well-known pathway to exemplify the occurrence of complex dynamics in metabolic networks is the glycolytic pathway of yeast. Arguably one of the most modeled pathways ever, minimal models of yeast glycolysis were studied since the 1960s [94, 273, 305–308] and give rise to a rich spectrum of
172
ralf steuer and bjo¨rn h. junker
dynamic phenomena. In particular, the observance of oscillations in glycolytic intermediates has triggered the development of an exceptional number of dynamic models [274, 307, 309], ranging from minimal two variables models [96, 273, 310], intermediate models [126], to detailed kinetic models of the pathway [101]. Consider the scheme already depicted in Fig. 5 within Section III. Neglecting the branching reaction, the system can be represented by only five metabolites, interconverted by three irreversible biochemical reactions. Gx þ 2 ATP%!2 TP þ 2 ADP
ð75Þ
TP þ 2 ADP%!2 ATP þ Px ATP%!ADP ðATPaseÞ
ð76Þ ð77Þ
Metabolites denoted with a subscript x are considered external and the total amount AT ¼ ½ATP( þ ½ADP( is conserved. The differential equations for the remaining two independent species are 2 3 " ! " ! n1 ðGx ; ATPÞ d TP 2 %1 0 6 4 n2 ðTP; ADPÞ 5 ¼ ð78Þ %2 þ2 %1 dt ATP n3 ðATPÞ The first reaction n1 ðGx ; ATPÞ describes the upper part of glycolysis, converting one (external) molecule of glucose (Gx ) into two molecules of triosephosphate (TP), using two molecules of ATP. The second reaction n2 (TP, ADP) describes the synthesis of two molecules ATP from each molecule of TP. The third reaction n3 ðATPÞ describes a (lumped) overall ATP utilization. To obtain a minimal kinetic model for the glycolytic pathway, we adopt rate function similar to [96], using n1 ðGx ; ATPÞ ¼
Vm1 G0x ½ATP( 1 þ ð½ATP(=KI Þn
ð79Þ
and n2 ðTP; ADPÞ ¼ k2 ½TP( 6 ½ADP(;
n3 ðATPÞ ¼
Vm3 ½ATP( Km3 þ ½ATP(
ð80Þ
Similar to Eq. (67), the first reaction (incorporating the enzyme phosphofructokinase) exhibits a Hill-type inhibition by its substrate ATP [126]. The overall ATP utilization n3 (ATP) is modeled by a saturable Michaelis–Menten function. The system is specified by five kinetic parameters (with Gx lumped into Vm1 ), the Hill coefficient n, and the total concentration AT ¼ ½ATP( þ ½ADP(. Note that the model is not intended to capture biological realism, rather it serves as a paradigmatic example to identify dynamic behavior in metabolic pathways.
computational models of metabolism
173
Figure 21. The nullclines of the minimal model of glycolysis (schematic). The graphic analysis allows to deduce the qualitative dynamics of the system. Each area in the phasespace is characterized by the signs of the local derivatives, corresponding to increasing or decreasing concentration of the respective variable. The gray arrows indicate the direction a trajectory will go. Note that the trajectories may only intersect vertically or horizontally with the nullclines. For simplicity, the nullclines are depicted schematically only, for the actual nullclines corresponding to the rate equations see Fig. 22C.
Interestingly, the simple model is already sufficient to exhibit a variety of dynamic regimes, including bistability and oscillations. To obtain a qualitative impression of the dynamics, Fig. 21 shows a schematic depiction of the nullclines of the system. The nullclines are defined as curves in phasespace on which the derivative of a variable vanishes. Specifically, the nullcline for TP is defined by d½TP(0 ¼0 dt
,
TP0 ¼
2n1 ðATP0 Þ k2 ½ADP(
ð81Þ
For ATP, we obtain d½ATP(0 ¼0 dt
,
TP0 ¼
2n1 ðATP0 Þ þ n3 ðATP0 Þ 2k2 ½ADP(
ð82Þ
Intersections of both nullclines correspond to the steady states of the system. A graphical analysis of the nullclines allows to deduce qualitative dynamic features, as exemplified in Fig. 21.
174
ralf steuer and bjo¨rn h. junker
(A)
(B) 2
2 dTP = 0
1.5
TP concentration
TP concentration
Vm3 = 0.1
1
0.5
dATP = 0
0 0
0.2
0.4
0.6
0.8
Vm3 = 0.5
1 dATP = 0
0.5
0 0
1
0.2
0.4
0.6
0.8
1
ATP concentration
ATP concentration
(D)
(C) 2
2
m3
= 1.5
dATP = 0
TP concentration
V TP concentration
dTP = 0
1.5
1.5 dTP = 0
1
0.5
0 0
0.2
0.4
0.6
0.8
1
Vm3 = 2.5
dATP = 0
1.5 dTP = 0
1
0.5
0 0
0.2
ATP concentration
0.4
0.6
0.8
1
ATP concentration
Figure 22. The nullclines corresponding to the minimal model of glycolysis. Depending on the value of the maximal ATP utilization Vm3 , the pathway either exhibits a unique steady state or allows for a bistable solution. Note that the nullcline for TP does not depend on Vm3 . The corresponding steady states are shown in Fig. 23. Parameters are Vm1 ¼ 3:1, KI ¼ 0:57, k2 ¼ 4:0, Km3 ¼ 0:06, and n ¼ 4 (the values do not correspond to a specific biological situation).
For the choice of parameters used here, the simple pathway gives rise to bistability and hysteresis. In particular, Fig. 22 depicts the nullclines for different values of the maximal ATP consumption rate Vm3 . The corresponding steady states, given as the solution of the equation 2n1 ðATP0 Þ % n3 ðATP0 Þ ¼ 0
ð83Þ
are shown in Fig. 23. Note that ½ATP0 ( ¼ 0, and hence ½TP0 ( ¼ 0, is always a steady-state solution. As the different dynamic regimes will be of importance in Section IX, we briefly step through the scenarios shown in Figs. 22 and 23. For small Vm3 , the solution ½ATP0 ( ¼ 0 is unstable. Though there exist another steady-state solution with ½ATP0 ( > 0, this solution is outside the physiologically feasible region, confined by 0 + ½ATP0 ( + AT . That is, the nullclines in Fig. 22A will not
175
computational models of metabolism 1.5 0
ATP = A
steady states ATP
0
T
1 stable states
0.5
unstable state
0 A
0
B
0.5
C
1
1.5
D
2
2.5
maximal ATP utilization Vm3
Figure 23. The steady-state ATP concentration as a function of maximal ATP utilization Vm3 for the minimal model of glycolysis. The letters denoted on the x axis correspond to the different scenarios shown in Fig. 22A–D. Bold lines indicate stable steady states. Note that the physiologically feasible region is confined to the interval ATP0 2 ½0; AT (. For low ATP usage (Vm3 small), there are three steady states, two of which are stable. However, both stable states are outside the feasible interval.
intersect for ½ATP( + 1 ¼ AT . The result is a ‘‘runaway’’ solution with ½ATP( ¼ AT and TP increasing toward infinity; that is, the pathway is not able to reach a steady state. Increasing the maximal ATP consumption rate, an additional unstable state appears and the state ½ATP0 ( ¼ 0 becomes stable. However, initially, the basin of attraction is very small and most initial conditions will again lead to the runaway solution described above. Only when Vm3 is increased further, a stable solution with 0 < ½ATP0 ( + AT is feasible. In this case, the system is bistable, see Fig. 22C. However, if Vm3 is increased still further, the stability of the intermediate state is lost again and the system again exhibits asingle, but now stable, state ½ATP0 ( ¼ 0 (see Fig. 22D). We note that the dynamics described above are characteristic for autocatalytic pathways, and very similar results are obtained in more elaborate models of the glycolytic pathway [94, 126, 294, 303, 308, 311, 312]. In addition to bistability and hysteresis, the minimal model of glycolysis also allows nonstationary solutions. Indeed, as noted above, one of the main rationales for the construction of kinetic models of yeast glycolysis is to account for metabolic oscillations—observed experimentally for several decades [297, 305] and probably the model system for metabolic rhythms. In the minimal model considered here, oscillations arise due to the inhibition of the first reaction by its substrate ATP (a negative feedback). Figure 24 shows the time courses of oscillatory solutions for the minimal model of glycolysis. Note that for a large
176
ralf steuer and bjo¨rn h. junker
3.5 3.5
3 3 2.5 2.5
2
TP
TP
2 1.5
1.5
1
1
0.5
0 0
0.5
0.2
0.4
0.6
0.8
0 0
1
0.2
0.4
ATP ATP
ATP
0.8
1
1
0.5
0 0
0.6
ATP
1
5
10
15
20
0.5 0 0
5
time t
10
15
20
time t
Figure 24. The nullclines (upper panels, gray lines) and time courses (lower panels) for oscillatory solutions of the minimal model of glycolysis. Left panels: Damped oscillations. The trajectory spirals intothe (stable) steady state ðATP0 ; TP0 Þ ¼ ð0:5; 1:0Þ. Parameters are Vm1 ¼ 16, KI ¼ 0:307, k2 ¼ 4:0, Km3 ¼ 0:556, Vm3 ¼ 2:22, and n ¼ 4. Right panels: Sustained oscillations. The trajectory shows a stable limit-cycle around the steady state ðATP0 ; TP0 Þ ¼ ð0:5; 1:0Þ. Parameters are Vm1 ¼ 40, KI ¼ 0:2395, k2 ¼ 4:0, Km3 ¼ 0:556, Vm3 ¼ 2:22, and n ¼ 4.
parameter region the limit-cycle solution coexists with a (stable) steady state ðATP0 ; TP0 Þ ¼ ð0; 0Þ. Oscillations of the glycolytic pathway are again considered in Section VIII. In particular, within Section VIII.C, the origin and the possible physiological relevance of oscillatory solutions are discussed. B.
Metabolic Control Analysis
An early systematic approach to metabolism, developed in the late 1970s by Kacser and Burns [313], and Heinrich and Rapoport [314], is Metabolic Control Analysis (MCA). Anticipating systems biology, MCA is a quantitative framework to understand the systemic steady-state properties of a biochemical reaction network in terms of the properties of its component reactions. As emphasized by Kacser and Burns in their original work [313], Flux is a systemic property and questions of its control cannot be answered by looking at one step in isolation—or even each step in isolation. An analysis must consequently be in terms of the quantitative relations between the parts as much as in terms of the gross structure or the molecular architecture of its catalysts.
Note that MCA is not a tool for dynamic modeling of biochemical systems. Rather, MCA is essentially a generic sensitivity analysis, predominantly
computational models of metabolism
177
concerned with small (local) perturbations of a steady state. Within the past decades, a vast number of detailed treatises have appeared [15,62,96]; here, we only summarize the basic nomenclature. MCA distinguishes between local and global (systemic) properties of a reaction network. Local properties are characterized by sensitivity coefficients, denoted as elasticities, of a reaction rate ni ðS; pÞ toward a perturbation in substrate concentrations (E-elasticities) or kinetic parameters (p-elasticities). The elasticities measure the local response of a reaction in isolation and are defined as the partial derivatives at a reference state S0 5 5 qni 55 qni 55 and pik ¼ ð84Þ Eij ¼ qSj 5S0 qpk 5S0
Both quantities are usually written as m ) r elasticity matrices E and p, respectively. In contrast to the local elasticities, the control coefficients describe the global or systemic properties of the system, that is, the response to the perturbation after all variable shave relaxed to the new state. !Xi dXi =dpk ¼ !nk !0 !nk dnk =dpk
CnXki :¼ CikX ¼ lim
ð85Þ
As the reaction rate may not be assessed directly, the definition invokes an auxiliary parameter pk (e.g., an enzyme concentration) that is assumed to act only on the rate nk . Note that X may stand for an arbitrary steady-state property—with the coefficients for concentrations CS and flux CJ as the most important examples. Aiming at a more systematic approach, the relationship between local and global properties are obtained by the implicit derivative of the steady-state condition Nm ¼ 0 [62, 96]. Assuming, for simplicity, the absence of massconservation relationships, we obtain qm dS qm þN ¼0 N qS dp qp
)
! " dS qm %1 qm ¼% N N dp qS qp
ð86Þ
Equation (86) describes the effect of a perturbation in parameters on the state variables S. The equation may be summarized as RS ¼ CS p, with RS denoting the response coefficient [96] and ! " qm %1 C ¼% N N ¼ %M %1 N qS S
ð87Þ
denoting the concentration control coefficient. Note that, since no mass conservation relationships are considered, the Jacobian matrix M is assumed
178
ralf steuer and bjo¨rn h. junker
to be invertible. Analogously, the response of the steady-state flux can be evaluated " # ! " dm qm qm dS qm qm %1 qm ¼ þ ¼ 1% N N dp qp qS dp qS qS qp
ð88Þ
resulting in the flux control coefficient CJ ¼ 1 þ
qm S C ¼ 1 þ ECS qS
ð89Þ
Taking into account mass conservation relationships, specified by the link matrix L defined in Eq. (13), the expressions for the control coefficient need to be modified. We obtain ) *%1 qm N 0 ¼ %LðM 0 Þ%1 N 0 CS ¼ %L N 0 L qS
ð90Þ
) *%1 qm qm L N0 L N 0 ¼ 1 þ ECS qS qS
ð91Þ
and CJ ¼ 1 %
1.
The Summation and Connectivity Theorems
The utility and success of Metabolic Control Analysis is mostly due to a number of simple relationships that interconnect the various coefficients and that bridge between local and global properties of the network. First, the summation theorems relate to the structural properties of the network and are independent of kinetic parameters [96]. Using Eq. (90) and (91), it is straightforward to verify that CS K ¼ 0 and
CJ K ¼ K
ð92Þ
where K denotes the right nullspace of N (see Section III.B). Second, the connectivity theorems relate the local properties to the systemic behavior. Again it is straightforward to verify that CS EL ¼ %L
and
CJ E L ¼ 0
ð93Þ
The four expressions are usually combined into a more compact matrix equation
computational models of metabolism !
" J
C ½K CS
EL ( ¼
!
K 0
0 %L
"
179 ð94Þ
Equation (94) fully specifies the flux and concentrationco efficients in terms of elasticities and stoichiometry [96]. 2.
Scaled Coefficients and Their Interpretation
Up to this point, we have only considered ordinary partial derivatives. However, within MCA, it is often preferred to consider normalized (scaled) partial derivatives instead. Apart from minor exceptions, all equations are invariant under the normalization provided all involved quantities are scaled appropriately [315]. The normalization restates the equations in a logarithmic space. In particular, the normalized elasticities are defined as )Eij ¼
S0j qni q ln ni ¼ n0i qSj q ln Sj
ð95Þ
Using matrix notation, we define DS0 and Dm0 to be diagonal matrices with elements S0 and m0 on the diagonal, respectively. The normalized (or scaled) matrices of elasticities E and control coefficients CS are then obtained by the transformations e !e ¼ Dm0 eDS01
ð96Þ
CS !CS ¼ DS01 CS Dm0
ð97Þ
and
Correspondingly, the normalized flux control coefficient CJ is defined as CJ ¼ Dm0 CJ Dm01 ¼ 1 þ eCS
ð98Þ
An advantage of the normalized (or scaled) coefficients is their straightforward interpretability in biochemical terms. For example, consider the scaled elasticity of a simple Michaelis–Menten equation: nðSÞ ¼
Vm S Km þ S
)
E¼
S0 qn 1 ¼ n0 qS 1 þ KS0
ð99Þ
m
The limiting cases are limS0 !0 E ¼ 1 and limS0 !1 E ¼ 0. The normalized elasticity can be interpreted as a measure of the kinetic order of a reaction.
180
ralf steuer and bjo¨rn h. junker
For small substrate concentration S0 ( Km , the reaction acts in the linear regime, with a kinetic order of E ) 1. For increasing concentrations, the kinetic order E is monotonously decreasing. For very large concentrations S0 * Km , we obtain E ) 0, that is, the reaction is fully saturated by the substrate. Equation (99) implies that it is often possible to specify intervals or approximate values for the scaled elasticities in terms of relative saturation, even when detailed kinetic information is not available. For example, as a rule of thumb, the substrate concentration can often be considered to be on the order of the Km value. As the scaled elasticities, by means of the control coefficients, can be directly translated into a systemic response, it is possible to utilize such heuristic arguments to acquire an initial approximation of global network properties. Heuristic values for the elasticities become even more apparent when reversible reactions are considered. Recall the reversible Michaelis–Menten equation, given in Eq. (43). With a straightforward calculation we obtain S0
ES ¼
2½0;1-
0
!
S0 qn Keq KmS ¼1 0 0 þ n0 qS 1 þ KSmS þ KPmP 1 K!eq |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflffl{zfflfflffl}
ð100Þ
2½0;1Þ
with ! :¼ PS0 . Keq denoting the mass-action ratio. Analogously,
P0 qn EP ¼ 0 ¼ n qP
1þ
P0 KmP S0 P0 KmS þ KmP
! Keq
1
! Keq
ð101Þ
The scaled elasticities of a reversible Michaelis–Menten equation with respect to its substrate and product thus consist of two additive contributions: The first addend depends only on the kinetic propertiesand is confined to an absolute value smaller than unity. The second addend depends on the displacement from equilibrium only and may take an arbitrary value larger than zero. Consequently, for reactions close to thermodynamic equilibrium ! ) Keq , the scaled elasticities become almost independent of the kinetic propertiesof the enzyme [96]. In this case, predictions about network behavior can be entirely based on thermodynamic properties, which are not organism specific and often available, in conjunction with measurements of metabolite concentrations (see Section IV) to determine the displacement from equilibrium. Detailed knowledge of Michaelis– Menten constants is not necessary. Along these lines, a more stringent framework to utilize constraints on the scaled elasticities (and variants thereof) as a determinant of network behavior is discussed in Section VIII.E.
computational models of metabolism 3.
181
A Brief Criticism of Metabolic Control Analysis
As one of the groundbreaking formal frameworks for a systematic theoretical analysis of metabolic networks, MCA has led to a variety of results related to the regulation and control of metabolic systems, with a comprehensive summary being out of the scope of this review. In particular, and similar to other first-order methodologies, such as Fourier analysis or principal component analysis, it combines conceptual simplicity with tremendously powerful real-world applicability. Probably even more remarkable, MCA is one of the rare contributions of computational or theoretical biology that attained widespread recognition also among more experimentally oriented scientists—indeed a major accomplishment in bridging an abyss of difficult communication. However, as also emphasized by Bailey [33]: This success, however, has a dark side. There are clear signs that MCA has been oversold in a certain sense that some relative novices in systems mathematics (not all of whom are biologists) view MCA as the end-all of metabolic systems theory—that MCA is seen as the ‘‘theory of everything’’ for metabolic engineering. This is, sadly, far from true.
Notwithstanding its merits, it should be recognized that MCA has clear limits in its scope of applicability. From a mathematical point of view, the control coefficients of MCA are merely logarithmic sensitivity coefficients at a particular steady state. MCA itself neither does address the stability of this steady state nor does it account for the time scales in which the system reacts to perturbations. More important, the applicability of MCA as a guide to metabolic engineering is often hampered by the fact that regulation on the transcriptional and post-transcriptional level is not considered. In such a scenario, the control coefficients provide a useful characteristic of a steady state—they do not predict, and do not even relate to, the long-term adaption in response to an engineered modification. Historically, MCA is also challenged by rivaling theories; see, for example, the debate between Kacser and Savageau [316, 317]. C.
Biochemical Systems Theory and Related Approaches
Given the difficulties to obtain the precise mechanism of an enzymatic reaction, an increasing number of authors opt for using heuristic rate laws to simulate metabolic networks [85, 89, 161, 318]. Such heuristic rate laws are required to capture the generic dependencies of typical reactions on their substrates and products, but these do not necessarily rely on a detailed mechanistic foundation and are usually assumed to be of identical functional form for all participating enzymes.
182
ralf steuer and bjo¨rn h. junker
In particular, for large-scale reaction networks, it can be reasonably argued that peculiar details of individual rate equations are not relevant for the largescale behavior of the network. For example, Rohwer et al. [164] assert that ‘‘classical enzyme kinetics, as developed in the 20th century, had as primary objective the elucidation of mechanism of enzyme catalysis. In systems biology, however, the precise mechanism of an enzyme is less important; what is required is a description that will adequately reflect the response of an enzyme to changes in substrate and product concentrations’’ (see also Ref. 165). The authors conclude that ‘‘for the pathway as a whole, the exact mechanisms (e.g., ordered against ping-pong) of catalysis is irrelevant’’ [164]. Although we do not necessarily agree that the exact mechanism is always irrelevant, an approximative scheme to represent enzyme-kinetic rate equations indeed often allows to deduce putative properties of the network in a quick and straightforward way. Consequently, the utilization of approximative kinetics in the analysis of metabolic networks provides a reasonable strategy toward a large-scale dynamic view on cellular metabolism. In the following we give a brief summary of the most common approaches. 1.
Biochemical Systems Theory
Although the importance of a systemic perspective on metabolism has only recently attained widespread attention, a formal frameworks for systemic analysis has already been developed since the late 1960s. Biochemical Systems Theory (BST), put forward by Savageau and others [142, 144–147], seeks to provide a unified framework for the analysis of cellular reaction networks. Predating Metabolic Control Analysis, BST emphasizes three main aspects in the analysis of metabolism [319]: (i) the importance of the interconnections, rather than the components, for cellular function; (ii) the nonlinearity of biochemical rate equations; (iii) the need for a unified mathematical treatment. Similar to MCA, the achievements associated with BST would warrant a more elaborate treatment, here we will focus on BST solely as a tool for the approximation and numerical simulation of complex biochemical reaction networks. A core constituent of BST is to represent all metabolic rate equations as power law functions. Using the power-law formalism, each reaction rate is written as a product nj ðSÞ ¼ aj
m Y
g
Skjk
ð102Þ
k¼1
where the parameters aj and gjk denote a rate constant and the kinetic orders, respectively. The kinetic orders gjk are usually non-integer and may be negative for inhibitory dependencies.
183
computational models of metabolism
To reconcile the nomenclature with Metabolic Control Analysis, we note that the kinetic orders corresponds to the (scaled) elasticities of the reaction Ejk ¼
& S0k qnj && ¼ gjk n0j qSk &S0
ð103Þ
For power-law functions the (scaled) elasticities do not depend on the substrate concentration, that is, unlike Michaelis–Menten rate equations, power-law functions will not saturate for increasing substrate concentration. To simulate the overall network behavior, the power-law formalism is applied in two different ways. Within a generalized mass-action model (GMA), each biochemical interconversion is modeled with a power-law term, resulting in a differential equation analogous to Eq. (5) r m Y dSi X g ¼ Nij aj Skjk dt j¼1 k¼1
ð104Þ
Equation (104) offers a concise representation of metabolic networks that is more accurate than a usual linear approximation but still amendable to analytical treatment. Importantly, the parameters retain their biochemical interpretability and may thus be chosen according to heuristic principles. In contrast to generalized mass-action models, an S-system model is obtained by lumping (or aggregating) all synthesizing and consuming reactions of each metabolite into a single power-law term, respectively. The mathematical structure of a S-System is independent of the complexity of the network. For any metabolite Si , we obtain dSi ¼ naggrþ ðSÞ dt
naggr ðSÞ;
where
naggr/ ðSÞ ¼ a/ j
m Y
g/
Skjk
ð105Þ
k¼1
The obvious advantage is that the steady-state solution of an S-system model is accessible analytically. However, while the drastic reduction of complexity can be formally justified by a (logarithmic) expansion of the rate equation, it forsakes the interpretability of the involved parameters. The utilization of basic biochemical interrelations, such as an interpretation of fluxes in terms of a nullspace matrix is no longer possible. Rather, an incorporation of flux-balance constraints would result in complicated and unintuitive dependencies among the kinetic parameters. Furthermore, it must be emphasized that an S-system model does not necessarily result in a reduced number of reactions. Quite on the contrary, the number of reactions r ¼ 2m usually exceeds the value found in typical metabolic networks.
184
ralf steuer and bjo¨rn h. junker
We note that BST and S-System models are objects of considerable controversy, a more detailed criticism is given in [96]. Some merits and drawbacks of BST are discussed below in connection with a closely related approach. 2.
Linear-Logarithmic Kinetics
The approximation of biochemical rate equations by linear-logarithmic (lin-log) equations [318] seeks to avoid several drawbacks of the power-law formalism. Using the lin-log framework, all reaction rates are described by their dependencies on logarithmic concentrations, based on deviations from a reference state m0 and S0 : ( )" * +# X E Sk T E0jk ln 0 nj ðSÞ ¼ n0j 0 1 þ ET S k k
ð106Þ
Note that written in this form Eq. (106) retains the linear dependency of the rate on the total enzyme concentration ET , typical for most Michaelis–Menten mechanisms. The dependence on the substrate concentrations is approximated by a sum of nonlinear logarithmic terms [85, 86, 318, 320]. It has been demonstrated that lin-log equations capture hyperbolic kinetics slightly better than either a linear approximation or the power-law approach [318, 320–322]. One reason for the improved performance is that for lin-log kinetics the elasticities (and kinetic orders) are not constant, but change with changing metabolite concentrations. For a monosubstrate reaction nðSÞ, and omitting the dependence on the enzyme concentration, we obtain ( * +) S nðSÞ ¼ n0 1 þ E0 ln 0 S
)
E¼
E0 . / 1 þ E0 ln SS0
ð107Þ
with E0 denoting the reference elasticity. Analogous to Michaelis–Menten kinetics, the elasticity tends to zero for increasing substrate concentration. Similar to generalized mass-action models, lin-log kinetics provide a concise description of biochemical networks and are amenable to an analytic solution, albeit without sacrificing the interpretability of parameters. Note that lin-log kinetics are already written in term of a reference state m0 and S0 . To obtain an approximate kinetic model, it is thus sometimes suggested to choose the reference elasticities according to simple heuristic principles [85, 89]. For example, Visser et al. [85] report acceptable result also for the power-law formalism when setting the elasticities (kinetic orders) equal to the stoichiometric coefficients and fitting the values for allosteric effectors to experimental data. Nonetheless, as a tool for analysis and simulation of large-scale metabolic networks the use of approximate kinetics may be criticized for several reasons:
computational models of metabolism
185
0 The approximate kinetic formats discussed above face inherent difficulties to account for fundamental physicochemical properties of biochemical reactions, such as the Haldane relation discussed in Section III.C.4—a major drawback when aiming to formulate thermodynamically consistent models. 0 Both formalisms are only valid in the vicinity of a reference state and have inherent limits when dealing with small or vanishing concentrations. It is sometimes argued that this is no serious problem because homeostatic mechanisms keep the intracellular concentration in a limited range [318]. However, this reasoning is not entirely convincing, because kinetic models are also—and should be—utilized to test and elucidate those mechanism, rather than presupposing them. 0 The benefits of using approximate kinetics are unclear since the two main arguments in their favor have lost their significance. Unlike the situation in the late 1960s, a gain in computational efficiency is no longer a critical advantage. Likewise owing to increased computer power, the analytical accessibility of solutions is no longer mandatory to obtain results on largescale dynamics—in particular when the requirement of analytical accessibility entails unrealistic consequences, such as the existence of unique steady states. 0 To use approximate ad hoc functions, such as power-law or lin-log kinetics, makes it difficult to incorporate available biochemical information. For example, in a worst case scenario, all kinetic constants have to be estimated de novo and cannot be obtained from or compared to existing literature and databases on reaction kinetics. 0 The necessity of developing approximate kinetics is unclear. It is sometimes argued that uncertainties in precise enzyme mechanisms and kinetic parameters requires the use of approximate schemes. However, while kinetic parameters are indeed often unknown, the typical functional form of generic rate equations, namely a hyperbolic Michaelis–Mententype function, is widely accepted. Thus, rather than introducing ad hoc functions, approximate Michaelis–Menten kinetics can be utilized—an approach that is briefly elaborated below. 0 Finally, and more profoundly, not all properties require explicit knowledge of the functional form of the rate equations. In particular, many network properties, such as control coefficients or the Jacobian matrix, only depend on the elasticities. As all rate equations discussed above yield, by definition, the assigned elasticities, a discussion which functional form is a better approximation is not necessary. In Section VIII we propose to use (variants of) the elasticities as bona fide parameters, without going the loop way via explicit auxiliary functions.
186
ralf steuer and bjo¨rn h. junker 3.
Convenience Kinetics and Related Approaches
Most problems associated with approximate kinetics are avoided when Michaelis– Menten-type rate equations are utilized. Though this choice sacrifices the possibility of analytical treatment, reversible Michaelis–Menten-type equations are straightforwardly consistent with fundamental thermodynamic constraints, have intuitively interpretable parameters, are computationally no more demanding than logarithmic functions, and are well known to give an excellent account of biochemical kinetics. Consequently, Michaelis–Menten-type kinetics are an obvious choice to translate large-scale metabolic networks into (approximate) dynamic models. It should also be emphasized that simplified Michaelis–Menten kinetics are common in biochemical practice—almost all rate equations discussed in Section III.C are simplified instances of more complicated rate functions. Along these lines, Liebermeister and Klipp [161] suggested the use of a rapid-equilibrium random-order binding scheme as a generic mechanism for all enzymes, independent of the actual reaction stoichiometry. While there will be deviations from the (unknown) actual kinetics, such a choice, still outperforms power-law or lin-log approximations [161]. To obtain the explicit functional form of the convenience kinetics rate equation, recall the Bi-Bi random-order mechanism already considered in Section III.C.5 and depicted again below:
Comparing with Eq. (44) and using the rapid equilibrium assumption with dissociation constants KS , the total enzyme concentration can be written as (* +* + * +* + ½A½B½P½QET ¼ ½E- 1 þ 1þ þ 1þ 1þ Ka Kb Kp Kq
1
)
ð108Þ
Generalizing the expression for the random order mechanism to convert ms substrates into mp products, such that all intermediate complexes occur, we obtain "
+ Y + mp * ms * Y ½Si ½Pj 1þ 1þ ET ¼ ½Eþ KSi KPj i¼1 j¼1
1
#
ð109Þ
computational models of metabolism
187
Proceeding analogously to Eq. (32) and assuming the overall reaction to be determined by the single catalytic step nck ¼ kþ ½ES1 . . . Sms - k ½EP1 . . . Pmp -, we obtain a rate equation of the form Q Q ½Pj kþ i K½SSi - k j K Pj 2 i Q 1 2 nck ¼ ET Q 1 ð110Þ mp ½P ½Si ms þ j¼1 1 þ KPj 1 i¼1 1 þ KS i
j
Note that the parameters in Eq. (110) are interdependent. To facilitate a thermodynamically consistent analysis, a more suitable representation is given in terms of the equilibrium constant Q ½Pj Q ET k þ i ½Si jK 2 Q 1 eq 2 nck ¼ Q 1Q 1 ð111Þ m ½P ½S m p i s i KSi þ j¼1 1 þ KPj 1 i¼1 1 þ KS i
j
with
Q kþ j KPj Q Keq :¼ k i KSi
ð112Þ
A thorough analysis of Eq. (111), including a decomposition of all parameters into a thermodynamically independent representation, is given in Ref. [161]. Here we only note that Eq. (111) is consistent with Eq. (47) and provides a generic functional form to describe (unknown) rate equations in large-scale metabolic networks. In particular, Eq. (111) allows us to make use of existing biochemical databases and known dissociation constants (see Section III.D). Sometimes it is useful to rewrite Eq. (111) in terms of a (measured or assumed) metabolic steady state S0 and m 0 —in particular, given the progress in experimental accessibility of system variables discussed in Sections IV and VI. In this respect, an appropriate reparameterization is obtained straightforwardly by adjusting the product ET kþ to yield nj ðS0 ; P0 Þ ¼ n0j . Although not without restrictions, the utilization of approximative kinetics, when chosen appropriately, provides a promising path towards large-scale kinetic models of metabolism. A number of recent studies already explore the automated construction of metabolic models, making use of existing databases and integrating kinetic, thermodynamic and proteomic information [161, 323]. Nonetheless, it must be emphasized that this path is not without pitfalls. Contrary to the assertion of Rohwer et al. [164], specific individual enzyme kinetic schemes may give rise to highly idiosyncratic behavior, not captured by any generic approximative scheme. In particular enzymes that consist of several subunits, such as the pyruvate dehydrogenase (PDH), are known to be capable of complex
188
ralf steuer and bjo¨rn h. junker
dynamics already on an individual enzyme level [324]. In this sense, while an ‘‘averaging effect’’ of large networks would drastically simplify the construction of large-scale approximate models, it must not be consistent with reality. For this and other reasons, it is also of importance to devise general strategies to elucidate the dynamics of complex reaction networks. One such approach, focusing on those dynamic properties that are a priori independent of the particular functional form of the rate equations, is discussed in the next section. VIII.
STRUCTURAL KINETIC MODELING
Also when resorting to heuristic rate equations or other approximative schemes, the construction of detailed kinetic models necessitates quantitative knowledge about the kinetic properties of the involved enzymes and membrane transporters. Notwithstanding the formidable progress in experimental accessibility of system variables, detailed in Sections IV and VI, for most metabolic systems such quantitative information is only scarcely available. In this section, we describe a recently proposed approach that aims overcome some of the difficulties [23, 84, 296, 325]: Structural Kinetic Modeling (SKM) seeks to provide a bridge between stoichiometric analysis and explicit kinetic models of metabolism and represents an intermediate step on the way from topological analysis to detailed kinetic models of metabolic pathways. Different from approximative kinetics described above, SKM is based on those properties that are a priori independent of the functional form of the rate equation. Specifically, SKM seeks to overcome several known deficiencies of stoichiometric analysis: While stoichiometric analysis has proven immensely effective to address the functional capabilities of large metabolic networks, it fails for the most part to incorporate dynamic aspects into the description of the system. As one of its most profound shortcomings, the steady-state balance equation allows no conclusions about the stability or possible instability of a metabolic state, see also the brief discussion in Section V.C. The objectives and main requirements in devising an intermediate approach to metabolic modeling are as follows, a schematic summary is depicted in Fig. 25: 0 Structural kinetic modeling keeps the advantages of the stoichiometric analysis, while incorporating dynamic aspects into the description of the system. 0 Extending stoichiometric analysis, SKM allows to investigate the quantitative effects of allosteric regulation on the stability and dynamics of a metabolic system. 0 The analysis of large-scale systems is computationally feasible and does not require extensive information about the involved enzymatic rate equations and their associated parameter values.
computational models of metabolism
189
Figure 25. Structural Kinetic Modeling seeks to keep the advantages of stoichiometric analysis, while incorporating dynamic properties into the description of the system. Specifically, SKM aims to give a quantitative account of the possible dynamics of a metabolic network.
0 SKM provides exact results. The analysis is not based upon any approximation. 0 The approach is motivated and founded upon the increasing experimental accessibility of system variables, such as flux and concentrations, and is consistent with experimental knowledge and thermodynamic constraints. The basic idea is very simple: In many scenarios the construction of an explicit kinetic model of a metabolic pathway is not necessary. For example, as detailed in Section IX, to determine under which conditions a steady state loses its stability, only a local linear approximation of the system at this respective state is needed, that is, we only need to know the eigenvalues of the associated Jacobian matrix. Similar, a large number of other dynamic properties, including control coefficients or time-scale analysis, are accessible solely based on a local linear description of the system. In close analogy to flux-balance analysis, we thus extend the constraint-based description of metabolic networks to incorporate (local) dynamic properties. Recall the expansion of the mass-balance equation into a Taylor series, already given in Eq. (68) & dS qm & ¼ NmðS0 Þ þ N && ðS S0 Þ þ 1 1 1 ð113Þ |fflfflffl{zfflfflffl} dt qS S0 |fflfflffl{zfflfflffl} ¼0 ¼:M
The first term describes the steady-state condition of the system and constrains the stoichiometrically feasible flux distributions – providing the foundation for
190
ralf steuer and bjo¨rn h. junker
Figure 26. The proposed workflow of structural kinetic modeling: Rather than constructing a single kinetic model, an ensemble of possible models is evaluated, such that the ensemble is consistent with available biological information and additional constraints of interest. The analysis is based upon a (thermodynamically consistent) metabolic state, characterized by a vector S0 and the associated flux m0 ¼ mðS0 Þ. Since based only on the an evaluation of the eigenvalues of the Jacobian matrix are evaluated, the approach is (computationally) applicable to large-scale system. Redrawn and adapted from Ref. 296.
flux-balance analysis discussed in Section V.B. Along similar lines, taking the next term of the expansion into account, the structure of the Jacobian matrix M constrains the possible dynamics of the system at a given state. The basis of Structural Kinetic Modeling thus consists of giving a parametric representation of the Jacobian matrix of a metabolic system at each possible point in parameter space, such that each element is accessible even without explicit knowledge of the functional form of the rate equations. Once this parametric representation is obtained, it allows to give a quantitative account of the dynamical capabilities of the metabolic system at a given state. In particular, we aim at a statistical evaluation of the Jacobian matrix, with each element of the Jacobian constraint by the available experimental information. Rather than evaluating a single model, we evaluate an ensemble of possible models—with each instance reflecting the available experimental and biochemical information. The proposed workflow is depicted in Fig. 26, a detailed description is provided in the following.
computational models of metabolism A.
191
Definitions: The Jacobian Matrix Revisited
The Jacobian matrix of any metabolic network can be written as product of two matrices [23, 84, 325]. Consider the metabolic balance equation, describing the time-dependent behavior of the concentration Si ðtÞ, r dSi ðtÞ X ¼ Nij nj ðSÞ dt j¼1
ð114Þ
within a metabolic network of m metabolites and r reactions. We assume the existence of a positive state S0 that fulfils the steady-state condition NmðS0 ; kÞ ¼ 0. Note that the state S0 is neither required to be unique, nor stable. If a metabolic system gives rise to multiple solutions (bi- or multistability; see also Section VII.A), the steady state S0 corresponds to one of the possible states. An obvious choice for the metabolic state S0 is often an experimentally observed concentration vector. In the following, we denote S0 and the associated flux vector m0 ¼ mðS0 Þ as the metabolic state of the system. The derivation follows the description given in [84, 293, 299]: Using a simple transformation, Eq. (114) can be rewritten as r n0 d Si ðtÞ X nj ðSÞ j ¼ Nij 0 0 dt Si n0j S j¼1 |ffliffl{zfflffl} |ffl{zffl} :¼$ij
ð115Þ
:¼mj ðSÞ
where S0i and n0j ¼ nj ðS0 Þ denote the metabolic state. The transformation makes use of the definitions $ij :¼
n0j S0i
Nij
and
mj ðSÞ ¼
nj ðSÞ n0j
ð116Þ
Using the variable substitution xi ðtÞ ¼ Si ðtÞ=S0i , the time-dependence of the new variables can be expressed as dx ¼ LlðxÞ dt
ð117Þ
with L corresponding to a rescaled stoichiometric matrix and lðxÞ a normalized rate equation. Note that at the steady state x0 ¼ 1, we obtain lðx0 Þ ¼ 1 and L 1 1 ¼ 0. The Jacobian matrix M x with respect to the normalized variables at the steady state x0 ¼ 1 is & ql&& l l M x ¼ Lhx where hx :¼ & ð118Þ qx x0 ¼1
192
ralf steuer and bjo¨rn h. junker
The scaled Jacobian M x can be straightforwardly transformed back into the original Jacobian M using a similarity transformation. Equation (118) provides the conceptual basis for all subsequent considerations: The nonzero elements of the matrices L and hlx define the new parameter space of the system, that is, the possible dynamic behavior of the system is evaluated in terms of these new parameters. Crucial to the analysis, the elements of both matrices have a well-defined and straightforward interpretation in biochemical terms, making their evaluation possible even in the face of incomplete knowledge about the detailed kinetic parameters of the involved enzymes and membrane transporters. Any further evaluation now rest on a careful interpretation of the two parameters matrices. 1.
The Matrix L
The elements of the matrix L are fully specified by the stoichiometry matrix N and the metabolic state of the system. Usually, though not necessarily, the metabolic state corresponds to an experimentally observed state of the system and is characterized by steady-state concentrations S0 and flux values mðS0 Þ. Alternatively, we may assume that there exists some (but possibly limited) knowledge about the typical concentrations involved. For each metabolite, we can then specify an interval Si . S0i . Sþ i that defines a physiologically feasible range of the respective concentration. Furthermore, the steady-state flux vector m0 is subject to the mass-balance constraint Nm0 ¼ 0, leaving only r rankðNÞ independent reaction rates. Again, an interval ni . n0i . nþ i can be specified for all independent reaction rates, defining a physiologically admissible flux space. The elements of the matrix L have the units of inverse time and determine the state at which the Jacobian is to be evaluated. Each element of L is—at least in principle—experimentally accessible and does not hinge upon a specific mathematical representation of any biochemical rate equations. 2.
The Saturation Matrix hlx
The interpretation of the elements of the matrix hlx is slightly more subtle, as they represent the derivatives of unknown functions lðxÞ with respect to the variables x at the point x0 ¼ 1. Nevertheless, an interpretation of these parameters is possible and does not rely on the explicit knowledge of the detailed functional form of the rate equations. Note that the definition corresponds to the scaled elasticity coefficients of Metabolic Control Analysis, and the interpretation is reminiscent to the interpretation of the power-law coefficients of Section VII.C: m Each element yxij of the matrix hlx measures the normalized degree of saturation, or likewise, the effective kinetic order, of a reaction nj with respect to a substrate Si at the metabolic state S0 . Importantly, the interpretation of the elements of hlx does again not hinge upon any specific mathematical representation of specific
computational models of metabolism
193
rate equations. Rather the definition encompasses almost all rate equations that satisfy some reasonable assumptions. A detailed discussion, including several examples of possible rate equations, is given in Section VIII.E, here we only summarize several paradigmatic cases. For mass-action kinetics, we obtain Y Y nj ðSÞ ¼ kj xi ð119Þ Si ) mðxÞ ¼ m
resulting in a saturation parameter yxij ¼ 1 8i (linear regime). Analogously, for power-law kinetics, we obtain Y g Y g nj ðSÞ ¼ kj xi ij ð120Þ Si ij ) mðxÞ ¼
highlighting the correspondence between saturation parameter and the kinetic m order yxij ¼ gij. The case of a single substrate Michaelis–Menten equation is slightly more complicated. We obtain nðSÞ ¼
Vm S Km þ S
)
mðx; S0 Þ ¼ x
K m þ S0 Km þ xS0
ð121Þ
The scaled derivative is thus ymx
& qm&& 1 2 ½0; 1¼ & ¼ qx x0 ¼1 1 þ KS0
ð122Þ
m
identical to the scaled elasticity Eq. (99) of Metabolic Control Analysis. The interpretation is analogous to Section VII.B.2: The limiting cases are limS0 !0 ymx ¼ 1 and limS0 !1 ymx ¼ 0. For small substrate concentration S0 ( Km the reaction acts in the linear regime. For increasing concentrations the saturation parameter ymx is monotonously decreasing. For very large concentrations S0 * Km , the saturation parameter approaches zero ymx ) 0. The dependency of ymx is again depicted in Fig. 27. Importantly, the discussion given above holds for a large class of possible rate functions—making the interpretation of the saturation matrix independent of a specific functional form. We note that for any rate equation that is consistent with the generic form given in Eq. (47), we can specify an interval for the saturation parameter. Specifically, for an irreversible rate equation of the form nðS; kÞ ¼ kv Sna =Fni ðS; kÞ
ð123Þ
where the dependence on other reactants than S is absorbed into the parameters kv and k, and with Fni ðS; kÞ denotes a polynomial of order ni in S with positive coefficients, the saturation parameter is confined to the interval ymx ¼ na
ani ;
where
a 2 ½0; 1-
ð124Þ
194
ralf steuer and bjo¨rn h. junker 1 µ
θµx
rate ν(S) / V
M
0.8 0.6
0.1 < θ < 0.9 x
> 0.9
θµ < 0.1 x
linear regime
0.4
high saturation
0.2 0 −2 10
−1
10
0
10
1
10
2
10
relative substrate concentration S0/KM Figure 27. Interpretation of the saturation parameter. Shown is a Michaelis–Menten rate equation (solid line) and the corresponding saturation parameter ymx (dashed line). For small substrate concentration S0 ( Km the reaction acts in the linear regime. For increasing concentrations the saturation parameter ymx is monotonously decreasing. For very large concentrations S0 * Km , we obtain ymx ) 0.
The limiting cases are limS0i !0 a ¼ 1 and limS0i !1 a ¼ 0. To evaluate the saturation matrix hlx , we restrict each element to a well-defined interval, specified in the following way: As for most biochemical rate laws na ¼ ni ¼ 1, the saturation parameter of substrates usually takes a value between zero and unity that determines the degree of saturation of the respective reaction. In the case of cooperative behavior with a Hill coefficient n ¼ na ¼ ni 3 1, the saturation parameter is restricted to the interval ½0; n- and, analogously, to the interval ½0; n- for inhibitory interaction with na ¼ 0 and n ¼ ni 3 1. Note that the sigmoidality of the rate equation is not specifically taken into account, rather the intervals for hyperbolic and sigmoidal functions overlap. Once the elements of the matrix hlx are specified, the Jacobian matrix of the metabolic network can be evaluated. A more detailed discussion, including a thermodynamically consistent parameterization, is given in Section VIII.E. 3.
An Alternative Derivation and the Relationship with MCA
To highlight the relationship of the matrices L and hlx to the quantities discussed in Section VII.A (Dynamics of Metabolic Systems) and Section VII.B (Metabolic Control Analysis), we briefly outline an alternative approach to the parameterization of the Jacobian matrix. Note the correspondence between the saturation parameter and the scaled elasticity: ymx
& & & qm&& S0 qnðSÞ&& q ln nðSÞ&& ¼ ¼E ¼ & ¼ qx x0 ¼1 n0 qS &S0 q ln S &S0
ð125Þ
Using an alternative definition of the saturation matrix, in analogy to the normalization described in Section VII.B.2
computational models of metabolism hlx ¼ Dm01
& qm && D0 qS&S0 S
the Jacobian matrix can be written as & qm && M ¼ N & ¼ NDm0 hlx DS01 qS S0
195 ð126Þ
ð127Þ
Using a similarity transform
M ! M x ¼ DS01 MDS0
ð128Þ
we recover the original definition Eq. (118) M x ¼ Lhlx
where
L ¼ DS01 NDm0
ð129Þ
Note that the discussion above assumes the absence of mass conservation relationships. Taking into account the link matrix L, the reduced Jacobian matrix M 0 in terms of the saturation matrix is M 0 ¼ N 0 Dm0 hlx DS01 L
ð130Þ
Equation (130) is often useful for a straightforward numerical implementation of a metabolic system. Explicit examples of the parameter matrices are given in Section VIII.C. Despite the obvious correspondence between scaled elasticities and saturation parameters, significant differences arise in the interpretation of these quantities. Within MCA, the elasticities are derived from specific rate functions and measure the local sensitivity with respect to substrate concentrations [96]. Within the approach considered here, the saturation parameters, hence the scaled elasticities, are bona fide parameters of the system—without recourse to any specific functional form of the rate equations. Likewise, SKM makes no distinction between scaled elasticities and the kinetic exponents within the power-law formalism. In fact, the power-law formalism can be regarded as the simplest possible way to specify a set of explicit nonlinear functions that is consistent with a given Jacobian. Nonetheless, SKM seeks to provide an evaluation of parametric representation directly, without going the loop way via auxiliary ad hoc functions. Closely related to the approach considered here are the formal frameworks of Feinberg and Clarke, briefly mentioned in Section II.A. Though mainly devised for conventional chemical kinetics, both, Chemical Reaction Network Theory (CRNT), developed by M. Feinberg and co-workers [79,80], as well as Stoichiometric Network Analysis (SNA), developed by B. L. Clarke [81–83], seek to relate aspects of reaction network topology to the possibility of various
196
ralf steuer and bjo¨rn h. junker
kinds of unstable dynamics [326]. However, within the present work, we do not seek to derive precise algebraic relations to characterize the stability of a given reaction network. Rather, as will be detailed below, we are interested in simple probabilistic statements that associate given metabolic states with specific dynamic behavior. B.
Rewriting the System: A Simple Example
To illustrate the parameterization and the workflow of SKM more clearly, we briefly consider a simple example. The simplest possible metabolic network consists of r ¼ 2 reactions and m ¼ 1 metabolite v2 ðSÞ
v1
ð131Þ
!S !
with a stoichiometric matrix N ¼ ½ 1 rate equations mðSÞ ¼
1 - and the (as yet unspecified) vector of (
n1 n2 ðSÞ
)
ð132Þ
A possible explicit differential equation that describes the dynamics of the system is dS ¼ N 1 mðSÞ ¼ c dt
Vmax S Km þ S
ð133Þ
with n1 ¼ c denoting a constant influx, Vmax the maximal reaction velocity and Km a Michaelis constant. Using the explicit equation, the pathway is characterized by the three kinetic parameters c, Vmax , Km . Once the parameters and an initial condition Sð0Þ are specified, the pathway can be integrated numerically to obtain the time-dependent behavior of the concentration SðtÞ. In contrast, SKM does not assume knowledge of thespecific functional form of the rate equations. Rather, the system is evaluated in terms of generalized parameters, specified by the elements of the matrices L and hmx . In this sense, the matrices L and hmx are bona fide parameters of the system: The pathway is described in terms ofan average metabolite concentration S0 , and a steady-state flux vector n0, together defining the metabolic state of the pathway. Additionally, we assume that the substrate only affects reaction n2 , the saturation matrix is thus fully specified by a single parameter ynS2 2 ½0; 1-. Note that the number of parameters is identical to the number used within the explicit equation. The structure of the parameter matrices is ( ) ( ) 0 n0 n0 l L¼ ; hx ¼ n 2 ð134Þ 0 0 y S S S
computational models of metabolism
197
Given the generalized parameter matrices, the Jacobian is specified according to Eq. (118): Mx ¼ Lhlx ¼
n0 n2 y S0 S
ð135Þ
Once the parametric representation of the Jacobian is obtained, the possible dynamics of the system can be evaluated. As detailed in Sections VII.A and VII.B, the Jacobian matrix and its associated eigenvalues define the response of the system to (small) perturbations, possible transitions to instability, as well as the existence of (at least transient) oscillatory dynamics. Moreover, by taking bifurcations of higher codimension into account, the existence of complex dynamics can be predicted. See Refs. [293, 299] for a more detailed discussion. We highlight several advantages of the generalized parameters, as compared to the more usual description in terms of Michaelis constants and maximal reaction velocities: 0 The generalized parameters are often intuitively accessible: Even without detailed knowledge of the kinetic mechanisms it is feasible to define a physiologically plausible range for each parameter. That is, it is possible to specify intervals S0 2 ½Smin ; Smax -, n0 2 ½nmin ; nmax -, and ynS2 2 ½0; 1- that define an physiologically admissible parameter space of the system. 0 The generalized parameters can be straightforwardly converted back into to the original kinetic parameters. Note that while this transformation is usually straightforward and almost always has a unique solution, the opposite does not hold: The estimation of the metabolic state from explicit kinetic parameters is often computationally demanding and must not give rise to a unique solution. 0 In terms of the generalized parameter matrices, the Jacobian is given as product of a simple matrix multiplication. Using explicit kinetic parameters, the estimation of the Jacobian can be tedious and computationally demanding, prohibiting the analysis of large ensembles of models. 0 The generalized parameters are invariant with respect to different functional forms of the rate equation. All results hold for a large class of biochemical rate functions [84]. For example, the Michaelis–Menten rate function used in Eq. (133) is not the only possible choice. A number of alternative rate equations are summarized in Table VI. Although in each case the specific kinetic parameters may differ, each rate equation is able to generate a specified partial derivative and is thereby consistent with results obtained from an analysis of the corresponding Jacobian. Note that, obviously, not each rate equation is capable to generate each possible Jacobian. However, vice versa, for each possible Jacobian there exists a class of rate equations that is consistent with the Jacobian.
198
ralf steuer and bjo¨rn h. junker TABLE VI Alternative Rate Equations that may be Assigned to the Pathway Specfied in Eq. (133)a
Rate Equation nðSÞ ¼
Saturation Parameter
Vmax S Km þ S
ymx ¼
1 0 1 þ KSm
Vmax Sn nðSÞ ¼ n Ks þ Sn
ymx
nðSÞ ¼ aSgS
ymx ¼ gS
. / nðSÞ ¼ n0 1 þ E ln SS0
ymx ¼ E
1 nðSÞ ¼ b 1
e
¼n 1þ
S K
2
ymx ¼
Km ¼ S0
1 1 2n S0 Km
0
Ks ¼ S
1 *
ymx ymx
+1 ymx n n ymx
gS ¼ ymx E ¼ ymx
S0 K
e
Kinetic Parameter
S0 K
— 1
a An analysis is terms of the normalized partial derivative (saturation parameter) is invariant with respect to the specific rate equation. Note that not all choices are necessarily equally plausible.
C.
Detecting Dynamics and Bifurcations: Glycolysis Revisited
Prior to an application to more detailed biochemical networks, we provide a brief example using the minimal model of the glycolytic pathway depicted in Fig. 5. As discussed in Section VII.A.4, the minimal model already gives rise to a variety of dynamic regimes, including multistability and oscillations. In contrast to the explicit analysis given in Section VII.A.4, we seek to evaluate the model without assuming knowledge of the detailed functional form of the rate equations. The system is represented by the set of differential equations given in Eq. (78) 2 3 ) ( ) ( n1 ðATPÞ d TP 2 1 0 1 4 n2 ðTP; ADPÞ 5 ¼ ð136Þ 2 þ2 1 dt ATP n3 ðATPÞ Starting with an evaluation of the stoichiometric matrix, we obtain the null space matrix K and the link matrix L, 2 3 1 0 ð137Þ K ¼ ½ 1 2 2 -T and L ¼ 4 0 1 5 0 1 The metabolic state of the minimal model is specified by one independent flux value n0 and 3 steady-state metabolite concentrations, 2 0 3 2 3 1 0 0 TP 0 0 ð138Þ Dm0 ¼ n0 4 0 2 0 5 and DS0 ¼ 4 0 0 5 ATP0 0 0 2 0 0 ADP0
computational models of metabolism
199
To specify the matrix hlx , we take into account the minimal model discussed in Section VII.A.4: The first reaction n1 ðATPÞ, including the lumped PFK reaction, depends on ATP only (with glucose assumed to constant). The cofactor ATP may activate, as well as inhibit, the rate (substrate inhibition). To specify the interval of the corresponding saturation parameter, we use Eq. (79) as a proxy and obtain n1 ðATPÞ ¼
VM ATP h in 1 þ ATP KI
)
ymx ¼ 1
an
ð139Þ
with
a :¼
h
ATP0 KI
1þ
h
in
ATP0 KI
in 2 ½0; 1-
ð140Þ
The overall influence of ATP on the rate n1 ðATPÞ is measured by a saturation parameter x 2 ð 1; 1-. Note that, when using Eq. (139) as an explicit rate equation, the saturation parameter implicitly specifies a minimal Hill coefficient nmin > x necessary to allow for the reverse transformation of the parameters. The interval x 2 ½0; 1- corresponds to conventional Michaelis–Menten kinetics. For x ¼ 0, ATP has no net influence on the reactions, either due to complete saturation of a Michaelis–Menten term or, equivalently, due to an exact compensation of the activation by ATP as a substrate by its simultaneous effect as an inhibitor. For x < 0, the inhibition by ATP supersedes the activation of the reaction by its substrate ATP. The parameterization of the remaining reactions is less complicated. For simplicity, the rate n2 ðTP; ADPÞ is assumed to follow mass-action kinetics, giving rise to saturation parameters equal to one. Finally, the ATPase represents the overall ATP consumption within the cell and is modeled with a simple Michaelis–Menten equation, corresponding to a saturation parameter y 2 ½0; 1-. The saturation matrix is thus specified by four nonzero entries: 2
0 x hlx ¼ 4 1 0 0 y
3 0 15 0
ð141Þ
We emphasize that, although the discussion is largely based on the explicit equations given in Section VII.A.4, the saturation matrix does not presuppose a specific functional form of the rate equations.
200
ralf steuer and bjo¨rn h. junker
Finally, using matrix notation and accounting for the mass-conservation relationship ATP þ ADP ¼ AT , the Jacobian is given as a product of the parameter matrices. M 0 ¼ N 0 Dm0 hlx DS01 L
ð142Þ
Specifically, we obtain M 0 ¼ n0
(
2 2
2 þ4
2 ) 0 0 4 1 2 0
x 0 y
32 0 ðTP0 Þ 1 54 0 0 0
1
0
3
ðATP0 Þ 1 5 ðADP0 Þ 1
as a parametric representation of the 2 4 2 Jacobian matrix. The Jacobian is specified by six parameters, corresponding to the number of parameters in the explicit model. 1.
Evaluating the Dynamics
We are interested in a brief description of the dynamics, in particular, to recover the results obtained in Section VII.A.4. Without any restrictions on the generality, the parameters n0 ¼ 1 and TP0 ¼ 1 are assumed to be unity, that is, time and concentrations are measured in arbitrary units. Utilizing the eigenvalues of the Jacobian, we classify the local dynamic behavior into different dynamic regimes. To this end, we evaluate the maximal real part lmax within the spectrum of eigenvalues, determining the stability of < the metabolic state; see also the diagram depicted in Fig. 20. Assuming a fixed metabolic state n0 ¼ 1, TP0 ¼ 1, ATP0 ¼ 0:5, and AT ¼ 1, Fig. 28 shows the largest real part of the eigenvalues lmax as a function of the feedback strength < x. The parameter space is subdivided into different dynamic regimes: (i) In the interval x 2 ½0; 1- the metabolic state loses its stability via a saddle-node bifurcation. Note that this interval corresponds to an absence (or only weak) inhibitory influence of ATP on the first reaction. (ii) For moderate negative feedback, the metabolic state is stable. In particular, for a certain negative feedback strength x < 0, the maximal real part lmax < of the eigenvalues exhibits a minimum—corresponding to an optimally fast response to perturbations. (iii) For increasing negative feedback, damped oscillations occur. The eigenvalues with largest real parts have nonzero (complex conjugate) imaginary parts. (iv) If the feedback strength x is increased further, the metabolic state loses stability via a Hopf bifurcation, indicating the presence of sustained oscillations. Figure 29 shows the bifurcation diagram for different values of the saturation parameter y of the ATPase reaction. Qualitatively, the plot shows the same
201
computational models of metabolism
and λ
max ℑ
10
max ℜ
eigenvalues λ
max ℑ
λ
5
max ℜ
λ
0 HO
SN −5 (ii)
(i) −10 1
(iii)
(iv)
0 −1 −2 −3 Influence ξ of ATP on ν ∈ [1,−∞)
−4
1
Figure 28. The eigenvalues of the Jacobian of minimal glycolysis as a function of the influence x of ATP on the first reaction n1 ðATPÞ (feedback strength). Shown is the largest real part of the eigenvalues (solid line), along with the corresponding imaginary part (dashed line). Different dynamic regimes are separated by vertical dashed lines, for lmax < > 0 the state is unstable. Transitions occur via a saddle-node (SN) and a Hopf (HO) bifurcation. Parameters are n0 ¼ 1, TP0 ¼ 1, ATP0 ¼ 0:5, AT ¼ 1, and y ¼ 0:8. See color insert.
saturation θ of ATPase
1
0.8 HO
SN 0.6
0.4
0.2
(i) 0 1
(ii) 0
(iii) −1
−2
(iv) −3
−4
Influence ξ of ATP on ν1 ∈ [1,−∞) Figure 29 Bifurcation diagram of the minimal model of glycolysis as a function of feedback strength x and saturation y of the ATPase reaction. Shown are the transitions to instability via a saddle-node (SN) and a Hopf (H0) bifurcation (solid lines). In the regions (i) and (iv), the largest real part with in the spectrum of eigenvalues is positive lmax > 0. Within region (ii), the < metabolic state is a stable node, within region (iii) a stable focus, corresponding to damped transient oscillations.
202
ralf steuer and bjo¨rn h. junker
dynamic regimes as depicted in Fig. 28. The region of instability increases with increasing saturation. A more detailed physiological interpretation of the different dynamic regimes is given in the next section. D. Yeast Glycolysis: A Monte Carlo Approach Going beyond the minimal model, we now consider a slightly more elaborate model of the yeast glycolytic pathway. As already noted in Section VII.A glycolysis is probably the most thoroughly studied pathways in biology, with a large variety of detailed metabolic models available [94, 101, 121, 126, 308, 310]. The earliest computer-based simulations of glycolysis were already published in the 1960s by Chance et al. [104], using a rather complex representation of glycolysis and oxidative phosphorylation to examine the Crabtree effect in aerobic mouse ascites cells. The model was later extended by Garfinkel and Hess [105] to encompass no less than 89 reactions among 69 metabolic intermediates. In the following, we focus on a medium-complexity representation of glycolysis, roughly corresponding to the model of Wolf et al. [126]. The discussion follows the analysis given in Ref. [84], a schematic depiction of the pathway is shown in Fig. 30. In particular, we seek to address two characteristic questions associated with the evaluation of many explicit models of yeast glycolysis, as well as show that these can be readily answered using the concept of structural kinetic modeling: First, we seek to elucidate whether the proposed reaction mechanism indeed facilitates sustained oscillations for the experimentally observed concentration and flux values. And, if yes, what are the kinetic conditions and requirements under which such sustained oscillations can be expected. Second, we seek to elucidate the possible functional role of the oscillations, or rather, elucidate the functional role of the mechanisms that give rise to sustained oscillations.
Figure 30. A medium-complexity model of yeast glycolysis [342]. The model consists of nine metabolites and nine reactions. The main regulatory step is the phosphofructokinase (PFK), combined with the hexokinase (HK) reaction into a single reaction n1 . As in the minimal model, we only consider the inhibition by its substrate ATP, although PFK is known to have several effectors. External glucose (Glcx ) and ethanol (EtOH) are assumed to be constant. Additional abbreviations: Glucose (Glc), fructose-1,6-biphosphate (FBP), pool of triosephosphates (TP), 1,3-biphosphoglycerate (BPG), and the pool of pyruvate and acetaldehyde (Pyr).
203
computational models of metabolism 1.
Defining the Structural Kinetic Model
Following the workflow depicted in Fig. 26, we start with an analysis of the stoichiometric matrix. The model consists of m ¼ 9 (internal) metabolites and r ¼ 9 reactions, interconnected according to the stoichiometry specified below: n0
n1
n2
n3
n4
n5
n6
n7
n8
Glc FBP TP BPG Pyr ATP NADH
þ1 0 0 0 0 0 0
1 þ1 0 0 0 2 0
0 1 þ2 0 0 0 0
0 0 1 þ1 0 0 þ1
0 0 0 1 þ1 þ2 0
0 0 0 0 1 0 1
0 0 0 0 1 0 0
0 0 1 0 0 0 1
0 0 0 0 0 1 0
NADþ ADP
0 0
0 þ2
0 0
1 0
0 2
þ1 0
0 0
þ1 0
0 þ1
The rank of the stoichiometric matrix is rankðNÞ ¼ 7, corresponding to two mass conservation relationships, namely, ATP þ ADP ¼ AT
and
NADþ þ NADH ¼ NT
ð143Þ
All feasible steady-state flux vectors mðS0 Þ are described by two basis vectors ki : 0 1 0 1 1 1 B1C B1C B C B C B1C B1C B C B C C B B2C 1 2 X C B B C Cc1 þ B 2 Cc2 ki ci ¼ B 1 mðS0 Þ ¼ B C B C B0C B2C i¼1 B C B C B1C B0C B C B C @1A @0A 0 2
ð144Þ
To evaluate model, we focus on the experimentally observed metabolic state of the pathway. However, in the case of sustained oscillations, the (unstable) steady state cannot be observed directly. We thus approximate the metabolic state by the average observed concentration and flux values, as reported in Refs. [101, 126]. See Table VII for numeric values. Note that the approximation of the unstable state by the average concentrations is justified by the fact that in most cases the actual unstable state is reasonably close
204
ralf steuer and bjo¨rn h. junker TABLE VII The Metabolic State at which the System is Evaluateda
FBP
TP
BPG
Pyr
ATP
NADH
NAD
ADP
5.1
0.12
0.0001
1.48
2.1
0.33
0.67
1.9
a
The concentrations correspond to the average values reported in Refs. 101, 126, with flux values c1 ¼ 20 mM min 1 and c2 ¼ 30 mM min 1 . Glucose (Glc) is assumed to be constant, all concentrations are reported in units of [mM].
to the average values. It can be explicitly ascertained that the result does not depend crucially on the exact knowledge of the metabolic state by repeating the analysis in the vicinity of the observed (average) state, see Ref. [84] for details. In general, it is recommended not to focus on one specific state only, but to include variability into the evaluation of the system. As the second step, the matrix of saturation parameters hlx has to be specified. For simplicity, and following the model of Wolf et al. [126], all reactions are assumed to be irreversible and dependent on their substrates only. The matrix hlx is then specified by 12 free parameters: 0
0
B yr2 B FBP B 0 B B 0 B l hx ¼ B 0 B B B 0 B @ 0 0
0 0 yr3 TP 0 0 0 yr7 TP 0
0 0 0 yr4 BPG 0 0 0 0
0 0 0 0 yr5 Pyr yr6 Pyr 0 0
yr1 ATP 0 0 0 0 0 0 yr8 ATP
0 0 0 0 yr5 NADH 0 yr7 NADH 0
0 0 yr3 NAD 0 0 0 0 0
0 0 0
1
C C C C r4 C yADP C 0 C C C 0 C C 0 A 0
The dependence yr1 ATP of n1 on ATP is modeled as in the previous section, using an interval yr1 2 ½ 1; 1- that reflects the dual role of the cofactor ATP as ATP substrate and as inhibitor of the reaction. All other reactions are assumed to follow Michaelis–Menten kinetics with yrS 2 ½0; 1-. No further assumption about the detailed functional form of the rate equations is necessary. Given the stoichiometry, the metabolic state and the matrix of saturation parameter, the structural kinetic model is fully defined. An explicit implementation of the model is provided in Ref. [84]. 2.
An Analysis of the Parameter Space
Evaluating the structural kinetic model, we first consider the possibility of sustained oscillations. Starting with the simplest scenario, all saturation parameters are set to unity, corresponding to bilinear mass-action kinetics and
205
computational models of metabolism
max
20
and λI
max
λR
(d)
10
(a)
0
(c)
−10
(b)
−20 1
0.5
0
−0.5
−1
−1.5
v1 ATP
negative feedback θ
ATP [mM]
(A)
v1
θATP=0.8
(B)
(C)
θv1 =−0.06 ATP
(D)
θv1 =−0.7 ATP
3
3
3
2
2
2
2
1
1
1
1
3
0 0
10 20 time [min]
30
0 0
10 20 time [min]
30
0 0
5 time [min]
v1
θATP=−1.0
0 0
5 time [min]
Figure 31. Dynamics of glycolysis. Upper panel: The eigenvalue with the largest real part lmax as a function of the feedback strength yr1 < ATP of ATP on the combined PFK–HK reaction. All other saturation parameters are unity ymx ¼ 1. Shown is lmax (solid line) together with the imaginary < part lmax (dashed line). At the Hopf bifurcation, a complex conjugate pair of eigenvalues I max lmax < / il= crosses the imaginary axis. Note the similarity to Fig. 28 (minimal glycolysis). Lower panel: Upon variation of yr1 ATP , four dynamic regimes can be distinguished. Shown are the corresponding time courses of ATP using an explicit kinetic model at the points (a, b, c, d) indicated in the plot. (a) A small negative real part lmax < , corresponding to slow relaxation to the stable steady state (yv1 ATP ¼ 0:8). (b) An optimal response to perturbations, as determined by a minimal largest eigenvalue lmax (yv1 R ATP ¼ 0:06). (c) Oscillatory return to the stable steady state. The metabolic state 0:7). (d) Sustained oscillations is stable, but with nonzero imaginary eigenvalues (yv1 ATP ¼ yv1 ATP ¼ 1:0. All different regimes can be deduced solely from the Jacobian and are only exemplified using the explicit kinetic model. See color insert.
already investigated in Refs. [84] and [126]. However, note that the inhibition term yr1 ATP still corresponds to an unspecified nonlinear saturable function. Figure 31 shows the largest eigenvalue of the Jacobian at the experimentally observed metabolic state as a function of the parameter yr1 ATP. Similar to Fig. 28 obtained for the minimal model, several dynamic regimes can be distinguished. In particular, for sufficient strength of the inhibition parameter, the system undergoes a Hopf bifurcation and the pathway indeed facilitates sustained oscillations at the observed state. Important for our analysis, the different regimes shown in Fig. 28 are detected solely based on knowledge of the eigenvalues of the Jacobian matrix.
206
ralf steuer and bjo¨rn h. junker
(A) 15 max R
eigenvalue λ
v8 θATP=0.1
5 0 −5
θv8 =0.5
−10
θATP=1.0
ATP v8
−15 −20
1
0.5 0 −0.5 −1 v1 negative feedback θ ATP
−1.5
saturation θv8 ATP
(B)
10
1 max
λR
0.8 0.6
SN
<0
HO
0.4 max
0.2 λR 0 1
>0
λmax >0 R
0.5 0 −0.5 −1 v1 negative feedback θ
−1.5
ATP
Figure 32. Bifurcation diagram of the medium-complexity model of glycolysis, analogous to Fig. 29. A: The largest real part of the eigenvalues as a function of the feedback strength yv1 ATP , depicted for increasing saturation of the overall ATPase reaction. B: The metabolic state is stable only for an intermediate value of the feedback parameter. For increasing saturation of the ATPase reaction, the stable region decreases.
Nonetheless, each regime corresponds to distinctive dynamic behavior, exemplified in Fig. 31 using an explicit kinetic model of the pathway. Corresponding to Fig. 29, a bifurcation diagram with the saturation yv8 ATP of the ATPase as an additional parameter is shown in Fig. 32. We encounter similar features as for the minimal model considered above: The stable region is confined to an intermediate value of the feedback strength yv1 ATP . In particular, with an unregulated PFK–HK reaction n1 , corresponding to an interval yv1 ATP 2 ½0; 1-, the system may lose its stability via a saddle-node bifurcation. For increasing saturation of the overall ATPase reaction, the stable region decreases significantly. We highlight one specific feature of the analysis presented here: Within SKM, the impact of the inhibition is decoupled from the steady-state concentrations and flux values the system adopts. Consequently, it is specifically evaluated whether an assumed inhibition or interaction is indeed a necessary condition for the observation of oscillations at the experimentally observed metabolic state. In contrast to this, using an explicit kinetic model and reducing the influence of a regulatory interaction, for example, by increasing the corresponding Michaelis constant, would concomitantly result in altered steadystate concentrations, thus not straightforwardly addressing the question. 3.
Sampling the Parameters
Adopting the more general approach, we now aim for an evaluation of the pathway with respect to all possible explicit kinetic models that comply with the experimentally observed metabolic state. To this end, rather than constructing a single model, we generate an ensemble of models that is consistent with a given metabolic state [84]. Using a straightforward Monte Carlo approach, the ensemble is obtained by sampling all saturation parameters ymS 2 ½0; 1- randomly from their predefined intervals. Subsequently, the Jacobian matrix is evaluated with respect
207
computational models of metabolism (B) percentage unstable models η
maximal eigenvalue λ R
(A) 60 40 20
λmax>0 R
0 λmax<0 −20 1
R
0.5
0
negative
−0.5
−1
feedback θv1 ATP
−1.5
1
0.8 0.6
SN
HO
0.4 0.2 0 1
0.5
0
−0.5
−1
−1.5
negative feedback θv1
ATP
Figure 33. The stability of yeast glycolysis: A Monte Carlo approach. A: Shown in the distribution of the largest positive real part within the spectrum of eigenvalues, depicted from above (contour plot). Darker colors correspond to an increased density of eigenvalues. Instances with lmax > 0 are unstable. B: The probability that a random instance of the Jacobian corresponds to an < unstable metabolic state as a function of the feedback strength yv1 ATP . The loss of stability occurs either via in a saddle-node (SN) or via a Hopf (HO) bifurcation.
to the elements of the matrix hlx , defining the spectrum or scope of dynamic behavior at the respective metabolic state. In this way, we obtain an unbiased picture of the possible dynamics at the metabolic state and are able to evaluate and compare the dynamic behavior under different preconditions, such as a varying strength of feedback inhibition. The workflow is summarized in Fig. 26. Applying the concept on the medium complexity model of glycolysis, random realizations of the Jacobian matrix are iteratively generated and the largest real part lmax of the eigenvalues is recorded for each realization. Figure 33 shows the R histogram of the largest real part within the spectrum of eigenvalues, with lmax >0 < implying instability. Note that in the absence of the inhibitory feedback yv1 , ATP the metabolic state is likely to be unstable, that is, most random realizations result in a Jacobian with at least one positive real part within its spectrum of eigenvalues. Increasing the feedback strength will result in a decreased probability of unstable models, until a minimal percentage of unstable models is reached. However, if the negative feedback is increased further, almost all models again undergo a Hopf bifurcation, with the concomitant loss of stability of the metabolic state. Extending the analysis, Fig. 34 takes the saturation of the ATPase as an additional parameter into account. Similar to the bilinear case considered above, increasing saturation increases the probability of unstable models. Note that in each case, a minimal percentage of unstable models (maximal stability) is obtained if ATP has no net influence on the combined PFK–HK reaction. 4.
The Possible Function of Glycolytic Oscillations
A prominent puzzle related to glycolytic oscillations in yeast is the question of their physiological significance. A number of diverse hypothesis have been
208
ralf steuer and bjo¨rn h. junker
0.8
0.8
saturation θv8
saturation θv8
η=0
0.6
η>0.99
0.4
0< η ≤ 0.99
0.2 0 1
ATP
(B) 1
ATP
(A) 1
0.5
0 −0.5 v1 −1 negative feedback θATP
−1.5
median(λmax) ≤ 0 R
) >0 median(λmax R
0.6 0.4 0.2 0 1
0.5
0 −0.5 v1 −1 negative feedback θATP
−1.5
Figure 34. Same as Fig. 33, but with the saturation of the ATPase as an additional parameter. A: Shown is the percentage Z of unstable models among the random realizations of the Jacobian. B: The shaded area indicated whether the median of the largest positive real part lmax > 0 is above or < below zero. Intriguingly, in both plots, a minimal percentage of unstable models (maximal stability) is obtained if ATP has no net influence on the combined PFK–HK reaction.
proposed, ranging from an ancient circadian oscillator (as the period can be rather variable) to an alleged increased yield for oscillatory dynamics. However, no conclusive results have been obtained. Considering the dynamic behavior of the pathway, as obtained from the ensemble of models, a more straightforward solution seems plausible. First, we note that the dynamic behavior shown in Fig. 31 is generic for almost any negative feedback mechanism: Weak feedback results in a small impact on the dynamics, corresponding to point (a) in Fig. 31. As the strength of the negative feedback is increased, the response time of the system becomes faster, the largest real part of the eigenvalue decreases [327]. The decrease of lmax upon increasing feedback strength will continue, until an < optimal fastest response time is attained; see point (b) in Fig. 31. At this point, the system overshoots. That is, in antagonizing the original perturbation and forcing the system to return to its steady state, the negative feedback results in an (albeit smaller) perturbation in the opposite direction. The pathway exhibits damped oscillations and the eigenvalue lmax < , as well as the response time, increases again, see point (c) in Fig. 31. Finally, upon a still increasing negative feedback, perturbations are no longer damped and the system exhibits sustained oscillations, corresponding to point (d) in Fig. 31. The negative feedback in glycolysis, induced by substrate inhibition of lumped PFK–HK reaction, thus fulfills an important functional role but concomitantly opens the possibility of sustained oscillations. In particular, because glycolytic oscillations have no obvious physiological role and are only observed under rather specific experimental conditions, it is plausible that they are merely an unavoidable side effect of regulatory interactions that are optimized for other purposes. The latter becomes even more important upon the realization that the glycolytic pathway has an inherent potential for instability. Coined with the term ‘‘turbo design’’ [311], two initial ATP-consuming steps are followed by
209
computational models of metabolism
ATP-generation steps, resulting in a net yield of ATP. This autocatalytic structure is not without danger for the functioning of the pathway. As has been demonstrated, in the absence of a safeguard mechanism (that is realized by a negative feedback), the turbo design may have lethal effects [311, 328]. In this sense, and closely related to the situation shown in Fig. 33, the negative feedback does not just speed up the response but is a necessary element of an otherwise unstable pathway, albeit with the potential to induce failure outside a (evolutionary optimized) region of parameters. This finding holds some possible implications for biotechnological modification of metabolic networks, as is discussed in Section IX. E.
Thermodynamics and Reversible Rate Equations
Up to now, the analysis was restricted to simplified rate equations, making use of several approximating assumptions, such as irreversible kinetics. Envisioning a path toward genome-scale kinetic models, we provide a more systematic parameterization of the Jacobian. In particular, we highlight several structural features of the Jacobian matrix and distinguish between kinetic, thermodynamic, and regulatory contributions [44]. Furthermore, several ambiguities and pitfalls in the parameterization are pointed out. The starting point is the generic reaction equation (47), already discussed in Section III.C.5. The equation is rewritten using the Haldane relation Vm nðS; P; IÞ ¼ hðI; K I Þ 1
1Q
Q
i Si
Pj j Keq
FðS; P; K m Þ
2
ð145Þ
and augmented with a multiplicative function hðI; K I Þ that accounts for (activating or inhibiting) allosteric effectors, depending on concentrations I and kinetic constants K I . We distinguish between forward and backward rates, such that nðS; P; IÞ ¼ hðI; K I Þ 1 ½nþ ðS; PÞ
n ðS; PÞ-
ð146Þ
with Q Vm i Si n ðS; PÞ ¼ FðS; P; K m Þ þ
and n ðS; PÞ ¼
Vm Keq
Q
j
Pj
FðS; P; K m Þ
ð147Þ
Note that the ratio Q Pj n 1 ! Qj ¼ g :¼ þ ¼ Keq i Si Keq n
ð148Þ
210
ralf steuer and bjo¨rn h. junker
only depends on thermodynamic properties. Consequently, the rate law specified in Eq. (145) can be written as a product of three basic contributions ngeneric ðS; P; IÞ ¼ hðI; K I Þ 1 nþ ðS; PÞ 1 fth ðS; PÞ
ð149Þ
with fth ðS; PÞ :¼ ð1 gÞ denoting the displacement from thermodynamic equilibrium. Aiming at a systematic parameterization of the Jacobian, we are interested in the partial derivatives of the rate equations & & qm && q ln m && ¼ Dm0 D 01 qS&S0 q ln S&S0 S
ð150Þ
where, for brevity, the vector S is assumed to include substrates, products, and allosteric effectors. Evaluating the logarithmic derivative, we obtain a sum of contributions & & & & q ln n && q ln h&& q ln nþ && q ln fth && ¼ þ þ q ln S&S0 q ln S&S0 q ln S &S0 q ln S &S0
ð151Þ
Using matrix notation and recalling the definition of the Jacobian M 0 given in Eq. (130), the sum of partial derivatives straightforwardly translates into an additive relationship for the Jacobian matrix M 0 ¼ M 0reg þ M 0kin þ M 0th
ð152Þ
For any arbitrary metabolic network, the Jacobian matrix can be decomposed into a sum of three fundamental contributions: A term M 0reg that relates to allosteric regulation. A term M 0kin that relates to the kinetic properties of the network, as specified by the dissociation and Michaelis–Menten parameters. And, finally, a term M 0th that relates to the displacement from thermodynamic equilibrium. We briefly evaluate each contribution separately. 1.
The Contribution from Allosteric Regulation
In Eq. (149) we assume that allosteric regulation affects the reaction rates as a multiplicative factor hðIÞ. Following Ref. [161], a generic functional form for an inhibitory effector is hI ðIÞ ¼ 1þ
1 1 2n ½IKI
ð153Þ
similar to the terms used in the models of glycolysis considered above. Correspondingly, activation is modeled by a Hill-type prefactor
computational models of metabolism
hA ðAÞ ¼
1 2n ½AKA
1þ
1 2n ½AKA
* +n ) ½Aor hA ðAÞ ¼ 1 þ KA (
211
ð154Þ
It is straightforward to verify that both cases result in a well-defined interval for the (normalized) partial derivatives, namely, & q ln hI && 2 ½0; n- and q ln I &I 0
& q ln hA && 2 ½0; nq ln A &A0
ð155Þ
for inhibition and activation, respectively. Note that in practice, the assumption of multiplicative prefactors is not necessarily restrictive. We may straightforwardly also include competitive (inhibitory) terms in the kinetic contribution, without affecting the validity of the analysis. To account for the contribution from regulatory interactions, we thus use the equations above as a proxy and utilize Eq. (155) to specify the intervals of the respective saturation parameters. 2.
The Contribution from Kinetics
In the previous section, we have already made use of the parameterization of the forward reaction Q V m i Si nþ ðS; PÞ ¼ ð156Þ FðS; P; K m Þ where FðS; P; K m Þ denotes a polynomial with positive coefficients. Following the discussion in Section VIII.A.2, the partial derivative with respect to a substrate concentration S is & q ln nþ && ¼1 q ln S &S0
& S0 qF && F 0 qS &S0
ð157Þ
Assuming a functional form of F ¼ 1 þ aS þ b, with a and b as auxiliary parameters, the (normalized) partial derivative is confined to the unit interval. Analogously, we evaluate the dependency on a product concentration P and obtain & q ln nþ && ¼ q ln P &P0
& P0 qF && 2 ½0; 1F 0 qP&P0
ð158Þ
Note that the arguments can be straightforwardly extended to include arbitrary polynomials with positive coefficients, see Ref. [84] for an explicit proof.
212
ralf steuer and bjo¨rn h. junker
Furthermore, we can account for additional competitive inhibition by metabolites I by assuming a polynomial of the form FðS; P; I; K m Þ. In this case, the intervals for competitive inhibition correspond to the intervals obtained for the products of a reaction. 3.
The Contribution from Thermodynamics
The last term to evaluate is the contribution of the displacement from equilibrium fth ¼ 1 g to the Jacobian matrix. Distinguishing between substrates and products, it is straightforward to verify that & & q ln fth && g q ln fth && g and ð159Þ ¼ ¼ 1 g q ln S &S0 1 g q ln P &P0
Importantly, the contribution from thermodynamics is not restricted to finite interval. With g 2 ½0; 1-, the normalized partial derivative may attain any (absolute) value between zero and infinity. In particular, for reactions close to equilibrium g 7 1, we obtain & & & & & q ln fth && q ln fth && && & lim ¼1 ¼ &lim g!1 q ln S & 0 g!1 q ln P & 0 & S P
ð160Þ
As already discussed in Section VII.B.2, reactions close to equilibrium are dominated by thermodynamics and the kinetic properties have no, or only little, influence on the elements of the Jacobian matrix. Furthermore, thermodynamic properties are, at least in principle, accessible on a large-scale level [329, 330]. In some cases, thermodynamic properties, in conjunction with the measurements of metabolite concentrations described in Section IV, are thus already sufficient to specify some elements of the Jacobian in a quantitative way. 4.
Parameterizing the Jacobian
Given the individual contributions, we are now in a position to obtain a consistent parameterization of the Jacobian matrix. Starting from Eq. (151), each element & & i n0j q ln n& n0 h qnj && & ¼ j yreg þ ykin þ yth ¼ ij ij ij & & 0 0 qSi S0 Si q ln S S0 Si
ð161Þ
of the matrix of (nonnormalized) partial derivatives in Eq. (150) is specified by the following quantities: (i) a net flux n0j through the reaction, (ii) the steady-state concentration S0i , and (iii) three (unknown) saturation parameters corresponding to the regulatory, kinetic, and thermodynamic contributions considered above.
computational models of metabolism
213
Note that each element is weighted by the associated net flux, reactions with low flux correspond to small entries in Jacobian (but may nonetheless be crucial for stability). We emphasize that all quantities that specify the elements of the matrix of partial derivatives are local quantities. The properties of the network only enter in terms of the (net)flux distribution m0 that obeys the flux balance equation Nm0 ¼ 0. That is, reactions ‘‘see’’ other reactions only via the flux distribution. The locality of kinetic properties also allows the straightforward specification of an explicit kinetic model that corresponds to a given Jacobian. 5.
Examples and Pitfalls in the Sampling of the Parameter
The parameterization of the Jacobian is not without pitfalls. In the following, we note some restrictions and guidelines to avoid possible misinterpretations of the results. When parameterizing the Jacobian, obviously, redundancies should be avoided. In particular, if a metabolite affects a reaction as a substrate, as well as an allosteric effector, the interaction should be coded into one parameter, rather than a sum of two parameters. Unfortunately, to detect and avoid higher order redundancies is no easy task and we are not aware of any straightforward solution. Likewise, isozymes may be treated as a single reaction. In fact, within a linear representation, two reactions of identical functional form but with different parameters may always be treated as a single reaction. For example, consider a reaction catalyzed by two enzymes with different parameters. The overall rate is the sum nðSÞ ¼ n1 ðSÞ þ n2 ðSÞ, with n1 ðSÞ ¼
Vm1 S Km1 þ S
and n2 ðSÞ ¼
Vm2 S Km2 þ S
ð162Þ
In this case, the saturation parameter ymS of the overall reaction is simply the m weighted sum of the individual saturation parameters yS i : & q ln n && n0 m n0 m ymS ¼ ð163Þ ¼ 10 yS 1 þ 20 yS 2 2 ½0; 1& q ln S S0 n n P The expression holds for an arbitrary number of reactions. For nðSÞ ¼ ni ðSÞ, we obtain & & & X n0 q ln ni & q ln n&& S0 Xqni && m i & yS ¼ ¼ ¼ ð164Þ 0 q ln S & 0 q ln S&S0 n0 i qS &S0 n S i Note that this expression also provides a conceptual foundation to approximate complex processes, like ATP utilization, by a single reaction—with a single saturation parameter sampled randomly from a specified interval.
214
ralf steuer and bjo¨rn h. junker
Slightly more complex are constraints with respect to the feasible intervals that are induced by interactions between metabolites. Until now, all saturation parameters were chosen independently, using a uniform distribution on a given interval. We emphasize that this choice indeed samples the comprehensive parameter spaces, and for all samples, there exists a system of explicit differential equations that are consistent with the sampled Jacobian. However, obviously, not all rate equations can reproduce all sampled values. In particular, competition between substrates for a single binding site will prohibit certain combinations of saturation values to occur. For example, consider an irreversible monosubstrate reaction with competitive inhibition (see Table II): n¼
VM K½S-m
ð165Þ
1 þ K½I-I þ K½S-m
Estimating the saturation parameter for both reactants, we obtain ynS
¼
1 þ K½I-I 1þ
½IKI
1þ
½IKI ½I½SKI þ K m
þ
½SKm
¼
@
¼
@
1 for ½S- ¼ 0 0 for ½S- ! 1
ð166Þ
and ynI
¼
0 1
for ½I- ¼ 0 for ½I- ! 1
ð167Þ
Obviously, both parameters cover their intervals ynS 2 ½0; 1- and ynI 2 ½0; 1independent of the concentration of the other reactant. However, both values are not independent. Evaluating the absolute values, we obtain & n& ½I& yI & & & ¼ KI < 1 & yn & 1 þ ½IS
ð168Þ
KI
Since both reactants compete for the same binding site, both saturation parameters are interrelated jynI j < jynS j. A similar situation occurs for two substrates that compete for the same binding site. Nonetheless, note that such constraints hinge upon detailed knowledge of the functional form of the rate equation. For example, for noncompetitive inhibition, no restriction occurs: Both saturation parameters may attain any value with in their assigned interval, independent of the saturation of the other reactant. We thus emphasize that choosing all saturation parameters
computational models of metabolism
215
independent of their assigned intervals is still a valid approach to evaluate the comprehensive parameter space, even though specific models may prohibit certain parameter combinations to occur. Alternatively, if such restrictions need to be incorporated, it is suggested to sample from the explicit rate equation. For example, the generic rate law in Eq. (111), discussed in Section VII.C.3, allows to generate an ensemble of saturation parameters that obey all relevant inequalities. F.
Complex Dynamics: A Model of the Calvin Cycle
To demonstrate the applicability of the described approach to a system of a reasonable complexity, we briefly consider a (parametric) model of the CO2 assimilating Calvin cycle. In particular, we seek to detect and quantify the possible dynamic regimes of the model—without specifying a set of explicit differential equations. The pathway is depicted in Fig. 35. The Calvin cycle, taking place in the chloroplast stroma of plants, is a primary source of carbon for all organisms and of central importance for a variety of biotechnological applications. The set of reactions, summarized in Table VIII, is adopted from the earlier models of
Figure 35. A model of the photosynthetic Calvin cycle, adapted from the earlier models of Petterson and Ryde-Petterson [113] and Poolman et al. [124, 125, 331]. The pathway consists of r ¼ 20 reactions and m ¼ 18 metabolites. For metabolite abbreviations, see Table VIII.
216
ralf steuer and bjo¨rn h. junker TABLE VIII A Model of the Photosynthetic Calvin Cyclea
Label
Enzyme
Reaction
1 2 3 4 5 6 7 8 9 10 11 12 13
Rubisco PGK G3Pdh TPI F. Aldo FBPase F. TKL S. Aldo SBPase S. TKL R5Piso X5Pepi Ru5Pk
CO2 þ RuBP ! 2 PGA PGA þ ATP $ BPGA þ ADP BPGA þ NADPH $ GAP þ NADP þ Pi GAP $ DHAP DHAP þ GAP $ FBP FBP ! F6P þ Pi F6P þ GAP $ E4P þ X5P E4P þ DHAP $ SBP SBP ! S7P þ Pi GAP þ S7P $ X5P þ R5P R5P $ Ru5P X5P $ Ru5P Ru5P þ ATP ! RuBP þ ADP
14 15 16
PGI PGM Starch
F6P $ G6P G6P $ G1P G1P þ ATP ! ADP þ 2 Pi þ starch
17 18 19
TPT TPT TPT
20
Light
Triose phosphate translocator PGA þ Picyt ! PGAcyt þ Pi GAP þ Picyt ! GAPcyt þ Pi DHAP þ Picyt ! DHAPcyt þ Pi ATP regeneration ADP þ Pi ! ATP
a
The set of reactions is adapted from Petterson and Ryde-Petterson [113] and Poolman et al. [124, 125, 331]. All reversible reactions are modeled as rapid equilibrium reactions, assuming mass-action kinetics. Metabolite abbreviations are phosphoglycerate (PGA), bisphosphoglycerate (BPGA), glyceraldehyde phosphate (GAP), dihydroxyacetone phosphate (DHAP), fructose 1,6-bisphosphate (FBP), fructose 6-phosphate (F6P), glucose 6-phosphate (G6P), glucose 1-phosphate (G1P), erythrose 4-phosphate (E4P), sedoheptulose 1,7-bisphosphate (SBP), sedoheptulose 7-phosphate (S7P), xylulose 5-phosphate (X5P), ribose 5-phosphate (R5P), ribulose 5-phosphate (Ru5P), ribulose 1,5-bisphosphate (RuBP), and inorganic phosphate (Pi).
Petterson and Ryde-Petterson [113] and Poolman et al. [124, 125, 331]—the latter model being only a minor modification of the former. As one of its characteristic features, the Calvin cycle leads to a net synthesis of its intermediates—with significant implications for the stability of the cycle. Obviously, the balance between withdrawal of triosephosphates (TP) for biosynthesis and triosephosphates that are required for the recovery of the cycle is crucial. The overall reaction of the Calvin cycle is 3 CO2 þ 6 NADPH þ 9 ATP ! 1TP þ 6 NADPþ þ 9ADP þ 8 Pi as depicted in Fig. 36. The autocatalytic structure of the cycle already suggests the possibility of nontrivial dynamic behavior [332].
computational models of metabolism
217
Figure 36. The Calvin cycle leads to an autocatalytic net synthesis of cycle intermediates. Upon three cycles, one triosephosphate is synthesized for export. The figure is inspired by a depiction of the Calvin cycle given on http://sandwalk.blogspot.com/2007/07/Calvin-cycleregeneration.html.
The construction of the structural kinetic model proceeds as described in Section VIII.E. Note that in contrast to previous work [84], no simplifying assumptions were used; the model is a full implementation of the model described in Refs. [113, 331]. The model consists of m ¼ 18 metabolites and r ¼ 20 reactions. The rank of the stoichiometric matrix is rank ðNÞ ¼ 16, owing to the conservation of ATP and total inorganic phosphate. The steadystate flux distribution is fully characterized by four parameters, chosen to be triosephosphate export reactions and starch synthesis. Following the models of Petterson and Ryde-Petterson [113] and Poolman et al. [124, 125, 331], 11 of the 20 reactions were modeled as rapid equilibrium reactions—assuming bilinear mass-action kinetics (see Table VIII) and saturation parameters ymx ¼ 1. þ Each reversible reaction is associatedwith a reversibility parameter g ¼ nn , entirely determined by the equilibrium value and the metabolite concentrations. For the irreversible reactions, we assume Michaelis–Menten kinetics, giving rise to 15 saturation parameters ymx 2 ½0; /1- for substrates and products, respectively. In addition, the triosephospate translocator is modeled with four saturation parameters, corresponding to the model of Petterson and RydePetterson [113]. Furthermore, allosteric regulation gives rise to 10 additional parameters: 7 parameters ymx 2 ½0; n- for inhibitory interactions and 3 parameters ymx 2 ½0; n- for the activation of starch synthesis by the metabolites PGA, F6P, and FBP. We assume n ¼ 4 as an upper bound for the Hill coefficient. Once the matrix of dependencies is specified, the model is evaluated at a given metabolic state, characterized by 18 metabolite concentrations and 4 independent flux values. The numerical values for concentrations and fluxes are adopted from Ref. [113], describing the pathway under conditions of light and
218
ralf steuer and bjo¨rn h. junker
CO2 saturation. All metabolite concentrations are measured in [mM], with PGA BPGA
GAP
DHAP
FBP
F6P
E4P
SBP S7P
X5P
0.59
0.01
0.27
0.024
1.36
0.04
0.13 0.22
0.04
0.001
and R5P
Ru5P RuBP
G6P
G1P
ATP
ADP Pi
0.06
0.02
3.12
0.18
0.39
0.11 8.1
0.14
as well as the four independent steady-state fluxes (in [mM min 1 ]): nstarch
n17
n18
n19
0.16
7.1
0.56
12.0
percentage of unstable models
Iterative sampling of the Jacobian, with all saturation parameters drawn randomly from the predefined intervals, allows several conclusions on the possible dynamics of the system. Figure 37 shows the percentage of unstable model as a function of the number of models sampled from the parameter space. We observe a percentage of Z ) 0:32 of stable models with lmax < 0. More frequent are < instances with either one (Z ¼ 0:66) or two (Z ¼ 0:02) real parts within the spectrum of eigenvalues larger than zero. A small number of instances exhibit three or more real parts larger than zero (Z ¼ 3:5 4 10 4 ). More specifically, Fig. 38 shows a bifurcation diagram for the model of the Calvin cycle with 0
10
λmax <0 R #(λ >0)=1
−2
R
10
#(λR>0)=2 #(λR>0)>2 −4
10
0
2
4
6
number of iterations
8
10 4
x 10
Figure 37. The percentage of unstable model as a function of the number of models sampled from the parameter space. Note that the values quickly converge. Using 105 samples, we observe a percentage of Z ) 0:32 with no real part lmax > 0 larger than zero (solid line), a percentage of < Z ¼ 0:66 with one (dashed line), and a percentage of Z ¼ 0:02 with two eigenvalues with real parts larger than zero (dash–dotted line). A small number of instances exhibit three or more eigenvalues with positive real parts (Z ¼ 3:5 1 10 4 ). In the simulation, there reversibility parameter was set to g ¼ 0:9 for all reversible reactions.
computational models of metabolism
219
Figure 38. A bifurcation diagram for the model of the Calvin cycle with product and substrate saturation as global parameters. Left panel: Upon variation of substrate and product saturation (as global parameter, set equal for all irreversible reactions), the stable steady state is confined to a limited region in parameter space. All other parameters fixed to specific values (chosen randomly). Right panel: Same as left panel, but with all other parameters sampled from their respective intervals. Shown is the percentage Z of unstable models, with darker colors corresponding to a higher percentage of unstable models (see color bar for numeric values). See color insert.
product and substrate saturation as global parameters. The analysis shows that the metabolic state will eventually lose its stability, that is, there are conditions under which the observed steady state is no longer stable. The existence of bifurcation of the HO and SN type indicate the presence of oscillatory and bistable dynamics, respectively. Both dynamic features have been observed for the Calvin cycle: Photosynthetic oscillations are known for several decades and have been subject to extensive experimental and numerical studies [276]. Furthermore, multistability was reported in a detailed kinetic model of the Calvin cycle and claimed to occur also in vivo [125]. A more detailed analysis of the transitions to instability is relegated to Section IX.A, here we focus first on the small percentage of models with three or more eigenvalues exhibiting a positive real part. Extending the previous analysis, we seek to make use of additional properties and features of the Jacobian matrix: Although oscillations and multistability are the most common (and certainly most relevant) dynamic phenomena with signatures thereof present in the Jacobian matrix, the analysis is not restricted solely to these scenarios. Following recent approaches to obtain knowledge of the global dynamics from local properties [293, 299, 333], the Jacobian matrix allows us, at least qualitatively, to deduce also the existence of quasiperiodic and chaotic regimes. In this respect, of particular interest are bifurcations of higher codimension, such as the Takens–Bogdanov (TB), the Gavrilov–Guckenheimer (GG), and the double Hopf (DH) bifurcation [292, 293]. In general, the stability of a steady state is lost either via a Hopf bifurcation (HO) or via a bifurcation of saddle-node (SN) type, both of
220
ralf steuer and bjo¨rn h. junker
codimension-1. The intersection of codimension-1 bifurcations results in a local bifurcation of codimension-2, with the number indicating that two parameters must be varied to locate the bifurcation (note that within a two-dimensional bifurcation diagram, a codimension-1 bifurcation is a line whereas a bifurcation of codimension-2 is a point in parameter space). Bifurcation of codimension-2 is associated with characteristic dynamic behavior. For details, we refer to Ref. [292] and the work of T. Gross [293, 299, 333], here we only briefly summarize the essential properties: The TB bifurcation, corresponding to an intersection of a SN and a Hopf bifurcation, indicates the possibility of spiking or bursting behavior. A Gavrilov–Guckenheimer bifurcation indicates that complex (quasiperiodic or chaotic) dynamics exist in the vicinity of the bifurcation. A double Hopf bifurcation indicates the existence of a chaotic parameter region [293]. Figure 39 shows several bifurcation diagrams of the model of the Calvin cycle at the metabolic state for selected regions of the parameter space. Not surprisingly, the system has a rich bifurcation structure, giving rise to various codimension-1 and codimension-2 bifurcations. In particular, the various scenarios shown in Fig. 39 point to the possibility of quasiperiodic or chaotic dynamics for the model of the photosynthetic Calvin cycle. However, given the consideration above, we note that such complex dynamics are confined to a rather small region in parameter space. Nonetheless, we emphasize that the method described here is a useful and computationally highly effective approach to locate and quantify regions in parameter space associated with unstable or complex dynamics, independent of the explicit functional form of the rate equations. IX.
STABILITY AND REGULATION IN METABOLISM
As argued in the previous sections, cellular metabolism is a highly dynamic process, and a description entirely in terms of flux balance constraints is clearly not sufficient to understand and predict the functioning of metabolic processes [334, 335]. Specifically, we seek to demonstrate that the dynamic properties of large-scale metabolic networks play a far more important role than currently anticipated. Understanding the dynamics of metabolic networks will prove critical to a further understanding of metabolic function and regulation and critical to our ability to manipulate cellular system in a desired way. At this point, our notion and implications of the term stability must be clarified. At the most basic level, and as utilized in Section VII.A, dynamic stability implies that the system returns to its steady state after a small perturbation. More quantitatively, increased stability can be associated with a decreased amount of time required to return to the steady state—as for example, quantified by the largest real part within the spectrum of eigenvalues. However, obviously, stability does not imply the absence of variability in metabolite concentrations. In the face of constant perturbations, the concentration and flux values will fluctuate around their
computational models of metabolism
221
Figure 39. Bifurcation diagrams for the model of the Calvin cycle for selected parameters. All saturation parameters are fixed to specific values, and two parameters are varied. Shown is the number of real parts of eigenvalues larger than zero (color coded), with blank corresponding to the stable region. The stability of the steady state is either lost via a Hopf (HO) or via saddle-node (SN) bifurcations, with either two or one eigenvalue crossing the imaginary axis, respectively. Intersections point to complex (quasiperiodic or chaotic) dynamics. See text for details. See color insert.
steady-state values, rather than attaining these values exactly [58, 233, 234], nor does the stability imply nonchanging metabolite concentrations: While dynamic stability is mandatory to ensure the existence of a metabolic state, the metabolite state will nonetheless often depend on (slowly) varying enzyme concentrations and other time-dependent factors, corresponding, for example, to circadian regulation. In this context, we see dynamic stability as closely associated with the potential of a
222
ralf steuer and bjo¨rn h. junker
metabolic system to be regulated: Oscillatory dynamics or other weakly damped intrinsic dynamics, entailing the existence of multiple resonance frequencies, will potentially interfere with regulation by time-dependent enzyme activities. Our hypothesis is that a stable steady state is an evolutionary preferred state of a metabolic system, given constant external conditions and constant enzyme concentrations. While the metabolic state indeed changes with external conditions and enzyme concentrations, strong intrinsic dynamics would be disadvantageous. While no conclusive evidence can be given for this hypothesis, we emphasize that almost all metabolic models to date adhere to this principle: For almost all metabolic models published so far, a stable steady state is assumed to be the generic solution that requires no further explanation, while any observed oscillation implies the necessity to construct elaborate models to account for the observed behavior. In the following, we seek to explore a slight modification of this point of view: We argue that even within a seemingly simple scenario, such as a metabolic system at a steady state, the dynamic properties play a crucial role to ensure and maintain the function and stability of the system. There is no particular reason to assume that large metabolic systems are generically stable; On the contrary, numerous numerical and theoretical studies demonstrate that instability and oscillations, rather than a stable steady state, is the generic dynamic behavior of large systems [336]. In particular, the probability of the instability of a system increases with the size of the system (number of variables). To support this assertion, we illustrate the relationship between stability and size using a simple hypothetical network, depicted in Fig. 40:
Figure 40. A simple example: Cellular metabolism is modeled as a linear chain of reactions, with long-range interactions mimicking the cellular environment and interactions within the metabolic network. The parameters are the number of metabolites m, the number of regulatory interactions, the probability p of positive versus negative interaction, as well as the maximal displacement gmax from equilibrium for each reaction. Each reaction is modeled as a reversible Michaelis–Menten equation according to the methodology described in Section VIII.
computational models of metabolism (A) 1 0.9 0.8 0.7 0 200 400 600 size of pathway (metabolites and interactions
fraction of unstable models
fraction of unstable models
(B) 1
0.5
0 0
100 200 300 400 number of interactions
500
1
0.95
0.9 0
0.2 0.4 0.6 0.8 maximal reversibility γ
1
fraction of unstable models
(D)
(C) fraction of unstable models
223
1
0.95
0.9
0.85 0 0.2 0.4 0.6 0.8 1 percentage of negative interactions
Figure 41. Evaluating the stability of the simple example pathway shown in Fig. 40: Metabolic states and the corresponding saturation parameters are sampled randomly and their stability is evaluated. For each sampled model, the largest positive part within the spectrum of eigenvalues is recorded. Shown is the probability of unstable models, as a function of (A) The size of the system. Here the number of regulatory interactions increases proportional to the length of pathway (number of metabolites). Other parameters are maximal reversibility gmax ¼ 1 and p ¼ 0:5. (B) An increasing number of regulatory interactions. The number of metabolites m ¼ 100 is constant. Maximal reversibility gmax ¼ 1 and p ¼ 0:5. (C) Maximal reversibility gmax ¼ n0þ =n0 and p ¼ 0:5. Other parameters are m ¼ 101, p ¼ 0:5, and 100 regulatory interactions. (D) Percentage of negative interactions p. Note that there is a bias because all metabolites have a negative element on the diagonal in the absence of long-range interactions.
Modeled is a linear chain of reaction, corresponding, for example, to the trace of a particular molecule through a metabolic network. For simplicity, the surrounding network is replaced by a network of interactions, inducing longrange interactions between metabolites and more distant reactions. The system is evaluated using the methodology described in Section VIII and the results are depicted in Fig. 41. Shown is the probability of instability of a (randomly selected) metabolic state, analogous to the seminal work of Robert May on random community matrices. Noteworthy, the probability of instability of a (randomly selected) state increases rapidly with increasing size of the system. In addition to the size of the system (number of metabolites), other parameters of interest are an increasing number of interactions, as well as the (average)
224
ralf steuer and bjo¨rn h. junker
displacement from equilibrium. As expected, the latter has a significant impact on network stability, with results depicted in Fig. 41B–D. Although any actual metabolic network will undoubtedly exhibit a more complicated topological structure, the simple example clearly shows that the static picture of pathways and flux distributions, as often encountered in textbook diagrams, is highly misleading. A flux distribution is no static entity. Its stability, and thus its existence over a prolonged period of time, depends crucially on the numerical values of kinetic parameters and regulatory interactions: Only an intricate network of mutual interactions ensures metabolic homeostasis, prevents depletion of metabolic intermediates, and allows an optimal response to changing environmental conditions [22]. A.
Identifying Stabilizing Sites in Metabolic Networks
The stability of a metabolic state is a systemic property, that is, the question whether a metabolic state will maintain its stability is not determined by a single reaction or parameter alone. Nonetheless, not all parameters and reactions are equally important. Rather the changes in kinetic parameters will have a differential impact on the stability of the state. Given the generic instability of metabolic networks discussed above, we are particularly interested in the identification of crucial parameters and reactions—those that predominantly contribute to network stability. Following the approach of Grimbs et al. [296], the relative impact of each reaction upon the dynamical properties of the system can be evaluated. In particular, with respect to biotechnological applications, we envision that intended modifications concomitantly bring about changes in the stability properties of the network. A knowledge of important parameter may thus guide strategies to ensure the viability of an intended functional state of the system. Using the model of the Calvin cycle described already in Section VIII.F, we note that the majority of randomly sampled models (Jacobians) already correspond to a situation in which the metabolic state is unstable. The distribution of largest real part within the spectrum of eigenvalues is shown in Fig. 42. Restricting the ensemble of possible models to those instances that give rise to a stable steady state, the important parameters are easily identified: In the simplest case, we compare the distribution of parameters within the ensemble of stable models to initial distribution within the full ensemble. An example is depicted in Fig. 43. Shown is the distribution of the saturation parameter yPGA, corresponding to the saturation of TPT with respect to PGA. Within the full ensemble, the parameter is sampled randomly from the unit interval. However, restricting the ensemble to stable models only, a marked shift in the distribution is observed: Stable models predominantly correspond to a situation where the TPT operates in the linear regime with respect to PGA.
225
computational models of metabolism 1000 λmax <0 Re
λmax >0 (unstable models) Re
histogram
800 600 400 200 0 −5
0 5 eigenvalue λmax
10
Re
Figure 42. The distribution of the largest real part within the spectrum of eigenvalues for the model of the Calvin cycle described in Section VIII. F. Only a minority of sampled models correspond to a stable steady state. See also Fig. 37 for convergence in dependence of the number of samples.
Providing a more formal analysis, several objective measures to rank the parameters according to their impact on the stability can be utilized. Possible measures of dependency are: 0 The (Pearson) correlation coefficient: The most common measure of dependency is the (Pearson) correlation coefficient. It holds the advantage that it is straightforward to estimate and distinguishes between positive and negative dependencies. The results obtained for the model of the
(A)
(B) 400
600 histogram
histogram
max
400 200 0 0
0.5 Parameter θ
PGA
1
λ <0 Re stable models 200
0 0
0.5 Parameter θ
1
PGA
Figure 43. Measuring the importance of parameters. A: The distribution of the saturation parameter yPGA (saturation of TPT with respect to PGA) for the full ensemble of models, chosen randomly from the unit interval. B: The distribution of the saturation parameter yPGA after restricting the ensemble to stable states only. Within the ensemble of stable models, instances with yPGA close to the linear regime are overrepresented.
226
ralf steuer and bjo¨rn h. junker
correlation coefficient
0.4
θP
θI
θS
θA
θTPT
0.2 0 −0.2 −0.4
0
5
10
15 Parameter
20
25
30
Figure 44. The correlation coefficient of the saturation parameters with stability, here identified with the largest real part within the spectrum of eigenvalues. Models (Jacobians) of the Calvin cycle are iteratively sampled and the correlation coefficient between each saturation parameter and the largest real part of the eigenvalues are evaluated. Negative values imply a negative correlation, that is, small saturation parameters correspond to a higher probability of instability.
Calvin cycle are shown in Fig. 44. Note that the correlation coefficient assumes linear dependencies and is sensitive to skewed distributions [337]. 0 The mutual information: Probably the most general measure to evaluate dependencies between variables is the mutual information. Based on information theory, the mutual information evaluates deviations from statistical independence, and it is zero if and only if both variables are statistically independent. An detailed account of the mutual information is given in Ref. [337]. 0 The Kolmogorov–Smirnov test: Closely related to the visual approach employed in Fig. 42, the KS test evaluates the equality of two distributions. Under the assumption that a saturation parameter has no impact on stability, its distribution within the stable subset is identical (in a statistical sense) to its initial distribution. Deviations between the two distributions, such as those shown in Fig. 42, thus indicate a dependency between the parameter and dynamic stability. As demonstrated in the work of Grimbs et al. [296] using a model of the human erythrocyte, the ranking obtained by any of the measures above is i) consistent independent of the specific measure and ii) corresponds closely to a ranking obtained from an explicit kinetic model of the pathway, and it is in excellent agreement with prior knowledge of the metabolic system. Consequently, we expect that the methods described here and in [296] provide a suitable starting point to locate crucial parameters and reactions in cellular metabolism, as well as in cases where the construction of explicit kinetic models is not (yet) feasible.
computational models of metabolism B.
227
The Robustness of Metabolic States
To illustrate the actual importance of dynamic properties for the functioning of metabolic networks, we briefly describe and summarize a recent computational study on a model of human erythrocytes [296]. Erythrocytes play a fundamental role in the oxygen supply of cells and have been subject to extensive experimental and theoretical research for decades. In particular, a variety of explicit mathematical models have been developed since the late 1970s [108, 111, 114, 123, 338–341], allowing us to test the reliability of the results in a straightforward way. Based on a previously constructed model, two crucial questions with respect to dynamic properties were investigated [296]: (i) What is the role of feedback regulation in maintaining the stability of the steady state. In particular, does the feedback regulation contribute to a stabilization of the in vivo metabolic state? (ii) Is it possible to distinguish between different metabolic states? That is, does knowledge of the metabolic state, characterized by the flux distribution and metabolite concentrations, indeed constrain the dynamic properties of the system? To investigate these two questions, a parametric model of the Jacobian of human erythrocytes was constructed, based on the earlier explicit kinetic model of Schuster and Holzhu¨tter [119]. The model consists of 30 metabolites and 31 reactions, thus representing a metabolic network of reasonable complexity. Parameters and intervals were defined as described in Section VIII, with approximately 90 saturation parameters encoding the (unknown) dependencies on substrates and products and 10 additional saturation parameters encoding the (unknown) allosteric regulation. The metabolic state is described by the concentration and fluxes given in Ref. [119] for standard conditions and is consistent with thermodynamic constraints. 1.
The Role of Feedback Mechanisms
To assess the functional role of feedback mechanisms, two distinct ensembles, both consistent with the in vivo metabolic state, were considered. The first ensemble, denoted as Creg , includes feedback regulation, with each parameter sampled randomly from the predefined intervals. The second ensemble, denoted as Cnoreg , corresponds to a hypothetical system in which no feedback regulation is active; all corresponding parameters are set to zero. By choosing the remaining parameters from their preassigned intervals, it is thus possible to directly compare both scenarios. As discussed already in Section VIII, we note that such a comparison would be a tedious task when using explicit kinetic models of the pathway: We are specifically interested in the behavior of the pathway with an identical functionality (as defined by the observed flux distribution), albeit with no active allosteric regulation. However, using an explicit kinetic model, a
228
ralf steuer and bjo¨rn h. junker with regulation
(A)
normal condition
no regulation
(B) 30 distribution
distribution
30
20
10
0 −0.1
energy drain
0
0.1
eigenvalue λmax R
20
10
0 −0.1
0
0.1
eigenvalue λmax R
Figure 45. The distribution of the real parts of the eigenvalues sampled from the parametric model of the human erythrocyte. (A) The ensemble of models with Creg and without regulation Cnoreg . (B) Comparing the distributions associated with two metabolic states: normal conditions (Cvivo ) and increased energy drain (CATP ). The data are adapted from Ref. 296.
change of the kinetic parameters that correspond to the influence of allosteric regulation would concomitantly result in an altered metabolic state—and thus not straightforwardly contributing to the question. Comparing both ensembles, the respective distribution of the largest real part within the spectrum of eigenvalues was evaluated for both ensembles Creg and Cnoreg . The result is shown in Fig. 45. Starting with the ensemble Cnoreg , neglecting any allosteric regulation, the in vivo state was found to be predominantly stable. That is, the vast majority (Z ) 81%) of possible models that are consistent with the observed state also correspond to a situation where the metabolic state is stable. Nonetheless, there is a non-negligible region in parameter space for which the observed state is unstable. Evaluating the ensemble for Creg, thus accounting for allosteric regulation, the percentage of unstable models decreases (with Z ) 91% stable models). Interestingly, the reduction does not entail a general shift towards more negative values of lmax Re , rather a specific suppression of unstable instances is observed. See Fig. 45A for details. However, nonzero allosteric regulation does not only result in a greater region of stability. As noted by Grimbs et al. [296], the ensemble of models Creg also exhibits a more complicated spectrum of bifurcations, including bifurcations of higher codimension. Thus, similar to the situation encountered in Section VIII.D.4, we again note the ambiguous role of feedback regulation: Feedback regulation may stabilize the system and ensure an optimal response to perturbation, while at the same time it may also induce instability and more complicated dynamics. Additional results show that those parameters that have
computational models of metabolism
229
a high correlation with the largest real part of the eigenvalues correspond to those reactions that are also allosterically regulated—an important finding to infer putative regulatory sites in metabolic networks [296]. 2.
The Robustness of Metabolic States
The second scenario considered by Grimbs et al. [296] relates to a comparison of metabolic states, based solely on knowledge of an observed flux distribution and associated metabolite concentrations. In particular, the energy metabolism of erythrocytes has to deal with large fluctuations in the ATP demand, for example under conditions of osmotic or mechanic stress [342, 343]. To investigate the effects of such an increased energy demand, a second state was sampled from the model, corresponding to a situation were the kinetic parameter ATP utilization was significantly increased. The resulting metabolic state has an approximately twofold increase in the net flux of the ATPase reaction. It is emphasized that both metabolic states, denoted as Cvivo and CATP , cannot be distinguished based on stoichiometric consideration alone. Both states satisfy the flux balance condition, and both states are consistent with thermodynamic constraints. Taking the kinetic model into account, both states are viable steady states of the system, albeit with different numerical values for concentrations and fluxes. Nonetheless, both states show drastic differences in their stability properties. To demonstrate this assertion an ensemble of models was sampled for both states and the largest real part within the spectrum of eigenvalues was evaluated. Figure 45B shows the distributions for both states. The state CATP , corresponding to an increased energy drain, shows a dramatic increase in the percentage of unstable models. Instead of Z ) 91%, now only Z ) 13:2% of the randomly sampled instances correspond to a stable steady state. Note that this result does not imply actual instability of the state; shown is the ensemble of all possible models, while both states are indeed steady states of the system. Nonetheless, the result shown in Fig. 45B has profound consequences for the robustness of the second metabolic state. In particular, the in vivo state does not rely on any fine-tuning of parameters to ensure the stability of the state. Indeed, with respect to the requirement of dynamic stability, the parameters could be chosen (almost) randomly, allowing evolution to optimize and adjust them with respect to other important criteria than stability. However, such a reasoning does not hold for the second state CATP . Even when assuming the state is actually stable, a small perturbation in parameters may result in a loss of stability. The situation is illustrated in Fig. 46. Though both states are stable with approximately similar distance to the nearest bifurcation, they differ drastically in the response to perturbations and parameter fluctuations. While in the first case (A), any (random) perturbation will result in an increase in average stability—without any specific tuning or mechanism—the second state (B) is prone to instability. Figure 47 confirms this
230
ralf steuer and bjo¨rn h. junker (A) (A)
(B) unstable region
(B)
eigenvalues
unstable region
stable region
stable region
Figure 46. The scenario shown in Fig. 45 has profound consequences for the robustness of the metabolic state. Note that, even though the (nearest) distance to the bifurcation is similar in both cases, the probability that a perturbation of a given size will result in a loss of stability is markedly different.
reasoning to indeed hold for the model of the erythrocyte. Shown is the probability that a (randomly chosen) model loses stability upon perturbations of the parameters. Plotted is the percentage of stable models versus the magnitude of perturbations. Clearly, starting with a set of 100% stable models only, the probability that the stable state is unstable increases with increasing perturbations. However, the drop in average stability is significantly more pronounced for the state CATP (increased ATP demand), indicating a strongly reduced robustness of the state. We emphasize that these results have several implications for the modeling and analysis of metabolic systems. First, stoichiometric properties alone are not (B)
1
percentage stable models
percentage stable models
(A)
0.9 0.8 0.7 Cvivo
0.6
CATP
0.5 0
0.2 perturbation ρ
0.4
1 0.9 0.8 0.7 Cvivo
0.6
CATP
0.5 0
0.2 perturbation ρ
0.4
Figure 47. The robustness of metabolic states. Shown is the probability that a randomly chosen state is unstable. Starting with initially 100% stable models, the parameters are subject to increasing perturbations of strength r, corresponding to a random walk in parameter space. (A) The initial states are chosen randomly from the parameter space. (B) The initial states are confined to a small region with 0:01 . lmax Re < 0. Note that the state CATP exhibits a rapid decay in stability. The data is adapted from [296].
computational models of metabolism
231
sufficient to characterize a metabolic state. Two metabolic states that both satisfy the flux balance conditions and thermodynamic constraints might nonetheless drastically differ in their stability and robustness properties. Second, the difference in stability properties can be evaluated using an ensemble of models consistent with the metabolic states. Parameters that predominantly contribute to stability are identified. In particular for biotechnological applications we expect dynamic properties to be of major importance. While a biotechnologically desired flux distribution might be stoichiometrically feasible, unanticipated changes in dynamic properties can lead to failure of network function. The approach described here and in [296], provides a suitable starting point to anticipate such changes in dynamic properties of metabolic networks in a straightforward and semi-automated way. Finally, the results of Grimbs et al. [296] highlight an additional feature that is of general importance for the analysis of cellular systems. Both states considered above are the consequence of the same system, with only one parameter changed to account for an increased energy utilization. Nonetheless, the steady state under normal conditions does not ‘‘reveal’’ its properties—the set of Km values appears random, as far as stability is concerned. However, investigating the same system under stress conditions reveals that the set of Km values is actually far from random. In this sense, what might be an unremarkable property under standard conditions might require extensive tuning in another condition. These findings emphasize the importance of describing and measuring a system under a variety of different conditions, including conditions that pose a challenge to the regulatory capabilities of the system. X.
EPILOGUE: TOWARD GENOME-SCALE KINETIC MODELS
Over the past decade, there has been a tremendous increase in the recognition that computational modeling can provide an important contribution to the understanding of cellular metabolism. Under the guise of what is now emerging as the field of Systems Biology, mathematical modeling and other computational approaches have reached the realms of mainstream molecular biology. To what extent this will persistently shape and contribute to the future developments in the field, remains yet to be seen. However, past experience tells us that the global behavior of systems composed of interacting units cannot be understood based on intuitive reasoning alone, even given that each unit and each interaction itself are readily intelligible without recourse on formal reasoning. Within this contribution, our aim was to summarize and describe the ways and means of modelmaking. In our opinion, and reflecting the structure of this contribution, the construction and analysis of metabolic models rests upon five main pillars: (i) The expertise and knowledge of classic biochemistry, defining the building blocks and their interactions (as described in Section III).
232
ralf steuer and bjo¨rn h. junker
(ii) Databases and increasingly automated large-scale reconstructions of cellular metabolism, as well as an analysis in terms of FBA (as described in Section V). (iii) High-throughput measurements of metabolic intermediates (described in Section IV). (iv) In vivo flux measurements to reveal the actual functional organization of metabolism (described in Section VI). And, finally, (v) a theoretical and computational framework to make ‘‘sense of the soup’’ [5] (outlined in Section VII). Within this review, particular emphasis was placed on intermediate and heuristic approaches to elucidate the functioning of metabolism. Given the complexity of even very simple metabolic networks, along with the persistent lack of quantitative information on kinetic properties and parameter, such heuristic and approximative approaches will undoubtedly play an increasing role in the construction of computational models. Modeling of metabolism must, and probably much more than currently anticipated, take into account the multiple levels of uncertainty [87, 88] that are associated with the construction of a metabolic model. It is unlikely that the textbook approach, with a complete set of parameters turned into a comprehensive kinetic model, will be realized for any metabolic system of reasonable complexity (that is, on an compartmental or cellular scale) any time soon. Fortunately, the situation is not hopeless: A model can be highly predictive even with only a fraction of the parameters known [344]. Parameter measurements will thus be increasingly guided by their ability to predict model behavior, rather than by a general quest for comprehensiveness. A topic of major importance, as detailed in Section IX, is the dynamic properties of metabolic systems. Surprisingly, and despite the extensive research in chemistry and biochemistry, the possible dynamics of large-scale metabolic systems is almost entirely uncharted territory. Concomitant to the success of flux balance analysis, and owed to the lack of reliable kinetic information, the analysis of kinetic properties has played a rather minor role in large-scale metabolic modeling so far. However, a metabolic network is a dynamic systems. As described in Section IX, there is only little reason to assume that metabolic systems are generically stable. Rather, the overwhelming stability exhibited by most cellular fluxes and concentrations is most likely achieved by a network of regulation and feedback mechanisms—a network whose function is dictated by the topology of the regulatory interactions, as well as by the specific numerical values of kinetic parameters. We anticipate that for a large number of biotechnological applications it will prove to be crucial to go beyond the static picture of metabolic pathways. Intended changes in function will concomitantly result in changes in the dynamic behavior of the system. While the desired flux distribution might be stoichiometrically feasible, a breakdown of regulatory interactions can still lead to a failure of network function. Indeed, recently the hypothesis has been put forward that metabolic stability—here described as the
computational models of metabolism
233
capacity of cellular regulatory networks to maintain metabolic homeostasis— is closely connected to senescence and aging [345, 346]. In Section VIII we have described a methodology that allows us to anticipate changes in dynamic properties, providing a suitable starting point to detect changes in dynamic properties of metabolic networks for which the construction of detailed kinetic models is not yet possible. To what extend these, and other, approaches, will lead to the emergence of a ‘‘computer-aided design’’ (CAD) of cellular metabolism can only be speculated upon. However, a workflow for the construction of large-scale models is clearly beginning to emerge [347, 348]. There is strong tendency toward automatization, a tendency toward defining standards for exchange and storage of models [196, 349], and, last but not least, a tendency toward integration of highthroughput data into the description of the system [58, 350, 351]. As a matter of course, the field of metabolic modeling is broader than could be covered within a single review. Certainly a number of important issues were not touched upon in this contribution. In particular, metabolism does not act isolated—a topic of outstanding interest are thus integrative models, combining metabolism, signaling, and transcriptional regulation. For a model to be truly predictive, it must include several hierarchies of cellular regulation and be able to account for changes on the transcriptional level. However, in the quest for integrated models it should not be kept in mind that for integration, an prior understanding of the parts is necessary. In this respect, models of increasing complexity, grounded in validated research data, will continue to emerge and improve our understanding of cellular metabolism. Acknowledgments We wish to thank Kai Schallau for helpful discussions and for preparation of figures. BHJ was supported by the German Federal Ministry of Education and Research (BMBF) through grants 0315044A and 0315295. RS thanks the colleagues at the Manchester Interdisciplinary Biocenter for discussion and support.
References 1. B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell, 4th ed., Garland Science (Taylor & Francis Group), New York, 2002. 2. O. Fiehn, Metabolomics—The link between genotype and phenotype. Plant Mol. Biol. 78, 155– 171 (2002). 3. R. Goodacre, S. Vaidyanathan, W. B. Dunn, G. G. Harrigan, and D. B. Kell, Metabolomics by numbers: Acquiring and understanding global metabolite data. Trends in Biotechnology. 22, 245–252 (2004). 4. L. W. Sumner, P. Mendes, and R. Dixon, Plant metabolomics: Large-scale phytochemistry in the functional genomics era. Phytochemistry 62, 817–836 (2003). 5. D. B. Kell, Metabolomics and systems biology: Making sense of the soup. Curr. Opin. Microbiol. 7(3), 296–307 (2004).
234
ralf steuer and bjo¨rn h. junker
6. A. R. Fernie, R. N. Trethewey, A. J. Krotzky, and L. Willmitzer, Metabolite profiling: From diagnostics to systems biology. Nat. Rev. Mol. Cell Biol. 5, 1–7 (2004). 7. H. Kitano, ed., Foundations of Systems Biology, MIT Press, Cambridge, 2001. 8. H. Kitano, Computational systems biology. Nature 420, 206–210 (2002). 9. H. V. Westerhoff and B. Ø. Palsson, The evolution of molecular biology into systems biology. Nat. Biotechnol. 22, 1249–1252 (2002). 10. T. Ideker and D. Lauffenburger, Building with a scaffold: Emerging strategies for high- to lowlevel cellular modeling. Trends Biotechnol. 21(6), 255–262 (2003). 11. M. Csete and J. Doyle, Bow ties, metabolism and disease. Trends Biotechnol. 22(9), 446–450 (2004). 12. R. Heinrich, S. M. Rapoport, and T. A. Rapoport, Metabolic regulation and mathematical models. Prog. Biophys. Mol. Biol. 32, 1–82 (1977). 13. J. J. Tyson and H. G. Othmer, The dynamics of feedback control circuits in biochemical pathways. Progr. Theor. Biol. 5, 1–62 (1978). 14. R. Heinrich, H.-G. Holzhu¨tter, and S. Schuster, A theoretical approach to the evolution and structural design of enzymatic networks; linear enzymatic chains, branched pathways and glycolysis of erythrocytes. Bull. Math. Biol. 49(5), 539–595 (1987). 15. D. Fell, Understanding the Control of Metabolism, Portland Press, London (1997). 16. R. Heinrich and S. Schuster, The modeling of metabolic systems. structure, control and optimality. BioSystems 47, 61–77 (1998). 17. G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium System, John Wiley & Sons, New York, 1977. 18. D. Kaplan and L. Glass, Understanding Nonlinear Dynamics, Springer-Verlag, New York 1995. 19. H. Ma and A.-P. Zeng, Reconstruction of metabolic networks from genome data and analysis of their global structure for various organisms. Bioinformatics 19(2), 270–277 (2003). 20. B. H. Junker, J. Lonien, L. E. Heady, A. Rogers, and J. Schwender, Parallel determination of enzyme activities and in vivo fluxes in Brassica napus embryos grown on organic or inorganic nitrogen source. Phytochemistry 68, 2232–2242 (2007). 21. S Rossell, CC van der Weijden, A Lindenbergh, A van Tuijl, C Francke, BM Bakker, and HV Westerhoff, Unraveling the complexity of flux regulation: A new method demonstrated for nutrient starvation in Saccharomyces cerevisiae. Proc. Natl. Acad. Sci. USA 103(7), 2166–2171 (2006). 22. L. J. Sweetlove and A. R. Fernie, Regulation of metabolic networks: Understanding metabolic complexity in the systems biology era. New Phytol. 168, 9–24 (2005). 23. R. Steuer, Computational approaches to the topology, stability and dynamics of metabolic networks. Phytochemistry 68(16–18), 2139–2151 (2007). 24. U. Sauer, Metabolic networks in motion: 13C-based flux analysis. Mol. Syst. Biol. 2, 62 (2006); doi:10.1038/msb4100109. 25. L. M. Raamsdonk, B. Teusink, D. Broadhurst, N. Zhang, A. Hayes, M. C. Walsh, J. A. Berden, K. M. Brindle, D. B. Kell, J. J. Rowland, H. V. Westerhoff, K. van Dam, and S. G. Oliver, A functional genomics strategy that uses metabolome data to reveal the phenotype of silent mutations. Nat. Biotechnol. 19, 45–50 (2001). 26. A. Cornish-Bowden and M. L. Ca´rdenas, Silent genes given voice. Nature 409, 571–572 (2001). 27. S. Schuster, Use and limitations of modular metabolic control analysis in medicine and biotechnology. Metab. Eng. 1, 232–242 (1999).
computational models of metabolism
235
28. R. van Wijk and W. W. van Solinge, The energy-less red blood cell is lost: Erythrocyte enzyme abnormalities of glycolysis. Blood 106(13), 4034–4042 (2005). 29. N. Jamshidi, S. J. Wiback, and B. Ø. Palsson, In silico model-driven assessment of the effects of single nucleotide polymorphisms (SNPs) on human red blood cell metabolism. Gen. Res. 12 (11), 1687–1692 (2002). 30. H. Ma, A. Sorokin, A. Mazein, A. Selkov, E. Selkov, O. Demin, and I. Goryanin, The Edinburgh human metabolic network reconstruction and its functional analysis. Mol. Syst. Biol. 3, 135 (2007). 31. V. Hatzimanikatis, C. A. Floudas, and J. E. Bailey, Optimization of regulatory architectures in metabolic reaction networks. Biotechnol. Bioeng. 52(4), 485–500 (1996). 32. V. Hatzimanikatis, M. Emmerling, U. Sauer, and J. E. Bailey, Application of mathematical tools for metabolic design of microbial ethanol production. Biotechnol. Bioeng. 58(2–3), 154–161 (1998). 33. J. E. Bailey, Mathematical modeling and analysis in biochemical engineering: Past accomplishments and future opportunities. Biotech. Prog. 14, 8–20 (1998). 34. G. N. Stephanopoulos, A. A. Aristidou, and J. Nielsen, Metabolic Engineering: Principles and Methodologies, Academic Press, Elsevier Science, New York, 1998. 35. G. N. Stephanopoulos, H. Alper, and J. Moxley, Exploiting biological complexity for strain improvement through systems biology. Nat. Biotechnol. 22, 1261–1267 (2004). 36. D. Garfinkel, L. Garfinkel, M. Pring, S. B. Green, and B. Chance, Computer applications to biochemical kinetics. Annu. Rev. Biochem. 39, 473–498 (1970). 37. J. A. Morgan and D. Rhodes, Mathematical modeling of plant metabolic pathways. Metab. Eng. 4, 80–89 (2002). 38. P. J. Mulquiney and P. W. Kuchel, Modeling Metabolism with Mathematica, CRC Press, London, UK, 2003. 39. R. Rios-Estepa and B. M. Lange, Experimental and mathematical approaches to modeling plant metabolic networks. Phytochemistry 68(16–18), 2351–2374 (2007). 40. K. Hashimoto M. Tomita, K. Takahashi, T. S. Shimizu, Y. Matsuzaki, F. Miyoshi, K. Saito, S. Tanida, K. Yugi, J. C. Venter, and C. A. Hutchison III, E-Cell: Software environment for whole-cell simulation. Bioinformatics 15(1), 72–84 (1999). 41. B. M. Slepchenko, J. C. Schaff, I. Macara, and L. M. Loew, Quantitative cell biology with the virtual cell. TRENDS Cell Biol. 13(11), 570–577 (2003). 42. N. Ishii, M. Robert, Y. Nakayama, A. Kanai, and M. Tomita, Toward large-scale modeling of the microbial cell for computer simulation. J. Biotechnol. 113, 281–294 (2004). 43. K Yugi and M. Tomita, Applications note: A general computational model of mitochondrial metabolism in a whole organelle scale. Bioinformatics 20(11), 1795–1796 (2004). 44. N. Jamshidi and B. Ø. Palsson, Formulating genome-scale kinetic models in the post-genome era. Mol. Syst. Biol. 4, 171 (2008). 45. H. Jeong, B. Tombor, R. Albert, Z. N. Oltvai, and A.-L. Baraba´si, The large-scale organization of metabolic networks. Nature 407, 651–654 (2000). 46. D. A. Fell and A. Wagner, The small world of metabolism. Nat. Biotechnol. 18, 1121–1122 (2000). 47. R. Albert and A.-L. Baraba´si, Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002). 48. A.-L. Baraba´si and Z. N. Oltvai, Network biology: Understanding the cell’s functional organization. Nat. Rev. Genet. 5, 101–113 (2004).
236
ralf steuer and bjo¨rn h. junker
49. J. Fo¨rster, I. Famili, P. Fu, B. O. Palsson, and J. Nielsen, Genome-scale reconstruction of the Saccharomyces cerevisiae metabolic network. Gen. Res. 13, 244–253 (2003). 50. B. Ø. Palsson, Systems Biology: Properties of Reconstructed Networks, Cambridge University Press, New York, 2006. 51. N. C. Duarte, S. A. Becker, N. Jamshidi, I. Thiele, M. L. Mo, T. D. Vo, R. Srivas, and B. Ø. Palsson, Global reconstruction of the human metabolic network based on genomic and bibliomic data. Proc. Natl. Acad. Sci. USA 104(6), 1777–1782 (2007). 52. A. Wagner and D. A. Fell, The small world inside large metabolic networks. Proc. R. Soc. Lond. B. 268, 1803–1810 (2001). 53. P. Holme, M. Huss, and H. Jeong, Subnetwork hierarchies of biochemical pathways. Bioinformatics 19(4), 532–538 (2003). 54. E. Ravasz, A. L. Somera, D. A. Mongru, Z. N. Oltvai, and A.-L. Baraba´si, Hierarchical organization of modularity in metabolic networks. Science 297, 1551–1555 (2002). 55. J. Gagneur, D. B. Jackson, and G. Casari, Hierarchical analysis of dependency in metabolic networks. Bioinformatics 19(8), 1027–1034 (2003). 56. R. Albert, H. Jeong, and A.-L. Baraba´si, Error and attack tolerance of complex networks. Nature 406, 378–382 (2000). 57. N. Lemke, F. Here´dia, C. K. Barcellos, A .N. dos Reis, and J. C. M. Mombach, Essentially and damage in metabolic networks. Bioinformatics 20(1), 115–119 (2004). 58. R. Steuer, On the analysis and interpretation of correlations in metabolomic data. Brief. Bioinform. 7(2), 151–158 (2006). 59. B. L. Clarke, Stability of complex reaction networks, in Advances in Chemical Physics, S. A. Rice and I. Prigogine, eds., John Wiley & Sons, New York, 1980, pp. 1–215. 60. B. L. Clarke, Stoichiometric network analysis. Cell Biophys. 12, 237–253 (1988). 61. A. Varma and B. Ø. Palsson, Metabolic flux balancing: Basic concepts, scientific and practical use. Bio/Technology (now Nat. Biotechnol.) 12, 994–998 (1994). 62. C. Reder, Metabolic control theory: A structural approach. J. Theoret. Biol. 135, 175–201 (1988). 63. J. S. Edwards and B. Ø. Palsson, The Escherichia coli mg1655 in silico metabolic genotype: Its definition, characteristics, and capabilities. Proc. Natl. Acad. Sci. USA 97(10), 5528–5533 (2000). 64. J. Stelling, S. Klamt, K. Bettenbrock, S. Schuster, and E. D. Gilles, Metabolic network structure determines key aspects of functionality and regulation. Nature 420, 190–193 (2002). 65. S. Schuster, T. Pfeiffer, F. Moldenhauer, I. Koch, and T. Dandekar, Exploring the pathway structure of metabolism: Decomposition into subnetworks and application to Mycoplasma pneumoniae. Bioinformatics 18(2), 351–361 (2002). 66. J. A. Papin, N. D. Price, J. S. Edwards, and B. O. Palsson, The genome-scale metabolic extreme pathway structure in Haemophilus influenzae shows significant network redundancy. J. Theor. Biol. 215, 67–82 (2002). 67. J. A. Papin, N. D. Price, S. J. Wiback, D. A. Fell, and B. Ø. Palsson, Metabolic pathways in the post-genome era. Trends Biochem. Sci. 28, 250–258 (2003). 68. N. D. Price, J. L. Reed, J. A Papin, S. J. Wiback, and B. O. Palsson, Network-based analysis of metabolic regulation in the human red blood cell. J. Theor. Biol. 225, 185–194 (2003). 69. I. Famili, J. Fo¨rster, J. Nielsen, and B. Ø. Palsson, Saccharomyces cerevisiae phenotypes can be predicted by using constraint-based analysis of a genome-scale reconstructed metabolic network. Proc. Natl. Acad. Sci. USA 100(23), 13134–13139 (2003).
computational models of metabolism
237
70. C. H. Schilling, J. S. Edwards, and B. Ø. Palsson, Towards metabolic phenomics: Analysis of genomic data using flux balances. Biotechnol. Prog. 15, 288–295 (1999). 71. B. Ø. Palsson, The challenges of in silico biology. Nat. Biotechnol. 18, 1147–1150 (2000). 72. B. Ø. Palsson, In silico biology through ‘‘omics’’. Nat. Biotechnol. 20, 649–650 (2002). 73. E. Almaas, B. Kovacs, T. Vicsek, Z. N. Oltvai, and A.-L. Baraba´si, Global organization of metabolic fluxes in the bacterium Escherichia coli. Nature 427, 839–843 (2004). 74. H.-G. Holzhu¨tter, The principle of flux minimization and its application to estimate stationary fluxes in metabolic networks. Eur. J. Biochem. 271(14), 2905–2922 (2004). 75. J. E. Bailey, Complex biology with no parameters. Nat. Biotechnol. 19, 503–504 (2001). 76. F. J. M. Horn and R. Jackson, General mass action kinetics. Arch. Rational Mech. Anal. 47(2), 81–116 (1972). 77. M. Feinberg, On chemical kinetics of a certain class. Arch. Rational Mech. Anal. 46(1), 1–41 (1972). 78. M. Feinberg and F. J. M. Horn, Dynamics of open chemical systems and the algebraic structure of the underlying reaction network. Chem. Eng. Sci. 29(3), 775–787 (1974). 79. M. Feinberg, The existence and uniqueness of steady states for a class of chemical reaction networks. Arch. Rational Mech. Anal. 132(4), 311–370 (1995). 80. G. Craciun, Y. Tang, and M. Feinberg, Understanding bistability in complex enzyme-driven reaction networks. Proc. Natl. Acad. Sci. USA 103(23), 8697–8702 (2006). 81. B. L. Clarke, Stability analysis of a model reaction network using graph theory. J. Chem. Phys. 60(4), 1493–1501 (1974). 82. B. L. Clarke, Stability of topologically similar chemical networks. J. Chem. Phys. 62(9), 3726– 3738 (1975). 83. B. D. Agunda and B. L. Clarke, Bistability in chemical reaction networks: Theory and application to the peroxidase reaction. J. Chem. Phys. 87(6), 3461–3469 (1987). 84. R. Steuer, T. Gross, J. Selbig, and B. Blasius, Structural kinetic modeling of metabolic networks. Proc. Natl. Acad. Sci. USA 103(32), 11868–11873 (2006). 85. D. Visser, R. van der Heijden, K. Mauch, M. Reuss, and S. Heijnen, Tendency modeling: A new approach to obtain simplified kinetic models of metabolism applied to Saccharomyces cerevisiae. Metab. Eng. 2(3), 252–275 (2000). 86. D. Visser, J. W. Schmid, K.Mauch, M. Reuss, and J. J. Heijnen, Optimal re-design of primary metabolism in Escherichia coli using linlog kinetics. Metab. Eng. 6(4), 378–390 (2004). 87. E. Klipp, W. Liebermeister, and C. Wierling, Inferring dynamic properties of biochemical reaction networks from structural knowledge. Gen. Inform. Ser. 15(1), 125–137 (2004). 88. W. Liebermeister and E. Klipp, Biochemical networks with uncertain parameters. IEE Proc. Syst. Biol. 152(3), 97–107 (2005). 89. K. Smallbone, E. Simeonidis, D. S. Broomhead, and D. B Kell, Something from nothing— bridging the gap between constraint-based and kinetic modeling. FEBS J. 274(21), 5576–5585 (2007). 90. J. L. Casti, Reality Rules: I. Picturing the World in Mathematics, The Fundamentals, John Wiley & Sons, New York, 1992. 91. M. Kollmann and V. Sourjik, In silico biology: from simulation to understanding. Curr. Biol. 17, R132–134 (2007). 92. L. M. Loew and J. C. Schaff, The virtual cell: A Software environment for computational cell biology. Trends Biotechnol. 19(10), 401–406 (2001).
238
ralf steuer and bjo¨rn h. junker
93. H. Kitano, News and views: International alliances for quantitative modeling in systems biology. Mol. Syst. Biol. 11, E1–E2 (2005). 94. E. E. Selkov, Self-oscillations in glycolysis. Eur. J. Biochem. 4, 79–86 (1968). 95. C. Giersch, Oscillatory response of photosynthesis in leaves to environmental perturbations: A mathematical model. Arch. Biochem. Biophys. 245(1), 263–270 (1986). 96. R. Heinrich and S. Schuster, The Regulation of Cellular Systems, Chapman & Hall, New York (1996). 97. W. Wiechert and R. Takors, Validation of metabolic models: Concepts, tools, and problems, in Metabolic Engineering in the Post Genomic Era, B. N. Kholodenko and H. V. Westerhoff, eds., Horizon Bioscience, 2004, pp. 277–320. 98. J. J. Tyson, K. C. Chen, and B. Novak, Sniffers, buzzers, toggles and blinkers: Dynamics of regulatory and signaling pathways in the cell. Curr. Opin. Cell Biol. 15, 221–231 (2003). 99. O. Wolkenhauer and M. Mesarovic, Feedback dynamics and cell function: Why systems biology is called systems biology. Mol Biosyst. 1(1), 14–16 (2005). 100. B. Chance, The kinetics of the enzyme-substrate compound of peroxidase. J. Biol. Chem. 151 (2), 553–577 (1943). 101. F. Hynne, S. Danø, and P. G. Sørensen, Full-scale model of glycolysis in Saccharomyces cerevisiae. Biophys. Chem. 94, 121–163 (2001). 102. H. Berry, Monte Carlo simulations of enzyme reactions in two dimensions: Fractal kinetics and spatial segregation. Biophys. J. 83(4), 1891–1901 (2002). 103. E. Levine and T. Hwa, Stochastic fluctuations in metabolic pathways. Proc. Natl. Acad. Sci. USA 104(22), 9224–9229 (2007). 104. B. Chance, D. Garfinkel, J. Higgins, and B. Hess, Metabolic control mechanisms: A solution for the equations representing interaction between glycolysis and respiration in ascites tumor cells. J. Biol. Chem. 235(8), 2426–2439 (1960). 105. D. Garfinkel and B. Hess, Metabolic control mechanisms. VII. A detailed computer model of the glycolytic pathway in ascites cells. J. Biol. Chem. 239, 971–983 (1964). 106. B. Wright, W. Simon, and T. Walsh, A kinetic model of metabolism essential to differentiation in Dictyostelium discoideum. Proc. Natl. Acad. Sci. 60, 644–651 (1968). 107. C. J. van den Berg and D. Garfinkel, A simulation study of brain compartments: Metabolism of glutamate and related substances in mouse brain. Biochem. J. 123, 211–218 (1971). 108. T. A. Rapoport, R. Heinrich, and S. M. Rapoport, The regulatory principles of glycolysis in erythrocytes in vivo and in vitro. a minimal comprehensive model describing steady states, quasi-steady states and time-dependent processes. Biochem J. (1976). 109. M. J. Achs and D. Garfinkel, Computer simulation of energy metabolism in anoxic perfused rat heart. Am. J. Physiol. Regul. Integr. Comp. Physiol. 232, R164–R174 (1977). 110. B. D. Hahn, A mathematical model of leaf carbon metabolism. Ann. Botany 54, 325–339 (1984). 111. H. G. Holzhu¨tter, G. Jacobasch, and A. Bisdorff, Mathematical modeling of metabolic pathways affected by an enzyme deficiency. A mathematical model of glycolysis in normal and pyruvate-kinase-deficient red blood cells. Eur. J. Biochem. 149(1), 101–111 (1985). 112. B. D. Hahn, A mathematical model of the Calvin cycle: Analysis of the steady state. Ann. Botany 57, 639–653 (1986). 113. G. Petterson and U. Ryde-Petterson, A mathematical model of the Calvin photosynthesis cycle. Eur. J. Biochem. 175, 661–672 (1988). 114. A. Joshi and B. O. Palsson, Metabolic dynamics in the human red cell. Part I—A comprehensive kinetic model. J. Theor. Biol. 141(4), 515–528 (1989).
computational models of metabolism
239
115. J. L. Galazzo and J. E. Bailey, Fermentation pathway kinetics and metabolic flux control in suspended and immobilized Saccharomyces cerevisiae. Enzyme Microb. Technol. 12(3), 162– 172 (1990). 116. B. E. Wright, M. H. Butler, and K. R. Albe, Systems analysis of the tricarboxylic acid cycle in Dictyostelium discoideum. (I). The basis for model construction. J. Biol. Chem. 267(5), 3101– 3105 (1992). 117. B. E. Wright and K. R. Albe, Carbohydrate metabolism in Dictyostelium discoideum: I. Model construction. J. Theor. Biol. 169(3), 231–241 (1994). 118. E. M. T. El-Mansi, G. C. Dawson, and C. F. A. Bryce, Steady-state modeling of metabolic flux between the tricarboxylic cycle and the glyoxylate bypass in Escherichia coli. CABIOS (now Bioinformatics) 10(3), 295–299 (1994). 119. R. Schuster and H. G. Holzhu¨tter, Use of mathematical models for predicting the metabolic effect of large-scale enzyme activity alterations. Application to enzyme deficiencies of red blood cells. Eur. J. Biochem. 229(2), 403–418 (1995). 120. M. Rizzi, M. Baltes, U. Theobald, and M. Reuss, In vivo analysis of metabolic dynamics in Saccharomyces cerevisiae: II. Mathematical model. Biotechnol. Bioeng. 55, 592–608 (1997). 121. B. Teusink, J. Passarge, C. A. Reijenga, E. Esgalhado, C. C. van der Weijden, M. Schepper, M. C. Walsh, B. M. Bakker, K. van Dam, H. V. Westerhoff, and J. L. Snoep, Can yeast glycolysis be understood in terms of in vitro kinetics of the constituent enzymes? Testing biochemistry. Eur. J. Biochem. 267(17), 5313–5329 (2000). 122. S. Vaseghi, A. Baumeister, M. Rizzi, and M. Reuss, In vivo dynamics of the pentose phosphate pathway in Saccharomyces cerevisiae. Metab Eng 1(2), 128–140 (1999). 123. P. J. Mulquiney and P. W. Kuchel, Model of 2,3-bisphosphoglycerate metabolism in the human erythrocyte based on detailed enzyme kinetic equations: Computer simulation and metabolic control analysis. Biochem. J. 342 (3), 597–604 (1999). 124. M. G. Poolman, D. A. Fell, and S. Thomas, Modeling photosynthesis and its control. J. Exp. Botany 51(GMP Special issue), 319–328 (2000). ¨ lcer, J. C. Lloyd, C. A. Raines, and D. A. Fell, Computer modeling and 125. M. G. Poolman, H. O experimental evidence for two steady states in the photosynthetic Calvin cycle. Eur. J. Biochem. 268, 2810–2816 (2001). 126. J. Wolf, J. Passarge, O. J. G. Somsen, J. L. Snoep, R. Heinrich, and H. V. Westerhoff, Transduction of intracellular and intercellular dynamics in yeast glycolytic oscillations. Biophys. J. 78, 1145–1153 (2000). 127. J. M. Rohwer and F. C. Botha, Analysis of sucrose accumulation in the sugar cane culm on the basis of in vitro kinetic data. Biochem. J. 358, 437–445 (2001). 128. C. Chassagnole, B. Raı¨s, E. Quentin, D. A. Fell, and J. P. Mazat, An integrated study of threonine-pathway enzyme kinetics in Escherichia coli. Biochem. J. 356 (2), 415–423 (2001). 129. C. Chassagnole, N. Noisommit-Rizzi, J. W. Schmid, K. Mauch, and M. Reuss, Dynamic modeling of the central carbon metabolism of Escherichia coli. Biotechnol. Bioeng. 79(1), 53–73 (2002). 130. M. H. N. Hoefnagel, M. J. C. Starrenburg, D. E. Martens, J. Hugenholtz, M. Kleerebezem, I. I Van Swam, R. Bongers, H. V. Westerhoff, and J. L. Snoep, Metabolic engineering of lactic acid bacteria, the combined approach: Kinetic modeling, metabolic control and experimental analysis. Microbiology, 148(4), 1003–1013 (2002). 131. A. D. Maher, P. W. Kuchel, F. Ortega, P. de Atauri, J. Centelles, and M. Cascante, Mathematical modeling of the urea cycle. A numerical investigation into substrate channelling. Eur. J. Biochem. 270(19), 3953–3961 (2003).
240
ralf steuer and bjo¨rn h. junker
132. B. H. Junker, Sucrose breakdown in the potato tuber. Ph.D. thesis, Universita¨t Potsdam, 2004. 133. E. Klipp, B. Nordlander, R. Kru¨ger, P. Gennemark, and S. Hohmann, Integrative model of the response of yeast to osmotic shock. Nat. Biotechnol. 23(8), 975–982 (2005). 134. H. Assmus, Mathematical modeling of potato tuber carbohydrate metabolism. Ph.D. thesis, Oxford Brookes University (2005). 135. X.-G. Zhu, E. de Sturler, and S. P. Long, Optimizing the distribution of resources between enzymes of carbon metabolism can dramatically increase photosynthetic rate: A numerical simulation using an evolutionary algorithm. Plant Physiol. 145(2), 513–526 (2007). 136. F. Wu, F. Yang, KC Vinnakota, and DA Beard, Computer modeling of mitochondrial tricarboxylic acid cycle, oxidative phosphorylation, metabolite transport, and electrophysiology. J. Biol. Chem. 282(34), 24525–24537 (2007). 137. C. De Maria, D. Grassini, F. Vozzi, B. Vinci, A. Landi, A. Ahluwalia, and G. Vozzi, Hemet: Mathematical model of biochemical pathways for simulation and prediction of hepatocyte metabolism. Comput. Methods Programs Biomed. (2008). 138. S. Klamt, S. Schuster, and E. D. Gilles, Calculability analysis in underdetermined metabolic networks illustrated by a model of the central metabolism in purple nonsulfur bacteria. Biotechnol. Bioeng. 77(7), 734–751 (2002). 139. I. H. Segel, Enzyme Kinetics: Behavior and Analysis of Rapid Equilibrium and Steady-State Enzyme Systems, Wiley-Interscience, New York, 1975 (new edition 1993). 140. N. C. Price and L. Stevens, Fundamentals of Enzymology, 2nd ed., Oxford University Press, Oxford, 1989. 141. R. Kopelman, Fractal reaction kinetics. Science 241(4873), 1620–1626 (1988); DOI:10.1126/ science.241.4873.1620. 142. M. A. Savageau, Enzyme kinetics in vitro and in vivo: Michaelis–Menten revisited. Principles Med. Biol. 4(1), 93–146 (1995); doi:10.1016/S1569-2582(06)80007-3. 143. R. Grima and S. Schnell. A systematic investigation of the rate laws valid in intracellular environments. Biophys. Chem. 124(124), 1–10 (2006). 144. M. A. Savageau, Development of fractal kinetic theory for enzyme-catalysed reactions and implications for the design of biochemical pathways. BioSystems 47(1–2), 9–36 (1998). 145. M. A. Savageau, Biochemical systems analysis. I. Some mathematical properties of the rate law for the component enzymatic reactions. J. Theor. Biol. 25, 365–369 (1969). 146. M. A. Savageau, Biochemical systems analysis. II. The steady-state solutions for an n-pool system using a power-law approximation. J. Theor. Biol. 25, 370–379 (1969). 147. E. O. Voit, Computational Analysis of Biochemical Systems. A Practical Guide for Biochemists and Molecular Biologists, Cambridge University Press, Cambridge, United Kingdom (2000). 148. D. E. Koshland, Application of a theory of enzyme specificity to protein synthesis. Proc. Natl. Acad. Sci. USA 44(2), 98–104 (1958). 149. D. A. Beard, S. Liang, and H. Qian, Energy balance for analysis of complex metabolic networks. Biophys. J. 83(1), 79–86 (2002). 150. A. Ku¨mmel, S. Panke, and M. Heinemann, Putative regulatory sites unraveled by networkembedded thermodynamic analysis of metabolome data. Mol. Syst. Biol. 2 (2006). 151. A. Hoppe, S. Hoffmann, and H.-G. Holzhu¨tter, Including metabolite concentrations into flux balance analysis: Thermodynamic realizability as a constraint on flux distributions in metabolic networks. BMC Syst. Biol. 1, 23 (2007). 152. L. Michaelis and M. L. Menten, Die Kinetik der Invertin-Wirkung (the kinetics of invertase activity). Biochem. Z. 49, 333–369 (1913).
computational models of metabolism
241
153. G. E. Briggs and J. B. S. Haldane, A note on the kinetics of enzyme action. Biochem. J. 19, 338– 339 (1925). 154. V. Henri, Theorie generale de l’action de quelques diastases. Comptes Rend. Acad. Sci. Paris 135, 916–919 (1902). 155. V. Henri, Lois Ge´ne´rales de l Action des Diastases, Hermann, Paris (1903). 156. A. Ciliberto, F. Capuani, and J. J. Tyson2, Modeling networks of coupled enzymatic reactions using the total quasi-steady state approximation. PLoS Comput. Biol. 3(3), e45 (2007). 157. V. Leskovac, Comprehensive Enzyme Kinetics, Springer, Netherlands, 2003. 158. W. W. Cleland. The kinetics of enzyme-catalyzed reactions with two or more substrates or products. I. Nomenclature and rate equations. Biochim. Biophys. Acta. 67, 104–137 (1963). 159. W. W. Cleland, The kinetics of enzyme-catalyzed reactions With two or more substrates or products. II. Inhibition: Nomenclature and theory. Biochim. Biophys. Acta. 67, 173–187 (1963). 160. W. W. Cleland, The kinetics of enzyme-catalyzed reactions with two or more substrates or products. III. Prediction of initial velocity and inhibition patterns by inspection. Biochim. Biophys. Acta. 67, 188–196 (1963). 161. W. Liebermeister and E. Klipp, Bringing metabolic networks to life: Convenience rate law and thermodynamic constraints. Theor. Biol. Med. Modeling 3(42) (2006). 162. D. A. Beard, K. C. Vinnakota, and F. Wu, Detailed enzyme kinetics in terms of biochemical species: Study of citrate synthase. PLoS ONE 3(3), e1825 (2008). 163. M. S. Chapman, Enzyme kinetics and thermodynamics. Online 1994. 164. J. M. Rohwer, A. J. Hanekom, C. Crous, J. L. Snoep, and J.-H. S. Hofmeyr, Evaluation of a simplified generic bi-substrate rate equation for computational systems biology. IEE Proc.-Syst. Biol. 153(5), 338–341 (2006). 165. A. J. Hanekom, Generic kinetic equations for modeling multisubstrate reactions in computational systems biology. Master’s thesis, Stellenbosch University (2006). 166. H. E. Umbarger, Evidence for a negative-feedback mechanism in the biosynthesis of isoleucine. Science 123, 848 (1956). 167. H. E. Umbarger and B. Brown, Isoleucine and valine metabolism in Escherichia coli. VIII. The formation of acetolactate. J. Biol. Chem. 233, 1156–1160 (1958). 168. J. Monod, J.-P. Changeux, and F. Jacob, Allosteric proteins and cellular control systems. J. Mol. Biol. 306–329 (1963). 169. A. Pi/’erard, Control of the activity of Escherichia coli carbamoyl phosphate synthetase by antagonistic allosteric effectors. Science 154, 1572–1573 (1966). 170. P. Datta, Regulation of branched biosynthetic pathways in bacteria. Science 165, 556–562 (1969). 171. M. Lynch, The evolution of genetic networks by non-adaptive processes. Nat. Rev. Genet. 8, 803–813 (2007); doi:10.1038/nrg2192. 172. E. Klipp, R. Herwig, A. Kowald, C. Wierling, and H. Lehrach, Systems Biology in Practice: Concepts, Implementation and Application, Wiley-VCH Verlag GmbH, Weinheim, 2005. 173. A. V. Hill, The possible effects of the aggregation of the molecules of hemoglobin on its dissociation curves. J. Physiol. (Lond) 40, 4–7 (1910). 174. J. Monod, J. Wyman, and J. P. Changeux, On the nature of allosteric transitions: A plausible model. J. Mol. Biol. 12, 88–118 (1965). 175. J.-P. Changeux and S. J. Edelstein, Allosteric mechanisms of signal transduction. Science, 308 (5727), 1424–1428 (2005). 176. D. E. Koshland, G. Ne´methy, and D. Filmer, Comparison of experimental binding data and theoretical models in proteins containing subunits. Biochemistry 5(1), 365–368 (1966).
242
ralf steuer and bjo¨rn h. junker
177. R. Alves, F. Antunes, and A. Salvador, Tools for kinetic modeling of biochemical networks. Nat. Biotechnol. 24(6), 667–672 (2006). 178. S. Hoops, S. Sahle, R. Gauges, C. Lee, J. Pahle, N. Simus, M. Singhal, L. Xu, P. Mendes, and U. Kummer, Copasi—A complex pathway simulator. Bioinformatics 22, 3067–3074 (2006). 179. A. Funahashi, N. Tanimura, M. Morohashi, and H. Kitano, CellDesigner: A process diagram editor for gene-regulatory and biochemical networks. BIOSILICO 1, 159–162 (2003). 180. P. Dhar, T. C. Meng, S. Somani, L. Ye, A. Sairam, Z. Hao, A. Krishnan, and K. Sakharkar, Cellware—Multi-algorithmic software for computational systems biology. Bioinformatics 20(8), 1319–1321 (2004). 181. Herbert M Sauro, Michael Hucka, Andrew Finney, Cameron Wellock, Hamid Bolouri, John Doyle, and Hiroaki Kitano, Next generation simulation tools: The systems biology workbench and BioSPICE integration. OMICS 7(4), 355–372 (2003). 182. H. Schmidt and M. Jirstrand, Systems biology toolbox for Matlab: A computational platform for research in systems biology. Bioinformatics 22(4), 514–515 (2006). 183. J. Wright and A. Wagner, The systems biology research tool: Evolvable open-source software. BMC Syst. Biol. 2, 55 (2008). 184. Y. Gibon, O. E. Blaesing, J. Hannemann, P. Carillo, M. Hoehne, J. H. M. Hendriks N. Palacios, J. Cross, J. Selbig, and M. Stitt, A robot-based platform to measure multiple enzyme activities in Arabidopsis using a set of cycling assays: Comparison of changes of enzyme activities and transcript levels during diurnal cycles and in prolonged darkness. Plant Cell 16, 3304––3325 (2004). 185. N. Le Nove`re, A. Finney, M. Hucka, U. S. Bhalla, F. Campagne, J. Collado-Vides, E. J. Crampin, M. Halstead, E. Klipp, P. Mendes, P. Nielsen, H. Sauro, B. Shapiro, J. L. Snoep, H. D. Spence, and B. L. Wanner, Minimum information requested in the annotation of biochemical models (MIRIAM). Nat. Biotechnol. 23, 1509–1515 (2005). 186. M. Kanehisa and S. Goto. KEGG: Kyoto Encyclopedia of Genes and Genomes. Nucleic Acids Res. 28(1), 27–30 (2000). 187. M. Kanehisa, S. Goto, M. Hattori, K. F Aoki-Kinoshita, M. Itoh, S. Kawashima, T. Katayama, M. Araki, and M. Hirakawa, From genomics to chemical genomics: new developments in KEGG. Nucleic Acids Res. 34(Database issue), D354–D357 (2006). 188. M. Kanehisa, M. Araki, S. Goto, M. Hattori, M. Hirakawa, M. Itoh, T. Katayama, S. Kawashima, S. Okuda, T. Tokimatsu, and Y. Yamanishi, KEGG for linking genomes to life and the environment. Nucleic Acids Res. 36(Database issue), D480–D484 (2008). 189. R. Caspi, H. Foerster, C. A. Fulcher, P. Kaipa, M. Krummenacker, M. Latendresse, S. Paley, S. Y. Rhee, A. G. Shearer, C. Tissier, T. C. Walk, P. Zhang, and P. D. Karp, The MetaCyc database of metabolic pathways and enzymes and the BioCyc collection of pathway/genome databases. Nucleic Acids Res. 36(Database issue), D623–D631 (2008). 190. I. Vastrik, P. D’Eustachio, E. Schmidt, G. Joshi-Tope, G. Gopinath, D. Croft, B. de Bono, M. Gillespie, B. Jassal, S. Lewis, L. Matthews, G. Wu, E. Birney, and L. Stein, Reactome: A knowledge base of biologic pathways and processes. Genome Biol 8(3), R39 (2007). 191. I. Schomburg, A. Chang, C. Ebeling, M. Gremse, C. Heldt, G. Huhn, and D. Schomburg, BRENDA, the enzyme database: Updates and major new developments. Nucleic Acids Res 32 (Database issue), D431–D433 (2004). 192. J. Barthelmes, C. Ebeling, A. Chang, I. Schomburg, and D. Schomburg, Brenda, amenda and frenda: The enzyme information system in 2007. Nucleic Acids Res. 35(Database issue), D511– D514 (2007).
computational models of metabolism
243
193. I. Rojas, Martin Golebiewski, Renate Kania, Olga Krebs, Saqib Mir, Andreas Weidemann, and Ulrike Wittig, Storing and annotating of kinetic data. In Silico Biol. 7(2 Suppl), S37–S44 (2007). 194. I. Rojas, M. Golebiewski, R. Kania, O. Krebs, S. Mir, A. Weidemann, and U. Wittig, SABIORK: A database for biochemical reactions and their kinetics. BMC Syst. Biol., 1(Suppl 1), S6 (2007). 195. R. N. Goldberg, Y. B. Tewari, and T. N. Bhat. Thermodynamics of enzyme-catalyzed reactions— A database for quantitative biochemistry. Bioinformatics 20(16), 2874–2877 (2004). 196. N. Le Nove`re, B. Bornstein, A. Broicher, M. Courtot, M. Donizelli, H. Dharuri, L. Li, H. Sauro, M. Schilstra, B. Shapiro, J. L. Snoep, and M. Hucka, Biomodels database: A free, centralized database of curated, published, quantitative kinetic models of biochemical and cellular systems. Nucleic Acids Res. 34, D689–D691 (2006); doi:10.1093/nar/gkj092. 197. B. G. Olivier and J. L. Snoep, Web-based kinetic modeling using JWS online. Bioinformatics 20, 2143–2144 (2004). 198. C. M. Lloyd, M. D. B. Halstead, and P. F. Nielsen, CellML: Its future, present and past. Prog. Biophys. Mol. Biol. 85(2–3); 433–450 (2004). 199. D. Nickerson, C. Stevens, M. Halstead, P. Hunter, and P. Nielsen, Toward a curated CellML model repository. Conf. Proc. IEEE Eng. Med. Biol. Soc. 1, 4237–4240 (2006). 200. M. Hucka, A. Finney, H. M. Sauro, H. Bolouri, J. C. Doyle, H. Kitano, A. P. Arkin, B. J. Bornstein, D. Bray, A. Cornish-Bowden, A. A. Cuellar, S. Dronov, E. D. Gilles, M. Ginkel, V. Gor, I. I. Goryanin, W. J. Hedley, T. C. Hodgman, J.-H. Hofmeyr, P. J. Hunter, N. S. Juty, J. L. Kasberger, A. ˜ ¤re, L. M. Loew, D. Lucio, P. Mendes, E. Minch, E. D. Kremling, U. Kummer, N. Le NovA Mjolsness, Y. Nakayama, M. R. Nelson, P. F. Nielsen, T. Sakurada, J. C. Schaff, B. E. Shapiro, T. S. Shimizu, H. D. Spence, J. Stelling, K. Takahashi, M. Tomita, J. Wagner, J. Wang, and S. B. M. L. Forum, The systems biology markup language (SBML): A medium for representation and exchange of biochemical network models. Bioinformatics 19(4), 524–531 (2003). 201. M. Hucka, A. Finney, B. J. Bornstein, S. M. Keating, B. E. Shapiro, J. Matthews, B. L. Kovitz, M. J. Schilstra, A. Funahashi, J. C. Doyle, and H. Kitano, Evolving a lingua franca and associated software infrastructure for computational systems biology: The systems biology markup language (SBML) project. Syst. Biol. (Stevenage) 1(1), 41–53 (2004). 202. U. Roessner, A. Luedemann, D. Brust, O. Fiehn, T. Linke, L. Willmitzer, and A. R. Fernie, Metabolic profiling allows comprehensive phenotyping of genetically or environmentally modified plant systems. Plant Cell 13, 11–29 (2001). 203. M. Stitt, R. M. Lilley, R. Gerhardt, and H. W Heldt, Metabolite levels in specific cells and subcellular compartments of plant leaves. Methods Enzymol. 174, 221–335 (1989). 204. J. Kehr, Single cell technology. Curr. Opin. Plant Biol. 6, 617–621 (2003). 205. Y. Gibon, H. Vigeolas, A. Tiessen, P. Geigenberger, and M. Stitt, Sensitive and high throughput metabolite assays for inorganic pyrophosphate, ADPGlc, nucleotide phosphates, and glycolytic intermediates based on a novel enzymic cycling system. Plant J. 30, 221–235 (2002). 206. http://www.genomicglossaries.com/content/omes.asp. 207. O. Fiehn, J. Kopka, P. Do¨rmann, T. Altmann, R. T. Trethewey, and L. Willmitzer, Metabolite profiling for plant functional genomics. Nat. Biotechnol. 18, 1157–1161 (2000). 208. O. Fiehn, Combining genomics, metabolome analysis, and biochemical modeling to understand metabolic networks. Comp. Funct. Genom. 2, 155–168 (2001). 209. D. B. Kell, Metabolomics and machine learning: Explanatory analysis of complex metabolome data using genetic programming to produce simple, robust rules. Mol. Biol. Rep. 29, 237–241 (2002).
244
ralf steuer and bjo¨rn h. junker
210. O. Fiehn and W. Weckwerth, Deciphering metabolic networks. Eur. J. Biochem. 270, 579–588 (2003). 211. S. G. Villas-Boas, U. Roessner, M. Hansen, J. Smedsgaard, and J. Nielsen, Metabolome Analysis: An Introduction, John Wiley & Sons, Hoboken, NJ, 2007. 212. C. Wagner, M. Sefkow, and J. Kopka, Construction and application of a mass spectral and retention time index database generated from plant GC/EI-TOF-MS metabolite profiles. Phytochemistry 62, 887–900 (2003). 213. A. Aharoni, C. H. R. de Vos, H. A. Verhoeven, C. A. Maliepaard, G. Kruppa, R. J. Bino, and D. B. Goodenowe, Nontargeted metabolome analysis by use of Fourier transform ion cyclotron mass spectrometry. Omics 6, 217–234 (2002). 214. B. Luo, K. Groenke, R. Takors, C. Wandrey, and M. Oldiges, Simultaneous determination of multiple intracellular metabolites in glycolysis, pentose phosphate pathway and tricarboxylic acid cycle by liquid chromatography–mass spectrometry. J. Chromatogr. A 1147, 153–164 (2007). 215. K. Arlt, S. Brandt, and J. Kehr, Amino acid analysis in five pooled single plant cell samples using capillary electrophoresis coupled to laser-induced fluorescence detection. J. Chromatogr. A 926, 319–325 (2001). 216. H. Jenkins, N. Hardy, M. Beckmann, J. Draper, A. R. Smith, J. Taylor, O. Fiehn, R. Goodacre, R. J. Bino, R. Hall, J. Kopka, G. A. Lane, B. M. Lange, J. R. Liu, P. Mendes, B. J. Nikolau, S. G. Oliver, N. W. Paton, S. Rhee S, U. Roessner-Tunali, K. Saito, J. Smedsgaard, L. W. Sumner, T. Wang, S. Walsh, E. S. Wurtele, and D. B. Kell, A proposed framework for the description of plant metabolomics experiments and their results. Nat. Biotechnol. 22, 1601–1606 (2004). 217. K. Morgenthal, W. Weckwerth, and R. Steuer, Metabolomic networks in plants: Transitions from pattern recognition to biological interpretation. BioSystems 83(2–3), 108–117 (2006). 218. L. W. Summer, E. Urbanczyk-Wochniak, and C. D. Broekling, Metabolomics data analysis, visualization, and integration, in Plant Bioinformatics—Methods and Protocols, D. Edwards, ed., Humana Press, Towota, NJ, 2007, pp. 409–436. 219. R. A. Notebaart, F. H. J. van Enckevort, C. Francke, R. J. Siezen, and B. Teusink, Accelerating the reconstruction of genome-scale metabolic networks. BMC Bioinform. 7, 296 (2006). 220. J. S. Edwards and B. Ø. Palsson, Systems properties of the Haemophilus influenzae Rd metabolic genotype. J. Biol. Chem. 274, 17410–17416 (1999). 221. M. L. Mo, N. Jamshidi, and B. Ø. Palsson, A genome-scale, constraint based approach to systems biology of human metabolism. Mol. BioSystems 3, 598–603 (2007). 222. H. Ma and I. Goryanin, Human metabolic network reconstruction and its impact on drug discovery and development. Drug Discov. Today 13(9–10), 402–408 (2008). 223. R. Steuer and G. Zamora Lo´pez, Global network properties. Analysis of Biological Networks, Wiley Series on Bioinformatics: Computational Techniques and Engineering. B. H. Junker and F. Schreiber, eds., John Wiley & Sons, Inc. 2008. 224. S. Klamt, J. Saez-Rodriguez, and E. D. Gilles, Structural and functional analysis of cellular networks with CellNetAnalyzer. BMC Syst. Biol. 1, 2 (2007). 225. S. A. Becker, A. M. Feist, M. L. Mo, G. Hannum, B. O. Palsson, and M. J. Herrgard, Quantitative prediction of cellular metabolism with constraint-based models: The COBRA toolbox. Nat. Protocols 2(3), 727–738 (2007); doi:10.1038/nprot.2007.99. 226. T. Pfeiffer, I. Sa´nchez-Valdenebro, J. C. Nun˜o, F. Montero, and S. Schuster, Metatool: For studying metabolic networks. Bioinformatics 15(3), 251–257 (1999). 227. A. von Kamp and S. Schuster, Metatool 5.0: Fast and flexible elementary modes analysis. Bioinformatics 22(15), 1930–1931 (2006).
computational models of metabolism
245
228. R. Schwarz, P. Musch, A. von Kamp, B. Engels, H. Schirmer, S. Schuster, and T. Dandekar, YANA - a software tool for analyzing flux modes, gene-expression and enzyme activities. BMC Bioinformatics 6, 135 (2005). 229. R. Schwarz, C. Liang, C. Kaleta, M. Ku¨hnel, E. Hoffmann, S. Kuznetsov, M. Hecker, G. Griffiths, S. Schuster, and T. Dandekar, Integrated network reconstruction, visualization and analysis using YANAsquare. BMC Bioinformatics 8, 313 (2007). 230. Robert Urbanczik, SNA—a toolbox for the stoichiometric analysis of metabolic networks. BMC Bioinformatics 7, 129 (2006). 231. S. L. Bell and B. Ø. Palsson, Expa: A program for calculating extreme pathways in biochemical reaction networks. Bioinformatics 21(8), 1739–1740 (2005). 232. K. Raman and N. Chandra, Pathwayanalyser: A systems biology tool for flux analysis of metabolic pathways. Nature Precedings, 2008. 233. R. Steuer, J. Kurths, O. Fiehn, and W. Weckwerth, Observing and interpreting correlations in metabolomic networks. Bioinformatics 19(8), 1019–1026 (2003). 234. D. Camacho, A. de la Fuente, and P. Mendes, The origin of correlations in metabolomics data. Metabolomics 1, 53–63 (2005). 235. C. H. Schilling, S. Schuster, B. O. Palsson, and R. Heinrich, Metabolic pathway analysis: Basic concepts and scientific applications in the post-genomic era. Biotechnol. Prog. 15, 296–303 (1999). 236. S. Schuster and C. Hilgetag, On elementary flux modes in biochemical reaction systems at steady state. J. Biol. Syst. 2, 165–182 (1994). 237. S. Schuster, T. Dandekar, and D. A. Fell, Detection of elementary flux modes in biochemical networks: A promising tool for pathway analysis and metabolic engineering. TIBTECH 17, 53–60 (1999). 238. S. Schuster, D. A. Fell, and T. Dandekar, A general definition of metabolic pathways useful for systematic organization and analysis of complex metabolic systems. Nat. Biotechnol. 18, 326– 332 (2000). 239. S. Klamt and J. Stelling, Two approaches for metabolic pathway analysis? Trends Biotechnol. 21(2), 64–69 (2003). 240. J. A. Papin, J. Stelling, N. D. Price, S. Klamt, S. Schuster, and B. Ø. Palsson, Comparison of network-based pathway analysis methods. Trends Biotechnol. 22(8), 400–405 (2004). 241. M. G. Poolman, D. A. Fell, and C. A. Raines, Elementary modes analysis of photosynthate metabolism in the chloroplast stroma. Eur. J. Biochem. 270, 430–439 (2003). 242. S. Klamt and E. D. Gilles, Minimal cut sets in biochemical reaction networks. Bioinformatics 20(2), 226–234 (2004). 243. S. Klamt, Generalized concept of minimal cut sets in biochemical networks. Biosystems 83(2–3), 233–247 (2006). 244. I. Koch, B. H. Junker, and M. Heiner. Application of Petri net theory for modeling and validation of the sucrose breakdown pathway in the potato tuber. Bioinformatics 21(7), 1219– 1226 (2005). 245. E. Grafahrend-Belau, F. Schreiber, M. Heiner, A. Sackmann, B. H. Junker, S. Grunwald, A. Speer, K. Winder, and I. Koch, Modularization of biochemical networks based on classification of Petri net t-invariants. BMC Bioinform. 9, 90 (2008). 246. J. Nielsen, Principles of optimal metabolic network operation. Mol. Syst. Biol. 3, 126 (2007). 247. F. Llaneras and J. Pico, Stoichiometric modeling of cell metabolism. J. Biosci. Bioeng. 105(1), 1–11 (2008).
246 248. W. Wiechert,
ralf steuer and bjo¨rn h. junker 13
C metabolic flux analysis. Metab. Eng. 3, 195–206 (2001).
249. J. E. Bailey, Lessons from metabolic engineering for functional genomics and drug discovery. Nat. Biotechnol. 17, 616–618 (1999). 250. S. Baginsky and W. Gruissem, Chloroplast proteomics: Potentials and challenges. J. Exp. Bot. 55, 1213–1220 (2004). 251. M. K. Hellerstein, In vivo measurement of fluxes through metabolic pathways: The missing link in functional genomics and pharmaceutical research. Annu. Rev. Nutr. 23, 379–402 (2003). 252. J. Schwender, F. Goffman, J. B. Ohlrogge, and Y. Shachar-Hill, Rubisco without the Calvin cycle improves the carbon efficiency of developing green seeds. Nature 432, 779–782 (2004). 253. A. R. Fernie, P. Geigenberger, and M. Stitt. Flux an important, but neglected, component of functional genomics. Curr. Opin. Plant Biol. 8, 174–182 (2005). 254. B. H. Junker, J. Lonien, L. E. Heady, A. Rogers, and J. Schwender, Parallel determination of enzyme activities and in vivo fluxes in Brassica napus embryos grown on organic or inorganic nitrogen source. Phytochemistry 68, 2232–2242 (2007). 255. M. Calvin and A. A. Benson, The path of carbon in photosynthesis. IV: The identity and sequence of the intermediates in sucrose synthesis. Science 109, 140–142 (1949). 256. A. R. Fernie, A. Roscher, R. G. Ratcliffe, and N. J. Kruger, Fructose 2,6-bisphosphate activates pyrophosphate:fructose-6-phosphate 1-phosphotransferase and increases triose phosphate to hexose phosphate cycling in heterotrophic cells. Planta 212, 250–263 (2001). 257. B. H. Junker, R. Wuttke, A. Tiessen, P. Geigenberger, U. Sonnewlad, L. Willmitzer, and A. R. Fernie, Temporally regulated expression of a yeast invertase in potato tubers allows dissection of the complex metabolic phenotype obtained following its constitutive expression. Plant Mol. Biol. 56, 91–110 (2004). 258. W. Wiechert and A. A. de Graaf, Bidirectional reaction steps in metabolic networks. Part I: Modeling and simulation of carbon isotope labelling experiments. Biotechnol. Bioeng. 55, 101– 117 (1997). 259. W. Wiechert, C. Siefke, A. A. de Graaf, and A. Marx, Bidirectional reaction steps in metabolic networks. Part II: Flux estimation and statistical analysis. Biotechnol. Bioeng. 55, 118–135 (1997). 260. W. Wiechert, M. Mo¨llney, N. Isermann, M. Wurzel, and A. A. de Graaf, Bidirectional reaction steps in metabolic networks. Part III: Explicit solution and analysis of isotopomer labeling systems. Biotechnol. Bioeng. 66, 69–85 (1999). 261. M. Mo¨llney, W. Wiechert, D. Kownatzki, and A. A. de Graaf, Bidirectional reaction steps in metabolic networks. Part III: Optimal design of isotopomer labeling experiments. Biotechnol. Bioeng. 66, 86–103 (1999). 262. W. Wiechert, M. Mo¨llney, S. Petersen, and A. A. de Graaf, A universal framework for 13C metabolic flux analysis. Metab. Eng. 3, 265–283 (2001). 263. G. Sriram, D. B. Fulton, and J. V. Shanks, Flux quantification in central carbon metabolism of Catharanthus roseus hairy roots by 13c labeling and comprehensive bondomer balancing. Phytochemistry 68, 2243–2257 (2007). 264. W.-N. P. Lee, L. O. Byerley, E. A. Bergner, and J. Edmond, Mass isotopomer analysis: Theoretical and practical considerations. Biol. Mass Spectrom. 20, 451–458 (1991). 265. T. Szyperski, Biosynthetically directed fractional 13C-labeling of proteinogenic amino acids— an efficient analytical tool to investigate intermediary metabolism. Eur. J. Biochem. 232, 433– 448 (1995). 266. U. Sauer, N. Zamboni, and E. Fischer, FiatFlux–a software for metabolic flux analysis from 13C-glucose experiments. BMC Bioinform. 6, e209 (2005).
computational models of metabolism
247
267. R. G. Ratcliffe and Y. Shachar-Hill, Measuring multiple fluxes through plant metabolic networks. Plant J. 45, 490–511 (2006). 268. I. G. L. Libourel and Y. Shachar-Hill, Metabolic flux analysis in plants: From intelligent design to rational engineering. Annu. Rev. Plant Biol. 59, 625–650 (2008). 269. A. P. Sims and B. F. Folkes, A kinetic study of the assimilation of [15N]-ammonia and the synthesis of amino acids in an exponentially growing culture of Candida utilis. Proc. Roy. Soc. Lond. B. Biol. Sci. 159, 479–502 (1964). 270. U. Roessner-Tunali, J. Liu, A. Leisse, I. Balbo, A. Perez-Melis, L. Willmitzer, and A. R. Fernie, Kinetics of labelling of organic and amino acids in potato tubers by gas chromatography–mass spectrometry following incubation in 13C labeled isotopes. Plant J. 39, 668–679 (2004). 271. J. Huege, R. Sulpice, Y. Gibon, J. Lisec, K. Koehl, and J. Kopka, GC-EI-TOF-MS analysis of in vivo carbon-partitioning into soluble metabolite pools of higher plants by monitoring isotope dilution after 13CO2 labelling. Phytochemistry 68, 2258–2272 (2007). 272. P. E. Rapp, An atlas of cellular oscillators. J. Exp. Biol. 81, 281–306 (1979). 273. E. E. Selkov, Stabilization of energy charge, generation of oscillation and multiple steady states in energy metabolism as a result of purely stoichiometric regulation. Eur. J. Biochem. 59, 151– 157 (1975). 274. S. Danø, P. G. Sørensen, and F. Hynne, Sustained oscillations in living cells. Nature 402, 320– 322 (1999). 275. A. Laisk and D. A. Walker, Control of phosphate turnover as a rate-limiting factor and possible source of oscillations in photosynthesis, a mathematical model. Proc. R. Soc. London B 227, 281–302 (1986). 276. U. Ryde-Petterson, Identification of possible two-reactant sources of oscillations in the Calvin photosynthesis cycle and ancillary pathways. Eur. J. Biochem. 198, 613–619 (1991). 277. A. Goldbeter and S. R. Caplan, Oscillatory enzymes. Annu. Rev. Biophys. Bioeng. 5, 449–476 (1976). 278. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms: The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, Cambridge, United Kingdom (1997). 279. U. Schibler and F. Naef, Cellular oscillators: Rhythmic gene expression and metabolism. Curr. Opin. Cell. Biol. 17, 223–229 (2005). 280. M. J. Berridge, M. D. Bootman, and P. Lipp, Calcium a life and death signal. Nature 395, 645–648 (1998). 281. D. Lloyd and D. Murray, The temporal architecture of eukaryotic growth. FEBS Lett. 580(12), 2830–2835 (2005). 282. D. Lloyd and D. B. Murray, Redox rhythmicity: Clocks at the core of temporal coherence. BioEssays 29(5), 465–473 (2007). 283. A. Goldbeter, Computational approaches to cellular rhythms. Nature 420, 238–245 (2002). 284. J. J. Tyson, Modeling the cell division cycle: cdc2 and cyclin interactions. Proc. Natl. Acad. Sci. USA 88, 7328–7332 (1991). 285. J. J. Tyson, K. Chen, and B. Novak, Network dynamics and cell physiology. Nat. Rev. Mol. Cell Biol. 2, 908–916 (2001). 286. U. S. Bhalla and R. Iyengar, Emergent properties of networks of biological signaling pathways. Science 283, 381–387 (1999). 287. J. E. Ferrell, Jr., and W. Xiong, Bistability in cell signaling: How to make continuous processes discontinuous, and reversible processes irreversible. Chaos 11(1), 227–236 (2001).
248
ralf steuer and bjo¨rn h. junker
288. M. Laurent and N. Kellershohn, Multistability: A Major Means of differentiation and evolution in biological systems. Trends Biochem Sci. 24(11), 418–422 (1999). 289. M. Freeman. Feedback control of intercellular signaling in development. Nat. Rev. 408, 313– 319 (2000). 290. S. H. Strogatz, Nonlinear Dynamics and Chaos, Addison Wesley, Reading, MA, 1994. 291. J. P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors. Rev. Mod. Phys. 57(3), 617–656 (1985). 292. Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, Berlin (1995). 293. T. Gross, Population Dynamics: General Results from Local Analysis, Der Andere Verlag To¨nning, Germany (2004). 294. V. Hatzimanikatis and J. E. Bailey, Studies on glycolysis—I. Multiple steady states in bacterial glycolysis. Chem. Eng. Sci. 52(15), 2579–2588 (1997). 295. P. de Atauri, M. J. Ramı´rez, P. W. Kuchel, J. Carreras, and M. Cascante, Metabolic homeostasis in the human erythrocyte: In silico analysis. Biosystems 83(2–3), 118–124 (2006). 296. S. Grimbs, J. Selbig, S. Bulik, H.-G. Holzhu¨tter, and R. Steuer, The stability and robustness of metabolic states: identifying stabilizing sites in metabolic networks. Mol. Syst. Biol. 3, 146 (2007). 297. J. Higgins, The theory of oscillating reactions. Ind. Eng. Chem. 59, 18–62 (1967). 298. R. Lefever and G. Nicolis, Chemical instabilities and sustained oscillations. J. Theor. Biol. 30 (2), 267–284 (1971). 299. T. Gross and U. Feudel, Generalized models as an universal approach to the analysis of nonlinear dynamical systems. Phys. Rev. E 73, 016205 (2006). 300. M. R. Roussel and D. Lloyd, Observation of a chaotic multioscillatory metabolic attractor by real-time monitoring of a yeast continuous culture. FEBS J. 274(4), 1011–1018 (2007). 301. M. R. Roussel, Slowly reverting enzyme inactivation: A mechanism for generating long-lived damped oscillations. J. Theor. Biol. 195, 233–244 (1998). 302. O. Delcroly and A. Goldbeter, Birhythmicity, chaos, and other patterns of temporal selforganization in a multiply regulated biochemical system. Proc. Natl. Acad. Sci. USA 79, 6917– 6921 (1982). 303. M. Markus and B. Hess, Transitions between oscillatory modes in a glycolytic model system. Proc. Natl. Acad. Sci. USA 81, 4394–4398 (1984). 304. K. Nielsen P. G. Sørensen, and F. Hynne, Chaos in glycolysis. J. Theor. Biol. 186(3), 202–206 (1997). 305. B. Hess and A. Boiteux, Oscillatory phenomena in biochemistry. Ann. Rev. Biochem. 40, 237– 258 (1971). 306. A. Goldbeter and R. Lefever, Dissipative structures for an allosteric model: Application to glycolytic oscillations. Biophys. J. 12, 1302–1315 (1972). 307. A. Boiteux, A. Goldbeter, and B. Hess, Control of oscillating glycolysis of yeast by stochastic, periodic, and steady source of substrate: A model and experimental study. Proc. Natl. Acad. Sci. 72(10), 3829–3833 (1975). 308. Y. Termonia and J. Ross, Oscillations and control features in glycolysis: Numerical analysis of a comprehensive model. Proc. Natl. Acad. Sci. USA 78, 2952–2956 (1981). 309. K. Nielsen, P. G. Sørensen, F. Hynne, and H.-G. Busse, Sustained oscillations in glycolysis: An experimental and theoretical study of chaotic and complex periodic behavior and of quenching of simple oscillations. Biophys. Chem. 72, 49–62 (1998).
computational models of metabolism
249
310. M. Bier, B. M. Bakker, and H. V. Westerhoff, How yeast cells synchronize their glycolytic oscillations: A perturbation analytic treatment. Biophys. J. 78, 1087–1093 (2000). 311. B. Teusink, M. C. Walsh, K. van Dam, and H. V. Westerhoff, The danger of metabolic pathways with turbo design. Trends Biochem. Sci. 23(5), 162–169, (1998); doi:10.1016/S0968-0004(98) 01205-5. 312. P. B. Iynedjian, Glycolysis, turbo design and the endocrine pancreatic cell. Trends Biochem. Sci. 23(12), 467–468 (1998). 313. H. Kacser and J. A. Burns, The control of flux. Symp. Soc. Exp. Biol. 27, 65–104 (1973). 314. R. Heinrich and T. A. Rapoport, A linear steady-state treatment of enzymatic chains. General properties, control and effector strength. Eur. J. Biochem. 42, 89–95 (1974). 315. J.-H. S. Hofmeyr, Metabolic control analysis in a nutshell. Online Proceedings ICSB 2001, http://www.icsb-2001.org (2001). 316. M. A. Savageau, Biochemical systems theory and metabolic control theory: 1. Fundamental similarities and differences. Math. Biosci. 86, 127–145 (1987). 317. M. A. Savageau, Dominance according to metabolic control analysis: Major achievement or house of cards (letter to the editor). J. Theor. Biol. 154, 131–136 (1992). 318. J. J. Heijnen, Approximative kinetics formats used in metabolic network modeling. Biotechnol. Bioeng. 91(5), 535–545 (2005). 319. D. A. Beard, H. Qian, and J. B. Bassingthwaighte, Stoichiometric foundation of large-scale biochemical system analysis, in Modeling in Molecular Biology, G. Ciobanu and G. Rozenberg, eds. Springer Natural Computing Series, Springer-Verlag, Berlin, 2004, pp. 1–19. 320. J. E. Bailey and V. Hatzimanikatis, Effects of spatiotemporal variations on metabolic control: Approximate analysis using (log)linear kinetic models. Biotechnol. Bioeng. (1997). 321. H. V. Westerhoff and K. van Dam, Thermodynamics and Control of Biological Free Energy Transduction, Elsevier, Amsterdam 1987. 322. J. Nielsen, Metabolic control analysis of biochemical pathways based on a thermokinetic description of reaction rates. Biochem. J. 321(1), 133–138 (1997). 323. W. Liebermeister and E. Klipp, Bringing metabolic networks to life: Integration of kinetic, metabolic, and proteomic data. Theor. Biol. Med. Modeling 3(42), (2006). 324. A.-P. Zeng, J. Modak, and W.-D. Deckwer, Nonlinear dynamics of eucaryotic pyruvate dehydrogenase multienzyme complex: Decarboxylation rate, oscillations, and multiplicity. Biotechnol. Prog. 18(6), 1265–1276 (2002). 325. R. Steuer, A. Nunes-Nesi, A. R. Fernie, T. Gross, B. Blasius, and J. Selbig, From structure to dynamics of metabolic pathways: Application to the plant mitochondrial TCA cycle. Bioinformatics 23(11), 1378–1385 (2007). 326. I. Stoleriu, Stability analysis of metabolic networks, in M. Kirkilionis, U. Kummer, and I. Stoleriu, eds., 2nd UniNet Workshop: Data, Networks and Dynamics, Logos Verlag, Berlin, 2006. 327. N. Rosenfeld, M. Elowitz, and U. Alon, Negative autoregulation speeds the response times of transcription networks. J. Mol. Biol. 323(5), 785–793 (2002). 328. B. M. Bakker, F. I. C. Mensonides, B. Teusink, P. van Hoek, P. A. M. Michels, and H. V. Westerhoff, Compartimentation protects trypanosomes from the dangerous design of glycolysis. PNAS 97(5), 2087–2092 (2000). 329. A. M. Feist, C. S. Henry, J. L. Reed, M. Krummenacker, A. R. Joyce, P. D. Karp, L. J. Broadbelt, V. Hatzimanikatis, and B. Ø. Palsson, A genome-scale metabolic reconstruction for Escherichia coli
250
ralf steuer and bjo¨rn h. junker k-12 mg1655 that accounts for 1260 ORFs and thermodynamic information. Mol. Syst. Biol. 3, 121 (2007).
330. C. S. Henry, L. J. Broadbelt, and V. Hatzimanikatis, Thermodynamics-based metabolic flux analysis. Biophys J. 92(5), 1792–1805 (2007). 331. M. G. Poolman, Computer modeling applied to the Calvin cycle. Ph.D. thesis, Oxford Brookes University, 1999. 332. A. Kun, B. Papp, and E. Szathma´ry, Computational identification of obligatorily autocatalytic replicators embedded in metabolic networks. Gen. Biol. 9(3), R51 (2008). 333. T. Gross, W. Ebenho¨h, and U. Feudel, Long food chains are in general chaotic. Oikos 109, 135–144 (2005). 334. H. J. Morowitz, W. A. Higinbotham, S. W. Matthysse, and H. Quastler, Passive stability in a metabolic network. J. Theor. Biol. 7(1), 98–111 (1964). 335. C.-S. Chin, V. Chubukov, E. R. Jolly, J. DeRisi, and H. Li, Dynamics and design principles of a basic regulatory architecture controlling metabolic pathways. PLoS Biol. 6(6), e146 (2008). 336. K. S. McCann, The diversity-stability debate. Nature (insight review) 405, 228–233 (2000). 337. R. Steuer, J. Kurths, C. O. Daub, J. Weise, and J. Selbig, The mutual information: Detecting and evaluating dependencies between variables. Bioinformatics 18(Suppl. 2), S231–S240 (2002). 338. F. I. Ataullakhanov, V. M. Vitvitsky, A. M. Zhabotinsky, A. V. Pichugin, O. V. Platonova, B. N. Kholodenko, and L. I. Ehrlich, The regulation of glycolysis in human erythrocytes. The dependence of the glycolytic flux on the ATP concentration. Eur. J. Biochem. 115(2), 359–365 (1981). 339. L. M. McIntyre, D. R. Thorburn, W. A. Bubb, and P. W. Kuchel, Comparison of computer simulations of the f-type and l-type non-oxidative hexose monophosphate shunts with 31PNMR experimental data from human erythrocytes. Eur. J. Biochem. 180(2), 399–420 (1989). 340. T. C. Ni and M. A. Savageau, Application of biochemical systems theory to metabolism in human red blood cells. Signal propagation and accuracy of representation. J. Biol. Chem. 271 (14), 7927–7941 (1996). 341. Y. Nakayama, A. Kinoshita, and M. Tomita, Dynamic simulation of red blood cell metabolism and its application to the analysis of a pathological condition. Theor. Biol. Med. Model. 2(1), 18 (2005). 342. N. Dariyerli, S. Toplan, M. C. Akyolcu, H. Hatemi, G, and Yigit, Erythrocyte osmotic fragility and oxidative stress in experimental hypothyroidis. Endocrine 25(1), 1–5 (2004). 343. M. Kodicek, Enhanced glucose consumption in erythrocytes under mechanical stress. 4, 153– 155 (1986). 344. R. N. Gutenkunst, J. J. Waterfall, F. P. Casey, K. S. Brown, C. R. Myers, and J. P. Sethna, Universally sloppy parameter sensitivities in systems biology models. PLoS Comput. Biol. 3(10), 1871–78 (2007). 345. L. Demetrius, Caloric restriction, metabolic rate, and entropy. J. Gerontol. A Biol. Sci. Med. Sci. 59(9), B902–B915 (2004). 346. L. Demetrius, Of mice and men. EMBO Rep. 6 Spec No, S39–S44 (2005). 347. J. L. Snoep, The silicon cell initiative: Working towards a detailed kinetic description at the cellular level. Curr. Opin. Biotechnol. 16, 336–343 (2005). 348. D. B. Kell, The virtual human: Towards a global systems biology of multiscale, distributed biochemical network models. IUBMB Life 59(11), 689 – 695 (2007). 349. N. Le Nove`re, A. Finney, M. Hucka, U. S. Bhalla, F. Campagne, J. Collado-Vides, E. J. Crampin, M. Halstead, E. Klipp, P. Mendes, P. Nielsen, H. Sauro, B. Shapiro, J. L. Snoep, H. D.
computational models of metabolism
251
Spence, and B. L. Wanner, Minimum information requested in the annotation of biochemical models (MIRIAM). Nat. Biotechnol. 23, 1509–1515 (2005). 350. M. D. Haunschild, B. Freisleben, R. Takors, and W. Wiechert, Investigating the dynamic behavior of biochemical networks using model families. Bioinformatics 21(8), 1617–1625 (2005). 351. F. Jourdan, R. Breitling, M. P. Barrett, and D. Gilbert, MetaNetter: Inference and visualization of high-resolution metabolomic networks. Bioinformatics 24(1), 143–145 (2008).
AUTHOR INDEX Numbers in parentheses are reference numbers and indicate that the author’s work is referred to although his name is not mentioned in the text. Numbers in italic show the page on which the complete references are listed. Abascal, J. L. F., 89(155), 101 Abramavicius, D., 66(103–104), 100 Abu-samha, M., 19(186), 51 Achs, M. J., 123(109), 238 Adamo, C., 20(193), 52 Adler, T. B., 14(122), 22(122), 37(122,275), 47, 56 Adrio, J. A., 34(257), 55 Agunda, B. D., 171(83), 195(83), 237 Aharoni, A., 150–151(213), 244 Ahlborn, H., 63(79), 64(79,90), 90(79,90), 96 (197), 99–100, 102 Ahluwalia, A., 124(137), 240 Ahmed, M., 5(53), 44 Ahmed, M. K., 62(61), 89(61), 99 Akyolcu, M. C., 202(342), 229(342), 250 Albe, K. R., 123(116–117), 239 Albert, R., 113(45,47,56), 152–153(45), 235–236 Alberts, B., 107(1), 110(1), 128(1), 233 Aliev, A. E., 31(239), 54 Alikhami, M. E., 13(103), 18(103), 28(103), 31(103), 46 Al-Laham, M. A., 20(193), 52 Allan, D. R., 5(44), 31(44), 40(44), 43 Allan, M., 16(140), 48 Allen, H. C., 96(182,186), 102 Allouche, A., 16(138), 48 Almass, E., 114(73), 156(73), 237 Alon, U., 208(327), 249 Alper, H., 112(35), 114(35), 156(35), 235 Al Rabaa, A., 13–14(112), 37(112), 47 Altmann, T., 149(207), 243 Alves, R., 144(177), 242
Amir, W., 61(33), 98 Andanson, J.-M., 3(25), 16(25), 18(25), 31(25), 42 Anderson, V., 11(100), 46 Andreades, S., 33(252), 55 Antion, D. J., 26(222), 53 Antunes, F., 144(177), 242 Aoki-Kinoshita, K. F., 145(187), 242 Araki, M., 145(187–188), 242 Aristidou, A. A., 112(34), 235 Arkin, A. P., 145(200), 243 Arlt, K., 151(215), 244 Arroyo, A., 34(263), 55 Asbury, J. B.: 2(11), 41; 62(53–55,57), 83(151), 85(53–55,151), 86(55,57), 98–99, 101 Aspiala, A., 32(241), 54 Assmus, H., 123(134), 240 Astumian, R. D., 13(105), 46 Ataullakhanov, F. I., 227(338), 250 Auer, B., 61(6), 66–67(98–99), 68(99), 72 (6,99), 73(98), 75(6,98), 76(6,99), 77(98), 78(6,98), 79(98), 81–84(98), 87(98), 90 (6,99), 92–94(6), 97, 100 Austin, A. J., 20(193), 52 Ayala, P. Y., 20(193), 52 Ayott, P., 16(136), 48 Baba, T., 32(242), 54 Baboul, A. G., 20(193), 52 Bach-Verge´s, M., 31(239), 54 Bacic, Z., 13(108), 47 Bader, J. S., 63(82), 99 Badger, R. M., 8(71), 11(71), 45
Advances in Chemical Physics, Volume 142, edited by Stuart A. Rice Copyright # 2009 John Wiley & Sons, Inc.
253
254
author index
Bagchi, B., 62(59), 86–88(59), 99 Baginsky, S., 158(250), 246 Bailey, J. E., 112(31–33), 114(75), 115(63,75), 116(33), 118(33), 123(115), 138(31), 143 (33), 158(249), 168(294), 171(294), 175 (294), 181(33), 184(320), 235, 237, 239, 246, 248–249 Bakke, J. M., 33(253), 55 Bakker, B. M., 111(21), 123(121), 172(310), 202(121,310), 209(328), 234, 239, 249 Bakker, H. J.: 16(142), 26(142), 49; 61(36,44–46,48), 62(48,56,58), 84(44–46,48), 85(46), 86(48,58), 90(160), 98–99, 101 Bako, I., 34(264), 36(264), 55 Balbo, I., 164(270), 247 Baldelli, S., 96(185), 102 Ball, P., 60(1–2), 97 Baltes, M., 123(120), 239 Bansil, R., 64(87), 90(87), 99 Baraba´si, A.-L., 113(45,47–48,54,56), 114(73), 152(45,47–48), 153(45,54), 156(73), 235–237 Barber-Armstrong, W., 34(260), 36(260), 55 Barcellos, C. K., 113(57), 153(57), 236 Barlow, S. J., 3(24), 16(24), 29(24), 42 Barnes, A. J., 3(27), 28(226), 42, 53 Barnes, I., 37(271), 56 Barone, V., 20(193), 22(206), 52 Barrett, M. P., 233(351), 251 Barron, L. D., 3(29), 39(29), 42 Barthelmes, J., 145(192), 242 Bassingthwaighte, J. B., 182(319), 249 Bauer, S. H., 8(71), 11(71), 45 Bauerecker, S., 66(111), 73(111), 76(111), 96 (111), 100 Baumeister, A., 123(122), 239 Baytekin, B., 2(15), 14(15), 38(15), 42 Baytekin, H. T., 2(15), 14(15), 38(15), 42 Beard, D. A., 124(136), 130(149), 137(162), 156 (149), 182(319), 240–241, 349 Becker, K. H., 37(271), 56 Becker, S. A., 155(225), 244 Beckmann, K., 20(191), 51 Beckmann, M., 151(216), 244 Be´gue´, J.-P., 34(256), 55 Behrens, M., 18(160), 25(160), 50 Belau, L., 5(53), 44 Belch, A., 61(7), 62(7,72), 67(7), 76(7), 89–90 (7,72), 94(7,72), 96(72), 97, 99
Bellisent Funel, M. C., 34(264), 36(264), 55 Benderskii, A. V., 96(202), 103 Benjamin, I., 96(193), 102 Benson, A. A., 158(255), 246 Berden, G., 32(247), 55 Berden, J. A., 111(25), 234 Berendsen, H. J. C., 73(135), 90(135), 101 Berens, P. H., 63(81), 99 Berger, R., 13(109), 47 Berger, T., 64(87), 90(87), 99 Bergersen, H., 19(186), 51 Bergner, E. A., 161(264), 246 Bergren, M., 62(72), 89–90(72), 94(72), 96(72), 99 Berkessel, A., 34(257), 55 Bernasconi, M., 64(91), 83(152), 90(91), 100–101 Berne, B. J., 63(82), 71(126), 77(126), 96 (171,189–190), 99, 101–102 Bernstein, E. R., 18(172), 19(172,184–185), 25 (172),28(184), 50–51 Bernstein, H. J., 62(65), 89(65), 99 Berridge, M. J., 165(280), 247 Berry, H., 124(102), 238 Bertie, J. E., 62(61–62), 72(142), 89(61), 90 (62), 99, 101 Besnard, M., 3(25), 16(25), 18(25), 31(25), 42 Bettenbrock, K., 114(64), 124(64), 154(64), 236 Bhalla, U. S., 145–146(185), 165(286), 233 (349), 242, 247, 250 Bhat, T. N., 145(195), 243 Bier, M., 172(310), 202(310), 249 Bino, R. J., 150(213), 151(213,216), 244 Birney, E., 145(190), 242 Bizzarri, A., 5(52), 44 Bjerkeseth, L. H., 33(253), 55 Blaesing, O. E., 144(184), 242 Blainey, P. C., 16(141), 37(141), 48 Blasius, B., 115(84), 144(84), 188(84,325), 191 (84,325), 197(84), 202(84), 205–206(84), 211(84), 217(84), 237, 249 Bleiber, A., 21(199), 23(199), 26(199), 52 Blume, D., 18(164), 50 Bochenkova, A. V., 29(232), 54 Boese, A. D., 5(57), 21(57), 44 Bo¨hm, M., 32(247), 55 Bohn, R. K., 12(102), 16(102), 29(102), 46 Boiteux, A., 171(305), 172(307), 175(305), 248 Bolouri, H., 145(200), 243
author index Bondarenko, G. V., 3(24), 16(24), 29(24), 42 Bongers, R., 123(130), 239 Bonn, M., 96(204), 103 Bonnet-Delpon, D., 34(256), 55 Bootman, M. D., 165(280), 247 Bordenyuk, A. N., 96(202), 103 Borho, N., 13(111), 14(115,122,125–126,128), 15(128), 16(144), 17(153–154), 22(122), 33(154), 37(122,125,274–275), 38 (111,126,154), 47–49, 56 Bornstein, B., 145(196,200–201), 233(196), 243 Borve, K. J., 19(186), 51 Botha, F. C., 123(127), 239 Botschwina, P., 9–10(80), 13–14(80), 18(80), 22(80), 27–29(80), 36(80), 40(80), 45 Bour, P., 90(158), 101 Bourde´ron, C., 9(85), 45 Boutellier, Y., 11(95), 46 Bowman, J. M., 22(208), 52 Bowron, D. T., 31(236–237), 54 Boyarkin, O. V., 9(82–83), 23(82,217), 26(83), 45, 53 Boyd, S. L., 21(197), 25(197), 52 Bozek, J. D., 2(12), 41 Bradley, R. J., 2(2), 41 Brandt, S., 151(215), 244 Bratos, S., 61(31–33), 71(127), 98, 101 Bray, D., 145(200), 243 Breitling, R., 233(351), 251 Brenner, V., 5(54), 14(54), 32(54), 44 Briggs, G. E., 131(153), 241 Brindle, K. M., 111(25), 234 Broadbelt, L. J., 212(329–330), 249–250 Broadhurst, D., 111(25), 234 Brock, C. Pratt, 2(7), 5(7), 41 Broekling, C. D., 151(218), 244 Broicher, A., 145(196), 233(196), 243 Broomhead, D. S., 115(89), 137(89), 181(89), 184(89), 237 Brown, B., 137(167), 241 Brown, E., 96(201), 103 Brown, G. G., 16(148), 49 Brown, K. S., 232(344), 250 Brown, M. G., 96(182,184), 102 Brubach, J. B., 62(63), 89(63), 99 Bruice, T. C., 29(232), 54 Bruner, B. D., 62(73), 95–96(73), 99 Brust, D., 147(202), 150(202), 243 Brutschy, B., 2(3), 19(3,175–177), 41, 50–51
255
Bryce, C. F. A., 123(118), 239 Bubb, W. A., 227(339), 250 Buch, V.: 15(133), 48; 62(71), 66(71,110–111), 67(110), 73(111), 76(71,110–111), 90 (71,110), 94(71), 96(71,110–111,175,191), 99–102 Buck, M., 34(258), 55 Buck, U.: 5(48), 11(96,98), 16(48), 18 (96,98,160,174), 20(189), 22(189), 23 (98,189), 25(160,174), 26(98), 37(275), 39 (98), 40(48), 43, 46, 50–51, 56; 66(111), 73 (111,134), 75(134), 76(111,134), 96(111), 100–101 Bukowski, R., 21(201), 52 Bulik, S., 168(296), 188(296), 190(296), 224(296), 226–231(296), 248 Buntkowsky, G., 8(73), 45 Burant, J. C., 20(193), 51 Burger, K., 36(267), 37(270), 56 Burnham, C. J., 96(166), 102 Burns, J. A., 176(313), 249 Busse, H.-G., 172(309), 248 Butler, M. H., 123(116), 239 Byerley, L. O., 161(264), 246 Cahn, R. S., 13(114), 47 Callam, C. S., 38(281), 56 Calvin, M., 158(255), 246 Camacho, D., 153(234), 245 Caminati, W., 16(145), 32(145), 38(282), 49, 56 Cammi, R., 20(193), 52 Campagne, F., 145–146(185), 233(349), 242, 250 Caplan, S. R., 165(277), 247 Cappa, C. D., 61–62(12), 77(12), 97 Capuani, F., 133(156), 241 C¸arc¸abal, P., 2(17), 38–39(17), 42 Ca´rdenas, M. L., 111(26), 234 Cardini, G., 38(280), 56 Carey, D. M., 62(68), 89(68), 99 Carey, P. R., 11(100), 46 Carillo, P., 144(184), 242 Carney, J. R., 11(99), 46 Carreras, J., 168(295), 248 Carrington, T. Jr., 22(209), 53 Carter, S., 22(208), 52 Casaes, R. N., 8(75), 17(75,157), 23(75), 28–29(157), 45, 49 Casari, G., 113(55), 153(55), 236
256
author index
Cascante, M., 168(295), 248 Case, D. A., 96(173), 102 Casey, F. P., 232(344), 250 Caspi, R., 145(189), 242 Casti, J. L., 115(90), 117(90), 237 Catalano, T., 2(12), 41 Cataliotti, R. S., 18(171), 50 Centelles, J., 123(131), 239 Ce´zard, C., 3(30), 7(69), 8(30,72), 12(30), 15 (30), 16(69), 18(69), 19(30,69), 20(72), 29–30(69), 31(72), 34–35(30), 36(30,72), 37(30), 40(30,69), 42–43, 45 Chakraborty, T., 13–14(113), 47 Challacombe, M., 20(193), 52 Chance, B., 113(36), 119(36,100), 123 (100,104), 202(104), 235, 238 Chandra, N., 155(232), 245 Chandrasekhar, J., 83(150), 101 Chang, A., 145(191–192), 242 Changeux, J.-P., 138(168), 142(174–175), 241 Chapman, M. S., 137(163), 241 Chapo, C., 8(75), 17(75,157), 23(75), 28–29 (157), 45, 49 Chaquin, P., 37(273), 56 Chassagnole, C., 123(128–129), 239 Cheatum, T. E. III, 96(173), 102 Cheeseman, J. R., 20(193), 51 Chelli, R., 38(280), 56 Chen, K. C., 117(98), 165(98,285), 168(98), 238, 247 Chen, S., 69(114), 100 Chen, S. H., 64(87), 90(87), 99 Chen, W., 20(193), 52 Chiavassa, T., 13(104), 32(104), 46 Chin, C.-S., 220(335), 250 Chirokov, I., 18(162), 50 Cho, M., 66(105,108–109), 67(105), 69(119), 73(132), 100–101 Choi, J., 66(105,109), 67(105), 100 Chowdhary, J., 96(192), 102 Christe, K. O., 34(261), 55 Christen, D., 38(278), 56 Christiansen, O., 96(165), 102 Chubukov, V., 220(335), 250 Chugh, B., 62(73), 95–96(73), 99 Cieplak, P., 20(187), 51 Ciliberto, A., 133(156), 241 Cioslowski, J., 20(193), 52 Clark, S. J., 5(44), 31(44), 40(44), 43
Clarke, B. L., 114(59–60), 115(81–83), 152 (159), 153(59–60), 155(59), 195(81–83), 236–237 Cleland, W. W., 135(158–160), 241 Clifford, S., 20(193), 52 Co, R. J., 2(12), 41 Cobley, R. V., 11(91), 14(91), 16(91), 28(91), 36 (91), 46 Cohen, R. C., 61–62(12), 77(12), 97 Collado-Vides, J., 145–146(185), 233(349), 242, 250 Colombo, G., 34(259), 55 Compton, D. A. C., 9(81), 35(81), 45 Consta, S., 22(213), 31(213), 53 Cook, W. B., 23(216), 53 Copasi,, 144(178), 154–155(178), 242 Corbett, C. A., 16(135), 48 Corcelli, S. A., 61(13), 62(53–55), 65(13), 69–70(120), 72(13,120,129,131), 75(13), 78(13,120,131), 83(120,151), 85(53–55,151), 86(55), 97–98, 100–101 Cornish-Bowden, A., 111(26), 145(200), 234, 243 Cossi, M., 20(193), 52 Courtot, M., 145(196), 233(196), 243 Coussan, S., 4(34), 11(95), 13(103–104), 17 (155), 18(34,103), 25(34), 28(103), 31 (103), 32(104,155), 40(155), 43, 46, 49 Coutinho-Neto, M. D., 22(210), 32(210), 53 Cowan, M. L., 62(73–74), 95–96(73–74), 99 Craciun, G., 115(80), 168(80), 171(80), 195 (80), 237 Crampin, E. J., 145–146(185), 233(349), 242, 250 Creitz, E. C., 4(37), 31(37), 43 Cremer, P. S., 97(208), 103 Cringus, D., 66(100), 100 Croft, D., 145(190), 242 Cross, J., 144(184), 242 Cross, J. B., 20(193), 52 Cross, P. C., 61(5), 97 Crous, C., 137(164), 182(164), 187(164), 241 Crousse, B., 34(256), 55 Csete, M., 108(11), 110–111(11), 234 Cuellar, A. A., 145(200), 243 Cui, Q., 20(193), 52 Czarnecki, M. A., 14(116), 47 Dandekar, T., 114(65), 154(65,237–238), 155 (228–229), 236, 245
author index Daniels, A. D., 20(193), 52 Dannenberg, J. J., 20(193), 52 Danø, S., 121(101), 123(101), 165(274), 172(101,274), 202–203(101), 238, 247 Dapprich, S., 20(193), 52 Dariyerli, 202(342), 229(342), 250 D’Arrigo, G., 62(66), 89(66), 99 Datta, P., 138(170), 241 Daub, C. O., 226(337), 250 Dauster, I., 5(48), 16(48), 40(48), 43 Davis, B. G., 38(286), 57 Davis, Paul H., 2(9), 41 Dawson, G. C., 123(118), 239 Dea`k, J. C., 62(75), 95(75), 99 De Atauri, P., 123(131), 168(295), 239, 248 De Bono, B., 145(190), 242 Decatur, S. M., 34(260), 36(260), 55 Decius, J. C., 61(5), 97 Deckwer, W.-D., 188(324), 249 De Graaf, A. A., 160(258–262), 161(260), 162 (258–260), 163(261–262), 246 De la Fuente, A., 153(234), 245 De la Moya Cerero, S., 5(43), 43 Delcroly, O., 171(302), 248 De Lucia, F. C., 12(102), 16(102), 29(102), 46 Demaison, J., 6(63), 8(63), 23(63), 44 De Maria, C., 124(137), 240 Demetrius, L., 233(345–346), 250 Demin, O., 111(30), 235 DeRisi, J., 220(335), 250 De Sturler, E., 124(135), 240 DeTuri, V. F., 5(38), 21(38), 43 D’Eustachio, P., 145(190), 242 DeVane, R., 95(163), 102 Devlin, J. P.: 15(133), 48; 66(111), 73(111), 76 (111), 96(111,175,191), 100, 102 de Vos, C. H. R., 150–151(213), 244 Dhar, P., 144(180), 242 Dharuri, H., 145(196), 233(196), 243 Dijkstra, A. G., 66(102), 94(162), 100, 102 Dimicoli, I., 2(20), 5(20,50), 42–43 Di Paolo, T., 9(85), 45 Diraison, M., 71(127), 101 Ditter, W., 9(84), 45 Dixon, J. R., 6(64), 23(64), 44 Dixon, R., 107(4), 149(4), 233 Dlott, D. D., 61(18–19,34–35), 62(18–19, 75–77), 95(18–19,75–77), 97–99
257
Domı´nguez-Go´mez, R., 34(263), 36(268), 55–56 Donizelli, M., 145(196), 233(196), 243 Dopfer, O., 15(131), 48 Do¨rmann, P., 149(207), 243 Dos Reis, A. N., 113(57), 153(57), 236 Douglass, K. O., 16(148), 49 Doye, J. P. K., 21(203), 52 Doyle, J., 108(11), 110–111(11), 145(200–201), 234, 243 Draper, J., 151(216), 244 Dronov, S., 145(200), 243 Du, Q., 96(177–178), 102 Duan, C., 32(243), 54 Duarte, N. C., 113(51), 236 Duncan, L. L., 2(7), 5(7), 41 Dunitz, J. D., 32(250), 55 Dunn, W. B., 107(3), 149(3), 151(3), 233 Dunning, T. H., 21(200), 52 Durig, J. R., 26(222), 53 Dwyer, D. S., 2(2), 41 Dwyer, J. R., 62(73), 95–96(73), 99 Dyczmons, V., 4(33), 5(58), 7–8(58), 9–10(80), 13–14(80), 18(80), 22(80), 23(58), 27(80), 28(33,80), 29(58,80), 36(33,80), 40(80), 42, 44–45 Dyke, T. R., 18(168), 50 Eaves, J. D., 61(23–24,43,47), 71(43,126), 77 (43,126), 83(23–24,43), 83(23–24,43,47), 86(24), 98, 101 Ebata, T., 2(21), 9(21), 18(166), 19 (21,166,178,181), 23(218), 25(166), 31–32 (21), 40(21), 42, 50–51, 53 Ebeling, C., 145(191–192), 242 Ebenlo¨h, W., 219–221(333), 250 Eberlin, M. N., 14(123), 38(123), 47 Eckmann, J. P., 165(291), 248 Edelstein, S. J., 142(175), 241 Edmond, J., 161(264), 246 Edwards, J. S., 114(63,66,70), 124(63), 151 (220), 154(63), 236–237, 244 Egorov, S. A., 63(83), 99 Ehara, M., 20(193), 52 Ehbrecht, M., 11(94), 29(94), 46 Ehrlich, L. I., 227(338), 250 Elguero, J., 21(195), 23(195), 52 Elizarova, T., 18(162), 50 El-Mansi, E. M. T., 123(118), 239 Elowitz, M., 208(327), 249
258
author index
Elsaesser, T.: 2(10), 41; 61(20–21), 62(73–74), 69(117), 83(20–21), 95–96(73–74), 97, 99–100 Emmeluth, C., 3(30), 5(58), 7(58), 8(30,58), 9–10(80), 12(30), 13–14(80), 15(30,130), 17(158), 18(80), 19(30), 22(80), 23(58), 26 (158), 27–28(80), 29(58,80), 32(245), 33 (130), 34(30,130), 35(30), 36(30,80), 37 (30), 40(30,80), 42, 44–45, 48–49, 54 Emmerichs, U., 61(44), 84(44), 98 Emmerling, M., 112(31), 138(31), 235 Engels, B., 155(228), 245 England, D. C., 33(252), 55 England-Kretzer, L., 11(87), 46 Enomoto, S., 5(47), 15(47), 43 Erlekam, U., 36(266), 56 Esgalhado, E., 123(121), 202(121), 239 Ettischer, I., 11(96), 18(96,174), 25(174), 46, 50 Everitt, K. F., 63(83), 66(107), 99–100 Eysel, H. H., 62(61), 89(61), 99 Faeder, J., 96–97(187), 102 Falk, M.: 5(40), 17(40), 26–27(40), 43; 61(9), 62(9,50), 89(9), 97–98 Famili, I., 113(49), 114(69), 154(69), 236 Fang, H. L., 9(81), 35(81,265), 45, 55 Fanourgakis, G. S.: 22(213), 31(213), 53; 96 (172), 203 Farkas, O., 20(193), 52 Fa´rnı´k, M., 9(78), 18(174), 25(174), 37(275), 45, 50, 56 Farrell, J. T., 11(90), 46 Fausto, R., 11(100), 46 Favero, P. G., 16(145), 32(145), 38(282), 49, 56 Fayer, M. D.: 2(8–9,11), 41; 62(53–55,57,59), 69(114,118), 70(121), 75(121), 78(121), 83 (121,151), 85(53–55,151), 86 (55,57,59,121), 87–88(59), 97(205,213), 98–101, 103 Fecko, C. J., 61(23–24,47), 83(23–24), 84(23–24,47), 86(24), 98 Fehrensen, B., 13(107), 47 Feinberg, M., 115(77–80), 171(79), 195(79), 237 Feist, A. M., 155(225), 212(329), 244, 249 Felker, P. M., 18(165), 50 Fell, D., 108(15), 113(15,46,52), 123(124–125), 153(52), 154(237–238,241), 168(125), 177 (15), 215–217(124–125), 219(125), 234– 236, 239, 245
Fenn, E. E., 62(59), 86–88(59), 99 Ferna´ndez, J. M., 18(162–164), 50 Fernie, A. R., 107(6), 111(22), 147(202), 149 (6), 150(202), 153(22), 158(253,256–257), 164(270), 188(325), 191(325), 224(22), 234, 243, 246–247, 249 Ferrell, J. E. Jr., 165–166(287), 168(287), 247 Ferro, Y., 16(138), 48 Feudel, U., 171(299), 191(299), 197(299), 219 (299,333), 220–221(333), 248, 250 Fiehn, O., 107(2), 111(2), 147(2,202), 149 (2,207–208,210), 150(202), 151(216),153 (233), 233, 243–245 Filabozzi, A., 62(63), 89(63), 99 Filmer, D., 142(176), 241 Finney, Andrew, 144(181), 145(185,200–201), 146(185), 233(349), 242–243, 250 Finney, J. L., 31(236–237), 54 Fioroni, M., 34(259), 36(267), 37(270), 55–56 Firanescu, G., 39(288), 57 Fischer, E., 163(266), 246 Fischer, M. R., 62(67), 89–90(67), 99 Flaherty, R. A., 23(216), 53 Florian, J., 23(215), 53 Florio, G. M., 11(99), 46 Floudas, C. A., 112(31), 138(31), 235 Foerster, H., 145(189), 242 Folkes, B. F., 163(269), 247 Ford, T. A., 61–62(92), 89(9), 97 Foresman, J. B., 20(193), 52 Fo¨rster, J., 113(49), 114(69), 154(69), 236 Forum, S. B. M. L., 145(200), 243 Fox, D. J., 20(193), 52 Francis, B. F., 76(142), 101 Francke, C., 111(21), 151(219), 234, 244 Frankowski, M., 36(266), 56 Fraser, G. T., 16(146), 34(146), 49 Frecko, C. J., 61(10), 77–78(10), 83(10), 97 Freeman, M., 165(289), 248 Freysz, E., 96(177–178), 102 Frieisleben, B., 233(350), 251 Frigden, T. D., 15(132), 48 Frisch, M. J., 20(193), 51 Fro¨chtenicht, R., 18(160), 25(160), 50 Frommhold, L., 63(80), 99 Fu, H. B., 18(172), 19(172,184–185), 25(172), 28(184), 50–51 Fu, P., 113(49), 236 Fujii, A., 5(47), 15(47), 18(167), 23(218), 43, 50, 53
author index Fujii, M., 5(55), 31(235), 32(55), 44, 54 Fujita, T., 29(231), 54 Fukuda, R., 20(193), 52 Fulcher, C. A., 145(189), 242 Fulton, D. B., 160(263), 246 Funahashi, A., 144(179), 145(201), 242–243 Furlan, A., 18(169), 50 Gaffney, K. J., 2(8–9), 41 Gagneur, J., 113(55), 153(55), 236 Gaigeot, M.-P., 22(212), 53 Galazzo, J. L., 123(115), 239 Gale, G. M., 61(29–33), 98 Gallot, G., 61(29–33), 98 Gamblin, D. P., 38(286), 57 Ganim, Z., 67(112), 100 Garcı´a Fraile, A., 5(43), 43 Garcı´a Martı´nez, A., 5(43), 43 Garfinkel, D., 113(36), 119(36), 123(104– 105,107,109), 202(104–105), 235, 238 Garfinkel, L., 113(36), 119(36), 235 Gauges, R., 144(178), 154–155(178), 242 Geerke, D. P., 38(279), 56 Geigenberger, P., 149(205), 158(253,257), 243, 246 Geise, H. J., 32(240), 54 Geissler, P. L., 61(12,23–24,43), 62(12), 71(43), 77(12,43), 83–84(23–24,43), 86(24), 89 (156), 97(206), 97–98, 101, 103 Gellini, C., 38(280), 56 Gennemark, P., 123(133), 240 George, W. O., 2(5), 6(64), 23(64), 24(5), 41, 44 Georges, R., 17(159), 50 Gerhards, M., 20(191), 51 Gerhardt, R., 147–148(203), 243 Gerschel, A., 62(63), 89(63), 99 Gervasio, F. L., 38(280), 56 Ghosh, A., 96(204), 103 Giardini-Guidoni, A., 5(49), 14(119), 43, 47 Gibon, Y., 144(184), 149(205), 164(271), 242– 243, 246 Giersch, C., 116(95), 123(95), 238 Gilbert, D., 233(351), 251 Gill, P. M. W., 20(193), 52 Gilles, E. D., 114(64), 124(64,138), 126(138), 145(200), 154(64,138,224,242), 155(224), 236, 240, 243–245 Gillespie, M., 145(190), 242 Gillies, C. W., 16(146), 34(146), 49 Gillies, E. D., 124(138), 126(138), 154(138), 240
259
Gillies, Z., 16(146), 34(146), 49 Ginkel, M., 145(200), 243 Giorgini, M. G., 18(170), 50 Glass, L., 108(18), 165(18), 234 Glew, D. N., 74(138), 101 Glidewell, C., 31(239), 54 Glusac, K., 97(213), 103 Go, M. K., 62(49), 89(49), 98 Goffman, F., 158(252), 160(252), 246 Goldberg, R. N., 145(195), 243 Goldbeter, A., 165(277–278,283), 171 (302,306–307), 172(307), 247 Goldstein, T., 14(129), 16(129), 37(129), 48 Golebiewski, Martin, 145(193–194), 243 Gomperts, R., 20(193), 52 Gonzalez, C., 20(193), 52 Gonzalez, J. M., 23(219), 37(219), 53 Goodacre,R.,107(3), 149(3),151(3,216),233,244 Goodenowe, D. B., 150–151(213), 244 Goodman, L., 23(215), 53 Gopinath, G., 145(190), 242 Gor, V., 145(200), 243 Gorbaty, Y. E., 3(24), 16(24), 29(24), 42 Gorbunov, R. D., 66(96), 100 Goryanin, V., 111(30), 145(200), 151(222), 235, 243–244 Goto, S., 145(186–188), 242 Goun, A., 97(213), 103 Gozzo, F. C., 14(123), 38(123), 47 Graener, H., 61(27), 98 Graf, S., 13(108), 47 Grafahrend-Belau, E., 155(245), 245 Graham Cooks, R., 14(123), 38(123), 47 Grana, A. M., 23(220), 53 Granovsky, A. A., 31(233), 54 Grassini, D., 124(137), 240 Grate, J. W., 34(255), 55 Graur, I. A., 18(162), 50 Green, J. L., 62(69), 89(69), 94(69), 99 Green, S. B., 113(36), 119(36), 235 Green, T., 96(198), 103 Greenfield, S. R., 69(114), 100 Gregurick, S. K., 22(207), 38(207), 52 Gremse, M., 145(191), 242 Griffin, G. B., 16(137), 48 Griffiths, G., 155(229), 245 Grigera, J. R., 73(135), 90(135), 101 Grima, R., 129(143), 240 Grimbs, S., 168(296), 188(296), 190(296), 224 (296), 226–231(296), 248
260
author index
Grimme, S., 21(198), 52 Groenke, K., 150(214), 244 Groenzin, H., 62(71), 66(71), 76(71), 90(71), 94 (71), 96(71), 99 Gross, T., 115(84), 144(84), 165(293), 171 (293,299), 188(84,325), 191 (84,293,299,325), 197(84,293,299), 202 (84), 205–206(84), 211(84), 217(84), 219 (293,299,333), 220(293,333), 221(333), 237, 248–250 Gruenloh, C. J., 11(99), 21(196), 25(196), 46, 52 Gruissem, W., 158(250), 246 Grunwald, S., 155(245), 245 Gua`rdia, E., 64(88–89), 90(88–89), 99–100 Guchhait, N., 13–14(113), 19(178), 47, 51 Guillot, B., 63(85), 96(85), 99 Guissani, Y., 71(127), 101 Gu¨nthard, H. H., 23(214), 53 Gutenkunst, R. N., 232(344), 250 Ha, T.-K., 21(194), 52 Ha¨ber, T.: 5(39,46), 6(65), 7(39), 8(65,76), 11 (65), 15(76,130), 17(65,158), 20(39), 23 (65), 26(65,158), 28–29(65), 31(39,65), 33–34(130), 35(76), 36(65,76), 37(269), 43–45, 48–49, 56 Hache, F., 61(29,31), 98 Hachiya, M., 18(167), 50 Hada, M., 20(193), 52 Hadad, C. M., 38(281), 56 Hagemeister, F. C., 11(99), 18(160), 19(180), 21 (196), 25(160,196), 46, 50–52 Hahn, B. D., 123(110,112), 238 Hahn, S., 66(105,108–109), 67(105,108), 100 Haiges, R., 34(261), 55 Haldane, J. B. S., 131(153), 241 Halkier, A., 96(165), 102 Hall, R., 151(216), 244 Hall, R. G., 34(254), 55 Hallam, H. E., 3(27), 28(226), 42, 53 Halstead, M., 145(185,198–199), 146(185), 233 (349), 242–243, 250 Ham, S., 66–67(108), 100 Hamm, P., 61(20–21), 69(115–116), 83(20–21), 96(167), 97, 100, 102 Handgraaf, J.-W., 22(212), 53 Hanekom, A. J., 137(164–165), 182(164–165), 187(164), 241 Hannemann, J., 144(184), 242 Hannum, G., 155(225), 244
Hansen, M., 149(211), 244 Hao, Z., 144(180), 242 Harder, E., 71(126), 77(126), 96(189–190), 101–102 Hardy, N., 151(216), 244 Hare, D. E., 62(70), 89(70), 94(70), 99 Harnes, J., 19(186), 51 Harpham, M. R., 97(210–211), 103 Harrigan, G. G., 107(3), 149(3), 151(3), 233 Harris, K. D. M., 31(239), 54 Harrison, R. J., 96(166), 102 Hartke, B., 21–22(204), 52 Hartmann, M., 18(160), 25(160), 50 Hasegawa, J., 20(193), 52 Hashimoto, K., 113(40), 116(40), 144(40), 235 Hatemi, H., 202(342), 229(342), 250 Hattori, M., 145(187–188), 242 Hatzimanikatis, V., 112(31–32), 138(31), 168 (294), 171(294), 175(294), 184(320), 212 (329–330), 235, 248–250 Haunschild, M. D., 233(350), 251 Hayashi, T., 66(104), 71–72(128), 77(128,146), 100–101 Hayes, A., 111(25), 234 Heady, L. E., 111(20), 158(254), 234, 246 Hearn, J. P. I., 11(91), 13(110), 14(91), 16 (91,110), 28(91), 36(91), 38(110), 46–47 Hecht, L., 3(29), 39(29), 42 Hedley, W. J., 145(200), 243 Hegge, J., 34(261), 55 Heijnen, J. J., 115(85–86), 181(85,318), 184 (85–86,318), 185(318), 237, 249 Heijs, D.-J., 94(162), 102 Heinemann, M., 130(150), 156(150), 240 Heiner, M., 155(244–245), 245 Heinrich, R., 108(12,14,16), 113(12,14,16), 116 (96), 123(108,126), 125(96), 128–129(96), 133–134(96), 137(96), 154(96,235), 165 (96), 167(96), 168(96,126), 171(96), 172 (96,126), 175(126), 176(314), 177–180 (96), 195(96), 202–205(126), 227(108), 234, 237–239, 245, 249 Heldt, C., 145(191), 242 Heldt, H. W., 147–148(203), 243 Helgaker, T., 96(165), 102 Hellerstein, M. K., 158(251), 246 Hendriks, J. H. M., 144(184), 242 Henri, V., 132(154–155), 241 Henrichs, U., 32(246), 55 Henry, C. S., 212(329–330), 249–250
author index Henseler, D., 5(51), 44 Hepp, M., 17(159), 50 Herbich, J., 19(176–177), 51 Herbst, E., 12(102), 16(102), 29(102), 46 Here´dia, F., 113(57), 153(57), 236 Herman, M., 6(63), 8(63), 17(159), 23(63), 44, 50 Hermansson, K., 67(113), 71(113,124), 72(130), 73(133), 77(113,124), 100–101 Hermsdorf, D., 39(288), 57 Herrgard, M. J., 155(225), 244 Herrmann, C., 22(205), 52 Herwig, K. W., 97(210), 103 Herwig, R., 140(172), 241 Hess, B., 123(104–105), 171(303,305), 175 (303,305), 202(104–105), 238, 248 Higgins, J., 123(104), 171(297), 175(297), 202 (104), 238, 248 Higinbotham, W. A., 220–221(334), 250 Hilgetag, C., 154(236), 245 Hill, A. V., 141(173), 241 Hirakawa, M., 145(187–188), 242 Hirst, J. D., 66(102), 100 Hobza, P., 2(3), 5(56), 19(3), 41, 44 Hochstrasser, R. M., 69(115–116), 100 Hodgman, T. C., 145(200), 243 Hoefnagel, M. H. N., 123(130), 239 Hoehne, M., 144(184), 242 Hoffmann, E., 155(229), 245 Hoffmann, S., 130(151), 240 Hofmeyr, J.-H. S., 137(164), 145(200), 179 (315), 182(164), 187(164), 241, 243, 249 Hoge, B., 34(261), 55 Hohmann, S., 123(133), 240 Hokmabadi, M. S., 62(67), 89–90(67), 99 Holme, P., 113(53), 153(113), 236 Holzhu¨tter, H.-G., 108(14), 114(74), 123 (111,119), 130(74,151), 156(74), 168(296), 188(296), 190(296), 224(296), 226(296), 227(111,119,296), 228–231(296), 234, 237–238, 240, 248 Honda, Y., 20(193), 52 Hoops, S., 144(178), 154–155(178), 242 Hopkins, G. A., 18(168), 50 Hoppe, A., 130(151), 240 Hore, D. K., 96(194), 102 Horn, F. J. M., 115(76,78), 237 Hornig, D. F., 61(8), 89(8), 97 Hossain, F., 6(64), 23(64), 44 Hougen, J. T., 16(143), 23(143), 49
261
Howard, B. J., 11(91), 13(110), 14(91,117, 129), 16(91,110,117,129), 18(163), 28(91), 31 (117), 36(91), 37(129), 38(110), 46–48, 50 Howard, D. L., 38(277), 56 Hratchian, H. P., 20(193), 52 Hu, Y. J., 18(172), 19(172,184–185), 25(172), 28(184), 50–51 Huang, X. C., 22(208), 52 Hucka, Michael, 144(181), 145(185,196,200– 201), 146(185), 233(196,349), 242–243, 250 Huege, J., 164(271), 247 Hugenholtz, J., 123(130), 239 Huggins, C. M., 74(136), 101 Huhn, G., 145(191), 242 Huisken, F., 11(94,97), 18(173), 23(173), 26 (97), 29(94), 46, 50 Hu¨nig, I., 2(17), 38–39(17), 42 Hunt, R. H., 23(216), 53 Hunter, P. J., 145(199–200), 243 Hurtmans, D., 17(159), 50 Huse, N., 62(73–74), 95–96(73–74), 99 Huss, M., 113(53), 153(53), 236 Hutchinson, C. A. III, 113(40), 116(40), 144 (40), 235 Hwa, T., 124(103), 238 Hynes, J. T., 61(40,42), 71(40,125), 75–76 (140), 77(40,125), 84(40,42), 85(153), 96 (140,195), 98, 101–102 Hynne, F., 121(101), 123(101), 165(274), 171 (304), 172(101, 274,309), 202–203(101), 238, 247–248 Ibanescu, B. C., 16(140), 48 Ideker, T., 108(10), 234 Ikeda, S., 96(176), 102 Imhof, P., 29(229), 54 Impey, R. W., 83(150), 101 Iogansen, A. V., 31(238), 54 Isermann, N., 160–162(260), 246 Ishida, M., 20(193), 52 Ishii, N., 113(42), 116(42), 235 Ishiuchi, S.-I., 5(55), 31(235), 32(55), 44, 54 Itoh, M., 145(187–188), 242 Iuchi, S., 96(174), 102 Iwaki, L. K., 62(75), 95(75), 99 Iwamoto, R., 38(276), 56 Iyengar, R., 165(286), 247 Iyengar, S. S., 20(193), 51 Iynedjian, P. B., 175(312), 249
262
author index
Jackson, D. B., 113(55), 153(55), 236 Jackson, M., 37(272), 56 Jackson, R., 115(76), 237 Jacob, F., 138(168), 241 Jacoby, C., 16(149), 19(182), 32(149,182), 49, 51 Jamshidi, N., 111(29), 113(44), 151(221), 235, 244 Jansen, T. I. C., 66(95,100–102), 71–72(128), 77(128,146), 95–96(164), 100–102 Janzen, C., 29(229), 54 Jaramillo, J., 20(193), 52 Jarmelo, S., 11(100), 46 Jassal, B., 145(190), 242 Jenkins, H., 151(216), 244 Jeong, H., 113(45,53,56), 152(45), 153(45,53), 235–236 Jetzki, M., 38(283–284), 39(284), 56–57 Ji, X., 63–64(79), 90(79), 99 Jirstrand, M., 144(182), 242 Johannesen, C., 3(29), 39(29), 42 Johns, J. E., 16(148), 49 Johnson, A., 107(1), 110(1), 128(1), 233 Johnson, B., 20(193), 52 Johnson, B. G., 23(215), 53 Johnson, M. A., 16(136), 48 Jokusch, R. A., 2(17), 38(17,285), 39(17), 42, 57 Jolly, E. R., 220(335), 250 Jones, B. F., 2(5), 24(5), 41 Jones, D., 3(27), 42 Jo¨nsson, P.-G., 27(224), 53 Jordan, K. D., 16(136), 48 Jørgensen, P., 96(165), 102 Jorgensen, W. L., 83(150), 101 Joshi, A., 123(114), 227(114), 238 Joshi-Tope, G., 145(190), 242 Jourdan, F., 233(351), 251 Joyce, A. R., 212(329), 249 Jungwirth, P., 96(191,201), 102–103 Junker, B. H., 111(20), 123(132), 155(244– 245), 158(254,257), 234, 240, 245–246 Juty, N. S., 145(200), 243 Kacser, H., 176(313), 249 Kagunya, W. W., 31(239), 54 Kahn, K., 29(232), 54 Kaipa, P., 145(189), 242 Kaleta, C., 155(229), 245 Kalkman, I., 32(247), 55 Kaloudis, M., 18(173), 23(173), 50
Kammrath, A., 16(137), 48 Kanai, A., 113(42), 116(42), 235 Kanehisa, M., 145(186–188), 242 Kania, Renate, 145(193–194), 243 Kanters, J. A., 3(32), 39(32), 42 Kaplan, D., 108(18), 165(18), 234 Karapetian, K., 16(136), 48 Kariuki, B. M., 31(239), 54 Karp, P. D., 145(189), 212(329), 242, 249 Karpfen, A., 6(60), 44 Karplus, M., 20(190), 51 Kasberger, J. L., 145(200), 243 Kasprzyk, C. R., 96(198), 103 Katayama, T., 145(187–188), 242 Katsumoto, Y., 2(21), 9(21), 19(21), 31–32(21), 40(21), 42 Kawashima, S., 145(187–188), 242 Kazimirski, J. K., 66(111), 73(111,134), 75 (134), 76(111,134), 96(111), 100–101 Keating, S. M., 145(201), 243 Kehr, J., 148(204), 151(215), 243–244 Keith, T., 20(193), 52 Kell, D. B., 107(3,5), 111(25), 115(89), 137(89), 149(3,5,209), 151(3,5,209,216), 181(89), 184(89), 232(5), 233(348), 233–234, 237, 243–244, 250 Kellershohn, N., 165(288), 248 Kempter, V., 16(138), 48 Keutsch, F. N., 2(4), 41 Keyes, T., 95(163), 102 Kholodenko, B. N., 227(338), 250 King, A. K., 14(117), 16(117), 31(117), 47 King, Bretta F., 6(66), 44 Kinoshita, A., 227(341), 250 Kint, S., 62(49), 89(49), 98 Kinzel, T., 9–10(80), 13–14(80), 18(80), 22 (80), 27–29(80), 36(80), 40(80), 45 Kitano, H., 108(7–8), 116(7,93), 144(179), 145 (200–201), 234, 238, 242–243 Kitao, O., 20(193), 52 Kitchin, S. J., 31(239), 54 Kjaergaard, H. G., 38(277), 56 Klamt, S., 114(64), 124(64,138), 126(138), 154 (64,138,224,239–240,242–243), 155(224), 236, 240, 244–245 Kleerebezem, M., 123(130), 239 Klein, M. L., 83(150), 101 Kleinermanns, K., 20(191), 29(229), 32(246– 247), 51, 54–55 Klene, M., 20(193), 52
author index Klipp, E., 115(87), 123(133), 135(161), 137 (161), 140(172), 145–146(185), 181(161), 186(161), 187(161,323), 210(161), 232 (87), 233(349), 237, 240–242, 249–250 Klompmaker, E. R., 26(221), 53 Klopper, W., 5(57), 21(57,202), 44, 52 Klug, D. D., 74–75(139), 101 Knoester, J., 66(95,102), 94(162), 100, 102 Knowles, P. J., 20–21(192), 51 Knox, E., 20(193), 52 Knuts, S., 72(130), 101 Kobus, M., 66(96), 100 Koch, H., 96(165), 102 Koch, I., 114(65), 154(65), 155(244–245), 236, 245 Koch, K. J., 14(123), 38(123), 47 Koch, M., 18(173), 23(173), 50 Kodicek, M., 229(343), 250 Koehl, K., 164(271), 247 Kollmann, M., 115(91), 117(91), 237 Kollmann, P. A., 20(187), 51 Komaromi, I., 20(193), 52 Kopelman, R., 129(141), 240 Kopka, J., 149(207), 150(212), 151(212,216), 164(271), 243, 244, 247 Korenowski, G. M., 62(68), 89(68), 99 Koshland, D. E., 129(148), 142(148,176), 240– 241 Kova´cs, A., 32(249), 55 Kovacs, B., 114(73), 156(73), 237 Kovitz, B. L., 145(201), 243 Kowald, A., 140(172), 241 Kownatzki, D., 160(261), 163(261), 246 Kozin’ski, M., 61(33), 98 Kraemer, D., 62(74), 95–96(74), 99 Krebs, Olga, 145(193–194), 243 Kremling, A., 145(200), 243 Krishnan, A., 144(180), 242 Kroemer, R. T., 2(17), 38–39(17), 42 Kroon, J., 3(32), 39(32), 42 Kropman, M. F., 62(56), 99 Krotzky, A. J., 107(6), 149(6), 234 Kruger, N. J., 158(256), 246 Kru¨ger, R., 123(133), 240 Kru¨gler, D., 32(247), 55 Krummenacker, M., 145(189), 212(329), 242, 249 Kruppa, G., 150–151(213), 244 Kubo, R., 60(3), 68(3), 97
263
Kuchel, P. W., 113(38), 123(38,123,131), 168 (295), 227(123,339), 235, 239, 248, 250 Kudin, K. N., 20(193), 51 Kuharski, R. A., 96(168), 102 Kuhn, L. P., 2(1), 41 Ku¨hn, O., 22(211), 53 Ku¨hnel, M., 155(229), 245 Kumar, R., 63(86), 66–67(98), 73(98), 75(98), 77(98,143), 78–79(98), 81–84(98), 87(98), 96(86), 99–101 Ku¨mmel, A., 130(150), 156(150), 240 Kummer, U., 144(178), 145(200), 154–155 (178), 242–243 Kun, A., 216(332), 250 Kunitski, M., 19(177), 51 Kunzmann, M. K., 39(287), 57 Kuo, I.-F. W., 96(188), 102 Kurths, J., 153(233), 226(337), 245, 250 Kusanagi, H., 38(276), 56 Kuyanov, K. E., 8(74), 45 Kuznetsov, Y., 155(229), 165(292), 219–220 (292), 245, 248 Kwac, K., 73(132), 101 Kyrychenko, A., 19(176–177), 51 Laage, D., 85(153), 101 Lacey, A. R., 62(69), 89(69), 94(69), 99 Ladanyi, B. M., 96(187,192), 97(187,210–211), 102–103 Laenen, R., 61(16–17,28), 84(16), 97–98 Lahmani, F., 4(35–36), 5(50,54), 7–8(35–36), 13(112–113), 14(54,112–113,120– 121,126), 19(35–36,120–121), 26(36), 32 (35,54,120), 37(113), 38(126), 43–44, 47–48 Laisk, A., 165(275), 247 Lan, Z., 62(62), 89–90(62), 99 Landi, A., 124(137), 240 Landis, C., 77(145), 101 Lane, G. A., 151(216), 244 Lange, B. M., 113(39), 144(39), 151(216), 235, 244 Larsen, P., 2(16), 8–11(16), 18(16), 23–25(16), 40(16), 42 Lascoux, N., 61(29–33), 98 Latendresse, M., 145(189), 242 Latini, A., 5(49), 14(119), 43, 47 Laubereau, A., 61(16–17,27–28), 84(16), 97–98 Lauffenburger, D., 108(10), 234 Laurent, M., 165(288), 248
264
author index
Lawrence, C. P., 61(15,38–39,41), 62(53–55), 66(107), 71(15,38–39), 72(131), 77 (15,38–39), 78(131), 79(15,38), 83 (41,147,151), 84(15,38–39,41), 85(53–55, 151), 86(55), 87(38,154), 97–98, 100–101 Le Barbu-Debus, K. L., 4(35–36), 5(54), 7–8(35–36), 13(112–113), 14(54, 112–113,120–121,126), 17(154), 19(35–36,120–121), 26(36), 32 (35,54,120), 33(154), 37(112), 38 (126,154), 43–44, 47–49 Lee, C., 144(178), 154–155(178), 242 Lee, H., 66(109), 100 Lee, J. J., 7(69), 16(69), 18–19(69), 29–30(69), 40(69), 45 Lee, K.-K., 66(109), 100 Lee, W.-N. P., 161(264), 246 Lefever, R., 171(298,306), 248 Lehn, J.-M., 2(6), 41 Lehrach, H., 140(172), 241 Leicknam, J.-Cl., 61(29,31–33), 71(127), 98, 101 Leisse, A., 164(270), 247 Lemaire, J., 15(132), 48 Lemke, N., 113(57), 153(57), 236 Le Nove`re, N., 145(185,196,200), 146(185), 233(196,349), 242–243, 250 Leone, S. R., 5(53), 44 Leskovac, V., 133–134(157), 137(157), 241 Leszczynski, J., 23(215), 53 Leutwyler, S., 5(51), 13(108), 18(169), 44, 47, 50 Levering, L. M., 96(186), 102 Levesque, D., 66(106), 71(106), 100 Levine, E., 124(103), 238 Levinger, N. E.: 2(9), 41; 97(210–212), 103 Lewis, J., 107(1), 110(1), 128(1), 233 Lewis, R., 2(5), 6(64), 23(64), 24(5), 41, 44 Lewis, S., 145(190), 242 Li, H., 220(335), 250 Li, I., 62(71), 66(71), 76(71), 90(71), 94(71), 96 (71), 99 Li, L., 145(196), 233(196), 243 Li, S., 72(129), 101 Li, X., 20(193), 52 Li, Y. S., 26(222), 53 Liang, C., 155(229), 245 Liang, S., 130(149), 156(149), 240 Liashenko, A., 20(193), 52 Libourel, I. G. L., 163(268), 247
Liebermeister, W., 115(87–88), 135(161), 137(161), 181(161), 186(161), 187 (161,323), 210(161), 232(87–88), 237, 241, 249 Liedl, R., 21(194), 52 Lievin, J., 6(63), 8(63), 23(63), 44 Lilley, R. M., 147–148(203), 243 Lim, M., 69(115), 100 Limbach, H.-H., 8(73), 45 Lin, Y.-S., 62(52,59), 85(52), 86(59), 87(52,59), 88(59), 98–99 Lindenbergh, A., 111(21), 234 Lindgren, J., 67(113), 71(113), 72(130), 77 (113), 100–101 Lindner, J., 61(37), 98 Lindner, K., 5(41), 43 Lindsey, R. V. Jr., 32(248), 55 Linke, T., 147(202), 150(202), 243 Lipari, G., 70(122), 84(122), 100 Lipp, P., 165(280), 247 Lipson, R. H., 22(213), 31(213), 53 Lisec, J., 164(271), 247 Lisy, J. M., 15(134), 16(135), 48 Liu, B., 38(286), 57 Liu, D., 96(186), 102 Liu, G., 20(193), 52 Liu, J. R., 151(216), 164(270), 244, 247 Liu, P., 96(189–190), 102 Liu, Y., 16(139), 17(156), 26(156), 40(156), 48–49 Llaneras, F., 156(247), 245 Lloyd, D., 165(281–282), 171(300), 247–248 Lloyd, J. C., 123(125), 145(198), 168(125), 215–217(125), 219(125), 239, 243 Lock, A. J., 90(160), 101 Loew, L. M., 113(41), 116(41,92), 145(200), 235, 237, 243 Loewenschuss, A., 8(70), 27(225), 45, 53 Lomas, J. S., 3(22), 16(22), 42 Long, S. P., 124(135), 240 Lonien, J., 111(20), 158(254), 234, 246 Loparo, J. J., 61(10,14,23–26,47), 65(14), 70 (121), 72(121), 75(121), 77(10), 78 (10,121), 79(14), 83(10,23–26,121,149), 84(23–24,47), 86(24,121,149), 97–98, 100–101 Lopez, J. M., 8(73), 45 Lotta, T., 32(241), 54 Loutellier, A., 4(34), 18(34), 25(34), 43
author index Lovas, F. J., 16(143,146), 23(143), 33(251), 34 (146), 49, 55 Lowary, T. L., 38(281), 56 Lucio, D., 145(200), 243 Luck, W. A. P., 9(84), 45 Luckhaus, D., 11(89), 13(107), 17(152,155), 32 (155,243), 35–36(89), 38(283), 40(155), 46–47, 49, 54, 56 Ludwig, R., 5(45), 40(45), 43 Luedemann, A., 147(202), 150(202), 243 Lugez, C. L., 16(143), 23(143), 49 Luo, B., 150(214), 244 Lutz, B. T. G., 29(230), 31(234), 54 Lynch, M., 138(171), 241 Ma, G., 96(186), 102 Ma, H., 110(19), 111(30), 113(19), 151(222), 153(19), 234–235, 244 MacAleese, L., 15(132), 48 Macara, I., 113(41), 116(41), 235 Maccaferri, G., 38(282), 56 MacLean, E. J., 31(239), 54 Macleod, N. A., 2(17), 3(29), 38(17), 39(17,29), 42 Macrae, C. F., 5(42), 43 Madsen, D., 61(20–21), 83(20–21), 97 Madura, J. D., 83(150), 101 Maeda, A., 12(102), 16(102), 29(102), 46 Maerker, C., 21(194), 52 Maher, A. D., 123(131), 239 Maienfisch, P., 34(254), 55 Maisano, G., 62(66), 89(66), 99 Maiti, N., 11(100), 46 Maitre, P., 15(132), 48 Maksyutenko, P., 9(83), 26(83), 45 Malick, D. K., 20(193), 52 Maliepaard, C. A., 150–151(213), 244 Mallamace, F., 62(66), 89(66), 99 Manby, F. R., 20–21(192), 51 Manca, C., 13(104), 32(104), 46 Mandado, M., 23(220), 53 Mandel, M., 26(221), 53 Ma¨nnle, F., 8(73), 45 Mannucci, B., 20(193), 52 Manthe, U., 22(210), 32(210), 53 Mantsch, H. H., 37(272), 56 Marcus, Y., 8(70), 27(225), 45, 53 Mark, A. E., 34(259), 36(267), 55–56 Markus, M., 171(303), 175(303), 248 Maroncelli, M., 18(168), 50
265
Martens, D. E., 123(130), 239 Martı´, J., 64(88–89), 90(88–89), 99–100 Martin, J. M. L., 5(57), 21(57), 44 Martin, R. L., 20(193), 52 Martinez, T. J., 16(135), 48 Marx, A., 160(259), 162(259), 246 Mas, E. M., 21(201), 52 Mate´, B., 18(162), 50 Matsuda, T., 38(276), 56 Matsuda, Y., 18(167), 50 Matsumoto, M., 28(227), 53 Matsumoto, Y., 19(181), 51 Matsuura, H., 29(231), 54 Matsuzaki, Y., 113(40), 116(40), 144(40), 235 Matthews, J., 145(201), 243 Matthews, L., 145(190), 242 Matthysse, S. W., 220–221(334), 250 Mauch, K., 115(85–86), 123(129), 181(85), 184 (85–86), 237, 239 Maxton, P. M., 18(165), 50 May, O., 16(140), 48 Mazat, J. P., 123(128), 239 Mazein, A., 111(30), 235 Mazurek, U., 14(124), 38(124), 48 McCann, K. S., 222(336), 250 McClellan, A. L., 74(137), 101 McGregor, P. A., 5(44), 31(44), 40(44), 43 McGuire, J. A., 96(203), 103 McIntyre, L. M., 227(339), 250 McMahon, T. B., 15(132), 48 McQuade, C. R., 12(101), 27(101), 46 McQuarrie, D. A., 63(78), 68(78), 99 Meerts, W. Leo, 32(247), 55 Meijer, E. J., 22(212), 53 Meijer, G., 36(266), 56 Melandri, S., 16(145), 32(145), 49 Mendes, P., 107(4), 149(4), 144(178), 145 (185,200), 146(185), 151(216), 153(234), 154–155(178), 233(349), 233, 242–245, 250 Meng, T. C., 144(180), 242 Mensonides, F. I. C., 209(328), 249 Menten, M. L., 130(152), 132(152), 240 Mermet, A., 62(63), 89(63), 99 Mesarovic, M., 117(99), 238 Meyer, R., 23(214), 53 Michaelis, L., 130(152), 132(152), 240 Michels, P. A. M., 209(328), 249 Middleton, W. J., 32(248), 55 Mielke, Z., 11(88), 18(88), 25–26(88), 46
266
author index
Migliardo, P., 62(66), 89(66), 99 Mikami, N., 5(47), 15(47), 18(166–167), 19 (166,178,181), 23(218), 25(166), 43, 50–51, 53 Miki, K., 89(156), 101 Milet, A., 96(191), 102 Millam, J. M., 20(193), 51 Miller, D. R., 3(28), 42 Miller, M. A., 21(203), 52 Miller, R. J. D., 62(74), 95–96(74), 99 Millie´, Ph., 5(54), 14(54), 32(54), 44 Mil’nikov, G. V., 22(211), 53 Minch, E., 145(200), 243 Mir, Saqib, 145(193–194), 243 Miyazaki, M., 5(47), 15(47), 43 Miyoshi, F., 113(40), 116(40), 144(40), 235 Mjolsness, E. D., 145(200), 243 Mo, M. L., 151(221), 155(225), 244 Mo´, O., 21(195), 23(195), 52 Modak, J., 188(324), 249 Moilanen, D. E., 62(59), 86–88(59), 97(212–213), 99, 103 Moldenhauer, F., 114(65), 154(65), 236 Møllendal, H., 16(147), 49 Møller, K. B., 61(40,42), 71(40,125), 77 (40,125), 84(40,42), 98, 101 Mo¨llney, M., 160(260–262), 161(260–261), 162 (260), 163(261–262), 246 Mombach, J. C. M., 113(57), 153(57), 236 Mongru, D. A., 113(54), 153(54), 236 Monney, A., 16(140), 48 Monod, J., 138(168), 142(174), 241 Mons, M., 2(20), 5(20,50), 19(179), 32(179), 42–43, 51 Montero, F., 155(226), 244 Montero, S., 18(162–164), 50 Montgomery, J. A. Jr., 20(193), 51 Mooij, W. T. M., 20(188), 51 Moore, P. B., 63(79), 64(79,90), 90(79,90), 96 (197–200), 99–100, 102–103 Morgan, J. A., 113(37)165(37), 235 Morgenthal, K., 151(217), 153(217), 244 Morino, I., 32(242), 54 Morita, A., 75–76(140), 96(140,195–196), 101–102 Morohashi, M., 144(179), 242 Morokuma, K., 20(193), 52 Morowitz, H. J., 220–221(334), 250 Morresi, A., 18(171), 50 Morris, D. G., 5(43), 43
Mortenson, P. N., 21(203), 52 Mosquera, R. A., 23(220), 53 Moulin, V., 37(273), 56 Moxley, J., 112(35), 114(35), 156(35), 235 Mucha, M., 96(201), 103 Muir, K. W., 5(43), 43 Mukamel, S., 65(92), 66(101,103–104), 69–70(92), 71–72(128), 77(128,146), 100–101 Mukhopadhyay, I., 9(82), 23(82), 45 Mu¨ller, H., 32(246), 55 Mu¨ller, H. S. P., 38(278), 56 Mu¨ller, M., 96(204), 103 Mulquiney, P. J., 113(38), 123(38,123), 227 (123), 235, 239 Mundy, C. J., 96(188), 102 Munoz-Caro, C., 36(268), 56 Muraya, K., 28(227), 53 Murphy, W. F., 62(65), 89(65), 99 Murray, D., 165(281–282), 247 Murto, J., 32(241), 54 Musch, P., 155(228), 245 Myers, C. R., 232(344), 250 Naef, F., 165(279), 247 Nair, P. M., 16(148), 49 Nakai, H., 20(193), 52 Nakajima, T., 20(193), 52 Nakamura, H., 22(211), 53 Nakatsuji, H., 20(193), 52 Nakayama, Y., 113(42), 116(42), 145(200), 227 (341), 235, 243, 250 Nanayakkara, A., 20(193), 52 Neipert, C., 95(163), 96(198–200), 102–103 Nelson, M. R., 145(200), 243 Ne´methy, G., 142(176), 241 Nemukhin, A. V., 31(233), 54 Nesbitt, D. J., 6(62), 9(78), 11(86,90), 17(151), 44–46, 49 Neugebauer, J., 22(205), 52 Neumark, D. M., 16(137), 48 Nguyen, P. H., 66(96), 100 Ni, T. C., 227(340), 250 Nibbering, E. T. J.: 2(10), 41; 61(20–21), 62(73–74), 69(117), 83(20–21), 95–96 (73–74), 97, 99–100 Nibler, J. W., 18(168), 50 Nickerson, D., 145(199), 243 Nicodemus, R. A., 61(14), 62(60), 65(14), 79 (14), 85(60), 97, 99
author index Nicolis, G., 108(17), 171(17,298), 234, 248 Nielsen, I. M. B., 96(165), 102 Nielsen, J., 112(34), 113(49), 114(69), 149 (211), 154(69), 156(246), 184(322), 235–236, 244–245, 249 Nielsen, K., 171(304), 172(309), 248 Nielsen, O. J., 37(271), 56 Nielsen, P. F., 145(185,198–200), 146(185), 233 (349), 242–243, 250 Nielson, N., 62(72), 89–90(72), 94(72), 96(72), 99 Nienhuis, H.-K., 61(45), 84(45), 98 Nienhuys, H.-K.: 16(142), 26(142), 49; 61(46), 62(56), 84–85(46), 98–99 Nikolau, B. J., 151(216), 244 Nino, A., 36(268), 56 Nishi, N., 28(227), 53 Nizkorodov, S. A., 11(86), 45 Noisommit-Rizzi, N., 123(129), 239 Nordlander, B., 123(133), 240 Nosenko, Y., 19(176–177), 51 Notebaart, R. A., 151(219), 244 Novak, B., 117(98), 165(98,285), 168(98), 238, 247 Nunes-Nesi, A., 188(325), 191(325), 249 Nun˜o, J. C., 155(226), 244 Nyman, G., 96(169–170), 102 Ochterski, J. W., 20(193), 52 Ohashi, N., 16(143), 23(143), 49 Ohlrogge, J. B., 158(252), 160(252), 246 Ohno, K., 29(231), 54 Ojama¨e, L., 67(113), 71(113,124), 77(113,124), 100 Okuda, S., 145(188), 242 ¨ lcer, H., 123(125), 168(125), 215–217(125), O 219(125), 239 Oldiges, M., 150(214), 244 Oliver, S. G., 111(25), 151(216), 234, 244 Olivier, B. G., 145(197), 243 Olovsson, I., 3(31), 42 Oltvai, Z. N., 113(45,48,54), 114(73), 152 (45,48), 153(45,54), 156(73), 235–237 Ortega, F., 123(131), 239 Ortiz, J. V., 20(193), 52 Ostroverkhov, V., 96(180–181), 102 Othmer, H. G., 108(13), 108(13), 171(13), 234 Oxtoby, D. W., 66(106), 71(106,123), 100
267
Paarman, A., 62(74), 95–96(74), 99 Padro´, J. A., 64(88–89), 90(88–89), 99–100 Paesani, F., 96(173–174), 102 Pahle, J., 144(178), 154–155(178), 242 Pakoulev, A., 61(18–19), 62(18–19,76), 95(18– 19,76), 97, 99 Palacios, N., 144(184), 242 Paley, S., 145(189), 242 Palsson, B. Ø., 108(9), 111(29), 113(44,49–51), 114(50,61,63,66–72), 118(50), 124 (50,61,68), 125(50), 151(50–51,220–221), 153(61), 154(50,61,63,68–69,71,235,240), 155(50,225,231), 156(50,67,72), 212(329), 234–237, 244–245, 249 Pang, Y., 61(18–19,34–35), 62(18–19,77), 95 (18–19,77), 97–99 Panja, S., 13–14(113), 47 Panke, S., 130(150), 156(150), 240 Paolantoni, M., 18(171), 50 Papin, J. A., 114(66–67), 154(240), 156(67), 236, 245 Papp, B., 216(332), 250 Park, K., 63(84), 99 Park, S., 97(205), 103 Parrinello, M., 64(91), 90(91), 100 Parsons, S., 5(44), 31(44), 40(44), 43 Passarge, J., 123(121,126), 168(126), 172(126), 175(126), 202(121,126), 203–205(126), 239 Passchier, W. F., 26(221), 53 Pate, B. H., 16(148), 49 Pate, C. B., 26(222), 53 Paton, N. W., 151(216), 244 Paul, J. B., 8(75), 13(110), 16(110), 17(75,157), 23(75), 28–29(157), 38(110), 45, 47, 49 Pearson, J. C., 12(102), 16(102), 29(102), 46 Pegram, K. M., 97(107), 203 Peng, C. Y., 20(193), 52 Perchard, J. P., 4(34), 11(88,95), 13(103), 18 (34,88,103), 25(34,88), 26(88), 28(103), 31 (103), 43, 46 Peremans, A., 4(34), 18(34), 25(34), 43 Perera, A., 2(13), 31(13), 36(13), 41 Perez-Melis, A., 164(270), 247 Perez-Ortega, A., 34(263), 55 Perry, A., 95(163), 96(197–200), 102–103 Perry, D. S., 9(82–83), 23(82,217), 26(83), 45, 53 Perttila, M., 34(262), 37(262), 55 Petersen, S., 160(262), 163(262), 246
268
author index
Petterson, G. A.: 20(193), 52; 123(113), 168 (113), 215–217(113), 238 Pfeiffer, T., 114(65), 154(65), 155(226), 236, 244 Piccirillo, S., 5(49), 14(119), 43, 47 Pichugin, A. V., 227(338), 250 Pickett, H. M., 33(251), 55 Pico, J., 156(247), 245 Pierard, A., 138(169), 241 Piletic, I. R., 97(212), 103 Piletic, R., 2(8–9), 41 Pimentel, G. C., 74(136–137), 101 Pires, M. M., 5(38), 21(38), 43 Piryatinski, A., 61(41), 83(41,147), 84(41), 98, 101 Piskorz, P., 20(193), 52 Pitzer, K. S., 2(14), 6(14), 41 Piuzzi, F., 2(20), 5(20,50), 42–43 Platonova, O. V., 227(338), 250 Plu¨tzer, C., 19(182), 32(182), 51 Poliakoff, M., 3(24), 16(24), 29(24), 42 Pomelli, C., 20(193), 52 Pommeret, S., 61(32–33), 98 Poolman, M. G., 123(214–125), 154(241), 168 (125), 215–217(124–125,331), 219(125), 239, 245, 250 Pople, J. A., 20(193), 52 Poulsen, J. A., 96(169–170), 102 Pratt, D. W., 16(150), 49 Pribble, R. N., 19(180), 51 Price, J. M., 6(64), 23(64), 44 Price, N. C., 128(140), 133(140), 137–138(140), 141–142(140), 240 Price, N. D., 114(66–68), 124(68), 154(68,240), 156(67), 236, 245 Prigogine, I., 108(17), 171(17), 234 Pring, M., 113(36), 119(36), 235 Probst, M., 71(124), 77(124), 100 Procacci, P., 38(280), 56 Provencal, R. A., 8(75), 17(75,157), 23(75), 28–29(157), 45, 49 Pshenichnikov, M. S., 61(22), 66(100), 83(22), 98, 100 Pyckhout, W., 32(240), 54 Qian, H., 130(149), 156(149), 182(319), 240, 249 Quack, M., 6(67), 7(68), 9(79), 13(107,109), 17 (152), 21(194,202), 39(289–290), 40(79), 44–45, 47, 49, 52, 57
Quastler, H., 220–221(334), 250 Quentin, E., 123(128), 239 Raamsdonk, L. M., 111(25), 234 Rabuck, A. D., 20(193), 52 Racine, S., 4(34), 18(34), 25(34), 43 Radnai, T., 34(264), 36(264), 55 Raff, M., 107(1), 110(1), 128(1), 233 Raghavachari, K., 20(193), 52 Raines, C. A, 123(125), 154(241), 168(125), 215–217(125), 219(125), 239, 245 Raı¨s, B., 123(128), 239 Raman, K., 155(232), 245 Ramı´rez, M. J., 168(295), 248 Ramos, A., 18(163), 50 Rapoport, S. M., 108(12), 113(12), 123(108), 227(108), 234, 238 Rapoport, T. A., 108(12), 113(12), 123(108), 176(314), 227(108), 234, 238, 249 Rapp, P. E., 164(272), 247 Ra¨sa¨nen, M., 32(241), 54 Ratcliffe, R. G., 158(256), 163(267), 246–247 Rath, N. S., 74(138), 101 Ratzer, C., 16(149), 32(149,247), 49, 55 Rauscher, C., 61(16–17), 84(16), 97 Ravasz, E., 113(54), 153(54), 236 Raveendran, P., 5(46), 43 Raymond, E. A., 96(182–184), 102 Record, M. T. Jr., 97(107), 203 Reder, C., 114(62), 153(62), 117(2), 236 Redington, R. L., 32(244), 54 Redington, T. E., 32(244), 54 Reed, A. E., 77(144), 101 Reed, J. L., 212(329), 249 Rees, F. S., 16(148), 49 Rega, N., 20(193), 52 Reichelt, A., 16(149), 32(149), 49 Reid, P. J., 16(141), 37(141), 48 Reiher, M., 14(122), 22(122,205), 37(122), 47, 52 Reijenga, C. A., 123(121), 202(121), 239 Reimers, J. R., 90(159), 101 Reuss, J., 5(52), 44 Reuss, M., 115(85–86), 123(120,122,129), 181 (85), 184(85–86), 237, 239 Rey, R., 61(40,42), 71(40,125), 77(40,125), 84 (40,42), 98, 101 Rezus, Y. L. A., 61(48), 62(48,58), 84(48), 86 (48,58), 98–99 Rhea, S. T., 62(75), 95(75), 99
author index Rhee, S. Y., 145(189), 151(216), 242, 244 Rhodes, D., 113(37)165(37), 235 Ricci, M. A., 64(87), 90(87), 99 Riccobono, J., 12(102), 16(102), 29(102), 46 Rice, Corey A., 8(72), 16(139), 20(72), 31(72), 36(72), 45, 48 Rice, S., 61(7), 62(7,72), 67(7), 76(7), 89–90 (7,72), 94(7,72), 96(72), 97, 99 Richmond, G. L., 62(71), 66(71), 76(71), 90 (71), 94(71), 96(71,182–184,194), 99, 102 Richter, W., 18(161), 50 Rick, S. W., 83(152), 101 Ridley, C., 95(163), 96(199), 102–103 Riehn, C., 19(177), 51 Riley, K. E., 5(56), 44 Rios-Estepa, R., 113(39), 144(39), 235 Rizio Speranza, M., 5(49), 43 Rizzi, M., 123(120,122), 239 Rizzo, T. R., 9(82–83), 13(107), 23(82,217), 26 (83), 45, 47, 53 Robb, M. A., 20(193), 51 Robert, M., 113(42), 116(42), 235 Roberts, K., 107(1), 110(1), 128(1), 233 Roberts, S. T., 61(10,14,24–26,47), 65(14), 70 (121), 72(121), 75(121), 77(10), 78 (10,121), 79(14), 83(10,24–26,121,149), 84(24,47), 86(24,121,149), 97–98, 100–101 Robertson, E. G., 19(179), 32(179), 51 Robertson, W. H., 16(136), 48 Roccatano, D., 34(259), 36(267), 37(270), 55–56 Roessner-Tunali, U., 147(202), 149(211), 150(202), 151(216), 164(270), 243–244, 247 Rogers, A., 111(20), 158(254), 234, 246 Rohwer, J. M.: 123(127), 239; 137(164), 182 (164), 187(164), 241 Rojas, I., 145(193–194), 243 Rønnow, T. E. C. L., 33(253), 55 Roscher, A., 158(256), 246 Rosenfeld, N., 208(327), 249 Rossell, S., 111(21), 234 Rossky, P. J., 96(168–170), 102 Roth, K., 8(75), 17(75,157), 23(75), 28–29 (157), 45, 49 Roth, W., 29(229), 54 Roubin, P., 13(104), 32(104), 46 Roussel, M. R., 171(300–301), 248 Rowland, J. J., 111(25), 234
269
Roy, R., 62(63), 89(63), 99 Rozenberg, M., 8(70), 27(225), 31(238), 45, 53–54 Rude, B. S., 2(12), 41 Rueda, D., 9(82), 23(82), 45 Ruelle, D., 165(291), 248 Ruszel, W. M., 95–96(164), 102 Ryde-Petterson, U., 165(276), 219(276), 247 Ryder, K. S., 5(43), 43 Sabo, D., 13(108), 47 Sackmann, A., 155(245), 245 Sadlej, J.: 15(133), 48; 73(134), 75–76(134), 101 Saeki, M., 5(55), 31(235), 32(55), 44, 54 Saenger, W., 5(41), 43 Saez-Rodriguez, J., 154–155(224), 244 Sahle, S., 144(178), 154–155(178), 242 Sairam, A., 144(180), 242 Saito, K., 113(40), 116(40), 144(40), 151(216), 235, 244 Sakai, M., 5(55), 31(235), 32(55), 44, 54 Sakharkar, K., 144(180), 242 Sakurada, T., 145(200), 243 Salvador, A., 144(177), 242 Salvador, P., 20(193), 52 Sa´nchez-Valdenbro, I., 155(226), 244 Sandorfy, C., 9(85), 45 Sanz, E., 89(155), 101 Sartakov, B. G., 8(74), 45 Sassi, P., 18(171), 50 Satta, M., 14(119), 47 Sauer, J., 13(106), 21(199), 23(199), 25(106), 26 (199), 47, 52 Sauer, U., 110(24), 112(31), 138(31), 163(266), 234–235, 246 Sauro, Herbert M., 144(181), 145(185,196, 200), 146(185), 233(196,349), 242–243, 250 Savageau, M. A., 129(142,144–146), 137 (142,144), 181(316–317), 182(142, 144–146), 227(340), 240, 2490250 Saven, J. G., 65(93–94), 100 Saykally, R. J.: 2(4,12), 8(75), 17(75,157), 23 (75), 28–29(157), 41, 45, 49; 61–62(12), 77(12), 97(206,209), 97, 103 Scalmani, G., 20(193), 52 Scatena, L. F., 96(182), 102 Sceats, M. G., 62(69), 89(69), 94(69), 99
270
author index
Schaal, H., 5(39), 7(39), 20(39), 31(39), 37 (269), 43, 56 Schachar-Hill, Y., 158(252), 160(252), 163 (267–268), 246–247 Schaefer, H. F. III, 8(75), 17(75,157), 23 (75,219), 28–29(157), 37(219), 45, 49, 53 Schaeffer, M. W., 18(165), 50 Schaff, J. C., 113(41), 116(41,92), 145(200), 235, 237, 243 Schaller, R. D.: 2(12), 41; 3(30), 8(30), 12(30), 15(30), 19(30), 34–37(30), 40(30), 42 Schalley, C. A., 2(15), 14(15), 38(15), 42 Scharge, T., 3(30), 8(30,76), 11(89), 12(30), 15 (30,76,130), 19(30), 33(130), 34(30,130), 35–36(30,76,89), 37(30), 40(30), 42, 45–46, 48 Schenter, G. K., 96(172), 203 Schepper, M., 123(121), 202(121), 239 Scherer, J. R., 62(49), 76(141–142), 89(49), 98, 101 Scherrenberg, R., 31(234), 54 Schettino, V., 38(280), 56 Schibler, U., 165(279), 247 Schiel, D., 18(161), 50 Schilling, C., 114(70), 154(235), 237, 245 Schilstra, M., 145(196,201), 233(196), 243 Schirmer, H., 155(228), 245 Schlegel, H. B., 20(193), 51 Schmid, J. W., 115(86), 123(129), 184(86), 237, 239 Schmidt, D. A., 89(156), 101 Schmidt, E., 145(190), 242 Schmidt, H., 144(182), 242 Schmidt, J. R., 66–67(98), 69(120), 70(120– 121), 72(120–121,129), 75(121), 73(98), 75(98), 77(98,143), 78(98,120–121), 79 (98), 81–82(98), 83(98,120–121,148), 84 (98), 86(121), 87(98), 100–101 Schmitt, M., 16(149), 19(182), 32 (149,182,246–247), 49, 51, 55 Schmitt, U., 6(65), 8(65), 9(79), 11(65), 17 (65,152,158), 23(65), 26(65,158), 28–29(65), 31(65), 36(65), 40(79), 44–45, 49 Schneider, W. F., 37(271), 56 Schnitzer, C., 96(185), 102 Schomburg, D., 145(191–192), 242 Schomburg, I., 145(191–192), 242 Schreiber, F., 155(245), 245
Schriver, A., 37(273), 56 Schriver-Mazzuoli, L., 37(273), 56 Schultz, M. J., 62(71), 66(71), 76(71), 90(71), 94(71), 96(71), 99 Schulz, F., 21–22(204), 52 Schuster, S., 108(14,16), 111(26), 113(14,16), 114(64–65), 116(96), 123(119), 124 (64,138), 125(96), 126(138), 128–129(96), 133–134(96), 137(96), 154(64–65,96,138, 235–238,240), 155(226–229), 165(96), 167–168(96), 171–172(96), 177–180(96), 195(96), 227(119), 234, 236–237, 239–240, 244–245 Schu¨tz, A., 3(30), 8(30), 12(30), 15(30), 19(30), 34–37(30), 40(30), 42 Schwabe, T., 21(198), 52 Schwarz, R., 155(228–229), 245 Schwarzer, D., 61(37), 98 Schwender, J., 111(20), 158(252), 160(252), 234, 246 Schwettman, H. A., 69(114), 100 Screen, J., 38(286), 57 Scuseria, G. E., 20(193), 51 Sefkow, M., 150–151(212), 244 Segel, I. H., 128(139), 137(139), 240 Sehested, J., 37(271), 56 Seifert, G., 61(27), 98 Selbig, J., 115(84), 144(84,184), 168(296), 188 (84,296,325), 190(296), 191(84,325), 197 (84), 202(84), 205–206(84), 211(84), 217 (84), 224(296), 226(296,337), 227–231 (296), 237, 242, 248–250 Selkov, A., 111(30), 235 Selkov, E. E., 111(30), 116(94), 123(94), 165 (272), 168(272), 171(94,272), 172(272), 175(94), 202(94), 235, 238, 247 Senent, M. L., 34(263), 36(268), 55–56 Senior, W. A., 62(51), 98 Sepio, N., 4(36), 19(36), 26(36), 32(36), 43 Sepiol, Katia, 14(121), 19(121), 47 Serrallach, A., 23(214), 53 Serrano-Gonza´les, H., 31(239), 54 Sethna, J. P., 232(344), 250 Seurre, N., 4(35–36), 7–8(35–36), 14(121,126), 19(35–36,121), 26(36), 32(35), 38(126), 43, 47–48 Shanks, J. V., 160(263), 246 Shapiro, B. E., 145(185,196,200–201), 146 (185), 233(196,349), 242–243
author index Shearer, A. G., 145(189), 242 Shelton, W. N., 23(216), 53 Shen, Y. R., 96(177–181,203), 102–103 Shi, Y. J., 22(213), 31(213), 53 Shimizu, T. S., 113(40), 116(40), 144(40), 145 (200), 235, 243 Shubert, V. A., 19(183), 51 Shultz, M. J., 96(185), 102 Sieben, A., 13(109), 47 Siebers, J.-G., 18(160,174), 20(189), 22–23 (189), 25(160,174), 50–51 Siefke, C., 160(259), 162(259), 246 Siezen, R. J., 151(219), 244 Signorell, R., 38(283–284), 39(284,287–288), 56–57 Silvestrelli, P. L., 64(91), 90(91), 100 Simeonidis, E., 115(89), 137(89), 181(89), 184 (89), 237 Simeonidis, K., 61(28), 98 Simon, W., 123(106), 238 Simonelli, D., 96(185), 102 Simons, J. P., 2(17), 3(29), 19(179), 32(179), 38 (17,285–286), 39(17,29), 42, 51, 57 Sims, A. P., 163(269), 247 Simus, N., 144(178), 154–155(178), 242 Singer, S. J., 38(281), 56 Singhal, M., 144(178), 154–155(178), 242 Skinner, J. L., 60(4), 61(6,13,15,38–39,41), 62 (52–55,59), 63(83–84,86), 65(4,13, 93–94), 66(98–99,107), 67(98–99), 68 (99), 69(120), 70(120–121), 71(15,38–39), 72(6,13,99,120–121,129,131), 73(98), 75 (6,13,98,121), 76(6,99), 77(15,38–39,98, 143), 78(6,13,98,120–121,131), 79(15, 38–39,98), 81–82(98), 83(41,98, 120–121,147–148,151), 84(15,38–39,41, 98), 85(52–55), 86(55,59,121), 87 (38,52,59,98,154), 88(59), 90(6,99,161), 92(6), 93(6,161), 94(6), 96(86), 97–101 Slepchenko, B. M., 113(41), 116(41), 235 Slipchenko, M. N., 8(74), 45 Smallbone, K., 115(89), 137(89), 181(89), 184 (89), 237 Smedsgaard, J., 149(211,216), 244 Smeyers, Y. G., 36(268), 56 Smith, A. R., 151(216), 244 Smith, F. A., 4(37), 31(37), 43 Smith, J. D., 61–62(12), 77(12), 97(206), 97, 103 Smith, T. L., 69(114), 100
271
Smits, M., 96(204), 103 Snoek, L. C., 2(17), 38(17,286), 39(17), 42, 57 Snoep, J. L., 123(121,126,130), 137(164), 145 (185,196–197), 146(185), 168(126), 172 (126), 175(126), 182(164), 187(164), 202 (121,126), 203–205(126), 233 (196,347,349), 239, 241–243, 250 Snow, M. S., 14(129), 16(129), 37(129), 48 Snyder, R. G., 76(141), 101 Soetens, J.-C., 3(25), 16(25), 18(25), 31(25), 42 Sokolic, F., 2(13), 31(13), 36(13), 41 Solca`, N., 15(131), 48 Somani, S., 144(180), 242 Somera, A. L., 113(54), 153(54), 236 Somesen, O. J. G., 123(126), 168(126), 172 (126), 175(126), 202(126), 203–205(126), 239 Sonnewald, U., 158(257), 246 Soper, A. K., 31(236), 54 Sorensen, C. M., 62(70), 89(70), 94(70), 99 Sørensen, P. G., 121(101), 123(101), 165(274), 171(304), 172(101,274,309), 202–203 (101), 238, 247–248 Sorokin, A., 111(30), 235 Sourik, V., 115(91), 117(91), 237 Space, B., 63(79), 64(79,90), 90(79,90), 95 (163), 96(197–200), 99–100, 102–103 Spangenberg, D., 29(229), 54 Speer, A., 155(245), 245 Spence, H. D., 145(185,200), 146(185), 233 (349), 242–243, 251 Speranza, M., 14(118–119), 47 Spry, D. B., 97(212–213), 103 Sridharan, M., 34(260), 36(260), 55 Sriram, G., 160(263), 246 Staemmler, M., 11(97), 26(97), 46 Stanca-Kaposta, E. C., 38(286), 57 Starrenburg, M. J. C., 123(130), 239 Starzyk, A., 34(260), 36(260), 55 Stefanov, B. B., 20(193), 52 Stein, L., 145(190), 242 Steinbach, C., 18(174), 25(174), 37(275), 50, 56 Steinel, Tobias: 2(11), 41; 62(53–55,57), 83 (151), 85(53–55,151), 86(55,57), 98–99, 101 Steiner, T., 3(23), 42 Steinsvoll, K., 33(253), 55 Stelling, J., 114(64), 124(64), 145(200), 154 (64,239–240), 236, 243, 245 Stenger, J., 61(20–21), 83(20–21), 97
272
author index
Stephanopoulos, G. N., 112(34–35), 114(35), 156(35), 235 Stephens, M. D., 65(94), 100 Stern, H. A., 96(171), 102 Sterrer, M., 96(204), 103 Steuer, R., 111–112(23), 114(58), 115(23,84), 116(23), 144(84), 151(58,216,222), 152 (223), 153(23,58,216,233), 168(296), 188 (23,84,296,325), 190(296), 191(23,84, 325), 197(84), 202(84), 205–206(84), 211 (84), 217(84), 221(58), 224(296), 226 (296,337), 227–231(296), 234, 236–237, 244–245, 248–250 Stevens, C., 145(199), 243 Stevens, L., 128(140), 133(140), 137–138(140), 141–142(140), 240 Stitt, M., 144(184), 147–148(203), 149(205), 158(253), 242–243, 246 Stock, G., 66(96), 100 Stohner, J., 7(68), 44 Stoleriu, I., 196(326), 249 Stolte, S., 5(52), 44 Straatsma, T. P., 73(135), 90(135), 101 Strain, M. C., 20(193), 52 Stratmann, R. E., 20(193), 52 Strogatz, S. H., 165(290), 248 Stuart, S. J., 83(152), 101 Suenram, R. D., 16(146), 33(251), 34(146), 49, 55 Suhm, M. A., 2(16), 3(30), 5(39,46, 48,58), 6 (61–62,65,67), 7(39,58,61,68–69), 8 (16,30,58,65,72,76), 9(16,77,79–80), 10 (16,80), 11(16,65,89–90,92–93), 12 (30,93), 13(80,111), 14(80,122,125–127), 15(30,76,127,130), 16(48,69,139), 17 (65,93,152–156,158), 18(16,69,77,80), 19 (30,69), 20(39,72), 21(194,202), 22(80), 23 (16,58,65,77), 24(16), 25(16,77,156,158), 26(61,65,93), 27(80,93), 28(65,77,80), 29 (58,65,69,80), 30(69,77), 31(39,65,72, 233), 32(155,245), 33(130,154), 34 (30,130), 35(30,76,89), 36(30,65,72, 76–77,80,89), 37(30,125,269,274–275), 38(111,126,154), 39(287,289), 40 (16,30,48,69,79–80,93,155–156), 42–49, 52, 54, 56–57 Sulpice, R., 164(271), 247 Sumner, L. W., 107(4), 149(4), 151(216,218), 233, 244 Sundlass, N., 83(148), 101
Superfine, R., 96(177), 102 Suzuki, T., 6(59), 38–39(59), 44 Sweetlove, L. J., 111(22), 153(22), 224(22), 234 Swofford, R. L., 35(265), 55 Szabo, A., 70(122), 84(122), 100 Szalewicz, K., 21(201), 52 Szathmary, E., 216(332), 250 Szczeponek, K., 96(176), 102 Szyperski, T., 161(265), 246 Tadjeddine, A., 4(34), 18(34), 25(34), 43 Takahashi, K., 113(40), 116(40), 144(40), 145 (200), 235, 243 Takahashi, S., 28(227), 53 Takamuku, T., 28(227), 53 Takors, R., 116(97), 119–120(97), 143(97), 150 (214), 233(350), 238, 244 Talbot, F. O., 38(285), 57 Tamimura, N., 144(179), 242 Tanaka, A., 28(227), 53 Tanaka, K., 32(242), 54 Tanaka, T., 32(242), 54 Tang, Y., 115(80), 168(80), 171(80), 195(80), 237 Tanida, S., 113(40), 116(40), 144(40), 235 Tarbuck, T., 62(71), 66(71), 76(71), 90(71), 94 (71), 96(71,183–184), 99, 102 Tassaing, T., 3(25), 16(25), 18(25), 31(25), 42 Taylor, J., 151(216), 244 Taylor, R., 5(42), 32(250), 43, 55 Tegenfeldt, J., 67(113), 71(113), 77(113), 100 Tejeda, G., 18(162–164), 50 Termonia, Y., 171(308), 175(308), 202(308), 248 Teso, Vilar, E., 5(43), 43 Teusink, B., 111(25), 123(121), 151(219), 175 (311), 202(121), 208(311), 209(311,328), 234, 239, 244, 249 Tewari, Y. B., 145(195), 243 Theobald, U., 123(120), 239 Theule´, P., 13(104), 32(104), 46 Thomas, S., 123(124), 215–217(124), 239 Thorburn, D. R., 227(339), 250 Thummel, R. P., 19(176–177), 51 Tiessen, A., 149(205), 158(257), 243, 246 Tissier, C. A., 145(189), 242 Tobias, D. J., 96(201), 103 Toennies, J. P., 18(164), 50 Toja, D., 5(49), 43 Tokimatsu, T., 145(188), 242
author index Tokmakoff, A., 61(10,14,23–26,43,47), 62(60), 65(14), 67(112), 69(114), 70(121), 71 (43,126), 72(121), 75(121), 77(14,43,126), 78(10,121), 79(14), 83(10,23–26,43,121, 149), 84(23–24,43), 85(60), 86 (24,47,121,149), 97–101 Tomasi, J., 20(193), 51 Tombor, B., 113(45), 152–153(45), 235 Tomita, M., 113(40,42–43), 116(40,42), 123 (43), 144(40), 145(200), 227(341), 235, 243, 250 Toplan, S., 202(342), 229(342), 250 Torii, H., 66(97), 76(97), 90(97), 95(97), 100 Toukan, K., 64(87), 90(87), 99 Toyota, K., 20(193), 52 Trethewey, R. N., 107(6), 149(6,207), 234, 243 Trivella, A., 13(104), 32(104), 46 Trosell, E., 12(102), 16(102), 29(102), 46 Trucks, G. W., 20(193), 51 Tschumper, G. S., 8(75), 17(75,157), 23 (75,219), 28–29(157), 37(219), 45, 49, 53 Tyson, J. J., 108(13), 117(98), 133(156), 165 (98,284–285), 168(98), 171(13), 234, 238, 241, 247 Ueberschaer, R., 39(288), 57 Umbarger, H. E., 137(166–167), 241 Uras-Aytemiz, N., 15(133), 48 Urbanczik, Robert, 155(230), 245 Urbanczyk-Wochniak, E., 151(218), 244 Va´cha, R., 96(191), 102 Vaidyanathan, S., 107(3), 149(3), 151(3), 233 Van Alsenoy, C., 32(240), 54 Van Dam, K., 111(25), 123(121), 175(311), 184 (321), 202(121), 208–209(311), 234, 239, 249 Van den Berg, C. J., 123(207), 238 Van den Broek, M. A. F. H., 16(142), 26(142), 49 Van den Enden, L., 32(240), 54 Van der Heijden, R., 115(85), 181(85), 184(85), 237 Van der Maas, J. H., 29(230), 31(234), 54 Van der Weijden, C. C., 111(21), 123(121), 202 (121), 234, 239 Van Duijneveldt, F. B., 3(32), 5(52), 20(188), 39 (32), 42, 44, 51 Van Duijneveldt-van de Rijdt, J. G. C. M., 3(32), 5(52), 20(188), 39(32), 42, 44, 51
273
Van Eijck, Bouke P., 20(188), 51 Van Enckevort, F. H. J., 151(219), 244 Van Gunsteren, W. F., 38(279), 56 Van Hoek, P., 209(328), 249 Vanhouteghem, F., 32(240), 54 Van Santen, R. A., 61(45), 84(45), 98 Van Solinge, W. W., 111(28), 235 Van Swam, I. I., 123(130), 239 Van Tuijl, A., 111(21), 234 Van Wijk, R., 111(28), 235 Varga, Z., 32(249), 55 Varma, A., 114(61), 124(61), 153–154(61), 236 Vaseghi, S., 123(122), 239 Vastrik, I., 145(190), 242 Vega, C., 89(155), 101 Vener, M. V., 13(106), 25(106), 47 Venkatesan, V., 23(218), 53 Venter, J. C., 113(40), 116(40), 144(40), 235 Verhoeven, H. A., 150–151(213), 244 Verlet, J. R. R., 16(137), 48 Verrall, R. E., 62(51), 98 Vicsek, T., 114(73), 156(73), 237 Viel, A., 22(210), 32(210), 53 Vigeolas, H., 149(205), 243 Vilesov, A. F., 8(74), 45 Villas-Boas, S. G., 149(211), 244 Vinci, B., 124(137), 240 Vinnakota, K. C., 124(136), 137(162), 240–241 Visser, D., 115(85–86), 181(85), 184(85–86), 237 Vitvitsky, V. M., 227(338), 250 Vliegenthart, J. A., 3(32), 39(32), 42 Vo¨hringer, P., 61(37), 98 Voit, E. O., 129(147), 182(147), 240 von Helden, G., 36(266), 56 Von Kamp, A., 155(227–228), 244–245 Von Rague´ Schleyer, P., 21(194), 52 Voth, G. A.: 20(193), 52; 96(173–174), 102 Vozzi, F., 124(137), 240 Vozzi, G., 124(137), 240 Vreven, T., 20(193), 51 Wagner, A., 113(46,52), 144(183), 152–153 (52), 155(183), 235–236, 242 Wagner, C., 150–151(212), 244 Wagner, J., 145(200), 243 Wales, D. J., 21(203), 52 Walk, T. C., 145(189), 242 Walker, D. A., 165(275), 247 Walker, D. S., 96(194), 102
274
author index
Wall, T. T., 61(8), 89(8), 97 Wallington, T. J., 37(271), 56 Walrafen, G. E., 61(11), 62(11,64,67), 89 (64,67), 90(67), 97, 99 Walsh, M. C., 111(25), 123(121), 175(311), 202 (121), 308–309(311), 234, 239, 249 Walsh, S., 151(216), 244 Walsh, T., 123(106), 238 Walsh, T. R., 21(203), 52 Walter, P., 107(1), 110(1), 128(1), 233 Waluk, J., 19(176–177), 51 Wanderlingh, F., 62(66), 89(66), 99 Wandrey, C., 150(214), 244 Wang, J., 145(200), 243 Wang, J. M., 20(187), 51 Wang, T., 151(216), 244 Wang, X.-G., 22(209), 53 Wang, Z., 61(18–19,34–35), 62(18–19,76–77), 95(18–19,76–77), 97–99 Wanner, B. L., 145–146(185), 233(349), 242, 251 Warner, H. E., 16(146), 34(146), 49 Wasserman, T. N., 7(69), 16(69), 17(155), 18– 19(69), 29–30(69), 32(155), 40(69,155), 45, 49 Watanabe, H., 29(231), 54 Watanabe, T., 18–19(166), 25(166), 50 Waterfall, J. J., 232(344), 250 Watson, T. M., 66(102), 100 Watts, R. O., 90(159), 101 Wawer, I., 8(73), 45 Waychunas, G. A., 96(180), 102 Weckwerth, W., 149(210), 151(217), 153 (217,233), 244–245 Wei, X., 96(180), 102 Weidemann, Andreas, 145(193–194), 243 Weimann, M., 6–7(61), 17(156), 26(61,156), 37 (275), 40(156), 44, 49, 56 Weinhold, F.: 6(66), 44; 77(144–145), 101 Weis, J.-J., 66(106), 71(106), 100 Weise, J., 226(337), 250 Wellock, Cameron, 144(181), 242 Weltner, W. Jr., 2(14), 6(14), 41 Werhahn, O., 18(173), 23(173), 50 Werner, H.-J., 20–21(192), 51 Westerhoff, H. V., 108(9), 111(21,25), 123 (121,126,130), 168(126), 172(126,310), 175(126,311), 184(321), 202 (121,126, 310), 203–205(126), 208(311), 209 (311,328), 234, 239, 249
Westphal, A., 16(149), 32(149), 49 Whalley, E.: 5(40), 17(40), 26–27(40), 43; 74–75(139), 101 Wheatley, R. J., 20(189), 22–23(189), 51 White, S. R., 63(81), 99 Wiback, S. J., 111(29), 235 Wickleder, C., 5(51), 44 Wiechert, W., 116(97), 119–120(97), 143(97), 157(248), 160(248,258–262), 161 (248,260), 162(248,258–260), 163 (248,261–262), 233(350), 238, 246, 251 Wiedemann, H., 26(223), 53 Wierling, C., 115(87), 140(172), 237, 241 Wiersma, D. A., 61(22), 83(22), 98 Willeke, M., 13(107,109), 47 Willmitzer, L., 107(6), 147(202), 149(6,207), 150(202), 158(257), 164(170), 234, 243, 246–247 Wilson, C. C., 31(239), 54 Wilson, E. B., 61(5), 97 Wilson, K. R.: 2(12), 5(53), 41, 44; 61–62(12), 63(81), 77(12), 97, 99 Winder, K., 155(245), 245 Wittig, Ulrike, 145(193–194), 243 Wo´jcik, M. J., 96(176), 102 Wolf, J., 123(126), 168(126), 172(126), 175 (126), 202–205(126), 239 Wolkenhauer, O., 117(99), 238 Wong, M. W., 20(193), 52 Woods, K. N., 26(223), 53 Woutersen, S., 61(36,44,46), 62(56), 84(44,46), 96(167), 98–99, 102 Wright, B., 123(106,116–117), 238, 239 Wright, J., 144(183), 155(183), 242 Wu, F., 124(136), 137(162), 240–241 Wu, G., 145(190), 242 Wugt Larsen, R., 2(16), 8–10(16), 11(16,93), 12 (93), 17(93), 18(16), 23–25(16), 26–27 (93), 40(16,93), 42, 46 Wu¨lfert, S., 18(169), 50 Wurtele, E. S., 151(216), 244 Wurzel, M., 160–162(260), 246 Wuttke, R., 158(257), 246 Wyman, J., 142(174), 241 Wyss, H. R., 62(50), 98 Xantheas, S. S., 96(166,172), 102 Xu, L., 144(178), 154–155(178), 242 Xu, L. H., 16(146), 34(146), 49
author index Xu, Y., 16(144), 49 Yamada, K. M. T., 32(242), 54 Yamada, Y., 2(21), 9(21), 19(21), 31–32(21), 40 (21), 42 Yamaguchi, T., 3(24,26), 16(24), 29(24), 42 Yamaguchi, Y., 28(227), 53 Yamanishi, Y., 145(188), 242 Ya´n˜ez, M., 9–10(80), 13–14(80), 18(80), 21 (195), 22(80), 23(195), 27–29(80), 36(80), 40(80), 45, 52 Yang, F., 124(136), 240 Yang, W. H., 62(67), 89–90(67), 99 Yazyev, O., 20(193), 52 Ye, L., 144(180), 242 Yeremenko, S., 61(22), 83(22), 98 Ygit, 202(342), 229(342), 250 Yi, J. T., 16(150), 49 Yoshida, H., 29(231), 54 Young, R. M., 16(137), 48 Yu, X., 14–15(128), 48 Yugi, K., 113(40), 116(40), 144(40), 235 Zakrzewski, V. G., 20(193), 52 Zamboni, N., 163(266), 246 Zamora Lo´pez, G., 152(223), 224 Zehnacker, A., 5(50), 14–15(127), 17(154), 33 (154), 38(154), 43, 48–49 Zehnacker-Rentien, A., 4(35–36), 5(50,54), 7–8(35–36), 13(112–113), 14(54, 112–113,120–121,126), 19(35–36,
275
120–121), 26(36), 32(35,54,120), 37(112), 38(126), 43–44, 47–48 Zeng, A.-P., 110(19), 113(19), 153(19), 188 (324), 234, 249 Zeuch, T., 5(48), 16(48), 40(48), 43 Zhabotinzky, A. M., 227(338), 250 Zhang, D., 14(123), 38(123), 47 Zhang, N., 111(25), 234 Zhang, P., 145(189), 242 Zhang, W., 96(173), 102 Zhang, Y. J., 97(208), 103 Zheng, J., 62(57), 69(118), 86(57), 99–100 Zheng, W. Q., 4(34), 11(95), 13(103), 18 (34,103), 25(34), 28(103), 31(103), 43, 46 Zhu, X.-G., 124(135), 240 Zhuang, W., 66(101,104), 71–72(128), 77 (128,146), 100–101 Zielke, P., 2(16), 3(30), 7(69), 8(16), 9(16,77), 10(16), 11(16,92), 12(30), 15(30), 16(69), 18(16,69,77), 19(30,69), 23(16,77), 24(16), 25(16,77), 28(77), 29(69), 30(69,77), 34–35(30), 36(30,77), 37(30), 40 (16,30,69), 42, 45–46 Ziemkiewicz, M., 11(86), 45 Zimmermann, D., 5(39,46), 7(39), 20(39), 31 (39), 43 Zinders, D., 69(114), 100 Zoranic, 2(13), 31(13), 36(13), 41 Zwier, T. S., 2(19), 11(99), 19(19,180,183), 21 (196), 25(195), 32(19), 42, 46, 51–52
SUBJECT INDEX Ab initio techniques: hydrogen bonds in alcohol clusters: electronic structure calculations, 20–21 ethanol gauche dimers, 28–29 vibrational line shapes: coupling frequency calculations, 76 single water molecules, inhomogeneous electric field, 71–72 water clusters, 72–74 Ad hoc functions, linear-logarithmic kinetics, 185 Alcohol clusters, hydrogen bonds: anion solvation, 16 basic principles, 2–3 bulky alcohols, 31 cation solvation, 15 C H and C O stretching dynamics, 11–12 chirality recognition, 13–15 chlorinated alcohols, 37 computational methods, 20–23 electronic structure calculations, 20–21 empirical force fields, 20 internuclear dynamics, 21–23 cooperativity, 6–7 crossed beam techniques, 18 energetics, 5–6 ester alcohols, 37–38 ethanol systems, 27–29 fluoroalcohols, 32–34 infrared absorption, 17–18 isomerism, 7–8 linear alcohols, 29–31 methanol systems, 23–27 microwave spectroscopy, 16 O H stretching dynamics, 8–9 polyols and sugars, 38–39 Raman scattering, 18
structures and topologies, 3–5 torsional dynamics, 12–13 trifluoroalcohols, 34–37 tunneling dynamics, 13 ultraviolet-infrared coupling, 19 unsaturated and aromatic alcohols, 32 visible ultraviolet-infrared coupling, 19 Allosteric regulation: enzyme kinetics, 137–143 structural kinetic metabolic modeling, thermodynamics and reversible rate equations, 210–211 Anabolism, cellular metabolic modeling, 110 Anharmonicity constants, hydrogen bonds in alcohol clusters: internuclear dynamics, 22–23 methanol systems, 25–27 O H stretching dynamics, 10–11 Anion solvation, hydrogen bonds in alcohol clusters, 16 Argon atoms, methanol clusters, hydrogen bond dynamics, 26–27 Aromatic alcohols, hydrogen bond dynamics, 32 Averaging effect, metabolomics, 147–148 B3LYP predictions, hydrogen bonds in alcohol clusters, linear alcohols, 29–31 Bidirectional fluxes, 13C-based metabolic flux analysis, 158–163 Bifurcations: computational metabolic modeling, Jacobian matrix and linear stability, 171 structural kinetic metabolic modeling: Calvin cycle model, 219–221 glycolysis example, 198–202 Biochemical reaction networks, cellular metabolic modeling, 115
Advances in Chemical Physics, Volume 142, edited by Stuart A. Rice Copyright # 2009 John Wiley & Sons, Inc.
277
278
subject index
Biochemical systems theory (BST) computational metabolic modeling, 181–188 linear-logarithmic kinetics, 184–185 Bistability, computational metabolic modeling, 167–168 glycolysis dynamics modeling, 174–176 Born-Oppenheimer approximation: hydrogen bonds in alcohol clusters, internuclear dynamics, 21–23 vibrational line shapes: ab initio calculations: single water molecules, inhomogeneous electric field, 71–72 water clusters, 72–74 classical approach, 63–64 simulation potential, 71 Bow-tie architecture, cellular metabolic modeling, 110 Bulky alcohols, hydrogen bond dynamics, 31 2-Butanol, hydrogen bonds in alcohol clusters, 31 t-Butyl alcohol, hydrogen bond dynamics, 31 Calvin cycle: stabilizing sites, metabolic networks, 224–226 structural kinetic metabolic modeling, 215–220 Capillary electrophoresis (CE), metabolomics measurements, 151 Carbon-carbon (C C) stretching dynamics, hydrogen bonds in alcohol clusters, torsional dynamics, 12–13 Carbon-hydrogen (C H) stretching dynamics, hydrogen bonds in alcohol clusters, 11–12 trifluoroalcohols, 37 Carbon labeling experiments (CLEs), 13C-based metabolic flux analysis, 157–164 basic principles, 160–161 dynamic flux analysis, 163–164 evaluation, 161–163 steady-state flux analysis, 158–163 Carbon-oxygen (C O) stretching dynamics, hydrogen bonds in alcohol clusters, 11–12 ethanol gauche dimers, 29 torsional dynamics, 12–13 Car-Parrinello simulation, vibrational line shapes, neat water chemistry, 90–94
Carrier molecules, cellular metabolic modeling, 110 Cation solvation, hydrogen bonds in alcohol clusters, 15 Cellular metabolic modeling, 109–119 computer-aided design, 233 intermediate approaches, 114–115 kinetic models, 112–113 linear chain reactions, 222–224 mathematical modeling, 115–119 stoichiometric analysis, 114 topological network analysis, 113–114 Chain-length alternation, hydrogen bonds in alcohol clusters, linear alcohols, 30–31 Chemical equilibrium, enzyme kinetics, computational metabolic modeling, 129–130 Chemical Reaction Network Theory (CRNT): computational metabolic modeling, Jacobian matrix and linear stability, 171 structural kinetic metabolic modeling, 195–196 Chirality discrimination, hydrogen bonds in alcohol clusters, 14 Chirality induction, hydrogen bonds in alcohol clusters, 14 Chirality recognition, hydrogen bonds in alcohol clusters, 13–15 bulk alcohols, 31 ethanol systems, gauche conformations, 27–29 fluoroalcohols, 33–34 trifluoroalcohols, 35–37 Chirality synchronization, hydrogen bonds in alcohol clusters, 14–15 Chlorinated alcohols, hydrogen bond dynamics, 37 Chromophore, vibrational line shapes: coupled chromophores and time-averaging approximation, 65–68 HOD/D2O system, hydrogen bonds, 82 pulse-echo experiments, 68–69 single chromophore, 64–65 Classical theory, vibrational line shapes, 63–64 Cleland plots, enzyme kinetics, multisubstrate reactions, 135–137 Competitive inhibition, enzyme kinetics, 138–143
subject index Complex systems theory: cellular metabolic modeling, 118–119 metabolic modeling, 164–176 bistability and hysteresis, 167–168 glycolysis minimal model, 171–176 Jacobian matrix and linear stability analysis, 168–171 pathway stability, 165–167 metabolomics, 147–148 structural kinetic metabolic modeling, Calvin cycle model, 215–220 Computational methods: hydrogen bonds in alcohol clusters, 20–23 electronic structure calculations, 20–21 empirical force fields, 20 internuclear dynamics, 21–23 metabolic modeling: basic principles, 107–109, 120–124 biochemical systems theory, 181–188 convenience kinetics approaches, 186–188 linear-logarithmic kinetics, 184–185 cellular metabolism, 109–119 intermediate approaches, 114–115 kinetic models, 112–113 mathematical modeling, 115–119 stoichiometric analysis, 114 topological network analysis, 113–114 complex system dynamics, 164–176 bistability and hysteresis, 167–168 glycolysis minimal model, 171–176 Jacobian matrix and linear stability analysis, 168–171 pathway stability, 165–167 enzyme and chemical reaction rates, 127–143 chemical equilibrium and thermodynamics, 129–130 equilibrium constants and Haldane relationship, 134–135 inhibition and allosteric control, 137–143 Michaelis-Menten kinetics, 130–133 reversible Michaelis-Menten kinetics, 133–134 steady-state kinetics, multisubstrate reactions, 135–137 fluxome measurement, 13C-based flux analysis, 157–164
279
basic principles, 160–161 dynamic flux analysis, 163–164 evaluation, 161–163 steady-state flux analysis, 158–163 genome-scale kinetic models, 231–233 guidelines for, 143–146 metabolic control analysis, 176–181 limitations, 181 scaled coefficients, 179–180 summation and connectivity theorems, 178–179 metabolomics, 146–151 high-throughput measurements, 149–151 organizational complexity, 147–148 targeted metabolite analysis, 148–149 stability and regulation mechanisms, 220–231 robustness, 227–231 feedback mechanisms, 227–229 stabilizing site identification, 224–226 stoichiometric matrix, 124–127 left null space, 125–126 right null space, 126 structural kinetic modeling, 188–220 complex system dynamics, Calvin cycle model, 215–220 example, 196–198 glycolysis dynamics and bifurcations, 198–202 Jacobian matrix, 191–196 matrix gamma (), 192 metabolic control analysis and, 194–196 saturation matrix, 192–194 thermodynamics and reversible rate equation, 209–215 allosteric regulation, 210–211 kinetics, 211–212 parameter sampling issues, 213–215 yeast glycolysis, Monte Carlo simulation, 202–209 topological and stoichiometric analysis, 151–157 elementary flux modes, 154–155 flux balance analysis, 156–157 network analysis, 152–153 Computer-aided design (CAD), genome-scale kinetic modeling, 233
280
subject index
Concentration control coefficient, metabolic control analysis, 177–181 Concerted models, enzyme kinetics, 142–143 Conformational isomerism, hydrogen bonds in alcohol clusters, 7–8 ethanol cluster systems, 27–29 linear alcohols, 29–31 trifluoroalcohols, 34–37 Connectivity theorem, metabolic control analysis, 178–181 Control coefficients, metabolic control analysis, 177–181 Convenience kinetics, computational metabolic modeling, 186–188 Cooperativity: enzyme kinetics, inhibition and allosteric regulation, 141–143 hydrogen bonds in alcohol clusters, 6–7 Coupled-cluster perturbation theory, hydrogen bonds in alcohol clusters, electronic structure calculations, 21 Coupling constant, hydrogen bonds in alcohol clusters, O H stretching dynamics, 9 Coupling frequency calculations, vibrational line shapes, 76 Crossed beam techniques, hydrogen bonds in alcohol clusters, 18 Cumomers, carbon labeling experiments, 162–163 Database sources, kinetic metabolic modeling, 145–146 Davydov splittings, hydrogen bonds in alcohol clusters, O H stretching dynamics, 9 Decoding relation, cellular metabolic modeling, 117–119 Degrees of freedom, vibrational line shapes, mixed quantum/classical approach, 64–65 Delocalization effects, vibrational line shapes, neat water chemistry, 94 Density functional theory (DFT), hydrogen bonds in alcohol clusters, electronic structure calculations, 20–21 Depolarized (VH) line shapes: HOD/D2O system, 61, 78–79 neat water, 89–94 transition polarizability calculations, 75–76
Deuteration analysis, hydrogen bonds in alcohol clusters, 10–11 2,2-Difluoroethanol, hydrogen bond dynamics, 34 Dimeric structures, hydrogen bonds in alcohol clusters, 4–5 chirality recognition, 13–15 ethanol systems, 28–29 fluoroalcohols, 33–34 linear alcohols, 30–31 methanol systems, 23–27 trifluoroalcohols, 34–37 Dissipative reactions, hydrogen bonds in alcohol clusters, O H stretching dynamics, 9 Donor-acceptor isomerism, hydrogen bonds in alcohol clusters, 7–8 Double harmonic approximation, hydrogen bonds in alcohol clusters, internuclear dynamics, 22–23 Double Hopf bifurcation, structural kinetic metabolic modeling, 219–220 Dual causality, cellular metabolic modeling, 118–119 Dynamic flux analysis, metabolic modeling, 163–164 Effective kinetic order, Jacobian matrix, structural kinetic metabolic modeling, 192–194 Eigenstate properties: computational metabolic modeling: Jacobian matrix and linear stability, 169–171 structural kinetic modeling, glycolysis, 200–202 vibrational line shapes, neat water chemistry, 93–94 Elasticities, metabolic control analysis, 177–181 Electronic structure calculations: hydrogen bonds in alcohol clusters, 20–21 vibrational line shapes, hydrogen bonding, 76–77 Elementary flux modes, computational metabolic models, 153–155 Empirical force field analysis, hydrogen bonds in alcohol clusters, 20 Encoding operations, cellular metabolic modeling, 117–119
subject index Endergonic processes, computational metabolic modeling, enzyme kinetics, 129–130 Energetics, hydrogen bonds in alcohol clusters, 5–6 Enzyme kinetics: cellular metabolic modeling, 114–115 computational metabolic modeling, chemical reaction rates, 127–143 chemical equilibrium and thermodynamics, 129–130 equilibrium constants and Haldane relationship, 134–135 inhibition and allosteric control, 137–143 Michaelis-Menten kinetics, 130–133 reversible Michaelis-Menten kinetics, 133–134 steady-state kinetics, multisubstrate reactions, 135–137 Equilibrium constant, computational metabolic modeling: enzyme kinetics, 129–130 Haldane relationship, 134–135 Ester alcohols, hydrogen bond dynamics, 37–38 Ethanol cluster systems, hydrogen bond dynamics, 27–29 Exergonic processes, computational metabolic modeling, enzyme kinetics, 129–130 Explicit kinetic models, metabolic reactions, 123–124 External metabolites, computational metabolic models, 120–124 Extreme pathways/extreme currents, computational metabolic modeling, elementary flux modes, 155 Feedback inhibition: enzyme kinetics, 138–143 metabolic state robustness, 227–229 Fluoroalcohols, hydrogen bond dynamics, 32–34 2-Fluoroethanol, hydrogen bond dynamics, 33–34 Flux balance analysis (FBA), computational metabolic models, 153, 156–157 pathway stability, 166–167
281
Flux control coefficient, metabolic control analysis, 178 Fluxome measurement, 13C-based flux analysis, computational metabolic modeling, 157–164 basic principles, 160–161 dynamic flux analysis, 163–164 evaluation, 161–163 Fourier transform, vibrational line shapes, 62–63 Franck-Condon effect, hydrogen bonds in alcohol clusters, methanol systems, 24–27 Frequency-dependent anisotropy decay, vibrational line shapes: HOD/D2O systems, 84–85 HOD/H2O/HOT/H2O systems, 86–89 Frequency distributions, vibrational line shapes, HOD/D2O system, hydrogen bonds, 81–82 Frequency-domain spectral analysis, hydrogen bonds in alcohol clusters, future research issues, 39–40 Full width at half maximum (FWHM), vibrational line shapes, 62 coupled chromophores, time-averaging approximation, 68 neat water chemistry, 89–94 Gamma matrix, Jacobian matrix, structural kinetic metabolic modeling, 192 Gas chromatography-mass spectrometry (GC-MS): carbon labeling experiments, 160–163 metabolomics measurements, 150–151 Gauche conformations, hydrogen bonds in alcohol clusters: ethanol systems, 27–29 linear alcohols, 29–31 trifluoroalcohols, 35–37 Gaussians, vibrational line shapes, neat water chemistry, 89–94 Gavrilov-Guckenheimer (GG) bifurcation, structural kinetic metabolic modeling, Calvin cycle model, 219–220 Generalized mass-action (GMA) model, biochemical systems theory, 183–188
282
subject index
Genome-scale reconstructions: computational metabolic models, stoichiometric network analysis, 151–157 kinetic models, 231–233 Gibbs free energy, computational metabolic modeling, enzyme kinetics, 129–130 Glycolysis, computational metabolic models: complex dynamics, 171–176 mathematical modeling, 121–124 stoichiometric matrix, 126–127 structural kinetic metabolic modeling, 198–202 yeast glycolysis, Monte Carlo simulation, 202–209 Haldane relationship: enzyme kinetic modeling, 134–135 linear-logarithmic kinetics, 185 structural kinetic modeling, thermodynamics and reversible rate equations, 209–215 Harmonic quantum correction factor, vibrational line shapes, 63–64 Hartree-Fock (HF) method, hydrogen bonds in alcohol clusters, 20–21 methanol systems, 23–27 Heavy atom tunneling, hydrogen bonds in alcohol clusters, 13 Heisenberg time-dependent operator, vibrational line shapes, 63 High-throughput measurements, metabolomics, 148–151 Hilbert space, vibrational line shapes, 63 mixed quantum/classical approach, single chromophore, 65 Hill equation: computational metabolic models, glycolysis dynamics, 172–176 enzyme kinetics, 141–143 structural kinetic metabolic modeling: Calvin cycle model, 217–220 thermodynamics and reversible rate equations, 210–211 HOD/D2O system, vibrational line shapes, 61–62 ab initio calculations, water clusters, 72–74 frequency-dependent anisotropy decay, 84–85
hydrogen bonding, 79–82 IR line shapes, 77–78 mixed quantum/classical approach, 64–65 Raman line shapes, 78–79 simulation potential, 71 three-pulse echoes and two-dimensional IR spectra, 83–84 two-pulse echoes, 83 HOD/H2O system, vibrational line shapes: frequency-dependent anisotropy decay, 86–89 line shapes, 85 three-pulse echoes and two-dimensional IR spectra, 85–86 Hopf bifurcation: computational metabolic modeling: Jacobian matrix and linear stability, 171 structural kinetic modeling, glycolysis, 200–202 structural kinetic metabolic modeling, Calvin cycle modeling, 219–220 HOT/H2O system, vibrational line shapes: frequency-dependent anisotropy decay, 86–89 line shapes, 85 three-pulse echoes and two-dimensional IR spectra, 85–86 Hu¨ckel molecular orbital theory, hydrogen bonds in alcohol clusters, O H stretching dynamics, 9 Hydrogen bonds: alcohol clusters: anion solvation, 16 basic principles, 2–3 bulky alcohols, 31 cation solvation, 15 C H and C O stretching dynamics, 11–12 chirality recognition, 13–15 chlorinated alcohols, 37 computational methods, 20–23 electronic structure calculations, 20–21 empirical force fields, 20 internuclear dynamics, 21–23 cooperativity, 6–7 crossed beam techniques, 18 energetics, 5–6 ester alcohols, 37–38 ethanol systems, 27–29 fluoroalcohols, 32–34 infrared absorption, 17–18
subject index
283
isomerism, 7–8 linear alcohols, 29–31 methanol systems, 23–27 microwave spectroscopy, 16 O H stretching dynamics, 8–9 polyols and sugars, 38–39 Raman scattering, 18 structures and topologies, 3–5 torsional dynamics, 12–13 trifluoroalcohols, 34–37 tunneling dynamics, 13 ultraviolet-infrared coupling, 19 unsaturated and aromatic alcohols, 32 visible ultraviolet-infrared coupling, 19 vibrational line shapes, 76–77 HOD/D2O system, 79–82 Hydrogen fluoride, cooperativity, 6–7 Hysteresis, computational metabolic modeling, 167–168 glycolysis dynamics modeling, 174–176
methanol systems, 25–27 topological features, 5 Isoptomers, carbon labeling experiments, 161–163 Isotope effects, hydrogen bonds in alcohol clusters, 9–11
Infrared (IR) spectroscopy: HOD/D2O vibrational line shapes, 61–62 hydrogen bonds in alcohol clusters, 17–18 future research issues, 39–40 methanol clusters, 26–27 O H stretching dynamics, 8–9 vibrational line shapes: HOD/D2O system, 77–78 HOD/H2O/HOT/H2O systems, 85 neat water chemistry, 89–94 Inhibition mechanisms, enzyme kinetics, 137–143 Inhomogeneous electric field, vibrational line shapes, ab initio calculations, single water molecules, 71–72 Inhomogeneous limit, vibrational line shapes, coupled chromophores, timeaveraging approximation, 66–68 In silico models, cellular metabolism, 116 Instanton techniques, hydrogen bonds in alcohol clusters, internuclear dynamics, 22–23 Internal metabolites, computational metabolic models, 120–124 Internuclear dynamics, hydrogen bonds in alcohol clusters, 21–23 Isomerism, hydrogen bonds in alcohol clusters: conformational isomerism and, 7–8
Kinetic modeling. See also Enzyme kinetics biochemical systems theory, 182–188 linear-logarithmic kinetics, 184–185 cellular metabolism, 112–113 computational metabolic modeling: convenience kinetics, 186–188 guidelines for, 143–146 mathematical principles, 122–124 software packages for, 144 dynamic flux analysis, 163–164 genome-scale reconstructions, 231–233 structural kinetic metabolic modeling, thermodynamics and reversible rate equations, 211–215 Kolmogorov-Smirnov test, stabilizing sites, metabolic networks, 226 Koshland, Nemethy, and Filmer (KNF) sequential model, enzyme kinetics, 142–143
Jacobian matrix, computational metabolic modeling: linear stability analysis and, 168–171 structural kinetic modeling, 190–196 glycolysis, 200–202 matrix gamma (), 192 metabolic control analysis and, 194–196 saturation matrix, 192–194 thermodynamics and reversible rate equations, 210–215 yeast glycolysis, Monte Carlo simulation, 205–209
Large-scale modeling, cellular metabolism, 116–119 Left nullspace, stoichiometric matrix, computational metabolic models, 125–126 Linear arrangement. See also Vibrational line shapes hydrogen bonds in alcohol clusters, 3–5, 29–31
284
subject index
Linear chain of reaction, metabolic modeling, stability and regulation mechanisms, 222–227 Linear-logarithmic kinetics, biochemical systems theory, 184–185 Linear stability analysis, computational metabolic modeling, Jacobian matrix, 168–171 Link stoichiometric matrix, computational metabolic models, 125–126 Liquid chromatography-mass spectrometry (LC-MS), metabolomics measurements, 150–151 Lone-pair arrangements, hydrogen bonds in alcohol clusters, 3–5 methanol systems, 25–27 Mass-action kinetics, cellular metabolic modeling, 115 Mathematical modeling: cellular metabolism, 112–119 computational metabolic models: basic principles, 120–124 Weichert and Takors scheme, 119–120 Maximal enzyme catalytic activity, kinetic metabolic modeling, 144–146 Medium-complexity modeling, yeast glycolysis, Monte Carlo simulation, 202–209 Metabolic control analysis (MCA): basic principles, 176–181 biochemical systems theory, 183–188 structural kinetic metabolic modeling, Jacobian matrix derivation, 194–196 Metabolic modeling: basic principles, 107–109, 120–124 biochemical systems theory, 181–188 convenience kinetics approaches, 186–188 linear-logarithmic kinetics, 184–185 cellular metabolism, 109–119 intermediate approaches, 114–115 kinetic models, 112–113 mathematical modeling, 115–119 stoichiometric analysis, 114 topological network analysis, 113–114 complex system dynamics, 164–176 bistability and hysteresis, 167–168 glycolysis minimal model, 171–176 Jacobian matrix and linear stability analysis, 168–171
pathway stability, 165–167 enzyme and chemical reaction rates, 127–143 chemical equilibrium and thermodynamics, 129–130 equilibrium constants and Haldane relationship, 134–135 inhibition and allosteric control, 137–143 Michaelis-Menten kinetics, 130–133 reversible Michaelis-Menten kinetics, 133–134 steady-state kinetics, multisubstrate reactions, 135–137 fluxome measurement, 13C-based flux analysis, 157–164 basic principles, 160–161 dynamic flux analysis, 163–164 evaluation, 161–163 steady-state flux analysis, 158–163 genome-scale kinetic models, 231–233 guidelines for, 143–146 metabolic control analysis, 176–181 limitations, 181 scaled coefficients, 179–180 summation and connectivity theorems, 178–179 metabolomics, 146–151 high-throughput measurements, 149–151 organizational complexity, 147–148 targeted metabolite analysis, 148–149 stability and regulation mechanisms, 220–231 robustness, 227–231 feedback mechanisms, 227–229 stabilizing site identification, 224–226 stoichiometric matrix, 124–127 left null space, 125–126 right null space, 126 structural kinetic modeling, 188–220 complex system dynamics, Calvin cycle model, 215–220 example, 196–198 glycolysis dynamics and bifurcations, 198–202 Jacobian matrix, 191–196 matrix gamma (), 192 metabolic control analysis and, 194–196 saturation matrix, 192–194 thermodynamics and reversible rate equation, 209–215 allosteric regulation, 210–211
subject index kinetics, 211–212 parameter sampling issues, 213–215 yeast glycolysis, Monte Carlo simulation, 202–209 topological and stoichiometric analysis, 151–157 elementary flux modes, 154–155 flux balance analysis, 156–157 network analysis, 152–153 Metabolite measurements: metabolite profiling technique, 147 metabolomics, 146–151 structural kinetic metabolic modeling, Jacobian parametrization, 213–215 Metabolomics, metabolic modeling, 107–109, 146–151 high-throughput measurements, 149–151 organizational complexity, 147–148 targeted metabolite analysis, 148–149 Methanol systems, hydrogen bond dynamics, 23–27 Michaelis constant: cellular metabolic modeling, 117–119 kinetic metabolic modeling, 144–146 structural kinetic metabolic modeling, 197–198 Michaelis-Menten rate equation. See also Reversible Michaelis-Menten kinetics cellular metabolic modeling, 117–119 computational metabolic modeling: convenience kinetics, 186–188 enzyme kinetics, 130–133 inhibition and allosteric regulation, 139–143 glycolysis dynamics, 172–176 metabolic control analysis, 179–181 pathway stability, 165–167 structural kinetic metabolic modeling, 197–198 linear-logarithmic kinetics, 184–185 structural kinetic metabolic modeling: Calvin cycle model, 217–220 thermodynamics and reversible rate equations, 210–215 yeast glycolysis, Monte Carlo simulation, 204–209
285
Microwave spectroscopy, hydrogen bonds in alcohol clusters, 16 Minimal modeling: cellular metabolism, 116–119 glycolysis, 121–124 glycolysis dynamics, 172–176 MIRIAM data standard, kinetic metabolic modeling, 146 Mixed quantum/classical approach, vibrational line shapes: coupled chromophores and time-averaging approximation, 65–68 HOD/D2O system, 77–78 single chromophore, 64–65 Molecular dynamics simulation, vibrational line shapes, neat water chemistry, 90–94 Møller-Plesset perturbation theory, hydrogen bonds in alcohol clusters, electronic structure calculations, 21 Monod, Wyman, and Changeux (MWC) concerted model, enzyme kinetics, 142–143 Monomer trans conformations, hydrogen bonds in alcohol clusters, trifluoroalcohols, 36–37 Monte Carlo simulation, yeast glycolysis metabolic modeling, 202–209 Multistability concepts, computational metabolic modeling, hysteresis, 167–168 Multisubstrate reactions, computational metabolic modeling, steady-state kinetics, 135–137 Mutual information measurement, stabilizing sites, metabolic networks, 226 Nonaqueous fractionation, metabolomics, 147–148 Noncompetitive inhibition, enzyme kinetics, 140–143 Non-Condon effect, vibrational line shapes: HOD/D2O system, 78–79 mixed quantum/classical approach, single chromophore, 65 transition dipole calculations, 74–75 Nonlinear experiments, vibrational line shapes, 68–69 Nullclines, glycolysis dynamics modeling, 173–176
286
subject index
ONIOM layered technique, hydrogen bonds in alcohol clusters, electronic structure calculations, 21 Open-chain structure, hydrogen bonds in alcohol clusters, 4–5 Ordinary differential equations (ODEs), cellular metabolic modeling, kinetic models, 113 Organizational complexity, metabolomics, 147–148 Oscillation parameters: metabolic modeling, stability and regulation mechanisms, 222–227 yeast glycolysis, Monte Carlo simulation, 207–209 Overtone effects, hydrogen bonds in alcohol clusters, 9–11 Oxygen-hydrogen (O H) stretching dynamics: hydrogen bonds in alcohol clusters, 8–9 chirality recognition, 13–15 ester alcohols, 37–38 ethanol gauche dimers, 28–29 fluoroalcohols, 33–34 isotope and overtone effects, 9–11 linear arrangement, 29–31 methanol systems, 23–27 polyols and sugars, 38–39 trifluoroalcohols, 35–37 tunneling dynamics, 13 unsaturated and aromatic alcohols, 32 vibrational line shapes, 60–61 ab initio calculations, water clusters, 73–74 classical approach, 63–64 frequency-dependent anisotropy decay, 84–85 mixed quantum/classical approach: coupled chromophores, time-averaging approximation, 65–68 single chromophore, 64–65 neat water chemistry, 93–94 simulation potential, 71 transition dipole calculations, 74–75 transition polarizability calculations, 75–76
Parameter space analysis, structural kinetic metabolic modeling: Jacobian parametrization, 212–215 yeast glycolysis, Monte Carlo simulation, 204–209 Pathway stability: metabolic modeling, complex system dynamics, 165–167 yeast glycolysis, Monte Carlo simulation, 206–209 Pauli repulsion, hydrogen bonds in alcohol clusters, 3–5 Pearson correlation coefficient, stabilizing sites, metabolic networks, 225–226 Perturbation theory, hydrogen bonds in alcohol clusters, electronic structure calculations, 21 Petri nets, computational metabolic modeling, elementary flux modes, 155 Photon echo experiments, vibrational line shapes, neat water chemistry, 95 Polarized (VV) line shapes: HOD/D2O system, 61, 78–79 neat water, 89–94 Raman experiments, 68 transition polarizability calculations, 75–76 Polyols, hydrogen bond dynamics, 38–39 Positional enrichment, carbon labeling experiments, 161–163 Power-law formalism, biochemical systems theory, 182–188 Predictions, cellular metabolic modeling, 117–119 2-Propanol, hydrogen bond dynamics, 37 Proteomics, metabolic modeling, 107 Pulse echo experiments, vibrational line shapes, 68–69 HOD/H2O/HOT/H2O systems, 85–86 three-pulse echoes and two-dimensional IR spectra, 83–84 two-pulse echoes, 83 Pump-probe techniques: vibrational line shapes, neat water chemistry, 95 vibrational line shapes, HOD/D2O systems, frequency-dependent anisotropy decay, 84–85
subject index Quantum chemistry, hydrogen bond dynamics in alcohol clusters, 39–40 Quasi-steady-state assumption (QSSA), enzyme kinetic modeling: Michaelis-Menten constant, 131–133 reversible Michaelis-Menten kinetics, 133–134 Raman line shapes: HOD/D2O system, 61–62, 78–79 HOD/H2O/HOT/H2O systems, 85 neat water chemistry, 90–94 transition polarizability calculations, 75–76 Raman scattering, hydrogen bonds in alcohol clusters, 18 Random Bi Bi Mechanism: convenience kinetics and, 186–188 enzyme kinetics, multisubstrate reactions, 135–137 Rapid equilibrium, enzyme kinetic modeling, Michaelis-Menten constant, 131–133 Rate constant, biochemical systems theory, 182–188 Rate law parametrization, kinetic metabolic modeling, 144–146 Red shift mechanics, hydrogen bonds in alcohol clusters: ethanol gauche dimers, 28–29 isotope and overtone effects, 10–11 O H stretching dynamics, 8–9 Response coefficient, metabolic control analysis, 177–181 Reversible Michaelis-Menten kinetics: computational metabolic modeling, metabolic control analysis, 180–181 enzyme kinetic modeling, 133–134 Haldane relationship, 134–135 Reversible rate equations, structural kinetic modeling, 209–215 allosteric regulation, 210–211 kinetics, 211–212 parameter sampling issues, 213–215 Right nullspace, stoichiometric matrix, computational metabolic models, 126 Ring structures, hydrogen bonds in alcohol clusters, 5 Robustness, metabolic states, 227–231 feedback mechanisms, 227–229
287
‘‘Runaway’’ solution, glycolysis dynamics modeling, 175–176 Saddle point regime, computational metabolic modeling, Jacobian matrix and linear stability, 170–171 Saturation matrix: Jacobian matrix, structural kinetic metabolic modeling, 192–194 structural kinetic metabolic modeling: glycolysis example, 199–202 yeast glycolysis, Monte Carlo simulation, 204–209 Scaled coefficients, metabolic control analysis, 179–181 ‘‘Scale-free’’ distribution, computational metabolic models, topological network analysis, 153 Self-averaging, vibrational line shapes, coupled chromophores, time-averaging approximation, 67–68 Self-organized systems, cellular metabolic modeling, 118–119 Sequential models, enzyme kinetics, 142–143 Simulation potential, vibrational line shapes, 71 ‘‘Small-world’’ metabolite properties, computational metabolic models, topological network analysis, 153 Solvation process, hydrogen bonds in alcohol clusters: anion solvation, 16 cation solvation, 15 SPC/E water simulation, vibrational line shapes: ab initio calculations, water clusters, 73–74 HOD/D2O system, 77–78 hydrogen bonds, 80–82 neat water chemistry, 90–94 three-pulse echo experiments, 83–84 S-system model, biochemical systems theory, 183–188 Stability mechanisms, metabolic modeling, 220–231 robustness, 227–231 feedback mechanisms, 227–229 stabilizing site identification, 224–226 Stabilizing sites, metabolic networks, 224–226 Stable node/focus regimes, computational metabolic modeling, Jacobian matrix and linear stability, 170–171
288
subject index
Steady-state condition, computational metabolic modeling, 122–124 elementary flux modes, 153–157 flux balance analysis, 153–163 glycolysis dynamics modeling, 174–176 multisubstrate reactions, 135–137 pathway stability, 166–167 Stoichiometric analysis: cellular metabolic modeling, 114 computational metabolic models: flux balance analysis and, 156–157 mathematical modeling, 121–124 matrix properties, 124–127 robustness of metabolic states, 229–231 structural kinetic metabolic modeling: glycolysis example, 198–202 yeast glycolysis, Monte Carlo simulation, 203–209 Stoichiometric network analysis (SNA), computational metabolic models, 151–157 elementary flux modes, 154–155 flux balance analysis, 156–157 Jacobian matrix and linear stability, 171 network analysis, 152–153 Structural kinetic modeling (SKM): cellular metabolism, 115 metabolic modeling, 188–220 complex system dynamics, Calvin cycle model, 215–220 example, 196–198 glycolysis dynamics and bifurcations, 198–202 Jacobian matrix, 191–196 matrix gamma (), 192 metabolic control analysis and, 194–196 saturation matrix, 192–194 thermodynamics and reversible rate equation, 209–215 allosteric regulation, 210–211 kinetics, 211–212 parameter sampling issues, 213–215 yeast glycolysis, Monte Carlo simulation, 202–209 Substrate binding, enzyme kinetics, 128–129 multisubstrate reactions, 135–137 Sugars, hydrogen bond dynamics, 38–39 Summation theorem, metabolic control analysis, 178–181
Supersonic jets, hydrogen bonds in alcohol clusters, infrared absorption, 17–18 Switching mechanisms, computational metabolic modeling, hysteresis, 167–168 Symmetry-adapted perturbation theory, hydrogen bonds in alcohol clusters, electronic structure calculations, 21 Symmetry breaking, hydrogen bonds in alcohol clusters, chirality induction, 14–15 Systems biology: cellular metabolic modeling, 111 genome-scale reconstructions, 231–233 metabolic modeling, 108 Systems Biology Markup Language (SBML), kinetic metabolic modeling, 146 Takens-Bogdanov (TB) bifurcation, structural kinetic metabolic modeling, Calvin cycle model, 219–220 Target(ed) analysis, metabolite measurement, 147–149 Taylor series expansion, computational metabolic modeling, Jacobian matrix and linear stability, 169–171 Tetrahedral structure, hydrogen bonds in alcohol clusters, 3–5 Tetrameric structures, hydrogen bonds in alcohol clusters, methanol systems, 25–27 Thermodynamic equilibrium, computational metabolic modeling: enzyme kinetics, 129–130 flux balance analysis and, 156–157 mathematical principles, 122–124 structural kinetic modeling, 209–215 allosteric regulation, 210–211 kinetics, 211–212 parameter sampling issues, 213–215 Three-body interaction, hydrogen bonds in alcohol clusters, 6–7 Three-pulse echo peak shift (3PEPS) experiment, vibrational line shapes, 68–69 HOD/H2O/HOT/H2O systems, 85–86 two-dimensional IR spectra, 83–84
subject index Time-averaging approximation (TAA), vibrational line shapes: coupled chromophores, 65–68 neat water chemistry, 90–94 ultrafast spectroscopy, 95 Time-correlation function (TCF): vibrational line shapes, 63 classical approach, 63–64 HOD/H2O/HOT/H2O systems, 86–89 pulse echo experiments, 68–69 vibrational line shapes, HOD/D2O systems, frequency-dependent anisotropy decay, 84–85 Topological network analysis: cellular metabolic modeling, 113–114 metabolic modeling, 151–157 elementary flux modes, 154–155 flux balance analysis, 156–157 network analysis, 152–153 Torsional dynamics, hydrogen bonds in alcohol clusters, 12–13 Transcriptomics, metabolic modeling, 107 Transition dipole calculations, vibrational line shapes, 74–75 Transition polarizabilities, vibrational line shapes, 75–76 Trifluoroalcohols, hydrogen bond dynamics, 34–37 Tunneling dynamics, hydrogen bonds in alcohol clusters, 13 ‘‘Turbo design’’ concept, yeast glycolysis, Monte Carlo simulation, 208–209 Two-dimensional infrared (2DIR) spectra, vibrational line shapes, 83–84 HOD/H2O/HOT/H2O systems, 85–86 neat water chemistry, 95 Two-pulse echo experiments, vibrational line shapes, 83 Ultrafast spectroscopy, vibrational line shapes: neat water chemistry, 95 pulse echoes and nonlinear experiments, 68–69 Ultraviolet-infrared (UV-IR) coupling, hydrogen bonds in alcohol clusters, 19 Uncompetitive inhibition, enzyme kinetics, 140–143
289
Unsaturated alcohols, hydrogen bond dynamics, 32 Unstable node/focus regime, computational metabolic modeling, Jacobian matrix and linear stability, 170–171 Van der Waals interactions, hydrogen bonds in alcohol clusters, linear alcohols, 29–31 Vibrational line shapes: ab initio calculations: single water molecule, inhomogeneous electric field, 71–72 water clusters, 72–74 basic principles, 60–61 classical approach, 63–64 simulation potential, 71 coupling frequency calculations, 76 echoes and nonlinear experiments, 69–70 HOD/D2O: frequency-dependent anisotropy decay, 84–85 hydrogen bonding, 79–82 IR line shapes, 77–78 Raman line shapes, 78–79 three-pulse echoes and two-dimensional IR spectra, 83–84 two-pulse echoes, 83 HOD/H2O and HOT/H2O: frequency-dependent anisotropy decay, 86–89 line shapes, 85 three-pulse echoes and two-dimensional IR spectra, 85–86 hydrogen bonding, 76–77 mixed quantum/classical approach: couple chromophores and time-averaging approximation, 65–68 implementation, 70–71 single chromophore, 64–65 neat water (H2O): line shapes, 89–94 ultrafast experiments, 95 Raman line shapes, 68 transition dipole calculations, 74–75 transition polarizabilities calculation, 75–76 Vibrational spectroscopy, basic principles, 60–62
290
subject index
Visible ultraviolet-infrared (VUV-IR) coupling, hydrogen bonds in alcohol clusters, 19 methanol systems, 25–27 Water chemistry. See also HOD/D2O system basic principles, 60 vibrational line shapes: coupling frequencies, 76 hydrogen bonding, 76–77
neat water (H2O), 89–94 ultrafast experiments, 95 single water molecules, inhomogeneous electric field, 71–72 water clusters, ab initio calculations, 72–74 Whole cell models, cellular metabolism, 116 Yeast glycolysis, Monte Carlo simulation, 202–209