PROTEIN
NMR SPECTROSCOPY
PROTEIN NMR SPECTROSCOPY PRINCIPLES AND PRACTICE SECOND EDITION
JOHN CAVANAGH WAYNE J. FAIRBROTHER ARTHUR G. PALMER, III MARK RANCE NICHOLAS J. SKELTON
AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO ACADEMIC PRESS IS AN IMPRINT OF ELSEVIER
Copyrighted Material
Elsevier Academic Press 30 Corporate Drive, Suite 400. Burlington. MA 01803, USA 525 B Street. Suite 1900, San Diego, California 92101-4495, USA 84 Theobald's Road. London WC1X 8RR. UK
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ISBN 13: 978-0-12-164491-8 ISBN 10: 0-12-164491-X For all information on all Elsevier Academic Press publications visit our Web site at www.books.elsevier.com Printed in the United States of America
PREFACE
The second edition of Protein NMR Spectroscopy: Principles and Practice reflects the continued rapid pace of development of biomolecular NMR spectroscopy since the original publication in 1996. While these developments will no doubt continue in the future, ensuring a ready need for additional monographs, the present time is auspicious for a new edition that incorporates important recent developments. The most notable change in the second edition is evident on the cover: Mark Rance has been added as an author. In writing the first edition of Protein NMR Spectroscopy: Principles and Practice, the original authors benefited greatly from many ‘‘behind-the-scenes’’ discussions of NMR theory, instrumentation, and experimental methods with Mark. After publication, the original authors continued to have frequent discussions with Mark concerning improvements for the second edition. Accordingly, the original authors were delighted that, when work on the second edition began in earnest, Mark agreed to abandon his advisory role and become a co-author. Many of the strengths of the second edition of Protein NMR Spectroscopy: Principles and Practice are derived directly from his contributions. The second edition of Protein NMR Spectroscopy: Principles and Practice includes two new Chapters: experimental techniques for investigating molecular conformational dynamics through spin relaxation are described in Chapter 8, and techniques applicable to larger
v
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PREFACE
proteins and molecular complexes are described in Chapter 9. As a result, Chapter 8 in the first edition now is renumbered Chapter 10. The other Chapters have been revised to incorporate new techniques, including methods to measure residual dipole couplings and to utilize transverse relaxation optimized spectroscopy, as well as our own improved understanding of NMR spectroscopy. As in the first edition of Protein NMR Spectroscopy: Principles and Practice, the second edition uses the small protein ubiquitin (MW ¼ 8.6 kD) to demonstrate the majority of the experimental aspects of NMR spectroscopy. In the second edition, the protein calbindin D28k (MW ¼ 30 kD), is used to demonstrate experimental techniques for proteins of molecular mass 420 kD. Details of sample preparation, resonance assignments, and structure determination of calbindin D28k have been reported [W. Lutz, E. M. Frank, T. A. Craig, R. Thompson, R. A. Venters, D. Kojetin, J. Cavanagh and R. Kumar (2003) Biochem. Biophys. Res. Commun. 303, 1186–1192; R. A. Venters, L. M. Benson, T. A. Craig, K. H. Paul. D. R. Kordys, R. Thompson, S. Naylor, R. Kumar and J. Cavanagh (2003) Anal. Biochem. 317, 59–66; D. J. Kojetin, R. A. Venters, D. R. Kordys, R. J. Thompson, R. Kumar and J. Cavanagh (2006) Nat. Struct. Mol. Biol. 13, 641–647]. Although we wish that the second edition will be free of errors or inaccuracies, we recognize that readers undoubtedly will find mistakes (and hopefully communicate them to A. G. P. at
[email protected]). An errata page will be maintained at http://www.palmer.hs.columbia. edu/protein_nmr_spectroscopy. We wrote the first edition of Protein NMR Spectroscopy: Principles and Practice to enable graduate students, postdoctoral scientists, and senior investigators to understand the unifying principles of NMR spectroscopy and to evaluate, implement and optimize experimental NMR techniques for their own research. We hope that the second edition continues to meet these objectives. John Cavanagh Wayne J. Fairbrother Arthur G. Palmer, III Mark Rance Nicholas J. Skelton
PREFACE TO THE F IRST EDITION
Concomitant developments of modern molecular biology and multidimensional nuclear magnetic resonance (NMR) spectroscopy have increased explosively the use of NMR spectroscopy for generating structural and dynamical information on small to medium-sized biological macromolecules. Efficient molecular biological techniques for incorporation of the stable, NMR active, 13C and 15N isotopes into overexpressed proteins have resulted in dramatic advances in the design and implementation of multidimensional heteronuclear NMR spectroscopic techniques. Consequently, the maximum size protein amenable to complete structural investigation has increased from 10 kDa using 1 H homonuclear NMR spectroscopy to 30 kDa using 13C and 15N heteronuclear NMR spectroscopy and perhaps to 40 or 50 kDa using 13 C and 15N heteronuclear NMR spectroscopy combined with fractional 2 H enrichment. Most recently, in vitro transcription techniques have expanded the application of 13C and 15N heteronuclear NMR spectroscopy to RNA molecules. Research programs for isotopically enriching DNA and carbohydrate molecules promise to further extend the reach of these powerful NMR techniques. The maturation of the field of structural biology has made the study of structure-function relationships of biological macromolecules by NMR spectroscopy an integral part of diverse chemical and biological research efforts. As an indication of the success of the technique, NMR
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TO THE
FIRST EDITION
spectroscopy increasingly is being utilized by chemical and biological scientists not specifically trained as NMR spectroscopists. At the same time, a bewildering number of complex 13C and 15N heteronuclear NMR experiments that make increasingly sophisticated use of the quantum mechanics of nuclear spin systems have been developed (for example, compare the two 1H radiofrequency pulses utilized in the COSY experiment with the 27 radiofrequency pulses applied at five different frequencies and four extended decoupling sequences utilized in the CBCA(CO)NH experiment). These developments have occurred largely after the publication of the seminal texts NMR of proteins and nucleic acids, by K. Wu¨thrich in 1986 and Principles of nuclear magnetic resonance in one and two dimensions, by R. R. Ernst, G. Bodenhausen and A. Wokaun in 1987. In our view, a definite need exists for a graduate-level textbook that not only describes the practical aspects of state-of-the-art techniques in biomolecular NMR spectroscopy, but also presents the fundamental principles used to develop these techniques. Only a thorough understanding of the unifying principles of NMR spectroscopy empowers a student or researcher to evaluate, implement and optimize new techniques that continue to emerge at a dizzying pace. In this spirit, Protein NMR Spectroscopy: Principles and Practice systematically explicates NMR spectroscopy from the basic theoretical and experimental principles, to powerful theoretical formulations of the quantum mechanics of nuclear spin systems, and ultimately to optimal experimental methods for biomolecular investigations. Although the text concentrates on applications of NMR spectroscopy to proteins, all of the theory and most of the experiments are equally relevant to nucleic acids, carbohydrates and small organic molecules. The text focuses on the NMR spectroscopy of diamagnetic molecules (without unpaired electron spins); issues germane specifically to paramagnetic molecules (with unpaired electron spins) are discussed in other sources (see Suggested Reading). This text will serve a wide audience of students and researchers reflective of the variety of disciplines that employ NMR spectroscopy, including biochemistry, biology, chemistry, and physics. Protein NMR Spectroscopy: Principles and Practice provides a comprehensive treatment of the principles and practice of biomolecular NMR spectroscopy. The theoretical basis of NMR spectroscopy is described in Chapters 1, 2, 4 and 5. Classical NMR spectroscopy of isolated spins is introduced through the Bloch equations in Chapter 1. The density matrix and product operator theoretical formalisms of NMR spectroscopy of coupled multi-spin systems are presented in Chapter 2. The major principles of multidimensional NMR
PREFACE
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FIRST EDITION
ix
spectroscopy, including frequency labeling of coherences, coherence transfer and mixing, and coherence pathway selection, are described in Chapter 4. The principles of nuclear spin relaxation and chemical exchange are developed by using the Bloch, Solomon and semiclassical theoretical descriptions in Chapter 5. The experimental techniques used in modern multidimensional NMR spectroscopy of biological macromolecules in solution are described in Chapters 3, 6, and 7. Theoretical and practical aspects of experimental NMR spectroscopy, including data acquisition and data processing, are introduced in Chapter 3. Widely used spectroscopic techniques, such as spin decoupling, water suppression, composite pulses, selective pulses and one-dimensional NMR spectroscopy, also are presented in Chapter 3. Multidimensional 1H homonuclear NMR spectroscopy is described theoretically and illustrated with experimental examples in Chapter 6. Multidimensional 13 C/15N heteronuclear NMR spectroscopy is described theoretically and illustrated with experimental examples in Chapter 7. Both Chapter 6 and 7 present the principal experimental techniques used to obtain resonance assignments, to measure internuclear distances, and to determine scalar coupling constants. Methods for the interpretation of NMR spectra, including resonance assignment strategies and protocols for structure calculations, are summarized in Chapter 8. These aspects of biomolecular NMR spectroscopy are evolving rapidly and detailed discussions could constitute an entire additional book. Consequently, Chapter 8 is intended to provide an overview of the subject and an entry into the primary literature. In order to provide continuity and consistency throughout the text, a single protein, ubiquitin (76 amino acid residues, Mr ¼ 8,565 Da), is used to demonstrate the experimental aspects of NMR spectroscopy. Unlabeled bovine ubiquitin was purchased from Sigma Chemical Company (product number U6253, St. Louis, MO). 15N-labeled and 13 C/15N-double-labeled human ubiquitin were purchased from VLI Research (Southeastern, PA). The human and bovine amino acid sequences are identical. NMR spectroscopy was performed using Bruker 500- and 600-MHz NMR spectrometers at a temperature of 300 K. Sample concentrations were 2.0 mM for unlabeled ubiquitin and 1.25 mM for labeled ubiquitin. Samples were prepared in aqueous (95% H2O/5% D2O or 100% D2O) 50 mM potassium phosphate buffer at pH 5.8. NMR samples in 100% D2O solutions were prepared from samples in 95% H2O/5% D2O by performing four cycles of lyophilizing and dissolving in D2O (99.999 atom%) in the NMR tube. A common lament of the scientist who wishes to understand a new discipline is ‘‘What books should I read?’’ We hope that Protein NMR
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FIRST EDITION
Spectroscopy: Principles and Practice provides an answer for students and researchers with an interest in biomolecular NMR spectroscopy. John Cavanagh Wayne J. Fairbrother Arthur G. Palmer, III Nicholas J. Skelton
ACKNOWLEDGEMENTS
In writing the second edition of Protein NMR Spectroscopy: Principles and Practice, we have benefited greatly from helpful discussions with Mikael Akke, Clemens Anklin, Volker Do¨tsch, George Grey, Christian Griesinger, Stephan Grzesiek, William Hull, Lewis Kay, James Keeler, Eriks Kupcˇe, Ann McDermott, Detlef Moskau, Daniel Nietlispach, Daniel Raleigh, A. J. Shaka, Steve Smallcombe, Ron Venters and Jonathan Waltho. The second edition of Protein NMR Spectroscopy: Principles and Practice also has benefited immensely from comments from numerous individuals who have learned or taught from the first edition. We hope that these individuals, anonymous only because they are too numerous to list, will recognize their suggestions incorporated into the revised text. We apologize in advance if we, through our own oversight, have failed to correct inaccuracies pointed out to us by readers. We thank Bruker Instruments, Inc. for providing Figures 3.2 and 3.3, Ad Bax for providing Figure 7.54, Janet Cheetham and Duncan Smith for providing data for Figures 7.43 and 7.46, Ron Venters for providing data for Figure 9.1 and Daniel Nietlispach for providing data for Figure 9.2. Figure 3.17 was prepared using the Azara program (generously provided by Wayne Boucher). We thank Joel Butterwick, Michael Grey, and Francesca Massi for assistance in preparing the new figures that have been added in Chapters 6, 7, 8, and 10. We are
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ACKNOWLEDGEMENTS
particularly indebted to James Keeler for permitting us to follow closely his lecture notes in preparing Section 4.3. We thank editors Noelle Gracy, Luna Han, Julie Ochs, and Anne Russum (Elsevier) for their continued assistance and encouragement throughout the writing of the second edition of Protein NMR Spectroscopy: Principles and Practice. Finally, we are grateful particularly for the patience, support and understanding of Patricia Bauer, the Cavanagh family, Jenni Heath, Pearl Tsang, and Cindy Skelton throughout many evenings and weekends devoted to this project.
CONTENTS
Preface
v
Preface to the First Edition
vii
Acknowledgements
C H A P T E R
xi
1
CLASSICAL NMR SPECTROSCOPY 1.1
Nuclear Magnetism
2
1.2
The Bloch Equations
7
1.3
The One-Pulse NMR Experiment
16
1.4
Linewidth
18
1.5
Chemical Shift
21
1.6
Scalar Coupling and Limitations of the Bloch Equations
23
References
27
xiii
xiv
CONTENTS
C H A P T E R
2
THEORETICAL DESCRIPTION OF NMR SPECTROSCOPY 2.1
Postulates of Quantum Mechanics 2.1.1 2.1.2 2.1.3 2.1.4
2.2
THE SCHRÖDINGER EQUATION EIGENVALUE EQUATIONS SIMULTANEOUS EIGENFUNCTIONS EXPECTATION VALUE OF THE MAGNETIC MOMENT
The Density Matrix 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5
DIRAC NOTATION QUANTUM STATISTICAL MECHANICS THE LIOUVILLE-VON NEUMANN EQUATION THE ROTATING FRAME TRANSFORMATION MATRIX REPRESENTATIONS OF THE SPIN OPERATORS
29 30 31 34 35 37 37 40 41 43 46
2.3
Pulses and Rotation Operators
50
2.4
Quantum Mechanical NMR Spectroscopy
54
2.4.1 2.4.2
55 56
2.5
EQUILIBRIUM AND OBSERVATION OPERATORS THE ONE-PULSE EXPERIMENT
Quantum Mechanics Of Multispin Systems 2.5.1 2.5.2 2.5.3 2.5.4
DIRECT PRODUCT SPACES SCALAR COUPLING HAMILTONIAN ROTATIONS IN PRODUCT SPACES ONE-PULSE EXPERIMENT FOR A TWO-SPIN SYSTEM
58 59 61 65 68
2.6
Coherence
70
2.7
Product Operator Formalism
77
2.7.1 2.7.2 2.7.3
OPERATOR SPACES BASIS OPERATORS EVOLUTION IN THE PRODUCT OPERATOR FORMALISM
78 80 84
2.7.3.1 2.7.3.2 2.7.3.3
84 85 86
2.7.4 2.7.5 2.7.6
FREE PRECESSION PULSES PRACTICAL POINTS
SINGLE-QUANTUM COHERENCE AND OBSERVABLE OPERATORS MULTIPLE-QUANTUM COHERENCE COHERENCE TRANSFER AND GENERATION OF MULTIPLE-QUANTUM COHERENCE
88 90 92
xv
CONTENTS
2.7.7 EXAMPLES OF PRODUCT OPERATOR CALCULATIONS 2.7.7.1 2.7.7.2 2.7.7.3 2.7.7.4 2.8
THE SPIN ECHO INSENSITIVE NUCLEI ENHANCED BY POLARIZATION TRANSFER REFOCUSED INEPT SPIN-STATE SELECTIVE POLARIZATION TRANSFER
93 93 96 98 99
Averaging of the Spin Hamiltonians and Residual Interactions
102
References
112
C H A P T E R
3
EXPERIMENTAL ASPECTS OF NMR SPECTROSCOPY 3.1
NMR Instrumentation
114
3.2
Data Acquisition 3.2.1 SAMPLING 3.2.2 OVERSAMPLING AND DIGITAL FILTERS 3.2.3 QUADRATURE DETECTION
124 124 126 132
3.3
Data Processing 3.3.1 FOURIER TRANSFORMATION 3.3.2 DATA MANIPULATIONS 3.3.2.1 ZERO-FILLING 3.3.2.2 APODIZATION 3.3.2.3 PHASING 3.3.3 SIGNAL-TO-NOISE RATIO 3.3.4 ALTERNATIVES TO FOURIER TRANSFORMATION 3.3.4.1 LINEAR PREDICTION 3.3.4.2 MAXIMUM ENTROPY RECONSTRUCTION
136 136 142 142 143 151 158 159 160 161
3.4
Pulse Techniques 3.4.1 OFF-RESONANCE EFFECTS 3.4.2 B1 INHOMOGENEITY 3.4.3 COMPOSITE PULSES 3.4.4 SELECTIVE PULSES 3.4.5 PHASE-MODULATED PULSES 3.4.6 ADIABATIC PULSES
165 165 172 174 179 181 189
xvi 3.5
CONTENTS
Spin Decoupling
201
3.5.1 3.5.2
SPIN DECOUPLING THEORY COMPOSITE PULSE DECOUPLING
201 204
3.5.3 3.5.4 3.5.5
ADIABATIC SPIN DECOUPLING CYCLING SIDEBANDS RECOMMENDATIONS FOR SPIN DECOUPLING
209 212 216
3.6
B0 Field Gradients
217
3.7
Water Suppression Techniques
221
3.7.1 3.7.2
223
3.7.3 3.7.4 3.8
PRESATURATION JUMP-RETURN AND BINOMIAL SEQUENCES SPIN LOCK AND FIELD GRADIENT PULSES POSTACQUISITION SIGNAL PROCESSING
One-Dimensional 1H NMR Spectroscopy 3.8.1 3.8.2
3.8.2.2 3.8.2.3 3.8.2.5
SAMPLE PREPARATION INSTRUMENT SETUP 3.8.2.1 TEMPERATURE CALIBRATION 237 TUNING SHIMMING 238 3.8.2.4 PULSE WIDTH CALIBRATION RECYCLE DELAY 257 3.8.2.6 LINEWIDTH MEASUREMENT
3.8.3 3.8.4 3.8.4.2
REFERENCING ACQUISITION AND DATA PROCESSING 3.8.4.1 ONE-PULSE EXPERIMENT HAHN ECHO EXPERIMENT
References
C H A P T E R
224 227 232 234 234 236 236 252 259 262 263 263 265 267
4
MULTI-DIMENSIONAL NMR SPECTROSCOPY 4.1
Two-Dimensional NMR Spectroscopy
273
4.2
Coherence Transfer and Mixing 4.2.1 THROUGH-BOND COHERENCE TRANSFER 4.2.1.1 COSY-TYPE COHERENCE TRANSFER 4.2.1.2 TOCSY TRANSFER THROUGH-BONDS
280 280 281 284
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CONTENTS
4.2.2 4.2.3 4.2.4 4.3
THROUGH-SPACE COHERENCE TRANSFER HETERONUCLEAR COHERENCE TRANSFER COHERENCE TRANSFER UNDER RESIDUAL DIPOLAR COUPLING HAMILTONIANS
Coherence Selection, Phase Cycling, and Field Gradients 4.3.1 COHERENCE LEVEL DIAGRAMS 4.3.2
4.3.3
4.3.4
PHASE CYCLES 4.3.2.1 SELECTION OF A COHERENCE TRANSFER PATHWAY 4.3.2.2 SAVING TIME 4.3.2.3 ARTIFACT SUPPRESSION 4.3.2.4 LIMITATIONS OF PHASE CYCLING PULSED FIELD GRADIENTS 4.3.3.1 SELECTION OF A COHERENCE TRANSFER PATHWAY 4.3.3.2 ARTIFACT SUPPRESSION 4.3.3.3 LIMITATIONS OF PULSED FIELD GRADIENTS FREQUENCY DISCRIMINATION 4.3.4.1 FREQUENCY DISCRIMINATION BY PHASE CYCLING 4.3.4.2 FREQUENCY DISCRIMINATION BY PULSED FIELD GRADIENTS 4.3.4.3 ALIASING. FOLDING, AND PHASING IN MULTIDIMENSIONAL NMR SPECTROSCOPY
289 290 291 292 293 295 298 305 307 310 311 311 313 314 315 320 322 323
4.4
Resolution and Sensitivity
326
4.5
Three- and Four-Dimensional NMR Spectroscopy
327
References
331
C H A P T E R
5
RELAXATION AND DYNAMIC PROCESSES 5.1 Introduction and Survey of Theoretical Approaches
334
5.1.1
RELAXATION IN THE BLOCK EQUATIONS
337
5.1.2
THE SOLOMON EQUATIONS
338
5.1.3
RANDOM-PHASE MODEL FOR TRANSVERSE RELAXATION
346
5.1.4
BLOCH, WANGSNESS, AND REDFIELD THEORY
350
xviii 5.2
CONTENTS
The Master Equation
351
5.2.1
INTERFERENCE EFFECTS
359
5.2.2
LIKE SPINS, UNLIKE SPINS, AND THE SECULAR APPROXIMATION
360
5.2.3
RELAXATION IN THE ROTATING FRAME
363
5.3
Spectral Density Functions
365
5.4
Relaxation Mechanisms
370
5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.6
INTRAMOLECULAR DIPOLAR RELAXATION FOR IS SPIN SYSTEM INTRAMOLECULAR DIPOLAR RELAXATION FOR SCALAR-COUPLED IS SPIN SYSTEM INTRAMOLECULAR DIPOLAR RELAXATION FOR IS SPIN SYSTEM IN THE ROTATING FRAME CHEMICAL SHIFT ANISOTROPY AND QUADRUPOLΛR RELAXATION RELAXATION INTERFERENCE SCALAR RELAXATION
371 378 381 383 385 387
5.5
Nuclear Overhauser Effect
388
5.6
Chemical Exchange Effects in NMR Spectroscopy
391
5.6.1 5.6.2
392
CHEMICAL EXCHANGE FOR ISOLATED SPINS QUALITATIVE EFFECTS OF CHEMICAL EXCHANGE IN SCALAR-COUPLED SYSTEMS References
C H A P T E R
401 402
6
EXPERIMENTAL 1H NMR METHODS 6.1 6.2
Assessment of the 1D 1H Spectrum
COSY-Type experiments 6.2.1 COSY 6.2.2.1 PRODUCT OPERATOR ANALYSIS 6.2.2.2 EXPERIMENTAL PROTOCOL 6.2.2.3 PROCESSING 6.2.2.4 INFORMATION CONTENT 6.2.2.5 QUANTITATION OF SCALAR COUPLING CONSTANTS IN COSY SPECTRA
406 409 410 410 412 415 418 420
xix
CONTENTS
6.2.2.6
EXPERIMENTAL VARIANTS
426
6.2.2 RELAYED COSY 6.2.2.1 PRODUCT OPERATOR ANALYSIS 6.2.2.2 EXPERIMENTAL PROTOCOL 6.2.2.3 PROCESSING 432 6.2.2.4 INFORMATION CONTENT 6.2.3 DOUBLE-RELAYED COSY
429 430 432
6.3
437
Multiple-Quantum Filtered COSY
432 433
6.3.1 2QF-COSY 6.3.1.1 PRODUCT OPERATOR ANALYSIS 6.3.1.2 EXPERIMENTAL PROTOCOL 6.3.1.3 PROCESSING 6.3.1.4 INFORMATION CONTENT 6.3.2 3QF-COSY 6.3.2.1 PRODUCT OPERATOR ANALYSIS 6.3.2.2 EXPERIMENTAL PROTOCOL AND PROCESSING 6.3.2.3 INFORMATION CONTENT 6.3.3 E.COSY 6.3.3.1 PRODUCT OPERATOR ANALYSIS 6.3.3.2 EXPERIMENTAL PROTOCOL 6.3.3.3 PROCESSING 6.3.3.4 INFORMATION CONTENT 6.3.3.5 EXPERIMENTAL VARIANTS
440 440 444 445 446 449 449 451 452 455 457 460 461 462 463
6.4 Multiple-Quantum Spectroscopy 6.4.1 2Q SPECTROSCOPY 6.4.1.1 PRODUCT OPERATOR ANALYSIS 6.4.1.2 EXPERIMENTAL PROTOCOL 6.4.1.3 PROCESSING 6.4.1.4 INFORMATION CONTENT 6.4.2 3Q SPECTROSCOPY 6.4.2.1 PRODUCT OPERATOR ANALYSIS 6.4.2.2 EXPERIMENTAL PROTOCOL AND PROCESSING 6.4.2.3 INFORMATION CONTENT
463 465 466 473 474 474 481 481 483 484
6.5 TOCSY 6.5.1 PRODUCT OPERATOR ANALYSIS 6.5.2 EXPERIMENTAL PROTOCOL 6.5.3 PROCESSING
486 486 493 496
xx
CONTENTS
6.5.4 INFORMATION CONTENT 6.5.5 EXPERIMENTAL VARIANTS 6.6 Cross-Relaxation NMR Experiments 6.6.1 NOESY 6.6.1.1 PRODUCT OPERATOR ANALYSIS 6.6.1.2 EXPERIMENTAL PROTOCOL 6.6.1.3 PROCESSING 6.6.1.4 INFORMATION CONTENT 6.6.1.5 EXPERIMENTAL VARIANTS 6.6.2 ROESY 6.6.2.1 PRODUCT OPERATOR ANALYSIS 6.6.2.2 EXPERIMENTAL PROTOCOL AND PROCESSING 6.6.2.3 INFORMATION CONTENT 6.6.2.4 EXPERIMENTAL VARIANTS
498 499 502 502 503 506 510 510 511 517 517 520 522 524
6.7
1H 3D Experiments
525
6.7.1 EXPERIMENTAL PROTOCOL 6.7.2 PROCESSING 6.7.3 INFORMATION CONTENT
526 527 527
6.7.4 EXPERIMENTAL VARIANTS
528
References
529
C H A P T E R
7
HETERONUCLEAR NMR EXPERIMENTS 7.1 Heteronuclear Correlation NMR Spectroscopy 7.1.1 BASIC HMQC AND HSQC EXPERIMENTS 7.1.1.1 THE HMQC EXPERIMENT 7.1.1.2 THE HSQC EXPERIMENT 7.1.1.3 THE CONSTANT-TIME HSQC EXPERIMENT 7.1.1.4 COMPARISON OF HMQC AND HSQC SPECTRA 7.1.2 ADDITIONAL CONSIDERATIONS IN HETERONUCLEAR CORRELATION EXPERIMENTS 7.1.2.1 PHASE CYCLING AND ARTIFACT SUPPRESSION 7.1.2.2 13C SCALAR COUPLING AND MULTIPLET STRUCTURE
535 536 536 540 543 544 546 546 548
xxi
CONTENTS
7.1.2.3 7.1.2.4
FOLDING AND ALIASING PROCESSING HETERONUCLEAR CORRELATION EXPERIMENTS
7.1.3 DECOUPLED HSQC, SENSITIVITY-ENHANCED HSQC AND TROSY EXPERIMENTS 7.1.3.1 THE DECOUPLED HSQC EXPERIMENT 7.1.3.2 SENSITIVITY-ENHANCED HSQC 7.1.3.3 TROSY EXPERIMENT 7.1.3.4 COMPARISON OF DECOUPLED HSQC, PEP-HSQC, AND TROSY EXPERIMENTS 7.1.3.5 RELAXATION INTERFERENCE AND TROSY 7.1.3.6 SPECTRA OF LARGER PROTEINS 7.1.4 WATER SUPPRESSION AND GRADIENT ENHANCEMENT OF HETERONUCLEAR CORRELATION SPECTRA 7.1.4.1 SOLVENT SUPPRESSION 7.1.4.2 GRADIENT-ENHANCED HSQC AND TROSY NMR SPECTROSCOPY 7.1.5 THE CONSTANT-TIME 1H-13C HSQC EXPERIMENT 7.2
Heteronuclear-Edited NMR Spectroscopy 7.2.1 3D NOESY-HSQC SPECTROSCOPY 7.2.1.1 3D 1H-15N NOESY-HSQC 7.2.1.2 3D 1H-13C NOESY-HSQC 7.2.2 3D TOCSY-HSQC SPECTROSCOPY 7.2.3 3D HSQC-NOESY AND HSQC-TOCSY EXPERIMENTS 7.2.4 HMQC-NOESY-HMQC EXPERIMENTS 7.2.4.1 3D 15N/15N HMQC-NOESY-HMQC 7.2.4.2 4D 13C/15N HMQC-NOESY-HMQC 7.2.4.3 4D 13C/13C HMQC-NOESY-HMQC 7.2.4.4 PROCESSING 4D HMQC-NOESY-HMQC SPECTRA 7.2.5 RELATIVE MERITS OF 3D AND 4D HETERONUCLEAR-EDITED NOESY SPECTROSCOPY 7.3 13C - 13C CORRELATIONS: THE HCCH-COSY AND HCCH-TOCSY EXPERIMENTS 7.3.1 7.3.2 7.3.3
HCCH-COSY CONSTANT-TIME HCCH-COSY HCCH-TOCSY
549 552 552 553 560 566 570 570 573 573 574 578 581 582 585 588 589 591 593 594 595 597 599 600 601 603 607 608
xxii 7.4
CONTENTS
3D Triple-Resonance Experiments 7.4.1
7.4.2 7.4.3 7.4.4
7.4.5
7.4.6 7.5
7.6
A PROTOTYPE TRIPLE-RESONANCE EXPERIMENT: HNCA 7.4.1.1 A SIMPLE HNCA EXPERIMENT 7.4.1.2 THE CT-HNCA EXPERIMENT 7.4.1.3 THE DECOUPLED CT-HNCA EXPERIMENT 7.4.1.4 THE GRADIENT-ENHANCED HNCA EXPERIMENT 7.4.1.5 THE GRADIENT-ENHANCED TROSY-HNCA EXPERIMENT A COMPLEMENTARY APPROACH: THE HN(CO)CA EXPERIMENT A STRAIGHT-THROUGH TRIPLE-RESONANCE EXPERIMENT: H(CA)NH BACKBONE CORRELATIONS WITH THE 13CO SPINS 7.4.4.1 HNCO 7.4.4.2 HN(CA)CO CORRELATIONS WITH THE Cβ/Hβ SPINS 7.4.5.1 CBCA(CO)NH 7.4.5.2 CBCANH 7.4.5.3 HNCACB ADDITIONAL CONSIDERATIONS FOR TRIPLE-RESONANCE EXPERIMENTS
613 614 618 625 626 627 628 629 632 637 637 638 641 642 645 650 654
Measurement of Scalar Coupling Constants
656
7.5.1 7.5.2
656 660
HNCA-J EXPERIMENT HNHA EXPERIMENT
Measurement of Residual Dipolar Coupling Constants References
C H A P T E R
665 673
8
EXPERIMENTAL NMR RELAXATION METHODS 8.1
Pulse Sequences and Experimental Methods
680
8.2
Picosecond-Nanosecond dynamics
685
8.2.1
EXPERIMENTAL METHODS FOR 15N LABORATORY-FRAME RELAXATION
686
CONTENTS
xxiii 8.2.2 8.2.3 8.2.4
EXPERIMENTAL METHODS FOR 15N RELAXATION INTERFERENCE EXPERIMENTAL METHODS FOR 13CH2D METHYL LABORATORY-FRAME RELAXATION EXPERIMENTAL METHODS FOR 13CO LABORATORY-FRAME RELAXATION
8.3 Microsecond-Second Dynamics 8.3.1 8.3.2 8.3.3 8.3.4 8.3.5
LINESHAPE ANALYSIS ZZ-EXCHANGE SPECTROSCOPY R1ρ ROTATING-FRAME RELAXATION METHODS CPMG RELAXATION METHODS CHEMICAL EXCHANGE IN MULTIPLE-QUANTUM SPECTROSCOPY 8.3.6 TROSY-BASED APPROACHES References
C H A P T E R
692 693 699 702 704 706 707 711 715 718 721
9
LARGER PROTEINS AND MOLECULAR INTERACTIONS 9.1 Larger Proteins 9.1.1 9.1.2
9.1.3 9.1.4 9.1.5
PROTEIN DEUTERATION RELAXATION IN PERDEUTERATED AND RANDOM FRACTIONALLY DEUTERATED PROTEINS SENSITIVITY FOR PERDEUTERΛTED PROTEINS 2H ISOTOPE SHIFTS EXPERIMENTS FOR 1HN, 15N, 13Cα, 13C, AND 13CO ASSIGNMENTS IN DEUTERATED PROTEINS
9.1.6
CONSTANT-TIME HNCA FOR DEUTERATED PROTEINS 9.1.5.2 HN(CA)CB FOR DEUTERATED PROTEINS 9.1.5.3 OTHER EXPERIMENTS FOR RESONANCE ASSIGNMENTS SIDE CHAIN 13C ASSIGNMENTS IN DEUTERATED
9.1.7
SIDE CHAIN 1H ASSIGNMENTS
725 726
728 729 732 733
9.1.5.1
PROTEINS
735 737 739 740 743
xxiv
CONTENTS
9.1.8
9.1.9 9.2
NOE RESTRAINTS FROM DEUTERATED PROTEINS 9.1.8.1 4D HN-HN 15N/15N-SEPARATED NOESY EXPERIMENT 13C/15N, 13C/15C AND 15N/15N-SEPARATED NOESY 9.1.8.2 EXPERIMENTS ON RANDOM FRACTIONALLY DEUTERATED PROTEINS SELECTIVE PROTONATION
Intermolecular Interactions 9.2.1 9.2.2
745
747 749 753
EXCHANGE REGIMES PROTEIN-LIGAND BINDING INTERFACES
753 755
9.2.2.1 9.2.2.2 9.2.2.3
756 757
CHEMICAL SHIFT MAPPING CROSS-SATURATION TRANSVERSE RELAXATION AND AMIDE PROTON SOLVENT EXCHANGE 9.2.3. RESONANCE ASSIGNMENTS AND STRUCTURAL RESTRAINTS FOR PROTEIN COMPLEXES 9.2.3.1 ASSIGNMENTS AND STRUCTURES OF PROTEINS IN PROTEIN-LIGAND COMPLEXES 9.2.3.2 ISOTOPE EDITED/FILTERED NOESY TO DEFINE INTERMOLECULAR INTERFACES 9.3
745
759 760 761 762
Methods for Rapid Data Acquisition
769
9.3.1 NONUNIFORM SAMPLING 9.3.2 GFT-NMR SPECTROSCOPY 9.3.3 PROJECTION-RECONSTRUCTION References
770 771 773 775
C H A P T E R
1 0
SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION, AND OTHER APPLICATIONS 10.1 Resonance Assignment Strategies 10.1.1
782
1H RESONANCE ASSIGNMENTS FOR UNLABELED
PROTEINS 10.1.2 HETERONUCI.EAR RESONANCE ASSIGNMENTS FOR ISOTOPICALLY LABELED PROTEINS
10.2 Three-Dimensional Solution Structures
782 792 796
CONTENTS
xxv
10.2.1 NMR-DERIVED STRUCTURAL RESTRAINTS 10.2.1.1 NOE DISTANCE RESTRAINTS 10.2.1.2 DIHEDRAL ANGLE RESTRAINTS FROM SCALAR COUPLING CONSTANTS 10.2.1.3 DIHEDRAL ANGLE RESTRAINTS FROM ISOTROPIC CHEMICAL SHIFTS 10.2.1.4 RESTRAINTS FROM RESIDUAL DIPOLAR COUPLING CONSTANTS 10.2.1.5 HYDROGEN BOND RESTRAINTS FROM AMIDE PROTON-SOLVENT EXCHANGE 10.2.1.6 HYDROGEN BOND RESTRAINTS FROM TRANS-HYDROGEN BOND SCALAR COUPLING CONSTANTS 10.2.2 STRUCTURE DETERMINATION 10.3 Conclusion References
796 797
Table of Symbols List of Figures List of Tables Suggested Reading Index
819 825 837 839 841
For the reader's easy reference, the Table of Constants and the Spin-1/2 Product Operator Equations are given on the inside back cover end pages.
798 804 804 805
806 806 813 814
CHAPTER
1 CLASSICAL NMR SPECTROSCOPY
The explosive growth in the field of nuclear magnetic resonance (NMR) spectroscopy that continues today originated with the development of pulsed Fourier transform NMR spectroscopy by Ernst and Anderson (1) and the conception of multidimensional NMR spectroscopy by Jeener (2, 3). Currently, NMR spectroscopy and x-ray crystallography are the only techniques capable of determining the three-dimensional structures of macromolecules at atomic resolution. In addition, NMR spectroscopy is a powerful technique for investigating time-dependent chemical phenomena, including reaction kinetics and intramolecular dynamics. Historically, NMR spectroscopy of biological macromolecules was limited by the low inherent sensitivity of the technique and by the complexity of the resultant NMR spectra. The former limitation has been alleviated partially by the development of more powerful magnets and more sensitive NMR spectrometers and by advances in techniques for sample preparation (both synthetic and biochemical). The latter limitation has been transmuted into a significant advantage by the phenomenal advances in the theoretical and experimental capabilities of NMR spectroscopy (and spectroscopists). The history of these developments has been reviewed by Ernst and by Wu¨thrich in their 1991 and 2002 Nobel Laureate lectures, respectively (4, 5). In light of subsequent developments, the conclusion of Bloch’s
1
2
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
initial report of the observation of nuclear magnetic resonance in water proved prescient: ‘‘We have thought of various investigations in which this effect can be used fruitfully’’ (6).
1.1 Nuclear Magnetism Nuclear magnetic resonances in bulk condensed phase were reported for the first time in 1946 by Bloch et al. (6) and by Purcell et al. (7). Nuclear magnetism and NMR spectroscopy are manifestations of nuclear spin angular momentum. Consequently, the theory of NMR spectroscopy is largely the quantum mechanics of nuclear spin angular momentum, an intrinsically quantum mechanical property that does not have a classical analog. The physical origins of the nuclear spin angular momentum are complex, but have been discussed in review articles (8, 9). The spin angular momentum is characterized by the nuclear spin quantum number, I. Although NMR spectroscopy takes the nuclear spin as a given quantity, certain systematic features can be noted: (i ) nuclei with odd mass numbers have half-integral spin quantum numbers, (ii ) nuclei with an even mass number and an even atomic number have spin quantum numbers equal to zero, and (iii ) nuclei with an even mass number and an odd atomic number have integral spin quantum numbers. Because the NMR phenomenon relies on the existence of nuclear spin, nuclei belonging to category (ii ) are NMR inactive. Nuclei with spin quantum numbers greater than 1/2 also possess electric quadrupole moments arising from nonspherical nuclear charge distributions. The lifetimes of the magnetic states for quadrupolar nuclei in solution normally are much shorter than are the lifetimes for nuclei with I ¼ 1/2. NMR resonance lines for quadrupolar nuclei are correspondingly broad and can be more difficult to study. Relevant properties of nuclei commonly found in biomolecules are summarized in Table 1.1. For NMR spectroscopy of biomolecules, the most important nuclei with I ¼ 1/2 are 1H, 13C, 15N, 19F, and 31P; the most important nucleus with I ¼ 1 is the deuteron (2H). The nuclear spin angular momentum, I, is a vector quantity with magnitude given by jIj ¼ ½I I1=2 ¼ h½IðI þ 1Þ1=2 ,
½1:1
in which I is the nuclear spin angular momentum quantum number and h is Planck’s constant divided by 2. Due to the restrictions of quantum mechanics, only one of the three Cartesian components of I can be
3
1.1 NUCLEAR MAGNETISM
TABLE 1.1 Properties of selected nucleia Nucleus 1
H H 13 C 14 N 15 N 17 O 19 F 23 Na 31 P 113 Cd 2
I
(T s)–1
Natural abundance (%)
1/2 1 1/2 1 1/2 5/2 1/2 3/2 1/2 1/2
2.6752 108 4.107 107 6.728 107 1.934 107 2.713 107 3.628 107 2.518 108 7.081 107 1.0839 108 5.961 107
99.99 0.012 1.07 99.63 0.37 0.038 100.00 100.00 100.00 12.22
a
Shown are the nuclear spin angular momentum quantum number, I, the magnetogyric ratio, , and the natural isotopic abundance for nuclei of particular importance in biological NMR spectroscopy.
specified simultaneously with I2 I I. By convention, the value of the z-component of I is specified by the following equation: Iz ¼ hm,
½1:2
in which the magnetic quantum number m ¼ (I, I þ 1, . . . , I 1, I). Thus, Iz has 2I þ 1 possible values. The orientation of the spin angular momentum vector in space is quantized, because the magnitude of the vector is constant and the z-component has a set of discrete possible values. In the absence of external fields, the quantum states corresponding to the 2I þ 1 values of m have the same energy, and the spin angular momentum vector does not have a preferred orientation. Nuclei that have nonzero spin angular momentum also possess nuclear magnetic moments. As a consequence of the Wigner–Eckart theorem (10), the nuclear magnetic moment, l, is collinear with the vector representing the nuclear spin angular momentum vector and is defined by l ¼ I, z ¼ Iz ¼ hm,
½1:3
in which the magnetogyric ratio, , is a characteristic constant for a given nucleus (Table 1.1). Because angular momentum is a quantized
4
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
property, so is the nuclear magnetic moment. The magnitude of , in part, determines the receptivity of a nucleus in NMR spectroscopy. In the presence of an external magnetic field, the spin states of the nucleus have energies given by E ¼ l B,
½1:4
in which B is the magnetic field vector. The minimum energy is obtained when the projection of onto B is maximized. Because |I| 4 Iz, l cannot be collinear with B and the m spin states become quantized with energies proportional to their projection onto B. In an NMR spectrometer, the static external magnetic field is directed by convention along the z-axis of the laboratory coordinate system. For this geometry, [1.4] reduces to Em ¼ Iz B0 ¼ mhB0,
½1:5
in which B0 is the static magnetic field strength. In the presence of a static magnetic field, the projections of the angular momentum of the nuclei onto the z-axis of the laboratory frame results in 2I þ 1 equally spaced energy levels, which are known as the Zeeman levels. The quantization of Iz is illustrated by Fig. 1.1.
a
b
+h
2h
+h/2 3h/2 –h/2 –h
FIGURE 1.1 Angular momentum. Shown are the angular momentum vectors, I, and the allowed z-components, Iz, for (a) a spin-1/2 particle and (b) a spin-1 particle. The location of I on the surface of the cone cannot be specified because of quantum mechanical uncertainties in the Ix and Iy components.
5
1.1 NUCLEAR MAGNETISM
At equilibrium, the different energy states are unequally populated because lower energy orientations of the magnetic dipole vector are more probable. The relative population of a state is given by the Boltzmann distribution, ,X I Nm Em Em ¼ exp exp N kB T kB T m ¼ I
mhB0 ¼ exp kB T
,X I
mhB0 1þ kB T
exp
m ¼ I
,X I m ¼ I
1 mhB0 1þ , 2I þ 1 kB T
mhB0 kB T
mhB0 1þ kB T
½1:6
in which Nm is the number of nuclei in the mth state and N is the total number of spins, T is the absolute temperature, and kB is the Boltzmann constant. The last two lines of [1.6] are obtained by expanding the exponential functions to first order using Taylor series, because at temperatures relevant for solution NMR spectroscopy, m hB0 =kB T 1. The populations of the states depend both on the nucleus type and on the applied field strength. As the external field strength increases, the energy differences between the nuclear spin energy levels become larger and the population differences between the states increase. Of course, polarization of the spin system to generate a population difference between spin states does not occur instantaneously upon application of the magnetic field; instead, the polarization, or magnetization, develops with a characteristic rate constant, called the spin–lattice relaxation rate constant (see Chapter 5). The bulk magnetic moment, M, and the bulk angular momentum, J, of a macroscopic sample are given by the vector sum of the corresponding quantities for individual nuclei, l and I. At thermal equilibrium, the transverse components (e.g., the x- or y-components) of l and I for different nuclei in the sample are uncorrelated and sum to zero. The small population differences between energy levels give rise to a bulk magnetization of the sample parallel (longitudinal) to the static magnetic field, M ¼ M0k, in which k is the unit vector in the z-direction.
6
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
Using [1.2], [1.3], and [1.6], M0 is given by I X
M0 ¼ h
mNm
m ¼ I
¼ Nh
I X
, m expðmhB0 =kB TÞ
m ¼ I
Nh
I X
I X
expðmhB0 =kB TÞ
m ¼ I
, mð1 þ mhB0 =kB TÞ
m ¼ I
I X
ð1 þ mhB0 =kB TÞ
m ¼ I
I X N 2 h2 B0 =kB Tð2I þ 1Þ m2 m ¼ I 2 2
N h B0 IðI þ 1Þ=ð3kB TÞ:
½1:7
By analogy with other areas of spectroscopy, transitions between Zeeman levels can be stimulated by applied electromagnetic radiation. The selection rule governing magnetic dipole transitions is m ¼ 1. Thus, the photon energy, E, required to excite a transition between the m and m þ 1 Zeeman states is E ¼ hB0 ,
½1:8
which is seen to be directly proportional to the magnitude of the static magnetic field. By Planck’s Law, the frequency of the required electromagnetic radiation is given by ! ¼ E=h ¼ B0 , –1
¼ !=2 ¼ B0 =2,
½1:9
in units of s or Hertz, respectively. The sensitivity of NMR spectroscopy depends upon the population differences between Zeeman states. The population difference is only on the order of 1 in 105 for 1 H spins in an 11.7-T magnetic field. As a result, NMR is an insensitive spectroscopic technique compared to techniques such as visible or ultraviolet spectroscopy. This simple observation explains much of the impetus to construct more powerful magnets for use in NMR spectroscopy. For the most part, this text is concerned with the NMR spectroscopy of spin I ¼ 1/2 (spin-1/2) nuclei. For an isolated spin, only two nuclear spin states exist and two energy levels separated by E ¼ hB0 are obtained by application of an external magnetic field. A single Zeeman
7
1.2 THE BLOCH EQUATIONS
transition between the energy levels exists. The spin state with m ¼ þ1/2 is referred to as the state, and the state with m ¼ 1/2 is referred to as the state. If is positive (negative), then the state has lower (higher) energy compared to the state.
1.2 The Bloch Equations Bloch formulated a simple semiclassical vector model to describe the behavior of a sample of noninteracting spin-1/2 nuclei in a static magnetic field (11). The Bloch model is outlined briefly in this section; many of the concepts and terminology introduced persist throughout the text. The evolution of the bulk magnetic moment, M(t), represented as a vector quantity, is central to the Bloch formalism. In the presence of a magnetic field, which may include components in addition to the static field, M(t) experiences a torque that is equal to the time derivative of the angular momentum, dJðtÞ ¼ MðtÞ BðtÞ: dt
½1:10
Multiplying both sides by yields dMðtÞ ¼ MðtÞ BðtÞ: dt
½1:11
The physical significance of this equation can be seen by using a frame of reference rotating with respect to the fixed laboratory axes. The angular velocity of the rotating axes is represented by the vector x. Without loss of generality, the two coordinate systems are assumed to be superposed initially. Vectors are represented identically in the two coordinate systems; however, time differentials are represented differently in the two coordinate systems. The equations of motion of M(t) in the laboratory and rotating frames are related by (12)
dMðtÞ dMðtÞ ¼ þ MðtÞ x dt rot dt lab ¼ MðtÞ ½BðtÞ þ x:
½1:12
8
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
The equation of motion for the magnetization in the rotating frame has the same form as in the laboratory frame, provided that the field B(t) is replaced by an effective field, Beff, given by Beff ¼ BðtÞ þ x=:
½1:13
For the choice x ¼ B(t), the effective field is zero, so that M(t) is time independent in the rotating frame. Consequently, as seen from the laboratory frame, M(t) precesses around B(t) with a frequency x ¼ B. For a static field of strength B0, the precessional frequency, or the Larmor frequency, is given by !0 ¼ B0 :
½1:14
Thus, in the absence of other magnetic fields, the bulk magnetization precesses at the Larmor frequency around the main static field axis (defined as the z-direction). As discussed by Levitt (13), the Larmor frequency has different signs for spins with positive or negative gyromagnetic ratios, e.g., 1H and 15N, and this fact historically has caused confusion in correctly determining the absolute sign of NMR parameters. The magnitude of the precessional frequency is identical to the frequency of electromagnetic radiation required to excite transitions between Zeeman levels [1.9]. This identity is the reason that, within limits, a classical description of NMR spectroscopy is valid for systems of isolated spin-1/2 nuclei. Before proceeding further, the nomenclature used to refer to the strength of a magnetic field needs to be clarified. In NMR spectroscopy, the magnetic field strength B normally appears in the equation ! ¼ –B that defines the precessional frequency of the nuclear magnetic moment. Conventionally, B is referred to as the magnetic field strength measured in frequency units. Strictly speaking, the strength of the magnetic field is B, measured in Gauss or Tesla (104 G ¼ 1 T); therefore, denoting B as the magnetic field strength is incorrect (and has the obvious disadvantage of depending on the type of nucleus considered). That said, however, measuring magnetic field strength in frequency units (s–1 or Hertz) is very convenient in many cases. Consequently, throughout this text, both terms, B and B, will be used to denote field strength in appropriate units. For example, common usage refers to NMR spectrometers by the proton Larmor frequency of the magnet; thus, a spectrometer with an 11.7-T magnet is termed a 500-MHz spectrometer, and a spectrometer with a 21.2-T magnet is termed a 900-MHz spectrometer.
1.2 THE BLOCH EQUATIONS
9
Precession of the bulk magnetic moment about the static magnetic field constitutes a time-varying magnetic field. According to Faraday’s law of induction, a time-varying magnetic field produces an induced electromotive force in a coil of appropriate geometry located in the vicinity of the bulk sample (14, 15). Equation [1.11] suggests that precession of the bulk nuclear magnetization can be detected by such a mechanism. However, at thermal equilibrium, the bulk magnetization vector is collinear with the static field and no signal is produced in the coil. The key to producing an NMR signal is to disturb this equilibrium state. This text has as its subject pulsed NMR experiments in which a short burst of radiofrequency (rf ) electromagnetic radiation, typically of the order of several microseconds in duration, displaces the bulk magnetization from equilibrium. Such rf bursts are referred to as pulses. After the rf field is turned off, the bulk magnetization vector, M(t), will not, in general, be parallel to the static field. Consequently, the bulk magnetization will precess around the static field with an angular frequency !0 ¼ B0 and will generate a detectable signal in the coil. The magnetic component of an rf field that is linearly polarized along the x-axis of the laboratory frame is written as Brf ðtÞ ¼ 2B1 cosð!rf t þ Þi ¼ B1 fcosð!rf t þ Þi þ sinð!rf t þ Þjg
½1:15
þ B1 fcosð!rf t þ Þi sinð!rf t þ Þjg, where B1 is the amplitude of the applied field, !rf is the angular frequency of the rf field, often called the transmitter or carrier frequency, is the phase of the field, and i and j are unit vectors defining the x- and y-axes, respectively. In the present context, the amplitude and phase of the rf field are assumed to be constant; time-varying amplitude- or phase-modulated rf fields are considered in Section 3.4. In the second equality in [1.15], the rf field is decomposed into two circularly polarized fields rotating in opposite directions about the z-axis. Only the field rotating in the same sense as the magnetic moment interacts significantly with the magnetic moment; the counterrotating, nonresonant field influences the spins to order (B1/2B0)2, which is normally a very small number known as the Bloch-Siegert shift (but see Section 3.4.1). Thus, the nonresonant term can be ignored and the rf field is written simply as Brf ðtÞ ¼ B1 fcosð!rf t þ Þi þ sinð!rf t þ Þjg:
½1:16
10
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
In the case of a time-dependent field such as this, the solution to [1.11] can be found by moving to a rotating frame, which makes the perturbing field time independent. This is referred to as the rotating frame transformation. The new frame is chosen to rotate at angular frequency !rf about the z-axis. The equation of motion for the magnetization in the rotating frame, Mr(t), is given by dMr ðtÞ ¼ Mr ðtÞ Br ðtÞ, dt
½1:17
in which the effective field, Br, in the rotating frame is given by Br ¼ B1 cosir þ B1 sinjr þ B0 kr ;
½1:18
here B0 is known as the reduced static field and is equivalent to the z-component of the effective field, B0 ¼ =,
½1:19 r
r
and ¼ B0 !rf ¼ !0 !rf is known as the offset, and i , j , and kr are unit vectors in the rotating frame. Equation [1.17] differs from [1.12] only because the quantities on both sides of the equality have been expressed in the rotating frame. The rf field is described by the amplitude B1 and the phase . In accordance with Ernst et al. (16), the phase angle has been defined such that for an rf field of fixed phase x, Bx ¼ B1 and By ¼ 0. The magnitude of the effective field is given by Br ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðB1 Þ2 þ ðB0 Þ2 ¼ B1 = sin
½1:20
and the angle through which the effective field is tilted with respect to the z-axis is defined by tan ¼
B1 B1 !1 ¼ , ¼ B0
½1:21
in which !1 ¼ B1.The direction of the effective field, as defined by and , depends on the strength of the rf field, Brf(t), the difference between the transmitter and Larmor frequencies, and the phase of the rf field in the laboratory frame, as illustrated in Fig. 1.2. Frequently, Brf(t) is referred to directly as the ‘‘B1 field.’’ In the rotating frame, upon application of the B1 field, Mr(t) precesses around the effective field Br
11
1.2 THE BLOCH EQUATIONS z
∆B0
Br q y f
B1
x
FIGURE 1.2 Orientations of B0, B1, and Br in the rotating reference frame. Angles and are defined by [1.21] and [1.18].
with an angular frequency !r, !r ¼ Br :
½1:22
If the rf field is turned on for a time period p, called the pulse length, then the effective rotation angle (or flip angle) is given by ¼ !r p ¼ Br p ¼ B1 p = sin ¼ !1 p = sin:
½1:23
If the transmitter frequency, !rf, is equal to !0, then the irradiation is said to be applied on-resonance. In the on-resonance case, the offset term, , equals zero, Br ¼ B1, and the effective field is collinear with the B1 field in the rotating frame. These results have an important implication: the influence of the main static magnetic field, B0, has been removed. The bulk magnetization Mr(t) precesses around the axis defined by the B1 field, with frequency !r ¼ Br ¼ B1 ¼ !1. Precession of the magnetization about the effective field in the rotating reference frame is illustrated in Fig. 1.3. As general practice in this text, the rotating frame will not be indicated explicitly, and unless otherwise stated, the rotating frame of reference will be assumed [i.e., M(t) will be written instead of Mr(t)]. Following an rf pulse, the bulk magnetization precesses about the static magnetic field with a Larmor frequency !0. As described previously, following an initial pulse, the magnetization would continue
12
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY Br
z a
z b
Mr(t)
Br y
y
Mr(t) x
x
FIGURE 1.3 Effect of applied rf field. (a) In the presence of an applied rf field with y-phase, the effective field, Br is in the y–z plane in the rotating reference frame, and the magnetization vector, Mr(t), precesses around Br. (b) If the rf field is applied on-resonance, then Br is oriented along the y-axis, and Mr(t) rotates in the x–z plane orthogonal to Br.
to evolve freely in the transverse plane forever. This, of course, is not the case because eventually thermal equilibrium must be re-established. Bloch defined two processes to account for the observed decay of the NMR signal (11). These two relaxation processes are responsible for the return of the bulk magnetization to the equilibrium state following some perturbation to the nuclear spin system. The first relaxation mechanism accounts for the return of the population difference across the Zeeman transition back to the Boltzmann equilibrium distribution, and is known as longitudinal, or spin–lattice, relaxation. Bloch assumed that spin–lattice relaxation is characterized by the first-order rate expression, dMz ðtÞ ¼ R1 ½M0 Mz ðtÞ, dt
½1:24
Mz ðtÞ ¼ M0 ½M0 Mz ð0Þ expðR1 tÞ,
½1:25
such that
in which R1 is the spin–lattice relaxation rate constant (the spin–lattice relaxation time constant, T1 ¼ 1/R1, is often encountered), and Mz(0)
13
1.2 THE BLOCH EQUATIONS
is the value of the component of the magnetization along the z-axis at t ¼ 0. As shown, the z-component, or longitudinal, magnetization returns to thermal equilibrium in an exponential fashion. A second relaxation process was introduced to account for the decay of the transverse magnetization in the x–y plane following a pulse. Transverse, or spin–spin, relaxation also is characterized by a first-order rate expression, dMx ðtÞ ¼ R2 Mx ðtÞ, dt
½1:26
dMy ðtÞ ¼ R2 My ðtÞ, dt and Mx ðtÞ ¼ Mx ð0Þ expðR2 tÞ, ½1:27 My ðtÞ ¼ My ð0Þ expðR2 tÞ, in which R2 is the spin–spin relaxation rate constant (the spin–spin relaxation time constant is T2 ¼ 1/R2) and Mx(0) and My(0) are the values of the transverse magnetization at t ¼ 0. The introduction of the concept of relaxation here is simply to assist in the initial description of the NMR phenomenon, and more detailed treatments of relaxation theory and processes will be presented in Chapter 5. Combining [1.11], [1.24], and [1.26] yields the famous Bloch equations in the laboratory reference frame: h i dMx ðtÞ ¼ MðtÞ BðtÞ R2 Mx ðtÞ x dt i h ¼ My ðtÞBz ðtÞ Mz ðtÞBy ðtÞ R2 Mx ðtÞ, h i dMy ðtÞ ¼ MðtÞ BðtÞ R2 My ðtÞ y dt i h ¼ Mz ðtÞBx ðtÞ Mx ðtÞBz ðtÞ R2 My ðtÞ, h i h i dMz ðtÞ ¼ MðtÞ BðtÞ R1 Mz ðtÞ M0 z dt i h i h ¼ Mx ðtÞBy ðtÞ My ðtÞBx ðtÞ R1 Mz ðtÞ M0 ,
½1:28
14
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
describing the evolution of magnetization in a magnetic field. In the rotating reference frame, the Bloch equations are given by dMx ðtÞ ¼ My ðtÞ þ !1 sinMz ðtÞ R2 Mx ðtÞ, dt dMy ðtÞ ¼ Mx ðtÞ !1 cosMz ðtÞ R2 My ðtÞ, dt h h i i dMz ðtÞ ¼ !1 sin Mx ðtÞ þ cosMy ðtÞ R1 Mz ðtÞ M0 : dt
½1:29
These equations can be written in a convenient matrix form as 2 3 2 3 R2 !1 sin 0 dMðtÞ 6 6 7 7 ¼4 R2 !1 cos 5MðtÞ þ R1 M0 4 0 5, ½1:30 dt 1 !1 sin !1 cos R1 in which 2
3 Mx ðtÞ MðtÞ ¼ 4 My ðtÞ 5: Mz ðtÞ
½1:31
In the absence of an applied rf field, !1 ¼ 0 and the Bloch equations become dMx ðtÞ ¼ My ðtÞ R2 Mx ðtÞ, dt dMy ðtÞ ¼ Mx ðtÞ R2 My ðtÞ, dt dMz ðtÞ ¼ R1 ½Mz ðtÞ M0 : dt
½1:32
Evolution in the absence of an applied rf field is referred to as free precession. In a common experimental situation in pulsed NMR spectroscopy, the B1 field is applied for a time p 1=R2 and 1/R1, and the Bloch equations simplify to 2 3 0 !1 sin dMðtÞ 6 7 ¼4 0 !1 cos 5MðtÞ: ½1:33 dt !1 sin !1 cos 0
15
1.2 THE BLOCH EQUATIONS
If neither B1 nor is time dependent, then the solution to [1.33] can be represented as a series of rotations (16, 17): Mð p Þ ¼ Rz ðÞRy ðÞRz ðÞRy ðÞRz ðÞMð0Þ, in which the rotation matrices are 2 3 1 0 0 6 7 Rx ðÞ ¼ 4 0 cos sin 5, 0 sin cos 2 3 cos 0 sin 6 7 1 0 5, Ry ðÞ ¼ 4 0 sin 0 cos 3 2 cos sin 0 7 6 Rz ðÞ ¼ 4 sin cos 0 5: 0
0
½1:34
½1:35
1
In [1.35], the notation Rx() designates a right-handed rotation of angle about the axis x. A positive rotation is counterclockwise when viewed down the axis x toward the origin, or clockwise when viewed from the origin along x. The rotation matrices and [1.34] will be used frequently to calculate the effect of rf pulses on isolated spins. For example, the effect of an x-phase ( ¼ 0) pulse is described by Mð p Þ ¼ 2
3 cos2 cos þ sin2 cos sin cos sinð1 cosÞ 4 5Mð0Þ: cos sin cos sin sin 2 2 cos sinð1 cosÞ sin sin sin cos þ cos ½1:36
The effective rotation angle, 12, and rotation axis, n12, that result from consecutive pulses with rotation angles 1 and 2, respectively, and rotation axes, n1 and n2, respectively, can be determined using the quarternion formalism to be (18) 12 1 2 1 2 ¼ cos cos sin sin n1 n2 , cos 2 2 2 2 2 12 1 2 1 2 n12 ¼ sin cos n1 þ cos sin n2 sin 2 2 2 2 2 1 2 sin n1 n2 : sin ½1:37 2 2
16
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
These equations can be applied iteratively to generate expressions for three or more rotations and are particularly useful in determining the effective rotations produced by composite pulses (see Section 3.4.2).
1.3 The One-Pulse NMR Experiment Experimental aspects of NMR spectroscopy are described in detail in Chapter 3. In this section, a brief overview of a simple NMR experiment is presented. In the Bloch model, the maximum NMR signal is detected when the bulk magnetic moment is perpendicular (transverse) to the static magnetic field. As noted previously, an rf pulse causes M(t) to precess about an axis defined by the direction of the effective magnetic field in the rotating frame; therefore, the properties of an rf pulse that cause rotation of M(t) from the z-axis through an angle of 908 are particularly important in pulsed NMR spectroscopy. An ideal one-pulse experiment that achieves a 908 rotation of M(t) will be considered. An rf pulse of duration p, strength B1, and tilt angle ¼ /2 is applied to the equilibrium magnetization state. If the rf pulse is applied along the y-axis of the rotating frame (setting ¼ /2 in [1.18]), then the magnetization following the pulse is given by (see [1.34]) 2 3 M0 sin
5, M p ¼ Ry ðÞM0 ¼ iM0 sin þ kM0 cos ¼ 4 ½1:38 0 M0 cos where M0 is the magnitude of the equilibrium magnetization and is the rotation angle. The maximum transverse magnetization is generated for a rotation angle of 908. The rf pulse used to achieve this state is conventionally called a 908 or (/2) pulse. A 908 pulse equalizes the populations of the and spin states. In contrast, a 1808 (or ) pulse generates no transverse magnetization. Instead, the bulk magnetization is inverted from its original state to yield M( p) ¼ M0k. In the Bloch vector model, the bulk magnetization following a 1808 pulse is aligned along the z-axis. This corresponds to a population inversion between the and states, such that the state now possesses excess (deficient) population of nuclei for positive (negative) . The populations of the Zeeman states and the net magnetization vectors following on-resonance pulses are illustrated in Fig. 1.4.
17
1.3 THE ONE-PULSE NMR EXPERIMENT
a
b
z
c
z
y
x
y
x
b a
N/2 – ∆N N/2 + ∆N
z
y
x
b a
N/2
N/2
b a
N/2 + ∆N N/2 – ∆N
FIGURE 1.4 On-resonance pulses. Shown are the magnetization vectors and spin states and (a) for thermal equilibrium, (b) following a 908 pulse with y-phase, and (c) following a 1808 pulse. The populations of each spin state are indicated for positive . The total number of spins is N and N ¼ NhB0 =ð4kB TÞ:
Following the pulse, the magnetization precessing during the so-called acquisition period, t, generates the signal that is recorded by the NMR spectrometer. The signal is referred to as a free induction decay (FID). The free-precession Bloch equations in the rotating frame [1.32] show that the free induction decay can be described in terms of two components, Mx ðtÞ ¼ M0 sin cosðtÞ expðR2 tÞ,
½1:39
My ðtÞ ¼ M0 sin sinðtÞ expðR2 tÞ, which can be combined in complex notation as Mþ ðtÞ ¼ Mx ðtÞ þ iMy ðtÞ ¼ M0 sin expðit R2 tÞ:
½1:40
As a consequence of relaxation, the components of the bulk magnetization vector precessing in the transverse plane following an rf pulse are damped by the exponential factor exp(–R2t). In practice, both parts of the complex signal are detected simultaneously by the NMR spectrometer as sþ(t) ¼ Mþ(t), with being an experimental constant
18
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
of proportionality. The complex time-domain signal is Fourier transformed to produce the complex frequency-domain spectrum, Z1 Sð!Þ ¼ sþ ðtÞ expði!tÞ dt 0
¼ ð!Þ þ iuð!Þ,
½1:41
in which vð!Þ ¼ M0
uð!Þ ¼ M0
R22
R2 , þ ð !Þ2
½1:42
R22
! : þ ð !Þ2
½1:43
The function v(!) represents a signal with an absorptive Lorentzian lineshape and the function u(!) represents a signal with the corresponding dispersive Lorentzian lineshape. The real part of the complex spectrum, v(!), normally is displayed as the NMR spectrum. This simple one-pulse NMR experiment is illustrated schematically in Fig. 1.5.
1.4 Linewidth The phenomenological linewidth is defined as the full-width at halfheight (FWHH) of the resonance lineshape and is a primary factor affecting both resolution and signal-to-noise ratio of NMR spectra. The homogeneous linewidth is determined by intrinsic molecular properties while the inhomogeneous linewidth contains contributions from instrumental imperfections, such as static magnetic field inhomogeneity or thermal gradients within the sample. For a Lorentzian lineshape [1.42], the homogeneous linewidth is given by FWHH ¼ R2/ in Hertz (or !FWHH ¼ 2R2 in angular units, s–1) and the inhomogeneous linewidth is FWHH ¼ R 2 =, in which R 2 ¼ R2 þ Rinhom , and Rinhom represents the broadening of the resonance signal due to instrumental imperfections. In modern NMR spectrometers Rinhom/ is on the order of 1 Hz (in the absence of significant temperature gradients in the sample). As will be discussed in detail in Chapter 5, values of R2 (and hence homogeneous linewidths) are proportional to the overall rotational correlation time of the protein, c , and thus depend on molecular mass and shape of the molecule, with larger molecules having larger
19
1.4 LINEWIDTH z
z
a
z
b
M0
c
y
y
y
M(0) x
M(t)
x
x
d
e
FIGURE 1.5 One-pulse NMR experiment. Shown are (a) the orientation along the z-axis of the net magnetization at equilibrium, (b) the orientation along the x-axis of the net magnetization at the start of acquisition following a 908 pulse with y-phase, (c) the precessing magnetization in the x–y plane, (d) the FID recorded for the precessing magnetization during the acquisition period, and (e) the real component of the complex frequency domain NMR spectrum obtained by Fourier transformation of the FID.
linewidths. As discussed in Section 6.1, observed linewidths significantly larger than expected based on the molecular mass of the protein imply that aggregation is increasing the apparent rotational correlation time or that chemical exchange effects (Section 5.6) contribute significantly to the linewidth. Given theoretical or experimental estimates of c, the theoretical equations presented in Chapters 5 and 7 can be used to calculate approximate values of resonance linewidths. The resulting curves are shown in Fig. 1.6. The principal uncertainties in the calculation are due to the following factors: (i) anisotropic rotational diffusion of nonspherical molecules, (ii) differential contributions from internal motions (particularly in loops or for side chains), (iii) cross-correlation effects, (iv) dipolar interactions with nearby 1H spins (which depend on detailed structures of the proteins), and (v) incomplete knowledge of fundamental parameters (such as chemical shift anisotropies).
20
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY 40
a 30
20
10
0 25
b 20
15
10
5
0 2
4
6
8
10
12
14
FIGURE 1.6 Resonance linewidths. Protein resonance linewidths are shown as a function of rotational correlation time. (a) Linewidths for 1H spins (solid), 1 H spins covalently bonded to 13C (dotted), and 1H spins covalently bonded to 15N nuclei (dashed). (b) Heteronuclear linewidths for proton-decoupled 13 C (solid), proton-coupled 13C (dashed), proton-decoupled 15N (dash-dot), and proton-coupled 15N spins (dotted). Calculations included dipolar relaxation of all spins, and CSA relaxation of 15N spins. For 1H–1H dipolar interactions, P 6 ˚ 6 (49). j rij ¼ 0:027 A
21
1.5 CHEMICAL SHIFT
The correlation time for Brownian rotational diffusion can be measured experimentally by using time-resolved fluorescence spectroscopy, light scattering, and NMR spin relaxation spectroscopy, or can be calculated by using a variety of hydrodynamic theories (that unfortunately require detailed information on the shape of the molecule) (19). In the absence of more accurate information, the simplest theoretical approach for approximately spherical globular proteins calculates the isotropic rotational correlation time from Stokes’ law: c ¼ 4 w r3H =ð3kB TÞ,
½1:44
in which w is the viscosity of the solvent, rH is the effective hydrodynamic radius of the protein, kB is the Boltzmann constant, and T is the temperature. The hydrodynamic radius can be very roughly estimated from the molecular mass of the protein, Mr, by assuming that the specific volume of the protein is V ¼ 0:73 cm3/g and that a hydration layer of rw ¼ 1.6 to 3.2 A˚ (corresponding to one-half to one hydration shell) surrounds the protein (20): 1=3 ½1:45 rH ¼ 3VMr =ð4NA Þ þ rw , in which NA is Avogadro’s number. Rotational correlation times in D2O solution are approximately 25% greater than in H2O solution because of the larger viscosity of D2O. The small protein ubiquitin is used as an example throughout this text. The protein sequence consists of 76 amino acid residues and Mr ¼ 8400. For ubiquitin, rH ¼ 16.5 A˚ is calculated from [1.45], and c ¼ 3.8 ns at 300 K is calculated from [1.44]. This estimate can be compared with a value of 4.1 ns determined from NMR spectroscopy (21). In light of the uncertainties, the results presented in Fig. 1.6 should be regarded as approximate guidelines. For example, 1H (in an unlabeled sample), 13C, and 15N linewidths are 6–9, 7, and 3 Hz, respectively, for ubiquitin. These values are consistent with values of 5, 6, and 2 Hz determined from Fig. 1.6.
1.5 Chemical Shift A general feature of NMR spectroscopy is that the observed resonance frequencies depend on the local environments of individual nuclei and differ slightly from the frequencies predicted by [1.14]. The differences in resonance frequencies are referred to as chemical shifts
22
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
and offer the possibility of distinguishing between otherwise identical nuclei in different chemical environments. The phenomenon of chemical shift arises because motions of electrons induced by the external magnetic field generate secondary magnetic fields. The net magnetic field at the location of a specific nucleus depends upon the static magnetic field and the local secondary fields. The effect of the secondary fields is called nuclear shielding and can augment or diminish the effect of the main field. In general, the electronic charge distribution in a molecule is anisotropic and the effects of shielding on a particular nucleus are described by the secondrank nuclear shielding tensor, represented by a 3 3 matrix. In the principal coordinate system of the shielding tensor, the matrix representing the tensor is diagonal, with principal components 11, 22, and 33. If the molecule is oriented such that the kth principal axis is oriented along the z-axis of the static field, then the net magnetic field at the nucleus is given by B ¼ ð1 kk ÞB0 :
½1:46
In isotropic liquid solution, collisions lead to rapid reorientation of the molecule and, consequently, of the shielding tensor. Under these circumstances, the effects of shielding on a particular nucleus can be accounted for by modifying [1.14] as ! ¼ ð1 ÞB0 ,
½1:47
in which is the average, isotropic, shielding constant for the nucleus: ¼ ð 11 þ 22 þ 33 Þ=3:
½1:48
The chemical shift anisotropy (CSA) is defined as ¼ 11 ð 22 þ 33 Þ=2,
½1:49
and the asymmetry of the tensor is defined as ¼
3ð 22 33 Þ : 2
½1:50
The parameters , , and constitute an equivalent description of the shielding tensor as the principal values. Variations in due to different electronic environments cause variations in the resonance frequencies of the nuclei. In effect, each nucleus experiences its own local magnetic field. Fluctuations in the local magnetic field as the
1.6 SCALAR COUPLING
AND
LIMITATIONS
OF THE
BLOCH EQUATIONS
23
molecule rotates results in the CSA relaxation mechanism described in Section 5.4.4. Resonance frequencies are directly proportional to the static field, B0; consequently, the difference in chemical shift between two resonance signals measured in frequency units increases with B0. In addition, the absolute value of the chemical shift of a resonance is difficult to determine in practice because B0 must be measured very accurately. In practice, chemical shifts are measured in parts per million (ppm, or ) relative to a reference resonance signal from a standard molecule: ref
¼ 106 ¼ ð ref Þ 106 , ½1:51 !0 in which and ref are the offset frequencies of the signal of interest and the reference signal, respectively. Chemical shift differences measured in parts per million are independent of the static magnetic field strength so that, for example, all else being equal, chemical shifts reported from experiments on a 500-MHz spectrometer will be the same as those determined on an 800-MHz spectrometer. Referencing of NMR spectra is discussed in detail in Section 3.6.3. Observed chemical shifts in proteins commonly are partitioned into the sum of two components: the so-called random coil chemical shifts, rc, and the conformation-dependent secondary chemical shifts, . The random coil chemical shift of a nucleus in an amino acid residue is the chemical shift that is observed in a conformationally disordered peptide (22–27). The secondary chemical shift contains the contributions from secondary and tertiary structures. This distinction is useful because secondary chemical shifts display characteristic patterns for secondary structural elements (28–32) and other motifs (33) that can provide important structural information and constraints for proteins (34–40). In addition, theoretical treatments (41–46) are becoming increasingly accurate in predicting protein chemical shifts and chemical shift anisotropies. Distributions of chemical shifts observed in proteins (47) are presented in Chapter 9.
1.6 Scalar Coupling and Limitations of the Bloch Equations A brief treatment of a phenomenon of great practical importance, which will be discussed throughout this text, will be used to illustrate the deficiencies of the Bloch theory. High-resolution NMR spectra of
24
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
liquids reveal fine structure due to interactions between the nuclei. However, the splitting of the resonance signals into multiplets is not caused by direct dipolar interactions between magnetic dipole moments. Such dipolar coupling, although extremely important in solids, is an anisotropic quantity that is averaged to zero to first order in isotropic solution (second-order effects are discussed in Chapter 5). Ramsey and Purcell suggested that the interaction is mediated by the electrons forming the chemical bonds between the nuclei (48). This interaction is known as spin–spin coupling or scalar coupling. The strength of the interaction is measured by the scalar coupling constant, nJab, in which n designates the number of covalent bonds separating the two nuclei, a and b. The magnitude of nJab is usually expressed in Hertz and the most important scalar coupling interactions in proteins have n ¼ 1 to 4. In the present text, n will be written explicitly only if the intended value of n is not clear from the context. Scalar coupling modifies the energy levels of the system, and the NMR spectrum is modified correspondingly. The prototypical example consists of two spin-1/2 nuclei (e.g., two 1H spins or an 1H spin and a 13C spin). The two spins are designated I and S. The resonance frequencies are !I and !S, respectively, !I ¼ I B0 ð1 I Þ,
!S ¼ S B0 ð1 S Þ:
½1:52
The magnetic quantum numbers are mI and mS; each spin has two stationary states that correspond to the magnetic quantum numbers 1/2 and 1/2. The complete two-spin system is described by four wavefunctions corresponding to all possible combinations of mI and mS,
1 1
1 1 1 ¼ 2 ¼ 2, 2 , 2, 2 , ½1:53
1 1
1 1 , , ¼ , ¼ , 3 4 2 2 2 2 where the first quantum number describes the state of the I spin and the second describes the S spin. In the absence of scalar coupling between the spins, the energies of these four states are the sums of the energies for each spin. Remembering that the state has a higher (lower) energy compared to the state for positive (negative) , the energies are found to be E1 ¼ 12h!I þ 12h!S ,
E2 ¼ 12h!I 12h!S ,
E3 ¼ 12h!I þ 12h!S ,
E4 ¼ 12h!I 12h!S :
½1:54
1.6 SCALAR COUPLING
AND
LIMITATIONS
OF THE
25
BLOCH EQUATIONS
The total magnetic quantum number m for each energy level is the sum of the individual terms m1 ¼ þ12 þ 12 ¼ þ1,
m2 ¼ þ12 12 ¼ 0,
m3 ¼ 12 þ 12 ¼ 0,
m4 ¼ 12 12 ¼ 1:
½1:55
The energy level diagram for a two-spin system with I 4 S 4 0 is shown in Fig. 1.7a. The observable transitions obey the selection rule m ¼ 1. Therefore, the allowed transitions occur between states 1–2, 3–4, 1–3, and 2–4 in Fig. 1.7; transitions between 2–3 or 1–4 are
ββ
ββ
a
4
b
4
βα
βα
3
3
αβ 2
αβ αα
2
αα
1
1 c
d
1-3 2-4 ωI
1-2 3-4 ωS
JIS
1-3
JIS
2-4
ωI
1-2
3-4
ωS
FIGURE 1.7 Energy levels for an AX spin system. Shown are the energy levels for an AX spin system in the (a) absence and (b) presence of scalar coupling interactions between the spins, assuming JIS 4 0 and I 4 S 4 0. The allowed transitions are indicated between arrows. The energies of the four spins states are defined by (a) [1.54] and (b) [1.56].
26
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
forbidden. The first two transitions involve a change in the spin state of the S spin while the latter two involve a change in the spin state of the I spin. Consequently, the NMR spectrum shown in Fig. 1.7c consists of one resonance line at !I, due to transitions 1–3 and 2–4, and one resonance line at !S, due to transitions 1–2 and 3–4. Introducing the scalar coupling between I and S, with a value of JIS, modifies the energy levels to E1 ¼ 12h!I þ 12h!S þ 12hJIS ,
E2 ¼ 12h!I 12h!S 12hJIS ,
E3 ¼ 12h!I þ 12h!S 12hJIS ,
E4 ¼ 12h!I 12h!S þ 12hJIS ,
½1:56
in which weak coupling has been assumed with 2jJIS j j!I !S j. These expressions are derived from the following equation (see Section 2.5.2): EðmI , mS Þ ¼ mI !I þ mS !S þ 2mI mS JIS :
½1:57
The term in JIS depends on the spin states of both nuclei but the terms in !I and !S depend on the spin state of a single nucleus. The energy level diagram for a scalar coupled two-spin system is shown in Fig. 1.7b, assuming that JIS 4 0. The resulting effect in the spectrum of the scalar coupled system is easily seen from the new values from the transition frequencies, !12 ¼ !S þ JIS , !13 ¼ !I þ JIS ,
!34 ¼ !S JIS , !24 ¼ !I JIS :
½1:58
Now the spectrum shown in Fig. 1.7d consists of four lines: two centered around the transition frequency, !S, of the S spin but separated by 2JIS, and two centered around the transition frequency of the I spin, !I, but separated by 2JIS. A weakly coupled two-spin system is referred to as an AX spin system and a strongly coupled two-spin system is referred to as an AB spin system, in which A and X or A and B represent the pair of scalar coupled spins. The Bloch vector model of NMR phenomena predicts that two resonance signals will be obtained for the two-spin system; in actuality, if the two spins share a nonzero scalar coupling interaction, then four resonance signals are obtained. The basic Bloch model can be extended to describe the evolution of a scalar coupled system by treating each resonance line resulting from the scalar coupling interaction as an independent magnetization vector in the rotating frame. Although additional insights can be gained from using this approach, many problems still arise: (i) strong coupling effects that occur when
1.6 SCALAR COUPLING
AND
LIMITATIONS
OF THE
BLOCH EQUATIONS
27
2JIS |!I – !S| cannot be described, (ii) the results of applying nonselective pulses to transverse magnetization in a homonuclear coupled system cannot be described without introducing additional ad hoc assumptions, and (iii) transfer of magnetization via forbidden transitions when the spin system is not at equilibrium cannot be explained. In principle, the Bloch picture is strictly only applicable to a system of noninteracting spin-1/2 nuclei. Despite these limitations, the Bloch model should not be abandoned completely. Many of the concepts and much of the terminology introduced by this model appear throughout the whole of NMR spectroscopy. Although the Bloch model is a valuable tool with which to visualize simple NMR experiments, more rigorous approaches are necessary to describe the gamut of modern NMR techniques. Much of the remaining theory presented in this text is devoted to developing methods of analysis that accurately predict the behavior of systems of two or more nuclear spins that interact via scalar coupling or other interactions.
References 1. R. R. Ernst, W. A. Anderson, Rev. Sci. Instrum. 37, 93–102 (1966). 2. J. Jeener, Ampe`re Summer School, Basko Polje, Yugoslavia (1971). 3. J. Jeener, in ‘‘NMR and More. In Honour of Anatole Abragam’’ (M. Goldman, M. Porneuf, eds.), pp. 1–379. Les Editions de Physique, Les Ulis, France, 1994. 4. R. R. Ernst, Angew. Chem., Int. Eng. Ed. 31, 805–930 (1992). 5. K. Wu¨thrich, Angew. Chem., Int. Eng. Ed. 42, 3340–3363 (2003). 6. F. Bloch, W. W. Hansen, M. Packard, Phys. Rev. 69, 127 (1946). 7. E. M. Purcell, H. C. Torrey, R. V. Pound, Phys. Rev. 69, 37–38 (1946). 8. T. Sloan, Phil. Trans. R. Soc. Lond., Ser. A 359, 379–389 (2001). 9. K. Rith, A. Schafer, Sci. Am. 281, 58–63 (1999). 10. E. Merzbacher, ‘‘Quantum Mechanics,’’ 2nd edn., pp. 1–621. Wiley & Sons, New York, 1970. 11. F. Bloch, Phys. Rev. 70, 460–474 (1946). 12. H. Goldstein, ‘‘Classical Mechanics,’’ 2nd edn., pp. 1–672. Addison-Wesley, Reading, MA, 1980. 13. M. H. Levitt, J. Magn. Reson. 126, 164–182 (1997). 14. D. I. Hoult, N. S. Ginsberg, J. Magn. Reson. 148, 182–199 (2001). 15. D. I. Hoult, B. Bhakar, Concepts Magn. Reson. 9, 277–297 (1997). 16. R. R. Ernst, G. Bodenhausen, A. Wokaun, ‘‘Principles of Nuclear Magnetic Resonance in One and Two Dimensions,’’ pp. 1–610. Clarendon Press, Oxford, 1987. 17. P. L. Corio, ‘‘Structure of High-resolution NMR Spectra,’’ pp. 1–548. Academic Press, New York, 1967. 18. C. Counsell, M. H. Levitt, R. R. Ernst, J. Magn. Reson. 63, 133–141 (1985). 19. R. C. Cantor, P. R. Schimmel, ‘‘Biophysical Chemistry,’’ pp. 1–1371. W. H. Freeman, San Francisco, 1980.
28
CHAPTER 1 CLASSICAL NMR SPECTROSCOPY
20. 21. 22. 23. 24. 25. 26.
R. M. Venable, R. W. Pastor, Biopolymers 27, 1001–1014 (1988). D. M. Schneider, M. J. Dellwo, A. J. Wand, Biochemistry 31, 3645–3652 (1992). R. Richarz, K. Wu¨thrich, Biopolymers 17, 2133–2141 (1978). D. Braun, G. Wider, K. Wu¨thrich, J. Am. Chem. Soc. 116, 8466–8469 (1994). A. Bundi, K. Wu¨thrich, Biopolymers 18, 285–297 (1979). G. Merutka, H. J. Dyson, P. E. Wright, J. Biomol. NMR 5, 14–24 (1995). S. Schwarzinger, G. J. A. Kroon, T. R. Foss, J. Chung, P. E. Wright, H. J. Dyson, J. Am. Chem. Soc. 123, 2970–2978 (2001). D. S. Wishart, C. G. Bigam, A. Holm, R. A. Hodges, B. D. Sykes, J. Biomol. NMR 5, 67–81 (1995). D. S. Wishart, B. D. Sykes, F. M. Richards, J. Mol. Biol. 222, 311–333 (1991). A. Pastore, V. Saudek, J. Magn. Reson. 90, 165–176 (1990). M. P. Williamson, Biopolymers 29, 1423–1431 (1990). A. Pardi, G. Wagner, K. Wu¨thrich, Eur. J. Biochem. 137, 445–454 (1983). S. Spera, A. Bax, J. Am. Chem. Soc. 113, 5490–5492 (1991). A. M. Gronenborn, G. M. Clore, J. Biomol. NMR 4, 455–458 (1994). G. Cornilescu, F. Delaglio, A. Bax, J. Biomol. NMR 13, 289–302 (1999). J. Kuszewski, J. Qin, A. M. Gronenborn, G. M. Clore, J. Magn. Reson., Ser. B 106, 92–96 (1995). D. S. Wishart, D. A. Case, Meth. Enzymol. 338, 3–34 (2001). D. S. Wishart, B. D. Sykes, Meth. Enzymol. 239, 363–392 (1994). D. S. Wishart, B. D. Sykes, F. M. Richards, Biochemistry 31, 1647–1651 (1992). P. Luginbu¨hl, T. Szyperski, K. Wu¨thrich, J. Magn. Reson., Ser. B 109, 229–233 (1995). R. D. Berger, P. H. Bolton, J. Biomol. NMR 10, 129–142 (1997). D. Sitkoff, D. A. Case, Prog. NMR Spectrosc. 32, 165–190 (1998). A. C. de Dios, J. G. Pearson, E. Oldfield, Science 260, 1491–1496 (1993). X. P. Xu, D. A. Case, J. Biomol. NMR 21, 321–333 (2001). E. Oldfield, Annu. Rev. Phys. Chem. 53, 349–378 (2002). X. P. Xu, D. A. Case, Biopolymers 65, 408–423 (2002). S. Neal, A. M. Nip, H. Zhang, D. S. Wishart, J. Biomol. NMR 26, 215–240 (2003). H. Zhang, S. Neal, D. S. Wishart, J. Biomol. NMR 25, 173–195 (2003). N. F. Ramsey, E. M. Purcell, Phys. Rev. 85, 143–144 (1952). A. G. Palmer, J. Cavanagh, P. E. Wright, M. Rance, J. Magn. Reson. 93, 151–170 (1991).
27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49.
CHAPTER
2 THEORETICAL DESCRIPTION OF NMR SPECTROSCOPY A rigorous treatment of the dynamics of nuclear spin systems and NMR spectroscopy is afforded by the quantum mechanical representation known as the density matrix formalism (1, 2). Instead of following only the evolution of the bulk magnetization vector as in the Bloch model, the evolution of the density matrix provides a complete description of the state of a spin system at any point during an NMR experiment. The next few sections present a detailed overview of the development of the density matrix theory and its application in the simplest pulsed NMR experiments.
2.1 Postulates of Quantum Mechanics A rather formal exposition of the mathematical concepts to be used through the remainder of the text is presented first. Commonly, in introductory quantum mechanics texts (3–5), quantum mechanical orbital angular momentum is introduced via the classical concepts of angular momentum. After establishing the relevant physics, the results are generalized to include the intrinsic angular momentum of electrons and nuclei. The intrinsic angular momentum does not have a classical analog; accordingly, in this text, orbital angular momentum will not be
29
30
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
discussed. Instead, the foundations of the theory of intrinsic angular momentum will be presented as postulates whose validity is established by comparison with experiment. NMR spectroscopy is a particularly powerful demonstration of the concepts.
2.1.1 THE SCHRo¨ DINGER EQUATION The evolution in time of a quantum mechanical system is governed by the Schro¨dinger equation: @ðtÞ i ¼ HðtÞ: ½2:1 @t h The operator H is termed the Hamiltonian of the system and incorporates the essential physics determining the evolution of the system. The Hamiltonian may be time dependent or time independent. Units in which h ¼ 1 will be assumed and factors of h will not be written explicitly; thus, @ðtÞ ¼ iHðtÞ: ½2:2 @t When desired, necessary factors of h can be reintroduced by dimensional analysis; equivalently, all energies are measured in angular frequency units with dimensions of s–1. The solution of the Schro¨dinger equation is called the wavefunction for the system, (t). The wavefunction contains all the knowable information about the state of the system and, consequently, is a function of the variables appropriate to the system of interest (e.g., spatial coordinates and spin coordinates). The probability density that the system is in the state described by (t) at time t is given by PðtÞ ¼ ðtÞðtÞ,
½2:3
in which *(t) is the complex conjugate of (t). If the wavefunction is known, then all the observable properties of the system can be deduced by performing the appropriate mathematical operations upon the wavefunction. Wavefunctions generally will be assumed to be normalized such that Z ðtÞðtÞ d ¼ 1, ½2:4 in which represents the generalized coordinates of the wavefunction (and may include sums over spin states). If necessary, wavefunctions can be normalized simply by defining Z 1=2 0 ðtÞ ¼ ðtÞ ðtÞðtÞ d : ½2:5
2.1 POSTULATES
OF
QUANTUM MECHANICS
31
If H is time independent, then [2.2] can be solved by the method of separation of variables. Defining (t) ¼ ()’(t), in which () contains the time-independent spatial and spin variables [for simplicity, () is frequently written as ] and ’(t) contains time-dependent terms, @ðtÞ ¼ iHðtÞ, @t d’ðtÞ ¼ iH ðÞ’ðtÞ, ðÞ dt Z Z ½2:6 d’ðtÞ ¼ i’ðtÞ ðÞ ðÞ d ðÞH ðÞ d, dt d’ðtÞ ¼ iE’ðtÞ, dt in which the energy of the system is defined by Z E¼ ðÞH ðÞ d: ½2:7 Solving [2.6] yields ’(t) ¼ C exp(–iEt). Using this result gives ðtÞ ¼ ðÞ expðiEtÞ,
½2:8
in which the integration constant C has been included in the normalization of (). If h is reintroduced explicitly, then ðtÞ ¼ ðÞ expðiEt=hÞ ¼ ðÞ expði!tÞ,
½2:9
in which E ¼ h!. As shown by [2.8] and [2.9], if H is time independent, then the time dependence of the wavefunctions is limited to a phase factor; this factor cancels when calculating probability densities using [2.3].
2.1.2 EIGENVALUE EQUATIONS The purpose of quantum mechanics, at least insofar as it is applied to NMR spectroscopy, is to calculate the results expected from experiments. In the language of quantum mechanics, every physically observable quantity, A, has associated with it a Hermitian operator A, that satisfies the eigenvalue equation: A f ð Þ ¼ fð Þ:
½2:10
This equation defines a set of eigenfunctions, fi(), and eigenvalues, i, for i ¼ 1 to N, that satisfy in turn A fi ð Þ ¼ i fi ð Þ:
½2:11
32
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
The number of eigenvalues and eigenfunctions, N, is determined by the system of interest and may be finite or infinite. The adjoint of an operator is defined by Ay ¼ AT*, in which T indicates transposition and indicates complex conjugation. The adjoint operator satisfies the eigenvalue equation, f ð ÞAy ¼ f ð Þ:
½2:12 y
Hermitian operators are self-adjoint, A ¼ A , and satisfy the relationship Z Z f ðÞAgðÞ d ¼ g ðÞA fðÞ d ½2:13 for well-behaved functions f() and g(). If f() is a normalized eigenfuntion of the operator A with eigenvalue , then the following relationships are obtained from [2.10] and [2.12]: Z Z f ð ÞA fð Þ d ¼ f ð Þfð Þ d ¼ , ½2:14 Z
f ð ÞAy fð Þ d ¼
Z
f ð Þfð Þ d ¼ :
½2:15
If the operator A corresponds to an observable quantity, then the eigenvalues of A must be real numbers. Thus, ¼ and equating [2.14] and [2.15] proves that A ¼ Ay and A is Hermitian. Consequently, operators corresponding to observable quantities in quantum mechanics must be Hermitian. The eigenfunctions of a Hermitian operator form a complete orthonormal set. The orthonormality condition is Z ½2:16 fi ðÞfj ðÞ d ¼ i, j , in which i, j is the Kronecker delta with values i, j ¼
0 1
for for
i 6¼ j : i¼j
½2:17
Unnormalized eigenfunctions can be normalized as in [2.5]; if necessary, the wavefunctions can be orthogonalized using a procedure known as the Gram–Schmidt process (5). A complete set of orthonormal functions, n, constitutes a set of basis functions for a vector space of dimension N, called the Hilbert space. Therefore, an arbitrary function defined in the
2.1 POSTULATES
OF
33
QUANTUM MECHANICS
vector space can be written as ðtÞ ¼
N X
cn
n,
½2:18
n¼1
in which the cn are complex numbers and may depend upon time. The eigenvalue equation [2.11] leads to the following interpretation of the relationship between an operator and its associated observable: the result of making a measurement of A upon a system is one of the eigenvalues of A. This statement illustrates the discrete nature of quantum mechanics: only a limited set of outcomes is possible for the measurement. In practice, however, the expectation value of A is measured experimentally. The expectation value is defined as the average magnitude of a particular property obtained following a large number of measurements of that property carried out over an ensemble of identically prepared systems. The expectation value of some property, hAi, is calculated mathematically as the scalar product of (t) and A(t), Z hAi ¼ ðtÞAðtÞ d: ½2:19 If the wavefunction for the system is an eigenfunction of the operator, (t) ¼ n, then Z Z Z hAi ¼ ðtÞAðtÞ d ¼ ½2:20 n A n d ¼ n n n d ¼ n : This result shows that if (t) is an eigenfunction of the operator A, then measuring A for each member of the ensemble yields the identical result n. In general, the wavefunction for the system will not be an eigenfunction of A, and [2.18] is used to express [2.19] in terms of the eigenfunctions of A. The derivation of hAi proceeds as follows: Z hAi ¼ ðtÞAðtÞ d # " # # " # Z "X Z "X N N N N X X ci i A cj j d ¼ ci i A cj j d ¼ i¼1
¼ ¼
N X N X
ci cj
i¼1 j¼1 N X cj cj j : j¼1
Z
j¼1 iA j
i¼1
d ¼
N X N X
ci cj j
Z
j¼1 i
j
d
i¼1 j¼1
½2:21
34
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
In obtaining [2.21], the orthonormality condition [2.16] has been used. The resulting equation for hAi has the following interpretation. When A is measured for a single member of the ensemble, the result obtained is one of the eigenvalues of A; however, which eigenvalue is obtained cannot be specified in advance of the measurement. For the ensemble as a whole, the result j is obtained in a proportion cj cj ; that is, cj cj is interpreted as the probability that the result j is obtained in a single measurement. Consequently, although the allowed values of A must be members of the discrete set of eigenvalues of A, the observable expectation value hAi can have any (continuous) value consistent with [2.21]. The time-independent Schro¨dinger equation is an eigenvalue equation for the Hamiltonian operator. Substitution of [2.8] into [2.2] yields @ðtÞ ¼ iHðtÞ, @t d exp½iEt ¼ iH ðÞ exp½iEt, ðÞ dt E ðÞ ¼ H ðÞ:
½2:22
The eigenvalues of this equation are the energies of the system and the eigenfunctions are termed the stationary states of the system.
2.1.3 SIMULTANEOUS EIGENFUNCTIONS Next, quantum mechanical restrictions on measurement of different observable quantities are presented. Two operators, A and B, corresponding to observable properties A and B, are considered as an example. The eigenfunctions and eigenvalues of A will be designated and a; the eigenfunctions and eigenvalues of B will be designated ’ and b. The operators are called compatible if the result of measuring A (or B) does not depend upon whether B (or A) is measured first. Compatible, or simultaneous, measurements of A and B are possible only if A and B have the same eigenfunctions (but not necessarily the same eigenvalues). An important theorem states that if AB ¼ BA, then A and B have the same complete set of eigenfunctions. The proof of this statement is straightforward: AB ¼ BA, AB i ¼ BA i , AB
i
¼ ai B i :
½2:23
2.1 POSTULATES
Next,
i
OF
35
QUANTUM MECHANICS
is expanded in the eigenfuctions of B: AB i ¼ ai B i , X X AB cij j ¼ ai B cij j , j
X
j
cij bj Aj ¼ ai
j
X
X
½2:24
cij bj j ,
j
cij bj Aj ai j ¼ 0:
j
By definition, cij 6¼ 0 for at least one value of j ¼ k. Thus, the bracketed term in the last equation is zero for some k: Ak ai k ¼ 0,
Ak ¼ ai k :
½2:25
Thus, k is an eigenfunction of A with eigenvalue ai and must be identical to i (to within a constant of proportionality). This equality is satisfied for all members of the set of eigenfunctions, and ; therefore, the general theorem must hold. The commutator of A and B is defined as ½A, B ¼ AB BA:
½2:26
The earlier result can be restated: if the commutator of two operators vanishes, then the operators have the same eigenfunctions. If the commutator does not vanish, then a Heisenberg uncertainty relationship can be established for the two operators (5).
2.1.4 EXPECTATION VALUE
OF THE
MAGNETIC MOMENT
As should now be clear, each operator for an observable quantity defines a set of basis vectors. Any complete orthonormal set can be used to expand an arbitrary wavefunction; consequently, a basis set can be chosen for computational convenience. In no case can the expectation value of an operator depend upon the choice of the basis functions. As an example of these ideas, the time-dependent expectation value of the magnetic moment l ¼ I of a single spin (I ¼ 1/2) will be calculated. Using [2.9] and [2.18], the wavefunction for the spin in the static magnetic field can be written as ¼ c
þ c
¼ a exp½i! t
þ b exp½i! t
,
½2:27
36
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
in which and are the stationary states and E ¼ h! and E ¼ h! are the energies of the states with m ¼ 1/2 and m ¼ –1/2, respectively, and a ¼ |a| exp[ia] and b ¼ |b| exp[ib] are complex numbers satisfying the normalization relation |a|2 þ |b|2 ¼ 1. U sing [2.19] yields Z Z hx i ¼ x d ¼ Ix d Z Z I d þ a b exp ið! ! Þt ¼ a a x Ix d Z Z þ ab exp ið! ! Þt Ix d þ b b Ix d hjaj jbj exp ifð! ! Þt þ g þ exp ifð! ! Þt þ g 2 ¼ hjaj jbj cos ð! ! Þt þ ¼ hjaj jbj cos½!0 t þ ,
¼
Z
½2:28
Z
y d ¼ Iy d Z Z I d þ a b exp ið! ! Þt ¼ a a y Iy Z Z I d þ b b þ ab exp ið! ! Þt y
y ¼
d
Iy
d
hjajjbj exp ifð! ! Þt þ g exp ifð! ! Þt þ g 2 ¼ hjaj jbj sin ð! ! Þt þ ¼ hjaj jbj sin½!0 t þ ,
¼ i
½2:29 Z
Z
z d ¼ Iz d Z h i ¼ a exp i! t þ b exp i! t Iz a exp½i! t þ b exp i! t d Z Z ¼ a a Iz d þ a b exp ið! ! Þt Iz Z Z þ ab exp ið! ! Þt Iz d þ b b
h z i ¼
¼
d
Iz
d,
h 2 jaj jbj2 , 2 ½2:30
37
2.2 THE DENSITY MATRIX
in which !0 ¼ ! – ! ¼ B0 is the Larmor frequency and ¼ b a is a phase angle. These results utilize the following equations for the angular momentum operators (note that only the equations for Iz are eigenvalue equations): Ix
Iy
Iz
¼
h 2
,
ih , 2 h ¼ , 2 ¼
Ix
Iy
Iz
¼
h 2
,
ih ¼ , 2 h ¼ , 2
½2:31
together with the orthonormality of the wavefunctions. Equations [2.31] are derived from the Pauli spin matrices as shown in Section 2.2.5. The three equations, [2.29]–[2.30], represent a vector of constant magnitude precessing about the z-axis with an angular velocity !0. This result is identical to the predicted motion of the magnetic moment obtained from the Bloch model.
2.2 The Density Matrix Calculations of scalar products and expectation values are frequent operations in quantum mechanics. Such calculations are facilitated by a formulation of quantum mechanics that focuses on the density matrix rather than on the wavefunction for a system. Additionally, the symbolic manipulations required are simplified by using a notational system introduced into quantum mechanics by Dirac (6).
2.2.1 DIRAC NOTATION The Dirac notation is a compact formalism for representing the scalar product. In this notation, a wavefunction, , is represented by the ket function, j i, and the conjugate wavefunction, *, is represented by the bra function, h j. In the Dirac notation, the scalar product of and ’ is written as the contraction of the bra h j and the ket j’i,
’
Z
’ d:
½2:32
38
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
Using the Dirac notation, an arbitrary wavefunction, , can be written as a superposition of a set of orthonormal time-independent kets, known as eigenkets or basis kets, ji ¼
N X
cn jni,
½2:33
n¼1
where jni are the basis kets (e.g., the and wavefunctions), cn are complex numbers, and N is the dimensionality of the vector space. For example, the wavefunction for a system consisting of a single spin-1/2 nucleus can be described by the linear combination of the kets for the and states of that nucleus (which are the eigenfunctions of the angular momentum operator). The coefficients, cn, can be regarded as amplitude factors that describe how much a particular basis ket contributes to the total wavefunction at any particular time. The basis kets are time independent; consequently, any time dependence in is contained in the complex coefficients. Premultiplying [2.33] by the bra, hmj, and applying the orthogonality condition yields, cm ¼ hm j i,
½2:34
so that ji ¼
N X n¼1
cn jni ¼
N X
hnjijni ¼
n¼1
N X
jnihnji:
½2:35
n¼1
The latter identity suggests that jnihnj is an operator acting on such that jnihn j i ¼ cn jni:
½2:36
Because [2.35] must hold for arbitrary , the useful Closure Theorem is obtained immediately, N X
jnihnj ¼ E,
½2:37
n¼1
in which E is the identity operator. The operator jnihnj is called a projection operator because it ‘‘projects out’’ from the component ket jni. The expectation value of some property, hAi, can be written in Dirac notation as Z hAi ¼ A d ¼ hjAji: ½2:38
39
2.2 THE DENSITY MATRIX
Now, using [2.33], hAi ¼
X
cm cn hmjAjni:
½2:39
nm
In contrast to [2.21], the kets jni are not necessarily the eigenfunctions of A; therefore, the scalar products hmjAjni do not necessarily vanish for m 6¼ n. Equation [2.21] is a special case derived from [2.39] if the kets jni are eigenfunctions of A. For a given basis set, the terms hmjAjni are constants, and the value of the observable A for a particular state of the system is determined by the products of the coefficients cm cn . Once the coefficients cm cn are known, the expectation value of any observable can be calculated. The term Amn ¼ hmjAjni is the (mn)th element of the N N matrix representation of the operator A in a given basis. The products cm cn can be regarded as the elements of a matrix representation of an operator P defined by Pnm ¼ hnjPjmi ¼ cm cn :
½2:40
Note that P can be explicitly written as a projection operator, P ¼ jihj. Substituting [2.40] into [2.39] yields X hAi ¼ cn cm hmjAjni ¼
nm X
hnjPjmihmjAjni ¼
nm
¼
X
Pnm Amn ¼
nm
X
X
hnjPAjni
n
ðPAÞnn
n
¼ TrfPAg,
½2:41
where Tr{} is the trace of a matrix defined as the sum of the diagonal elements of the matrix. The equality on line 2 of [2.41] is a consequence of the Closure Theorem [2.37]; the equality on line 3 results from the definition of matrix multiplication of the matrix representations of the operators. Equation [2.41] states that the expectation value of some observable of a system, say, for example, the amount of x-magnetization, is calculated as the trace of the product of P and A. P is the operator that is defined by the coefficients cm cn and so describes the state of the system at any particular point in time, and A is the operator corresponding to the required observable. For the sake of completeness and formality, P is a Hermitian operator such that hnjPjmi ¼ hmjPjni :
½2:42
40
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
The trace of a product of matrices is invariant to cyclic permutations of the matrices. Thus, TrfABCg ¼ TrfCABg ¼ TrfBCAg:
½2:43
A corollary of this theorem is that the trace of a commutator is zero: Tr ½A, B ¼ TrfAB BAg ¼ TrfABg TrfBAg ¼ 0: ½2:44
2.2.2 QUANTUM STATISTICAL MECHANICS The preceding analysis is applicable to a system in a so-called pure state in which the entire system is described by the same wavefunction. The wavefunction for a macromolecule in an NMR solution is an enormously complicated function of the degrees of freedom of the molecule and includes contributions from the spin, rotational, vibrational, electronic, and translational properties of the molecule. Determining the complete wavefunction for the molecule is both unfeasible and unnecessary because the properties of the nuclear spins are of primary interest in NMR spectroscopy. Accordingly, the system is divided into two components: the spin system and the surroundings (i.e., all other degrees of freedom). For historical reasons, the surroundings are termed the lattice. As a result of this division, the spin wavefunctions for different molecules in the NMR sample are no longer identical, but rather depend upon the ‘‘hidden’’ lattice variables. Such a system is called a mixed state and the effects of the lattice are incorporated by using statistical mechanics (2, 7). Each subensemble comprising the sample can be described by a wavefunction, , and a probability density, P(), that represents the contribution of the subensemble to the mixed state. The statistical value of the expectation value for a mixed state is then obtained by averaging over the probability distribution, Z ¼ PðÞhjAji d hAi XZ ¼ PðÞcn cm d hmjAjni ¼
nm X
cn cm hmjAjni:
½2:45
nm
The factors cn cm will vary from system to system, but the matrix elements hmjAjni will not. An overbar has been used to denote the statistical ensemble average in [2.45].
41
2.2 THE DENSITY MATRIX
The ensemble average of coefficients, cn cm , forms a matrix that is referred to as the density matrix. The density matrix is the matrix representation of an operator , referred to as the density operator, such that cn cm ¼ hnjPjmi ¼ hnj jmi ¼ nm :
½2:46
Because P is a Hermitian operator, so is . An expression similar to [2.41] for the expectation value of the property A in an ensemble of spins in a mixed state can be written as ¼ Trf Ag ¼ TrfA g: hAi
½2:47
The overbar will now be dropped for convenience, but an ensemble average is implied. To evaluate the expectation value of an observable, the matrix representation of the appropriate operator and, most importantly, the form of the density operator must be known. The time evolution of the system, say as it passes through a particular sequence of rf pulses and delays, is described by the time evolution of the density operator.
2.2.3 THE LIOUVILLE–VON NEUMANN EQUATION A differential equation that describes the evolution in time of the density operator must be derived. Using the Dirac notation, the timedependent Schro¨dinger equation [2.2] is written as X dcn ðtÞ dt
n
jni ¼ i
X
cn ðtÞHjni:
½2:48
cn ðtÞhkjHjni:
½2:49
n
Multiplying both sides by hkj yields X dcn ðtÞ n
dt
hk j ni ¼ i
X n
The set of basis kets is orthonormal; therefore, hk j ni ¼ 0 unless n ¼ k, and [2.49] reduces to X dck ðtÞ cn ðtÞhkjHjni: ¼ i dt n
½2:50
42
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
Equation [2.50] can be used to find a differential equation for the matrix elements of the density operator, dhkj jmi dck cm dc dc ¼ ck m þ k cm ¼ dt dt dt X dt X ¼i ck cn hnjHjmi i cn cm hkjHjni ¼i
n X
n
hkj jnihnjHjmi i
X
n
hkjHjnihnj jmi
n
¼ i½hkj Hjmi hkjH jmi,
½2:51
in which H is assumed to be identical for all members of the ensemble and the complex conjugate of [2.50] is written as " # X dck ðtÞ ¼ i cn ðtÞhkjHjni dt n X ¼i cn ðtÞhkjHjni ¼i
n X
cn ðtÞhnjHjki:
½2:52
n
The last line of [2.52] is obtained using the Hermitian property of H [2.13]. Equation [2.51] is written in operator form as d ðtÞ ¼ i ½H, ðtÞ: dt
½2:53
This is known as the Liouville–von Neumann equation and describes the time evolution of the density operator. The solution to [2.53] is straightforward if the Hamiltonian is time independent: ðtÞ ¼ expðiHtÞ ð0Þ expðiHtÞ:
½2:54
The exponential operator exp(A) used in [2.54] is defined by its Taylor series expansion: expðAÞ ¼
1 X 1 k 1 A ¼ E þ A þ AA þ , k! 2 k¼0
½2:55
in which E is the identity operator. The operators A and exp(A) necessarily commute. Using these results, [2.54] can be shown to be a
2.2 THE DENSITY MATRIX
43
solution to [2.53] by simple differentiation: d ðtÞ ¼ iH expðiHtÞ ð0Þ expðiHtÞ þ expðiHtÞ ð0ÞiH expðiHtÞ dt ¼ if expðiHtÞ ð0ÞH expðiHtÞ H expðiHtÞ ð0Þ expðiHtÞg ¼ if ðtÞH H ðtÞg ¼ i½H, ðtÞ: ½2:56 For completeness, some additional properties of the exponential operator are given here. First, in the eigenbase of A, the matrix representation of the exponential operator is hmj expðAÞjni ¼ hmjEjni þ hmjAjni þ ð1=2ÞhmjAAjni þ . . . ¼ m,n 1 þ Amm þ ð1=2ÞA2mm þ . . . ¼ m,n expðAmm Þ ¼ m,n expðm Þ,
½2:57
in which m ¼ Amm are the eigenvalues of A. Thus, the exponential matrix is diagonal in the eigenbase of A and the diagonal elements are the exponentials of the eigenvalues of A. Second, the Baker– Campbell–Hausdorff (BCH) relationship states that 1 ð½B, ½B, A þ ½½B, A, AÞ þ . . . : expfAg expfBg ¼ exp A þ B þ 12 ½B, A þ 12 ½2:58 An extremely important corollary to this theorem states that exp(AþB) ¼ exp(A) exp(B) if and only if [A, B] ¼ 0 (5).
2.2.4 THE ROTATING FRAME TRANSFORMATION The solution to the Liouville–von Neumann equation is straightforward if the Hamiltonian is time independent. A pulse sequence generally consists of two distinct parts: pulses (during which one or more rf fields are applied) and delays (during which no rf fields are present). For the present treatment, the time-dependent effects of the coupling between the spin system and the lattice will be neglected; these effects give rise to spin relaxation phenomena that will be discussed in Chapter 5. With this simplification, the Hamiltonian governing the delays is time independent; however, the rf fields comprising the pulses remain time-dependent perturbations. The simplest solution to this complication is to find a transformation that renders the rf Hamiltonian time independent and then apply [2.54]. The transformation that renders
44
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
H time independent is the quantum mechanical equivalent of the rotating frame transformation in the Bloch picture. A similarity transformation applied to the laboratory frame density operator generates a transformed density operator r, such that r ¼ U U1 ,
½2:59
in which U is an operator. A unitary operator is defined by the relationship U–1 ¼ Uy. If U is a unitary operator, then [2.59] is called a unitary transformation. The equation of motion for r is described by d r ðtÞ ¼ i ½He , r ðtÞ, dt
½2:60
in which He is a transformed Hamiltonian. The form of He can be established as follows: d r d U U1 d 1 dU dU1 ¼U U þ U1 þ U ¼ dt dt dt dt dt 1 dU dU U1 U U1 þ U U1 U ¼ iU½ , HU1 þ dt dt 1 dU dU U1 r þ r U : ¼ iU½ , HU1 þ dt dt
½2:61
The common technique of inserting E ¼ U–1U has been utilized. To proceed, the following identities are established: dE d UU1 dU 1 dU1 ¼ ¼ U þU ¼ 0, dt dt dt dt
½2:62
dU 1 dU1 U ¼ U , dt dt
½2:63
which yields
and U½ , HU1 ¼ Uð H H ÞU1 ¼ U U1 UHU1 UHU1 U U1 ¼ r , UHU1 : ½2:64
45
2.2 THE DENSITY MATRIX
Substituting [2.63] and [2.64] into [2.61] yields d r dU 1 r dU1 U þ rU ¼ iU½ , HU1 þ dt dt dt 1 r dU dU1 r þ r U ¼ i , UHU1 U dt dt 1 r dU ¼ i , UHU1 þ r , U dt 1 dU ¼ i r , UHU1 iU : dt
½2:65
This system obeys [2.60] if the effective Hamiltonian, He , is written as He ¼ UHU1 iU
dU1 : dt
½2:66
If a unitary transformation can be found that renders He time independent, then the solution to [2.60] can be obtained by straightforward adaptation of [2.54]: r ðtÞ ¼ expðiHe tÞ r ð0Þ expðiHe tÞ:
½2:67
The general procedure for solving [2.53] is as follows: find a unitary transformation that renders H time independent; transform (0) and H to r(0) and He; solve [2.60] for r(t); and, finally, transform r(t) back to (t). Spin operator calculations involving unitary transformations frequently involve propagator expressions of the general form BðÞ ¼ expðiAÞB expðiAÞ,
½2:68
in which A and B are Hermitian operators and is a real parameter. A series representation of B() is given by one form of the BCH formula: BðÞ ¼
1 X ðiÞk k¼0
k!
Ak fBg,
½2:69
in which Af g ¼ [A, ] is a commutation superoperator and A0 :¼ E. Thus, the propagator expression is evaluated as BðÞ ¼ B þ i½A, B þ
i ðiÞ2 h A, ½A, B þ . . . : 2
½2:70
46
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
Although this expression is an infinite series of terms, a compact closedform solution often can be obtained if recursive relationships can be identified after evaluating a small number of terms. For example, in evaluating the propagator exp(i Iz)Ix exp(i Iz) using the BCH formula, the series expansion can be separated into two parts that represent the series expansions of cos and sin , leading to the compact solution I x cos þ Iy sin . The BCH formula provides an alternative approach to that outlined in Section 2.3 for determining the effects of propagators on the density operator.
2.2.5 MATRIX REPRESENTATIONS
OF THE
SPIN OPERATORS
To proceed, the matrix representation of the angular momentum operators that uses the ji and ji states of the spin as basis functions must be presented. As shown by [1.1], the intrinsic spin angular momentum has units of h. Consequently, the spin angular momentum operators also have units of h, as shown directly by [2.31]. However, in the remainder of this text, the spin operators will be defined as dimensionless quantities by mapping I ! I=h. The constant of proportionality h will be reintroduced explicitly as necessary. As will be seen, a dimensionless set of spin angular momentum operators is particularly useful for analyzing the evolution of the density operator, which is itself dimensionless. The Pauli spin matrices form a complete basis set for a single spin-1/2 system (5): 1 0 1 1 0 i 1 1 0 , Iy ¼ , Iz ¼ : ½2:71 Ix ¼ 2 1 0 2 i 0 2 0 1 Each of these operators is Hermitian. The spin operators satisfy the commutation relation ½2:72 Ix , Iy ¼ iIz and any cyclic permutation of [2.72], i.e., [Iz, Ix] ¼ iIy and [Iy, Iz] ¼ iIx. The eigenkets are represented by the 2 1 column vectors, 1 0 j i ¼ , ¼ , ½2:73 0 1 and the eigenbras are represented by the 1 2 row vectors,
hj ¼ 1 0 , ¼ 0 1 :
½2:74
47
2.2 THE DENSITY MATRIX
Arbitrary kets and bras, expressed as a linear combination of the eigenkets or eigenbras, have the representations c 1 0 ji ¼ c ji þ c ¼ c þ c ¼ , c 0 1 ½2:75
hj ¼ c hj þ c ¼ c 1 0 þ c 0 1 ¼ c c : Thus, the matrix representation of a ket is the column vector whose elements are the coefficients from the expansion in terms of basis kets. The results of operator manipulations can be expressed using matrix algebra. For example, 1 0 2 1 1 0 Iy ji ¼ 2 i 1 1 Iz ji ¼ 2 0
Ix ji ¼
1 1 1 0 1 ¼ ¼ ; 2 1 2 0 0 i 1 1 0 i ¼ ¼ ; 2 2 0 0 i 0 1 1 1 1 ¼ ¼ ji; 2 0 2 1 0
1 0 Ix ¼ 2 1 1 0 Iy ¼ 2 i 1 1 Iz ¼ 2 0
1 0 1 1 1 ¼ ¼ ji; 2 0 2 0 1 i 0 1 i i ¼ ¼ ji; 2 2 0 1 0 0 0 1 0 1 ¼ ¼ ; 2 1 2 1 1 ½2:76
express the results of the Cartesian spin operators acting on the ji and ji kets. These results should be compared with [2.31]. Similarly, the orthogonality relations are obtained as 1 h j i ¼ 1 0 ¼ 1, 0
0 ¼ 0, j ¼ 1 0 1 ½2:77
1 j ¼ 0 1 ¼ 0, 0
0 j ¼ 0 1 ¼ 1: 1 The matrix representations of operators and wavefunctions depend upon the particular basis set employed. Matrix representations using different basis sets can be interconverted using unitary transformations. If 0 is the representation of a wavefunction in one (primed) basis set and is the representation in another (unprimed basis), then j0 i ¼ Uji,
½2:78
48
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
in which U is a unitary operator with matrix elements in the unprimed basis given by Uij ¼ hijUj ji ¼ hi j j0 i:
½2:79
The representation of an operator in the two basis sets is then given by the similarity transformation A0 ¼ UAU1 :
½2:80
Using these results, the expectation value of A0 is
0 0 0 A ¼ hjU1 UAU1 Uji ¼ hjAji,
½2:81
which justifies earlier assertions that the results of calculating the expectation value of an operator do not depend on the choice of basis set. In order to clarify these ideas, the transformation between a basis set consisting of the eigenfunctions of Iz and a basis set consisting of the eigenfunctions of Ix is presented. The eigenfunction equations for Ix are defined as 1 Ix ’1 ¼ ’1 , 2
1 Ix ’2 ¼ ’2 , 2
½2:82
in which 1 and ’2 are the (as yet unspecified) eigenfunctions. An arbitrary wavefunction can be written as ½2:83 ¼ c ji þ c in the basis functions of Iz and as 0 ¼ c1 ’1 þ c2 ’2
½2:84
in the basis functions of Ix. Application of [2.78] yields the matrix equation 0 ¼ U, c1 U11 ¼ c2 U21
U12 U22
c c
:
½2:85
Using [2.82] and [2.84], 1 1 1 c1 I x0 0 ¼ c1 ’1 c2 ’2 ¼ , 2 2 2 c2
½2:86
49
2.2 THE DENSITY MATRIX
in which the prime has been added to Ix to emphasize that the eigenbase of Ix is being utilized. Using [2.75] and [2.78], 1 1 0 0 1 I x ¼ UIx U U ¼ UðIx Þ ¼ U c þ c ji 2 2 U U c 1 11 12 ¼ , ½2:87 2 U21 U22 c in which the results in [2.76] have been used. Equating [2.86] and [2.87] yields c1 U11 U12 c ¼ : ½2:88 c2 U21 U22 c Satisfying the simultaneous system of equations in [2.85] and [2.88] requires that U11 ¼ U12 and U21 ¼ U22. The columns of U must be normalized and orthogonal, because U is unitary. Thus, U211 þ U222 ¼ 1 and U211 U222 ¼ 0. Finally, the determinant of U must equal þ1, so that U represents a proper rotation. Thus, 2U11U22 ¼ 1. These additional constraints give 1 1 1 U ¼ pffiffiffi , ½2:89 2 1 1 from which the explicit relationships are obtained: ’1 ¼ p1ffiffiffi ji þ , 2 ’2 ¼ p1ffiffiffi ji : 2
½2:90
Using [2.80], the operator, Iz for example, has a matrix representation in the basis set of the Ix eigenfunctions of 1 1 1 1 1 1 1 1 0 1 1 1 0 1 ¼ I z ¼ UIz U ¼ 4 1 1 0 1 1 1 4 1 1 1 1 0 1 1 : ¼ 2 1 0 ½2:91 In a particularly important application of these ideas, the matrix representation of the Hamiltonian operator, H, is calculated in some convenient basis. The matrix U is then determined such that the new
50
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
matrix representation of the operator, H0 , given by H0 ¼ UHU1 ,
½2:92
is a diagonal matrix. The transformed basis functions given by [2.78] then represent the eigenfunctions of the Hamiltonian operator and the diagonal elements of H0 are the energies associated with the stationary states of the system.
2.3 Pulses and Rotation Operators The simple case of applying an rf pulse to a single spin-1/2 nucleus in a static field B0 will be considered first. The pulse is applied as a linearly polarized transverse rf field with magnitude 2B1 and angular frequency !rf. Remembering from the Bloch approach that this field can be decomposed into two counter-rotating fields, only one of which has a significant effect on the spin, the Hamiltonian for the pulse is written as (8) H ¼ l BðtÞ ¼ Hz þ Hrf ¼ !0 Iz þ !1 ½Ix cosð!rf t þ Þ þ Iy sinð!rf t þ Þ,
½2:93
where I is the spin angular momentum operator along the axis , !0 ¼ B0, and !1 ¼ B1. The first term in [2.93] takes into account the precession of the spin under the influence of the static field, that is, the Zeeman Hamiltonian, and the second term represents the pulse. The choice of U that removes the time dependence from [2.93] is U ¼ expði!rf Iz tÞ:
½2:94
Application of this unitary transformation, using [2.66], gives the effective Hamiltonian, He ¼ !0 Iz þ !1 expði!rf Iz tÞ½Ix cosð!rf t þ Þ þ Iy sinð!rf t þ Þ expði!rf Iz tÞ þ i expði!rf Iz tÞi!rf Iz expði!rf Iz tÞ: ½2:95 Using the rotation properties of the spin angular momentum operators presented in Table 2.1 (these properties will be derived later), Ix cosð!rf tÞ þ Iy sinð!rf tÞ ¼ expði!rf Iz tÞIx expði!rf Iz tÞ Ix sinð!rf tÞ þ Iy cosð!rf tÞ ¼ expði!rf Iz tÞIy expði!rf Iz tÞ,
½2:96
2.3 PULSES
AND
51
ROTATION OPERATORS
TABLE 2.1 Rotation Properties of Angular Momentum Operators u, va x y z
x
y
z
Ix Iy cos þ Iz sin
Iz cos – Iy sin
Ix cos – Iz sin
Iy Iz cos þ Ix sin
Ix cos þ Iy sin
Iy cos – Ix sin
Iz
a
The table entries (u, v) are the results of the unitary transformation exp(–i Iv)Iu exp(i Iv).
the second term in [2.95] is simplified to !1(Ix cos þ Iy sin). The third term in [2.95] is simplified to !rfIz because an operator commutes with an exponential operator of itself. The effective Hamiltonian can be written as He ¼ !0 Iz þ !1 ðIx cos þ Iy sinÞ !rf Iz ¼ ð!0 !rf ÞIz þ !1 ðIx cos þ Iy sinÞ ¼ Iz þ !1 ðIx cos þ Iy sinÞ:
½2:97
This is now a time-independent effective Hamiltonian and the solution in the form of [2.67] describes evolution of the density operator in the rotating frame. Note the strong similarity between [2.97] and [1.18]. For completeness, the isotropic chemical shift Hamiltonian is given by H¼ !0 Iz ,
½2:98
in which is the isotropic shielding constant [1.48], rather than the density operator, and can be incorporated into the definition of ¼ !0(1 ) !rf. If ¼ 0 and ¼ 0, then the Hamiltonian for an on-resonance x-pulse becomes He ¼ !1 Ix
½2:99
and, as follows from [2.67], ð p Þ ¼ expðiHe p Þ ð0Þ expðiHe p Þ ¼ expði!1 Ix p Þ ð0Þ expði!1 Ix p Þ:
½2:100
For simplicity, the superscript has been omitted from the rotating frame density operator; in general, context is sufficient to establish whether a
52
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
rotating frame or laboratory frame density operator is intended. If ¼ !1 p, is defined to be the flip angle of the pulse of length p, then ðp Þ ¼ expðiIx Þ ð0Þ expðiIx Þ:
½2:101
The matrix representation of the exponential operators in [2.101] must be derived so that the effect on the density operator can be calculated. If the exponential rotation operators are defined as Rx ðÞ ¼ expðiIx Þ,
½2:102
ðtÞ ¼ Rx ðÞ ð0ÞR1 x ðÞ:
½2:103
then [2.101] becomes
The rotation operators can be expanded as 1 2 2 R1 x ðÞ ¼ E þ iIx 2 Ix þ . . . :
½2:104
Using the Pauli spin matrices given in [2.71], the following relationships are easily derived: 1 I2x ¼ I2y ¼ I2z ¼ E, 4 I2n ¼
1 E, 4n
¼ I2nþ1
1 I : 4n
½2:105
½2:106
½2:107
Substituting the results contained in [2.105]–[2.107] into [2.104] and grouping together even and odd powers of iIx yields 2 4 3 5 1 Rx ðÞ ¼ E 1 2 þ 4 þ þ 2iIx 3 þ 5 þ 2 3!2 2!2 4!2 5!2 ¼ E cosð=2Þ þ 2iIx sinð=2Þ: ½2:108 Expanding Ix in terms of the raising and lowering operators, I þ ¼ Ix þ iIy ,
I ¼ Ix iIy ,
½2:109
yields 2Ix ¼ ðI þ þ I Þ T:
½2:110
2.3 PULSES
AND
ROTATION OPERATORS
53
T is known as the inversion operator and has the effect of changing the spin quantum number from þ1/2 to 1/2 and vice versa. This leads to R1 x ðÞ ¼ E cosð=2Þ þ i T sinð=2Þ:
½2:111
By similar reasoning, Rx ðÞ ¼ E cosð=2Þ i T sinð=2Þ:
½2:112
The rotation matrix corresponding to a pulse of flip angle, , applied along the x-axis can now be calculated. The elements of the matrix representations of the pulse rotation operators R1 x ðÞ and Rx() are constructed from the basis eigenfunctions using the expressions ½R1 x ðÞrs ¼ hrjfE cosð=2Þ þ i T sinð=2Þg jsi, ½Rx ðÞrs ¼ hrjfE cosð=2Þ i T sinð=2Þg jsi:
½2:113
For example, if h1j ¼ hj and j2i ¼ ji, then matrix element ½R1 x ðÞ12 is ½R1 ½2:114 x ðÞ12 ¼ hjfE cosð=2Þ þ i T sinð=2Þg ¼ i sinð=2Þ: The matrix representations of the pulse operators are c is c is 1 and Rx ðÞ ¼ , Rx ðÞ ¼ is c is c
½2:115
where c ¼ cos(/2) and s ¼ sin(/2). Similar analysis for a pulse with y-phase ( ¼ /2) generates a rotation matrix of the form c s c s 1 Ry ðÞ ¼ and Ry ðÞ ¼ : ½2:116 s c s c Finally a rotation about the z-axis (which in practice is difficult to achieve experimentally with rf pulses) has the matrix representation c þ is 0 c is 0 1 Rz ðÞ ¼ and Rz ðÞ ¼ : 0 c is 0 c þ is ½2:117 The rotation induced by the general Hamiltonian given by [2.97], which includes off-resonance effects and arbitrary pulse phases, can be written as R ð, Þ ¼ expðin IÞ ¼ E cosð=2Þ i2n I sinð=2Þ,
½2:118
54
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
in which n is a unit vector along the direction of the effective field in the rotating frame Br, given by [1.18], is given by [1.23], n I ¼Ix cos sin þ Iy sin sin þ Iz cos ,
½2:119
and is defined by [1.21]. Rather than derive a matrix representation of [2.118], the following identity will be established: 1 R ð, Þ ¼ Rz ðÞRy ð ÞRz ðÞR1 y ð ÞRz ðÞ:
½2:120
The proof of [2.120] depends upon the following useful relationship: ½2:121 U fðAÞU1 ¼ f UAU1 , in which f(A) is an arbitrary function acting on the operator A. Equation [2.121] can be verified by expanding f(UAU–1) as a Taylor series. Using [2.121], 1 R ð, Þ ¼ Rz ðÞRy ð ÞRz ðÞR1 y ð ÞRz ðÞ 1 ¼ Rz ðÞ exp½iRy ð ÞIz R1 y ð ÞRz ðÞ
¼ Rz ðÞ exp½iðIz cos þ Ix sin ÞR1 z ðÞ ¼ exp½iRz ðÞðIz cos þ Ix sin ÞR1 z ðÞ ¼ exp½iðIz cos þ Ix cos sin þ Iy sin sin Þ ¼ exp½in I,
½2:122
which completes the desired proof. Thus, the operator for rotation about an arbitrary angle can be represented as a series of rotations about the y and z axes. The five rotations used to represent R(, ) in [2.120] are not mutually independent; the rotation R(, ) can be reduced to three independent rotations using the Euler decomposition of the general three-dimensional rotation (8).
2.4 Quantum Mechanical NMR Spectroscopy Theoretical analysis of an NMR experiment requires calculation of the signal observed following a sequence of rf pulses and delays. The initial state of the spin system is described by the equilibrium density operator. Evolution of the density operator through the sequence of pulses and delays is calculated using the Liouville–von Neumann equation [2.53]. The Hamiltonian consists of the appropriate spin interaction terms that govern evolution of the density operator.
2.4 QUANTUM MECHANICAL NMR SPECTROSCOPY
55
In isotropic solution, the Zeeman, chemical shift, scalar coupling, and rf pulse terms are the dominant interactions. The expectation value of the observed signal at the desired time is calculated using [2.47] as the trace of the product of the density operator and the observation operator corresponding to the observable magnetization. The equilibrium density and observation operators are described in the following section.
2.4.1 EQUILIBRIUM
AND
OBSERVATION OPERATORS
The lattice is assumed to always be in thermal equilibrium at a temperature T (equivalently, the lattice is assumed to have infinite heat capacity). At thermodynamic equilibrium, the nuclear spin states are assumed to be in thermal equilibrium with the lattice. Consequently, the values of P() (see Section 2.2.2) are constrained such that the populations of the stationary states (given by the diagonal elements of the density matrix) have a Boltzmann distribution. Furthermore, the density matrix is diagonal at equilibrium because the members of the different subensembles described by P() are uncorrelated. The form of the equilibrium density operator that satisfies these requirements is eq ¼ eH=kB T =Tr eH=kB T : ½2:123 In the eigenbase of the Hamiltonian operator, the matrix elements of eq are given by X N eq hijeH=kB T jii mn ¼ hmjeH=kB T jni i¼1
¼ m,n eEn =kB T
X N
eEi =kB T ,
½2:124
i¼1
which is a diagonal matrix whose elements are the required Boltzmann probabilities. In the high-temperature approximation, En kBT and the equilibrium density operator can be approximated by eq ¼ eH=kB T =Tr eH=kB T E H=kB T =Tr E H=kB T E H=kB T =TrfEg E=N H=ðNkB T Þ, ½2:125 in which N is the dimensionality of the Hilbert space and is equal to 2M for M spin-1/2 nuclei (i.e., M ¼ 2 for an IS two-spin system). The term
56
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
E/N is a constant that does not affect the NMR experiment; accordingly, this term is normally not written explicitly and the high-temperature approximation to the equilibrium density operator is simply written in terms of the Zeeman Hamiltonian as eq ¼ H=ðNkB T Þ ¼
M X h!0k Ikz , Nk BT k¼1
½2:126
in which h has been included explicitly. By convention, the complex magnetization recorded during the acquisition period of an NMR experiment is given by (2)
þ ½2:127 M ðtÞ ¼ Nh Tr ðtÞFþ , in which N is the number of spins per unit volume, Fþ ¼
M X k¼1
Ikþ ¼
M X
Ikx þ iIky :
½2:128
k¼1
The operator F – could have been chosen as the observation operator equally well.
2.4.2 THE ONE-PULSE EXPERIMENT The simplest NMR experiment consists of a single pulse followed by acquisition of the FID. For a single spin, [2.126] indicates that eq / Iz. The effect of a pulse with x-phase and a rotation angle pulse applied to Iz is calculated using [2.115] as " #" #" # c is 1 0 c is 1 Rx ðÞIz R1 x ðÞ ¼ 2 is c 0 1 is c " #" # is c is 1 c ¼ 2 is c is c " # 2ics 1 c2 s2 ¼ 2 2ics s2 c2 " # i sin 1 cos ¼ 2 i sin cos ¼ Iz cos Iy sin ,
½2:129
2.4 QUANTUM MECHANICAL NMR SPECTROSCOPY
57
in which the last line is obtained by using the Pauli spin matrices, [2.71]. If ¼ 1808, the final matrix would be equal to 1 1 0 1 1 0 1 ½2:130 ¼ ¼ Iz , Rx ð ÞIz Rx ð Þ ¼ 2 0 1 2 0 1 corresponding to population inversion. If ¼ 908, then the final matrix becomes 1 0 i 1 0 i 1 ½2:131 Rx ð =2ÞIz Rx ð =2Þ ¼ ¼ ¼ Iy : 2 i 0 2 i 0 Simply, a 908 pulse applied with x-phase to Iz magnetization generates Iy magnetization. These results are identical to the results obtained using the Bloch model. The Iy magnetization will evolve during acquisition under the Zeeman Hamiltonian (in the rotating frame) given as Hz ¼ ð!0 !rf ÞIz ¼ Iz :
½2:132
This is a time-independent Hamiltonian; therefore, ðtÞ ¼ expðiHz tÞ ð0Þ expðiHz tÞ ¼ expðiIz tÞ ð0Þ expðiIz tÞ ¼ U ð0ÞU1
½2:133
and U ¼ expðitIz Þ ¼
expðit=2Þ 0
0 : expðit=2Þ
Performing the matrix manipulations for (0) ¼ Iy yields 0 i expðitÞ 1 ðtÞ ¼ 2 i expðitÞ 0 0 i½cosðtÞ i sinðtÞ 1 ¼ 2 i½cosðtÞ þ i sinðtÞ 0 0 i cosðtÞ þ sinðtÞ 1 ¼ 2 i cosðtÞ þ sinðtÞ 0 0 sinðtÞ 0 cosðtÞ 1 i þ : ¼ 2 sinðtÞ 2 cosðtÞ 0 0
½2:134
½2:135
58
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
Using the Pauli spin matrices, [2.135] can be written as ðtÞ ¼ U ð0ÞU1 ¼ expðitIz ÞIy expðitIz Þ ¼ Ix sinðtÞ Iy cosðtÞ: ½2:136 Note that at t ¼ 0, (t) ¼ Iy and at t ¼ /2, (t) ¼ Ix. Magnetization with a positive resonance offset in the rotating frame precesses in the sense x ! y ! x! y. The preceding results yield the form of the detectable magnetization for a one-pulse sequence for a single isolated spin:
Nh2 !0 Mþ ðtÞ ¼ Tr Ix sinðtÞ Iy cosðtÞ Ix þ iIy 2kB T Nh2 !0 2 Tr Ix sinðtÞ þ i Tr Ix Iy sinðtÞ ¼ 2kB T Tr Iy Ix cosðtÞ i Tr I2y cosðtÞ Nh2 !0 sin ðtÞ i cosðtÞ 8kB T Nh2 !0 exp iðt =2Þ , ¼ 8kB T
¼
½2:137
in which all constants have been reintroduced. This signal has the form expected from the analysis of the same system using the Bloch model in the absence of relaxation (the factor exp[–i /2] is a time-independent phase factor).
2.5 Quantum Mechanics of Multispin Systems In this section, the use of the density operator approach to perform calculations on larger, scalar coupled spin systems will be illustrated; as discussed in Chapter 1, Section 1.6, the Bloch model fails to properly account for the evolution of such spin systems. The problem is to establish the matrix representation of wavefunctions and operators in a two-spin (in general, N-spin) system and derive an appropriate operator algebra. The central results will be derived using the direct product basis; transformations to other basis sets can be performed using similarity transformations as described previously. Additional details can be found in the monograph by Corio (8).
2.5 QUANTUM MECHANICS
OF
59
MULTISPIN SYSTEMS
2.5.1 DIRECT PRODUCT SPACES The wavefunctions in the product basis are given by the direct products of the wavefunctions for individual spins: N
k ¼ jm1 i jm2 i jmN i jmi i jm1 , m2 , . . . , mN i,
½2:138
i¼1
in which mi takes on all possible values, yielding 2N wavefunctions for spin-1/2 nuclei. The total magnetic quantum number associated with a wavefunction in the product basis is Mk ¼
N X
mi :
½2:139
i¼1
The direct product of two matrices is given by (illustrated for two 2 2 matrices) B11 B12 A11 B A12 B A11 A12
¼ A B¼ A21 B A22 B A21 A22 B21 B22 2 3 A11 B11 A11 B12 A12 B11 A12 B12 6A B 7 6 11 21 A11 B22 A12 B21 A12 B22 7 ¼6 7 4 A21 B11 A21 B12 A22 B11 A22 B12 5 A21 B21
A21 B22
A22 B21
A22 B22 ½2:140
Thus, for example, the four wavefunctions in the product basis of a two-spin system are 2 3 2 3 1 0 6 7 6 7 1 1 1 0 607 617
¼ 6 7;
¼ 6 7; 1 ¼ ji ¼ 2 ¼ ¼ 405 405 0 0 0 1
3
0 2 3 0 6 7 0 1 607 ¼ ¼
¼ 6 7; 415 1 0 0
4
0 2 3 0 6 7 0 0 607 ¼ ¼
¼ 6 7: 405 1 1 1 ½2:141
Next, consider the operator corresponding to the sum of the components Iz and Sz in a two-spin system. Clearly, the matrix
60
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
representation of Iz þ Sz in a two-spin system must be a 4 4 matrix because the vector space is spanned by four wavefunctions; thus, 1 1 0 1 1 0 1 0 þ ¼ : ½2:142 Iz þ Sz 6¼ 0 1 2 0 1 2 0 1 A more formal analysis indicates that matrix representations of the operators in the two-spin system can be calculated from the direct product of the one-spin operators with the identity operator. The results for a two-spin system are ¼ I ð1spinÞ E Ið2spinÞ
and S ð2spinÞ ¼ E S ð1spinÞ ;
½2:143
where ¼ x, y, or z. In general, for an N-spin system, the representations of the angular momentum operators for the kth spin are given by ðNspinÞ ¼ E1 E2 Ek1 Ið1spinÞ
Ekþ1 EN : Ik k
½2:144
Returning to the previous example,
Izð2spinÞ ¼ Izð1spinÞ E ¼
1 1 2 0
2
1 16 0 0 ¼ 6 1 240 0
0 1
1 0
0 1 0 0
0 0 1 0
3 0 0 7 7, 0 5 1 ½2:145
Szð2spinÞ ¼ E Szð1spinÞ ¼
2
1 0
1 0 0 6 1 1 1 0 0 1 0 0 ¼ 6
1 2 0 1 240 0 1 0 0 0
3 0 0 7 7: 0 5 1 ½2:146
The combination representation:
of
Izð2spinÞ þ Szð2spinÞ 2
Izð2spinÞ þ Szð2spinÞ
1 60 ¼6 40 0
0 0 0 0
gives
the
0 0 0 0
3 0 0 7 7: 0 5 1
correct
matrix
½2:147
From now on, the (2spin) superscript will be implied. The fundamental rule of the operator algebra in direct product spaces is (as illustrated for
2.5 QUANTUM MECHANICS
OF
61
MULTISPIN SYSTEMS
a two-spin system) ABij ðA BÞ jii j ¼ Ajii B j ,
½2:148
in which A is an operator that acts on the i spin and B is an operator that acts on the j spin. Also note that AB A B ¼ ðA EÞðE BÞ:
½2:149
Thus, for example, 1 1 Iz ðIz EÞ ji ¼ Iz ji E ¼ ji ¼ : 2 2 As a second example, 2Iz Sz 2ðIz Sz Þ ji ¼ 2 Iz ji Sz 1 1 1 ¼ 2 ji ¼ : 2 2 2 In matrix notation, 1 1 0 1 2Iz Sz 2Iz Sz ¼
0 2 0 1
0 1
2
1 16 0 ¼ 6 24 0 0
0 1 0 0
½2:150
½2:151
0 0 1 0
3 0 07 7, 05 1 ½2:152
so that [2.151] also can be written as 2 32 3 2 3 1 0 0 0 0 0 16 0 1 0 0 76 1 7 16 1 7 76 7 ¼ 6 7 ¼ 1 : 2Iz Sz 6 4 5 4 5 4 5 0 2 0 0 1 0 2 0 2 0 0 0 1 0 0
½2:153
As will be discussed in Section 2.7.1, the factor of 2 in the operator 2IzSz is introduced as a normalization factor.
2.5.2 SCALAR COUPLING HAMILTONIAN The free-precession laboratory frame Hamiltonian for N scalar coupled spins is H0 ¼ Hz þ HJ ¼
N X i¼1
xi Iiz þ 2p
N1 X
N X
i¼1 j¼iþ1
Jij Ii Ij ,
½2:154
62
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
in which !i is the Larmor frequency of the ith spin and Jij is the scalar coupling constant between the ith and jth spins. The eigenfunctions of this Hamiltonian are used as the basis functions for the construction of the matrix representation of the density operator. For completeness, the effects of strong coupling must be taken into account. The product wavefunctions given by [2.138] are eigenfunctions of H only if 2 Jij/|!i – !j| 1; this condition is known as the weak coupling regime. If the weak coupling condition does not hold, then the spins are said to be strongly coupled. In the strong coupling regime, the wavefunctions in the product basis with the same total magnetic quantum number become mixed and are no longer completely independent. A proper basis set is obtained by taking linear combinations of the subset of wavefunctions with the same value of m. Construction of wavefunctions for strongly coupled spin systems with N 4 2 is facilitated by use of group theoretical methods (8). Strong coupling effects are particularly important in the analysis of coherence transfer in isotropic mixing experiments; group theoretical analyses are also important for treatment of identical spins (such as the three protons in a methyl group). To illustrate these ideas, a scalar coupled two-spin system, which was treated in the weak coupling limit in Chapter 1, Section 1.6, will be analyzed. The two spins will be labeled I and S. The free-precession Hamiltonian laboratory frame for the IS spin system is H ¼ !I Iz þ !S Sz þ 2 JIS I S,
½2:155
in which the scalar coupling constant is JIS. A system of two coupled spins has the following four eigenfunctions: 2 ¼ cos þ sin , 1 ¼ ji, ½2:156 3 ¼ cos sin , 4 ¼ , where is known as the strong coupling parameter and is defined as tanð2 Þ ¼
2 JIS !I !S
½2:157
for 2 in the range 0 to radians. If the spins have the same resonance frequency, then ¼ /4 and the wavefunctions become 1 ¼ ji, 2 ¼ 21=2 þ , ½2:158 3 ¼ 21=2 , 4 ¼ :
2.5 QUANTUM MECHANICS
OF
MULTISPIN SYSTEMS
63
The wavefunctions of [2.158] are symmetric or antisymmetric under the exchange of identical particles, as is required by the postulates of quantum mechanics (5). The energies of the four eigenstates are E1 ¼ 12 !I þ 12 !S þ 12 JIS , E3 ¼ D 12 JIS ,
E2 ¼ D 12 JIS ,
E4 ¼ 12 !I 12 !S þ 12 JIS ,
½2:159
where
D¼
1=2 1 ð!I !S Þ2 þ ð2 JIS Þ2 : 2
½2:160
In the strongly coupled spectrum, the energies of the stationary states and the positions of the resonance signals in the spectrum are altered, compared to the weakly coupled spin system (see [1.56]). In addition, the intensities of the lines in the multiplet are no longer of equal intensity; specifically, the two outer lines reduce progressively in intensity as the strong coupling effect becomes more pronounced. The results given in [2.156]–[2.160] are derived by diagonalizing the Hamiltonian matrix in the product basis; these results can be easily verified. For example, if 2 is an eigenfunction of H, then H2 ¼ E2 2 ¼ ð!I Iz þ !S Sz þ 2 JIS I SÞðcos ji þ sin jiÞ ¼ 12 !I cos 12 !I sin 12 !S cos þ 12 !S sin 12 JIS cos 12 JIS sin þ JIS cos þ JIS sin ¼ 12 ð!I cos !S cos JIS cos þ 2 JIS sin Þ þ 12 ð!I sin þ !S sin JIS sin þ 2 JIS cos Þ ¼ 12 ð!I !S JIS þ 2 JIS tan Þ cos þ 12 ð!I þ !S JIS þ 2 JIS =tan Þ sin : ½2:161
64
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
In order for 2 to be an eigenfunction, the two terms in parentheses following the last equal sign must be identical. Thus, !I !S JIS þ 2 JIS tan ¼ !I þ !S JIS þ 2 JIS =tan , 2 tan ð!I !S Þ 2 JIS 1 tan2 ¼ 0, 2 tan
2 JIS ½2:162 , ¼ 1 tan2 ð!I !S Þ 2 JIS tanð2 Þ ¼ , ð!I !S Þ which completes the demonstration, because is defined according to [2.157]. By inspection, E2 ¼ 12ð!I !S JIS þ 2 JIS tan Þ,
½2:163
which is easily shown to be equal to [2.159] by solving [2.157] for tan . By comparing [2.138] and [2.156], the transformation matrix U that converts the product basis into the strong coupling basis (and diagonalizes the Hamiltonian) is given by 2
1
0
0
6 0 cos
6 U¼6 4 0 sin
cos
0
0
0
sin
0
3
07 7 7: 05
½2:164
1
In the limit of weak
¼ 0 and the wavefunctions of scalar coupling, the two energy levels and are independent. The weak coupling Hamiltonian simplifies to H ¼ !I Iz þ !S Sz þ 2 JIS Iz Sz :
½2:165
To calculate evolution of the density operator under the weak coupling Hamiltonian, the effect of the operation ðtÞ ¼ exp½i2Iz Sz ð0Þ exp½i2Iz Sz
½2:166
for ¼ JISt must be calculated. The derivation is similar to the derivation of the rotation operators; thus, exp½i2Iz Sz ¼ E þ i2Iz Sz 12 2 ð2Iz Sz Þ2 þ . . . :
½2:167
2.5 QUANTUM MECHANICS
OF
MULTISPIN SYSTEMS
65
Using the matrix representation given in [2.152], the following relationship is easily derived ð2Iz Sz Þ2n ¼ E:
½2:168
Substituting the results contained in [2.168] into [2.167] and grouping together even and odd powers of iIzSz yields
2 4 3 5 expði2Iz Sz Þ ¼ E 1 2 þ 4 þ þ 4iIz Sz 3 þ 5 þ 2 3!2 2!2 4!2 5!2 ¼ E cos þ 4iIz Sz sin : 2 2 ½2:169 Again using [2.152], the matrix representation of the operator becomes 2 3 c þ is 0 0 0 6 0 c is 0 0 7 7, exp½i2Iz Sz ¼ 6 ½2:170 4 0 0 c is 0 5 0 0 0 c þ is where c ¼ cos(/2) and s ¼ sin(/2).
2.5.3 ROTATIONS
IN
PRODUCT SPACES
For a homonuclear system of N spins, the matrix representation of the pulse operator can be calculated from N N 1 R1 x ðÞ ¼ Rjx ðÞ ¼ E cos þ i sin Tj , 2 2 j¼1 j¼1
½2:171
in which ¼ B1 p. In [2.171], the effect of the scalar coupling term of the Hamiltonian has been ignored; this simplification requires that the length of the rf pulse, p, satisfy 2 Jij p 1. For a two-spin system, þ i sin T þ i sin T2 : ð Þ ¼ E cos E cos ½2:172 R1
1 x 2 2 2 2 The elements of the matrix representation of R are constructed from the basis eigenfunctions using the expressions
N h j þ i sin Tj jsi: ð Þ ¼ r E cos R1
x rs 2 2 j¼1
½2:173
66
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
For example, using the strong coupling eigenbasis [2.158], the matrix element ½R1 x ðÞ12 is calculated as
N R1 x ðÞ 12 ¼ h1 j E cos þ i sin Tj j2 i 2 2 j¼1 N ¼ hj E cos þ i sin Tj cos þ sin 2 2 j¼1
¼ hj E cos2 þ i cos sin T1 þ i cos sin T2 sin2 T1 T2 2 2 2 2 2 2 cos þ sin ¼ i cos sin sin þ i cos sin cos
2 2 2 2 ¼ i cos sin ðcos þ sin Þ: 2 2 ½2:174
This result is calculated using the property that the inversion operator Tj changes the spin state of spin j from to and vice versa. As another 1 example, Rx ðÞ 14 is given by 1 N Rx ðÞ 14 ¼ h1 j E cos þ i sin Tj j4 i 2 2 j¼1 N ¼ hj E cos þ i sin Tj 2 2 j¼1
2 ¼ hj E cos þ i cos sin T1 þ i cos sin T2 sin2 T1 T2 2 2 2 2 2 2 2 ¼ sin : 2 ½2:175
Repeating these calculations for every element of the matrix representation of the pulse operator yields 2
c2 6 icsu 6 R1 x ðÞ ¼ 4 icsv s2
icsu 1 s2 u2 s2 uv icsu
icsv s2 uv 1 s2 v2 icsv
3 s2 icsu 7 7, icsv 5 c2
½2:176
where c ¼ cos(/2), s ¼ sin(/2), u ¼ cos þ sin , and v ¼ cos – sin . Because the rotation operators are unitary, Rx() is the adjoint
2.5 QUANTUM MECHANICS
of R1 x ðÞ,
OF
67
MULTISPIN SYSTEMS
2
icsu c2 6 icsu 1 s2 u2 Rx ðÞ ¼ 6 4 icsv s2 uv s2 icsu
3 s2 icsu 7 7: icsv 5 c2
icsv s2 uv 1 s2 v2 icsv
½2:177
The same calculation can be performed using rotation matrices that concentrate on each spin in the two-spin system individually rather than both at the same time. This approach can be particularly useful in heteronuclear NMR experiments. The matrix representations of the rotation operators are obtained from the direct products of the singlespin rotation operators derived previously in [2.115–2.117]. For example, for spin I, 2 3 c 0 is 0 6 0 c is 1 0 c 0 is 7 7,
¼6 Rx ðÞ½I ¼ Rx ðÞ E ¼ 4 is 0 is c 0 1 c 0 5 0 is 0 c
c s 1 Ry ðÞ½I ¼ Ry ðÞ E ¼
s c 0
2
c 60 0 ¼6 4s 1 0
0 c 0 s
½2:178 3 s 0 0 s 7 7, c 0 5 0 c ½2:179
and for spin S,
3 c is 0 0 6 is c 1 0 c is 0 0 7 7, Rx ðÞ½S ¼ E Rx ðÞ ¼
¼6 4 0 1 is c 0 0 c is 5 0 0 is c
1 Ry ðÞ½S ¼ E Ry ðÞ ¼ 0
2
2
c 6s 0 c s
¼6 40 1 s c 0
s c 0 0
0 0 c s
½2:180 3 0 0 7 7: s 5 c ½2:181
The result Rx() ¼ Rx()[I ]Rx()[S ] is obtained by matrix multiplication and agrees with [2.176] in the weak coupling limit where ¼ 0.
68
CHAPTER 2 THEORETICAL DESCRIPTION
2.5.4 ONE-PULSE EXPERIMENT
FOR A
OF
NMR SPECTROSCOPY
TWO-SPIN SYSTEM
To compute the observable magnetization following a pulse and subsequent free precession, the evolution of the density operator, beginning with the equilibrium matrix representation of the density operator for a two-spin system, must be determined. Using [2.125], the initial density matrix is written as 2
!I þ !S 16 0 ð0Þ !I Iz þ !S Sz ¼ 6 0 24 0
0 !I !S 0 0
0 0 !I þ !S 0
3 0 7 0 7, 5 0 !I !S ½2:182
in which a common divisor of 2kBT has not been written for convenience and weak coupling has been assumed. A pulse x (with rotation angle and x-phase) rotates an initial state of the density operator according to the now well-known general equation, ðtÞ ¼ Rx ðÞ ð0ÞR1 x ðÞ:
½2:183
For simplicity, an ideal 908 pulse with x-phase will be assumed. Using [2.182], [2.183], [2.176], and [2.177], ðtÞ ¼ Rx ð =2Þ ð0ÞR1 x ð =2Þ 2 32 !I þ !S 1 i i 1 0 0 6 7 6 !I !S 0 1 6 i 1 1 i 76 0 ¼ 6 76 4 5 4 8 i 1 1 i 0 0 !I þ !S 1 0 0 3 1 6 i 1 1 i 7 6 7 6 7 4 i 1 1 i 5 1 i i 1 2 3 i!I 0 0 i!S 0 0 i!I 7 16 6 i!S 7 ¼ 6 7 ¼ !I Iy !S Sy 2 4 i!I 0 0 i!S 5 2
1 i 1 i
0
i i
i!I i!S
0
0 0 0
3 7 7 7 5
!I !S
0 ½2:184
2.5 QUANTUM MECHANICS
OF
69
MULTISPIN SYSTEMS
because [2.143] yields the results that 2 3 2 0 0 i 0 0 i 0 7 6i 0 0 16 1 0 0 0 i 7 and Sy ¼ 6 Iy ¼ 6 0 5 24 i 0 0 240 0 0 0 i 0 0 0 0 i
3 0 0 7 7: i 5 0
½2:185
This is exactly the expected result: each term in the initial density operator is transformed identically by the nonselective pulse. Following the pulse, the density operator evolves under the free-precession Hamiltonian. Combining [2.134] with [2.170] yields the matrix representation of the exponential operator, exp½iðI Iz þ S Sz þ 2 JIS Iz Sz Þt ¼ 2 iðI þS þ JIS Þt=2 e 0 0 6 iðI S JIS Þt=2 0 e 0 6 6 iðI þS JIS Þt=2 4 0 0 e 0
0
0
3
0
7 7 7: 5
0 0 e
iðI S þ JIS Þt=2
½2:186 Performing the matrix multiplications yields exp½iðI Iz þS Sz þ2 JIS Iz Sz Þt Iy Sy exp½iðI Iz þS Sz þ2 JIS Iz Sz Þt 2 3 0 eiðS þ JIS Þt eiðI þ JIS Þt 0 ið þ J Þt 0 0 eiðI JIS Þt 7 i6 6 e S IS 7 ¼ 6 ið þ J Þt 7: iðS JIS Þt 5 24 e I IS 0 0 e eiðI JIS Þt eiðS JIS Þt
0
0
½2:187 This result is the final density operator (t). The observable signal is found by forming the product with operator Fþ / I þ þ S þ, 3 2 32 0 1 1 0 0 eiðS þ JIS Þt eiðI þ JIS Þt 0 6 7 ið þ J Þt 0 0 eiðI JIS Þt 7 i6 6 e S IS 76 0 0 0 1 7 6 7 6 iðI þ JIS Þt 7 24 e 0 0 eiðS JIS Þt 54 0 0 0 1 5 0 0 0 0 0 eiðI JIS Þt eiðS JIS Þt 0 2 iðI þ JIS Þt iðS þ JIS Þt 3 0 0 0 e e 6 7 iðS þ JIS Þt iðS þ JIS Þt e 0 i60 e 7 ¼ 6 7: 5 24 0 eiðI þ JIS Þt eiðI þ JIS Þt 0 0
0
0
eiðI JIS Þt þ eiðS JIS Þt ½2:188
70
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
The trace of this matrix is proportional to the observed complex magnetization:
Mþ ðtÞ / eiðI þ JIS Þt þ eiðI JIS Þt þ eiðS þ JIS Þt þ eiðS JIS Þt :
½2:189
The spectrum consists of four signals arranged into two doublets. One doublet consists of the frequencies I JIS and the other doublet consists of the frequencies S JIS.
2.6 Coherence So far the density operator has been represented in terms of a Cartesian basis of the spin angular momentum operators Ix, Iy, and Iz. Product operators in the Cartesian basis will be used most often in this text because the Cartesian basis affords the simplest treatment of pulses during a pulse sequence (9–11). For a system of two spin-1/2 nuclei, 16 Cartesian product operator terms are required: ð1=2ÞE Ix 2Ix Sz 2Iy Sz 2Ix Sx 2Iy Sy
Iy 2Iz Sz 2Ix Sy
Iz 2Iz Sx 2Iy Sx
Sx 2Iz Sy
Sy
Sz ½2:190
The matrix representations of these two-spin product operators, derived using [2.149], are shown in Table 2.2. The density operator also can be expressed in the shift operator basis, which provides additional insight into the density matrix theory. For a single spin-1/2 nucleus, the shift basis consists of the operators I þ ¼ Ix þ iIy ¼
0
1
,
0 0 pffiffiffi 1 1 0 I0 ¼ 2Iz ¼ pffiffiffi , 2 0 1
0 0 I ¼ Ix iIy ¼ , 1 0 1 1 1 0 pffiffiffi E ¼ pffiffiffi , 2 2 0 1
½2:191
formed by taking linear combinations of the Cartesian operators. As discussed in Section 2.7.1, the factors of 2–1/2 appearing in the matrix representations of the operators are normalization factors. Operators in the shift basis are transformed to the Cartesian basis by Ix ¼ 12 I þ þ I ,
Iy ¼ 2i1 I þ I ,
Iz ¼ p1ffiffi2 I0 :
½2:192
TABLE 2.2 Product Operators in the Cartesian Basis for a Two-Spin System
1 1 16 0 E¼ 6 2 240 0 2
0 16 0 Ix ¼ 6 4 2 1 0 2
0 16 1 Sx ¼ 6 4 2 0 0 2
0 16 0 2Ix Sx ¼ 6 4 2 0 1
0 1 0 0
0 0 1 0
3 0 07 7 05 1
0 0 0 1
1 0 0 0
3 0 17 7 05 0
1 0 0 0
0 0 0 1
3 0 07 7 15 0
0 0 1 0
0 1 0 0
2
0 1 0 0
0 0 1 0
3 0 0 7 7 0 5 1
0 0 16 0 0 Iy ¼ 6 4 2 i 0 0 i
i 0 0 0
3 2 0 0 0 60 0 1 i 7 7 2Ix Sz ¼ 6 0 5 241 0 0 0 1
1 16 0 Iz ¼ 6 240 0 2
2
0 16 i Sy ¼ 6 4 0 2 0
i 0 0 0 0 0 0 i
3 2 1 0 6 0 1 07 72I S ¼ 6 05 y y 24 0 0 1
0 0 1 0
2
1 16 0 Sz ¼ 6 240 0
3 2 0 0 61 1 0 7 7 2Iz Sx ¼ 6 i 5 240 0 0 3 2 0 1 0 60 1 1 0 7 7 2I S ¼ 6 0 0 5 x y 240 0 0 i
0 1 0 0
3 2 3 0 0 1 0 0 0 6 7 0 0 7 7 2Iz Sz ¼ 1 6 0 1 0 0 7 1 0 5 2 4 0 0 1 0 5 0 1 0 0 0 1
1 0 0 0
3 2 0 0 60 1 1 7 7 2I S ¼ 6 0 5 y z 24 i 0 0
3 0 i 0 0 0 i7 7 0 0 05 i 0 0
0 0 0 1
3 2 0 0 6i 1 0 7 7 2I S ¼ 6 1 5 z y 2 4 0 0 0
3 i 0 0 0 0 07 7 0 0 i5 0 i 0
0 0 0 i i 0 0 0
3 2 i 0 60 1 0 7 7 2I S ¼ 6 0 5 y x 240 0 i
0 0 i 0
1 0 0 0
0 i 0 0
2.6 COHERENCE
2
3 i 0 7 7 0 5 0
71
72
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
The 16 operators in the shift basis for a system of two spin-1/2 nuclei are I I0 Sþ S S0 ð1=2ÞE Iþ þ þ ½2:193 I S0 I S0 I0 S0 I0 S I0 S IþSþ 2I S 2Iþ S 2I Sþ The matrix representations of these operators are shown in Table 2.3. These operators are constructed from the direct products of the respective operators for each individual spin. For example, 2 3 2 3 0 0 0 1 0 0 0 0 60 0 0 07 1 60 0 0 07 7 7, I ¼ pffiffiffi 6 I þS þ ¼ 6 4 0 0 0 0 5, 4 5 1 0 0 0 2 0 0 0 0 0 1 030 2 0 0 0 0 60 0 1 07 þ 7 I S ¼6 4 0 0 0 0 5: 0 0 0 0 ½2:194 The physical meaning of the shift operator basis is illustrated by examining the matrix representations in Table 2.3. First, consider the I operator. For illustration, the matrix is written in [2.195] with the spin states of the system along the side and top of the matrix to indicate the spin states connected by each matrix element, 3 0 0 0 0 0 0 0 07 1 6 6 7 : I ¼ pffiffiffi 6 7 4 5 1 0 0 0 2 0 1 0 0 2
½2:195
The only nonzero matrix elements present correspond to the transitions ! and ! . The lowering operator, I , is associated with a change in the spin angular momentum quantum number of m ¼ –1 and a change in the state of the I spin from (þ1/2) ! (1/2). In the case of the I þS þ operator, the only nonzero matrixelement corresponds to m ¼ þ2 and a change in spin state from ! ji. In this instance, both the I spin and the S spin change state from to . Similarly for the I þS operator, the nonzero matrix element corre sponds to the transition ! with m ¼ 0. In this example, both spins change spin states in opposite senses.
TABLE 2.3
2
1 1 16 0 E¼ 6 2 240 0 2
0 1 6 0 þ I ¼ pffiffiffi 6 4 2 0 0 2
0 1 6 0 þ S ¼ pffiffiffi 6 4 2 0 0 2
1 0 0 0
3 2 0 0 60 1 17 7 I ¼ pffiffiffi 6 05 241 0 0
0 0 0 1
0 0 0 0
3 2 0 0 60 1 07 þ 7 I S0 ¼ pffiffiffi 6 05 240 0 0
0 0 0 0
1 0 0 0
3 2 0 0 60 1 1 7 7 I S0 ¼ pffiffiffi 6 0 5 241 0 0
0 0 0 1
0 0 0 0
3 0 07 7 05 0
1 0 0 0
0 0 0 0
3 2 0 0 61 1 07 7 S ¼ pffiffiffi 6 15 240 0 0
0 0 0 0
0 0 0 1
3 2 0 0 60 1 07 þ 7 I S ¼ pffiffiffi 6 05 0 240 0 0
1 0 0 0
0 0 0 0
3 2 0 0 61 1 0 7 7 I S ¼ pffiffiffi 6 1 5 0 240 0 0
0 0 0 0 0 0 0 1
3 0 07 7 05 0
0 0 0 0
0 0 0 0
3 1 07 7 05 0
0 0 1 0
0 0 0 0
3 0 07 7 05 0
0 0 0 0
2
0 1 0 0
0 60 þ I S ¼6 40 0
0 0 1 0
1 16 0 S0 ¼ 6 240 0
2
3 0 0 0 0 7 7 1 0 5 0 1
2
0 0 0 0
1 16 0 I0 ¼ 6 240 0
3 0 0 7 7 0 5 1
2
3 0 07 7 05 1
0 1 0 0
0 60 þ I S ¼6 40 0
0 1 0 0
3 0 07 7 05 0
3 1 0 0 0 1 6 0 1 0 0 7 7 I0 S0 ¼ 6 4 4 0 0 1 0 5 0 0 0 1
2
0 60 I S ¼6 40 1
0 0 0 0
0 0 0 0
3 0 07 7 05 0
73
0 60 þ þ I S ¼6 40 0
2
0 0 1 0
0 1 0 0
2.6 COHERENCE
Product Operators in the Shift Basis for a Two-Spin System
74
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
The preceding examples illustrate the concept of coherence, which is one of the most fundamental aspects of NMR spectroscopy. As has been stated previously, a diagonal matrix element of the density operator, nn ¼ cn cn , is a real, positive number that corresponds to the population of the state described by the basis function jni. Formally, an off-diagonal element of the density operator, nm, represents coherence between eigenstates jni and jmi, in the sense that the time-dependent phase properties of the various members of the ensemble are correlated with respect to jni and jmi. Those matrix elements that denote m ¼ 1 are called single-quantum coherence; those denoting m ¼ 2 are called double-quantum coherence and, not surprisingly, those denoting m ¼ 0 are called zero-quantum coherence. To make these ideas more concrete, consider the following example. The coefficients cn for the two-level system for a spin-1/2 can be written in polar notation in terms of an amplitude and a phase factor for the and states, c ¼ jc j expði Þ,
½2:196
c ¼ c exp i :
½2:197
Any wavefunction can be expressed as ¼ c ji þ c ¼ jc j expði Þji þ c exp i ,
½2:198
thus, for a pure state, the matrix elements of the projection operator P ¼ jihj are hjPji ¼ jc j2 , hjP ¼ jc jc exp i , ½2:199
2
P ¼ c , Pji ¼ jc jc exp i : Because the state is pure, all members of the ensemble are identical and the terms ( – ) do not vary between members of the ensemble. For a mixed, macroscopic state, however, hj ji ¼ jc j2 , hj ¼ jc jc exp ið Þ , ½2:200
2
¼ c , ji ¼ jc jc exp ið Þ : If no relationship exists between the macroscopic phase properties of the state (across the ensemble) and the phase properties of the state (across the ensemble), then ( ) takes on all values in the range 0 to 2 , and exp½ið Þ ¼ exp½ið Þ ¼ 0. In this case,
2.6 COHERENCE
75
hj ji ¼ hj ji ¼ 0, and there is no coherence between the two states. Therefore, as has been stated previously, the equilibrium density matrix is diagonal. The application of an rf pulse to the equilibrium density operator induces exchange of population (i.e., transitions) between stationary states for which m ¼ 1 and causes perturbations of the equilibrium population distribution. In the case of a spin-1/2 nucleus, an rf pulse that redistributes populations across the $ transition creates a phase relationship across that transition such that exp ið Þ 6¼ 0 (averaged over the ensemble and assuming that the rotation angle is not a multiple of 1808). The density operator following the pulse is said to represent a coherent superposition between the two states; more commonly, this phenomenon is referred to simply as coherence. Coherence describes correlation of quantum mechanical phase relationships among a number of systems (separate nuclei) that persists even after the rf field is removed. Coherence is a phenomenon associated with an NMR transition and is not a transition itself; evolution of coherence does not change the populations of the spin states. Nonzero off-diagonal elements of the density matrix denote the existence of coherence. Both shift and Cartesian basis operators are useful for describing NMR spectroscopy. The Cartesian operators are a convenient basis for describing the effects of rf pulses on the density operator, and the shift operators are a convenient basis for describing the evolution of coherence in an NMR experiment. Only two eigenstates, ji and ji, exist for a single spin-1/2 nucleus; consequently, coherences associated with the ji $ ji transitions with m ¼ 1 are conveniently represented by the raising and lowering operators I þ and I . Four eigenstates exist for a two-spin system. Figure 2.1 illustrates the appearance of double- and zero-quantum coherence where eigenstates are connected in which m ¼ 2 and m ¼ 0, respectively. Double-quantum coherence is associated with transitions in which the spin states can change from $ . The change in eigenstate is identical for both of the spins involved, and this is often called a ‘‘flip-flip’’ transition. On the other hand, zero-quantum coherence is associated with transitions in which the spin states change $ , i.e., in the opposite sense to each other; these are often called ‘‘flip-flop’’ terms. The two-spin case will be seen to be the most commonly encountered as far as this text is concerned; however, spin systems consisting of three or more scalar coupled spins are evidently important and display additional features. Some of the salient features of larger spin systems will be briefly discussed using a weakly coupled three-spin system as an exemplar. In the two-spin case, each of the m ¼ 1 transitions involves
76
CHAPTER 2 THEORETICAL DESCRIPTION 4
OF
NMR SPECTROSCOPY
bb ba 3
ab 2 1
aa
FIGURE 2.1 Multiple-quantum transitions for IS spin system. Shown are the zero-quantum flip-flop transitions between states ji and ji and the doublequantum flip-flip transitions between states ji and ji.
the spin state of one nucleus changing while the spin state of the other nucleus remains constant. The spectrum can be conveniently labeled with the spin states of the coupled spins as shown in Fig. 2.1. Similar considerations apply to the three-spin system, although the appearance of the spectrum is a little more complex. The three spins are denoted I, R, and S and have three scalar coupling constants, JIS, JIR, and JSR. The wavefunctions for the scalar coupled three-spin system are denoted jmI , mS , mR i in the product basis, and the energies of the eight levels can be calculated by generalizing [1.57] or by direct application of [2.154] and [2.7]. The energy level diagram for a three-spin system is shown in Fig. 2.2a. The single-quantum transitions that connect pairs of eigenstates in which the spin state of one of the three nuclei changes are represented as solid or dashed arrows. Each of the indicated transitions has m ¼ –1 and, just as in the two-spin case, is associated with a resonance line of a specific multiplet (in this case the multiplets are quartets) in the one-dimensional NMR spectrum. Schematic NMR spectra are shown in Fig. 2.2b,c. As seen by comparing b and c in Fig. 2.2, the appearance of the spectrum depends on the relative chemical shifts of the I, S, and R spins and on the relative sizes of JIS, JIR, and JSR; however, Fig. 2.2a is sufficient for illustrative purposes. The two transitions 1–2 and 2–4 share a common eigenstate (2 in this case); consequently, these two transitions are referred to as connected transitions. The spin state of one of the three spins remains unchanged across connected transitions (e.g., the I spin state is ji for the connected
77
2.7 PRODUCT OPERATOR FORMALISM a
4
2
8
bbb
6
bab
b J RS
JRS
abb
7
aab 3
bba
aba 5
1
JIR
aaa
3-4 1-2 7-8 5-6
JIR JRS
6-8 5-7 2-4 1-3
3-7 4-8
1-5 2-6
JIR
c JRS
JRS JRS
JIR
wS
wR
wI
baa
FIGURE 2.2 Spin states and spectrum for a three-spin IRS system. (a) The eight spin states and the allowed single-quantum transitions between states are shown. (- - -) Single-quantum transitions of the I spin, (—) single-quantum transitions of the R spin, and (– - –) single-quantum transitions of the S spin. (b) A schematic spectrum for an IRS spin system is shown for the special case that JIR 5 JRS and JIS ¼ 0. (c) A schematic spectrum for an IRS spin system is shown for the special case that JIR ¼ JRS and JIS ¼ 0.
transitions 1–2 and 2–4). The values of mi for the stationary states are m1 ¼ þ3/2, m2 ¼ þ1/2, and m4 ¼ 1/2. The states represented by m1 and m4 are at opposite ends of the transition pathway under discussion and differ in their value of m by 2. In this case, the two connected transitions are said to be progressively connected. On the other hand, levels 6 and 7 in the connected transitions 5–6 and 5–7 do not differ in their values of m (e.g., m6 ¼ m7 ¼ 1/2). In this instance, the transitions are said to be regressively connected. In contrast to the single- and two-spin systems, single-quantum transitions exist in three-spin systems for which all three nuclei change spin state. For example, the transition connecting eigenstate 2 with eigenfunction , and eigenstate 7 with eigenfunc tion , has m ¼ 1/2 1/2 ¼ 1.
2.7 Product Operator Formalism Although the density matrix theory provides a rigorous description of the evolution of a nuclear spin system, the requisite matrix calculations quickly become cumbersome as the number of spins and
78
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
eigenstates increases unless implemented numerically on a computer. Unfortunately as well, the density matrix formalism provides little physical insight into NMR experiments. The design of new experiments and the optimization of existing experiments are facilitated if the spectroscopist has an intuitive feel for the evolution of the important components of the density operator at each point in the experiment. The aim of the theoretical analysis of NMR spectroscopy is prediction of the outcome of experiments. The Hamiltonian is an operator, and as has been stated previously, physically observable quantities such as energy, position, and angular momentum are represented in quantum mechanics by operators. Therefore, concentration on the operators themselves, rather than on the solutions to the Schro¨dinger equation, proves to be a powerful approach. As an illustration, the analysis of the one-pulse experiment in Section 2.4.2 indicates that the equilibrium density operator can be expressed in terms of the Cartesian Iz spin operator. This operator is partially converted into the Iy operator by a pulse with x-phase and rotation angle ; subsequent evolution under the Zeeman Hamiltonian converts the Iy operator into a linear combination of Ix and Iy spin operators. In this case, the evolution of the density operator is represented by the interconversion of single spin operators. Increasingly, due to the continued development of stronger magnets, spin systems of interest in heteronuclear and 1H NMR spectroscopy of proteins are weakly coupled. A simplified formalism, referred to as the product operator formalism, that treats each weakly coupled system independently can be used to analyze evolution of the density operator (9–11). The product operator formalism retains much of the rigor of the full density matrix treatment while facilitating manual computation and offering considerable insight into complex NMR experiments.
2.7.1 OPERATOR SPACES In general, an arbitrary density operator can be represented as a linear combination of a complete set of orthogonal basis operators, Bk: ðtÞ ¼
K X
bk ðtÞBk ,
½2:201
k¼1
in which bk(t) are complex coefficients and K is the dimensionality of the Liouville operator space spanned by the basis operators. For a system of N spin-1/2 nuclei, K ¼ 4N. Liouville operator space, and its attendant operator algebra, can be regarded as an elaboration of the ideas of the
79
2.7 PRODUCT OPERATOR FORMALISM
Hilbert vector space and vector algebra (2, 12). The orthogonality condition is n o
Tr Byj Bk ¼ Bj j Bk ¼ jk Bk j Bk : ½2:202 Unnormalized basis operators, Bk, can be normalized using B0k ¼ Bk =hBk j Bk i1=2 :
½2:203
The expectation value of an operator A can be written, by substitution of [2.201] into [2.47], ( ) K K X X hAiðtÞ ¼ Trf ðtÞAg ¼ Tr bk ðtÞBk A ¼ bk ðtÞ TrfBk Ag: ½2:204 k¼1
k¼1
Note that Tr{AB} used in [2.204] and Tr{AyB} used in [2.202] in general are not equal unless A is a Hermitian operator. The time evolution of the density operator can be expressed, by substitution of [2.201] into [2.53] and [2.54], as K X d ðtÞ ¼ i½H, ðtÞ ¼ i bk ðtÞ½H, Bk , ½2:205 dt k¼1 ðtÞ ¼ expfiHtg ð0Þ expfiHtg ¼
K X
bk ð0Þ expfiHtgBk expfiHtg:
k¼1
½2:206 The usefulness of [2.204]–[2.206] is that the evolution of the density operator and expectation values can be calculated from a limited number of trace operations Tr{BkBj} and transformation rules for exp{–iHt}Bk exp{iHt}. A transformation of the density operator is formally described as a rotation of an initial density operator 1 to a new operator 2 under the effect of a particular Hamiltonian, H. The notation to be employed has the form 1
Ht
!
2 ,
½2:207
which represents the formal expression: 2 ¼ expfiHtg 1 expfiHtg:
½2:208
If H and are expressed in terms of the angular momentum operators, then the solutions to [2.208] are given by the expressions derived in
80
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
Section 2.3. These solutions can be applied as a recipe by using a simple set of rules, which are presented in Section 2.7.3. The Hamiltonians of most interest in solution NMR consist of one or more of four interactions: (1) rf pulse, (2) chemical shift, (3) scalar coupling, and (4) residual dipolar coupling. Most importantly, in the weak coupling regime, the chemical shift, scalar, and residual dipolar coupling interactions commute with each other. Note that throughout this analysis relaxation of the spins back to equilibrium is not considered.
2.7.2 BASIS OPERATORS The choice of basis operators is determined by the problem at hand at any specific time. For example, the angular momentum operators Ix, Iy, and Iz, which represent the x-, y-, and z-components of the spin angular momentum of the system, are particularly useful for calculating the effects of rf pulses, whereas the shift operators, I þ and I , are particularly suited to evaluating evolution under the free-precession Hamiltonian. For a single spin, the state of a magnetization vector can be specified by the amounts of x, y, and z magnetization. In the same way, the quantum mechanical state of the system can be described by specifying the magnitudes of the operators that are present at any time. Formally, the state of the system is specified by the density operator and the density operator is expressed as a linear combination of operators. In most cases, Cartesian basis operators (E/2, Ix, Iy, Iz), will be employed. Other basis sets, such as the single-element (I , I, I þ, I ) basis operators, defined using the Dirac notation as
I ¼ jihj, I þ ¼ ji , ½2:209 I ¼ hj, I ¼ , and the shift basis operators ([2.191]), are also useful. The Cartesian and single-element basis sets are related by Iz ¼ 12 I I , Ix ¼ 12 ðI þ þ I Þ, ½2:210 1 1 Iy ¼ 2i1 ðI þ I Þ, 2E ¼ 2 I þ I : Levitt notes that the three Cartesian operators form a threedimensional space in which the density operator, represented by a vector I, rotates at a frequency j xj ¼ ðx xÞ1=2 about a vector x, with Cartesian components !x ¼ !1 cos,
!y ¼ !1 sin,
!z ¼ ,
½2:211
81
2.7 PRODUCT OPERATOR FORMALISM Iz
w
I
Iy
Ix
FIGURE 2.3 Geometrical representations of rotations in an operator space: precession of the angular momentum operator about the effective field direction in angular momentum operator space.
where is the resonance offset, is the phase of an applied rf pulse, and !1 ¼ B1, and B1 is the rf field strength (13). This geometrical interpretation of the Cartesian operator space is illustrated in Fig. 2.3. The identity operator E is independent of rotation. Single-transition shift operators can be defined in terms of the Cartesian components or as products of kets and bras, I þ ðrsÞ ¼ Ix ðrsÞ þ iIy ðrsÞ ¼ jrihsj, I ðrsÞ ¼ Ix ðrsÞ iIy ðrsÞ ¼ jsihrj:
½2:212
As noted by Ernst (2), the indices are ordered such that Mr 4 Ms, as defined by [2.139], to ensure that the raising and lowering operators increase and decrease the magnetic quantum numbers, respectively: I þ ðrsÞjsi ¼ jri,
I ðrsÞjri ¼ jsi: ½2:213 For the one-spin case, the eigenstates are ji and , and the density þ operator can be expanded in terms of the basis operators I , I , I , and I . In this case, for example, I ji ¼ hj i ¼ because the eigenstate ji is associated with
m ¼ þ1/2 and is associated with þ m ¼ 1/2. Similarly I ¼ ji ¼ ji. The potential of the product operator approach becomes evident in the case of two weakly scalar coupled spins. Each pair of spins has four
82
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
eigenstates, ji, , , and , where the first symbol in each represents the state of the I spin and the second symbol represents the S spin. The single-element operator basis set contains four so-called population terms: ¼ I S , jihj ¼ I S , ½2:214 ¼ I S : ¼ I S , The basis set contains eight terms representing the single-quantum transitions associated with the two spins (remembering the definitions of [2.209]):
hj ¼ I S , ji ¼ I S þ , ¼ I S , ¼ I S þ ,
½2:215 hj ¼ I S , ji ¼ I þ S , ¼ I S : ¼ I þ S , In these cases, one spin remains ‘‘untouched’’ and the transition involves only the change in spin state of the other spin. These operators describe the single-quantum coherences associated with the single-quantum transitions. The basis set contains four terms representing transitions in which both spins change their spin state simultaneously. These coherences are classified as double-quantum coherence if both spins change spin states in the same sense,
hj ¼ I S , ji ¼ I þS þ , ½2:216 or zero-quantum coherence if both opposite sense, ¼ I þS ,
spins change spin states in the ¼ I S þ :
½2:217
Each of these product operators has a simple interpretation in terms of energy levels and transitions shown in Fig. 2.4. For N scalar coupled spins-1/2, a full operator set contains 4N elements. The relationship between the Cartesian and single-element operators [2.210] can be seen as, for example, Iz ¼ 2ðIz Þ 12 E ¼ 12 I I S þ S , ½2:218 Sx ¼ 2 12 E ðSx Þ ¼ 12 I þ I S þ þ S , 1 2Iz Sy ¼ 2ðIz Þ Sy ¼ 2i I I S þ S :
83
2.7 PRODUCT OPERATOR FORMALISM
a
IbSb IbSa
a b I S IaSa
IbS+
b
c I–Sb
IbS–
I+Sb
IaS+
I–Sa I+Sa
IaS–
d
e I–S–
I+S+
I–S+ I+S–
FIGURE 2.4 Single-transition-basis operators for IS spin system. (a) Populations, (b) S-spin and (c) I-spin single-quantum coherences, (d) double-quantum coherences, and (e) zero-quantum coherences.
The operators have physical interpretations; for example, E ¼ I S þ I S þ IS þ IS
½2:219
denotes equal populations of all energy levels, and Iz ¼ 12 I S þ I S I S I S
½2:220
denotes equal polarization across the two single-quantum transitions of the I spin.
84
CHAPTER 2 THEORETICAL DESCRIPTION
2.7.3 EVOLUTION
IN THE
OF
NMR SPECTROSCOPY
PRODUCT OPERATOR FORMALISM
The goal of the product operator formalism is to derive the evolution of a spin system through a particular pulse sequence as conveniently as possible. Effects of pulses and delays in terms of Cartesian product operators are extremely simple, because each factor of the product is rotated independently. Rotation operator equations similar to [2.136] can be derived by the matrix derivations established previously; however, this approach is rather laborious. Instead, the rules for transformations of product operators can be established using the following useful theorem: if three operators satisfy the commutation relationship (and its cyclic permutations) ½A, B ¼ iC,
½2:221
expði CÞA expði CÞ ¼ A cos þ B sin :
½2:222
then
Equation [2.222] can be verified by differentiating exp(–i C)A exp(i C) twice with respect to , applying the commutation relations and solving the resulting harmonic differential equation. The evolution indicated by [2.222] can be illustrated succinctly by Fig. 2.5. 2.7.3.1 Free Precession During periods of free precession, the effects of chemical shift evolution and scalar coupling evolution must be considered. For a spin I, the chemical shift Hamiltonian has the form H ¼ IIz, where I is the offset of spin I. Evolution during a delay, t, is described by Ix Iy Iz
I Iz t
) Ix cosðI tÞ þ Iy sinðI tÞ,
½2:223
I Iz t
) Iy cosðI tÞ Ix sinðI tÞ,
½2:224
)Iz :
½2:225
I Iz t
For a weakly coupled two-spin system, I and S, the scalar coupling Hamiltonian has the form H ¼ 2 JISIzSz, where JIS is the scalar coupling constant. Evolution of the single-spin operators during a delay, t, is described by Ix
2 JIS Iz Sz t
) Ix cosð2 JIS tÞ þ 2Iy Sz sinð2 JIS tÞ,
½2:226
85
2.7 PRODUCT OPERATOR FORMALISM C
B
A
FIGURE 2.5 Operator rotations. The rotations induced by an operator C acting on an operator A are illustrated. The operators satisfy the commutation relationship [2.221] and the rotations are represented mathematically by [2.222].
2 JIS Iz Sz t
Iy Iz
2 JIS Iz Sz t
) Iy cosð2 JIS tÞ 2Ix Sz sinð2 JIS tÞ,
½2:227
) Iz :
½2:228
Equations [2.226] and [2.227] demonstrate that single-spin operators evolve into two-spin operators under the influence of the scalar coupling interaction. The analogous evolution of the two-spin operators, 2I Sz, is given by 2Ix Sz 2Iy Sz 2Iz Sz
2 JIS Iz Sz t
) 2Ix Sz cosð2 JIS tÞ þ Iy sinð2 JIS tÞ,
½2:229
2 JIS Iz Sz t
) 2Iy Sz cosð2 JIS tÞ Ix sinð2 JIS tÞ,
½2:230
) 2Iz Sz :
½2:231
2 JIS Iz Sz t
Evolution of the S and 2IzS operators is obtained by exchanging the I and S labels in [2.223]–[2.231]. 2.7.3.2 Pulses On-resonance rf pulses applied along a specific axis induce rotations in a plane orthogonal to that axis. The Hamiltonian
86
CHAPTER 2 THEORETICAL DESCRIPTION
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NMR SPECTROSCOPY
expression describing the pulses can be written as Ht ¼ Ix or Iy, for an x-pulse or y-pulse, respectively, and is the flip angle of the pulse. Pulses of arbitrary phase or that include the effects of resonance offset can be obtained using composite rotations as in [2.120]. The transformations for a pulse of phase x are given by I x
Ix !Ix ,
½2:232
I x
Iy !Iy cos Iz sin, I x
Iz !Iz cos Iy sin,
½2:233 ½2:234
and, for a pulse of phase y, I y
Ix !Ix cos Iz sin, I y
Iy !Iy ,
½2:235 ½2:236
I y
Iz !Iz cos Ix sin:
½2:237
These transformations of the product operators are illustrated geometrically in Fig. 2.6. 2.7.3.3 Practical Points The preceding rules enable description of a wide variety of pulsed NMR experiments. Before examining some very useful specific examples, some formalities and practical points will be presented. Cascades. During a period of free precession for the two-spin system I and S, the evolution of the density operator is represented as I Iz tþS Sz tþ2 JIS Iz Sz t
1 ) 4: ½2:238 Because each term in this Hamiltonian commutes with the others, the one evolution period can be divided into a series of rotations or a cascade, I Iz t
2 JIS Iz Sz t
S Sz t
1 ) 2 ) 3 ) 4: ½2:239 The order in which the rotations due to shift and coupling evolution are applied is unimportant. Likewise, the effect of a nonselective pulse applied to the I and S spins is written as 1
ðIx þSx Þ
) 3:
½2:240
87
2.7 PRODUCT OPERATOR FORMALISM Iz
a
Iy Ωt
Ix
b
c
Iz
Iz
Iy
a
Iy
a
Ix
Ix
d
e
2IzSz
2IzSz
2IySz
Iy
t J IS 2p
Ix
2p
t J IS
2IxSz
FIGURE 2.6 Transformations of product operators. The product operator transformations given in [2.225]–[2.237] are represented geometrically. (a) Transformations under the chemical shift Hamiltonian, (b) rotations induced by a pulse of x-phase, (c) rotations induced by a pulse of y-phase, and (c)–(d) transformations under the scalar coupling Hamiltonian.
88
CHAPTER 2 THEORETICAL DESCRIPTION
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NMR SPECTROSCOPY
Because I and S operators commute, a nonselective pulse can be represented by a pulse on I first, followed by a pulse on S (or by a pulse acting on S followed by a pulse acting on I), Ix
1 ! 2
Sx
) 3 :
½2:241
Rotations of products. The effect of a pulse applied selectively to the S spin of a product term such as 2IxSz is obtained using the rule that rotations only affect operators of the same spin. In other words, the Ix part of the product operator remains untouched by the pulse to the S spin and the Sz term is rotated normally. The result obtained is 2Ix Sz
Sx
) 2Ix Sz cos 2Ix Sy sin:
½2:242
Rotations involving the same operator. Operators are unaffected by rotations about themselves because an operator and the exponential of an operator commute. For example, Ix
Ix ) Ix , 2Iy Sz
½2:243
S Sz t
) 2Iy Sz :
2.7.4 SINGLE-QUANTUM COHERENCE OBSERVABLE OPERATORS
½2:244
AND
The single-quantum coherence term Ix can be expressed using [2.210] and [2.218] as Ix ¼ 12 ðI þ S þ I S Þ þ 12 ðI þ S þ I S Þ:
½2:245
This operator, involving a transverse Cartesian component, results from the sum of the single-quantum transitions of the I spin. Evolution under the free-precession Hamiltonian yields expfiHtgIx expfiHtg ¼ 12 ðI þS exp½iðI þ JÞt þ I S exp½iðI þ JÞtÞ þ 21 ðI þ S exp½iðI JÞt þ I S exp½iðI JÞtÞ:
½2:246
89
2.7 PRODUCT OPERATOR FORMALISM
The trace of this result with the observation operator, ignoring for convenience the constants of proportionality, yields Trfexp½iHtIx exp½iHtFþ g ¼ 12 exp½iðI þ JÞt þ
1 2
exp½iðI JÞt: ½2:247
Both terms comprising the detected signal are positive, indicating an in-phase component of the x-magnetization. The frequencies of the two components of the in-phase signal are separated by the scalar coupling between the two spins. An operator with a single transverse Cartesian component is observable. Another example of a single-quantum coherence operator is 2Ix Sz ¼ 21 ðI þS þ I S Þ 12ðI þS þ I S Þ:
½2:248
Evolution under the free-precession Hamiltonian yields expfiHtg2Ix Sz expfiHtg ¼ 21 ðI þS exp½iðI þ JÞt þ I S exp½iðI þ JÞtÞ 12 ðI þ S exp½iðI JÞt þ I S exp½iðI JÞtÞ:
½2:249
The trace of this result with the observation operator yields Trfexp½iHt2Ix Sz exp½iHtFþ g ¼ 12 exp½iðI þ JÞt 12 exp½iðI JÞt:
½2:250
In this case, the contributions from the S spin are of opposite sign, indicating an antiphase x-component of the magnetization on the I spin, in which the two components of the signal have opposite sign. Formally, the antiphase terms are not directly observable in the sense that, at a particular instant, a term such as 2IxSz does not contribute to the observed x-magnetization. However, antiphase operators will evolve under the influence of the scalar coupling interaction, provided that I and S have a nonzero scalar coupling constant, into an in-phase operator that is detectable. Analogous terms for the I spin product operator involving y-components are similar except that the phase of the magnetization is shifted by 908. The preceding detailed calculations are not necessary in practice, because the form of the observable signal can be determined by inspection of the coherences present at the start of the acquisition period. For a system of N spins, the operators Iix and Iiy (1 i N) are
90
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
observable and generate in-phase resonance signals. Operators of the form I1z . . . Iði1Þz Iix Iðiþ1Þz . . . Ikz and I1z . . . Iði1Þz Iiy Iðiþ1Þz . . . Ikz evolve into Iix and Iiy and generate antiphase resonance signals if J1i . . . , J(i–1)i, J(iþ1)i, . . . , Jki are all nonzero, for k N. The resonance signals generated by operators containing Iix and Iiy have a 908 relative phase difference. If the signals arising from Iix are phased to have absorptive lineshapes, then the signals arising from Iiy will have dispersive lineshapes. In spin systems with N 3, single-quantum coherences with 41 transverse factors in their operator representations exist and are referred to as combination lines, combination operators, or N-spin 1 coherences (14, 15). For example, in an N
¼ 3 spin system, the singlequantum combination coherence has m ¼ 1 and is þ represented by the operator I 1 I2 I3 . In the weak coupling limit, combination operators are orthogonal to the detection operator and consequently are not directly observable during the acquisition period. In the strong coupling limit, the product wavefunctions are not eigenfunctions of the Hamiltonian [2.154]; eigenfunctions are obtained by diagonalizing the Hamiltonian as described in Section 2.5.2. The appearance of combination lines in strongly coupled NMR spectra is discussed by Bain (16) and references therein.
2.7.5 MULTIPLE-QUANTUM COHERENCE For a system of two spin-1/2 nuclei, multiple-quantum coherence states are represented by product operators in which both spins have transverse components. For example, 2Ix Sy ¼ 2i1 ðI þ S þ I S Þ 2i1 ðI þ S I S þ Þ:
½2:251
The first term on the right-hand side, (I þS þ – I S ), is pure doublequantum coherence (|m| ¼ 2), whereas the second term, (I þS I S þ), is pure zero-quantum coherence (m ¼ 0). The multiplequantum coherence term 2IxSy is a superposition of both double- and zero-quantum coherence. Multiple-quantum coherences can be prepared by suitable combinations of pulses and free-precession periods. Such terms have more than one transverse operator component and are not observable directly; however, multiple-quantum coherences possess some unique properties of considerable utility. Multiple-quantum coherences can be expressed conveniently in terms of Cartesian and/or shift operators. Pure double-quantum (DQ)
2.7 PRODUCT OPERATOR FORMALISM
91
coherence is represented by suitable combinations of bilinear product operators, 1 þ þ 2 ðI S þ 1 þ þ 2iðI S
I S Þ ¼ 12 ð2Ix Sx 2Iy Sy Þ ¼ DQx ,
I S Þ ¼ 12 ð2Ix Sy þ 2Iy Sx Þ ¼ DQy :
½2:252 ½2:253
Pure double-quantum coherence precesses at the sum of the two chemical shifts involved, e.g., during a delay, t, DQx DQy
I Iz tþS Sz t
) DQx cos½ðI þ S Þt þ DQy sin½ðI þ S Þt,
½2:254
) DQy cos½ðI þ S Þt DQx sin½ðI þ S Þt:
½2:255
I Iz tþS Sz t
Similarly, pure zero-quantum (ZQ) coherence is represented by 1 þ 2 ðI S
þ I S þ Þ ¼ 12 ð2Ix Sx þ 2Iy Sy Þ ¼ ZQx ,
½2:256
þ 1 2i ðI S
I S þ Þ ¼ 12 ð2Iy Sx 2Ix Sy Þ ¼ ZQy ,
½2:257
and evolution occurs at the difference of the chemical shifts of the spins involved, ZQx ZQy
I Iz tþS Sz t
) ZQx cos½ðI S Þt þ ZQy sin½ðI S Þt,
I Iz tþS Sz t
) ZQy cos½ðI S Þt ZQx sin½ðI S Þt:
½2:258 ½2:259
Two-spin multiple-quantum coherence such as that noted previously, does not evolve under the influence of the scalar coupling of the two spins involved in the coherence (the active coupling). This principle can be rationalized using [1.57] and the energy level diagram of Fig. 1.7b. Double-quantum coherence connects the states ji and ji. The difference in energy between these two states does not depend on the active scalar coupling constant. Similar considerations are relevant for zero-quantum coherence that connects states ji and ji. However, multiple-quantum coherence can evolve under the influence of a scalar coupling to a third passive spin. For example, consider the three-spin system I, S, R, where the couplings present are JIS, JIR, and JSR. A multiple-quantum coherence term can be identified by the appearance of more than one transverse Cartesian operator in the product; therefore, the operator, 4IySxRz, is a multiple-quantum coherence with
92
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
respect to I and S. This operator evolves under the JIR and JSR scalar coupling interactions but not under the JIS scalar coupling interaction. Evolution under the JIR scalar coupling interaction is given by 4Iy Sx Rz
2 JIR tIz Rz
) 4Iy Sx Rz cosð JIR tÞ 2Ix Sx sinð JIR tÞ:
½2:260
Evolution of multiple-quantum coherences under the scalar coupling interaction proceeds at the sum and difference frequencies of the passive scalar coupling constants in a manner analogous to the chemical shift evolution. For example, consider the zero-quantum term ZQIS y ¼ 1 ð2I S 2I S Þ evolving under the passive coupling effects J and y x x y IR 2 JSR for a time t, ZQIS y
2 JIR tIz Rz þ2 JSR tSz Rz
IS ) ZQIS y cosð KIS tÞ 2ZQx Rz sinð KIS tÞ,
½2:261 in which KIS ¼ |JSR –JIR| is known as the zero-quantum splitting, and 1 2ZQIS x Rz ¼ 2ð 2Ix Sx þ 2Iy Sy ÞRz :
2.7.6 COHERENCE TRANSFER AND GENERATION MULTIPLE-QUANTUM COHERENCE
½2:262
OF
Coherence transfer is a vital effect in multidimensional NMR spectroscopy, and, most notably, an effect that cannot be described in the Bloch model. Suppose that an antiphase component, 2IxSz, of the density operator has been generated in some manner. As will be discussed later, antiphase operators can be produced by the use of a spin echo pulse sequence. The effect of applying a 90y pulse to both spins is 2 Iy
2 Sy
2Ix Sz ! 2Iz Sz ! 2Iz Sx :
½2:263
The original antiphase coherence on the I spin (containing a single transverse operator) is converted to antiphase coherence on the S spin. Coherence has been transferred from one spin to another under the influence of the rf pulse. In contrast, application of a 90x to both spins gives 2Ix Sz
2 ðIx þSx Þ
) 2Ix Sy :
½2:264
93
2.7 PRODUCT OPERATOR FORMALISM
This operator represents multiple-quantum coherence (containing more than one transverse operator). The same result would be obtained if a pulse is applied to the S spin alone, 2 Sx
2Ix Sz ! 2Ix Sy :
½2:265
The two examples represented by [2.264] and [2.265] represent the generation of multiple-quantum coherences in homonuclear and heteronuclear spin systems, respectively.
2.7.7 EXAMPLES
OF
PRODUCT OPERATOR CALCULATIONS
Some simple examples using product operators to follow evolution during spin echo and polarization transfer pulse sequences will be presented. Although these examples may appear trivial, each one plays an important part as a component of more complicated pulse sequences. These pulse sequence elements will be encountered in many of the multidimensional NMR experiments discussed in Chapters 6 and 7. 2.7.7.1 The Spin Echo The spin echo pulse sequence must be examined in three cases: (a) one spin, (b) two coupled spins of the same nuclear type (homonuclear case), and (c) two coupled spins of different nuclear types (heteronuclear case). Starting from equilibrium magnetization proportional to Iz, an initial 90x pulse yields 2 Ix
Iz ! Iy :
½2:266
The spin echo pulse sequence for an isolated spin is written as t 180x t :
½2:267
Evolution during the period of free precession, t, yields Iy
I Iz t
) Iy cosðI tÞ þ Ix sinðI tÞ:
½2:268
The 180x pulse converts this density operator to Ix
Iy cosðI tÞ þ Ix sinðI tÞ ! Iy cosðI tÞ þ Ix sinðI tÞ:
½2:269
The 180x pulse inverts the Iy term but does not affect the Ix term. The final part of the spin echo sequence is another delay of
94
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
duration t, I Iz t
Iy cosðI tÞ þ Ix sinðI tÞ ! Iy cos2 ðI tÞ Ix cosðI tÞ sinðI tÞ þ Ix sinðI tÞ cosðI tÞ þ Iy sin2 ðI tÞ: ½2:270 2
2
Using the identity cos þ sin ¼ 1, [2.270] can be written as Iy cosðI tÞ þ Ix sinðI tÞ
I Iz t
) Iy:
½2:271
The overall effect of the spin echo segment, – t – 180x – t –, is seen to take an initial state Iy and generate a final state Iy. Apart from a sign change, no net evolution of the chemical shift occurs during the spin echo sequence: evolution under the chemical shift Hamiltonian is refocused. If a 180y pulse had been used for refocusing, then the sign inversion would not have occurred. The same result can be demonstrated more elegantly as follows. The density operator at the end of the pulse sequence is given by (t) ¼ U (0)U–1, with U ¼ exp½iI tIz exp½i Ix exp½iI tIz ,
½2:272
in which each factor in U represents the propagator for one segment of the spin echo sequence. Applying the identity of [2.121] yields U ¼ exp½iI tIz exp½i Ix exp½iI tIz ¼ exp½iI tIz exp½i Ix exp½iI tIz exp½i Ix exp½i Ix ¼ exp½iI tIz exp iI tei Ix Iz ei Ix exp½i Ix ¼ exp½iI tIz exp½iI tIz exp½i Ix ¼ exp½i Ix :
½2:273
Therefore, the net evolution during the spin echo sequence is given by Ix
Iy ! Iy ,
½2:274
in agreement with [2.271]. Considerable simplification of propagators for pulse sequences containing 1808 pulses is often possible by use of [2.121]. The same spin echo pulse sequence can be applied to a homonuclear IS two-spin system. The pulses are assumed to be nonselective and affect both the I and the S spins equally. As for the isolated spin, the chemical shift evolution of the I and the S spins is refocused over the spin echo sequence and can be neglected. Therefore, evolution during the pulse
95
2.7 PRODUCT OPERATOR FORMALISM
sequence is due to the scalar coupling interaction only. The initial 90x pulse generates the Iy operator from the equilibrium operator Iz (the similar S spin term is omitted for clarity). The coupling develops during t, Iy
2 JIS Iz Sz t
) Iy cosð JIS tÞ þ 2Ix Sz sinð JIS tÞ:
½2:275
The 1808 pulse, regarded as a 1808 pulse on one spin followed by a 1808 pulse on the other spin, yields Ix
Iy cosð JIS tÞ þ 2Ix Sz sinð JIS tÞ ! Iy cosð JIS tÞ þ 2Ix Sz sinð JIS tÞ Sx
! Iy cosð JIS tÞ 2Ix Sz sinð JIS tÞ: ½2:276 The 180x pulse applied to the I spin does not affect the S spin and vice versa. Evolution during the second delay, t, yields Iy cosð JIS tÞ 2Ix Sz sinð JIS tÞ 2 JIS Iz Sz t
) Iy cos2 ð JIS tÞ 2Ix Sz sinð JIS tÞ cosð JIS tÞ
2Ix Sz cosð JIS tÞ sinð JIS tÞ Iy sin2 ð JIS tÞ, which, using the identities 2 sin cos , reduces to
cos(2 ) ¼ cos2 – sin2
½2:277 and
sin(2 ) ¼
Iy cosð JIS tÞ 2Ix Sz sinð JIS tÞ 2 JIS Iz Sz t
) Iy cosð2 JIS tÞ 2Ix Sz sinð2 JIS tÞ:
½2:278
The overall effect of the – t – 180x – t– pulse sequence on the initial Iy magnetization is given by Iy
t ðIx þSx Þ t
) Iy cosð2 JIS tÞ 2Ix Sz sinð2 JIS tÞ:
½2:279
The density operator has evolved under the scalar coupling Hamiltonian for the entire spin echo period, 2t. The result obtained for initial Sz magnetization is obtained by exchanging I and S operators in [2.279]. Setting the delay, t, to be equal to 1/(4JIS) generates the purely antiphase term 2IxSz, while having t ¼ 1/(2JIS) serves to produce Iy. The generation of an antiphase state by this method is a common feature in many pulse sequences. If the two scalar coupled spins belong to different nuclear species, or if sufficiently selective rf pulses can be obtained (see Chapter 3,
96
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
Section 3.4.4), then the rf pulses in the spin echo sequence can be applied to only one of the scalar coupled spins (the I spins in the following example). For example, the spin echo sequence can be applied selectively to the proton spins in an 1H–15N scalar coupled spin system. Again, I spin chemical shift is refocused and can be ignored. As before, following the 90x pulse on the I spin, evolution occurs as follows: Iy
2 JIS Iz Sz t
) Iy cosð JIS tÞ þ 2Ix Sz sinð JIS tÞ
Ix
! Iy cosð JIS tÞ þ 2Ix Sz sinð JIS tÞ:
½2:280
Only the Iy term is inverted by the 180x pulse; the S spin is unaffected. The second delay generates Iy cosð JIS tÞ þ 2Ix Sz sinð JIS tÞ 2 JIS Iz Sz t
) Iy cos2 ð JIS tÞ 2Ix Sz sinð JIS tÞ cosð JIS tÞ þ 2Ix Sz cosð JIS tÞ sinð JIS tÞ þ Iy sin2 ð JIS tÞ,
½2:281
which reduces to Iy cosð JIS tÞ þ 2Ix Sz sinð JIS tÞ
2 JIS Iz Sz t
) Iy:
½2:282
So, for the heteronuclear spin echo, Iy
t Ix t
) Iy
½2:283
and both the chemical shift and the heteronuclear coupling are refocused. In essence, the S spins have been decoupled from the I spins by use of the echo sequence. The scalar coupling interaction evolves over the entire duration of the spin echo sequence in a homonuclear spin echo because the 1808 echo pulse affects both the I and the S spins equally. By analogy, the heteronuclear scalar coupling interaction evolves over the duration of a spin echo sequence if 1808 pulses are applied to both the I and the S spins simultaneously (usually using two rf transmitter channels). The overall effect of the – t – 180x ðI, SÞ – t– pulse sequence on initial Iy magnetization is given by Iy
t Ix , Sx t
) Iy cosð2 JIS tÞ 2Ix Sz sinð2 JIS tÞ:
½2:284
2.7.7.2 Insensitive Nuclei Enhanced by Polarization Transfer Pulse sequence elements can be combined to produce more complex sequences
97
2.7 PRODUCT OPERATOR FORMALISM
designed to perform specific tasks. An important experiment that takes advantage of the basic schemes is the INEPT (Insensitive Nuclei Enhanced by Polarization Transfer) sequence (17). The INEPT sequence is a crucial component of many multidimensional NMR experiments. The aim of the INEPT sequence is to transfer magnetization from a sensitive nucleus with a high magnetogyric ratio (usually protons) to a less sensitive nucleus with a lower magnetogyric ratio (e.g., nitrogen or carbon) by means of the scalar coupling interaction. By doing this, the detected signal from the heteronucleus will be increased. The INEPT sequence is written: I spin:
90x t
180x t 90y , 180x
S spin:
90x acquire:
½2:285
Up to the final pair of 908 pulses, the sequence is a spin echo in which both spins have been affected by 180x pulses, so that chemical shift is refocused during the echo, but scalar coupling evolves fully. Beginning with equilibrium magnetization KIIz, in which KI ¼ h!I/(4kBT ) [2.126], KI Iz
2 Ix t ðIx þSx Þ t
) KI fIy cosð2 JIS tÞ 2Ix Sz sinð2 JIS tÞg:
½2:286
A 90y pulse is applied to the I spin, and a 90x pulse is applied to the S spin, KI fIy cosð2 JIS tÞ 2Ix Sz sinð2 JIS tÞg 2
ðIy þSx Þ ) KI fIy cosð2 JIS tÞ 2Iz Sy sinð2 JIS tÞg:
½2:287
If the delay t ¼ 1/(4JIS), then the final signal is given by [ h!I/(4kBT )]2IzSy. The antiphase I spin magnetization has been transferred to the S spin. The antiphase term is scaled by a factor of h!I/(4kBT ). This is an important advantage over simply recording a spec trum on the S spin following the application of a simple 908 pulse. In this case, after a 90x pulse is applied to the S spin, the observable magnetization is given by [ h!S/(4kBT)]Sy, which is an in-phase doublet. The intensity ratio between the INEPT and conventional experiment is given by INEPT I ¼ Conventional S
½2:288
The advantage of performing the INEPT experiment becomes enormous as the magnetogyric ratio of the S spin decreases. INEPT procedures are used with great effect in multidimensional heteronuclear NMR
98
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
experiments. An additional advantage of the INEPT experiment, sometimes overlooked, is that the repetition rate of the experiment is set by the relaxation time constants of the I spin rather than the S spin. Typically, the I spin is proton, and the time constants can be notably shorter than the relaxation time constants are for the S spins (see Chapter 5). 2.7.7.3 Refocused INEPT NMR spectroscopy is a relatively insensitive technique because, as has been noted in Chapter 1 (Section 1.1), the differences in populations between stationary states of a nuclear spin are very small numbers. Maximizing the sensitivity of NMR experiments consequently is a major concern. The amplitudes of the resonance signals in a scalar coupled heteronuclear spin system can be increased dramatically by decoupling the spins involved in the scalar coupling interaction. Decoupling reduces the effect of the scalar coupling constant with the result that the signal normally observed as a multiplet is collapsed into a singlet resonance at the Larmor frequency of the unperturbed spin. Sensitivity is increased because the amplitude of the singlet is given by the sum of the amplitudes of the multiplet components. As will be discussed in Chapter 3 (Section 3.5), decoupling can be achieved by the application of a suitable rf field on one of the spins in a heteronuclear scalar coupled spin system. As the following example indicates, increased sensitivity is not obtained necessarily by application of a decoupling field to an arbitrary coherence. If a decoupling field is applied to the I spins during detection of the S spins following the INEPT sequence introduced in the previous section, the resonance signal disappears completely. Following the INEPT sequence, the density operator is proportional to 2IzSy. Decoupling prevents evolution of this operator into observable in-phase single-quantum Sx coherence under the influence of the scalar coupling Hamiltonian (Section 2.7.4). Viewed another way, the 2IzSy operator represents an antiphase doublet, in which one multiplet component of the doublet is positive and the other component is negative. Collapsing the doublet by decoupling the I spins results in the mutual cancellation of the doublet components of opposite sign. Constructive interference between multiplet components is obtained only if the decoupling field is applied to an in-phase operator (with respect to the decoupled spin). Re-examination of the INEPT experiment indicates that the antiphase coherence 2IzSy can be converted into in-phase coherence by an appropriate extension to the INEPT pulse sequence. The ensuing
99
2.7 PRODUCT OPERATOR FORMALISM
refocused INEPT experiment (18) can now be written as I spin: S spin:
90x t 180x t 90y 180x decouple, 180x
90x
180x
acquire:
½2:289
By setting ¼ 1/(4JIS), the final echo component of the sequence yields KI 2Iz Sy
ðIx þSx Þ
) KI Sx :
½2:290
This is an in-phase doublet and can now be decoupled to give enhanced sensitivity in the spectrum. As will be discussed in Chapter 7 (Section 7.1.1.3), the value of required for optimal sensitivity in a refocused INEPT sequence depends upon the nature of the spin system and must be adjusted appropriately for spin systems other than the twospin system considered in this example. The Sx operator obtained following the refocused INEPT sequence can be converted to an Sz operator by application of a 90y pulse to the S spin: KI Sx
2 Sy
) KI Sz :
½2:291
The equilibrium magnetization for the S spin is proportional to KSSz; therefore, the remarkable result is obtained that the Boltzmann population difference for the I spin has been transferred to the S spin by the refocused INEPT pulse sequence. 2.7.7.4 Spin-State Selective Polarization Transfer Numerous modern NMR experiments, including Transverse Relaxation Optimized Spectroscopy (TROSY; Chapter 7) and measurements of scalar and residual dipolar coupling constants (Chapter 7), utilize pulse sequence elements that transfer polarization between operators in the single-element basis, corresponding to transfer of coherence between individual transitions or between spin states. Although useful in their own right, these methods also provide additional insights into the behavior of spin systems that continue to lead to new applications. For example, the pulse sequence element I spin:
180x
180x ,
S spin:
90x t180x t90y ,
½2:292
100
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
with t ¼ 1/(4JIS), is called the spin-state-selective coherence transfer (S3CT) sequence element (19). The propagator corresponding to this sequence is given by U ¼ exp i Sy expði Ix Þ expðiHtÞ exp½i ðIx þ Sx Þ 2 expðiHtÞ exp i Sx 2
¼ exp i Sy expði Sx Þ exp i 2Iz Sz exp i Sx 2 2 2
¼ exp i Sy exp i Sx exp i Sx exp i 2Iz Sz exp i Sx 2 2 2 2 2
¼ exp i Sy exp i Sx exp i 2Iz Sy 2 2 2
¼ exp i Sz exp i Sy exp i 2Iz Sy : 2 2 2 ½2:293 This propagator can be expressed using single transition basis operators as
U ¼ exp i Sz exp i Sy exp i 2Iz Sy 2 2 2 1 ¼ exp i Sz exp i ðSy þ 2Iz Sy Þ 2 2 ¼ exp i Sz exp i I Sy : 2
½2:294
Ignoring the z-rotation, which can be compensated by phase shifting subsequent pulses in an actual experiment, the propagator corresponds to a selective inversion across the ji $ ji transition in the two-spin energy level system (Fig. 2.4). In contrast, a conventional 1808 pulse corresponds to simultaneous inversion across both the ji $ ji and the ji $ ji transitions. The effect of this sequence element on the Sz operator can be described in either the Cartesian or the single-transition operator bases by Sz I þ I Sz
I Sy I
) 2Iz Sz , Sy ) I þ I Sz :
½2:295
101
2.7 PRODUCT OPERATOR FORMALISM bb
a
ba I–S+ I– S a
ab aa pIaSy bb
b
ba I–S–
I–Sb
ab pIaSy
aa
FIGURE 2.7 Energy level diagram for S3CT pulse sequence. The S3CT pulse sequence element is equivalent to a selective inversion across the ji $ transition, as indicated by the curved arrows. (a) The S3CT sequence converts zero-quantum coherence to single-quantum coherence I S . (b) The S3CT sequence converts double-quantum coherence to single-quantum coherence I –S.
The second line shows that the S3CT sequence selectively inverts the component of Sz magnetization that is coupled to an I spin in the state. A pictorial representation of the effect of the S3CT sequence on coherences is illustrated using zero-quantum and double-quantum coherences as an example in Fig. 2.7. The selective inversion is seen to transfer I –Sþ coherence into I –S coherence and I S coherence into I S coherence. An explicit calculation yields I S þ
I S
I Sy
) I S ,
I Sy
½2:296
)I S ,
which is necessary to obtain the correct signs of the operators. For completeness, the corresponding selective inversion across the
102
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
ji $ ji transition is obtained by inverting the phase of the final y-pulse. The propagator U ¼ exp(i I Sx) is obtained by shifting the phases of all pulses by /2.
2.8 Averaging of the Spin Hamiltonians and Residual Interactions The presentations in the preceding sections of this chapter have focused on evolution of the density operator under the isotropic components of the nuclear spin Hamiltonian. At this point, a more formal analysis is presented of the nuclear spin Hamiltonian and the effects of isotropic and nonisotropic averaging in solution. The nuclear spin Hamiltonians important in NMR spectroscopy of diamagnetic molecules are described most generally in the form H ¼ uT Cv,
½2:297
in which u and v are vectors, the superscript T indicates the transpose, and C is a general second-rank Cartesian tensor. The Cartesian tensor C is represented by a 3 3 matrix and can be decomposed into the sum of irreducible tensors of rank 0, 1, and 2: C ¼ Cð0Þ þ Cð1Þ þ Cð2Þ ,
½2:298
in which Cð0Þ ¼ 13 TrfCgE, Cð1Þ ¼ ðC CT Þ=2 is traceless and antisymmetric, and C(2) ¼ (C þ CT)/2 – C(0) is traceless and symmetric. The vector u normally will be an angular momentum operator, the vector v will be an angular momentum operator or a magnetic field vector, and C will depend on the particular magnetic spin interaction being considered. For example, the chemical shielding Hamiltonian (introduced in Chapter 1, Section 1.5) is described, in the laboratory reference frame, by uT ¼ (Ix, Iy, Iz), v ¼ (0, 0, B0)T, and C ¼ r, in which 2 3 11 12 13 r ¼ 4 21 22 23 5 ½2:299 31 32 33 is the Cartesian nuclear shielding tensor for the I spin (which should not be confused with the density operator in this context). The observation that the nuclear spin Hamiltonian must be invariant to rotation has profound consequences for NMR spectroscopy because this constraint limits the types of interactions that can couple to the nuclear spin angular momentum operators. The antisymmetric tensor
2.8 AVERAGING
OF THE
103
SPIN HAMILTONIANS
does not affect the energy levels of the nuclear spin systems and consequently does not contribute to the observed resonance frequencies. This component of the nuclear spin Hamiltonian will not be considered further in this text. The Hamiltonian can then be written as H ¼ 13 TrfCgu v þ uT Cð2Þ v:
½2:300
In the principal axis reference frame of the tensor, C(2) is diagonal with elements dk ¼ CkkTr{C}/3, in which Ckk for k ¼ {x, y, z} are the principal values of C. In this frame, uT Cð2Þ v ¼ dx ux vx þ dy uy vy þ dz uz vz :
½2:301
Equation [2.301] is expressed in terms of the Cartesian components of u, v, and C(2), which facilitates a physical interpretation of spin interactions. However, the effects of rotation are more easily considered by expressing the Hamiltonian using spherical, rather than Cartesian, tensors. Thus, [2.301] can be reformulated as pffiffiffiffiffiffiffiffi uT Cð2Þ v ¼ 3=2 dz A02 þ 12 dz ðA22 þ A2 ½2:302 2 Þ, in which ¼ ðdx dy Þ=dz , Aq2 are the 2q þ 1 components of the irreducible spherical tensor operator of second rank: A02 ¼ p1ffiffi6 ð3uz vz u vÞ, 1 A 1 2 ¼ 2 ðu vz þ uz v Þ,
½2:303
1 A 2 2 ¼ 2u v ,
u ¼ ux iuy and v ¼ vx ivy, and u and v are expressed in the principal axis frame. In obtaining [2.302], the relationship dz ¼ (dx þ dy) has been invoked because the tensor C(2) is traceless. For the chemical shielding tensor, dz ¼ 23 [1.49]. The expression [2.302] can be written equivalently in the form uT Cð2Þ v ¼
2 X
ð1Þq F2q Aq2
½2:304
q¼2
by making the identifications F20 ¼
pffiffiffiffiffiffiffiffi 3=2 dz ,
F2 1 ¼ 0,
1
F2 2 ¼ 12 dz ¼ 2 dx dy :
½2:305
104
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
The F2q also are the 2q þ1 components of an irreducible spherical tensor of second rank. With these definitions, the nuclear spin Hamiltonian is given by H ¼ 13 TrfCgu v þ
2 X
ð1Þq F2q Aq2 :
½2:306
q¼2
The advantage of writing the Hamiltonian in this form is that [2.306] is valid in any reference frame provided that the tensors are expressed in the suitable frame of interest. Thus, the Hamiltonian in the principal axis frame is obtained by using Aq2 and F2q expressed in the principal axis frame, and the Hamiltonian in the laboratory frame is obtained by using Aq2 and F2q expressed in the laboratory reference frame. The tensors F2qðlabÞ are obtained in the laboratory frame from the tensors F2qðPASÞ in the principal axis frame by using the transformation properties of the irreducible spherical tensors (20): F2qðlabÞ ¼
2 X
D2kq ðLP , LP , LP ÞF2kðPASÞ ,
½2:307
k¼2
in which D2mn ðLP , LP , LP Þ are the Wigner rotation matrices, given in Table 2.4 and {LP, LP, LP} are the Euler angles specifying the relative orientation of the laboratory and principal axis reference frames. Using this relationship, the nuclear spin Hamiltonian is expressed in the laboratory reference frame as H ¼ 13 TrfCgu v þ
2 X q¼2
ð1Þq Aq2
2 X
D2kq ðLP , LP , LP ÞF2kðPASÞ , ½2:308
k¼2
in which the vectors u and v and the tensors Aq2 are understood to be expressed in the laboratory reference frame. This equation makes use of the observation that u, v, and Aq2 usually are much simpler to express in the laboratory reference frame but the tensors F2q have their simplest form in the principal axis system of the interaction. The full form of the Hamiltonian in [2.308] is important for the development of nuclear spin relaxation theory, and is discussed in Chapter 5. For the consideration of first-order spectra — that is, of resonance frequencies and intensities — the nuclear spin Hamiltonian can be treated as a weak perturbation to the Zeeman Hamiltonian. As a result, only the components of [2.308] that commute with the Zeeman Hamiltonian need to be retained. This simplification is called truncation
TABLE 2.4 2 Reduced Rotation Matrices dmn ð Þ
m, na
2
2
cos4 ð=2Þ
1
12 sinðcos þ 1Þ
0
1
2 a
pffiffiffiffiffiffiffiffi 3=8 sin2
1 2
sinðcos 1Þ
sin4 ð=2Þ
0
1
2
pffiffiffiffiffiffiffiffi 3=8 sin2
12 sinðcos 1Þ
sin4 ð=2Þ
pffiffiffiffiffiffiffiffi 3=2 sin cos
12 ð2 cos 1Þðcos 1Þ
12 sinðcos 1Þ
pffiffiffiffiffiffiffiffi 3=2 sin cos
pffiffiffiffiffiffiffiffi 3=8 sin2
1
1 2
1 2 ð2
sinðcos þ 1Þ
cos 1Þðcos þ 1Þ
pffiffiffiffiffiffiffiffi 3=2 sin cos
12 ð2 cos 1Þðcos 1Þ 1 2
sinðcos 1Þ
1 2
3 cos2 1
pffiffiffiffiffiffiffiffi 3=2 sin cos pffiffiffiffiffiffiffiffi 3=8 sin2
The Wigner rotation matrices are given by D2mn ð, , Þ ¼ d2mn ðÞ expðim in Þ:
1 2 ð2
cos 1Þðcos þ 1Þ
12 sinðcos þ 1Þ
1 2
sinðcos þ 1Þ
cos4 ð=2Þ
106
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
of the Hamiltonian, with the result H ¼ 13TrfCgu v þ A02 F20ðlabÞ ¼ 13TrfCgu v þ A02
2 X
D2k0 ðLP , LP , 0ÞF2kðPASÞ
k¼2
rffiffiffi n 0 3 1 ¼ 3TrfCgu v þ A2 dz D200 ð0, LP , 0Þ: 2 o 1 2 þ pffiffiffi D20 ðLP , LP , 0Þ þ D220 ðLP , LP , 0Þ : 6
½2:309
The third Euler angle is unnecessary for determining the truncated Hamiltonian, and has been arbitrarily set to zero, because only D2m0 ð, , Þ are required to obtain F20ðlabÞ using [2.307]. This simplification results because the truncated Hamiltonian commutes with the Zeeman Hamiltonian and consequently is unaffected by a rotation around the z-axis of the laboratory reference frame. Also, D2m0 ð, , 0Þ ¼ Ym 2 ð, Þ, D20m ð0, , Þ ¼ Ym 2 ð, Þ,
½2:310
in which Ym 2 ð , Þ are the modified spherical harmonic functions used in Chapter 5 (20). If operators of the form uþv– and u–vþ commute with the Zeeman Hamiltonian, then A02 is given in [2.303]. If operators of the form uþv– and u–vþ do not commute with the Zeeman Hamiltonian, then these are truncated as well and A02 ¼
pffiffiffiffiffiffiffiffi 2=3 uz vz :
½2:311
For example, this simplified expression for A02 is obtained if u ¼ I and v ¼ S refer to different heteronuclear spins. In solution NMR spectroscopy, the Hamiltonians given in [2.308] and [2.309] must be averaged over the rotational distribution of molecules in solution. The angular dependence of the Hamiltonian is expressed by the angular dependence of the Wigner rotation matrices. This means that the D2mn ðLP , LP , LP Þ in [2.308] and [2.309] must be replaced by average values hD2kq ðLP , LP , LP Þi. The rotational average of Dlmn ð, , Þ is defined in general by Z
l Dmn ð, , Þ ¼ Dlmn ð, , Þpð, , Þ sin d d d, ½2:312
2.8 AVERAGING
OF THE
SPIN HAMILTONIANS
107
in which p(, , ) is the probability distribution for the set of Euler angles {, , }. In isotropic solution, all molecular orientations are equally likely. Consequently, p(, , ) ¼ 1/(8 2) and Dlmn ð, , Þ ¼ 0. Therefore, the second term in [2.308] and [2.309], depending on the traceless tensor C(2), is zero as a result of averaging over the random distribution of molecular orientations. The first term is a scalar product and is invariant to rotation. The rotationally averaged nuclear spin Hamiltonian in isotropic phase is given by H ¼ 13 TrfCgu v:
½2:313
This is the form of the nuclear spin Hamiltonians considered thus far in this text. For example, the chemical shift Hamiltonian [2.98] has the form of [2.313] with uEv ¼ B0Iz and 13 Trf g ¼ r is the isotropic chemical shielding. The strong scalar coupling Hamiltonian [2.154] has the form of [2.313] with u ¼ (Ix, Iy, Iz)T; v ¼ (Sx, Sy, Sz)T; C ¼ J, where J is the scalar coupling tensor; and 13 TrfJg ¼ 2 JIS . The dipole–dipole and quadrupole tensors are traceless and consequently do not contribute to the rotationally averaged nuclear spin Hamiltonian in isotropic phase. The isotropy of a solution of molecules is destroyed if the molecules are subject to a potential of mean force, W(, , ), that depends on the orientation of a molecular fixed frame, relative to the laboratory reference frame. The time-dependent Euler angles {, , } describe the relative orientation of these two frames of reference. The probability distribution is given by the Boltzmann equation: exp½Wð, , Þ=kB T pð, , Þ ¼ R exp½Wð, , Þ=kB T sin d d d 1 ð1 Wð, , Þ=kB T Þ: ½2:314 8 2 The second equality is obtained by assuming that the potential of mean force is weak and that Z Wð, , Þ sin d d d ¼ 0: ½2:315 Because the probability p(, , ) is unaffected by adding a constant to W(, , ), the zero of potential energy always can be chosen to satisfy this constraint. The Wigner rotation matrices form a complete set; therefore, the probability density also can be expressed as (20) pð, , Þ ¼
l X
l 1 X ð 2l þ 1 Þ Dmn ð, , Þ Dlmn ð, , Þ: 2 8 l m,n¼l
½2:316
108
CHAPTER 2 THEORETICAL DESCRIPTION zA
OF
zLAB
zPAS
xPAS
NMR SPECTROSCOPY
bAP
A → LAB
aAP xA yPAS
yLAB
{aLA, bLA, g LA} gAP yA
xLAB
FIGURE 2.8 Coordinate transformations. The principal axis system (PAS) of the nuclear spin interaction is oriented with fixed angles { , } with respect to the molecule-fixed alignment reference frame, A. The alignment frame is oriented with respect to the laboratory (LAB) reference frame by time-dependent Euler angles {LA, LA, LA}.
In order to perform the averaging over the probability distribution, the principal axis system of the spin Hamiltonian must first be transformed to the molecule-fixed frame, which will be referred to as the alignment frame, and then the alignment frame must be transformed to the laboratory reference frame. The principal axis system is oriented with time-independent Euler angles {AP, AP, AP} with respect to the alignment frame. The alignment frame is oriented with time-dependent Euler angles {LA, LA, LA} with respect to the laboratory frame. The relationships between the three frames are illustrated in Fig. 2.8. Thus, [2.307] is generalized to F2qðlabÞ ¼
2 2 X X
D2jq ðLA , LA , LA ÞD2kj ðAP , AP , AP ÞF2kðPASÞ
½2:317
j¼2 k¼2
and the rotationally averaged truncated nuclear spin Hamiltonian becomes, by extension of [2.309] (21), H ¼ 13 TrfCgu v þ A02
2 X 2 X
hD2j0 ðLA , LA , 0ÞiD2kj ðAP ,AP , AP ÞF2kðPASÞ
j¼2 k¼2
¼ 13 TrfCgu v þ A02
2 X 2 X
Sj D2kj ðAP ,AP , AP ÞF2kðPASÞ ,
j¼2 k¼2
½2:318
2.8 AVERAGING
OF THE
SPIN HAMILTONIANS
in which the order parameters Sm are defined as
Sm ¼ D2m0 ðLA , LA , 0Þ , Sm ¼ 1m Sm :
109
½2:319
The order parameters in [2.319] are identical to the corresponding coefficients in the expansion of the probability density [2.316]. The set of five order parameters in [2.318] transform under rotation like a second-rank tensor and constitute the irreducible representation of the 3 3 Saupe order matrix defined by (22)
½2:320 Sij ¼ 32 cos i cos j ij =2, in which k, for k ¼ {x, y, z}, is the angle between the kth axis of the alignment frame and the z-axis of the laboratory frame. The spherical and Saupe order parameters are related by S0 ¼ Szz , pffiffiffiffiffiffiffiffi S 1 ¼ 2=3 Sxz iSyz , pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi S 2 ¼ 1=6 Sxx Syy i 2=3Sxy :
½2:321
The Saupe order matrix is a traceless, real, and symmetric Cartesian tensor of rank 2. Consequently, the alignment frame always can be defined such that the order matrix is diagonal with principal values Sxx, Syy, and Szz. In this frame, S 1 ¼ 0 and ( 2 X 0 1 H¼ 3 TrfCgu vþA2 Szz D2k0 ðAP ,AP ÞF2kðPASÞ k¼2
) qffiffi 2 kðPASÞ X 2 2 1 þ 6 Sxx Syy Dk2 ðAP ,AP ,AP ÞþDk2 ðAP ,AP ,AP Þ F2 : k¼2
½2:322 This expression is frequently expressed as (23) ( 2 X 0 1 H ¼ 3 TrfCgu v þ A2 Aa D2k0 ðAP , AP ÞF2kðPASÞ k¼2
) qffiffi 2 X þ 38 Ar D2k2 ðAP , AP , AP Þ þ D2k2 ðAP , AP , AP Þ F2kðPASÞ , k¼2
½2:323
110
CHAPTER 2 THEORETICAL DESCRIPTION
OF
NMR SPECTROSCOPY
in which Akk ¼ (2/3)Skk are the principal values of the alignment tensor, Aa ¼ (3/2)Azz is the axial component of the tensor, and Ar ¼ Axx – Ayy is the rhombic component of the tensor. If the Hamiltonian is axially symmetric with ¼ 0, then the above expression simplifies to pffiffiffiffiffiffiffiffi H ¼ 13 TrfCgu v þ A02 3=2 z Aa 12 3 cos2 1 þ 34 Ar sin2 cos2 , ½2:324 in which {AP, AP, AP} ¼ {0, , } and { , } are the polar angles describing the orientation of the z-axis of the principal axis system of the Hamiltonian with respect to the alignment frame. As an explicit example, the Hamiltonian for the dipole–dipole Hamiltonian is traceless and symmetric, with dz ¼ 2ð0 =4 ÞI S hr3 IS . Consequently, [2.322] is given by H ¼ DIS ð3Iz Sz I SÞ,
½2:325
in which 1 DIS ¼ Dmax Szz 2 3 cos2 1 þ Sxx Syy 12 sin2 cos2 IS 1 ¼ Dmax Aa 2 3 cos2 1 þ 34 Ar sin2 cos2 IS
½2:326
is the residual dipolar coupling constant (RDC), measured in units of Hertz, and Dmax IS ¼
0 I S h : 4 2 r3IS
½2:327
If the residual dipole coupling is weak, 2 DIS/|!I !S| 1, then the Hamiltonian is further truncated to H ¼ 2 DIS Iz Sz :
½2:328
This Hamiltonian has the same functional form as does the weak scalar coupling Hamiltonian. As a consequence, if alignment occurs, then the apparent scalar coupling constant observed experimentally is given by JIS þ DIS. The alignment of a molecule with an anisotropic magnetic susceptibility tensor in the presence of a static magnetic field is a simple, easily calculable example of the effects of an orienting potential. A molecule in a magnetic field, B, has an induced magnetic dipole moment that is proportional to the magnetic susceptibility tensor .
2.8 AVERAGING
OF THE
SPIN HAMILTONIANS
111
The potential energy function is (24) W¼
1 T B B: 20
½2:329
Using similar derivations as used for the nuclear spin Hamiltonians yields for the traceless symmetric component of the potential 2 B20 X pffiffiffi D2k0 ðLA , LA , LA Þ k2 , ½2:330 0 6 k¼2 pffiffiffiffiffiffiffiffi in which 20 ¼ 2=3 , 2 2 ¼ yy xx =2, ¼ zz – ( xx þ yy)/2, and { xx, yy, zz} are the principle values of . The isotropic component of the potential does not contribute to the probability density, as noted previously, and has not been included in [2.330]. Thus, a molecule in solution has a preferential orientation with respect to the static magnetic field. Integration of [2.312] using [2.314] and [2.330] gives
WðLA , LA , LA Þ ¼
B20 D200 ¼ , 150 kB T
D202 ¼ D202
B20 xx yy pffiffiffi ¼ , 10 60 kB T
½2:331
from which B20 , 150 kB T B20 xx yy Sxx Syy ¼ , 100 kB T
Szz ¼
½2:332
The resulting residual dipolar coupling constant is 3 I S hB20 1 2 2 3 cos
1 þ
cos2 : sin DIS ¼ xx yy 2 4 60 2 kB Tr3IS ½2:333 For diamagnetic molecules, the achievable alignment is weak because is very small. For example, a benzene molecule has ¼ 1.3 1033 m3 and xx yy ¼ 0. For B0 ¼ 18.8 T (800 MHz), T ¼ 300 K, a C–H bond length of 0.11 nm, and Dmax CH ¼ 45.1 kHz, a maximum value of DCH ¼ 0.26 Hz is obtained when ¼ 0, and a minimum value of DCH ¼ –0.13 Hz is obtained when ¼ /2. In this case, Szz ¼ 5.9 10–6,
112
CHAPTER 2 THEORETICAL DESCRIPTION
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which is indicative of a very small net alignment. The dependence of DIS on B20 can be used to separate DIS from the scalar coupling constant JIS. The effects of partial alignment of small molecules have been observed for the chemical shift, dipole–dipole, and quadrupole nuclear spin Hamiltonians (25). Alignment has been obtained using electric fields, magnetic fields, thermotropic liquid crystals, and lyotropic liquid crystals. Applications to proteins and other biological macromolecules, including the use of liquid crystalline and other media to obtain larger degrees of alignment (23), are discussed in Chapter 7.
References 1. K. Blum, ‘‘Density Matrix Theory and Applications,’’ 2nd edn., pp. 1–323. Plenum Press, New York, 1996. 2. R. R. Ernst, G. Bodenhausen, A. Wokaun, ‘‘Principles of Nuclear Magnetic Resonance in One and Two Dimensions,’’ pp. 1–610. Clarendon Press, Oxford, 1987. 3. I. N. Levine, ‘‘Quantum Chemistry,’’ 3rd edn., pp. 1–566. Allyn and Bacon, Boston, 1983. 4. D. A. McQuarrie, ‘‘Quantum Chemistry,’’ pp. 1–517. University Science Books, Mill Valley, CA, 1983. 5. E. Merzbacher, ‘‘Quantum Mechanics,’’ 2nd edn., pp. 1–621. Wiley & Sons, New York, 1970. 6. P. A. M. Dirac, ‘‘The Principles of Quantum Mechanics,’’ 4th edn., pp. 1–314. Oxford University Press, New York, 1967. 7. A. Abragam, ‘‘Principles of Nuclear Magnetism,’’ pp. 1–599. Clarendon Press, Oxford, 1961. 8. P. L. Corio, ‘‘Structure of High-Resolution NMR Spectra,’’ pp. 1–548. Academic Press, New York, 1967. 9. O. W. Sørensen, G. W. Eich, M. H. Levitt, G. Bodenhausen, R. R. Ernst, Prog. NMR Spectrosc. 16, 163–192 (1983). 10. K. J. Packer, K. M. Wright, Mol. Phys. 50, 797–813 (1983). 11. F. J. M. van de Ven, C. W. Hilbers, J. Magn. Reson. 54, 512–520 (1983). 12. J. Jeener, Adv. Magn. Reson. 10, 1–51 (1982). 13. M. H. Levitt, in ‘‘Pulse Methods in 1D and 2D Liquid-Phase NMR’’ (W. S. Brey, ed.), pp. 111–147. Academic Press, San Diego, 1988. 14. J. A. Pople, W. G. Schneider, H. J. Bernstein, ‘‘High-Resolution Nuclear Magnetic Resonance,’’ pp. 1–501. McGraw-Hill, New York, 1959. 15. M. H. Levitt, ‘‘Spin Dynamics,’’ pp. 1–686. John Wiley and Sons, Chichester, 2001. 16. A. D. Bain, Can. J. Chem. 77, 1810–1812 (1999). 17. G. A. Morris, R. Freeman, J. Am. Chem. Soc. 101, 760–762 (1979). 18. D. P. Burum, R. R. Ernst, J. Magn. Reson. 39, 163–168 (1980). 19. M. D. Sørensen, A. Meissner, O. W. Sørensen, J. Biomol. NMR 10, 181–186 (1997). 20. D. M. Brink, G. R. Satchler, ‘‘Angular Momentum,’’ pp. 1–170. Clarendon Press, Oxford, 1993. 21. S. Moltke, S. Grzesiek, J. Biomol. NMR 15, 77–82 (1999).
2.8 AVERAGING 22. 23. 24. 25.
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A. Saupe, Naturforschg. 19a, 161–171 (1964). A. Bax, G. Kontaxis, N. Tjandra, Meth. Enzymol. 339, 127–174 (2001). J. D. Jackson, ‘‘Classical Electrodynamics,’’ pp. 1–848. Wiley, New York, 1975. P. C. M. van Zijl, B. H. Ruessink, J. Bulthuis, C. MacLean, Acc. Chem. Res. 17, 172–180 (1984).
CHAPTER
3 EXPERIMENTAL ASPECTS OF NMR SPECTROSCOPY
Theoretical descriptions of the basics of NMR spectroscopy have been presented in Chapters 1 and 2. This chapter discusses aspects of experimental NMR spectroscopy that are essential for acquisition of one-dimensional spectra and that serve as building blocks for more complex multidimensional techniques to be introduced in Chapters 4, 6, and 7.
3.1 NMR Instrumentation Figure 3.1 illustrates a block diagram of a pulsed Fourier transform NMR spectrometer. The main subsystems of an NMR spectrometer are the superconducting magnet, probe, pulse programmer and rf transmitter, receiver, and data acquisition and processing computer. A brief description of each of these components follows. Necessary adjustments of the spectrometer for routine use are described in subsequent sections of this chapter.
114
115
3.1 NMR INSTRUMENTATION Magnet Field-Frequency Lock Shim Coils
Probe
Receiver
RF RF Transmitter
Frequency Synthesizer
Pulse Programmer
Computer
Preamplifier
Phase-Sensitive Detector
Audio Filters
ADC
FIGURE 3.1 Block diagram of an NMR spectrometer. The major components — including the magnet, rf electronics, receiver, and computer — and important subsystems are illustrated.
A schematic of a superconducting magnet system is illustrated in Fig. 3.2. The magnet consists of a superconducting solenoid immersed in liquid helium. The liquid helium dewar is surrounded by a thermal radiation shield, a vacuum space, and an outer dewar filled with liquid nitrogen. The room-temperature bore of the magnet is centered on the z-axis of the solenoid and houses the room-temperature shim coils and the probe. NMR spectroscopy requires enormous magnetic fields with extremely high homogeneity over macroscopic volumes.
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Vacuum vessel Magnet Spinner Sample lift NMR tube rf coil Cryoshims Shim coils
Vibration damper
OF
NMR SPECTROSCOPY He turret N2 turret Radiation shield He gas N2 gas N2 vessel He vessel Liquid N2 Superconducting coil Liquid He
Probe Vacuum space Inner bore Magnet stand
FIGURE 3.2 Cutaway diagram of a superconducting magnet. The probe, sample spinner, and room-temperature shim coils are positioned coaxially in the roomtemperature bore of the magnet. The solenoid and cryoshim coils are immersed in liquid helium. The helium dewar is surrounded by a thermal radiation shield, vacuum space, and a liquid nitrogen dewar. Diagram courtesy of Bruker Instruments, Inc.
Present-generation magnets have field homogeneities on the order of 1 part in 109. As of 2005, the largest commercially available magnets have magnetic field strengths of 21.1 T with 1H Larmor frequencies of 900 MHz. In the absence of other effects (such as increased contributions to the linewidth from chemical shift anisotropy, discussed in Section 5.4.4), the resolution in an NMR spectrum increases linearly with B0 and the sensitivity increases as B3=2 (1). Thus, a 900-MHz spectrometer 0 should have 50% greater resolution and 84% greater sensitivity than a 600-MHz spectrometer. The impetus for continued development of higher field magnets is therefore obvious. For high-resolution NMR spectroscopy, the temporal stability and spatial homogeneity of the magnetic field are critical. Temporal stability is affected both by the inevitable slow decay of the magnetic field, typically less than 10 Hz/hr for modern magnets, and by perturbations due to local influences on the field. The latter include external magnetic fields and moving metal objects, such as elevators. The stability of the static magnetic field is maintained using the field-frequency lock system.
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The lock circuitry is essentially a specifically tuned (usually to deuterium) NMR spectrometer that operates in parallel to the main spectrometer. The lock system continually measures the resonance frequency of deuterium, or other lock nuclei, in the sample. If the frequency begins to drift, then the electric current in a room-temperature electromagnet, called the Z0 coil, housed in the bore of the superconducting magnet is adjusted to return the frequency of the lock nucleus to its nominal value. In most cases, deuterated solvents provide a convenient method for introducing the necessary deuterium nuclei into the sample. The spatial homogeneity of the magnetic field is optimized by adjusting the currents in a set of room-temperature electromagnets called shims. Procedures for shimming are discussed in Section 3.8.2.3. The coupling that exists between the lock system and the magnet shimming is sometimes overlooked. As noted above, the lock system works by adjusting the electrical current in the Z0 coil to provide a small magnetic field that compensates for variations in the main field. If the Z0 coil (which is physically part of the shim stack) is imperfectly constructed, then the Z0 coil will contribute an additional spatial inhomogeneity to the static magnetic field. This additional contribution to inhomogeneity gets progressively worse as the lock current in the Z0 coil increases. Normally, inhomogeneity of the Z0 field is treated simply as another contribution to overall inhomogeneity of the B0 field, and is corrected during the shimming process. However, if the lock current appreciably changes after the shimming has been done, as would be the case during a long experiment on a magnet with a high drift rate, inhomogeneity due to the Z0 coil can be reintroduced, degrading the spectral lineshape as a function of time. The probe, illustrated in Fig. 3.3 is positioned coaxially in the roomtemperature bore of the magnet. Probe design strongly affects the sensitivity of the spectrometer, the homogeneity of the B1 rf fields, the susceptibility to rf heating of the sample, and the quality of the solvent suppression. In its simplest manifestation, the probe consists simply of an rf circuit containing one or more wire coils in proximity to the NMR sample. In principle, quadrature detection of the precessing magnetization can be obtained by using two orthogonal coils (one for detecting the x-magnetization and the other for detecting the y-magnetization). Orthogonal crossed coils tuned to the same frequency are difficult to construct and optimize; modern high-resolution probes utilize a single Helmholtz coil design and obtain quadrature detection as described in Section 3.2.2. In most designs, the same coil is used for applying rf pulses and for detecting subsequent evolution of the magnetization. Depending on the probe, rf circuits may be tuned to a single frequency, may be
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Glass tubing Observe coil
Decoupling coil
rf circuitry
Heater Dewar
Decoupling channel Lock channel
Observe channel
Connector plate
VT air connector rf connectors Heater connection
FIGURE 3.3 Probe assembly. Illustration of the major components of a highresolution NMR spectroscopy rf probe. Diagram courtesy of Bruker Instruments, Inc.
double tuned to be simultaneously sensitive to two different nuclei, or may be tunable over a wide frequency range (as in a so-called broadband probe). For example, a triple-resonance 1H–13C–15N probe contains two coils. One coil is double-tuned to 1H and 2H (for the lock system) and the other coil is double-tuned for 13C and 15N. The characteristics of the probe rf circuit are given by the quality factor, Q, and complex impedance, Z: Q ¼ !L=R,
½3:1
3.1 NMR INSTRUMENTATION
Z ¼ R þ i½!L 1=ð!CÞ,
119 ½3:2
in which ! is the resonance frequency, L is the inductance, R is the resistance, and C is the capacitance of the coil circuit (for simplicity, the effects of the sample magnetism on the impedance have been omitted). The main task for the user is to tune the resonant frequency and match the impedance of the probe prior to use; other operating characteristics of the probe are difficult to alter without major reconstruction or retrofitting (Section 3.6.2.1). In conventional NMR probes, the signal-to-noise (S/N) ratio is limited ultimately by thermal noise in the rf coil and preamplifier. The main factors contributing to the basic S/N ratio of NMR measurements are contained in the following expression (2, 3): 3=2 N e 3=2 d B0 K S=N / pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , fðTc Rc þ Ta ½Rc þ Rs þ Ts Rs Þ
½3:3
in which N is the number of observed nuclei in the sample, e and d are the magnetogyric ratios of the excited and detected nuclei, respectively, B0 is the static magnetic field strength, K is a factor dependent on the coil design, f is the receiver bandwidth (in Hertz), Rc and Tc are the resistance and temperature, respectively, of the coil, Ta is the noise temperature of the preamplifier, Ts is the sample temperature, and Rs is the resistance induced by the sample in the coil. Probes have been introduced commercially that provide increased S/N by cooling the rf coil and preamplifier to reduce the noise contributions from these sources. These probes are referred to as ‘‘cryogenic probes.’’ In 2006, current-generation cryogenic probes operate at coil temperatures of 20 K and provide increases in sensitivity of approximately a factor of four for samples with low-conductivity buffers; the sensitivity increase is less than the theoretical maximum due to the need to thermally insulate the sample from the rf coil, which adversely affects the filling factor (the ratio of the sample volume to the effective volume of the rf coil). As indicated by [3.3], an unfortunate consequence of lowering Tc , Rc , and Ta is that the S/N ratio is much more sensitive to the sample resistance, Rs . The fact that Rs is proportional to the sample conductivity leads to the result that the achievable S/N in a cryogenic probe is highly dependent on the conductivity of the sample buffer. The buffer conductivity is dependent on both the concentration and the mobility of the ions in solution, and thus improvements in S/N can be achieved by use of low-mobility ions (3, 4). Improvements in S/N can sometimes be achieved simply by using a smaller diameter sample tube, especially if the
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same absolute amount of sample material is retained (5); this approach works by reducing the coupling between the conductive sample and the rf coil, and can also be beneficial in noncryogenic probes. Employing sample tubes with asymmetric geometry and a ‘‘squashed’’ rf coil design also has been shown to reduce the sample resistance (6), providing another potential means of rendering cryogenic probe performance less susceptible to sample buffer conditions. Another possibility is to switch solvents: protein NMR experiments are normally performed using samples in aqueous buffer, but solvents with low conductivity have proved to be quite useful in cryogenic probe applications (7). The rf transmitter consists of frequency synthesizers, amplifiers, and associated electronics for producing pulses of highly monochromatic rf electromagnetic radiation with defined phases and amplitudes. Typically, one transmitter subsystem is dedicated to 1H frequencies; one or more additional transmitters are used to generate rf frequencies for heteronuclear spectroscopy. The amplitude of the rf field measured in frequency units is given by !1 ¼ B1; therefore, proportionally higher power amplifiers are required for low- nuclei. Typical 1H amplifiers have peak output powers of 50–100 W; broadband amplifiers for heteronuclear spectroscopy have peak output powers in the range of 300–1000 W. The pulse programmer implements the pulse program necessary to perform an NMR experiment by controlling the timing, durations, amplitudes, and phases of the rf pulses. Radiofrequency pulses with arbitrary phase angles are generated by applying a phaseshifted rf field that is linearly polarized along a fixed axis in the laboratory reference frame (see [1.18]). Careful consideration and consistent treatment of the signs of NMR frequencies and phases are important for proper implementation of experiments and interpretation of spectral data. Levitt and co-workers have identified a number of sign inconsistencies that are introduced in the hardware, the pulse programming language, and the data processing software among different commercial NMR spectrometers (8, 9). These inconsistencies frequently are unnoticed, because the errors introduced by each mutually cancel in many, but not all, circumstances. The following discussion focuses on the convention employed for the signs of the phases of rf pulses, because the choice of convention directly impacts the practical implementation of certain pulse sequences, such as those employed in Transverse Relaxation Optimized Spectroscopy (TROSY; Chapter 7) or other spin-state-selective experiments. Levitt emphasizes that the sense of the phase shift of the rf nutation axis (phase shift of the rf pulse) and the sense of the nutation itself (rotation by the pulse) must be distinguished. The convention chosen for the sense of the nutation
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has no practical consequences and both ‘‘positive’’ and ‘‘negative’’ conventions are found in the literature; a positive (right-handed) rotation convention is mathematically convenient and conforms to the usage of Ernst and co-workers (10) as well as of this text. On the other hand, the convention chosen for the sense of the phase shift of the nutation axis is very important, and must be taken into account when implementing pulse sequences on a particular NMR spectrometer. The effect of the sense of the nutation axis phase shift is illustrated by the two-pulse experiment: 90x ––90 ,
½3:4
in which ¼ 1/(4) and is the resonance offset (measured in Hz). If is positive, the magnetization vector initially collinear with the þz-axis (i.e., the direction of the static external field B0) is rotated to the z-axis if the pulse phase ¼ þy. In contrast, if ¼ y, then the magnetization is rotated to the þ z-axis. The resulting orientation of the magnetization is obviously quite different in these two cases, and the sense of the nutation axis phase shift is absolutely critical to determining the final outcome in this example. Standard conventions for the sense of the nutation axis phase shift have not been adopted by commercial spectrometer manufacturers. Thus, if the two-pulse experiment is performed by coding the pulse sequence with explicit phase ¼ þy, different results will be obtained using spectrometers from different manufacturers; however, all spectrometers manufactured by a given company appear to maintain a consistent convention. The origin of the different sign conventions stems from the control of the rf carrier phase. The rf generation scheme of a transmitter (xmtr) channel of an NMR spectrometer includes a carrier wave of the general form sxmtr ðtÞ ffi cos !xmtr t þ xmtr , ½3:5 in which j!xmtr j is a positive number representing the oscillation frequency of the carrier wave as generated by the frequency synthesis scheme, and xmtr is a phase constant. The pulse programming software specifies the phase xmtr to control the phase of an rf pulse derived from the signal sxmtr ðtÞ. However, the relationship of this ‘‘raw’’ phase to the rotating-frame nutation axes is dependent on a number of hardwarespecific design elements, such as the rf mixing schemes employed to generate the final transmitted rf pulse from sxmtr ðtÞ (9); thus, the hardware phase changes implemented via xmtr do not necessarily correspond to the desired sense of the nutation axis phase shifts that
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determines spin dynamics. In addition, most, if not all, commercial NMR spectrometer software packages lack a mechanism for accounting for the sign of the magnetogyric ratio of the spins being observed or manipulated. For example, the sense of nutation around a rotatingframe axis during application of an rf pulse is reversed for spins with negative versus positive gyromagnetic ratios, due to the relationship !1 ¼ B1 . The preceding issues become particularly relevant when translating the results of a spin dynamics calculation or a published pulse sequence into the actual pulse sequence code for a particular spectrometer. For example, if the spin dynamics calculations call for a pulse with þy phase, then the spectroscopist needs to know whether to specify a pulse with þy or y phase in the pulse programming language of the NMR spectrometer. In many experiments, the lack of rigor in specifying the sense of the nutation axis phase shift has no significant consequence. If a (þy, y) two-step phase cycle is employed to select for or against a particular spin coherence or polarization — for example, in a heteronuclear single-quantum correlation (HSQC) experiment — then uncertainty as to the sense of the phase shift only results in an inconsequential 1808 phase shift of the detected signal. In other cases, however, such as the TROSY experiment, proper translation of the spin dynamic phases into the required hardware phases is essential. Typically, when a pulse sequence is reported in the literature, the pulse phases correspond to those employed on the NMR spectrometer used for testing the experiment. To code that pulse program for a spectrometer from another vendor, any difference in the sense of nutation axis phase shifts, related to different vendors’ spectrometers, must be taken into account. Oftentimes, the proper phase shifts can be determined simply by running a test experiment; if the expected result is not obtained, then the usage of þy and y pulses should be reversed. Useful information about the sense of the nutation axis phase shifts can be gained from the simple two-pulse experiment used as an illustration at the beginning of this discussion, applied to any sample in H2O. Place the 1H transmitter frequency upfield (i.e., to the right) of the water resonance by some offset , and apply a 908 þx pulse. According to the sign analysis provided by Levitt (8, 9), the water magnetization will precess from the y-axis toward the x-axis in the rotating frame. After the delay period , the magnetization will be aligned approximately along the x-axis. At that point, a þy pulse should return the water magnetization to the þz-axis, and thus subsequent detection of the free induction decay (FID) should show a minimal signal, relatively speaking. On the other hand, if the sense of
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þ y and y pulse phases has been reversed on the spectrometer in use, the water magnetization will be placed approximately along the z-axis after the second pulse. Subsequent detection of the FID will reveal an enormous signal from the water magnetization as it traverses through the x–y plane as a result of radiation damping. The specific phase convention employed by two spectrometer manufacturers has been documented by Roehrl et al. (11). The receiver includes the preamplifier, phase-sensitive detector, and analog-to-digital converter. The preamplifier provides an initial stage of amplification of the NMR signal prior to further detection and processing. The noise figure of the preamplifier is a critical parameter fixing the signal-to-noise level of the spectrometer, because subsequent amplification and detection stages in the receiver unavoidably amplify the preamplifier noise along with the signal. To minimize losses, the preamplifier is located as close to the probe as practical. The phasesensitive detector achieves quadrature detection of the signal as described in Section 3.2.2. The detector also includes audio filters designed to restrict the frequency bandwidth of the receiver. As discussed in Section 3.2.1, the filters reduce the amount of noise aliased into the spectrum. Unavoidably, the intensity of signals with frequencies near the cutoff of the filters will be attenuated. In addition, the time constants of the audio filters are one of the significant sources of phase errors in NMR spectroscopy (12). The analog-to-digital converters (ADCs) convert the amplified analog signal to binary numbers for subsequent digital processing. Current-generation NMR spectrometers use 16-bit digitizers as a compromise between conversion speed and dynamic range. A 16-bit digitizer can represent numbers between 28 and 281 (32,768 to 32,767). Clearly, the magnitude of the analog signal must not exceed the dynamic range of the ADC (or of earlier amplification stages). Similarly, if the magnitude of the analog signal is too small (approximately less than 1/2 bit), then the analog signal rarely registers on the ADC. In this case, extremely long acquisitions will be required to detect the signal and the results will contain distortions from digital quantization noise (i.e., the signal will only be observed to take on a limited number of digital values, see Section 3.2.2). The data acquisition and processing system which may consist of multiple computers, control the operation of the various spectrometer components. In particular, the data acquisition computer must implement a pulse programming language to permit the user to control the pulse programmer. The processing computer must permit digital signal processing of the recorded time-domain signal to produce the frequencydomain spectrum.
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3.2 Data Acquisition In modern pulsed Fourier transform NMR spectrometers, transverse magnetization is generated by a series of one or more rf pulses. The evolution in time of the magnetization generates a time-varying voltage in the probe coil. The voltage is amplified and digitized by the receiver and recorded by the NMR spectrometer. The resulting voltage-versustime signal is called an interferogram or free induction decay; the latter term refers specifically to the signal recorded during the acquisition period, whereas the term ‘‘interferogram’’ may refer either to the FID or to the signals detected indirectly during evolution periods of multidimensional NMR experiments. The digitized time-domain signal is (generally) Fourier transformed to generate the frequency-domain NMR spectrum. As discussed in the following sections, representation of a continuous time-varying signal by a discretely sampled, digitized sequence has profound consequences for NMR spectroscopy. Most of the considerations discussed here for acquisition and data processing of the observable magnetization signal apply equally well to acquisition and processing of the signals recorded indirectly during the evolution periods of multidimensional NMR experiments. Issues particularly important for multidimensional NMR spectroscopy are discussed in Chapter 4.
3.2.1 SAMPLING The continuous NMR signal, s(t), is sampled at evenly spaced time intervals and is represented as s(kt) for k ¼ 0, 1, . . . , in which the sampling interval is t. The Nyquist frequency, fn ¼ 1=ð2tÞ,
½3:6
defines the highest frequency sinusoidal signal that is sampled at least twice per period if the sampling rate is t1. The Nyquist frequency plays a central role in digital signal processing applications, including NMR spectroscopy, because of the sampling theorem (13): If a continuous function in time, s(t), is bandwidth limited to frequencies smaller in magnitude than some value fc , then the continuous function is completely determined by the discretely sampled sequence, s(kt), provided that the sampling interval t is such that fn fc.
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The sampling theorem requires that the sampling interval be t 1/(2fc) or that the sampling rate be greater than or equal to 2fc. If the conditions of the sampling theorem are met, then the continuous function is given identically by sðtÞ ¼
1 X
sðktÞ sinc½2fn ðt ktÞ,
½3:7
k¼1
with sinc(x) ¼ sin(x)/x. If a signal is recorded with a sampling interval t, then the frequency range accurately represented is given by fn fn :
½3:8
The total frequency interval is termed the spectral width, SW, and is given by SW ¼ 2fn ¼ 1=t:
½3:9
If the signal is not bandwidth limited, then the signal components with frequencies || 4 fn appear artifactually within the frequency range fn fn. The spurious representation of frequencies greater than the Nyquist frequency is called folding or aliasing. As discussed in Section 4.3.4.3, conventional NMR usage ascribes slightly different meanings to the two terms. As a result of the sampling theorem, the frequency spectrum represented by the signal sequence must be periodic with a period equal to 2fn ¼ 1/t. Thus, a frequency, 4 fn appears in the spectrum of a complex data sequence at an aliased frequency, a, given by, ¼ 2mfn þ a ,
½3:10
in which m is an integer. Equation [3.10] indicates that frequencies greater than fn (or less than fn) are ‘‘wrapped around’’ and appear at the other edge of the spectrum. By way of illustration, Fig. 3.4 shows two cosine waves with frequencies 0 and 1 ¼ 0 þ 2fn. The discretely sampled points are identical for each sine wave; thus, both signals will be represented identically in the sampled data, and the frequency 1 will be aliased to the frequency 0. In general, aliased or folded peaks have systematically altered phases because the frequency-dependent phase error is a function of 0 but the phase correction applied is a function of a. This property can be used to identify aliased peaks. Because a depends upon the spectral width, folded or aliased peaks will change their apparent positions in the spectrum if the spectral width is changed.
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1.0
S(t)
0.5
0.0
–0.5
–1.0 0
2
1
3
4
5
t / ∆t
FIGURE 3.4 The Nyquist theorem. Sine waves with frequencies 0 (solid line) and 0 þ 2fn (dashed line) are illustrated. The two sine waves are sampled digitally at the Nyquist frequency t. The digital representations of the two sine waves are identical (solid dots). As a consequence, the two sine waves appear to have the same frequency in the digitally sampled data, and the high-frequency component is aliased to the lower frequency.
Folding and aliasing are used to advantage in multidimensional NMR spectroscopy to minimize the spectral width in the indirectly detected dimensions (Section 4.3.4.3).
3.2.2 OVERSAMPLING
AND
DIGITAL FILTERS
At first glance, the sampling theorem would appear to present a fatal flaw for Fourier transform NMR. Because the noise in the continuous signal would be expected to be nearly white (i.e., to have an infinite bandwidth), an infinite amount of noise power would be aliased into the frequency-domain NMR spectrum. To avoid this catastrophe, the receivers in NMR spectrometers incorporate analog filters to limit the bandwidth of the signal. All filters have a passband, a stopband, and a transition band. Ideally, the passband would cover the entire spectral range of interest, and the stopband would eliminate any noise from being aliased into the region of interest. If the sampling rate is chosen to be the minimum allowed by the sampling theorem (Section 3.21.), then the passband of the analog filter must be at least as large as the observed
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spectral width to avoid attenuation of the NMR signals. Unfortunately, because the transition band between the pass- and stopbands of any filter is finite, some noise unavoidably will be aliased into the spectrum. The regions near the edges of the spectral window will be most strongly affected. In principle analog filters can be designed that have extremely sharp cutoffs (i.e., extremely narrow transition bands), which would minimize the problem of aliased noise. However, filters have two deleterious effects on the NMR spectrum: (i) the transient response of the filter to the incoming signal distorts the initial points of the FID and (ii) the group delay of the filter retards the phase evolution of the resonance signals and results in frequency-dependent phase errors in the NMR spectrum (see Section 3.3.2.3) (12). As a result, filters with sharp cutoffs result in large, highly undesirable, phase distortions of the NMR spectrum. The simplest and most effective solution is to employ a sampling rate significantly higher than the minimal rate required to avoid aliasing, a technique that is referred to as oversampling. The cutoff frequency of the analog filters is chosen such that any significant noise that is aliased into the spectral window will fall in the wings of the spectrum, outside the spectral region containing NMR signals. In addition, simpler analog filters that have more favorable phase characteristics can be employed, because the criterion for the width of the transition band can be relaxed. In principle, data storage requirements are increased when oversampling is used, because more data points must be acquired to obtain the same digital resolution. Although this was a significant concern in the past, advances in storage technology have all but eliminated the need to compromise acquisition parameters to minimize data sizes. Oversampling also has important consequences for quantization noise. When the NMR signal is fed to the analog-to-digital converter, the continuous range of analog signal amplitudes is transformed to a discrete and finite set of numbers. The minimum step size between distinct numbers is determined by the number of bits in the ADC; the greater the number of bits, the smaller the step size for a given maximum allowable signal amplitude. The current generation of commercial solution-state NMR spectrometers typically employs 16-bit ADCs. Because the continuous-valued analog signal can only be converted to a finite number of distinct digital values, errors are introduced. If the analog signal at a given sampling point n is y(n) and the nearest output level of the ADC is yq(n), then the digitized signal will have a quantization error of e(n) ¼ y(n) yq(n). For most practical purposes, the quantization error can be modeled as a random variable uniformly
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distributed over the appropriate error range , in which is the step size in the ADC. The basic assumptions underlying this model are that (1) the error e(n) is uniformly distributed over the range , (2) the error e(n) and the error e(m) for m 6¼ n are uncorrelated, and (3) the error e(n) is uncorrelated with the ADC input y(n). Although these assumptions do not hold in general, they are valid when the quantization step size is small and the signal sequence y(n) traverses several quantization levels between two successive samples. In this model, the quantization error has the appearance of a noise source that is added to the NMR signal, which is therefore referred to as quantization noise. The effect of quantization noise is expressed in the form of a signal-to-noise (power) ratio, S=N ¼ 10 logðPs =Pn Þ, where Ps is the signal power and Pn is the quantization noise power, which is given by Pn ¼ 2 =12. If the allowed signal range is 0:5Ym yðnÞ 0:5Ym , and a b-bit digitizer is b employed, pthen ffiffiffiffiffi ¼ Ym 2 . In this case, S=N ¼ 6:02b þ 10:8 20 log Ym = Ps . For each additional bit used in the digitizer, the signal-to-quantization noise (power) ratio improves by 6 dB; the last term in this expression also indicates that a decrease in the S/N ratio will occur if the signal strength is not optimized to the full range of the digitizer. High-resolution digitizers are particularly important for sampling high dynamic range NMR data, i.e., data consisting of a superposition of signals of vastly different strength. If a weak signal is thought of as a weak modulation of a very strong signal, then the modulations will be subject to substantial error when digitized if the digitizer step size is on the order of, or larger than, the amplitude of the modulations. For this reason, the signal fed into the ADC should be maximized, while ensuring that the signal will never exceed the allowable range of the ADC for any FID anticipated during the experiment. Once quantization noise has been minimized by making full use of the available bits in the digitizer, additional improvements can be obtained by oversampling. The spectrum of the quantization noise is evenly distributed up to the Nyquist frequency; therefore, an increase in the sampling rate will result in a reduction in the quantization noise as it is spread over the broader frequency range. In quantitative terms, p theffiffiffi quantization root-mean-square noise will be reduced by a factor of 2 for each doubling of the sampling rate. In addition, the analog filter bandwidth is increased in step with the increase in the observed spectral window when oversampling. As a result, the noise level of the FID is increased. If the NMR data have a very high dynamic range and very low system noise (i.e., the noise level of the analog signal fed to the ADCs), then the additional noise fluctuations result in a greater chance
129
3.2 DATA ACQUISITION
that the signal y(n) will traverse at least several quantization levels between two successive samples. This effect thereby improves the validity of the statistical model for quantization noise. Sampling rates of several hundred kilohertz at 16-bit resolution are readily attainable using modern ADCs. To take full advantage of the benefits of oversampling, the ADC must be run at or near its maximum sampling rate. As a result, the region containing the actual NMR signals will typically be only a small fraction of the total spectral width, and a very large number of data points will need to be sampled to maintain the same digital resolution as for the case in which oversampling has not been employed. To avoid the need to manipulate and archive such large data sets, which would be particularly burdensome in multidimensional experiments, a method for reducing the data size while preserving the benefits of oversampling is desired. For this purpose, digital filters are ideally suited (14–16). Generally speaking, a digital filter performs the same function on a digital signal as an analog filter does on an analog signal. The oversampled NMR data can be processed with a low-pass (or in some cases a bandpass) digital filter to suppress all of the noise outside of the spectral region of interest while simultaneously reducing the effective sampling rate by a factor D. The process of reducing the sampling rate by an integer factor D (down-sampling by D) is called decimation. Digital filters provide several advantages over analog filters. The performance of an analog filter can be affected by temporal variations in the characteristics of the electronic components from which it is constructed; a digital filter, on the other hand, is absolutely stable by its inherent nature. A digital filter allows for great flexibility in reconfiguring the digital signal processing operations by simply changing the algorithms employed, whereas reconfiguration of an analog filter requires a new circuit to be constructed. Digital filters allow for the possibility of adaptive filtering, where the filter performance is altered ‘‘on the fly’’ in response to changes in the characteristics of the time-domain signal. An important criterion in designing digital filters for NMR data processing applications is that the filter should not introduce phase distortions within the passband of the filter. This requirement dictates the need for using a so-called finite impulse response (FIR) or nonrecursive filter. A general form for such a filter is given by yðnÞ ¼
ðM1Þ=2 X k¼ðM1Þ=2
cðkÞxðn kÞ,
½3:11
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OF
NMR SPECTROSCOPY
where x(n) is the time-domain input FID data, y(n) is the filtered output data, and the M values (M odd) of c(k) are the coefficients that definite the characteristics of the digital filter. Absence of phase distortion in the passband of the filter requires that the filter coefficients be symmetric about index 0, c(k) ¼ c(k). A vast literature describes methods for designing the filter coefficients; input parameters for the filter design include the width of the passband, the allowable amplitude modulation in the passband, the width of the transition band, and the minimum attenuation level in the stopband. In typical NMR applications, filters with a few hundred coefficients are employed. The convolution calculation, [3.11], can be executed either in the spectrometer host computer or with dedicated digital signal processing hardware in the spectrometer receiver unit. Down-sampling of the FID to reduce the spectral window to a reasonable size can be accomplished simply by applying [3.11] only to every Dth input data point, which results in a decimation factor of D. A significant challenge in implementing the digital filter algorithm [3.11] is posed by the question as to how to ‘‘prime’’ the filter (17). Inspection of [3.11] reveals that for the first (M 1)/2 output values (assuming for the moment that no decimation is performed), the required input data set is incomplete; x(1) is the earliest available data point from the FID, whereas values back to n ¼ ðM 1Þ=2 are required. In many applications of digital filter technology, the problem of missing data is rather inconsequential, because the transient distortion that results can be ignored. However, in NMR applications, faithful reproduction of the early part of the FID is critical for spectral fidelity, especially for minimizing baseline distortions. Therefore, the question of how to prime the filter cannot be ignored. This problem has been addressed in at least two fundamentally different ways in currently available commercial NMR spectrometers. The details of the different methods are proprietary to a large extent; however, the general principle behind each method can be recognized. In one approach, an approximation method is employed to introduce pseudo data in front of the FID to fill in for the missing real data. Various prescriptions exist for defining these pseudo data. In one approach, pseudo data points are calculated using the same window function that was employed in the design of the digital filter coefficients c(k) (17). The general method of filling in the required real data by approximate pseudo data yields a filtered output signal that appears like a normal FID, and which can therefore be subjected to conventional, postacquisition data processing. In the second approach, an alternate form
131
3.2 DATA ACQUISITION
of [3.11] is used: yðnÞ ¼
M1 X
dðkÞxðn kÞ,
½3:12
k¼0
in which the filter coefficients d(k) are symmetric about the midpoint (M 1)/2 and the missing data points for 2 M n 0 are simply substituted with zeros. Consequently, the effective origin of the FID signal is delayed by (M 1)/2 points (prior to decimation) and the step response of the digital filter appears at the beginning of the filtered output signal. As a result, unconventional data processing steps are required to obtain the desired NMR spectrum. The most straightforward procedure is simply to Fourier transform the data as usual, and correct for the time shift in the origin of the FID by applying a large first-order phase correction of the NMR spectrum. The exact value of the required phase correction constant is determined by the details of the digital filter design and can be obtained from a lookup table. If the baseline of the FID is not centered about zero, baseline correction involving use of the step response of the digital filter, which requires exact knowledge of the digital filter coefficients, is required prior to Fourier transformation. The problem arising from the step response of the digital filter is completely analogous to the situation that arises when analog filters are employed. The main practical difference is that the analog filters normally employed are of relatively low order, and thus the duration of the transient response is usually limited to a small number of points at the beginning of the FID. Digital filters, on the other hand, are designed to have very sharp cutoffs, and thus the step response affects a proportionally larger region of the FID. The tail end of the FID will also be affected by the filter step response as the filter runs out of real data. However, this issue is normally inconsequential, because the signal has typically decayed into the noise, and in any case is attenuated by the application of a window function prior to Fourier transformation. To this point the discussion of digital filters has focused on their use to suppress signals above a specified cutoff frequency, i.e., in low-pass applications. However, a major advantage of digital filter technology is the versatility in filter design, and a bandpass filter can be designed as easily as can be a low-pass filter. A bandpass filter sometimes is employed in commercial spectrometers to eliminate quadrature images, as discussed in the next section.
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3.2.3 QUADRATURE DETECTION As has already been described, the frequencies of resonance signals in NMR spectroscopy are measured as offset frequencies from an rf reference frequency. Offset frequencies can be positive (resonance frequency greater than the reference) or negative (resonance frequency less than the reference). Characterization of a sinusoidal signal requires that both the sign and absolute magnitude of the offset frequency be determined. A single detector measures the trigonometric projection of the harmonic signal onto a reference axis. Thus, a single detector might measure the cosinusoidally varying component of the signal. The sign of the frequency cannot be determined from such a data sequence. As is well known, both the cosine and sine components of a harmonic signal must be recorded in order to determine the sign of the frequency. Sampling a signal in a manner such that both the sine and cosine components are recorded is known as quadrature detection. In the earliest days of Fourier transform NMR spectroscopy, singlechannel detection was the norm and the problem of determination of the sign of the offset was solved by placing the rf reference frequency at one edge of the frequency spectrum. In this case, all the resonance offset frequencies have the same sign, so that quadrature detection is unnecessary. Almost without exception, modern NMR spectrometers record the signal in quadrature during acquisition of the FID; consequently, the rf reference can be set in the middle of the frequency spectrum. The latter approach offers some significant experimental advantages: (i) the frequency range that must be excited by the rf pulses is reduced by half, which reduces rf transmitter power requirements by a factor of four; (ii) the required sampling rate is halved, which simplifies data acquisition hardware; and (iii) aliasing of noise into the spectrum is minimized. As illustrated in Fig. 3.5, quadrature detection during acquisition is accomplished by dividing the signal produced in a single coil into two channels. The high-frequency (MHz) signals in the two channels are mixed with rf reference frequencies to generate audio frequency (kHz) signals. The two rf reference frequencies are 908 out of phase; therefore, the output of one channel consists of a cosine modulated signal at the frequency !0 !ref and the other channel consists of a sine modulated signal at the same frequency. The two channels constitute the quadrature pair for frequency discrimination. If the signal produced at the output of the probe and preamplifier is sinusoidally modulated as cos !0t, and for simplicity, the initial signal phase is assumed to be zero without loss of
133
3.2 DATA ACQUISITION 0° Reference
S(t)
Mixer
Filter
Digitizer
Re{S(t)}
Mixer
Filter
Digitizer
Im{S(t)}
Splitter
90° Reference
FIGURE 3.5 Experimental scheme for quadrature detection. The incoming signal recorded by the probe and preamplifier is split into two parallel channels. The signal in each channel is mixed with a reference signal, passed through a low-pass audio filter and digitized. Orthogonal components of the signal are obtained by shifting the relative phase of the reference signals by 908.
generality, then the process of detection can be represented by cosð!0 tÞ
Splitter
) cosð!0 tÞ i cosð!0 tÞ
Mixers
) cosð!0 tÞ cosð!rf tÞ i cosð!0 tÞ sinð!rf tÞ
¼
1 2
cos½ð!0 þ !rf Þt þ 12 cos½ð!0 !rf Þt
12i sin½ð!0 þ !rf Þt þ 12 i sin½ð!0 !rf Þt Audiofilters 1 ) 2 cos½ð!0 !rf Þt 1 ¼ 2 cos½t þ 12 i sin½t ¼ 12 exp½it,
½3:13
þ 12 i sin½ð!0 !rf Þt
pffiffiffiffiffiffiffi in which i ¼ 1 is used as a mathematical mechanism to distinguish between the signals in the two detection channels. The signal in the first (real) channel is modulated as cos t ¼ [exp(it) þ exp(it)]/2; therefore, the frequency-domain spectrum will contain two signals with positive amplitudes at frequencies þ and . The signal in the second (imaginary) channel is modulated as sin t ¼ [exp(it) exp(it)]/ (2i); therefore, the frequency-domain spectrum will consist of a positive amplitude signal at a frequency of þ, and a negative amplitude signal
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OF
NMR SPECTROSCOPY
at . Combining the frequency-domain signals from the two channels as shown by [3.13] cancels the signals at a frequency of and yields a final frequency-domain spectrum containing a single signal with a frequency of þ. The signals present at each step of the detection process are illustrated in Fig. 3.6. If the sensitivities of the two quadrature detection channels are not identical, the signal at will
a
g
b
c
h
i
d
e
j
k
f
l
FIGURE 3.6 Quadrature detection. (a) The FID sampled by a single coil in the probe generates a signal modulated as cos(!0t) that yields (g) a frequencydomain spectrum consisting of signals at !0. To obtain quadrature detection, the signal is split into two channels. (b) The first channel is mixed with a reference signal modulated as cos(!rft) to generate a new FID. (h) The frequency-domain spectrum consists of signals at frequencies (!0 !rf) and (!0 þ !rf). (c) The second channel is mixed with a reference signal modulated as sin(!rft) to generate a new FID. (i) The frequency domain spectrum consists of signals at frequencies (!0 !rf) and (!0 þ !rf); however, because the sine function is odd, the symmetric signals are inverted relative to each other. (d, e) The filters remove the high frequency components of the two signal channels. (j, k) The resulting frequency domain spectrum contains only frequencies (!0 !rf). (f) The two channels are combined to yield a single complex data set. The frequency-domain spectrum, obtained by summing (j) and (k) or by transforming the complex data set (f), contains a single resonance signal at a frequency ¼ !0 !rf. For clarity, all sine-modulated signals have been phase shifted by 908 in the frequency domain, which is equivalent to multiplication by i when forming the complex signal s(t) ¼ sx(t) þ isy(t), in which sx(t) and isy(t) are the outputs of the two quadrature channels.
3.2 DATA ACQUISITION
135
not be identically nulled. The final frequency-domain spectrum will contain a small signal at a frequency of that is called a quadrature image. Cyclically ordered phase sequence (CYCLOPS) cycling frequently is used to reduce quadrature images (Chapter 4, Section 4.3.2.3). Techniques for quadrature detection during the evolution periods of multidimensional NMR experiments are discussed in Section 4.3.4. It should be emphasized here that the detection scheme described here, in which the NMR signal is converted to the audio frequency range in a single mixing step, is an oversimplification of the actual design used in most current spectrometers. In order to optimize the performance of various electronic components, and to reduce the need for broadband frequency response, the mixing scheme is frequently implemented in two steps, with the NMR frequency first being converted down (or sometimes up) to some intermediate frequency (IF), and in a second step the quadrature detection scheme is implemented to convert the IF signal down to the audio frequency range. A very recent development in spectrometer design has eliminated the need for a hardware quadrature detection scheme. This design has become feasible by the development of ultra-fast, high dynamic range analog-to-digital converters and fast digital signal processing hardware. In general terms, the NMR signal is converted down to some intermediate frequency range, such as 20 MHz, and converted directly to a digital signal, instead of being down-converted again to the audio frequency range. Digital signal processing techniques are then employed to down-sample the signal and select the spectral band of interest. Although CYCLOPS phase cycling (Section 4.3.2.3) is very effective at eliminating quadrature images, it requires increasing the number of phase cycle steps by a factor of four, which frequently results in unacceptably long data acquisition times in multidimensional NMR experiments. A clever alternative can be implemented on modern commercial spectrometers equipped with digital filter technology (Section 3.2.2). The basis of this technique is quite simple. The NMR experiment is performed in a normal fashion until the start of the acquisition period. At that point, the receiver reference frequency is shifted past one end of the range of NMR signals, so that all NMR signals have the same sign, relative to the receiver reference. The FID is then recorded, using oversampling to ensure that no aliasing occurs, even with the shifted receiver reference frequency. Thus, any quadrature images that may exist will occur on the opposite side of the receiver reference, in an otherwise blank spectral region. At this point, instead of using a low-pass digital filter, a bandpass filter is used to select the
136
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OF
NMR SPECTROSCOPY
spectral region containing the NMR signals, and reject the region containing the quadrature images. The resulting NMR spectrum is completely free of quadrature images, without any need for increasing the length of the phase cycles. As an added bonus, any DC offset in the FID also is eliminated and baseline correction prior to Fourier transformation is unnecessary. Even though this procedure, referred to as Digital Quadrature Detection (DQD) by one of the commercial instrument manufacturers, is very effective, it should not substitute for careful calibration of the amplitude and phase balance of the quadrature channels in the receiver.
3.3 Data Processing The representation of the NMR signal as a discrete sampling sequence in digital form means that powerful numerical digital signal processing techniques can be used to extract the information content of the signal. The most common processing approach is to convert the time-domain signal into a frequency-domain spectrum by applying a Fourier transform. Various processing algorithms can be applied prior to or after the Fourier transformation to optimize the resulting spectrum. In addition, alternative techniques for spectral analysis, generally first applied in electronic or optical signal processing fields, are increasingly being applied to NMR spectroscopy to obviate the drawbacks to Fourier transformation (Section 3.3.4). A comprehensive monograph of data processing in NMR spectroscopy has been published (18).
3.3.1 FOURIER TRANSFORMATION The Fourier transformation as applied in NMR spectroscopy defines a relationship between one function in the time domain and another function in the frequency domain (19): Sð!Þ ¼ F sðtÞ ¼
Z
Z
1
sðtÞei!t dt,
1 1
½3:14 i2t
SðÞ ¼ FfsðtÞg ¼
sðtÞe 1
dt,
137
3.3 DATA PROCESSING
in which ! ¼ 2. The two functions s(t) and S(!) [or s(t) and S()] are said to form a Fourier transform pair. The inverse Fourier transformations are defined by Z 1 1 1 sðtÞ ¼ F Sð!Þ ¼ Sð!Þei!t d!, 2 1 Z1 ½3:15 1 i2t SðÞe d: sðtÞ ¼ F SðÞ ¼ 1
Fourier transformation and inverse Fourier transformation are linear operations and satisfy the following relationships, F csðtÞ ¼ F sðtÞ , ½3:16 F sðtÞþrðtÞ ¼F sðtÞ þF rðtÞ ,
½3:17
in which c is a complex constant. For completeness, some important theorems concerning Fourier transformations are now listed. Proofs of these theorems can be found in standard texts (19). 1. Similarity, 1 1 F sðatÞ ¼ Sð!=aÞ ¼ Sð=aÞ: jaj jaj
½3:18
F sðt Þ ¼ ei! Sð!Þ ¼ ei2 SðÞ:
½3:19
2. Time shifting,
3. Frequency shifting, F sðtÞei!0 ¼ Sð! !0 Þ, F sðtÞei20 ¼ Sð 0 Þ: 4. Derivative theorem, k d F k sðtÞ ¼ ði!Þk Sð!Þ ¼ ði2Þk SðÞ: dt
½3:20
½3:21
5. Convolution. If the convolution integral of two functions r(t) and s(t) is defined as Z1 rðtÞ sðtÞ ¼ rðÞsðt Þ d, ½3:22 1
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OF
NMR SPECTROSCOPY
then F rðtÞ sðtÞ ¼ Rð!ÞSð!Þ ¼ RðÞSðÞ,
½3:23
6. Correlation. If the correlation integral of two functions r(t) and s(t) is defined as Z1 Corr½rðtÞ, sðtÞ ¼ rðt þ ÞsðÞ d, ½3:24 1
then FfCorr½rðtÞ, sðtÞg ¼ Rð!ÞS ð!Þ ¼ RðvÞS ðvÞ,
½3:25
in which S*(!) and S*() are the complex conjugates of S(!) and S(), respectively. 7. Parseval’s theorem, Z Z1 2 sðtÞ dt ¼ 1
1
Sð!Þ2 d! ¼
1
Z
1
SðÞ2 d:
½3:26
1
These theorems have important practical consequences for NMR spectroscopy. The similarity theorem demonstrates that broadening of a function in one dimension results in narrowing of the function in the other dimension. The time-shifting theorem demonstrates that delaying acquisition (intentionally or due to instrumental delays) in the time domain results in a frequency-dependent phase shift in the frequency domain. The frequency-shifting theorem permits the apparent frequencies in the frequency domain to be shifted after acquisition. The convolution and correlation theorems provide efficient means of calculating the convolution and correlation of two functions. In most cases, the convolution or correlation of two functions is obtained more efficiently by Fourier transforming both functions, multiplying their transforms, and inverse Fourier transforming the product than by direct integration. As discussed later, apodization of the free induction decay in the time domain is performed to convolute the signal in the frequency domain with a more desirable lineshape function. Parseval’s theorem demonstrates that the signal energy is identical in the two domains and implies that the information content of the signal is identical in the time and frequency domains.
139
3.3 DATA PROCESSING
The most important operation for pulsed Fourier transform NMR spectroscopy in liquids is the Fourier transform of the time-domain signal for a damped oscillator, which is given by sðtÞ ¼ I0 exp½ði!0 l0 Þðt þ t0 Þ þ i0
½3:27
for t 0; s(t) ¼ 0 for t 5 0. In [3.27], I0 is the initial signal amplitude, !0 is the frequency, l0 is the decay constant (usually the transverse relaxation rate constant), 0 is the initial signal phase, and t0 is the value of the initial sampling delay. The initial sampling delay may arise from instrumental delays or may be intentionally set. The Fourier transform of s(t) is Z1 Sð!Þ ¼ I0 exp½ði!0 l0 Þt0 þ i0 exp ½ið!0 !Þ l0 t dt 0
1 exp ½ið!0 !Þ l0 t ¼ I0 exp½ði!0 l0 Þt0 þ i0 ið!0 !Þ l0 0 ¼ I0 exp½ði!0 l0 Þt0 þ i0
1 ið!0 !Þ l0
¼ I0 exp½ði!0 l0 Þt0 þ i0
1 ið!0 !Þ l0 ið!0 !Þ l0 ið!0 !Þ l0
¼ I0 exp½ði!0 l0 Þt0 þ i0
ið!0 !Þ þ l0 ð!0 !Þ2 þ l20
¼ I0 exp½ði!0 l0 Þt0 þ i0 ½Að!Þ þ iDð!Þ, ½3:28 in which the absorption, A(!), and dispersion, D(!), lineshapes can be expressed as Að!Þ ¼
Dð!Þ ¼
l20
l0 , þ ð!0 !Þ2
½3:29
l20
ð!0 !Þ : þ ð!0 !Þ2
½3:30
The Lorentzian lineshapes are illustrated in Fig. 3.7. The linewidth of the absorptive Lorentzian is defined as the full-width at half-height (FWHH) and is given by !FWHH ¼ 2l0 or FWHH ¼ l0/.
140
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
a
b
FIGURE 3.7 (a) Adsorptive and (b) dispersive Lorentzian lineshapes. The Fourier transform of an exponentially damped sinusoid generates a frequencydomain signal with real and imaginary components described by the adsorptive and dispersive Lorentzian functions, respectively.
The maximum and minimum cusps of the dispersive lineshape are separated by exactly the absorptive linewidth. Note that for large frequency offsets, the decay of the absorptive Lorentzian lineshape is proportional to 1/(!0 !)2, but the decay of the dispersive Lorentzian lineshape is proportional to 1/(!0 !). Accordingly, absorptive-phase lineshapes yield much more highly resolved NMR spectra and are greatly preferred to dispersive lineshapes. Because the free induction decay is sampled digitally, the experimental frequency domain spectrum is calculated using the discrete Fourier transform X N1 sð jtÞei2jk=N , SðÞ ¼ Sðk=NtÞ ¼ F sð jtÞ ¼
½3:31
j¼0
in which N is the number of (complex) data points, t is the sampling interval, k ¼ N/2, . . . , 0, . . ., N/2, and the digitized signal, corresponding to the continuous signal [3.27], is described by sð jtÞ ¼ I0 exp½ði!0 l0 Þð jt þ t0 Þ þ i0 :
½3:32
141
3.3 DATA PROCESSING
The inverse transform is given by X 1 N1 sð jtÞ ¼ F1 Sðk=NtÞ ¼ Sðk=NtÞei2jk=N : N k¼0
½3:33
The frequency range represented by the Fourier transformed signal is 1/(2t) 1(2t) in discrete steps of ¼ 1/(Nt). In terms of the Nyquist frequency, fn fn Equation [3.31] shows that the discrete Fourier transform of the N input signal points yields N þ 1 frequency domain data points. In fact, S(fn) ¼ S( fn), so that only N unique points are obtained in the frequency-domain function. Most Fourier transformation algorithms provide as output the N points for k ¼ N/2, . . . , N/2 1; i.e., the point S(fn) is not returned. Consequently, the zero frequency point in the frequency domain spectrum is not k ¼ N/2 but rather k ¼ N/2 þ 1. The discrete Fourier transform [3.31] can be expressed as Sð!k Þ ¼ I0 exp½ði!0 l0 Þt0 þ i0
1 exp½Ntði!0 i!k l0 Þ , ½3:34 1 exp½tði!0 i!k l0 Þ
in which !k ¼ 2k/(Nt) and the series [3.31] has been summed using the identity N1 X
xj ¼
j¼0
1 xN : 1x
½3:35
Equation [3.34] is equivalent to [3.28] if t ! 0 and N ! 1 while Ntl0 1. This limit represents quasi-continuous sampling of the time-domain signal until it has completely decayed. If s(jt) is a real function, then Sð½N k=NtÞ ¼
N1 X
sð jtÞei2jðNkÞ=N
j¼0
¼
N1 X
sð jtÞei2jþi2jk=N
j¼0
¼
N1 X
sð jtÞei2jk=N
j¼0
¼ S ðk=NtÞ:
½3:36
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CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
Equation [3.36] demonstrates that unique values of S(k/Nt) are obtained only for k ¼ 0, . . . , N/2 1. Thus, for an N-point real timedomain signal, a unique N/2-point complex frequency-domain spectrum is obtained. Incidentally, S(0) ¼ S*(1/t) and is consequently a real number. The discrete Fourier transform is never calculated numerically by using [3.31]. Direct calculation of the Fourier transformation is an order N2 process, which means that the computational burden increases as the square of the number of data points. Instead, the discrete Fourier transformation is calculated using the Fast Fourier Transformation (FFT) algorithm, which is an order N log2 N process. The time savings afforded by the FFT algorithm are enormous: for a data sequence of 256 complex points, the FFT algorithm is on the order of 32 times more rapid; for a data sequence of 4096 complex points, the FFT is on the order of 300 times more efficient. From the standpoint of the spectroscopist, the use of the FFT algorithm has one important consequence: the number of data points, N, must be an integral power of two (i.e., N ¼ 2m, with m an integer). If the acquired number of data points satisfy 2m–1 5 N 5 2m, then the data sequence must be extended to 2m points. The data may be extended by zero-filling (Section 3.3.2.1) or by linear prediction (Section 3.3.4).
3.3.2 DATA MANIPULATIONS Direct Fourier transformation of a recorded NMR signal rarely yields an optimal frequency-domain spectrum. Instead, a number of digital signal processing techniques are applied prior to (and after) Fourier transformation in order to maximize the information available from the spectrum. 3.3.2.1 Zero-Filling Zero-filling or zero-padding is the process of appending a sequence of zeros to a data sequence prior to Fourier transformation. For example, as described in Section 3.3.1, FFT algorithms require that the number of data points, N, be equal to an integral power of two. If 2m1 5 N 5 2m for an integer m, then, prior to Fourier transformation, zero-filling is used to generate a new data sequence of 2m points in which all points greater than N have the value zero. NMR data obey the causality principle because s(t) ¼ 0 for t 5 0; that is, the signal does not precede its cause (i.e., the pulse sequence). Somewhat surprisingly, as a consequence of causality, the real and imaginary components of the complex frequency spectrum have a
143
3.3 DATA PROCESSING
deterministic relationship relations (20):
embodied
in
the
Im Sð!0 Þ d!0 , 0 ! ! 1 Z 1 1 Re Sð!0 Þ d!0 : Im Sð!Þ ¼ 1 ! !0 1 Re Sð!Þ ¼
Z
Kramers–Kronig
1
½3:37
The mathematical operation indicated is called the Hilbert transform and permits the complex spectrum to be reconstructed given only the real component. The Hilbert transform finds frequent application in NMR spectroscopy. In many cases, particularly in multidimensional spectroscopy, the imaginary portion of the spectrum is discarded to reduce data storage requirements. Subsequently, the imaginary component of the spectrum can be regenerated by using [3.37]. The resulting complex spectrum can be phased normally. However, as noted by Ernst, the Kramers–Kronig relations do not hold for discretely sampled NMR data unless the data sequence is extended by a factor of two by zero-filling, because the periodicity in the signal implicit in the discrete Fourier transform renders the real and imaginary components of the spectrum independent (20). Thus, if 2m1 5 N 2m, then real improvement in the information content of an NMR spectrum is obtained by zero-filling to obtain a sequence of 2m þ 1 data points. Additional zero-filling results only in cosmetic interpolation between data points in the frequency domain; no additional information is obtained. 3.3.2.2 Apodization Direct Fourier transformation of an interferogram rarely yields a spectrum that is satisfactory in all respects. Most commonly, the spectrum will exhibit a number of shortcomings: truncation artifacts, low signal-to-noise ratios, limited resolution, or undesirable peak shapes. The properties of the spectrum can be improved by convoluting the spectrum with a more satisfactory lineshape function: Sf ð!Þ ¼ Hð!Þ Sð!Þ:
½3:38
Because convolution in the frequency domain is equivalent to multiplication in the time domain, common practice is to multiply the interferogram prior to Fourier transformation by the time-domain filter
144
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function, h(t), that represents the Fourier transform of the desired frequency-domain lineshape function, Sf ð!Þ ¼ F hðtÞsðtÞ : ½3:39 This process is variously termed windowing, apodization, or filtering in the time domain (10, 13). The digital signal processing literature contains a wealth of theoretical and empirical studies of apodization; nonetheless, relatively simple approaches have proved of greatest value in NMR spectroscopy. Of the theoretical results, only two will be mentioned: 1. Reduction of truncation artifacts requires that the time-domain signal be smoothly reduced to zero. The resulting frequency-domain lineshape is thereby broadened. The minimum truncation ripple for a given degree of broadening is given by the Dolph–Chebycheff window: ( ) 1 z0 cosð!t=2Þ 1 cos 2ðN 1Þ cos hðtÞ ¼ F , ½3:40
cosh 2ðN 1Þ cosh1 ðz0 Þ in which N þ 1 is the number of sample points, t is the sampling period, z0 ¼ ½cosðt=4Þ1 ,
½3:41
and is the broadening parameter measured in radians/sec. The Dolph–Chebycheff window is not normally used because of the complexity of [3.40]; however, it serves as a benchmark for evaluating the efficacy of other filter functions. 2. Maximum signal-to-noise ratio is obtained in a spectrum if a matched filter function is applied prior to Fourier transformation. The matched filter h(t) is equal to the envelope function of the signal, se(t). The envelope function is the function describing the decay of the signal (stripped of its harmonic content). The acquisition time for the interferogram in NMR spectroscopy is limited to times t tmax. Because the Fourier transformation algorithm assumes that data extends to t ¼ 1, the input signal for Fourier transformation can be represented as the product of the signal (extending to t ¼ 1) and the rectangle function: s0 ðtÞ ¼ sðtÞrðtÞ,
145
3.3 DATA PROCESSING
rðtÞ ¼
1 0
for for
0 t tmax : t 4 tmax
The resulting frequency spectrum is given by S0 ð!Þ ¼ F sðÞrðÞ ¼ Sð!Þ sinc ðtmax !Þ,
½3:42
½3:43
in which sinc(x) ¼ sin(x)/x. As shown in Fig. 3.8, convolution of S(!) with the sinc function produces severe oscillating truncation artifacts. The truncation artifacts can be reduced by apodization with a filter function that reduces the amplitude of the signal smoothly to zero at tmax. Figure 3.8 shows the lineshapes obtained for the cosine, Hamming, and Kaiser filter functions, respectively: hðtÞ ¼ cosðt=2tmax Þ,
½3:44
hðtÞ ¼ 0:54 þ 0:46 cosðt=tmax Þ,
½3:45
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðtÞ ¼ I0 1 t2 =t2max =I0 fg:
½3:46
In [3.46], I0{} is the zero-order modified Bessel function and is a parameter that determines the degree of apodization of the signal. Typical values of are , 1.5, and 2; increasing values of reduce the truncation ripples while increasing the degree of line broadening of the resonance signal. The cosine window perhaps is the window function most frequently applied to truncated NMR signals. Although the Hamming and Kaiser windows are used relatively infrequently, both are expected to more closely approach the performance of the Dolph– Chebycheff window (10). The Kaiser window has the added advantage that can be adjusted to optimize the trade-off between apodization and line broadening in particular circumstances. The Hanning window function is given by hðtÞ ¼ 0:5 þ 0:5 cosðt=tmax Þ
½3:47
and is equivalent, as may be shown using standard trigonometric identities, to the cosine-squared bell window function: hðtÞ ¼ cos2 ðt=2tmax Þ:
½3:48
The performance of the Hanning function is inferior to the cosine, Hamming, and Kaiser window functions; consequently, the cosinesquared bell window function is not recommended for routine use.
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a
f
b
g
c
h
d
i
e
j
FIGURE 3.8 Window functions for apodization. In each case, the time-domain representation of the window function was zero-filled by a factor of two and Fourier transformed to yield the frequency-domain representation. (a) A uniform square wave input yields a (f) sinc function on output. Other apodization functions illustrated include (b, g) cosine bell, (c, h) Hamming function, (d, i) Kaiser window with ¼ , and (e, j) Kaiser window with ¼ 2.
147
3.3 DATA PROCESSING
The signals recorded during an NMR experiment are the sums of exponentially decaying sinusoidal functions. If sufficient data have been recorded to minimize truncation artifacts (tmax 4 3T2), then optimal sensitivity is obtained using the matched exponential filter function hðtÞ ¼ expðltÞ,
½3:49
in which l is the line-broadening parameter. For matched filtering, l l0 ¼ R2 (i.e., 2l is the full-width at half-height of the Lorentzian lineshape measured in s1). Matched exponential filtering has the effect of doubling the linewidth in the frequency domain. Matched exponential filtering has two drawbacks. First, because different resonance signals in the spectrum frequently have different linewidths, l cannot be optimized for all lines simultaneously; thus, h(t) invariably is an approximation to the desired matched filter. Second, the lineshape in the frequency domain is Lorentzian; consequently, the absorption lineshape decays as 1/!2. The resulting tails degrade resolution in the spectrum and hinder accurate integration of peak intensities. Despite these drawbacks, exponential filtering generally can be recommended for application to the FID recorded during the acquisition dimension of NMR experiments because the signal is rarely truncated severely during acquisition. Exponential filtering is applied to indirectly detected evolution periods very infrequently because the interferograms are almost always severely truncated. Certain experiments, such as Correlation Spectroscopy (COSY) and multiple-quantum spectroscopy, yield antiphase peak shapes in the acquisition dimension. In these cases, the time-domain signal is initially zero and is sinusoidally modulated due to evolution of the scalar coupling interaction. Consequently, the exponential filter is not an appropriate matched filter. Instead, phase-shifted sine-bell functions frequently are applied:
t þ t0 hðtÞ ¼ sin , ½3:50 tmax þ t0 in which t0/(tmax þ t0) is the initial phase of the sine bell. As noted previously, the natural lineshape in solution NMR spectroscopy is Lorentzian. The spectrum can be given a new lineshape by use of the filter function, hðtÞ ¼ s0e ðtÞ=se ðtÞ ¼ s0e ðtÞ exp ðltÞ,
½3:51
in which se0 (t) ¼ F –1{S0 (!)}, and S0 (!) is the desired lineshape. Lineshape transformations frequently are used to enhance the resolution in a
148
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spectrum; however, resolution enhancement emphasizes later portions of the free induction decay. As a consequence, truncation artifacts may become more prominent (unless t max is very large or the filter function is suitably apodized) and the signal-to-noise ratio in the spectrum may be reduced. As a corollary, resolution can be enhanced only if signal has been recorded for long times. If the data are truncated, then little resolution enhancement is possible using digital filtering (but see Section 3.3.4). The Lorentzian-to-Gaussian transformation is obtained using
The resulting lineshape is Gaussian with a full-width-at-half-height equal to l'"wFwHH = 2A. g or l'" ])FWHH = }'g/n. The Gaussian lineshape decays exponentially; consequently, the tails do not degrade the resolution as much, compared to the Lorentzian lineshape, and accurate integration of the signal intensity is facilitated. In principle, the lineshape can be arbitrarily narrowed by decreasing Jog; in practice, O,S}, < }'g < 2.0}, will provide adequate resolution enhancement without degrading the signalto-noise ratio drastically. Maximum resolution enhancement with minimization of truncation artifacts can be obtained by using one of the filter functions recommended for removal of truncation artifacts (i.e., the Kaiser or Hamming functions) for se/(t). The main disadvantage to this approach is that signal-to-noise ratios may be severely reduced. Examples of the results of matched filtering and resolution enhancement are given in Fig. 3.9 for a single-resonance signal and in Fig. 3.10 for the one-dimensional I H NMR spectrum of ubiquitin. In both examples, the highest signal-to-noise ratios are obtained for exponential matched filtering. The highest degree of resolution enhancement results in the smallest signal-to-noise ratios. The effect of apodization on peak integrals is considered next. The integrated apodized spectrum is given by
The spectrum will be assumed to consist of a single resonance, so that A(S) represents the integrated area of that peak. Substituting the
149
3.3 DATA PROCESSING
d
a
1.0
e
b
1.9
f
c
1.8
g
0.25
h
FIGURE 3.9 Digital resolution enhancement. (a) The unapodized FID and its (e) Fourier transform are illustrated. (b) A matched exponential window function and the resulting FID are shown together with (f) the resulting frequencydomain spectrum. (c) A Lorentzian-to-Gaussian transformation and the resulting FID are shown together with (g) the resulting frequency-domain spectrum. (d, h) Maximum resolution enhancement is obtained by multiplying the FID with an increasing exponential and apodizing with a Kaiser window function. The signal-to-noise ratio in (e) is arbitrarily assigned a value of unity; relative signal-to-noise ratios for (f, g, h) are shown in the figure.
150
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a
b
c
d
10
9
8 1
7
6
H (ppm)
FIGURE 3.10 Digital resolution enhancement of ubiquitin 1H NMR spectrum. The amide region from 6 to 10 ppm is illustrated for spectra obtained by Fourier transformation of (a) the unapodized FID, (b) an exponential window function, (c) a Lorentzian-to-Gaussian transformation, and (d) maximum resolution enhancement obtained by multiplying the FID with an increasing exponential and apodizing with a Kaiser window function. The window functions are similar to those used in Fig. 3.9. The signal-to-noise ratios for the resonance at 6.52 ppm are (a) 61, (b) 150, (c) 110, and (d) 73.
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3.3 DATA PROCESSING
definition of the convolution function yields Z 1 Z 1 AðSf Þ ¼ Hð!0 ÞSð! !0 Þd!0 d!: 1
½3:54
1
Assuming that the order of integrations can be reversed, Z 1 Z 1 0 0 0 Hð! ÞSð! ! Þ d! d! AðSf Þ ¼ 1
Z ¼
1
1
Z Hð! Þ 0
1
1
Sð! ! Þ d! d!0 : 0
1
The part in square brackets on the last line is just the integral of the function S(!), regardless of the value of !0 (because the integration limits are infinity); thus, the variable !0 can be dropped from the inner integral, which then gives
Z
1 0
AðSf Þ ¼
0
Z
Hð! Þ d! 1
1
Sð!Þ d! 1
¼ AðHÞAðSÞ:
½3:56
The final results states that the integrated area of the apodized resonance peak is equal to the product of the area of the nonapodized peak and the area of the Fourier transform of the window function. Therefore, relative peak integrals within a spectrum are independent of the window function applied (assuming that AðHÞ 6¼ 0). Equation [3.56] also demonstrates that apodization does not alter peak integrals if A(H) ¼ 1, which is the case if h(0) ¼ 1, that is, if the initial value of the apodization function is unity. 3.3.2.3 Phasing The digitized signal represented by [3.27] gives rise to a spectrum given by [3.28] that displays a frequency-dependent phase error. The spectrum can be written as Sð!Þ ¼ I0 exp½ði!0 l0 Þt0 þ i0 ½Að!Þ þ i Dð!Þ ¼ I0 exp½l0 t0 fcos ð!0 t0 þ 0 ÞAð!Þ sinð!0 t0 þ 0 Þ Dð!Þ þ i½sinð!0 t0 þ 0 ÞAð!Þ þ cos ð!0 t0 þ 0 Þ Dð!Þg
½3:57
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by straightforward application of the time-shifting theorem [3.19]. The real part of the spectrum is seen to be a mixture of absorptive and dispersive lineshapes, as shown in Fig. 3.11. The factor I0 exp[l0t0] in [3.57] affects only the intensity of the resonance signal and is not written explicitly in the following equation. Mathematically, the absorptive and dispersive components of the spectrum can be separated by constructing a new data set by using the following prescription, S0 ð!Þ ¼ exp½ið!0 t0 þ 0 ÞSð!Þ ¼ cosð!0 t0 þ 0 Þ Re Sð!Þ þ sinð!0 t0 þ 0 Þ Im Sð!Þ
þ i sinð!0 t0 þ 0 Þ Re Sð!Þ þ cosð!0 t0 þ 0 Þ Im Sð!Þ ¼ Að!Þ þ i Dð!Þ:
½3:58
In practice, [3.58] cannot be used to phase a spectrum containing multiple resonances because !0 is different for each resonance. Instead, the phased spectrum is calculated as S0 ð!Þ ¼ uð!Þ þ ivð!Þ Að!Þ þ i Dð!Þ,
½3:59
uð!Þ ¼ cos½ð!Þ Re Sð!Þ þ sin½ð!Þ Im Sð!Þ , vð!Þ ¼ sin½ð!Þ Re Sð!Þ þ cos½ð!Þ Im Sð!Þ ,
½3:60
in which
and (!) is a frequency-dependent phase correction function that contains one or more parameters that are adjustable to minimize the phase error in the spectrum. In most processing software, (!) ¼ 0 þ 1(! !pivot)/(2 SW) is a linear function of frequency, in which 0 is called the zero-order phase correction, 1 is called the first-order phase correction, and !pivot is called the pivot frequency. The zero-order phase correction is frequency independent, while the first-order phase correction contributes to a linearly frequency-dependent phase. The frequency-dependent phase correction is zero at !pivot and has values of 1 [1/2 þ !pivot/(2 SW)] and 1 [1/2 !pivot/(2 SW)] at the two edges of the spectrum; thus, the total difference in phase correction from one edge of the spectrum to the other is 1. On modern NMR spectrometers, phasing is performed interactively by adjusting 0 and 1 until the lineshapes in the real part of the spectrum are absorptive. An example of the use of zero- and first-order phase corrections is given in Fig. 3.12.
153
3.3 DATA PROCESSING a
f
b
g
c
h
d
i
e
j
FIGURE 3.11 Phase dependence of lineshapes. Real (a–e) and imaginary (f–j) Lorentzian lineshapes are shown for phases of (a, f) 08, (b, g) 458, (c, h) 908, (d, i) 1358, and (e, j) 1808.
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CHAPTER 3 EXPERIMENTAL ASPECTS
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NMR SPECTROSCOPY
a
d
b
e
c
f
FIGURE 3.12 Phase corrections. Real (a–c) and imaginary (d–f) components of two signals of differing phase are shown. (a, d) The upfield resonance has been phased using a zero-order phase correction; however the downfield resonance has a phase error. (b, e) The downfield resonance has been phased using a zeroorder phase correction; however, the upfield resonance has a phase error. (c, f) Both signals have been phased simultaneously by applying zero- and first-order phase corrections.
More detailed analyses of the discrete Fourier transform [3.34] indicate that the baseline of the frequency-domain spectrum displays a nonzero offset and curvature unless the initial signal phase is adjusted to be a multiple of /2 and the sampling delay is adjusted such that t0 ¼ 0, 1/(2SW) or 1/SW (21, 22). For example, if 0 ¼ 0 and t0 ¼ 0, and assuming Ntl0 1, then [3.34] becomes Sð!k Þ ¼ I0
1 : 1 exp½tði!0 i!k l0 Þ
½3:61
155
3.3 DATA PROCESSING
The exponential function can be expanded to second order to yield: 1 tði!0 i!k l0 Þ þ t2 ði!0 i!k l0 Þ2 =2 1 I0 tði!0 i!k l0 Þ½1 þ tði!0 i!k l0 Þ=2 1 I0 ½1 tði!0 i!k l0 Þ=2 tði!0 i!k l0 Þ I0 1 I0 þ : t ði!0 i!k l0 Þ 2
Sð!k Þ I0
½3:62
Within a constant of proportionality, the first term is identical to the desired Lorentzian lineshape, given by [3.28]. However, the second term contributes a baseline offset that depends on the initial signal intensity, I0. This offset is eliminated by multiplying the first point of the FID by a factor of 1/2. As a second important example, if 0 ¼ 0 and t0 ¼ 1/(2SW) ¼ t/2, and assuming Ntl0 1, then [3.34] becomes Sð!k Þ ¼ I0
exp½ði!0 l0 Þt=2 : 1 exp½tði!0 i!k l0 Þ
½3:63
The spectrum described by this expression contains a linear first-order phase error that can be corrected by multiplying by the complex phase factor exp(i!kt/2) to yield exp½ði!0 i!k l0 Þt=2 1 exp½tði!0 i!k l0 Þ 1 ¼ I0 2 sinh½ði!0 i!k l0 Þt=2 I0 1 : t ði!0 i!k l0 Þ
Sð!k Þ ¼ I0
½3:64
The last line of this expression is obtained by expanding sinh(x) to second order. Within a constant of proportionality, the first term is identical to the desired Lorentzian lineshape, given by [3.28]. The necessary zero- and first-order phase corrections are given by 1 ¼ 20 ¼ 360
t0 SW ¼ 180 ,
½3:65
in which the pivot is assumed to be set at the downfield edge of the spectrum (22). Unlike [3.62], no adjustment of intensity of the first point
156
CHAPTER 3 EXPERIMENTAL ASPECTS
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NMR SPECTROSCOPY
a
b
c
FIGURE 3.13 Baseline distortions from phase corrections. (a) An FID recorded with an initial sampling delay of zero generates a properly phased spectrum without baseline distortions. (b) An FID recorded with an arbitrary nonzero initial sampling delay generates a spectrum with baseline distortions after phase correction. (c) An FID recorded with an initial sampling delay adjusted to one-half of the sampling time generates a spectrum without baseline distortions after phase correction.
of the FID is necessary. An example of the baseline distortions observed if the signal phase is not correctly adjusted is given in Fig. 3.13. In some cases, baseline distortions may be present even if sampling delays are properly taken into account. These distortions result from corruption of the first few points in the FID. If the receiver gain is set too high, then the magnitude of the analog signal being detected may exceed the dynamic range of the ADC or earlier stages of the signal amplifiers. Invariably, points at the beginning of the FID are affected and all appear with the same maximum value in the ADC. The FID is then said to be
157
3.3 DATA PROCESSING
‘‘clipped.’’ Fourier transformation of the FID is essentially the Fourier transformation of the superposition of the uncorrupted FID and a square function. The resulting frequency-domain spectrum exhibits ‘‘sinc-wiggles,’’ or truncation artifacts. The second problem, which is referred to as ‘‘baseline roll,’’ arises from the transient response of the audio filters to the incoming signal (12). The digitized signal is the superposition of the uncorrupted FID and a set of points corresponding to the transient response of the filters. In practice, only the first few points of the FID are affected by clipping or transient filter response. An error or distortion in the first sampled point of the FID gives rise to a constant baseline offset in the frequency-domain spectrum. Distortions in the second and subsequent points give rise to increasingly severe baseline effects. For example, distortion of the second point causes curvature in the baseline. Distortion of the third point results in a baseline with one node and two antinodes that resemble the superposition of the spectrum and a sine wave. Examples of these baseline distortions are shown in Fig. 3.14. Sizable reduction in baseline roll can be achieved by adjusting the time between the observed pulse and the start of sampling so that sampling occurs close the crossing point of the
a
b
c
d
FIGURE 3.14 Baseline distortions from a corrupt FID. (a) An uncorrupted FID generates a spectrum without baseline distortion. Baseline distortions are observed if the (b) first point of the FID, (c) first two points of the FID, and (c) first three points of the FID are corrupted. For this figure, corrupted time-domain data points were set identically to zero.
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CHAPTER 3 EXPERIMENTAL ASPECTS
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filter ringing pattern (12). The use of a so-called Hahn echo pulse sequence (Section 3.6.4.2) can alleviate many baseline distortions in 1Hdetected NMR spectroscopy. Linear prediction algorithms also can be used to correct the first few points of the FID to eliminate baseline distortions (Section 3.3.4).
3.3.3 SIGNAL-TO-NOISE RATIO The frequency difference between adjacent points in the frequencydomain spectrum following discrete Fourier transformation is ¼ 1/ (Nt) ¼ 1/tmax. The digital resolution in the final spectrum depends on the total acquisition time, and not on the sampling interval. Increasing the resolution in a spectrum requires that tmax be increased either by recording additional data points or by increasing t. Increasing t has the effect of reducing the spectral width, which may not be feasible. Increasing tmax is productive only if the signals of interest have sufficiently long T2 values. Once the signals have decayed to zero, increasing tmax increases the noise in the spectrum without increasing the resolution between resonance signals. NMR spectroscopy is an insensitive technique and optimization of the signal-to-noise ratio has long been of concern. The initial impetus for the development of pulsed Fourier transform NMR spectroscopy was its increased sensitivity. For a simple one-pulse experiment, the sensitivity, defined as the signal-to-noise ratio per unit acquisition time, is given by (10) hs hi tmax 1=2 1 S ¼ 1=2 , ½3:66 Tc h2 in which hs hi ¼
I0
Z
tmax
2 1 h ¼ tmax
tmax
se ðtÞhðtÞ dt, 0
Z
tmax
h2 ðtÞ dt,
½3:67
0
where Tc is the total time between acquisitions (acquisition time plus the recycle delay), I0 is the initial value of the signal, h(t) is the apodization function, and is the square root of the noise power spectral density. As indicated by [3.66], the sensitivity depends on the ratio of the average
159
3.3 DATA PROCESSING
weighted signal amplitude and the root-mean-square amplitude of the apodization function. If h(t) is chosen to be equal to the envelope function se(t) (see matched filtering in Section 3.3.2.2), then, 1=2 S ¼ I0 s2e ðtmax =Tc Þ1=2 1=:
½3:68
Optimal sensitivity depends, therefore, upon the root-mean-square amplitude of the resonance signal. Because I0 / Ns, and / N1=2 s , in which Ns is the number of transients that are signal averaged, S / N1=2 s . More detailed analyses of the determinants of I0 and yield the result that (23), 1=2 3=2 S / NQ 5=2 B3=2 ðtmax =Tc Þ1=2 , 0 T2 T
½3:69
in which N is the number of nuclear spins, T is the temperature, and Q is the quality factor of the probe coil. Not surprisingly, the greatest sensitivity is obtained for nuclei with large values for and long T2 relaxation times.
3.3.4 ALTERNATIVES
TO
FOURIER TRANSFORMATION
The Fourier transformation is fast, numerically stable, and produces phase-sensitive frequency-domain spectra in a convenient representation. Nonetheless, the Fourier transformation is not without disadvantages; principally, for short data records, the resolution in the frequencydomain spectrum is reduced and truncation artifacts can become large (unless strong window functions are applied, which correspondingly reduces the resolution in the spectrum). As discussed in Chapter 4, the time required to acquire a multidimensional NMR data set is proportional to the number of points acquired in the indirectly detected dimensions. Therefore, data records in the indirectly detected dimensions are almost always truncated, and in the case of three- and fourdimensional data sets, severely so. Accordingly, extensive efforts have been made to develop alternative methods of producing frequencydomain spectra from truncated time-domain interferograms that are more satisfactory than is Fourier transformation. The various methods proposed include linear prediction (24), maximum entropy reconstruction (25), maximum likelihood (26) and Bayesian analysis (27); of these, linear prediction and maximum entropy reconstruction are the most frequently utilized. The review by Stephenson provides a detailed
160
CHAPTER 3 EXPERIMENTAL ASPECTS
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NMR SPECTROSCOPY
introduction to both linear prediction and maximum entropy methods in NMR spectroscopy (28). 3.3.4.1 Linear Prediction Linear prediction algorithms model the time-domain signal (the interferogram) as (13) M X sðktÞ ¼ am sð½k mtÞ þ "m ,
½3:70
m¼1
in which M is the prediction order or the number of signal poles, "m is the prediction error (distinct from the random noise), am is the mth linear prediction coefficient, and k ¼ 0, . . . , N – 1. In essence, the kth data point is modeled as a linear function of the previous M points. Although [3.70] could be postulated as a description of an arbitrary signal, a close connection exists between [3.70] and the damped sinusoidal signal [3.27]. Linear prediction algorithms attempt to find the set of coefficients am that minimizes, in the least-squares sense, the prediction error. Once the linear prediction coefficients have been determined, the frequency domain spectrum can be calculated directly from the prediction coefficients or, more commonly, the linear prediction results can be used to calculate an extension to the interferogram prior to conventional processing by Fourier transformation. Linear prediction algorithms can predict the future behavior of a sinusoid over many periods; in contrast, polynomial expansions frequently fail to extend a sinusoid for more than a period accurately. Implicit in the formulation of the linear prediction method are a number of important issues: 1. The maximum number of resonance signals that can be modeled is given by the prediction order 2. Linear prediction algorithms generally require M N/2. 3. The optimal prediction order is difficult to determine rigorously. 4. Random noise is not incorporated into the linear prediction model. As a consequence, linear prediction methods generally work best for data with relatively high signal-to-noise ratios. 5. The FID usually cannot be extended by more than a factor of two without severely distorting the signal lineshapes. 6. Two- (and higher) dimensional data sets are processed by linearly predicting each (t1, !2) interferogram independently. Differences in numerical results from interferogram to interferogram can distort two-dimensional lineshapes.
3.3 DATA PROCESSING
161
The most common use for linear prediction methods is to extend the time-domain data for 13C and 15N resonance signals detected during indirect evolution periods of multidimensional NMR experiments. If all 1H dimensions are Fourier transformed first, then the heteronuclear dimensions generally contain relatively few signals in each interferogram, which simplifies the linear prediction problem. In some experimental situations, notably constant-time experiments or extremely truncated data, the signal interferogram is undamped by relaxation (limited damping by inhomogeneity broadening can be corrected by multiplication with an increasing exponential). If the phase has been properly adjusted by adjusting the initial value of the evolution period to 0 or 1/(2SW) (Section 3.3.2.3), then the complex signal satisfies the relationship s(t) ¼ s*(t), in which the asterisk indicates complex conjugation. This relationship can be used to generate a data sequence of length 2N from a sequence of length N in which the data points extend from N, Nþ1, . . . , N 1, N (if the initial value of t is 0, then the extended sequence contains 2N 1 points). The longer data sequence can be used as the input for linear prediction, after which the points from N to 1 are discarded. This technique has been called mirror image linear prediction and frequently allows higher resolution estimates of the frequency-domain spectrum to be obtained (29). At the present time, the algorithms based on singular value decomposition, such as the linear predictive and hyperbolic singular value decomposition (LPSVD and HSVD) algorithms, appear to be the most robust (30). An example in which the HSVD algorithm was used to linear predict the t1 interferograms in a two-dimensional constant-time 1 H–13C HSQC spectrum of ubiquitin (Chapter 7, Section 7.1.3.1) is shown in Figs. 3.15 and 3.16. Linear prediction also can be used to correct the magnitude of incorrectly sampled points (for example, the initial points of the FID may be corrupted by pulse breakthrough and filter response; see Section 3.3.2.3). For these applications, computationally less demanding algorithms, such as the Burg or Levinson–Durbin methods, are satisfactory (28, 31). 3.3.4.2 Maximum Entropy Reconstruction Maximum entropy methods reconstruct the frequency-domain NMR spectrum directly (no subsequent Fourier transformation is necessary) by determining the spectrum S[k/(Nt)] for k ¼ 0, 1, . . . , N 1 that maximizes the entropy
162
CHAPTER 3 EXPERIMENTAL ASPECTS
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NMR SPECTROSCOPY
a
e
b
f
c
g
d
h
FIGURE 3.15 Linear prediction using the HSVD algorithm. A constant-time 1 H–13C HSQC spectrum of ubiquitin was linear predicted in the t1 (13C) dimension following conventional processing in the acquisition dimension. The t1 interferogram at an F2 (1H) shift of 4.52 ppm was used for illustration. Shown are the (a) complete 240-point interferogram, (b) truncated 64-point interferogram, (c) truncated interferogram extended to 128 points by HSVD linear prediction, and (d) truncated interferogram extended to 240 points by mirror-image HSVD linear prediction. The spectra obtained by Fourier transformation of interferograms (a)–(d) are shown in (e)–(h).
function W¼
N1 X
S ½k=ðNtÞ ln S½k=ðNtÞ
½3:71
½sð jtÞ s^ð jtÞ2 = 2j ¼ M,
½3:72
k¼0
subject to the constraint C¼
M1 X j¼0
3.3
163
DATA PROCESSING
\::,
ce e
(\IE <00. -3 0
~
0
d
><;t Lf)
L........-
4.6
4.2
4.6
~
...._.i-~
4.2
1H (ppm) 3.l6 Linear prediction of an HSQC spectrum. A constant-time 'H- 13 C HSQC spectrum of ubiquitin was linear predicted in the II (13C) dimension following conventional processing in the acquisition dimension. Shown are regions of the two-dimensional spectra obtained from (a) 240 I I points, (b) 64 I I points, (c) 64 I, points extended to 128 points by HSVD linear prediction, and (d) 64 I, points extended to 240 points by mirror-image HSVD linear prediction. FIGURE
in which M is the number of experimental time domain points, s(j!::>.t) is thejth experimental time domain point, s(j!::>.t) = g;--J {S[k/CN!::>.t)]j is the inverse Fourier transform of the reconstructed spectrum, and CJj is the experimental noise level in the time domain data. The entropy and constraint equations can be combined into a single objective or cost
164
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
function, V ¼ W þ l C,
½3:73
in which l is a Lagrange multiplier. The objective function is maximized by iteratively refining an initial guess for S[k/(Nt)] using standard numerical algorithms. In essence, maximum entropy reconstruction amounts to the following prescription: from the (possibly infinite) set of candidate frequency-domain spectra whose time-domain interferograms reproduce the experimental interferogram within experimental uncertainty as described by the constraint function [3.72], select the single spectrum for which the entropy defined by [3.71] is a maximum. The preceding formulations have a number of important corollaries: 1. Unlike linear prediction algorithms, the functional forms of the NMR signals (damped sinusoids) are not assumed a priori by the maximum entropy methods; however, experimental uncertainty is considered explicitly by maximum entropy algorithms. 2. The spectral estimates are constrained to be positive by [3.71]. Negative intensity in NMR spectra is treated by representing the spectrum as S½k=ðNtÞ ¼ S þ ½k=ðNtÞ S ½k=ðNtÞ,
½3:74
in which S þ[k/(Nt)] and S [k/(Nt)] represent subspectra with positive and negative intensities (32). 3. Because M can be less than N, a high-resolution frequencydomain spectrum can be calculated from a limited number of data points. 4. Maximum entropy reconstruction can be performed for multidimensional data in straightforward fashion by generalizing [3.71] and [3.72] to include multiple summations (one summation for each dimension). 5. Justification of the use of [3.71] for NMR spectroscopy is not straightforward. Laue et al. discuss the rationale for using [3.73] in spectral reconstruction (33). Maximum entropy reconstruction of the truncated t1 interferograms in a two dimensional constant-time 1H–13C HSQC spectrum of ubiquitin (Chapter 7, Section 7.1.3.1) is shown in Fig. 3.17. Examination of Figures 3.15–3.17 demonstrate that for severely truncated data, both linear prediction and maximum entropy reconstruction can improve the resolution of the frequency-domain spectrum compared with Fourier
165
3.4 PULSE TECHNIQUES
b
62
13
C (ppm)
54
a
4.6
4.2 1H
(ppm)
FIGURE 3.17 Maximum entropy reconstruction of an HSQC spectrum. A constant-time 1H–13C HSQC spectrum of ubiquitin was processed by maximum entropy reconstruction in the t1 (13C) dimension following conventional processing in the acquisition dimension. Reconstruction was based on 64 t1 points. Shown are (a) a one-dimensional slice at an F2 shift of 4.52 ppm and (b) the same region of the two-dimensional spectrum shown in Fig. 3.16.
transformation. In neither case are the resulting spectra as highly resolved and distortion-free as is the spectrum obtained by Fourier transformation of the untruncated interferogram.
3.4 PulseTechniques Achieving optimal results for NMR spectroscopy of large complex macromolecules requires careful attention to experimental details. An extensive set of techniques has been developed in NMR spectroscopy to compensate for the deficiencies of simple rf pulses and to manipulate spin systems in particular ways. A number of these techniques are described in the following sections.
3.4.1 OFF-RESONANCE EFFECTS In practical situations, the nuclei in a molecule will possess a range of chemical shifts. Because an rf pulse can be applied at only one frequency (the transmitter frequency, !rf), some nuclei will have
166
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
resonant frequencies that are close to !rf while other nuclei will have resonant frequencies that are very different from !rf. Consequently, all nuclei cannot be expected to respond to the effect of an rf pulse in an ideal fashion. Those nuclei on-resonance with the pulse will respond ideally; other nuclei, near resonance with !0 !rf, will precess around an effective field, Br, that will be similar to that of B1. Further and further off resonance, where !0 6¼ !rf and the magnitude of the offset increases, the effective field, Br, will be very different from B1. In many cases, the offset of some nuclei may be comparable to the strength of the pulse measured in frequency units (!1 ¼ B1). In these circumstances, the effective field is tilted away from the x–y plane toward the z-axis (see Fig. 1.2). For the off-resonance case, a pulse of y-phase applied to equilibrium z-magnetization yields ([1.34]) Mx ¼ M0 sin sin, My ¼ M0 ð1 cos Þ sin cos, Mz ¼ M0 ðcos2 þ cos sin2 Þ,
½3:75
in which and are defined by [1.23] and [1.21]. In practical terms, two effects must be noted that yield both phase and intensity anomalies for resonance lines that have large offsets from the transmitter frequency. The phase anomalies can be described in terms of a phase shift, , where tan ¼
My ð1 cos Þ cos : ¼ sin
Mx
½3:76
ð1 cos Þ sin
: sin
!1
½3:77
Using [1.21], tan ¼
This equation is a convenient form in which to view the phase problems that arise because the dependence on resonance offset is indicated directly. The amplitude of the resonance signal also changes with offset as given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mx,y ¼ Mx2 þ My2 : ½3:78 The magnetization, phase angle, and effective rotation angle for an off-resonance 908 pulse are illustrated in Fig. 3.18; the resulting
167
3.4 PULSE TECHNIQUES 1.0
a
Amplitude
0.5
0.0
–0.5
–1.0 400
b
Angle (°)
300
200
100
0 0
1
2
3
4
Ω / g B1
FIGURE 3.18 Off-resonance effects for 908 and 1808 pulses. (a) Shown are (—) the magnitude of the transverse magnetization, Mx,y ¼ ðM2x þ M2y Þ1=2 , following a nominal 908 pulse and (- - -) the Mz magnetization following a nominal 1808 pulse. (b) Shown are the (—) effective rotation angle and (- - -) phase of the transverse magnetization following a nominal 908 pulse.
lineshapes are shown in Fig. 3.19. As the offset, , increases, the amplitude of the transverse magnetization remains approximately constant and the phase of the transverse magnetization increases linearly until the offset equals the rf field strength, ¼ !1. At larger offsets,
168
CHAPTER
3 EXPERIMENTAL ASPECTS OF NMR SPECTROSCOPY
1.0
.'!l
'c
0.5
::J
?:
~
:0
~
0.0
-0.5
a
2
3
4
Q/ y81 FIGURE 3.19 Resonance Iineshapes for off-resonance 90° pulses. The lineshapes are calculated using the magnitudes and initial phases of the resonance signals given in Figure 3.18.
the signal amplitude decreases until it disappears. At this point, the bulk magnetization vector rotates about the effective field such that it is aligned along the z-axis once more at the end of the pulse. At even larger offsets, the signal amplitude oscillates from positive to negative and the magnitude of the signal decreases. For offsets Q ::: w" nearly ideal results are obtained for a 90° pulse if the linear phase dependence is compensated. Expanding [3.77] in Taylor series about Q = 0 yields
[3.79] in which "90 is the length of the on-resonance 90° pulse. Thus, an offresonance 90° pulse can be treated as an ideal pulse followed by an evolution period given by
[3.80] An alternative derivation of this result begins with [2.120]. For Qlwl, and ex = nl2 + 8, where 8 = n (1 - sin 8)/(2 sin 8). Using these expressions, the propagator for an Q
> 0, 8 = nl2 - S, where tanCS) =
169
3.4 PULSE TECHNIQUES
off-resonance pulse with x-phase is Rx ð , Þ ¼ Ry ðÞRz ð ÞR1 y ð Þ 1 ¼ Ry ð ÞRy ð=2ÞRz ð=2ÞRz ð"ÞR1 y ð=2ÞRy ð Þ
¼ Ry ð ÞRx ð=2ÞRx ð"ÞR1 y ð Þ ¼ Rx ð=2ÞRz ð ÞRx ð"ÞR1 y ð Þ,
½3:81
and the identity [2.121] has been used. If and " are small, which will be the case if is not too large, then the rotation operators that depend on these values commute with each other. Thus, the order of the operators [3.81] can be rearranged to yield Rx ð , Þ ¼ Rx ð=2ÞRz ð ÞRx ð"ÞR1 y ð Þ Rx ð=2ÞR1 y ð ÞRx ð"ÞRz ð Þ Rz ð ÞRx ð=2ÞRx ð"ÞRz ð Þ Rz ð ÞRx ð ÞRz ð Þ:
½3:82
The off-resonance pulse is approximately described as an on-resonance pulse with x-phase and rotation angle (which will be denoted x) preceded and followed by a z-rotation with an angle . The rotation angle =!1 ¼ 2 90 = ¼ corr . This analysis yields a slightly more general result: the off-resonance pulse is approximated by the sequence corr– x– corr. The magnitude of the z-rotation is seen to be identical to the angular deviation between the effective field and the x-axis. In contrast, Fig. 3.18 demonstrates that a 1808 inversion pulse has a highly nonideal off-resonance excitation profile. Magnetization will not be fully inverted, and a notable amount of transverse magnetization will be generated. Because nonideal effects of 1808 pulses cause significant problems in NMR spectra, considerable effort has been devoted to improving the performance of 1808 pulses by use of EXORCYCLE phase cycles (Chapter 4, Section 4.3.2.3), composite pulses (Section 3.4.2), and static magnetic field gradients (Chapter 4, Section 4.3.3.2). The property of being able to create a null in the frequency excitation profile of a pulse can be used to advantage in many experiments, including water suppression methods and in some multidimensional NMR experiments. In some instances one set of spins must be excited while leaving others unperturbed. Using [3.75], a null in the excitation profile for a pulse with on-resonance rotation angle ¼ 908 is
170
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
achieved at an offset from resonance equal to pffiffiffiffiffi ¼ 15!1 :
½3:83
For a 1808 pulse, a null is achieved when pffiffiffi ¼ 3!1 :
½3:84
Significant off-resonance effects can be observed even for nonresonant spins with !1 if rf pulses are applied to coherent states of the density operator, as opposed to thermal equilibrium magnetization. Offresonance effects can lead to phase errors and frequency shifts in multidimensional NMR experiments, particularly when long, lowamplitude continuous or shaped rf pulses are employed. Off-resonance, or nonresonant, effects, are commonly called Bloch–Siegert shifts. The phrase ‘‘Bloch–Siegert shifts’’ of the second kind is used to distinguish these effects from Bloch–Siegert shifts (of the first kind) arising from the counter-rotating component of the rf field [1.15] (34). The phase shift of transverse magnetization during an off-resonance pulse train arises from a z-rotation and is given by (35, 36) NR ¼ !1 ðtÞ2 p =ð2Þ, ½3:85 in which the angle brackets represent an average over the pulse train. If p is not a constant, but varies with the evolution period of a multidimensional NMR experiment, then the off-resonance effect is manifested as a frequency shift, !NR, given by ½3:86 !NR ¼ !1 ðtÞ2 =ð2Þ: In addition to the nonresonant phase shift, a rotation occurs about the axis along which the rf field is applied and is given by NR ¼ NR !1 ðtÞ =: ½3:87 The nonresonant rotation is smaller than the phase shift by a factor of at least !1/ and is eliminated for amplitude- or phase-modulated pulse trains with a net rotation angle of zero [h!1(t)i ¼ 0], such as decoupling sequences. Nonresonant effects are common when performing frequencyselective homonuclear decoupling. For example, 13CO and 13C spins can be decoupled by applying a selective 1808 pulse to the 13CO or C
spins at the midpoint of the desired evolution period or by applying a frequency-selective decoupling sequence, such as SEDUCE-1, during the
3.4 PULSE TECHNIQUES
171
evolution period. These techniques will be illustrated for a t1/2– 1808(CO)–t1/2 evolution period for 13C magnetization. For a weak rectangular 1808 pulse applied on-resonance with the 13CO spins, h!1(t)2i ¼ !12 and the nonresonant phase shift of the aliphatic spins is given by [3.85] pffiffiffi ½3:88 NR ¼ !21 p =ð2 Þ ¼ !1 =ð2 Þ ¼ =ð2 3Þ ¼ 52 , in which !1 ¼ /31/2, is the frequency difference between 13CO and C spins [3.84], and the last equality obtains for ¼ . In this special case, the exact nonresonant phase shift can be calculated directly. Transverse 13C magnetization rotates by 2 around the effective field during the off-resonance pulse. Consequently, transverse magnetization does not acquire any additional phase during the p duration of the pulse. If however, the pulse were applied with zero amplitude, transverse magnetization would simply precess around the z-axis and acquire a phase shift p. The difference between the two results is (37) pffiffiffi ½3:89 NR ¼ p ¼ =!1 ¼ 3 ¼ 48 : 13
The slight difference between [3.88] and [3.89] arises because the condition [3.84] only minimally satisfies the requirement !1 used to derive [3.85]. Shaped, rather than rectangular, pulses frequently are used in order to further minimize perturbations of the aliphatic 13C spins during the application of 1808 pulses to the 13CO spins. A single lobe of the sinc shape most commonly is employed (Section 3.4.4). In this case, the nonresonant phase shift of the aliphatic spins is given by 2 2 !1 ðtÞ p 0:4514 !max p 1 NR ¼ ¼ ¼ 39:8 , ½3:90 2ð Þ 2
is the peak amplitude of the rf field during the pulse, and in which !max 1 the last equality is obtained assuming !max ¼ =31=2 . The sinc pulse has 1 max an average field strength h!1 ðtÞi 0:589!1 ; thus, the rf field strength and pulse length of a sinc pulse are calculated by multiplying either the pulse length or the rf field strength of a 1808 rectangular pulse by a factor of 1/0.589 ¼ 1.698. Four techniques have been proposed for compensating for the consequent nonresonant phase shifts. In the simplest approach, the phase errors are corrected during processing of the acquired spectrum. In the second approach, the phases of the aliphatic 13C pulses subsequent to the evolution period are shifted by NR to account for the nonresonant
172
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
phase shift of the aliphatic spins. In the third approach, compensatory rf fields are produced by modulating the amplitude of the selective pulse with a cosine function, cos( t), for t ¼ 0 to p (35). For a rectangular pulse, the resulting rf field is (assuming x-phase) given by [1.15] Brf ðtÞ ¼ 2B1 cosð tÞ cosð!rf tÞi ¼ B1 cos½ð!rf þ Þti þ B1 cos½ð!rf Þti, ½3:91 in which !rf is the transmitter frequency of the 13C spins. Because the cosinusoidal modulation generates two effective rf fields, the amplitude of the B1 field must be doubled (i.e., !1 ¼ 2 /31/2). The component of the field with frequency !rf þ is resonant with the 13CO spins and generates the 1808(CO) pulse. The component of the field with frequency !rf is not resonant with any spins of interest. The nonresonant phase shifts of the 13C spins caused by the two field components are equal and opposite; therefore, the net phase shift is zero at the frequency !rf. Near resonance, the nonlinear off-resonant phase shift [3.85] is converted to a linear phase shift: NR ¼ !1 ðtÞ2 p =2 , ½3:92 in which is the resonance offset of the 13C spins. Because [3.85] depends nonlinearly on the resonance offset, nonresonant phase shifts can be corrected by these three approaches only if the spins of interest resonate in a narrow frequency range near . In the fourth approach, compensatory rf pulses are incorporated into the pulse sequence. For example, if the simple t1/2–1808(CO)–t1/2 evolution period is replaced with the sequence t1 =2180 ðCOÞt1 =2180 ð13 CÞ180 ðCOÞ,
½3:93
in which is a short fixed delay, then the 1808(13C) pulse applied to the aliphatic spins serves to refocus the phase evolution occurring during the two flanking 1808(CO) pulses. This method does not require that the spectral region of interest be narrow and consequently is generally applicable. In addition, the initial sampling delay can be adjusted to zero by setting t1(0) ¼ (Section 3.3.2.3).
3.4.2 B1 INHOMOGENEITY The inhomogeneity of the rf field describes the variation in the amplitude of the B1 field as a function of position in the sample.
173
3.4 PULSE TECHNIQUES
Thus, B1 ðrÞ ¼ B1 ð0Þ þ B1 ðrÞ,
½3:94
in which B1(0) is the amplitude of the B1 field at the center of the sample, and B1(r) characterizes the inhomogeneity of the field. The net evolution of the Iz operator during a pulse of length t averaged over the sample volume, V, is given by
Iz
½B1 ðrÞtSy
) Iz
Z
Z cosðB1 ðrÞtÞdr=VþIx V
sinðB1 ðrÞtÞdr=V V
. Z Z ¼Iz cosðB1 ð0ÞtÞ cosðB1 ðrÞtÞdrsinðB1 ð0ÞtÞ sinðB1 ðrÞtÞdr V V
V
. Z Z þIx sinðB1 ð0ÞtÞ cosðB1 ðrÞtÞdrþcosðB1 ð0ÞtÞ sinðB1 ðrÞtÞdr V V
¼ Iz cosðB1 ð0ÞtÞþIx sinðB1 ð0ÞtÞ
V
Z cosðB1 ðrÞtÞdr=V V
Iz sinðB1 ð0ÞtÞIx cosðB1 ð0ÞtÞ
Z sinðB1 ðrÞtÞdr=V:
½3:95
V
Because the cosine and sine functions are oscillatory, the integrals in [3.95] tend to zero, provided that B1(r) is nonuniform throughout the sample and t is sufficiently long to ensure that B1(r)t is spatially randomized between 0 and 2. For modern NMR probes, |B1(r)/ B1(0)| 10–20% and significant dephasing of a signal can be achieved in a few milliseconds. Spin lock purge pulses dephase coherences orthogonal to the rf field (Iy or Iz coherences) while preserving the coherence locked along the rf field (Ix coherence). Purge pulses can be used to eliminate artifacts in NMR spectra arising from undesired operators orthogonal to the operator of interest (see Section 7.2.4.3 for an example). However, product operators containing two spin operators orthogonal to the rf field (e.g., homonuclear 2I1zI2z or 2I1zI2y operators) contain components that behave as zero-quantum coherences in the tilted rotating reference frame of the rf field (see Sections 5.2.3 and 5.4.3 for additional discussion
174
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
of tilted rotating reference frames) and consequently are dephased inefficiently by the inhomogeneous rf field (38, 39).
3.4.3 COMPOSITE PULSES The usual goal of applying an rf pulse to the sample is to achieve a rotation of coherences around a defined axis by a specified angle. The actual performance of an rf pulse can be degraded by any number of factors, including finite rise times of the pulse, amplitude droop during the pulse, phase instability during the pulse, spatial inhomogeneity of the rf field across the sample, resonance offset effects, or relaxation during the pulse. Certain of these effects are minimized by improvements in NMR spectrometer hardware, such as digital frequency synthesis. Some of these effects can be reduced by phase cycling and signal averaging. Other effects can be compensated by replacement of a single rf pulse by an extended pulse train that is designed to achieve the same ideal rotation as the single pulse, but which is more resistant to nonideal influences. These pulse trains are called composite pulses and have been extensively developed for reduction of the effects of rf field inhomogeneity and resonance offset. While composite pulses have been designed to meet a large number of objectives, for multidimensional NMR spectroscopy of biomolecules, three objectives are of the most relevance: 908 rotation of z-magnetization into the transverse plane, inversion of z-magnetization, and refocusing of transverse magnetization. In general, the design of composite pulses that compensate for both rf inhomogeneity and resonance offset effects is more difficult than is the design of composite pulses that compensate for one of the two effects individually. Shorter composite pulse trains generally give less satisfactory compensation; however, long composite pulse trains may be impractical (due to amplifier droop and evolution of the spin system during the extended pulse sequence). The initial state of the magnetization is important to the design of the composite pulse; a composite 1808 pulse that produces accurate inversion of z-magnetization may not necessarily produce accurate refocusing of transverse magnetization. The detailed mathematical derivations of composite pulses will not be presented in this section; the interested reader is referred to the comprehensive review by Levitt (40). The design of composite pulses is closely related to the design of spin decoupling techniques discussed in Section 3.4.5. Numerous composite pulses have been developed for effecting 908 and 1808 rotations. Composite 1808 pulses are utilized in multidimensional NMR spectroscopy much more frequently than are
175
3.4 PULSE TECHNIQUES
composite 908 pulses because of the nearly ideal performance of a 908 pulse (Fig. 3.18). The most successful composite pulses are designed to invert z-magnetization and refocus scalar coupling evolution. Composite pulses often do not perform appreciably better than a single 1808 pulse for refocusing chemical shift evolution because many composite pulses introduce offset-dependent phase shifts of the refocused coherences that are difficult to compensate in multidimensional NMR experiments (40). Early efforts at designing composite pulses relied on some combination of guesswork/insight, average Hamiltonian theory, Fourier analysis, and numerical integration of the Bloch equations. Increasingly, new composite pulse sequences are obtained by computer optimization of initial trial pulse sequences generated by the aforementioned techniques. Calculating the performance of a particular composite pulse is considerably simpler than is designing the pulse sequence. A windowless composite pulse consisting of a series of N consecutive pulses without intervening delays is described by P~ ð Þ ¼
N Y
Pi ð i Þ,
½3:96
i¼1
in which i, and i are the pulse phase and nominal (on-resonance) rotation angle of the ith pulse, respectively. The net rotation induced by the composite pulse is given by ~ ð Þ ¼ R
N Y
Rz ði ÞRy ði ÞRz ð i ÞRy ði ÞRz ði Þ,
½3:97
i¼1
in which i, i, and i represent the values of the phase, tilt angle, and effective rotation angle for the ith pulse element. The rotation matrices are given by [1.35]. The effective rotation angle and tilt angle are described by [1.23] and [1.21]. The magnetization following the composite pulse is ~ ð ÞMð0 Þ, Mð0þ Þ ¼ R
½3:98
in which M(0) and M(0þ) are the magnetizations before and after the pulse, respectively. Various strategies have been devised to develop composite pulses that have reduced sensitivity to off-resonance effects, but this normally comes at the expense of poorer performance in the presence of rf inhomogeneity. The inverse is also true: composite pulses optimized for tolerance to rf inhomogeneity generally exhibit increased sensitivity to resonance offset.
176
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
As an example, product operator calculations indicate that the z-magnetization obtained from a simple, on-resonance pulse of length 2 1808 is 2 x
Iz ! Iy sin 2 þ Iz cos 2
¼ Iy sin 2 Iz ð1 2 cos2 Þ:
½3:99
The pulse sequence x(2 )y x with 2 1808 is a commonly used composite 1808 pulse. The evolution of Iz is given by,
x
Iz ! Iy sin þ Iz cos
2 y
! Ix cos sin2 Iy sin þ Iz cos cos2
x
! Ix cos sin2 Iy sin2 ð1 þ cos2 Þ=2 Iz ðsin2 cos2 cos2 Þ ¼ 2Ix cos2 sin 2Iy cos3 sin Iz ð1 2 cos4 Þ:
½3:100
Because |cos |51, cos4 5 cos2 and the composite pulse sequence yields more accurate inversion of Iz than does a simple pulse with rotation angle 2 1808. Therefore, the composite pulse is compensated for rf inhomogeneity or ‘‘mis-setting’’ of the pulse lengths. An approach similar to that used in deriving [3.81] is useful in deriving a propagator for the composite pulse. The present calculation assumes resonance offsets are negligible and considers only variation in B1. The rotation angle ¼ /2 þ ", in which " ¼ ð=2Þð!1 =!01 1Þ, and !01 is the nominal value of the rf field strength corresponding to a /2 rotation. The propagator for the composite pulse is first premultiplied by U0U01, where U0 is the propagator for the ideal (/2)x()y(/2)x rotation: U ¼ Rx ð ÞRy ð2 ÞRx ð Þ ¼ U0 U1 0 Rx ð ÞRy ð2 ÞRx ð Þ ¼ Rx ð=2ÞRy ðÞRx ð=2ÞRx ð=2ÞRy ðÞRx ð=2ÞRx ð ÞRy ð2 ÞRx ð Þ ¼ Rx ð=2ÞRy ðÞRx ð=2ÞRx ð=2ÞRy ðÞRx ð=2Þ Rx ð=2ÞRx ð"ÞRy ðÞRy ð2"ÞRx ð=2ÞRx ð"Þ ¼ Rx ð=2ÞRy ðÞRx ð=2ÞRx ð"ÞRy ð2"ÞRx ð"Þ ¼ Rx ðÞRz ðÞRx ð"ÞRy ð2"ÞRx ð"Þ:
½3:101
177
3.4 PULSE TECHNIQUES
When " is small, then the rotation operators that depend on " commute with each other and U Rx ðÞ Rz ð 2"Þ Rx ðÞRz !1 =!01 Rz !1 =2!01 Rx ðÞRz !1 =2!01 Rz !1 =!01 Rx ðÞ:
½3:102
The penultimate line demonstrates that, for small variations in the B1 field strength, the composite pulse acts like an ideal pulse applied with phase !1 =2!01 The last line demonstrates that the composite pulse acts like an ideal pulse applied with x-phase followed by a z-rotation with a rotation angle of !1 =!01 . The result that the composite pulse represents an ideal rotation around a phase-shifted rotation axis is a property shared by all time-symmetric composite pulses (41). As already noted, the z-rotation is unimportant for inversion of longitudinal magnetization, but results in phase shifts when used to refocus transverse magnetization. The quarternion formalism also is useful in analyzing the performance of composite pulses by iterative application of [1.37]. Using this approach, the effective rotation angle, eff, of a time-symmetric composite pulse consisting of three pulses, 0 0 0 , is given by cosð eff Þ ¼ 2ðcos cosð 0 =2Þ cos sin sinð 0 =2Þ2 1,
½3:103
and inversion of Iz magnetization is described by
eff
Iz ! Iz cosð eff Þ:
½3:104
For the composite pulse x(2 )y x, cosð eff Þ ¼ 2 cos4 1 ¼ 2 cos4 !1 =2!01 1,
½3:105
which is identical to the product operator calculation of [3.100]. For more detailed analysis, numerical calculations of the performance of a composite pulse are more convenient. Plots showing the theoretical performance of some selected composite pulses are shown in Fig. 3.20. Design of a composite 1808 pulse that improves inversion of z-magnetization is considerably easier than is design of a composite 1808 pulse for refocusing transverse magnetization. Inclusion of the composite pulse (most commonly, 90x180y90x or 90x240y90x) into a pulse sequence whenever inversion of a z-operator is required frequently will improve the performance of the pulse sequence. In addition, composite
178
CHAPTER 3 EXPERIMENTAL ASPECTS OF NMR SPECTROSCOPY 1.5
1.0
0.5 1.5
~
1.0
0.5 1.5
1.0
0.5 -1
o
FIGURE 3.20 Excitation profiles for 180 0 pulses. Resulting (a, d, g) M.n (b, e, h) My, and (c, f, i) M z magnetization components obtained following application of (a, b, c) rectangular 180~ pulse, (d, e, f) 90 x 180y 90 x composite 180 0 pulse, and (g, h, i) 36001801201800180'20 composite 180 pulse to (a, d, g) M x , (b, e, h) My, 0
and (c, f, i) M z magnetization. Results are shown as contour plots with resonance offset effects displayed on the x-axis and effects of pulse miscalibration (or rf inhomogeneity) displayed on the y-axis. (a-f) Results are calculated relative to an ideal y-phase pulse that inverts M x and M z magnetization and does not affect My magnetization. (g, h, i) Results are calculated relative to an ideal x-phase pulse that inverts My and M z magnetization and does not affect M x magnetization.
3.4 PULSE TECHNIQUES
179
pulses are more useful for 13C and 15N than for 1H because the larger chemical shift dispersion of the heteronuclei, relative to available rf field strengths, results in larger off-resonance effects. Amplifier droop, sample heating, and other instrumental imperfections may limit the total number or overall duration of composite pulses that can be included in a particular pulse sequence. Determination of which 1808 pulses can be replaced profitably by composite pulses in a particular sequence frequently is an empirical process. The development of composite pulses, particularly for broadband inversion and excitation applications, continues to be an active area of research. The continuing development of higher field magnets puts an ever-increasing demand on the broadband performance of inversion and excitation pulses. While intuition has historically played an important role in the design of composite pulses, increasing reliance is now made on sophisticated computer optimization methods and formalisms such as optimal control theory. Examples of the current state-of-the-art are found in the work of Shaka and co-workers regarding broadband inversion pulses (BIPs) (42) and in the work of Khaneja, Luy, Glaser, and co-workers on composite pulses designed by optimal control theory [(43) and references cited therein].
3.4.4 SELECTIVE PULSES One of the original objectives in the development of pulsed NMR spectroscopy was to replace the time-consuming swept selective excitation of continuous-wave spectroscopy by broadband excitation using strong rf pulses. As spectrometer hardware has become more sophisticated and as the experimental systems studied have become more complex, the incorporation of selective pulses that excite resonances within a narrow frequency range into otherwise broadband pulse sequences has been shown to have some advantages: selective excitation of the solvent resonance can improve solvent suppression, spins with moderately different Larmor frequencies can be manipulated independently (i.e., carbonyl and C carbon spins), and the digitization requirements for multidimensional spectroscopy can be reduced (44, 45). The simplest selective pulse is obtained by reducing the amplitude and lengthening the duration of a rectangular pulse, i.e., an rf field of constant amplitude, B1, that is turned on at a time t and turned off at time t þ p. As shown by [3.75], a rectangular pulse produces excitation over the range 2!1 2!1. Such a simple approach is unsatisfactory because the excitation profile has lobes that extend over an extensive
180
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
frequency range, the phase of the transverse magnetization varies strongly with offset, and the excitation profile is not uniform in the neighborhood of the rf carrier frequency. The performance of selective (or soft) rectangular pulses can be improved by utilizing more complex amplitude or phase modulation of the rf field. Just as for composite pulses, integration of the Bloch equations to ascertain the excitation profile of a given selective pulse shape is straightforward. The inverse problem, determining the necessary pulse shape from the desired excitation profile, is much more difficult because of the nonlinear character of the Bloch equations. Accordingly, many selective pulses have been developed initially using approximate techniques (such as Fourier analysis of the desired excitation profile) and subsequently refined. The development of selective excitation pulses and their incorporation into multidimensional pulse sequences remain active areas of research and significant future progress can be expected. The following characteristics of a selective pulse are desirable: 1. Uniform excitation or inversion profile over the desired frequency range. 2. Minimal perturbation of resonances outside the desired frequency range. 3. Phase of the transverse magnetization generated by excitation pulses should be a smooth (preferably linear) function of offset. 4. The pulse should have a short duration to minimize relaxation effects and phase evolution during the pulse. 5. The pulse modulation scheme should be simple to implement experimentally. These objectives are not satisfied simultaneously for known pulse shapes. For example, some selective pulses that yield uniform excitation profiles include short high-power pulse segments; these pulse waveforms can be difficult to implement without special rf transmitters. Consequently, an optimal selective pulse cannot be determined without reference to the particular application. For the purpose of calculations (and for implementation in spectrometer hardware), the continuous selective pulse shape is approximated by a large number, N, of rectangular pulses of length t. The amplitudes and phases of the rectangular pulses are adjusted to mimic the desired shape. Thus, the total duration of the selective pulse is p ¼ Nt, N Y Pi ð i Þ: P~ p ¼ i¼1
½3:106
181
3.4 PULSE TECHNIQUES
The net rotation induced by the selective pulse is given by N Y ~ p ¼ R Rz ði ÞRy ði ÞRz ð i ÞRy ði ÞRz ði Þ,
½3:107
i¼1
in which i, i, and i represent the values of the phase, tilt angle, and effective rotation angle for the ith period t. The rotation matrices are given by [1.35]. The effective rotation angle and tilt angle are described by [1.23] and [1.21]. The magnetization following the selective pulse is ~ p Mð0 Þ, Mð0þ Þ ¼ R ½3:108 in which M(0) and M(0þ) are the magnetization vectors before and after the pulse, respectively. In contrast to [3.98], the duration p for a selective pulse may not be short relative to either relaxation or evolution of scalar coupling interactions; accordingly, [3.108] describes the effect of a selective pulse on an uncoupled, isolated spin in the absence of relaxation effects. The pulse shapes and excitation profiles for some common selective pulses are shown in Figs. 3.21–3.23. The most common, and simplest, applications of selective pulses in protein NMR spectroscopy are (i) decoupling of carbonyl and 13C spins by using selective 1808 pulses (Chapter 7, Section 7.1.2.2) and (ii) selective excitation of the solvent resonance to obtain solvent suppression without deleterious effects of presaturation (Section 3.5.1). More sophisticated uses of selective pulses in multidimensional NMR spectroscopy are described by Kessler et al. (45).
3.4.5 PHASE-MODULATED PULSES Many NMR experiments require that rf pulses be applied at more than one frequency, for a given nuclear species, during the course of a pulse sequence. One of the most common situations in protein NMR spectroscopy in which multiple-frequency irradiation is employed is in the implementation of triple-resonance (1H, 13C, 15N) experiments. These experiments often require the 13C transmitter frequency to be centered in different spectral regions at various times during the pulse sequence, such as when a selective 1808 pulse is applied in the carbonyl region for decoupling purposes during an aliphatic carbon evolution period. Another common example requiring variable-frequency capability is the implementation of frequency sweeps during adiabatic pulses. In a typical spectrometer configuration, only one transmitter channel
182
CHAPTER 3 EXPERIMENTAL ASPECTS OF NMR SPECTROSCOPY 1.0
a
d
b
e
0.5
0.0 1.0 Q)
'0
:::J
:!:::
0.. E 0.5
oct:
0.0 1.0
C
0.5
0.0
IL-_ _l - _ - - - l_ _--L_ _~
a
0.5
0.5
1.0
FIGURE 3.21 (a) Gaussian, (b) half-Gaussian, (c) sinc, (d) SEDUCE-I, (e) RSNOB, and (f) RE-BURP pulse shapes. Amplitudes are proportional to exp(-a/ 2 ) and sin(na/)/(nat) for the Gaussian and sinc pulses, respectively. The Gaussian pulse shapes are truncated at the 4% amplitude level; as a result, the length of the pulse is twice that of a rectangular pulse with the same rotation angle. The SEDUCE-I, R-SNOB, and RE-BURP pulse shapes are truncated at the 0%, 5% and 7.5% amplitude level, respectively. The sinc pulse is truncated at the first zero of the sinc function.
is dedicated for each nuclear species involved in the experiment, which therefore requires that a transmitter channel be frequency-agile if multi frequency excitation or frequency sweeping is desired. Although many commercial spectrometers have a limited capability for automatically combining the outputs of two rf channels into a single transmitter output, allowing for dual-frequency excitation for a specific
183
3.4 PULSE TECHNIQUES 1
a
e
i
b
f
j
c
g
k
d
h
l
0
–1 1
Amplitude
0
–1 1
0
–1 1
0
–1 –20
0
20 –20
0
20 –20
0
20
Ω / g B1
FIGURE 3.22 Selective 908 pulses. Resulting (a–d) Mx, (e–h) My, and (i–l) Mz magnetization components obtained following application of (a, e, i) rectangular, (b, f, j) Gaussian, (c, g, k) half-Gaussian 908, and (d, h, l) SEDUCE-1 pulses to equilibrium Mz magnetization. Simulations used a peak field strength of !1/2 ¼ 9.5 kHz, corresponding to pulse lengths of 26.2, 52.5, 52.5, and 56.9 s for rectangular, Gaussian, half-Gaussian, and SEDUCE-1, respectively. All pulses have x-phase.
184
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
1
Amplitude
0
–1 –5
a
b 0
5 –5
c 0
5 –5
0
5
1
0
–1 –5
d
e 0
5 –5
f 0
5
0
0.5
1.0
1.5
Ω / g B1
FIGURE 3.23 Selective 1808 pulses. Resulting Mz magnetization obtained following application of (a) rectangular, (b) Gaussian, (c) sinc, (d) R-SNOB, and (e) RE-BURP 1808 pulses to equilibrium Mz magnetization. Simulations used a peak field strength of !1/2 ¼ 9.5 kHz, corresponding to pulse lengths of 52.5, 105, 89, and 246 s, and 659 s for rectangular, Gaussian, sinc, R-SNOB, and RE-BURP pulses, respectively. (f) The excitation profiles for (solid) Gaussian, (dashed) R-SNOB, and (dotted) RE-BURP pulse are superposed and expanded to more clearly indicate the transition band of the pulse.
nuclear species, this capability is not commonly used. Under suitable conditions, multifrequency excitation can be achieved using a single rf channel simply by appropriate shifts of the transmitter frequency during the pulse sequence. Most modern NMR spectrometers have the capability to shift a transmitter frequency under pulse sequence control within a few microseconds. In many experimental situations, frequency shifts must be done phase-coherently. A phase-coherent frequency shift means that no phase step occurs at the instant when the frequency transition occurs. Phase-coherent frequency shifts can be obtained with modern frequency synthesizers, at least for the limited size of frequency jumps that would normally be desired. One obvious limitation of employing explicit frequency shifts is that at any point in the pulse
185
3.4 PULSE TECHNIQUES
fN
f1
∆t
∆f
FIGURE 3.24 Phase-modulated pulse. Each rectangular element of the shaped pulse has duration ; the phase shift between rectangular elements is . The initial and final phases, 1 and N, respectively, are given by [3.110].
sequence, only a single transmitter frequency is obtainable for a given nuclear species. For example, simultaneous selective irradiation of the carbonyl and aliphatic 13C resonances would not be possible. An alternative method for achieving frequency agility of a transmitter channel is the use of phase-modulated pulses (46, 47). The concept is based on the fundamental equation ! ¼ d=dt: a shift in the effective frequency is achieved by modulating the phase of the reference rf. In practice, a phase-modulated pulse, also referred to as a phase-incremented pulse (PIP) or a shifted laminar pulse (SLP), is implemented as a windowless series of short rectangular pulses, as illustrated in Fig. 3.24. Each component pulse has a constant phase and amplitude, and typically a constant duration . The phase of the rf is incremented between each adjacent pair of pulses. For a fixed frequency shift, the phase increment is a constant, given by ¼ ð2f Þð Þ
½3:109
where f is the desired frequency shift (in Hertz) and is the duration of the component pulses. Thus, the phase of a component pulse is specified by a linear phase ramp for a fixed frequency shift, and is given by i ¼ 0 þ =2 þ ði 1Þ, . . . 1 i N ¼ 0 þ ði 1=2Þ,
½3:110
186
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
in which 0 is a constant that determines the phase of the initial component pulse, and the entire pulse consists of N elements of equal duration . The term =2 in the first line of [3.110] is referred to as the universal phase shift (48) and arises due to the quantized nature of the phase ramp. The value of the phase constant 0 must be determined from the specific nature of the phase-modulated pulse and the context in which it is employed; rules for choosing 0 are discussed below. Phasemodulated pulses of the type depicted in Fig. 3.24 give rise to multiple, asymmetric excitation sidebands in addition to the central excitation frequency band. The sidebands are located at frequencies (in Hertz) of j/ in which j is a nonzero integer, relative to the centerband (48); typical choices for the value of place all sidebands well outside the spectral range of interest. A detailed analysis (48, 49) of the propagator for the phase-modulated pulse indicates that the effective rf field for the centerband excitation is scaled by the factor l: l¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2½1 cosðÞ=jj:
½3:111
As stated previously, the value of the initial phase 0 for the phase ramp given by [3.110] is determined by the specific characteristics of the overall pulse and the context in which the pulse is to be used. Several of the most commonly encountered situations are discussed here. If the phase-modulated pulse is intended to achieve an inversion of Iz magnetization, then the choice of 0 is unrestricted. For a 908 pulse intended to transform I ! Iz , where I lies in the x–y plane (i.e., ¼ 0 specifies Ix and ¼ /2 specifies Iy magnetization), 0 ¼ =2. Intuitively, this requirement is rationalized by realizing that, for example, a Iy ! Iz transformation requires the initial component pulse must be a rotation about the x-axis in the rotating frame. For a 908 pulse intended to transform Iz ! Iy , 0 ¼ ðN 1Þ; in this case, the last component pulse must be a rotation about the x-axis in the rotating frame in order to leave the final magnetization along the y-axis. For a 1808 refocusing pulse that uses a time-symmetric amplitude modulation, such as the RE-BURP (50) or R-SNOB (51) pulses, the value of 0 ¼ N/2 must be chosen such that the middle element of the overall pulse has zero phase, assuming an effective rotation about the x-axis in the rotating frame is desired. This result can be arrived at via the following analysis (41). Let A and B represent the propagators for the first and second halves, respectively, of the time-symmetric refocusing pulse, with A being generated from the propagator for the ith component
187
3.4 PULSE TECHNIQUES
pulse, Ai: A¼
N Y
Ai ¼ A1 A2 A3 AN :
½3:112
i¼1
By definition of a time symmetric pulse, the propagator B can be expressed as: B ¼ AN A3 A2 A1 1 1 1 ¼ expfiIz gA1 N A3 A2 A1 expfiIz g
¼ expfiIz gA1 expfiIz g:
½3:113
The second line of [3.113] is obtained because the rotation axes of the component pulses of A lie in the x–y plane, in the absence of offresonance effects. By definition, an 1808 refocusing pulse applied along the x-axis will invert the y-component of the magnetization. Using [3.113], the result is obtained that
B1 A1 Iy ðABÞ ¼ Iy A1 Iy A ¼ BIy B1 ¼ expfiIz gA1 expfiIz gIy expfiIz gA expfiIz g ¼ expfiIz g A1 Iy A expfiIz g: ½3:114
The last line of [3.114] indicates that the transformed vector A1 Iy A is unaffected by a 1808 rotation about the z-axis. Therefore, the transformed vector must lie along the z-axis. Thus, the composite pulse represented by A will rotate an arbitrary vector from the transverse plane into the x–z plane. Intuitively, the last component pulse of A must be applied along the x-axis for the previous statement to be true. Applying a phase ramp to shift the excitation frequency does not alter this analysis; consequently, as previously stated, the center element of a time-symmetric refocusing pulse must have x-phase, assuming the desired net result is a rotation about the x-axis in the rotating frame. To complete the discussion of the choice of the phase constant 0, the final example considers the case that multiple, phase-modulated pulses are combined to achieve some desired result, such as a frequencyshifted composite pulse. In this situation, the phase constant 0 for each
188
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
element of the composite pulse must take into account an inherited phase from the immediately preceding element (49). For a frequency-swept rf pulse that is generated using phase modulation, the phase increment between adjacent component pulses can be variable. Once the time dependence of the frequency sweep is specified, the phase modulation function is determined from the fundamental relationship: Zt !sweep ð Þ d, ½3:115 ðtÞ ¼ 0
where !sweep ðtÞ is the functional form of the desired frequency sweep. For example, if a linear frequency sweep is desired, !sweep ðtÞ ¼ kt, and the phase modulation function will be 1 ðtÞ ¼ 0 þ kt2 2
½3:116
and the phase constant 0 is chosen using the principles discussed previously. The frequency-swept pulse is generated by subdividing the pulse into a series of component pulses, typically of constant duration , with phases calculated according to [3.115]; amplitude modulation is achieved simply by scaling the amplitudes of the component pulses according to the time dependence of the amplitude profile. A few comments are made here to facilitate the practical implementation of phase-modulated pulses. The sign of the phase increment in [3.109] and the sign of the calculated phase modulation function in [3.116] will depend on the convention employed in the spectrometer operating software. This is due to the fact that, relative to the effects of an rf pulse stipulated to be x-phase, a y pulse on a spectrometer from one manufacturer may correspond to a y pulse on a spectrometer from another manufacturer. The appropriate sign of for a given sign of f can be ascertained by examining an example waveform, which is usually a text file containing a list of pulse amplitudes and phases, or by performing an appropriate experiment using a simple phase-modulated pulse to empirically determine the sign convention. The sign convention determined by either approach is applicable for the construction of all other phase-modulated pulses on that spectrometer. Software programs, such as the Pulsetool program provided by Varian, Inc., that allow the simulation of spin dynamics using the Bloch equations are very useful for testing and optimizing phase-modulated pulses.
3.4 PULSE TECHNIQUES
189
Generally speaking, an absolute requirement for generating a phasemodulated pulse is that the spectrometer hardware be capable of rapidly (i.e., 51 s) executing small-angle phase shifts of the rf frequency. In generating the phase modulation, the exact phases stipulated by [3.111] or [3.115] can only be approximated, due to the finite phase resolution of the hardware; however, modern commercial spectrometers typically have a minimum phase step of 0.258 or less and the resulting quantization error is usually negligible. Although not an absolute requirement, the duration of the component pulses of a phasemodulated pulse is usually a constant, . In choosing a value for , the upper limit is determined by the desire to keep sideband excitation frequencies outside the spectral region of interest (vide supra). The value chosen for also should be an integer multiple of the minimum step size of the hardware unit that generates the pulses. Frequency-shifted pulses generated using phase modulation have the desirable property that the frequency shifts are achieved phase coherently; depending on the spectrometer hardware, explicit frequency shifts using the rf frequency synthesizer may or may not be perfectly phase coherent. In addition, phase modulation can be used to generate multiple excitation frequencies simultaneously. If the individual frequency-selective pulses are represented by histograms of N component pulses, each with a prescribed phase and amplitude (as shown in Fig. 3.25), then the simultaneous excitation pulse profile is generated simply by calculating the vector sum of the individual waveforms at each time segment (52).
3.4.6 ADIABATIC PULSES The rf pulses discussed up to this point have been employed to effect a specific rotation of a magnetization vector about the axis of the effective field Beff in the rotating frame. If the pulse is applied onresonance, then the angle between the effective field and the magnetization vector is 908. Another possibility for manipulating magnetization that has several advantages is an adiabatic pulse (53). In such a pulse, Beff initially is aligned approximately with the magnetization vector, and subsequently the axis of Beff is caused to rotate adiabatically during the pulse. Under these conditions, the magnetization vector follows the same trajectory as Beff to some desired endpoint. Essentially, the magnetization is locked along the direction of Beff and caused to undergo the desired reorientation via tight control of the effective field direction during the course of the adiabatic pulse. Manipulating magnetization in such a fashion is commonly referred to as adiabatic following, or
190
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
a
b
c
d
FIGURE 3.25 Vector representations of individual rf pulse elements comprising three separate phase-modulated pulses (a–c), and their combination to form a single composite pulse (d). The length of the vectors represents the rf pulse amplitude while the rotation from vertical represents the rf phase of each element. Waveform (a) represents a non-phase-modulated pulse, while waveforms (b, c) have phase modulations to generate frequency shifts. Each element of the composite pulse (d) is generated by addition of the corresponding vectors from each of the component elements in (a–c).
adiabatic rapid passage, where the term ‘‘rapid’’ refers to the fact that the adiabatic pulse should be executed in a time that is short relative to the relaxation time constants T1 and T2. The conditions required for such a pulse can be determined from the following analysis. For simplicity, the treatment considers uncoupled spins. Under the influence of an arbitrary effective field Beff, the rate of change of the squared amplitude of a magnetization vector is given by d 2 dM M ¼ 2ME ¼ 0: dt dt
½3:117
The right-hand equality is obtained because M and dM=dt are orthogonal, according to [1.10]. Thus, the amplitude of the
3.4 PULSE TECHNIQUES
191
magnetization vector is unchanged under the influence of an arbitrary effective field, ignoring relaxation and scalar coupling effects. Consequently, the magnetization vector can be reoriented, but not changed in size. Therefore, the question arises as to how the relative orientation of the magnetization vector and the effective field axis will change as a function of time. If the effective field is fixed, independent of time in the rotating frame, then the analysis in Chapter 1 demonstrates that the magnetization vector precesses around the effective field while maintaining a fixed angle with respect to the effective field axis. If the effective field is time dependent, then the time dependence of the angle between M and Beff, denoted as ffðM, Beff Þ, can be examined by considering the second derivative of M, starting from [1.10]: d 2M dBeff : ¼ 2 ðM Beff Þ Beff þ M 2 dt dt
½3:118
This equation differs from the case where Beff is time independent (and hence ffðM, Beff Þ is a constant) by the addition of the second term on the right. Thus, if the following condition is true, M dBeff ðM Beff Þ Beff , ½3:119 dt then ffðM, Beff Þ will be approximately conserved. Any arbitrary vectors u and v satisfy the condition ju vj jujjvj; therefore, the inequality expressed in [3.119] will be satisfied if dBeff ðM Beff Þ Beff jMjjBeff j2 : jMj ½3:120 dt Using the definition !eff ¼ jBeff j, the inequality [3.120] simplifies to d!eff 2 ½3:121 dt !eff : An additional condition that must be met in order for a magnetization vector to undergo adiabatic passage is obtained by considering the following analysis. In the frame of reference rotating at the instantaneous frequency of the applied rf field B1, which is sometimes referred to as the frequency modulation (FM) frame, the effective field is given by Beff ¼ B1 ðtÞi þ ððtÞ= Þk, in which the rf field is assumed to be applied along the x-axis in the FM frame. The effective field makes an angle with respect to the z-axis of the FM frame, so that tan ¼ B1 ðtÞ=ðtÞ: A second rotating frame is defined with axes (xr, yr, zr), where the xr-axis
192
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
is collinear with the effective field Beff, the yr-axis is collinear with the yaxis of the FM frame, and the rotation is about the y-, yr-axis. The rotation of this second frame represents the motion of the effective field in the FM frame, and thus the angular velocity is given by d=dt. Following the rotating coordinates procedure (54) that leads to [1.13], the effective field in the second rotating frame is given by Breff ¼ Beff ir þ ð1= Þðd=dtÞjr . The magnetization vector will precess about the axis defined by Breff . For the magnetization vector to approximately preserve its orientation with respect to Beff, Breff must be approximately collinear with the xr-axis, which requires that 1 d dt jBeff j, ½3:122 d Beff :¼ !eff : dt Equation [3.122] is commonly referred to as the adiabatic condition, and indicates that for a magnetization vector to remain approximately aligned with the effective field, the rate of change in the direction of the effective field must be much slower than the precession frequency of the magnetization vector around the instantaneous effective field direction. To further illustrate the physical basis for an adiabatic sweep, the sweep is considered to consist of n discrete pulses of equal duration t, which is, of course, how an adiabatic sweep normally is implemented on an NMR spectrometer. The ith pulse is defined by the values of i and i ¼ !eff,i t, in which i is the tilt angle, !eff,i is the magnitude of the effective field, and i is the net rotation angle around the axis defined by the effective field during t. By construction, t is chosen short enough that i is small. For mathematical convenience in the following discussion, the first pulse is defined as the first pulse element with !1 4 0 and the nth pulse is defined as a pulse with !1 ¼ 0; consequently, 1 4 0 and n ¼ . Using [1.34], evolution during the initial time period t is described by MðtÞ ¼ Ry ð1 ÞRz ð1 ÞRy ð1 ÞM0 :
½3:123
The evolution during the second time period t is described by Mð2tÞ ¼ Ry ð2 ÞRz ð2 ÞRy ð2 ÞMðtÞ ¼ Ry ð2 ÞRz ð2 ÞRy ð2 ÞRy ð1 ÞRz ð1 ÞRy ð1 ÞM0 ¼ Ry ð2 ÞRz ð2 ÞRy ð2 ÞRz ð1 ÞRy ð1 ÞM0 ,
½3:124
193
3.4 PULSE TECHNIQUES
in which i ¼ i i1 and 0 0. Continuing this process leads to the result ( ) n Y MðntÞ ¼ Ry ðÞ Rz ði ÞRy ði Þ M0 i¼1 ( ) n Y ¼ Rz ði ÞRy ði Þ Ry ðÞM0 : ½3:125 i¼1
If the average values of i and i are defined as and , respectively, then i ¼ þ i and i ¼ þ i . Using these expressions, ( ) n Y MðntÞ ¼ Rz ðÞRz ði ÞRy ðÞRy ði Þ Ry ðÞM0 ( ¼
i¼1 n Y
) Rz ðÞRy ðÞ Ry ðÞM0
i¼1
n ¼ Rz ðÞRy ðÞ Ry ðÞM0,
½3:126
in which the second line is obtained by assuming all the rotations in the curlyP braces are P small enough to commute with each other and noting that i ¼ i ¼ 0. The product rotation in the square brackets on the final line of [3.126] can be expressed as a single rotation of angle
around an axis ^n by using [1.37], in which ¼ cos cos ; cos 2 2 2 ^ sin n ¼ sin sin i sin cos j 2 2 2 2 2 sin k: ½3:127 cos 2 2 Using this result, MðntÞ ¼ R^n ðn ÞRy ðÞM0 : The angle between the z-axis and n^ is given by ! sin =2 ¼ tan1 cos =2 sin =2
½3:128
½3:129
in which the second equality is obtained because and are small. Thus, the adiabatic sweep is modeled theoretically by an initial ideal
194
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
inversion around the y-axis followed by a rotation around the axis n^ by an angle n . An ideal inversion occurs if is sufficiently small, because then the magnetization never precesses significantly away from the zaxis, regardless of the value of : 2 is the maximum deviation between the magnetization vector and the z-axis. Assuming that 1 is sufficient to ensure ideal inversion yields: d , ½3:130 , dt h!eff i, t t in which h!eff i ¼ =t. Thus, the condition for an ideal inversion derived from this approach is equivalent to the adiabatic condition derived in [3.122]. Using [3.129] and [3.128], the angle , which determines the departure of the final state of the magnetization from an ideal inversion, also is given by
1 d : ¼ h!eff i ½3:131 dt The quantity inside the square brackets is closely related to the adiabaticity factor, which plays an important role in determining the quality of adiabatic pulses. The adiabaticity factor is discussed more fully later. A pictorial model for the evolution of magnetization during an adiabatic sweep is provided by the following simplified case: the orientation of the effective field makes n instantaneous jumps of equal angular increments, , in the y–z plane of the FM reference frame at equally spaced time intervals, t; the effective field is assumed to be of constant amplitude, !eff ; the magnetization precesses about the current direction of the effective field by an equal angle ’ ¼ !eff t during each time interval; and the magnetization vector is initially taken to be collinear with the starting position of the effective field along the þzaxis. A graphical representation of this simple model is provided by the gnomonic projection diagram in Fig. 3.26a, as originally presented by Powles (55). The horizontal line in Fig. 3.26a represents the great circle trajectory of the intersection point of the effective field vector on a sphere of radius M, where M is the amplitude of the magnetization vector. This diagram shows that the deviation of the magnetization vector from the effective field direction is constrained. In the arbitrarily chosen case for ¼ 0:7 radians (408), the maximum deviation of the tip of the magnetization vector is ymax 2:9Mjj. The maximum angular deviation, defined as , of the magnetization vector from the effective field direction satisfies the relation tan ¼ ymax =M. Therefore,
3.4
195
PULSE TECHNIQUES
~ 3
11
7
2
3
4
5
6
7
8
12
9
10
11
12
MM
b
---.. .2.. _. _..
... .. ...
..
'
.::
.
__
._-:
,
",-
.
, ,
!
.
0-';.
. w·- •
,
..x·
I
I I I
~::
...., " "
..
.. - .
'.'
FIGURE 3.26 Magnetization trajectories during an adiabatic pulse. (a) Projection view of a simplified adiabatic sweep in which a constant amplitude effective field rotates in angular steps of 6.8 and during the intervening intervals the magnetization vector precesses about the current effective field direction by a constant angle 6.¢. The horizontal line is a gnomonic projection of the path along a great circle followed by the tip of the effective field vector, while the arcs represent the path of the tip of the magnetization vector (adapted from Powles (55). (b) Perspective view of the trajectory (bold solid line) followed by the tip of the magnetization vector during an adiabatic sweep. The bold dashed line represents the path of the effective field. The numerical simulation was performed using an effective field with constant amplitude, the rate of change in the direction of the effective field was constant, and the precession frequency of the magnetization about the effective field was eight times faster compared to the rate of change of the direction of the effective field (i.e., the adiabaticity factor Q = 8).
196
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
for to remain very small, the condition ymax M must be satisfied, which requires jj 0:3 radians. Because ¼ 0:7 radians, the condition for to remain very small can be written as jj , or equivalently, d=dt ðd=dtÞ ¼ !eff , which is the result given by the second adiabatic condition, [3.122]. Inspection of Fig. 3.26a demonstrates that the adiabatic condition would result regardless of the choice of the scale M and the precession angle . A physically more accurate picture of the evolution of magnetization during an adiabatic sweep is shown in Fig. 3.26b. This figure provides a perspective view of the trajectories of the effective field and the magnetization vector, calculated via a numerical simulation of the Bloch equations (ignoring relaxation effects). The trajectory of the tip of the precessing magnetization trajectory is shown as a solid line on the unit sphere; the pathway followed by the effective field (the FM frame) is shown as a heavy dashed line. For the simulation, the amplitude of the effective field is assumed constant, the directory of the effective field vector is assumed to rotate in the y–z plane at a constant angular velocity d/dt, and the precession frequency !e of the magnetization vector about the effective field is set to !e ¼ 8d=dt. Under these conditions, the magnetization vector follows the effective field, as expected. The adiabatic condition, [3.122], limits the rate of change of the effective field, but otherwise does not specify the exact manner in which the effective field should be controlled in order to accomplish the desired result. The effective field is specified by the applied rf field !1 (¼ B1) and the resonance offset ; thus, the amplitude and orientation of the effective field are modulated by the time dependences of !1 and . Determining optimal modulation functions for the amplitude and frequency of the applied rf field has been an active area of research. A parameter for characterizing the adiabaticity of a pulse that has proved to be useful in design efforts is the dimensionless quantity Q (56): !eff , ½3:132 Q¼ jd=dtj which is referred to as the adiabaticity factor. To strictly satisfy the adiabatic condition, [3.122], Q should be much greater than one. Using the relationship tan ¼ !1 =, [3.132] becomes
3=2 !21 þ 2 : Q ¼ !1 d d!1 dt dt
½3:133
197
3.4 PULSE TECHNIQUES
A critical point in an adiabatic pulse occurs when the rf field sweeps through the resonance frequency, i.e., when ¼ 0, because at this point the adiabaticity factor reaches a minimum: Q0 ¼
!21 : jd=dtj¼0
½3:134
One of the principal applications of an adiabatic pulse is to perform broadband inversion, whereby magnetization is rotated from the þz-axis (in the laboratory frame) to the z-axis, or vice versa. In an adiabatic inversion pulse, the frequency of the rf field is far off-resonance at the beginning of the pulse, is swept through a frequency band of interest, and is far off-resonance on the other side of the frequency band at the end of the pulse. If the adiabatic condition [3.122] is maintained, i.e., if the rate of change of the effective field axis is significantly smaller than is the size of the effective field at all points during the frequency sweep, then the nuclear magnetization vector will track the effective field vector. If the nuclear magnetization vector is aligned with the effective field at the beginning of the sweep, then the magnetization vector will be inverted at the end of the sweep. A principal advantage of an adiabatic pulse is its insensitivity to B1 inhomogeneity. As long as !1 is sufficiently large to satisfy the adiabatic condition Q0 1, the magnetization vector remains nearly collinear with the effective field (in common terminology, the magnetization is said to be spin-locked to the effective field). This insensitivity to B1 inhomogeneity has made adiabatic pulses particularly valuable in in vivo spectroscopy and imaging applications, where the nature of the rf transmitter coils inherently results in poor rf homogeneity. Adiabatic pulses also have superior performance with respect to off-resonance effects, compared to conventional pulses or composite pulses. The orientation of the effective field Beff is determined by the amplitude and frequency of the rf field. The adiabatic sweep can therefore be accomplished by a suitable modulation of either or both of these parameters. Design criteria include maximum rf field amplitude, desired bandwidth, frequency selectivity, total pulse width, power efficiency, and sensitivity to rf inhomogeneity; in the case of coupled spins, the size of the scalar couplings should also be taken into account. The simplest scheme is the so-called CHIRP pulse that utilizes a linear frequency sweep with a constant rf amplitude. The main drawback of the CHIRP scheme is that sudden switching on (off ) of the rf field at the beginning (end) of the pulse can cause a violation of the adiabatic condition. A modulation scheme in which the rf
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amplitude has a sine dependence and the frequency sweep has a cosine dependence on time yields an effective field with constant amplitude. A popular adiabatic pulse, the hyperbolic secant or sech/tanh pulse, uses the modulation scheme (56, 57) B1 ðtÞ ¼ B01 sechð tÞ,
½3:135
ðtÞ ¼ tanhð tÞ,
in which and are real constants. The hyperbolic pulse exhibits a sharp transition between the regions where the magnetization vector experiences an inversion and the regions where the magnetization vector is unaffected. Thus, the hyperbolic secant pulse has the attractive properties of broad bandwidth and high selectivity. This pulse also exhibits constant adiabaticity over its bandwidth, allowing rf power to be used most efficiently. A potential drawback to the hyperbolic secant pulse is the high peak rf power required to achieve a broad bandwidth, which can be detrimental to the probe. To minimize the rf peak power, the so-called WURST scheme has proved to be quite effective (58). The WURST family of adiabatic pulses is closely related to CHIRP, in the sense that a linear frequency sweep is employed. To eliminate the violations of the adiabatic condition at the beginning and end of the CHIRP pulse, the WURST scheme provides for a smooth transition to and from the peak rf field using an amplitude modulation given by the function n !1 ðtÞ ¼ !max 1 sinð tÞ , ½3:136 1
!max 1
is the peak rf field strength, t is the orientation of the where effective field at time t (=2 t =2), and n is an index that determines the steepness of the rounding function at the beginning and end of the frequency sweep. The index n typically is chosen using the rule of thumb n p F=2, where F is the total sweep range (in Hertz) and p is the duration of the sweep. The linear frequency sweep rate is given by d=dt ¼ 2F= p . The root-mean-square field strength of a WURST pulse is determined by the index n according to the function
ðn 1Þ!! ð2n 1Þ!! 1=2 rms max 12 , n even, n 6¼ 0, !1 ¼ !1 þ n!! ð2nÞ!!
!rms 1
¼
!max 1
4 ðn 1Þ!! ð2n 1Þ!! 1=2 þ 1 , n!! ð2nÞ!!
n odd,
n 6¼ 1, ½3:137
199
3.4 PULSE TECHNIQUES
where the expression n!! refers to the double 8 < nðn 2Þ 5 3 1, n!! nðn 2Þ 6 4 2, : 1,
factorial n > 0 odd n > 0 even n ¼ 1, 0:
½3:138
The index n is also related to the bandwidth of the inversion profile of a WURST pulse: the higher the value of n, the broader the bandwidth. The on-resonance adiabicity factor Q0, [3.122], can be rearranged to give max 2 ! p F ¼ 1 : ½3:139 2Q0 Adiabatic pulses are generally straightforward to implement on modern NMR spectrometers. Such instruments are normally equipped with hardware for performing the necessary amplitude modulation of the rf pulse. The only caveat here is that if the peak rf amplitude drives the rf power amplifier into its nonlinear region of operation, the shape function must be appropriately modified in order to obtain the desired output. The frequency sweep is normally achieved by a corresponding phase modulation, as described in Section 3.4.5. Thus, an adiabatic pulse is normally implemented by concatenating a series of individual rectangular pulses with the prescribed amplitudes and phases. The number of incremental pulses, N, should satisfy N F p to avoid aliasing. An example of the amplitude and frequency modulation functions and the inversion profile for a WURST pulse with n ¼ 20 and Q0 ¼ 5 is shown in Fig. 3.27. Unlike the plots shown for composite pulses, Fig. 3.20, the inversion performance of the WURST sequence is plotted as a function of the absolute resonance offset frequency, instead of a relative offset, because a simple linear relationship between the inversion bandwidth and the rf field strength B1 does not exist. In the form just described, adiabatic pulses are most often used to effect the inversion transformation Mz $ Mz ; in this case, the pulse is referred to as adiabatic full passage (AFP). In an adiabatic half-passage (AHP), the pulse is terminated at the point where the magnetization crosses the transverse plane. Because the point in time at which the magnetization vector passes through the transverse plane is dependent on when the frequency sweep passes through a given resonance, an AHP pulse will have a narrow bandwidth when used as a 908 excitation pulse. The simple AHP pulse has proved to be very useful in rotating-frame relaxation experiments for orienting the magnetization vector along a local effective field direction at the beginning of spin lock period, and
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Amplitude
1.0
OF
NMR SPECTROSCOPY
a
0.5
0.0 6p
b Phase
4p
2p
0.0 0.0
0.5 tp
1.0
0
50
1.0
Mz
0.5 0.0
–0.5
c –1.0 –50
Ω (kHz)
FIGURE 3.27 Pulse shape and excitation profile for a WURST-20 adiabatic sweep. (a) The amplitude profile calculated using !max 1 =2 ¼ 9512 Hz and n ¼ 20 is shown. (b) The time dependence of the pulse phase, used to generate the linear frequency sweep, is shown. A 50,000-Hz frequency sweep and pulse length of 440 s were assumed. In practical implementation on an NMR spectrometer, the phases shown would be converted to the range to . (c) The inversion profile for Mz magnetization is shown.
for restoring the magnetization to the z-axis after the spin lock period (59). The mode of action of simple adiabatic pulses means that plane rotations cannot be achieved. A plane rotation transforms all magnetization vectors, regardless of initial orientation, through the same angle about a single rotation axis. Thus, the simplest forms of an adiabatic pulse
201
3.5 SPIN DECOUPLING
are not useful as refocusing 1808 pulses. However, schemes have been invented to create composite adiabatic pulses that accomplish all of the rotations possible with conventional pulses (60, 61). Pairs of adiabatic pulses also are useful for refocusing magnetization (53). In the preceding discussion involving geometrical pictures of adiabatic following, scalar couplings were assumed to be negligible. In fact, most of this analysis remains valid so long as the rf field (in frequency units) is much larger, compared to the scalar couplings. Elegant treatments of the effects of adiabatic pulses on coupled spin systems have been presented (62–64).
3.5 Spin Decoupling As has been discussed extensively in Chapter 2, if two spin-1/2 nuclei I and S are scalar coupled with a coupling constant JIS, evolution of transverse magnetization of the I spin leads to splitting of the resonance signal into two multiplet components at frequencies !I þ JIS and !I JIS (I þ JIS/2 and I JIS/2, in Hertz). Removing the resonance splitting caused by the scalar coupling to the S spin when recording the FID for the I spin, a procedure called spin decoupling, increases the signal intensity and simplifies the I spin spectrum. Product operator calculations demonstrate that if a series of 1808 pulses are applied selectively at the frequency of the S spin, then the effect of the scalar coupling interaction is refocused and the net evolution of the I spin yields a single resonance signal at !I. In the limit that the pulse spacing approaches zero, decoupling is obtained by applying a continuous rf field at the resonance frequency of the S spin while observing the I spin.
3.5.1 SPIN DECOUPLING THEORY This simple picture of spin decoupling suffers from the defect that pulses are assumed to be infinitely short, so that evolution under the scalar coupling and Zeeman Hamiltonians may be ignored, when applying the product operator formalism. This assumption is not valid if an rf field is applied for an extended period of time. Furthermore, as discussed in Section 3.4.1, the performance of a nominal 1808 pulse in inverting magnetization is a strong function of resonance offset and rf inhomogeneity. These effects are arduous to analyze for multiple pulse sequences using the product operator formalism. A more sophisticated theoretical approach is necessary to design and evaluate methods for spin decoupling. The principles of the modern theory of spin decoupling
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were developed by Waugh (65), extending the work of other pioneers in the field. The theory of spin decoupling has been reviewed elegantly by Shaka and Keeler (66), and only a brief description of the basic concepts is presented here. Reference to the energy level diagram for a weakly coupled IS spin-1/2 system (Fig. 1.7) provides some insight into the principles upon which modern decoupling theory is based. The NMR spectrum that would result from the transitions indicated by the arrows in Figure 1.7b is shown in Fig. 1.7d. From the energy levels shown in Fig. 1.7b, the frequencies of the four single-quantum transitions in the absence of spin decoupling are [1.58] fðI1 Þ ¼ !13 ¼ !I JIS , fðI2 Þ ¼ !24 ¼ !I þ JIS , fðS1 Þ ¼ !12 ¼ !S JIS, fðS2 Þ ¼ !34 ¼ !S þ JIS ,
½3:140
in which the four observable transitions are labeled I1, I2, S1, and S2. By observation, a frequency sum rule is seen to exist: fðI1 Þ þ fðS2 Þ ¼ fðI2 Þ þ fðS1 Þ:
½3:141
This sum rule is obeyed even if a spin decoupling pulse sequence is being applied to one of the spins. Equation [3.141] can be rearranged to give gðI1 Þ gðI2 Þ ¼ gðS1 Þ gðS2 Þ,
½3:142
where the change of symbol f ! g is simply meant to indicate that the transition frequencies in the presence of a decoupling sequence applied to spin S will not necessarily be the same as for the undecoupled case [3.140]. By definition, the goal of spin decoupling is to achieve the result that gðI1 Þ gðI2 Þ; in order to do so, [3.142] indicates that the decoupling sequence applied to the S spins must force the two components of the S spin doublet to precess at the same frequency, so that gðS1 Þ gðS2 Þ. In principle, even a coherent, continuous-wave (cw) rf field applied to the S spins can result in good decoupling, as long as the rf field strength is sufficiently strong and the rf frequency is set close to the S spin resonance. However, cw decoupling is very inefficient for achieving broadband decoupling, because the rf field strength would need to be increased to impractical levels in order to overcome the effect of resonance offset. The challenge for achieving efficient (i.e., minimal rf power) broadband decoupling performance is to design a pulse sequence that results in the two S spin transition frequencies being essentially equal over as wide a chemical shift range as possible. The decoupling condition, that the two S spin vectors be forced to have the same precession frequency, does not impose a requirement for any particular
3.5 SPIN DECOUPLING
203
trajectory under the influence of the decoupling pulse sequence. However, in practice, imposing the condition of cyclicity greatly facilitates the design of broadband decoupling sequences. A pulse sequence element is cyclic if a spin vector evolves such that it returns to its initial orientation at the end of the pulse sequence element. The simplest pulse sequence element that is cyclic is a 3608 pulse. If the S spin is subjected to a decoupling sequence that causes the two component S spin vectors to return to their initial position after each application of the primitive pulse sequence element, then the condition gðS1 Þ gðS2 Þ will be achieved. For broadband decoupling performance, the combination of two optimized 1808 pulses, each referred to here as an R element, to give a net 3608 pulse sequence RR, exploits the extensive efforts that have been made to design broadband inversion (i.e., 1808) pulses. This strategy, pioneered by Levitt and Freeman, is generally referred to as composite pulse decoupling, where the decoupling sequence is composed of a continuous series of composite 1808 pulses. A limitation implied in the preceding presentation is that the I spin FID is sampled stroboscopically with the cyclic perturbations of the S spin magnetization. In practice however, the FID must be sampled more rapidly. As a result, sampling points will occur at times when the trajectories of the two S spin magnetizations have diverged, which leads to small-amplitude modulations of the I spin FID. These modulations result in the appearance of cycling sidebands symmetrically located about the I spin resonance in the frequency spectrum. These sidebands occur at the cycling frequency and its harmonics. Although cycling sidebands are typically small in amplitude, their presence can significantly degrade the quality of an NMR spectrum in a high dynamic range situation, where weaker peaks of interest can be difficult to identify amid sideband peaks associated with nearby stronger resonances. The exact pattern and amplitude of cycling sidebands are strongly dependent on the details of the decoupling pulse sequence, and can be a significant factor in the choice of decoupling sequences for particular applications. Cycling sidebands can be attenuated by various techniques, such as applying the decoupling sequence in an asynchronous manner, meaning that the relationship between the start of the acquisition period and the decoupling cycle is not fixed. Optimizing the efficiency of spin decoupling sequences is very important for studies of biomolecules, due to the need to decouple over ever-widening frequency bandwidths as magnetic field strengths increase. Most commonly in the NMR spectroscopy of proteins, the 1 H signal is recorded and the heteronuclear spins are decoupled; therefore, the broad chemical shift range of species such as 13C also
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presents a significant challenge. The situation is exacerbated by the solution conditions required for studies of the biomolecules, because lossy buffers (e.g., composed of a high concentration of ionic or polar species) make samples very susceptible to dielectric heating from the electric field component of the rf irradiation. This heating effect becomes worse at higher frequencies. In composite pulse decoupling, optimization of decoupling performance is generally approached by developing improved broadband 1808 pulse sequence elements and finding a prescription for concatenating such elements to minimize the effects of their imperfections. In the following section, an outline of the principles used to develop improved spin decoupling sequences is presented. One measure of the efficiency of a decoupling sequence is the ratio of the scalar coupling observed in the decoupled NMR spectrum versus that in the undecoupled spectrum as a function of the resonance offset and rf field strength B1 . Using this ratio, designated lc(), with the so-called l-approximation, gives (66) 1 @ ½3:143 lc ðÞ ¼ , ts @ where is the net rotation angle experienced by a spin vector under the influence of one complete cycle of the decoupling sequence, calculated as a function of the offset frequency , and ts is the duration of a complete cycle. For continuous rf irradiation, the scaling factor is lc ðÞ ¼ cos ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , ðB1 Þ2 þ2
½3:144
which simply reflects the projection of the scalar coupling Hamiltonian into the tilted reference frame of the effective field for the S spin. Perfect decoupling, lc() ¼ 0, is obtained only for on-resonance irradiation. For off-resonance irradiation, effective decoupling requires that B1 , which is a condition that is extremely difficult to achieve experimentally. For example, if |/(B1)| ¼ 1, then lc() ¼ 0.707, which provides marginal reduction in the coupling constant.
3.5.2 COMPOSITE PULSE DECOUPLING The preceding discussion suggested that efficient spin decoupling can be achieved by applying a series of composite 1808 pulses. In the limit that the pulse spacing approaches zero, a phase-modulated rf field is obtained that can be described as the repetition of some
205
3.5 SPIN DECOUPLING
phase-modulated sequence R representing the composite 1808 pulse. Coherent averaging theory (65) demonstrates that this approach in fact improves decoupling and also leads to the conclusion that, given a sequence element R, improved sequence elements can be developed by recursive expansion of R into supercycles. For example, if R is a composite 1808 pulse and R is obtained by inverting the phases of the pulses comprising R, then the cyclic supercycles RRR R RRR R RRR R
½3:145
RRR R RRR R R RRR RR RR yield progressively better decoupling performance. In addition, if R is a composite pulse sequence, then Rp R p performs better than R. The element Rp is obtained by cyclically permuting a 908 pulse with the element R. For example, if R is R ¼ 90x 180x 270x ¼ 123,
½3:146
123 12 3 12 3 RRR R ¼ 123
½3:147
then
improves upon R. If this element is cyclically permuted from left to right, 123 12 3 12 3 1 Rp ¼ 23 23 12 42 31 ¼ 24
½3:148
23 12 42 312 42 31 24 23 1: Rp R p ¼ 24
½3:149
and
The composite pulse given in [3.146] is the primitive element for the WALTZ family of decoupling sequences (67). Equation [3.147] is the WALTZ-4 sequence and [3.149] is the WALTZ-8 sequence. The WALTZ-16 sequence is obtained by a cyclic permutation of [3.149] from right to left, 23 12 42 3 342 31 24 23 342 31 24 23 34 23 12 42 3: ¼ 34 RR RR
½3:150
Each expansion of the basic element results in an improvement in the decoupling efficiency by approximately an order of magnitude. Even though the expanded sequences are significantly longer in duration than is the primitive element, fortunately the basic condition for decoupling
206
CHAPTER 3 EXPERIMENTAL ASPECTS
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requires only that the primitive element be repeated rapidly compared to the coupling constant JIS. The decoupling scaling factors for the WALTZ family of decoupling sequences are shown in Fig. 3.28. WALTZ-16 yields lc 5 0.0006 for resonance offsets |/ (B1)| 1, an improvement of three orders of magnitude compared with continuous-wave decoupling. The search for new decoupling sequences at present is heavily dependent on computer optimization. An early example of this approach in which the led to the GARP-1 decoupling sequence (68), RRR R, element R is given by R ¼ 30:5 55:2 257:8 268:3 69:3 62:2 85:0 91:8 134:5 256:1 66:4 45:9 25:5 72:7 119:5 138:2 258:4 64:9 70:9 77:2 98:2 133:6 255:9 65:5 53:4: ½3:151 The pulse lengths are given in degrees and overbars indicate 1808 phase shifts. GARP-1 yields lc 5 0.002 for resonance offsets |/(B1)| 2.5. The decoupling scaling factor for the GARP-1 decoupling sequence is shown in Fig. 3.29a. The WALTZ-16 and GARP-1 sequences are two of the most commonly used techniques for spin decoupling in macromolecules. However, the design of decoupling sequences has been an active area of development, and a number of alternatives have been proposed. For example, in the SUSAN-1 sequence (69), the composite pulse element R is given by R ¼ 27:7 59:7 37:6 17:7 41:1 80:0 43:7 34:3 68:1 81:7 60:5 49:3 0:1 37:1 110:3 163:4 66:2 110:5 0:81 145:5 148:0, ½3:152 is incorporated and it, along with its phase-inverted counterpart R, . The into the MLEV-16 supercycle RRRR RRRR RRRR RR RR scaling factor for SUSAN-1 is shown in Fig. 3.29b, where it exhibits a somewhat broader bandwidth compared to GARP-1; SUSAN-1 provides a scaling factor lc5 0.007 for |/(B1)| 3.1, and is compensated for rf inhomogeneity over a range of 0:1ðB1 Þ. When comparing scaling factors for various decoupling sequences, the results depend heavily on the type and length of the supercycles employed. In addition to the MLEV-type supercycles described here, another scheme that has proved to be very effective is the five-step cycle
207
3.5 SPIN DECOUPLING 0.06
a
0.00 0.006
lc
b
0.000 0.0006
c
0.0000 0
0.5
1.0
Ω / (g B1)
FIGURE 3.28 Scalar coupling scaling factor for (a) WALTZ-4, (b) WALTZ-8, and (c) WALTZ-16 decoupling sequences.
of the general form ½0, 2 , 22 þ 120 , 32 þ 60 , 42 þ 120 (70). The composite pulse sequence consists of the concatenation of five primitive R pulse sequence elements, with the phases of all pulses in the second element being incremented by an amount 2 , the third element
208
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
a 0.006
lc
0
b 0.006
0 0
1
2
3
Ω / (gB1)
FIGURE 3.29 Scalar coupling scaling factor for (a) GARP-1 and (b) SUSAN-1 decoupling sequences.
incremented by 22 þ 120 , and so on. A specific set of phases commonly employed in practice is [08, 1508, 608, 1508, 08], called the Tycko 5-step phase cycle. This phase cycle works well to improve the inversion performance of composite 1808 pulses. Very often, a combination of the Tycko 5-step cycling with the MLEV-type cycling leads to improved decoupling performance, where the 5-step cycle is applied first and then the MLEV cycling is applied subsequently; this
3.5 SPIN DECOUPLING
209
is sometimes referred to as a T5M4 supercycle, in the case where MLEV is employed. 4 cycling (RRR R) An example of the use of the T5M4-type phase cycling for improving decoupling performance is the MPF family of sequences (71), where M refers to the MLEV supercycle, P to the 5-step Tycko cycle, and F to frequency-switched pulses. A distinctive design feature of this family is the use of coherent frequency switching in the optimization of broadband inversion composite pulses. For example, one family member consists of composite pulses containing seven rectangular pulses, with flip angles of 1688, 1908, 1928, 1748, 1928, 1908, and 1688, applied at offset frequencies of 4.27, 2.71, 1.37, 0.0, 1.37, 2.71, and 4.27, respectively, in units of B1 , the rf field strength of the pulses. The inversion performance of this sequence is improved significantly by application of the Tycko 5-step phase cycle. Adding a 4-step MLEV cycle leads to a decoupling sequence with a notable improvement in bandwidth, |/(B1)| 4.6, compared to WALTZ-16, GARP-1, and SUSAN-1. This decoupling sequence is referred to as MPF7 (7 refers to the number of pulses in the primitive sequence). The scaling factor for MPF7 is shown in Fig. 3.30a. Another member of the MPF family of sequences is MPF9, which employs nine pulses in the primitive element; the improvement in bandwidth is shown in Fig. 3.30b.
3.5.3 ADIABATIC SPIN DECOUPLING As developments in magnet technology have allowed progressively higher NMR resonance frequencies to be achieved, the demands on the performance of spin decoupling sequences have also increased. For example, increasing the 1H resonance frequency from 500 to 900 MHz results in the requirement for an increase in rf power of a factor of (900/500)2 ¼ 3.24 to maintain equivalent decoupling performance using any of the sequences previously described. Sample heating becomes an increasingly serious problem at high magnetic fields and the conventional hard-pulse methods approach their practical limits. Decoupling sequences that are based on adiabatic inversion pulses are an approach to overcome this limitation. Adiabatic rapid passage represents a way to achieve spin inversion completely different from the composite pulse techniques employed in the WALTZ, GARP, and other conventional spin decoupling sequences, and allows rf power to be used more efficiently. Once a suitable, adiabatic inversion pulse has been chosen, a broadband decoupling sequence can be constructed via the use of the same phase cycles and supercycles already presented.
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CHAPTER 3 EXPERIMENTAL ASPECTS
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NMR SPECTROSCOPY
0.006
a
lc
0 0.006
b
0 0
1
2
3
4
5
6
Ω / (g B1)
FIGURE 3.30 Scalar coupling scaling factor for (a) MPF-7 and (b) MPF-9 decoupling sequences.
Several different decoupling schemes based on different adiabatic inversion pulses have been described. The simplest sequence uses the CHIRP pulse, which consists simply of a linear frequency sweep of the decoupling rf field. An effective decoupling sequence based on this simple pulse is known as CHIRP-95 (72). The adiabatic condition [3.122] typically is violated at the beginning and end of the CHIRP pulse, because sudden switching of the rf field on or off means that the nuclear
211
3.5 SPIN DECOUPLING
magnetization vector does not begin or end in perfect alignment with the effective field. An 80-step phase cycle (5-step Tycko sequence 16-step MLEV supercycle) is quite effective in minimizing the deleterious effects caused by violations of the adiabatic condition. A more complex alternative to the CHIRP pulse is the so-called hyperbolic secant pulse, in which the pulse amplitude and frequency modulation functions are given by [3.135]. Such a pulse is attractive due to its excellent inversion profile. A prominent decoupling sequence based on the hyperbolic secant pulse [3.135] is known as STUD (73). A significant drawback to this approach is the very high peak rf field amplitude necessary for the hyperbolic secant pulse, which leads to a risk in causing arcing in probe circuitry, especially with cryogenically cooled probes. The so-called WURST decoupling sequences provide a popular and practical compromise approach (58). The WURST adiabatic pulse uses a linear frequency sweep and an amplitude modulation function given in [3.136]. The effective decoupling bandwidth, F , of a WURST sequence is related to the sweep width by the quality factor , where ¼ F =F; the value of is dependent on the index n in the WURST shape function, and typically is in the range 0.6–0.9, as determined from theoretical simulations. Inserting the quality factor into [3.139] results in the following expression for the effective decoupling bandwidth F ¼
ðB1 Þ2 p : 2Q0
½3:153
This equation shows the key result that the effective bandwidth of an adiabatic decoupling sequence is proportional to the square of the applied rf field strength, as opposed to the linear relationship that holds for other methods, such as composite pulse decoupling. The quadratic dependence on the rf field strength is the reason for the high efficiency of adiabatic decoupling sequences. The bandwidth also depends inversely on the adiabaticity factor Q0, and linearly on the primitive pulse length p. Increased bandwidths can be obtained by pushing the lower limit of Q0 ¼ 1 and lengthening the pulse duration. However, effective decoupling also requires p 51=JIS , where JIS is the size of the relevant scalar coupling, and minimizing p also is important for reducing cycling sidebands. As with all decoupling sequences, the use of phase cycles and supercycles significantly enhances the decoupling performance of WURST-based sequences. The Tycko 5-step cycle nested inside an MLEV-4 cycle works well. Unlike composite pulse decoupling sequences, in which the only adjustable experimental parameter is the value of the rf field strength B1,
212
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OF
NMR SPECTROSCOPY
the decoupling performance of a WURST sequence will depend on the choice of experimental parameters {Q0, p, n, B1, F}, and can be tailored to satisfy the requirements of a particular application. For example, extremely wide decoupling bandwidths can be achieved by minimizing Q0 and maximizing p and B1 within practical limits, at the expense of increased cycling sidebands; alternatively, cycling sidebands can be minimized by shortening the pulse length p, at the expense of reduced bandwidth. Based on the maximum value of the scalar coupling constant JIS of interest, the pulse duration to satisfy the condition is chosen 1 p < 1=JIS . Increasing the factor p JIS reduces the size of the cycling sidebands, while reducing this factor, by maximizing p, increases the decoupling bandwidth. After choosing the adiabaticity factor Q0 and rf field strength, F is determined using [3.139] and the index n is chosen using the rule of thumb n ¼ p F=2. The linear frequency sweep is conveniently generated by a phase modulation of the WURST pulse, as discussed in Section 3.4.5, and the amplitude is modulated according to [3.136]. Plots of the scaling factors for a typical WURST decoupling sequence are shown in Fig. 3.31; the parameters defining the WURST sequence are given in the figure caption. Unlike the plots of scaling factors shown in Figures 3.28–3.30, the scaling factors for the WURST sequence are plotted as a function of the absolute, rather than relative, resonance offset frequency, because the relationship between the decoupling bandwidth and the rf field strength B1 is not simply linear for adiabatic decoupling. The scaling factor shown in Fig. 3.31a corresponds to a WURST sequence generated using an adiabaticity factor Q0 ¼ 1.2, near the limit at which the sequence can be considered to be adiabatic. By increasing the rf field strength modestly, such that Q0 ¼ 1.5, the scaling factor is significantly improved, as shown in Fig. 3.31b. These results appear to suggest that, if only a modest bandwidth needs to be decoupled, an adiabatic sequence could be used with a very low rf field strength to minimize rf heating. However, this scheme does not work, because the adiabatic condition cannot be maintained with arbitrarily weak rf fields. The practical limit corresponds roughly to a 2-kHz rf field: above this, adiabatic sequences work well, while below this boundary, composite pulse schemes are advisable.
3.5.4 CYCLING SIDEBANDS Although the achievable bandwidth for a given decoupler rf field strength is an important criterion in choosing a decoupling sequence, all decoupling sequences generate cycling sidebands, due to
213
3.5 SPIN DECOUPLING 0.006
a
lc
0 0.006
b
0 0
5
10
15
20
25
Ω / ( g B1)
FIGURE 3.31 Scalar coupling scaling factors for a WURST-40 decoupling sequence, composed of a WURST pulse calculated with a pulse length of 1.25 ms and a frequency sweep of 64 kHz, and assembled into a T5M4 supercycle. (a) The rf field strength of 3.127 kHz, corresponding to an adiabicity factor Q0 ¼ 1.2. (b) The rf field strength of 3.5 kHz, corresponding to Q0 ¼ 1.5.
nonstroboscopic sampling of the FID relative to the decoupling sequence, and minimizing cycling sidebands also is an important criterion. Figures 3.32–3.34 show typical cycling sideband patterns for the decoupling sequences discussed in the preceding sections. Over its
214
CHAPTER 3 EXPERIMENTAL ASPECTS 0.2
OF
NMR SPECTROSCOPY
a
0.1
0
1.0 intensity (%)
b 0.5
0
3
c 2
1 0
–1000
0 frequency (Hz)
1000
FIGURE 3.32 Numerical simulations showing the cycling sidebands for (a) WALTZ-16, (b) GARP-1, and (c) SUSAN-1 decoupling sequences. In each case, the rf field strength was !1/2 ¼ 2 kHz and the decoupler transmitter was placed 500 Hz away from the heteronuclear resonance frequency. The simulations were performed for an IS spin system and a scalar coupling of JIS ¼ 221 Hz, using the COMPUTE procedure (122).
bandwidth, WALTZ-16 clearly provides the minimum sideband intensity. Comparison of GARP-1 versus SUSAN-1, shown in Fig. 3.32, indicates that the increase in bandwidth of SUSAN-1 over GARP-1 comes at the expense of a significant increase in sideband
3.5
215
SPIN DECOUPLING
2
a
C
II
,
d.
c
,
T
'I
I
1.1
I
1
I
r
'ijj
c
(j)
E
2
b
d
I
o -1000
II
,
.I
II
I
'I o
I 1000
, -1000
1111 1111 '11 II'
o
I 1000
frequency (Hz)
FIGURE 3.33 Cycling sidebands are shown for (a) MPF-7 with a field strength of w)/2rr = 2.0 kHz, (b) MPF-7 with a field strength of w,/2rr = 2.8 kHz, (c) MPF-9 with a field strength of wd2rr = 2.0 kHz and a resonance offset of 500 Hz, and (d) MPF-9 with a field strength of w,/2rr = 2.0 kHz and a resonance offset of 11.2 kHz, near the edge of the bandwidth.
intensity. As a general rule, shorter decoupling seq uences generate smaller cycling sidebands. This principle is illustrated in Fig. 3.33, where Fig. 3.33a shows the cycling pattern for MPF7 using a 2-kHz rf field, while Fig. 3.33b shows the result obtained when the rf field strength is increased to 2.78 kHz. Fig. 3.33c shows that the wider bandwidth of MPF9 is accompanied by larger sideband intensity. For many decoupling sequences, the cycling sideband intensity also depends on resonance offset. Simulations indicate that W ALTZ-16, GARP-l, and SUSAN-l exhibit only a very modest offset dependence, whereas the ultra-broadband sequences, such as CHIRP-95, MPFn, and WURST, typically have a much stronger offset dependence. This is illustrated by comparing the sideband patterns calculated for MPF9 . using a 2-kHz rf field strength. Fig. 3.33c is calculated using an offset of 500 Hz, and Fig. 3.33d is calculated using an offset of 11.2 kHz. Clearly, the sideband intensity increases significantly as the edge of the bandwidth is approached in Fig. 3.33d. The cycling sideband patterns of the WURST decoupling sequence described in Fig. 3.31 are shown in Fig. 3.34. Again, the dependence of sideband intensity on resonance
216
CHAPTER 3 EXPERIMENTAL ASPECTS
2
OF
NMR SPECTROSCOPY
a
1
0
intensity (%)
–1
–2
2
b
1
0
–1
–2 –1000
0
1000
frequency (Hz)
FIGURE 3.34 Cycling sidebands are shown for (a) WURST decoupling with a root-mean-square (rms) field strength of !1/2 ¼ 2.0 kHz and a resonance offset of 500 Hz, and (b) WURST decoupling with an rms field strength of !1/2 ¼ 2.0 kHz and a resonance offset of 20 kHz.
offset is demonstrated by comparing Figs. 3.34a, calculated for a resonance offset of 500 Hz, and 3.34b, calculated for a resonance offset of 20 kHz.
3.5.5 RECOMMENDATIONS
FOR
SPIN DECOUPLING
A general conclusion of the decoupling discussion is that no ‘‘best’’ spin decoupling sequence exists; rather, decoupling sequences should be chosen to match the requirements of a given application. For example,
3.6 B0 FIELD GRADIENTS
217
for the modest bandwidth of backbone 15N nuclei in proteins, the WALTZ-16 or GARP-1 sequence is perfectly suitable, even on 900-MHz spectrometers. GARP-1 is also quite useful for decoupling 13C if minimizing cycling sidebands over a limited range of the full 13C bandwidth is desired. For broader decoupling requirements, selection of an adiabatic sequence is appropriate, with the compromise between bandwidth and sideband intensity being taken into account when choosing the parameters for the adiabatic sequence. Some applications require good decoupling performance over a specified frequency bandwidth with minimal perturbation of spins resonating outside the chosen range. Decoupling 13CO and 13C spins from each other is the most common application in protein NMR spectroscopy. Suitable families of decoupling sequences have been developed in which the rf field is windowed or in which the amplitude and phase of the rf field are modulated to obtain the desired selectivity. For example, the SEDUCE-1 selective decoupling sequence is obtained by replacing each rectangular pulse in the WALTZ-16 pulse sequence with the SEDUCE-1 amplitude-modulated selective pulse, described in Section 3.4.4 (74). Incidentally, spin decoupling is somewhat related to the problem of obtaining an isotropic effective Hamiltonian for generating Total Correlation Spectroscopy (TOCSY) coherence transfer via strong coupling. These applications are discussed in Section 4.2.1.2.
3.6 B0 Field Gradients The development of actively shielded probes that can produce highpower field gradient pulses has resulted in new classes of experiments that utilize field gradients for coherence selection, for water suppression, and for mitigation of radiation damping effects. For many years, the disadvantage of using field gradient pulses was that the strong field gradients required disturbed the field frequency lock. However, with the advent of gradient coils that are actively shielded and with advances in lock blanking methods, these problems have been overcome. This section describes basic aspects of field gradient pulses. Applications in solvent suppression are discussed in Section 3.5. Applications in coherence selection, artifact suppression, and frequency discrimination are described in Section 4.3.3. A field gradient pulse is a time period during which the B0 field is made deliberately inhomogeneous. In the simplest and most common application, the B0 field is varied linearly in the z-direction.
218
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
Conceptually, the NMR sample can be envisaged as a column of thin slices (called isochromats) along the z-direction. The spins in each slice experience different magnetic fields and thus different Larmor resonance frequencies. Initially, the spins in each isochromat are phase coherent. As the gradient is applied, the phase coherence between slices is lost due to Larmor precession. After a sufficiently long time, complete dephasing occurs and the net magnetization of the sample becomes zero. The dephasing process caused by the gradient pulse results because the coherences acquire spatially dependent phase. However, the coherence can be refocused by another appropriately applied gradient to generate gradient echoes (75). More formally, the magnetic field produced by the gradient pulse, Bg(z) varies linearly along the z-axis, and can be written Bg ðzÞ ¼ zGz ,
½3:154
where Gz is the gradient strength given in T/m or, more usually, G/cm. With the origin of the z-axis being taken as the center of the sample, the Larmor frequency at any point in the sample, !(z), is given by
!ðzÞ ¼ B0 þ Bg ðzÞ ¼ ½B0 þ zGz ¼ !0 zGz : ½3:155 In the rotating frame, the !0 term vanishes. If the gradient is applied for a time t, then the spatially dependent phase at any position in the sample, (z), is given by ðzÞ ¼ zGz t:
½3:156
Consider the case of applying a gradient pulse of strength Gz for a time t to a system consisting of in-phase single-quantum coherence, e.g., Ix. At any point in the sample the evolution of Ix is given by zGz tIz
Ix ! cosðzGz tÞIx sinðzGz tÞIy :
½3:157
The net x-magnetization across the whole sample is determined by summing (integrating) over all the slices through the sample: Z rmax=2 1 2 sinðGz trmax =2Þ Mx ðtÞ ¼ cosðzGz tÞ dz ¼ ¼ sincðGz trmax =2Þ, rmax rmax=2 Gz trmax ½3:158 where the sample extends over a region rmax/2. This equation represents the decay of the x-magnetization oscillations during the
219
3.6 B0 FIELD GRADIENTS
Mx(t) / M0
1.0
0.5
0.0
–0.5 0
10
20
30
40
50
g Gzrmaxt
FIGURE 3.35 Dephasing of transverse magnetization by pulsed field gradients. The relative amplitude of transverse magnetization is plotted versus Grmaxt to show the effect of applying a pulsed field gradient in the z-direction to transverse magnetization. The solid line represents decay of the oscillatory signal according to [3.158]. The dotted line shows the asymptotic decay Mx(t)/M0 ¼ 2/(Grmaxt).
gradient pulse. First and most obviously, [3.158] demonstrates that application of a stronger gradient causes the magnetization to decay at a faster rate. Second, [3.158] shows that magnetization from nuclei with higher magnetogyric ratios decays faster. In addition, the strengths of gradients required to suppress coherences to a defined level can be estimated. Figure 3.35 shows a plot of Mx(t) versus Grmaxt for this equation. The overall decay for long times is given by 2/(Grmaxt). For example, in the case just noted, the amount of x-magnetization will have decayed to a fraction of its initial value after a time of the order 2/(Grmax ). Consequently, if the requirement was to suppress 1H x-magnetization to 1/1000th of its original value, then a gradient pulse of 2.5 ms with strength 30 G/cm (0.3 T/m) over a sample region rmax ¼ 1 cm would be sufficient. Of course, this is an idealized calculation and, in practice, fine tuning of the duration and the gradient strength is always required. Practically, field gradients can be generated in two ways. First, the shim coils can be used to produce a spatially inhomogeneous field, but this method only produces gradient amplitudes of 1 G/cm.
220
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
Gradients produced with the shim coils are called homospoil pulses. Second, and much more preferable, specialized actively shielded gradient coils can be built directly into a high-resolution NMR probe. Gradients of the strengths previously noted are easily obtainable by this setup. Probe manufacturers now also offer the option of threeaxis gradient systems containing three coils that can apply simultaneous gradients along the x-, y-, and z-directions. Importantly, an x- or y-gradient refers to a magnetic field in the z-direction whose magnitude depends linearly on the x- or y-position in the sample. Dephasing of magnetization by three orthogonal gradients is more rapid than for a single z-gradient, although the efficacy of the orthogonal gradients depends on the sample geometry. For a rectangular solid geometry, dephasing depends on the function 1/( 3GxGyGzVt3), where Gi is the strength of the gradient in direction i and V is the sample volume. Other sample geometries and nonideal effects not considered in this text have been discussed by Kingsley (76). Field gradient pulses also can be applied with an amplitude envelope that is a smooth function of time, rather than rectangular. A rectangular field gradient pulse has infinite slope at time zero and induces currents that generate torque on the gradient coils. If the gradient pulse is strong enough, then the gradient coils can be damaged physically. Shaped field gradient pulses reduce the induced currents and torques on the gradient coils. Generalizing the preceding discussion to include shaped three-axis field gradient pulses, the spatially dependent phase generated by a shaped gradient pulse of duration t applied to a (possibly multiple-spin heteronuclear) coherence is ðr, tÞ ¼ sBg ðrÞt
X
pi i ,
½3:159
i
in which i, and pi are the gyromagnetic ratios and coherence level of each nuclear species, i, contributing to the coherence. The shape factor, s, is defined as Z 1 t AðtÞ dt, ½3:160 s¼ t 0 in which the amplitude profile of the gradient pulse is given by |A(t)| 1. Opposing gradients, i.e., those that either increase or decrease as the z-coordinate increases, have values of s that are opposite in sign. The overall amplitude of the gradient at a point r ¼ (x, y, z) is represented
3.7 WATER SUPPRESSION TECHNIQUES
221
by Bg(r). For convenience, the dependence of on (r, t) and Bg on r will be implied in the following discussion. A magic-angle gradient pulse has gradient strengths adjusted such that G2z ¼ ðG2x þ G2y Þ=2 and the effective gradient direction makes an angle with respect to the z-axis of ¼ 54.78, the magic angle. Such gradients are useful in suppressing effects of the dipolar demagnetizing field, which result from the interaction between the strong water magnetization and the B0 field (77, 78), and provide superior water suppression in certain NMR experiments (79, 80). The multiplequantum echo (also called the multiple-spin echo) signal is observable in a sample prepared in H2O using the pulse sequence 90 x G1 90 x G2 acquire,
½3:161
in which G2 ¼ 2G1 for ¼ {x, y, z} and the total gradient strength is given by Ga ¼ ðG2ax þ G2ay þ G2ax Þ1=2 for a ¼ {1, 2}. The intensity of the multiple-quantum echo signal obtained from this sequence is proportional to (3 cos2 1)/2. Thus, a magic-angle gradient can be calibrated by using this same pulse sequence (80). An initial spectrum is acquired with G1 and G2 set to the desired values, but with the z-gradients equal to zero. Normally, the x- and y-gradients are set nominally equal to each other for convenience, although this is not essential. Next the magnitudes of the x-, y-, and z-gradients are adjusted, while keeping the total gradient strengths G1 and G2 constant, until the multiple-quantum signal is nulled. The resulting values of the gradients satisfy the magicangle condition. This procedure also provides a simple method for calibrating the z-axis gradient strength relative to the x- or y-gradient strength (80).
3.7 Water SuppressionTechniques An overwhelming majority of the studies of biological macromolecules using high-resolution NMR spectroscopy are performed in aqueous solutions. The concentration of 1H nuclei in water is approximately 110 M, in contrast to typical concentrations of macromolecules of 0.1–2 mM. Thus, the equilibrium magnetization of the water 1H spins is approximately 104–105 greater than is the equilibrium magnetization of a single 1H spin in a macromolecule. Detection of the solute signal in the presence of the solvent signal presents a difficult problem because, inevitably, the dynamic range of the electronic components of the spectrometer is limited. If, for example, the signal
222
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
from a 1 mM protein solution is to be digitized with 4-bit precision utilizing a 16-bit analog-to-digital converter, then the water signal must be reduced by at least a factor of 50 prior to acquisition to avoid overflowing the receiver ADC. In addition, even if the water and solute signal are digitized adequately, the solute resonances may be obscured by the tails of the large, broad solvent peak. This problem is particularly severe for multidimensional spectra in which changes in the phase of the solvent signal from experiment to experiment lead to severe distortions of the final spectrum. NMR spectroscopy in aqueous solution also suffers from radiation damping of the solvent signal (81, 82). Following an rf pulse, the water magnetization precesses in the transverse plane and induces a timevarying oscillating current in the coil. The current, in turn, induces an electromagnetic field of the same frequency that acts to rotate the water magnetization back toward the þz-axis. Following a 908 pulse, the time constant for radiation damping is given approximately by (81) RD ¼ ð2M0 QÞ1 :
½3:162
Thus, radiation damping is important for the water magnetization, and not directly for the solute magnetization, because the (unsuppressed) solvent magnetization is orders of magnitude larger. Although water 1H spins have T1 relaxation times of seconds, for a typical high-Q NMR probe, radiation damping following a 908 pulse will return the transverse magnetization to the þz-axis in tens of milliseconds or less. Radiation damping is even more severe in cryogenic probes, in which the 1H coil and preamplifier are cooled to low temperatures to increase sensitivity (and thereby increase Q). Radiation damping severely interferes with the expected evolution of the water magnetization through a pulse sequence. A straightforward method of reducing the resonance signal from H2O is to use D2O as the solvent. Deuterium oxide with a deuteron content of up to 99.999% is commercially available and offers theoretical reduction in the water 1H signal by a factor of 105. The main limitation to this approach is that signals from exchangeable 1H nuclei in the macromolecule are reduced as well. For example, signals from the amide groups in proteins usually are absent in spectra acquired using D2O solutions. Because scalar coupling and dipolar interactions of the backbone amide 1HN spins with amide 15N spins and with 1H spins are critical for backbone resonance assignments and secondary structure analyses (Chapter 9), at least some protein spectra must be acquired in H2O rather than D2O solution. A secondary problem is that inevitable
3.7 WATER SUPPRESSION TECHNIQUES
223
differences in sample conditions between samples in H2O and D2O (principally, pH differences or isotope effects) can complicate comparisons of data acquired in the two solvents. The usual course of action will be to acquire a series of experiments from H2O solution initially (with 5– 10% D2O present for the field frequency lock system), and then to transfer the sample to D2O for a second series of experiments. Even in D2O solution, the HDO resonance resulting from residual 1H spins frequently is further suppressed by one of the techniques described in the following discussions. The general subject of solvent suppression techniques can be divided into three stages: (1) techniques for reducing the solvent signal detected during a single transient acquisition to within the dynamic range of the spectrometer, (2) techniques for further reduction of the solvent signal that results from a single, signal-averaged, experiment, and (3) postacquisition digital signal processing to improve the solvent suppression. The commonly used methods for solvent suppression (dynamic range reduction) in biological samples are (1) presaturation of the solvent resonance during the recycle delay between transient acquisitions, (2) (semi) selective excitation of macromolecule resonances, and (3) dephasing of the solvent magnetization using rf or field-spoiling pulses. Experimental techniques for solvent suppression have been reviewed (83, 84) and are discussed briefly in the following sections.
3.7.1 PRESATURATION The most commonly used solvent suppression technique is presaturation of the solvent signal during the recycle delay using a weak rf field. Presaturation is very simple to implement and is very effective. The main disadvantages are that signals that resonate very close to the solvent signal (principally the 1H spins in proteins) may be partially saturated by the rf field and that saturation transfer may partially saturate exchangeable 1H spins. As shown in Fig. 3.36, the rates of exchange between 1HN spins in proteins and water 1H spins is pH dependent; accordingly, saturation transfer to the 1HN spins in proteins is particularly deleterious above pH 7. The quality of the water suppression obtained by presaturation depends very critically on the homogeneity of the magnetic field and hence on the quality of the shimming. Solvent signals originating outside of the main sample volume also degrade solvent suppression; consequently, modern probes designed for use with water solutions have rf coils with shielded leads. Irradiation times on the order of 1–2 s using rf fields with amplitudes of approximately 50 Hz usually provide adequate water suppression.
224
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
6
log(kintr) (min–1)
4
2
0
–2 0
2
4
6
8
10
pH / pD N
FIGURE 3.36 Intrinsic backbone H exchange rates (123). The intrinsic exchange rate, kintr, is shown for exchange of a backbone HN with (—) H2O or (- - -) D2O as a function of pH or pD. The pD values are corrected for isotope effects; uncorrected pH meter readings would be 0.4 units smaller.
Short irradiation times minimize the effects of saturation transfer, but require higher power rf fields and worsen saturation of signals resonating near the irradiation frequency. For each new sample, the shimming, irradiation frequency, and irradiation power must be adjusted to optimize water suppression.
3.7.2 JUMPRETURN
AND
BINOMIAL SEQUENCES
A variety of techniques have been developed that selectively excite the resonances from the solute while leaving equilibrium magnetization of the water 1H spins relatively undisturbed. Of these techniques, only the 1–1 ‘‘jumpreturn’’ and 1–3–3– 1 binomial sequence have achieved much popularity for protein NMR spectroscopy (85, 86). These techniques can be incorporated into Nuclear Overhauser Effect Spectroscopy (NOESY) pulse sequences in order to observe cross-relaxation between 1H nuclei that exchange rapidly with solvent. In the jumpreturn technique, the final read pulse in a pulse sequence is replaced by the pulse element 90x ––90x . The carrier is
3.7 WATER SUPPRESSION TECHNIQUES
225
placed on the solvent resonance and ¼ 1/(4max), in which max is the offset from the carrier at which excitation is maximized. Jumpreturn and binomial sequences provide nice illustrations of the technique of [2.121] in the analysis of propagators. For the jumpreturn sequence, the propagator is given by U ¼ exp½ið=2ÞIx exp½iIz exp½ið=2ÞIx
¼ exp ieið=2ÞIx Iz eið=2ÞIx
¼ exp iIy , in which the final two lines are obtained by applying [2.121]. Evolution during the jumpreturn element is given by Iy
Iz ! Iz cosðÞ þ Ix sinðÞ:
½3:164
The resonance offset of the solvent 1H spins is zero because the rf carrier is placed on the solvent resonance. No transverse solvent magnetization is generated and, in theory, complete suppression of the solvent signal is obtained. The amplitude of the detected signal for coherences that are not resonant with the rf carrier is proportional to Ix and depends upon the delay through the factor sin(). The spectrum that results has opposite phase above and below the carrier position because the sine is an odd function; however, no linear phase correction is required. In the 1 33 1 technique, the read pulse is replaced by the pulse element x 3 x 3 x x , in which 8 ¼ 908 and ¼ 1/ (2max). The propagator is U ¼ exp½i Ix exp½iIz exp½i3 Ix exp½iIz exp½i3 Ix exp½iIz exp½i Ix :
½3:165
Clearly, for ¼ 0, U ¼ E and no transverse magnetization is generated. The 1 33 1 sequence offers better theoretical suppression of the solvent resonance than the jump–return sequence. The evolution of the density operator for coherences that are not resonant with the rf carrier is quite complicated; for simplicity, consider only the offset at which maximum excitation occurs. In this case, ¼ and the propagator is
226
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
given by U ¼ exp½i Ix exp½iIz exp½i3 Ix exp½iIz exp½i3 Ix exp½iIz exp½i Ix
¼ exp½i Ix exp i3 eiIz Ix eiIz exp i3 ei2Iz Ix ei2Iz
exp i ei3Iz Ix ei3Iz exp½i3Iz ¼ exp½i Ix exp½i3 Ix exp½i3 Ix exp½i Ix exp½i3Iz ¼ exp½i8 Ix exp½i3Iz ¼ exp½ið=2ÞIx exp½iIz ,
½3:166
in which the last four lines are obtained by applying [2.121]. The evolution of the equilibrium density operator is given by Iz
ð=2ÞIx
Iz ! Iz ! Iy :
½3:167
Unfortunately, for other offsets, the amplitude of the detected signal depends upon the delay and a strong linear phase gradient exists across the spectrum. The linear phase gradient leads to baseline distortions that are a particular problem in multidimensional NMR spectroscopy. Consequently, the jump–return sequence has been much more widely applied in multidimensional NMR spectroscopy. Excitation profiles for the jump–return and binomial sequences are shown in Fig. 3.37. Jump–return and binomial pulse sequences are sensitive to pulse and phase imperfections. Optimal water suppression requires careful adjustment of the spectrometer. Pulse lengths and phases usually are adjusted slightly around the theoretical values to maximize water suppression (typically by 0.1–0.3 s and 1–38). In practice, suppression factors of 50–100 are generally obtained and are adequate when combined with postacquisition signal processing techniques. These sequences are most successful when used as the read pulses for relatively simple experiments with few pulses and no extended rf mixing sequences, such as NOESY and Heteronuclear Multiple-Quantum Coherence (HMQC) experiments. These sequences have an advantage over presaturation and the techniques described in the following sections in that the net excitation of the water signal is zero, or nearly so. Because the water signal remains nearly at equilibrium, saturation transfer from the water to the solute is minimized.
227
3.7 WATER SUPPRESSION TECHNIQUES 1.0
a
Mx,y
0.8 0.6 0.4 0.2 0.0 100 b 0
Phase
–100 –200 –300 –400 –500 –600 0.0
0.5
1.0
∆n / ∆nmax
1.5
2.0
FIGURE 3.37 Binomial excitation profiles. (a) Magnitude and (b) phase of the transverse magnetization, Mx,y ¼ ðM2x þ M2y Þ1=2 excited by the (—) jump–return and (- - -) 1 3 3 1 pulse sequences are plotted as function of resonance offset. Resonances on opposite sides of the carrier have inverted phases (not shown).
3.7.3 SPIN LOCK
AND
FIELD GRADIENT PULSES
Recent technological advances in NMR spectrometer hardware have resulted in the development of new methods of solvent suppression using combinations of selective pulses, spin lock purge pulses, and field gradient pulses. These techniques can generate nearly ideal excitation
228
CHAPTER 3 EXPERIMENTAL ASPECTS
a 1H
Selective
b
SL y
(p/2)–x 1H
(p/2)f1 Selective
NMR SPECTROSCOPY
py
(p/2)x (p/2)f
OF
SL y
∆
(p/2)–x (p/2)f2 Selective
Receiver
∆
(p/2)f3 py ∆
∆
Receiver
FIGURE 3.38 Water suppression using spin lock (SL) purge pulses incorporated into Hahn echo pulse sequences. Rectangular or shaped selective pulses are applied to the solvent magnetization and minimally perturb the solute magnetization. Thin and thick solid bars represent 908 and 1808 nonselective pulses, respectively. SL represents a spin lock purge pulse of 1–2 ms duration. Delays () are adjusted as described in Section 3.6.4.2 for the basic Hahn echo experiment. The phase cycles are (a) ¼ (x, x), receiver ¼ (x, x); (b) 1 ¼ (x, x, x, x), 2 ¼ (x, x, x, x), 3 ¼ (x, x, x, x), receiver ¼ (x, x, x, x). Basic phase cycles can be elaborated using CYCLOPS and EXORCYCLE as described for the Hahn echo pulse sequence in Section 3.6.4.2.
profiles with high degrees of water suppression. Large linear phase gradients are avoided and the techniques can be implemented into nearly all NMR experiments. Using these methods, the water signal can be saturated within a few milliseconds. Saturation transfer effects are much smaller than for presaturation; however, unless the recycle delay is very long, some attenuation of the water resonance and consequent saturation transfer to the solute molecule is unavoidable. In addition, techniques have been developed that attempt to maintain the water magnetization close to its equilibrium value and avoid presaturation effects as completely as possible (87, 88). For illustrative purposes, a variety of these water suppression techniques have been incorporated into homonuclear Hahn echo pulse sequences (Section 3.6.4.2) in Figs. 3.38 and 3.39. For probes that lack capability to apply field gradient pulses, water suppression must be achieved using only spin lock purge pulses. In these applications, the water signal is saturated by dephasing due to B1 rf inhomogeneity (Section 3.4.6). In the simplest approach, shown in Fig. 3.38a, a selective 908 pulse is used to rotate the water magnetization into the transverse plane. A hard (nonselective) 908 pulse rotates solute
229
3.7 WATER SUPPRESSION TECHNIQUES 1H x
1H x
y –y
–y
∆
y
acq.
acq.
τ τ τ τ τ ∆
∆
G1
G1
∆
Grad
–y
Grad G1
1H x
G1 y
1H x
y
–y
–y
∆1
∆1+∆2
acq.
–x
∆2
y
y
–y
–y
∆1
∆1+∆2
acq.
∆2
Grad
Grad G1
G1G2
G2
G1
G1G2
G2
FIGURE 3.39 Water suppression using field gradient pulses incorporated into Hahn echo pulse sequences. Thin and thick solid rectangular bars represent 908 and 1808 nonselective pulses, respectively. Thin and thick solid rounded bars represent 908 and 1808 selective pulses of 1–2 ms duration, respectively. Selective pulses are applied to the solvent magnetization and minimally perturb the solute magnetization. Open rectangular bars are the 3–9–19 pulse element 3 x ––9 x ––19 x ––19 x ––9 x ––3 x , in which 13 ¼ 908 and ¼ 1/(2max). Delays () are long enough to permit recovery following gradient pulses. Delays 1 and 2 are adjusted as described in Section 3.6.4.2 for the basic Hahn echo experiment and must be long enough to permit recovery following gradient pulses. Basic phase cycles can be elaborated using CYCLOPS and EXORCYCLE as described for the Hahn echo pulse sequence in Section 3.6.4.2.
magnetization into the transverse plane and rotates the water magnetization back to the longitudinal axis. Finally, a long spin lock purge pulse is applied phase shifted by 908 relative to the other pulses. Assuming that the selective and nonselective 908 pulses are applied with x-phase, the solute magnetization is described by an operator Iy, and the solvent magnetization is described by an operator Sz prior to the purge pulse. If the purge pulse is applied with y-phase, then the solute operator commutes with the rotation operator for the pulse
230
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
and no evolution occurs (other than relaxation). Solute magnetization is said to be spin locked by the pulse. However, solvent magnetization is orthogonal to the spin lock pulse and is dephased by the inhomogeneity of the rf field as described in Section 3.4.6 (hence the name ‘‘spin lock’’ purge pulse). The original implementation of this technique used a rectangular soft pulse for the selective 908 pulse; subsequent elaborations of the principle have used shaped selective pulses and have extended the sequence as shown in Fig. 3.38b to improve the water suppression (89). If the probe is capable of applying field gradient pulses, then additional control over and manipulation of the solvent signal are possible. Commonly utilized experimental techniques for solvent suppression fall into two categories. The first approach is similar in spirit to the spin lock purge pulse techniques in that a strong field gradient pulse is used to preferentially dephase the solvent signal (90). The second approach is similar in spirit to jump–return and binomial sequences in that the water magnetization is returned to the z-axis prior to acquisition (88). An example of the former technique, which has been designated WATERGATE (90), is illustrated in Fig. 3.39a. Following the initial nonselective pulse, a strong gradient pulse dephases both solvent and solute magnetization. Solute magnetization is unaffected by the selective pulses (assuming the solute resonances of interest are outside the bandwidth of the selective pulses). The nonselective 1808 pulse inverts the coherence order of the solute magnetization; therefore, the second gradient pulse rephases the solute magnetization to form a spin echo. In contrast, the combination of the selective 908 pulses and the nonselective 1808 pulse leaves the coherence order of the solvent magnetization unchanged; therefore, the second gradient pulse continues to dephase the solvent magnetization and no echo is formed. The ‘‘soft–hard–soft’’ pulse sequence element can be replaced by a binomial 1808 pulse; this has the advantage that the selective 908 pulse does not need to be calibrated. The sequence shown in Fig. 3.39b uses the ‘‘3–9–19’’ pulse for this purpose. The 3–9–19 pulse element is 3 x ––9 x ––19 x – –19 x ––9 x ––3 x , in which 13 ¼ 908 and ¼ 1/(2max). The excitation profiles for the two pulse sequence elements are shown in Fig. 3.40. The periodicity of the excitation profile of the binomial sequence often results in reduced amplitude of downfield 1HN or Trp 1H"1 resonances in proteins. Superior water suppression is obtained using a double-gradient echo technique called ‘‘excitation sculpting’’ (91). A pulse sequence for this experiment is shown in Fig. 3.39c.
231
3.7 WATER SUPPRESSION TECHNIQUES
a
b
4
3
2
1
0
Offset (kHz)
FIGURE 3.40 Water pulse excitation profiles shown for the (a) 3–9–19 pulse and (b) soft–hard–soft 90 x –180 x –90 x pulse sandwich. The 3–9–19 pulse used a delay ¼ 278 s and the soft 908 pulses had a length of 1 ms. Data were acquired on a 500 MHz NMR spectrometer.
An example of the second technique, which has been called ‘‘water flip-back’’ (88), is shown in Fig. 3.39d. The sequence is similar to the sequence of Fig. 3.39c except that a selective 908 pulse is inserted prior to the initial nonselective 908 pulse. Similar modifications are possible for the pulse sequences of Figs 3.39a and 3.39b. Following the nonselective 908 pulse, the solute magnetization is described by an operator Iy, and the solvent magnetization is described by an operator Sz. The field gradients flanking the 1808 pulses allow formation of a spin echo for solute magnetization (92). Any transverse solvent magnetization created
232
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
from nonideal performance of the selective 1808 pulse and the nonselective 1808 pulse is dephased by the field gradient pulses. The bulk of the water magnetization is aligned along the z-axis and is never saturated in this experiment. Radiation damping effects are minimized because any transverse solvent magnetization is dephased by the gradient pulses. In practice, 60–80% of the equilibrium water magnetization can be maintained along the z-axis and 430% increases in sensitivity are obtained for exchangeable solute 1H spins due to reduced saturation transfer effects (87, 88). A subtle difference between the WATERGATE and flip-back experiments concerns the strength of the gradient pulses. In the WATERGATE experiment, the bulk solvent signal must be dephased and strong (20–30 G/cm) gradients must be employed. In the flip-back experiments, only the fraction of the solvent signal that is not returned to the z-axis must be dephased and relatively weaker (5 G/cm) gradients are satisfactory. WATERGATE and flip-back experiments frequently need to rotate the water magnetization from the þz-axis to the transverse x–y plane (‘‘flip-down’’) or from the transverse plane to the þz-axis (‘‘flipup’’) using relatively long selective pulses. At high static magnetic fields and especially with high Q probes, selective flip-down and flip-up pulses must be calibrated independently, because the effects of radiation damping oppose the former and reinforce the latter. Examples of various water suppression techniques are given in Fig. 3.41. As described in Section 3.6.4.2, improved water suppression frequently is obtained if Hahn echo sequences are incorporated into the experimental pulse sequence. An example of the improved water suppression obtained is given in Fig. 3.54. The purpose of the selective pulses in Figs. 3.38 and 3.39 is to independently manipulate magnetization components from solute and solvent 1H spins. If the solute is isotopically enriched with 13C or 15N, then evolution under the heteronuclear scalar coupling Hamiltonian can be used to produce solute magnetization that is orthogonal or inverted relative to the solvent 1H spins (and uncoupled solute 1H spins). Spin lock purge pulses or field gradient pulses can be applied to preferentially dephase the solvent magnetization without recourse to selective pulses (93). Such techniques are commonly utilized in multidimensional heteronuclear NMR spectroscopy (Chapter 7).
3.7.4 POSTACQUISITION SIGNAL PROCESSING Although numerous techniques have been proposed for postacquisition water suppression, the only technique in common usage at present
233
3.7 WATER SUPPRESSION TECHNIQUES
a
b
c ~
10
8
6
4 1H
2
0
(ppm)
FIGURE 3.41 Water suppression using (a) presaturation, (b) jump–return, and (c) excitation sculpting techniques. Additional postacquisition water suppression can be obtained using a digital low-pass filter as illustrated in Fig. 3.42.
is the convolution difference low-pass filter technique (94). In this approach, the low-frequency components are filtered from the signal by constructing the data sequence m m .X X s~ðktÞ ¼ bj sð½k jtÞ bj ½3:168 j¼m
j¼m
in which bj are filter coefficients. The cosine-bell filter function, bj ¼ cos[j/(2m þ 2)], works well in practice. Typical values of m range between 8 and 32. The filtered data set is then obtained as sðktÞ s~ðktÞ If the experimental data consists of points s(kt) for
234
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
k ¼ 0, . . . , N 1, the filtered data set consists of points m, . . . , N m 1. The missing m points at the end of the data sequence are generally unimportant because the original data set can be truncated; however, the first m points must be reconstructed to avoid large distortions in the resulting filtered spectrum. In the original approach, the first points were obtained by linear extrapolation (94); other approaches have included linear prediction of the first m points (95) or acquisition of data points for k5 0 using a Hahn echo pulse sequence (96). In most applications, the carrier is positioned at the frequency of the water signal. Consequently, the water resonance has an offset frequency of zero and the above protocol filters the water signal from the spectrum. If the carrier is positioned elsewhere in the spectrum, the water frequency is shifted digitally to zero by multiplication by a complex exponential function as shown by [3.20]. The convolution filter is applied to suppress the water signal and the original frequency reference is restored by multiplication by the complex conjugate of the original exponential function. The use of the digital low-pass filter for water suppression is illustrated in Fig. 3.42. As shown, the water signal is almost completely removed, although some distortion of the baseline is obtained near the location of the water signal. In all likelihood, improved digital filtration techniques will be developed (or adapted) for use in NMR spectroscopy.
3.8 One-Dimensional 1H NMR Spectroscopy Nearly all experimental investigations begin with one-dimensional NMR experiments to assess the suitability of the sample for more detailed characterization by multidimensional NMR spectroscopy. In addition, the procedures used to acquire a one-dimensional experiment, including temperature calibration, tuning, shimming, and measurement of pulse lengths, are equally necessary for multidimensional experiments.
3.8.1 SAMPLE PREPARATION Sophisticated NMR experiments only rarely can compensate for an ill-behaved or ill-prepared NMR sample. Accordingly, in nearly all investigations, preliminary experiments must be performed to determine sample conditions that satisfy the following criteria: 1. The protein must be in a native, functional conformation (unless unfolded or intermediate states of the protein are the focus of investigation). Ideally, the pH and solvent composition should be
3.8
ONE-DIMENSIONAL IH
NMR
235
SPECTROSCOPY
a
b
10
8
6
4
2
o
1H (ppm) 3.42 Low-pass filter. An example of postacquisition water suppression using a low-pass digital filter is illustrated for ubiquitin. (a) Jump-return spectrum of ubiquitin displays a large-amplitude residual solvent peak with dispersive tails that obscure nearby resonances. (b) Following filtering in the time domain, the water signal has been suppressed dramatically. Only a small distorted region exists at the position of the water signal. FIGURE
close to physiological conditions so that the structural and dynamic observations reflect a functionally relevant state of the protein. 2. Solubility must be sufficient to permit spectra with satisfactory signal-to-noise ratios to be acquired in reasonable time periods. For conventional probes and sample tubes, 0.4-0.6 mI of a protein solution of approximately I mM concentration (in a 5-mm NMR tube) will be required for studies at 500 MHz. Sample volumes can be reduced to approximately 0.2 ml in low-volume NMR tubes.
236
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
If cryogenically cooled probes are available, sample concentrations can be reduced, depending on the ionic strength of the sample buffer. 3. Protein samples used for NMR spectroscopy must be free of contaminants arising from the NMR tube or the protein preparation. The protein should be monodispersed (unaggregated) at the concentrations required for NMR spectroscopy and stable for time periods longer that the desired NMR experiments. 4. Ideally, NMR spectroscopy should be performed under sample conditions that yield optimal spectra (maximal resonance dispersion and minimal linewidths). Temperature, pH, concentration, and buffer composition affect dramatically the solubility, aggregation state, and stability of proteins, and must be optimized empirically. Sample preparation for NMR spectroscopy has been the subject of extensive review articles (97, 98).
3.8.2 INSTRUMENT SETUP Temperature calibration, tuning, shimming, and pulse length calibration are preludes to data acquisition for all one-dimensional and multidimensional NMR experiments. Furthermore, information about longitudinal and transverse relaxation rate constants is useful for setting recycle delay times and anticipating the overall sensitivity of particular NMR experiments. 3.8.2.1 Temperature Calibration Most commercial spectrometers have some means of controlling the probe temperature, and this hardware provides a coarse estimate of the actual sample temperature. More accurate schemes have been developed that make use of the temperature-dependent chemical shifts of methanol and ethylene glycol to calibrate the actual temperature of a sample in the probe (99). Over the range 250–320 K, the difference in chemical shift between the methyl and hydroxyl resonances of 100% methanol, (in ppm), is given by T ðKÞ ¼ 403:0 29:53 23:87ðÞ2 :
½3:169
Over the range 300–370 K, the difference in chemical shift between the methylene and hydroxyl resonances of 100% ethylene glycol, (in ppm), is given by T ðKÞ ¼ 466:0 101:6:
½3:170
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
237
The chemical shift of the 1H spins in H2O has a slight dependence on pH (0.02 ppm/pH unit), and a more dramatic dependence on temperature. The chemical shift of the water resonance is given by (100, 101) ðH2 OÞ ¼ 7:83 T=96:9 ppm,
½3:171
in which the temperature is measured in Kelvin and pH ¼ 5.5. 3.8.2.2 Tuning In order to efficiently deliver rf energy into the sample volume and to sensitively detect precessing transverse magnetization, the probe circuitry must be tuned so that the resonant frequency of the circuit is equal to the rf frequency. In addition, the impedance of the coil must be matched to the impedance of the spectrometer electronics ([3.2]). The probe circuitry is tuned and matched by adjusting two capacitors mounted near the coil. The principle is simple: the coil is driven by an rf input and the response is observed as the fraction of reflected power (i.e., the rf power not transmitted into the sample volume). The capacitors are adjusted interactively to optimize the response. Two methods are commonly used to tune and match the probe; the two methods differ in the way in which the rf is applied and the response is detected. Increasingly, the capability to tune and match the probe is built into NMR spectrometers. In the first method, illustrated in Fig. 3.43a, a sweep generator, a 50- load, an oscilloscope, and the probe are connected to the terminals of an rf bridge. The sweep generator generates an rf field with a frequency that is cyclically varied in time. As the rf frequency ‘‘sweeps’’ through the resonance frequency of the coil circuit, the sides of the rf bridge become more balanced, and a dip (or peak, depending on the input polarity) is observed on the scope. The horizontal position of the dip indicates the resonance frequency of the coil (typically, an rf reference source is also displayed on the oscilloscope in order to calibrate the display); the depth of the dip is a measure of the match between the impedance of the circuit and the 50- load. The tuning and matching capacitors are adjusted until the resonance frequency of the coil equals the desired value and the impedance is optimally matched (as indicated by a maximum in the depth of the dip). In the second method, illustrated in Fig. 3.43b, a fixed-frequency rf source and a voltage standing-wave ratio (VSWR) rf power meter are connected to the probe through a directional coupler. The signal measured on the power meter is proportional to the power reflected from the probe; as the tuning and matching of the probe improve, the amount of reflected power decreases, because more power is being transmitted
238
CHAPTER 3 EXPERIMENTAL ASPECTS
a
NMR SPECTROSCOPY
b Sweep Generator
50 Ω Load
OF
RF Source Forward
RF Bridge
Probe
Probe
Directional Coupler Reflected
Oscilloscope
RF Power Meter
FIGURE 3.43 Tuning circuits employing the (a) sweep generator and (b) monochromatic rf source. Details on the use of these circuits are given in the text.
into the sample volume. Accordingly, the tuning and matching capacitors are adjusted to minimize the reflected power. If the probe is very poorly tuned, or if one is searching for a new nuclear resonance signal on a broadband probe, the sweep generator configuration is superior because tuning and matching responses are monitored independently. Once the probe is nearly tuned and matched, the second approach typically is more sensitive. Minimum pulse lengths for a given amplifier power level are obtained only if the probe is properly tuned and matched. A historical record of pulse lengths previously measured for the same sample and probe should be maintained (pulse lengths for different samples are a function of differences in ionic strength). A measured pulse length that deviates by more than a few percent from previous values indicates that either the probe has been improperly tuned or that a problem has developed with the transmitter circuitry. 3.8.2.3 Shimming The homogeneity of the static magnetic field is of paramount importance for high-resolution NMR spectroscopy. The natural magnetic field of a superconducting magnet is not sufficiently homogeneous for high-resolution spectroscopy. Accordingly, a necessary task prior to performance of an NMR experiment is the careful
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
239
adjustment of the magnetic fields produced by a set of auxiliary roomtemperature electromagnets to compensate the inhomogeneity of the main static field. This process is known as shimming. Two articles devoted to the process of shimming form the basis for the present discussion (102, 103). The spatial variation of the static magnetic field within the bore of the magnet satisfies the Laplace equation and can be described by an expansion in orthogonal spherical harmonic functions, rn Pnm ðcos Þ cos½mð’ ’nm Þ: B0 ðr, , ’Þ ¼
1 X n X n¼0 m¼0
cnm
r n a
Pnm ðcos Þ cos½mð’ ’nm Þ,
½3:172
in which a is the radius of the magnet bore, cnm and ’nm are constants, and Pnm(cos ) are the associated Legendre polynomials. The values of n and m for which cnm are nonzero and the particular values of cnm and ’nm are determined (depending on one’s point of view) by the empirical variation in the magnetic field or by the solution to the Laplace equation subject to the (complicated) appropriate boundary conditions inside the magnet bore. If the field were perfectly homogeneous, the only nonzero cnm in [3.172] would be c00, the amplitude of the static field. In principle, the inhomogeneity in the static magnetic field can be compensated identically by generation of additional magnetic fields with spatial variation described by the spherical harmonic functions and amplitude given by cnm. Each additional magnetic field would negate one of the terms in [3.172] and the number of auxiliary fields needed would be determined by the desired degree of compensation. In practice, the auxiliary fields are produced by specially designed electromagnets and the magnitudes of the fields are proportional to the applied electric currents. The shim coils are conventionally described using Cartesian rather than spherical coordinates; the spherical harmonic functions corresponding to shim coils frequently encountered on NMR spectrometers are given in Table 3.1. The functions of degree m40 appear in pairs with x and y interchanged in order to reproduce the phase dependence, ’nm, in [3.172]. In reality, shim magnets cannot be designed or fabricated to produce a pure spherical harmonic magnetic field; consequently, the field produced by the shim coils is described by B0 ðr, , ’Þ ¼
N X k¼1
bk Bk ðr, , ’Þ,
½3:173
240
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
TABLE 3.1 Shim Coil Spherical Harmonic Functions Order (n) 1 2 3 4 5 1 1 2 2 3 3 2 2 3 3 3 3
Degree (m)
Shim name
Functions
0 0 0 0 0 1 10 1 10 1 10 2 20 2 20 3 30
z z2 z3 z4 z5 x y zx zy z2 x z2 y x2 y2 xy z(x2 y2) zxy x3 y3
z 2z2 (x2 þ y2) z[2z2 3(x2 þ y2)] 8z[z2 3(x2 þ y2)] þ 3(x2 þ y2)2 48z3[z2 5(x2 þ y2)] þ 90z(x2 þ y2)2 x y zx zy x[4z2 (x2 þ y2)] y[4z2 (x2 þ y2)] x2 y2 xy z(x2 y2) zxy x(x2 3y2) y(3x2 y2)
in which N is the number of shim coils, Bk(r, , ’) is the spatial field produced by the kth shim coil, and bk is the amplitude of the field determined by the electric current applied to the electromagnet. Each Bk(r, , ’) is given by an expansion similar to [3.172]; as a result, the terms in [3.173] are not orthogonal and no one-to-one correspondence exists between the terms in [3.173] and the terms in [3.172]. In order to overcome this difficulty, some types of shim systems rely on a matrix design, in which the desired ideal field profiles are generated by supplying current to more than one physical coil. The potential advantage of a matrix shim system is that the difficulties of manufacturing a set of shim coils that generate spatial field profiles with the desired, pure spherical harmonic functional forms can be overcome; this is done by allowing the ideal shim profiles to be approximated more accurately by an appropriately weighted superposition of a subset of the field profiles generated by the actual physical shim coils. The necessary values of the matrix elements that determine the distribution of current in the physical coils are established and set by the manufacturer. Thus, current adjustments are applied to multiple physical coils when adjusting the value of an idealized shim in a matrix
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
241
shim system. From a practical point of view, the main reason one may need to know whether a matrix design is in use is for hardware debugging purposes: in a matrix shim system, a bad current supply or broken coil can affect more than one of the apparent shim components Bk(r, , ’). Empirical adjustment of the shim coils is performed using the magnitude of the field frequency lock signal, the decay envelope of the FID, and the lineshape in the frequency domain spectrum as measures of the homogeneity of the magnetic field. The lock signal is the simplest parameter to observe; because the integral of the resonance signal is constant, the lock signal increases in magnitude as the field homogeneity improves and the deuterium lineshape becomes narrower. However, the magnitude of the lock signal does not indicate the quality of the lineshape obtained. In addition, the lock system must be properly calibrated if proper shimming is to be obtained. The phase of the lock receiver must be adjusted to yield a purely dispersive signal. Drifts in the magnetic field are detected most sensitively if the lock signal is dispersive, because the dispersive lineshape has a null at the exact resonance frequency [3.30]. The power level must be adjusted to yield a signal with adequate signal-to-noise ratio without saturating the deuterium resonance. If the FID can be displayed in real time, the magnetic field can be shimmed by observing the shape of the decay envelope. Optimal homogeneity implies that the decay envelope is exponential with a maximal decay time constant. Unlike the lock signal, the FID is a direct indication of the quality of the lineshape. Shimming using the FID is easiest if the decay is dominated by a single resonance line; thus, for samples in water, the magnet can be shimmed using the water resonance. In this case, the probe should be detuned while shimming, to avoid radiation damping that distorts the shape of the FID (Section 3.5). After the magnet is shimmed, the probe is retuned (after tuning, the shims may need additional minor adjustments). The ultimate measures of the homogeneity of the magnetic field are the lineshape and resolution in the spectrum and quality of the solvent suppression. When beginning a research project on a new biomolecule, a pair of closely spaced resonances, preferably representing a scalar coupled multiplet, should be identified; the pair can be used subsequently to monitor shimming. That is, if a particular multiplet is resolvable when the magnet is properly shimmed, then the quality of the shimming can quickly be checked by examining the degree of resolution of the multiplet lineshape. External samples also can be used to check the homogeneity of the magnetic field. Samples of basic pancreatic trypsin inhibitor (BPTI),
242
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
ubiquitin, or tryptophan are useful in this regard. Unlike resolution test samples used by instrument manufacturers, these molecules can be dissolved in aqueous solutions with buffers and ionic strength matched to the conditions of the sample of interest. For BPTI, resolution can be evaluated by examining the tyrosine 23 1H" multiplet at 6.3 ppm. For ubiquitin, resolution can be evaluated by examining the leucine 50 1H multiplet at 0.17 ppm or the phenylalanine 45 1H multiplet at 7.33 ppm. For tryptophan, resolution can be evaluated by examining the 1H2 triplet at 7.24 ppm. In a high-resolution spectrum, the 4J coupling to 1H"3 should be resolvable almost to baseline and the small (0.5 Hz) 6J coupling to 1H1 should be discernible. The exact manual or semiautomatic protocol that is optimal for empirical shimming (by the operator) of a particular magnet by hand depends upon the complement of shim coils provided by the spectrometer manufacturer. Accordingly, the protocol given in Fig. 3.44 should be taken as a guideline appropriate for the shim set given in Table 3.1. The problem of empirical shimming is simplified somewhat because a given shim field is contaminated primarily by shim fields of lower orders with the same parity (i.e., z3 contains contributions primarily from z, and z4 contains contributions primarily from z2). In addition, the resonance lineshape obtained is frequently indicative of the order of the shim coil that must be adjusted. Misadjustment of the evenorder shims result in asymmetric lineshapes; misadjustment of the oddorder shims result in symmetrical, but non-Lorentzian, lineshapes. Furthermore, the effect on the lineshape is observed closer to the baseline for higher order shims. The problem of shimming reduces to implementation of a reasonable strategy for optimizing the coefficients bk in [3.173] to approximate to [3.172]. Unfortunately, trial-and-error manual adjustments or interative automated routines, in addition to being time-consuming, are prone to finding local optima in the shim values rather than global optima that maximize magnet homogeneity. The traditional methods for magnet shimming just described are being supplanted by gradient-based techniques that were initially developed for shimming magnets used for magnetic resonance imaging (104, 105). These so-called gradient shimming techniques are based on the simple principle of mapping both the magnetic field homogeneity and the magnetic field profiles generated by the various roomtemperature shim coils, and then calculating the optimal values for each shim current needed to minimize the residual field inhomogeneity. If the spatial dependence of the magnetic field inhomogeneity, expressed for simplicity using Cartesian coordinates, is given by Bðx, y, zÞ ¼ Bðx, y, zÞ B0 [where Bðx, y, zÞ is the actual uncorrected
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
Start
Tune Probe
Lock Spectrometer
Lock Phase
Z, Z2
243
Adjust
Z, Z2 Z3, Z Z4
,Z
XZ2, XZ Z, X
2
X, XZ Y, YZ
Z4, Z3, Z
XY, X2-Y2
Z5, Z3, Z
Z, Z2
Repeat Once
XZ, X, Z
YZ2, YZ Z, Y
Z, Z2
Z, Z2 ZXY, XY
YZ, Y, Z
X2-Y2
Z, Z2
Z(X2-Y2), XY, X2-Y2
XY, X
X3, X Y3, Y
Repeat Once
Stop
FIGURE 3.44 Shimming protocol.
magnetic field and B0 is the desired homogeneous field value], and the spatial profile of the magnetic field generated by a given shim coil is given by Si(x, y, z), then the basic goal of any shimming process is to minimize the residual function Bresid ðx, y, zÞ, X ci Si ðx, y, zÞ, ½3:174 Bresid ðx, y, zÞ ¼ Bðx, y, zÞ i
244
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
by adjusting the values of the coefficients ci. The summation is over the number of shim components contained in the shim system. The innovation supplied by the gradient shimming-based techniques is to actually map the functions B(x, y, z) and Si(x, y, z), such that a straightforward and deterministic mathematical procedure (106) can be used to minimize Bresid. Gradient shimming methods are noniterative (or rapidly convergent), are insensitive to local minima in [3.174], and provide a robust, convenient, and time-saving method for optimizing magnet homogeneity. The initial application of gradient shimming in NMR spectroscopy utilized three-axis pulsed field gradient hardware to generate threedimensional maps of the magnetic field and shim coil field profiles (106). In principle, the shim system itself could be used for the imaging experiments required for gradient shimming. However, the original method required that field gradients be switched on and off rapidly (typically less than 1 ms), and that no significant distortions were generated by gradient-induced eddy currents. The shim power supplies are normally designed for high stability in continuous operation, rather than in a pulsed mode, and the shim coils are not shielded, so that problems with eddy currents in pulsed operation would be significant. In addition, typically the shim currents cannot be rapidly switched under pulse sequence control. Some time ago, modified procedures were reported that allowed gradient shimming to be performed with the normal spectrometer shim system (107), and the spectrometer manufacturers have continued to develop and implement such techniques. The present discussion of gradient shimming will focus on the use of pulsed field gradient hardware, although most of the general principles remain the same if the shim system itself is employed. Furthermore, for the sake of clarity, the following discussion will be restricted to one-dimensional gradient shimming of the z-shims (108, 109). The z-shims are typically the ones most in need of routine adjustment from sample to sample, and thus the use of z-axis gradient shimming is currently the most common application of imaging-based shimming techniques. The basic task in gradient shimming is to measure, or map, the magnetic field variation within the sample volume of interest. This mapping is accomplished using the principles of magnetic resonance imaging. In the case of z-axis gradient shimming, the sample volume can be imagined to consist of thin disks stacked vertically on top of one another, with the z-axis running through the center of the disks, perpendicular to the plane of the disks. The z-axis is also defined to be collinear with the direction of the external magnetic field. Mapping of
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
245
the field along the z-axis therefore consists of measuring the average value of the magnetic field within each disk. This is accomplished by measuring an NMR signal for the sample contained within each imaginary disk. The sample should have one component that gives rise to a single resonance and completely dominates the NMR spectrum; for aqueous protein samples in H2O, the 1H resonance of the solvent water provides the requisite signal. If an FID is recorded while a substantial, linearly dependent (with respect to the z-coordinate) magnetic field gradient is applied along the z-axis, the exact resonance frequency of the water 1H nuclei contained within each disk will depend on the position of the disk along the z-axis: !ðzÞ ¼ B0 Gz ¼ !0 Gz,
½3:175
in which B0 is the strength of the external magnetic field, is the magnetogyric ratio for the nucleus being observed, and G is the strength of the applied magnetic field gradient. Equation [3.175] assumes that the applied magnetic field gradient is large enough to completely dominate the inhomogeneity in the external field, so that the actual field profile B(x, y, z) can be approximated by the single value B0. Without loss of generality, the center of the rf coil is assumed to define the origin of the z-axis. Due to the magnetic field gradient, the water resonance will appear as a broad distribution of frequencies in the NMR spectrum. A typical spectrum is shown in Fig. 3.45. For the case of a sample that extends well beyond the confines of the rf detection/transmitter coil, the lineshape of the NMR signal shown in Fig. 3.45 is essentially an ‘‘image,’’ or mapping, of the rf field distribution of the coil: the signal dies away at the edges of the spectrum, due to the drop-off in the rf field strength and the detection sensitivity at the extremes of the effective volume of the coil. If the volume of the sample is restricted and does not completely fill the active volume of the rf coil, then the NMR lineshape would in principle be an image of the physical location of the sample. While the horizontal axis in Fig. 3.45 is in frequency units, this can be directly converted to distance units along the z-axis, via [3.175], and assuming that the rf transmitter frequency coincides with the Larmor frequency of the water resonance in the absence of the field gradient. Thus, application of the magnetic field gradient allows spatial resolution of the water signal along the z-axis. In practice, the lineshape profile depends on the linearity of the applied z-axis magnetic field gradient as well as the rf field distribution within the coil volume. Assuming that the rf field distribution is
246
CHAPTER 3 EXPERIMENTAL ASPECTS
–40
–20
0 Frequency (kHz)
OF
20
NMR SPECTROSCOPY
40
FIGURE 3.45 The 1H NMR spectrum of an H2O sample, recorded at 600 MHz using the pulse sequence shown in Fig. 3.46a, with ¼ 0. The spectrum is plotted in magnitude mode. The spectrum corresponds approximately to a 1D image projection of the effective transmitter/receiver coil volume along the z-axis, which is defined as the direction of the static external magnetic field B0. The sample was contained in a standard 5-mm-diameter NMR tube; the left-hand side of the spectrum corresponds to the lower half of the effective sample volume (i.e., toward the bottom of the NMR tube). The absence of a flat plateau across the spectrum results from nonlinearity of the z-axis pulsed field gradient.
reasonably homogeneous, nonlinearity of the z-axis magnetic field gradient can result in distortions of the expected flat top profile and give a misleading impression of an unequal distribution of sample within the rf coil. The data shown in Fig. 3.45 were taken on a 9-year-old probe; current probes might exhibit a much flatter profile. In any case, linearity of the field gradient is not required for gradient shimming to work; the necessary condition is that the applied magnetic field increases monotonically as a function of the position along the z-axis. Having achieved spatial resolution via the application of the z-axis magnetic field gradient during acquisition of the NMR signal, the next step is to measure the actual spatial dependence of the static magnetic field within the active rf coil volume. This is accomplished via the measurement of a so-called phase map. Assume an rf pulse is applied to the sample to create transverse magnetization, and that a period of free precession is allowed to occur. The precession frequency, for a given
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
247
nucleus, will be directly proportional to the value of the local magnetic field strength. If the length of the precession period is given by , then the phase accumulated by the spin isochromats is given by ðzÞ ¼ !ðzÞ þ 0 ¼ ½B0 þ BðzÞ þ 0 ,
½3:176
in which the axial field variation, BðzÞ, resulting from inhomogeneity of the static external magnetic field over the active sample volume, is included explicitly. In this case, BðzÞ is the average field inhomogeneity over the x–y plane located at the given position along the z-axis and is the term to be measured and ultimately minimized via shimming adjustments. The constant term ’0 accounts for the net contribution from precession during periods other than . If the experiment is repeated twice, using different values of the free precession period 1 and 2, then the axial field variation can be determined from [3.176] to be BðzÞ ¼ B0 þ
½’2 ðzÞ ’1 ðzÞ ; ¼ B0 þ ð 2 1 Þ
½3:177
in which ðzÞ ¼ 2 ðzÞ 1 ðzÞ and ¼ 2 1 . Typically, 2 > 1 0. Thus, by mapping the z-dependence of the phase of the NMR signal, the desired map of the magnetic field inhomogeneity can be determined. Suitable pulse sequences commonly employed to perform the basic experiment just described are shown in Fig. 3.46. Both sequences in Fig. 3.46 result in the formation of an echo signal during the detection period, the purpose of which will be explained in the following discussion. The first sequence, Fig. 3.46a, is based on the use of a spin echo. Immediately after the initial 908 excitation pulse, a z-axis pulsed field gradient is applied, which is then followed by the variable precession period . A 1808 refocusing pulse is then applied, which will lead to the formation of a spin echo. At this point in the sequence the ‘‘read’’ gradient is turned on, and the FID is recorded during this period. The timings are arranged such that the maximum of the echo formation occurs in the middle of the data acquisition period, and that the signal is not truncated at the beginning or end of this period. The pulse sequence in Fig. 3.46b is essentially the same as that in Fig. 3.46a except that a gradient-recalled echo is employed instead of a spin echo; the gradient-recalled echo is achieved by inverting the sign of the initial pulsed field gradient, relative to the ‘‘read’’ gradient. Although either sequence can be used to achieve the desired result, the spin echo experiment has at least one practical advantage. The magnetic field inhomogeneity causes dephasing of the NMR signal during the full
248
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a rf tacq
+ d
t
d
d
tacq
+ d
t
d
d
tacq
2 grad
b rf tacq 2 grad
FIGURE 3.46 Pulse sequences for z-axis gradient shimming. Narrow and wide pulses on the rf channel represent 908 and 1808 flip angles, respectively. Gradient pulses, represented by the gray boxes, are applied along the z-axis; the first gradient pulse in sequence b is applied with the sign opposite to that used for the second (‘‘read’’) gradient. The total acquisition time, tacq, should be set such that the entire, observable echo signal is recorded. The delay ð1 ms) allows for dissipation of transient effects when the gradient pulses are turned on and off. FIDs are recorded for two different values of the delay to allow the desired measurement of the signal phase evolution.
period of the gradient-recalled echo experiment, whereas the effects of the field inhomogeneity are minimized in the spin echo experiment due to the refocusing effect of the 1808 pulse. A common variation of these pulse sequences is to reverse the order of the initial gradient pulse and the variable precession period, in order to eliminate potential problems with gradient ring-down effects during the variable precession period. However, the sequences as drawn in Fig. 3.46 have the advantage that radiation damping of the water magnetization is suppressed by the application of the gradient pulse immediately following the initial excitation rf pulse. The pulse sequences shown in Fig. 3.46 record the entire echo signal, from the initial buildup, through the echo maximum, and past where the echo has decayed away. If the time origin t ¼ 0 is placed at the position
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
249
of the echo maximum, then the NMR signal is effectively recorded for the entire period 1 5 t 5 þ1 (remembering that data collection is arranged such that the signal decays into the noise at the beginning and end of the detection period). Thus, the time-domain signal is no longer causal, and Fourier transformation results in an NMR spectrum that has no dispersive component (110). The absence of a dispersive component has two important consequences for application to gradient shimming. First, the phase of the NMR signal as a function of the position along the z-axis, which provides the phase maps ’i ðzÞ needed in [3.177], can be determined by taking the arctangent of the ratio of the real and imaginary components of the NMR spectrum. Second, a magnitude calculation (given by the square root of the sum of the squares of the real and imaginary components) applied to the frequency-domain NMR spectrum will result in a true image of the active coil volume or spin density, as shown in Fig. 3.45. In addition, collection of the full echo signal may result in increased sensitivity, by up to a factor of two depending on the signal loss incurred by delaying acquisition of the FID to allow an echo to form (110). A typical spectrum collected with the pulse sequence in Fig. 3.46b is shown in Fig. 3.45. Several parameters must be optimized in performing the experiments shown in Fig. 3.46. A practical issue concerns the length of the variable delay 2. If the sequences as drawn in Fig. 3.46 are employed, only a small decay of the signal intensity occurs between the two spectra acquired for 1 and 2. If the variable precession period is inserted before the first gradient pulse (or the gradient-recalled echo sequence, Fig. 3.46b) is employed, a substantial reduction in intensity will be observed for the spectrum recorded with the longer 2 value relative to that for 1, due to signal decay from the combined effects of spin relaxation, radiation damping, translational diffusion, convection, and magnetic field inhomogeneity. On the one hand, a precession period should be chosen in order to maximize the phase difference ðzÞ and therefore optimize the accuracy of the measurement of the field inhomogeneity. On the other hand, decay of signal intensity reduces the precision of the phase information that is extracted from the spectra. Maximizing the signal-to-noise ratio of the spectra is important for obtaining the best results. If a spin echo is used (Fig. 3.46a) and/or radiation damping is suppressed, very large values of can be chosen; otherwise, a reasonable rule-of-thumb in balancing the desire to maximize without giving up too much in spectral sensitivity is to choose 2 such that the spectrum collected at 2 is approximately half the intensity of the spectrum collected at the short time point 1 0. In addition, the phase values are determined from taking the inverse
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of sinusoidal functions, and are therefore restricted to the range 08–3608. At some point, as the precession period is increased, a so-called phase-wrap will occur, where the actual value of falls outside the 08– 3608 range and is aliased or wrapped back into that range; such an occurrence would obviously lead to an incorrect determination of the residual magnetic field B. The simplest solution to this problem is to restrict the duration ; however, this restriction compromises the accuracy of the measurement. Instead, algorithms can be devised for unwrapping the phases, thereby freeing up the user to optimize according to the above criteria. Along the same line of reasoning, the strength of the pulsed magnetic field gradients used in the pulse sequences (Fig. 3.46) must be large enough to provide adequate spatial resolution of the field map and to dominate the initial static field inhomogeneity, but not so large as to lead to an unacceptable loss in spectra sensitivity, which would result from spreading the spectrum over too wide a frequency range. Finally, the width of the spectral window used for the field mapping must be chosen. The wider the window, the larger the sample volume that can be mapped. On the other hand, the window should not be extended to the point where the signal has decayed essentially to zero, as the resultant errors in the phase map will be propagated into errors in establishing optimal values for the shim currents. The preceding discussion has focused on mapping the magnetic field inhomogeneity, BðzÞ, within the active sample volume. However, according to [3.175], the field profile generated by each of the shim coils considered in the gradient shimming process, Si ðzÞ, must be determined (remembering that we are limiting our discussion here to z-axis shimming). Because the actual spatial dependence of the shim fields are mapped, the profiles are not required to have ideal shapes (i.e., deviations of the Z2 shim profile from the ideal z2 dependence are easily tolerated). Mapping the field profiles generated by the shim coils is done in exactly the same fashion as discussed previously for the magnetic field inhomogeneity. Spectra similar to those of Fig. 3.45 are collected, using a pulse sequence such as one of those shown in Fig. 3.46, for two different shim settings for each of the shim coils being considered. Using the resulting phase maps and [3.177], the shim profiles are established. Typical results are shown in Fig. 3.47 for the Z1–Z6 shim coils of a modern NMR spectrometer. The horizontal axis is labeled in frequency units, but this is equivalent to a z-axis scale, via [3.175]. The ideal shapes can be recognized for most of the shim profiles, i.e., the Z1 shim gives a linear dependence, Z2 a parabolic dependence, Z3 a z3 dependence, and so on; the results shown in Fig. 3.47 also indicate that impurities exist
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
251
40 Z3
Field (arb. units)
20
Z2 Z6
0
Z5 Z1
–20 Z4 –40 –30
–20
–10
0 Frequency (kHz)
10
20
30
FIGURE 3.47 Typical z-axis shim maps determined using the pulse sequence of Fig. 3.46a, with values of 0 and 15 ms. Shim maps are shown for the Z1–Z6 shim coils.
in the various profiles. Practical experience has been that the Z6 shim usually deviates significantly from the ideal result. Once the shim profiles have been determined, calculating the set of weighting coefficients ci in [3.174] (in other words, the ‘‘shim values’’) that leads to a minimization of the residual magnetic field inhomogeneity is a straightforward mathematical exercise (106). In principle, mapping the field profiles of the shim coils needs to be done only when some major change has occurred, such as switching probes; in practice, very little time is required to map the z-axis shims and the procedure therefore can be performed whenever a sample is changed. An important point to note is that the shimming procedure relies on the shim maps covering the same region as covered by the maps of the field inhomogeneity, which means that the same offset frequency must be used (111). As already outlined, only a single mapping of the magnetic field inhomogeneity is needed in principle. However, due to inherent, practical limitations, the procedure usually is iterated until some acceptable level of field correction is achieved. The entire process typically is highly automated by the spectrometer software, and normally only a few iterations are needed to achieve optimal results.
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Several variations or extensions of the z-axis gradient shimming procedure as already outlined have been implemented. Although the current discussion has focused on z-axis gradient shimming, full threedimensional (3D) mapping can be performed (106), which allows all shims to be optimized, instead of just the z-axis shims. Full 3D gradient shimming is particularly useful when shimming a new probe, or when trying to obtain optimum lineshape in highly demanding situations such as water suppression. As mentioned already, modified procedures have been devised that allow the use of the room-temperature shims to perform the imaging experiments; this capability is particularly useful because many probes, notably all current-generation cryogenic probes, only have z-axis pulse field gradient capability. Gradient shimming also can be performed by observing the deuterium resonance for samples dissolved in a deuterated solvent (111); the current generation of commercial NMR spectrometers typically can perform deuterium gradient shimming under software control, without the need to reconfigure the cabling to the deuterium channel of the probe (the rf channel used for deuterium gradient shimming and the separate lock channel both feed into a single port of the NMR probe). When using a deuterium signal for gradient shimming, the phase evolution period must be lengthened because the 2H nucleus has a smaller magnetogyric ratio than does the 1H nucleus. Finally, a modification of the basic pulse sequences shown in Fig. 3.46 has been reported to be highly effective in minimizing the deleterious effects of sample convection on the gradient shimming process (112). As in any gradient-based experiment, sample convention severely attenuates NMR signals, due to incomplete refocusing of magnetization as individual molecules move from one location in the sample to another. Sample convection results from the presence of temperature gradients along the sample, and thus is most likely to occur for sample temperatures far from ambient. Other methods for shimming have been reported that make use of applied field gradients and do not require any special hardware. Conover described a method referred to as Z1 profile shimming, which relies on imaging the magnetic field with the Z1 gradient and manually correcting distortions in expected shape of the profile by adjusting the higher order shims (102). An automated procedure referred to as 3D profile edge shimming has been demonstrated to be quite effective (113). 3.8.2.4 Pulse Width Calibration For a spatially homogeneous rf field applied on resonance, the nominal rotation angle is ¼ B1 p. The pulse duration yielding a particular rotation angle must be determined empirically each time the spectrometer is to be used for an experiment.
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
253
Even for the same sample in the same spectrometer, small variations in p may be observed from day to day; however, large increases in p usually indicate mistuning of the probe or equipment failure. Modern NMR pulse sequences frequently use pulses with reduced amplitudes for solvent saturation, extended rf mixing or spin-locking periods, and selective pulses. Accordingly, the mathematical relationship between attenuation, transmitter power, pulse lengths, and B1 field strength is useful for approximating the pulse lengths at some attenuation, given the measured pulse lengths at another attenuation. Power levels commonly are measured on the decibel scale: P ¼ Pref 10dB=10 :
½3:178
For the dBm scale, Pref ¼ 1 mW, and for the dBW scale, Pref ¼ 1 W. Thus, an rf amplifier with a maximum power output rated at 20 dBW produces an output of P ¼ 1020/10 W ¼ 100 W; a preamplifier with a noise figure of 20 dBm produces a noise output of P ¼ 1020/10 mW ¼ 0.01 mW. The attenuation difference between two different power levels P1 and P2 is given by, dB ¼ 10 log10 ðP1 =P2 Þ:
½3:179
The B1 field produced depends upon the voltage in the coil, V, not on the power. Because P ¼ V2 =R,
½3:180
in which R is the resistance (typically 50 in a tuned probe), dB ¼ 20 log10 ðV1 =V2 Þ:
½3:181
The difference between voltage and power is critical: changing the attenuation by 3 dB changes the power by a factor of two, but the attenuation must be changed by 6 dB to change the voltage by a factor of two. Because the B1 field strength depends on the voltage, doubling the strength of the B1 field (which is equivalent to halving the 908 pulse length) requires that the power output of the transmitter be quadrupled. The voltage V in [3.180] is the root-mean-square voltage produced by the rf field. The peak-to-peak voltage, Vpp, is more easily measured using an oscilloscope. For a time-dependent voltage, V(t) ¼ (Vpp/2) cos!rft, the two quantities are related by Z .Z 2 V2pp 2 V2pp 2 2 : ½3:182 V ¼ cos d d ¼ 4 0 8 0
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Combining this result with [3.180] yields the following useful relationship for power measured in watts, voltage measured in volts, and an assumed 50- resistance, P ¼ V2pp =400:
½3:183
If direct observation of the signal from a particular nucleus is feasible (e.g., 1H), then the length of a 3608 pulse is determined by searching for the null in the signal observed after application of the pulse. First, a free induction decay is acquired with a pulse–acquire sequence using a short pulse length. The spectrum is Fourier transformed and phased. The result of this experiment is
Ix
Iz ! Iz cos Iy sin ,
½3:184
where the pulse applied for a time t produces a net rotation of Iz by
radians. The intensity of the signal will depend upon the rotation angle
(and the time t) in a sinusoidal manner. Determining the length of time required to produce a specific rotation is most accurately accomplished when the observed signal is at a null, i.e., where the pulse produces a rotation by a multiple of radians. After every pulse–acquire experiment, the system should be allowed to reach equilibrium, so that Iz is at a maximum at the start of the next experiment. For this reason, measurement of a 2 pulse is most accurate: as the null is approached, the magnetization will be rotated almost back to the þz-axis and will therefore require less time to return to equilibrium. In addition, a 2 pulse is less sensitive than is a pulse to resonance offset effects. If the approximate pulse length is not known, care must be exercised to ensure that the null corresponds to a 2 rotation, rather than to a different multiple of rotations. Once the length of time required for a 2 pulse has been determined, the length of time required for other pulses can be calculated from the proportionality between and t. As described in Section 3.4.1, when the offset is large compared to the B1 field strength, the magnetization does not behave as predicted by [3.184]. Thus, when calibrating the weak pulses required for spin lock or composite pulse decoupling schemes, only the magnitude of signals on resonance should be considered. However, the best solvent suppression is achieved when the H2O signal is on-resonance, in which case protein peaks that are close to resonance will be obscured by the incompletely suppressed solvent signal. Thus, weak B1 fields are best calibrated by shifting the transmitter frequency, after the presaturation pulse and prior to the excitation pulse, to be on-resonance with a well-resolved peak,
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
255
such as that of an upfield-shifted methyl group. In this way, the pulse length required to achieve a null can be accurately determined without influence of offset effects. In principle, pulse lengths for the heteronuclear channels can be measured in an analogous way; however, this is rarely practical because of the insensitivity of these nuclei. Further, the heteronuclear experiments described in this book make use of indirect detection of 13C and 15 N spins by transfer of coherence to the directly bonded 1H nuclei. Acquisition of such spectra may require different probe and preamplifier configurations than are required for a direct-detect experiment. Because the pulse lengths depend on the exact rf circuitry utilized, pulse lengths for an indirect-detect experiment should be determined by a method that employs the same hardware configuration as employed in the experiments to follow. If the protein sample contains a well-resolved 1H signal scalar coupled to the desired heteronucleus, then the protein sample may be used directly for calibration, although the amount of signal averaging to achieve acceptable signal-to-noise ratios may make such calibrations time consuming. In many cases, heteronuclear pulse lengths can be obtained more readily on a test sample of higher concentration that contains a single labeled moiety for each of the heteronuclei to be calibrated. Small peptides or amino acids are useful for this purpose. Pulse sequences for indirect calibration rely on the coherence transfer properties of an IS (I ¼ 1H and S ¼ 13C or 15N) spin system. Three pulse schemes for indirect measurements of heteronuclear pulse lengths are given in Fig. 3.48. If the resonance for the I spin attached to the S spin is well resolved, then the pulse sequence of Fig. 3.48a is satisfactory. The product operator analysis of the pulse sequence yields Iz ! 2Ix Sz cos 2Ix Sy sin ,
½3:185
in which ¼ 1/(2JIS) and JIS is the scalar coupling constant between the I and S spins. For ! 0, an antiphase lineshape is obtained; the signal is nulled when ¼ /2, because the 2IxSy multiple-quantum operator does not evolve into observable coherence during acquisition. The experiment is first run with 0 so that the phase parameters to give an antiphase doublet can be determined; is then systematically increased until a null is found for the intensity of the doublet. If the spectrum is very crowded, overlapping antiphase components may lead to a large degree of cancellation of the signals and difficulty in accurately determining the pulse length.
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CHAPTER 3 EXPERIMENTAL ASPECTS
a
OF
NMR SPECTROSCOPY
(p /2)f 1H
∆
Receiver
t ax
X
b
(p /2)f1 1H
∆
Receiver
t a f2
bx
X
c
px
(p /2)f1 1H
∆
t
t
a f2
bx
∆
Receiver
X
FIGURE 3.48 Pulse sequences for indirect calibration of heteronuclear pulse lengths. In all sequences, ¼ 1/(2JXH) and is long enough to encompass the maximum length of the X pulses to be used. Pulses on X nuclei are applied during the delays so that the phase of the recorded signal does not depend on the length of the X pulses. Phase cycles are (a) ¼ {x, y, x, y}, receiver ¼ {x, y, x, y}; (b) and (c) 1 ¼ 2{x, y, x, y}, 2 ¼ {x, x, x, x, x, x, x, x}, receiver ¼ {x, y, x, y, x, y, x, y}. Additional details are given in the text.
If the signal from the S-bound 1H spin is not well resolved from signals of 1H spins not bound to heteronuclei (for example, in a peptide containing a single site of 15N incorporation), the signal can be more clearly discerned by using the sequence of Fig. 3.48b. Product operator
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
257
analysis yields Iz ! 2Ix Sz sin sin 2Ix Sy sin cos :
½3:186
This experiment yields antiphase observable signals proportional to sin
sin . The maximum signal is obtained for ¼ /2, and a null is observed for ¼ . This sequence incorporates an isotope filter, by alternating the phase of the first pulse on the heteronuclear channel, so that signals from 1H spins not coupled to an S spin are suppressed. Initially, both and are set equal to an estimate of the /2 pulse length for the heteronuclear channel, resulting in an antiphase doublet for the S-bound 1H spin. The value of is then systematically increased, until a null is obtained. Alternatively, in-phase signals can be obtained by using the more elaborate pulse sequence of Fig. 3.48c, which yields Iz ! Iy sin sin 2Ix Sy sin cos : 13
½3:187 15
Due to the wide spectral widths encountered in C and N spectra, offset effects can be very severe. Thus, having the S nucleus onresonance is very important if accurate pulse lengths are to be determined. In cases where the chemical shifts of the S nucleus are not known, the pulses may be roughly measured by estimating the S nucleus frequency. A short HSQC or HMQC experiment (Section 7.1) is acquired to ascertain the exact chemical shift (these experiments are reasonably tolerant to imperfect pulse lengths). With this information in hand, the pulse length can be remeasured accurately. 3.8.2.5 Recycle Delay The optimal combination of the recycle delay between transients and the pulse rotation angle depends upon the rate at which equilibrium magnetization recovers after a perturbation. For signal averaging in a one-pulse experiment with a delay between acquisitions of T (equal to the sum of the recycle delay and acquisition time), the initial amplitude of the FID is proportional to "¼
1 eT=T1 sin : 1 eT=T1 cos
½3:188
Equation [3.188] is plotted in Fig. 3.49. For a time T, the maximum signal is obtained for a rotation angle, e, known as the Ernst angle, cos e ¼ exp ðT=T1 Þ:
½3:189
Thus, if T 5 1, then e 5 908 and " 5 1. Essentially complete recovery of the equilibrium longitudinal magnetization occurs for T 4 3T1. Recycle delays of T1 5 T 5 1.5T1 yield superior sensitivity per unit time
258
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OF
NMR SPECTROSCOPY
1.0
0.8
e
0.6
0.4
0.2
0
30
60
90
a (°)
FIGURE 3.49 Ernst angle. The fractional signal intensity, ", is shown as a function of pulse length, , in a pulse–acquire NMR experiment. Results are shown for normalized recycle delays, T/T1, equal to (—) 0.2, (– –) 0.5, (- - -) 1.0, (– - –) 1.5, (– – ) 2.0, and (– –) 3.0. The optimal curve yielding the highest value of " is shown also (- - - -).
and are frequently used in multidimensional experiments, because the reduced recycle delay permits increased signal averaging that offsets the loss of sensitivity due to incomplete relaxation recovery. The majority of multidimensional homonuclear and heteronuclear NMR experiments used for protein NMR spectroscopy initially excite equilibrium 1H magnetization. The recovery of an equilibrium magnetization state, and therefore the repetition rate of the experiment, depends on the relaxation properties of the 1H nuclei. In order to avoid steady-state artifacts and low amounts of signal per unit of measuring time, the sum of acquisition time and recycle delay usually should be greater than 1.3T1 (Fig. 3.49). The (nonselective) 1H T1 relaxation rate constant can be estimated from a one-dimensional inversion recovery experiment: recycle180 90 acquire,
½3:190
in which the 908 pulse and receiver are phase cycled in 908 increments. In this case, the recycle delay should be greater than 3T1 to ensure complete
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
259
relaxation between transients (which may require repeating the experiment following an initial approximate determination of T1). The magnitude of the transverse magnetization following the 908 pulse varies with the delay, , as MðÞ ¼ Mð0Þ½1 2 expð=T1 Þ:
½3:191
The delay is systematically varied until the signal is nulled. The value of T1 is approximately given by T1 ¼ null = ln 2:
½3:192
An example of a nonselective inversion recovery experiment for a ubiquitin sample in D2O solution is shown in Fig. 3.50. As is evident, the upfield methyl 1H spins in ubiquitin have shorter T1 values than do the 1
H and aromatic 1H spins, and recover to equilibrium more rapidly. Approximate values of T1 determined using [3.192] are 0.7, 1.4, and 1.8 s for methyl, , and aromatic 1H spins, respectively. A small number of persistent 1HN spins with T1 1.0 s are evident in the spectra. 3.8.2.6 Linewidth Measurement Linewidths, or transverse relaxation rate constants, inevitably determine the sensitivity, or even practicality, of multidimensional NMR experiments, which frequently include extended periods of evolution of transverse coherences. Accordingly, some estimation of the linewidths of the spin resonances is useful in initial assessment of a protein sample. Transverse relaxation of the amide 1HN spins can be estimated from one-dimensional jump–return Hahn echo experiments in which the jump–return technique is used both to avoid saturation of the water signal and to decouple the 1HN and 1H
spins (114). The pulse sequence shown in Fig. 3.51 is executed twice for a short, 1, and long, 2, value of the echo delay, . The ratio of signal intensities in the two spectra is used to obtain R2 from R2 ¼
1 Ið 2 Þ ln : 2ð2 1 Þ Ið 1 Þ
½3:193
Spectra for ubiquitin acquired with 1 ¼ 1 ms and 2 ¼ 51 ms are shown in Fig. 3.52 and yield an average R2 ¼ 21.6 s1 for the envelope of amide resonances. This simple experiment can be elaborated using gradient purge pulses (Section 4.3.3) and selective pulses for decoupling (Section 3.4.4).
260
CHAPTER
3 EXPERIMENTAL ASPECTS OF NMR SPECTROSCOPY
a
b
c
d
e
4
8
o
'H (ppm) FIGURE 3.50 IH inversion recovery spectra of ubiquitin. Partially relaxed nonselective inversion recovery spectra of ubiquitin are shown (a-e) for recovery delays of 3 f.LS and 0.5, 1.0, 1.5, and 2.0 s. Spectra were recorded with a 12-s recycle delay using a ubiquitin sample in 100% D 2 0 solution.
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY f1
f2
∆
f4
f3
t
261
t
2∆
FIGURE 3.51 Pulse sequence for 1D jump–return Hahn echo. The delay ¼ 1/(4max). The phase cycle is 1 ¼ x, 2 ¼ x, 3 ¼ x, y, x, y, 4 ¼ x, y, x, y, receiver ¼ x, x, x, x.
b
a
10
9
8 1H
7
6
(ppm)
FIGURE 3.52 Estimating 1H R2 relaxation rate constants. (a) Spectrum of ubiquitin recorded for ¼ 1 ¼ 1 ms. (b) Spectrum of ubiquitin recorded for ¼ 2 ¼ 51 ms. R2 is estimated from the ratio of signal intensities using [3.193].
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3.8.3 REFERENCING The concept of the chemical shift was introduced in Section 1.5. To facilitate comparisons of resonance positions between different samples and spectrometers, chemical shifts are measured by reference to a standard compound using units of parts per million, as described by [1.51]. Tetramethylsilane (TMS) is the universal reference for 1H NMR. In studies of organic molecules, TMS can be added directly to the solvent (e.g., deuterated chloroform spiked with TMS is commercially available), thereby providing an internal reference from which the chemical shifts of sample resonances can be determined. The situation is less straightforward in studies of proteins and other biological molecules because TMS is not soluble in aqueous solutions. Instead, either a different internal reference species or an external reference must be used. The ideal internal chemical shift marker should not interact with the protein under investigation, and should have a single resonance whose chemical shift varies with temperature and pH in a known manner. The resonance should be well resolved from the resonances of the protein, because the reference signal will have a long T1, and t1 noise emanating from it may obscure cross-peaks in 2D spectra. The IUPAC–IUBMB– IUPAB Inter-Union Task Group on the Standardization of Data Bases of Protein and Nucleic Acid Structures Determined by NMR Spectroscopy has recommend that the methyl 1H resonance of 2,2dimethyl-2-silapentane-5-sulfonic acid (DSS; ¼ 0.00) be used as the primary chemical shift reference for aqueous solutions of biomolecules (115). Concentrations of 10–20 M DSS are suitable as internal chemical shift reference standards. If DSS binds to the molecule of interest, then alternative internal standards can be used, such as dioxane ( ¼ 3.75 ppm); other suitable reference standards are discussed elsewhere (116). Once the 1H shifts are referenced, the heteronuclear chemical shifts are indirectly referenced by using the following relationship (117, 118): H X 0 ¼ 0 ,
½3:194
H where X 0 is the absolute frequency of 0 ppm for the X spin, 0 is the 1 absolute frequency of 0 ppm for the H spin, and is the relative frequency for the X spin, compared to 1H. Values of are given in Table 3.2. As an example, if the absolute frequency of DSS is measured to be 500.1366624 MHz at 0.0 ppm, then the absolute zero frequency
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
263
TABLE 3.2 Indirect Chemical Shift Referencesa Compound
2
H C 15 N 31 P
0.153506088 0.251449530 0.101329118 0.404808636
13
Secondary referencb DSS (internal) DSS (internal) Liquid NH3 (external) (CH3O)3PO (internal)
a
Reported values are from the IUPAC recommendations (115). DSS, 2,2-Dimethyl-2-silapentane-5-sulfonic acid.
b
of
15
N is determined from [3.194] to be
N 0 ¼ 500:1366624 ¼ 0:101329118 500:1366624 ¼ 50:67840688 MHz: ½3:195 Therefore, to perform an HSQC experiment with the 15N region centered in the middle of the amide nitrogen resonances (say at 115.0 ppm), the required experimental offset would be 50:67840688ð1 þ 115:0 106 Þ ¼ 50:6842349 MHz:
3.8.4 ACQUISITION
AND
½3:196
DATA PROCESSING
Acquisition of 1H spectra is the first step of any investigation by NMR spectroscopy. Such spectra can be acquired by a number of approaches. Two techniques, the one-pulse experiment and the Hahn echo experiment, are discussed in this section. These two experiments can be elaborated to avoid presaturation of the solvent resonance (Section 3.7) or to decouple scalar coupling interactions with heteronuclear spins (Section 3.5). 3.8.4.1 One-Pulse Experiment The basic NMR experiment is the socalled one-pulse experiment in which an rf pulse of rotation angle is applied to the system and the resulting transverse signal is detected: recyclepulseacquire:
½3:197
As discussed in Section 4.3.2.3, CYCLOPS phase cycling commonly is used during the one-pulse experiment to suppress quadrature images.
264
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NMR SPECTROSCOPY
a
b
c
8
4 1H 1
0
(ppm)
FIGURE 3.53 Hahn echo H NMR spectra acquired with a spectral width of 12,500 Hz and a filter width of 30,000 Hz. (a) Echo delays 1 ¼ 140 s and 2 ¼ 174 s are adjusted to eliminate phase errors in the spectrum, (b) 2 is misadjusted to be 10 s shorter than optimal, and (c) 2 is misadjusted to be 10 s longer than optimal.
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
265
In CYCLOPS phase cycling, the phase of the pulse and the phase of the receiver are shifted in 908 steps between transients. As a minimum, the following parameters must be adjusted in setting up the one-pulse experiment: recycle delay, rf carrier position, pulse duration, sampling interval or spectral width, number of digitized data points in the time domain, number of transients to be acquired, and receiver amplifier gain. The rf carrier frequency is set to the center of the spectrum or to the frequency of the solvent resonance. Normally the sampling interval, t, is adjusted such that the Nyquist frequency is larger than the maximum resonance frequency arising from the sample; however, in some instances, resonances may be deliberately aliased in order to minimize the sampling rate (Section 7.1.2.5). The number of data points acquired, N, is chosen such that Nt 4 3T2 in order to minimize truncation artifacts. The receiver gain is adjusted so that the signal arising from the FID does not underflow or overflow the dynamic range of the receiver. The number of transients acquired depends upon the signal averaging required to achieve the desired signal-to-noise ratio in the spectrum; the number of transients must be a multiple of four if CYCLOPS phase cycling is employed. 3.8.4.2 Hahn Echo Experiment The simple one-pulse experiment may not be satisfactory for detailed examination of the one-dimensional spectrum of a biological macromolecule. As was noted in Section 3.3.2.3, baseline distortions can hamper the interpretation of NMR spectra. Removal of baseline distortions can be achieved by the use of a so-called Hahn echo pulse sequence (119, 120). This technique, originally introduced for wideline NMR spectroscopy of spin-1/2 nuclei in anisotropic media (120), is well suited to application in high-resolution 1 H-detected NMR spectroscopy of biomolecules. In NMR spectroscopy, the entire FID must be recorded to obtain spectra free of distortions. Accurate detection of the early part of the FID is crucial in wideline NMR spectroscopy because the initial signal decays rapidly (the signal typically consists of a broad distribution of resonance frequencies and resonance linewidths can be of the order of several kilohertz). In the simple pulse–acquire detection scheme, a dead time follows the high-power rf pulse as the receiver is saturated and ringing effects are introduced in the tuned circuits. The FID decays markedly during this period; consequently, when the receiver is eventually actuated, the first part of the FID is absent. Rance and Byrd used the Hahn echo pulse sequence 90 1 180 2 acquire
½3:198
266
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
to create a spin echo at 2. By having the echo form beyond the receiver dead-time period, the Hahn echo sequence avoids distortions due to finite receiver recovery time. A similar situation is encountered in high-resolution 1H NMR spectroscopy. Baseline roll in high-resolution spectra, as was discussed in Section 3.3.2.3, commonly is caused by the transient response of the spectrometer to the incoming signal that distorts the first few points of the FID. The Hahn echo sequence allows these distortions to be avoided by the same mechanism as used for wideline spectra. The delays in the Hahn echo sequence can be calculated as 1 ¼ 2 2 90 = þ tgate ,
½3:199
in which 2 is greater than the filter response time, 90 is the length of the 908 pulse, and gate is the receiver gating delay. In each case, 2 is adjusted empirically by small amounts to ensure that acquisition is initiated at the exact top of the echo. As shown in Fig. 3.53, phase errors in the spectrum are eliminated when 2 is adjusted accurately. Once 2 is optimized with respect to the phase of the spectrum, the value of 2 is reduced by enough sample times to allow the filter transient response to decay prior to the top of the echo. Data points acquired prior to the top of the spin echo are discarded before Fourier transformation (96). A wide spectral width typically is used for oversampling of the FID (121). The basic phase cycle incorporates CYCLOPS phase cycling for the 908 pulse () and EXORCYCLE phase cycling for the 1808 pulse ( ) to yield a 16-step phase cycle: ¼ {x x x x y y y y x x x x y y y y}, ¼ 4 {x y x y}, and receiver ¼ {x x x x y y y y x x x x y y y y}. CYCLOPS and EXORCYCLE are discussed in Section 4.3.2.3. The Hahn echo sequence is slightly longer than a simple pulse– acquire sequence. Relaxation during the 1 and 2 delays can reduce the signal intensity; however, the delays are sufficiently short that relaxation effects generally can be safely ignored. Similarly, evolution of the homonuclear scalar coupling occurs during the 1 and 2 delays. For typical values of JHH 5 15 Hz, cos[JHH( 1 þ 2)] 4 0.9999 and sin[JHH( 1 þ 2)] 5 0.016; consequently, scalar coupling effects also can be ignored. In addition to its other benefits, the Hahn echo sequence also has demonstrated significantly improved water suppression due principally to the refocusing properties of the 1808 pulse, which reduces broadening at the base of the residual water peak due to signal originating outside of the homogeneous sample volume. Example onedimensional NMR spectra acquired with pulse–acquire and Hahn echo pulse sequences are shown in Fig. 3.54.
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
267
a
b
8
4 1H
0
(ppm)
FIGURE 3.54 (a) Pulse–acquire and (b) Hahn echo 1H NMR spectra of ubiquitin. The Hahn echo spectrum has better water suppression and a flatter baseline compared to the pulse–acquire spectrum. Both spectra were acquired with identical spectral and filter widths.
References 1. 2. 3. 4. 5. 6. 7.
A. Abragam, ‘‘Principles of Nuclear Magnetism,’’ pp.1–599. Clarendon Press, Oxford, 1961. H. Kovacs, D. Moskau, M. Spraul, Prog. NMR Spectrosc. 46, 131–155 (2005). A. E. Kelly, H. D. Ou, R. Withers, V. Do¨tsch, J. Am. Chem. Soc. 124, 12013–12019 (2002). A. N. Lane, S. Arumugam, J. Magn. Reson. 173, 339–343 (2005). R. W. Dykstra, J. Magn. Reson. 84, 388–391 (1989). T. M. de Swiet, J. Magn. Reson. 174, 331–334 (2005). P. F. Flynn, D. L. Mattiello, H. D. W. Hill, A. J. Wand, J. Am. Chem. Soc. 122, 4823–4824 (2000).
268
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
8. M. H. Levitt, J. Magn. Reson. 126, 164–182 (1997). 9. M. H. Levitt, O. G. Johannessen, J. Magn. Reson. 142, 190–194 (2000). 10. R. R. Ernst, G. Bodenhausen, A. Wokaun, ‘‘Principles of Nuclear Magnetic Resonance in One and Two Dimensions,’’ pp. 1–610. Clarendon Press, Oxford, 1987. 11. M. H. A. Roehrl, G. J. Heffron, G. Wagner, J. Magn. Reson. 174, 325–330 (2005). 12. D. I. Hoult, C.-N. Chen, H. Eden, M. Eden, J. Magn. Reson. 51, 110–117 (1983). 13. S. L. Marple, ‘‘Digital Signal Processing,’’ pp. 1–492. Prentice-Hall, Englewood Cliffs, NJ, 1987. 14. J. G. Proakis, D. G. Manolakis, ‘‘Introduction to Digital Signal Processing,’’ pp. 1–944. Macmillan, New York, 1988. 15. A. V. Oppenheim, R. W. Schafer, ‘‘Discrete-Time Signal Processing,’’ pp.1–879. Prentice Hall, Englewood Cliffs, NJ, 1989. 16. D. Moskau, Concepts Magn. Reson. 15, 164–176 (2002). 17. J. G. Wurl, Patent 5,652,518 (United States, 1997). 18. J. C. Hoch, A. S. Stern, ‘‘NMR Data Processing,’’ pp. 1–196. John Wiley and Sons, New York, 1996. 19. R. N. Bracewell, ‘‘The Fourier Transform and Its Applications,’’ 2nd edn., pp. 1–474. McGraw-Hill, New York, 1986. 20. E. Bartholdi, R. R. Ernst, J. Magn. Reson. 11, 9–19 (1973). 21. G. Zhu, D. A. Torchia, A. Bax, J. Magn. Reson., Ser. A 105, 219–222 (1993). 22. A. Bax, M. Ikura, L. E. Kay, G. Zhu, J. Magn. Reson. 91, 174–178 (1991). 23. R. K. Harris, ‘‘Nuclear Magnetic Resonance Spectroscopy,’’ pp. 1–260. Longman Scientific and Technical, Harlow, U.K., 1986. 24. H. Barkhuijsen, R. de Beer, W. M. M. J. Bove´e, D. van Ormondt, J. Magn. Reson. 61, 465–481 (1985). 25. S. Sibisi, J. Skilling, R. G. Brereton, E. D. Laue, J. Staunton, Nature (London) 311, 446–447 (1984). 26. R. E. Hoffman, A. Kumar, K. D. Bishop, P. N. Borer, G. C. Levy, J. Magn. Reson. 83, 586–594 (1989). 27. J. J. Kotyk, N. G. Hoffman, W. C. Hutton, G. L. Bretthorst, J. J. H. Ackerman, J. Magn. Reson. 98, 483–500 (1992). 28. D. S. Stephenson, Prog. NMR Spectrosc. 20, 515–626 (1988). 29. G. Zhu, A. Bax, J. Magn. Reson. 90, 405–410 (1990). 30. H. Barkhuijsen, R. de Beer, D. van Ormondt, J. Magn. Reson. 73, 553–557 (1987). 31. D. Marion, A. Bax, J. Magn. Reson. 83, 205–211 (1989). 32. E. D. Laue, M. R. Mayger, J. Skilling, J. Staunton, J. Magn. Reson. 68, 14–29 (1986). 33. E.D. Laue, J. Skilling, J. Staunton, J. Magn. Reson. 63, 418–424 (1985). 34. F. Bloch, A. Siegert, Phys. Rev. 57, 522–527 (1948). 35. M. A. McCoy, L. Mueller, J. Magn. Reson. 99, 18–36 (1992). 36. M. A. McCoy, L. Mueller, J. Magn. Reson. 98, 674–679 (1992). 37. M. Sattler, J. Schleucher, C. Griesinger, Prog. NMR Spectrosc. 34, 93–158 (1999). 38. M. H. Levitt, G. Bodenhausen, R. R. Ernst, J. Magn. Reson. 53, 443–461 (1983). 39. A. L. Davis, G. Estcourt, J. Keeler, E. D. Laue, J. J. Titman, J. Magn. Reson., Ser. A 105, 167–183 (1993). 40. M. H. Levitt, Prog. NMR Spectrosc. 18, 61–122 (1986). 41. C. Counsell, M. H. Levitt, R. R. Ernst, J. Magn. Reson. 63, 133–141 (1985). 42. M. A. Smith, H. Hu, A. J. Shaka, J. Magn. Reson. 151, 269–283 (2001). 43. K. Kobzar, T. E. Skinner, N. Khaneja, S. J. Glaser, B. Luy, J. Magn. Reson. 170, 236–243 (2004).
3.8 ONE-DIMENSIONAL 1H NMR SPECTROSCOPY
269
44. J. Keeler, in ‘‘Multinuclear Magnetic Resonance in Liquids and Solids-Chemical Applications’’ (P. Granger, R. K. Harris, eds.), pp. 201–238, Kluwer Academic Publishers, Netherlands, 1990. 45. H. Kessler, S. Mronga, G. Gemmecker, Magn. Reson. Chem. 29, 527–557 (1991). 46. J. Boyd, N. Soffe, J. Magn. Reson. 85, 406–413 (1989). 47. S. L. Patt, J. Magn. Reson. 96, 94–102 (1992). 48. S. Zhang, D. G. Gorenstein, J. Chem. Phys. 105, 5659–5664 (1996). 49. S. Zhang, D. G. Gorenstein, J. Magn. Reson. 154, 73–79 (2002). 50. H. Geen, R. Freeman, J. Magn. Reson. 93, 93–141 (1991). Kupcˇe, J. Boyd, I. D. Campbell, J. Magn. Reson., Ser. B 106, 300–303 (1995). 51. E. Kupcˇe, R. Freeman, J. Magn. Reson., Ser. A 105, 234–238 (1993). 52. E. Kupcˇe, Meth. Enzmol. 338, 82–111 (2001). 53. E. 54. I. I. Rabi, N. F. Ramsey, J. Schwinger, Rev. Modern Phys. 26, 167–171 (1954). 55. J. G. Powles, Proc. Phys. Soc. (London) 71, 497–500 (1958). 56. J. Baum, R. Tycko, A. Pines, Phys. Rev. A32, 3435–3447 (1985). 57. M. S. Silver, R. I. Joseph, D. I. Hoult, J. Magn. Reson. 59, 347–351 (1984). Kupcˇe, R. Freeman, J. Magn. Reson., Ser. A 115, 273–276 (1995). 58. E. 59. F. A. A. Mulder, R. A. de Graaf, R. Kaptein, R. Boelens, J. Magn. Reson. 131, 351–357 (1998). 60. R. A. de Graaf, K. Nicolay, Concepts Magn. Reson. 9, 247–268 (1997). 61. A. Tannus, M. Garwood, NMR Biomed. 10, 423–434 (1997). 62. C. Zwahlen, S. J. F. Vincent, L. E. Kay, J. Magn. Reson. 130, 169–175 (1998). 63. M. R. Bendall, J. Magn. Reson., Ser. A 116, 46–58 (1995). Kupcˇe, R. Freeman, J. Magn. Reson. 127, 36–48 (1997). 64. E. 65. J. S. Waugh, J. Magn. Reson. 50, 30–49 (1982). 66. A. J. Shaka, J. Keeler, Prog. NMR Spectrosc. 19, 47–129 (1987). 67. A. J. Shaka, J. Keeler, T. Frenkiel, R. Freeman, J. Magn. Reson. 52, 335–338 (1983). 68. A. J. Shaka, P. B. Barker, R. Freeman, J. Magn. Reson. 64, 547–552 (1985). 69. N. Sunitha Bai, N. Hari, R. Ramachandran, J. Magn. Reson., Ser. A 106, 241–244 (1994). 70. R. Tycko, A. Pines, Chem. Phys. Lett. 111, 462–467 (1984). 71. T. Fujiwara, T. Anal, N. Kurihara, K. Nagayama, J. Magn. Reson., Ser. A 104, 103–105 (1993). 72. R. Fu, G. Bodenhausen, J. Magn. Reson., Ser. A 117, 324–325 (1995). 73. M. R. Bendall, J. Magn. Reson., Ser. A 112, 126–129 (1995). 74. M. McCoy, L. Mueller, J. Magn. Reson., Ser. A 101, 122–130 (1993). 75. A. Wokaun, R. R. Ernst, Mol. Phys. 36, 317–341 (1976). 76. P. B. Kingsley, J. Magn. Reson., Ser. B 109, 243–250 (1995). 77. M. Levitt, Concepts Magn. Reson. 8, 77–103 (1996). 78. W. S. Warren, W. Richter, A. H. Andreotti, B. T. Farmer, Science 262, 2005–2009 (1993). 79. D. L. Mattiello, W. S. Warren, L. Mueller, B. T. Farmer, J. Am. Chem. Soc. 118, 3253–3261 (1996). 80. P. C. M. van Zijl, M. O’Neil Johnson, S. Mori, R. E. Hurd, J. Magn. Reson., Ser A 113, 265–270 (1995). 81. N. Bloembergen, R. V. Pound, Phys. Rev. 95, 8–12 (1954). 82. S. Bloom, J. Appl. Phys. 28, 800–805 (1957). 83. P. J. Hore, Meth. Enzymol. 176, 64–77 (1989). 84. M. Gue´ron, P. Plateau, M. Decorps, Prog. NMR Spectrosc. 23, 135–209 (1991). 85. P. J. Hore, J. Magn. Reson. 55, 283–300 (1983). 86. P. Plateau, M. Gue´ron, J. Am. Chem. Soc. 104, 7310–7311 (1982). 87. Y.-C. Li, G. T. Montelione, J. Magn. Reson., Ser. B 101, 315–319 (1993).
270 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123.
CHAPTER 3 EXPERIMENTAL ASPECTS
OF
NMR SPECTROSCOPY
S. Grzesiek, A. Bax, J. Am. Chem. Soc. 115, 12593–12594 (1993). V. Sklena´r, A. Bax, J. Magn. Reson. 75, 378–383 (1987). M. Piotto, V. Saudek, V. Sklena´r, J. Biomol. NMR 2, 661–665 (1992). T.-L. Hwang, A. J. Shaka, J. Magn. Reson., Ser. A 112, 275–279 (1995). A. Bax, S. S. Pochapsky, J. Magn. Reson. 99, 638–643 (1992). B. A. Messerle, G. Wider, G. Otting, C. Weber, K. Wu¨thrich, J. Magn. Reson. 85, 608–613 (1989). D. Marion, M. Ikura, A. Bax, J. Magn. Reson. 84, 425–430 (1989). F. Ni, J. Magn. Reson. 99, 391–397 (1992). J. P. Waltho, J. Cavanagh, J. Magn. Reson., Ser. A 103, 338–348 (1993). N. J. Oppenheimer, Meth. Enzymol. 176, 78–89 (1989). W. U. Primrose, in ‘‘NMR of Macromolecules, a Practical Approach’’ (G. C. K. Roberts, ed.), pp. 7–34. IRL Press, Oxford, 1993. D. S. Raiford, C. L. Fisk, E. D. Becker, Anal. Chem. 51, 2050–2051 (1979). L. P. M. Orbons, G. A. van der Marel, J. H. van Boom, C. Altona, Eur. J. Biochem. 170, 225–239 (1987). A. J. Hartel, P. P. Lankhorst, C. Altona, Eur. J. Biochem. 129, 343–357 (1982). W. W. Conover, in ‘‘Topics in Carbon-13 NMR Spectroscopy’’ (G. C. Levy, ed.), vol. 4, pp. 37–51. John Wiley & Sons, NY (1984). G. N. Chmurny, D. I. Hoult, Concepts Magn. Reson. 2, 131–149 (1990). A. A. Maudsley, H. E. Simon, S. K. Hilal, J. Phys. Chem. E17, 216–220 (1984). M. G. Prammer, J. C. Haselgrove, M. Shinnar, J. S. Leigh, J. Magn. Reson. 77, 40–52 (1988). P. C. M. van Zijl, S. Sukumar, M. O’Neil Johnson, P. Webb, R. E. Hurd, J. Magn. Reson., Ser. A 111, 203–207 (1994). H. Barjat, P. B. Chilvers, B. K. Fetler, T. J. Horne, G. A. Morris, J. Magn. Reson. 125, 197–201 (1997). G. A. Gray, B. K. Fetler, Varian ‘‘Magnetic Moments’’ (Online, Winter VIII, 1996). M. Weiger, D. Moskau, R. Kerssebaum, W. E. Hull, Bruker SpinReport, 3–27 (2005). P. T. Callaghan, ‘‘Principles of Nuclear Magnetic Resonance Microscopy,’’ pp. 1–492. University Press, Oxford, 1991. S. Sukumar, M. O’Neil Johnson, R. E. Hurd, P. C. M. van Zijl, J. Magn. Reson. 125, 159–162 (1997). C.-L. Evans, G. A. Morris, A. L. Davis, J. Magn. Reson. 154, 325–328 (2002). P. B. Chilvers, G. A. Morris, J. Magn. Reson. 133, 210–215 (1998). V. Sklena´r, A. Bax, J. Magn. Reson. 74, 469–479 (1987). J. L. Markley, A. Bax, Y. Arata, C. W. Hilbers, R. Kaptein, B. D. Sykes, P. E. Wright, K. Wu¨thrich, Pure Appl. Chem. 70, 117–142 (1998). D. S. Wishart, C. G. Bigam, J. Yao, F. Abildgaard, H. J. Dyson, E. Oldfield, J. L. Markley, B. D. Sykes, J. Biomol. NMR 6, 135–140 (1995). D. H. Live, D. G. Davis, W. C. Agosta, D. Cowburn, J. Am. Chem. Soc. 106, 1939–1941 (1984). A. Bax, S. Subramanian, J. Magn. Reson. 67, 565–569 (1986). D. G. Davis, J. Magn. Reson. 81, 603–607 (1989). M. Rance, R. A. Byrd, J. Magn. Reson. 52, 221–240 (1983). M. Delsuc, J. Lallemand, J. Magn. Reson. 69, 504–507 (1986). M. Ede´n, Y. K. Lee, M. H. Levitt, J. Magn. Reson., Ser. A 120, 56–71 (1996). G. P. Connelly, Y. Bai, M. F. Jeng, S. W. Englander, Proteins: Struct. Funct. Genet. 17, 87–92 (1993).
CHAPTER
4 MULTIDIMENSIONAL NMR SPECTROSCOPY
In the simplest pulsed NMR experiment, transverse magnetization excited by an rf pulse is sampled and stored at regular intervals during the acquisition period to generate a digital representation of the FID. Fourier transformation of the digitized signal yields the conventional one-dimensional spectrum (Chapter 3). In more complex onedimensional NMR experiments, perturbations, which usually take the form of applied rf fields, are imposed on the spin system during the acquisition period or during a preparation period that precedes the acquisition period. Comparison of the spectra obtained in the presence and absence of the perturbations then yields information on the properties of the spin system affected. For example, weak irradiation of a particular spin during the acquisition period of a spin tickling experiment (1, 2) alters the natural multiplet patterns of spins that are scalar coupled to the irradiated spin. As another example, selective saturation of the resonance of a particular spin during the preparation period of a nuclear Overhauser effect NOE difference experiment (3, 4) alters the normal intensities of nearby, dipolar-coupled spins. Unfortunately, one-dimensional NMR techniques, such as spin tickling, selective decoupling (2, 5–7), and NOE difference experiments, which yield extremely useful information in small molecules, are of limited applicability to the complex, highly overlapped spectra of
271
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a
b
10
5 1H
0
(ppm)
FIGURE 4.1 One-dimensional 500-MHz 1H NMR spectra of (a) a hexapeptide at 280 K and (b) ubiquitin at 300 K. Samples were prepared in 90%/10% H2O/ D2O. The two data sets were recorded at different temperatures; therefore, the water resonance signal appears at different chemical shifts in the two spectra.
biological macromolecules. By way of illustration, Fig. 4.1a shows the one-dimensional 1H spectrum of a hexapeptide. Virtually all of the 1H (multiplet) resonances are resolved; consequently, the assignment of each resonance to a particular 1H spin in the molecule is straightforward and perturbations to the spectrum that result from selective irradiation of particular spins are easily detected. On the other hand, Fig. 4.1b shows a one-dimensional 1H spectrum of the protein ubiquitin (Mr ¼ 8565).
273
4.1 TWO-DIMENSIONAL NMR SPECTROSCOPY
Several hundred 1H resonances are crowded into approximately the same spectral region as the few peptide resonances. Because many resonances are degenerate, the signals are impossible to assign by using one-dimensional techniques, and selective perturbations to particular spins are difficult to achieve or detect experimentally. To effectively utilize the information available from NMR spectroscopy of biological macromolecules, a general method is required for improving resolution, facilitating resonance assignments, and detecting the effects of interactions between spins in complex NMR spectra. The explosive growth in the application of NMR spectroscopy to biological macromolecules in the past three decades attests to the success of multidimensional experiments in achieving these objectives.
4.1 Two-Dimensional NMR Spectroscopy Initially, multidimensional NMR spectroscopy is introduced here by concentrating on two-dimensional spectroscopy. The overall structure of two-dimensional NMR experiments is presented, the separation of interactions into more than one frequency dimension is discussed, and techniques for selection of coherence transfer pathways are introduced. All of the methods and principles presented can be extended into higher dimensions in a straightforward manner. Specific multidimensional NMR experiments are discussed in Chapters 6–8. In two-dimensional spectroscopy, two new elements, known as the evolution and mixing periods, are introduced into the NMR experiment between the preparation and acquisition periods. Thus, a general scheme for recording two-dimensional spectra is segmented into the four distinct parts illustrated in Fig. 4.2. The evolution period contains a variable time delay that is increased during the course of a two-dimensional NMR experiment from an initial value to a final value in m (usually equal) increments. For each of the m values of the incrementable delay, the same pulse sequence is executed twice (necessary for quadrature detection, as discussed in Section 4.3.4) and two FIDs, each consisting of n digitized complex data points, are preparation
evolution (t1)
mixing
acquisition (t2)
FIGURE 4.2 General scheme for two-dimensional NMR spectroscopy. The twodimensional NMR experiment is divided into four defined components: preparation, evolution, mixing, and acquisition.
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recorded. Each FID consists of p co-added transients for signal averaging or phase cycling. Thus, a total of 2 m p separate transients are recorded during the two-dimensional experiment, and a data array, which takes the form of either an m (complex) n (complex) matrix or a 2m (real) n (complex) matrix, is generated as a function of the two separate, independent time periods. Herein, the acquisition time (when the receiver is turned on and actually detects signal) is designated t2, while the indirect evolution time is designated t1. The rows of the data matrix represent data collected for a fixed t1 value and different t2 values, while the columns represent data collected for a fixed t2 value and varying t1 values. Fourier transformation with respect to these two time domains generates a two-dimensional spectrum with two independent frequency dimensions, F1 (from t1) and F2 (from t2). Most importantly, the signal eventually recorded during t2 is modulated by events occurring during the evolution time t1. As an example, consider a single isolated spin with a Larmor frequency and a simple pulse sequence consisting of two 90x pulses separated by the variable period, t1. Using the product operator formalism introduced in Chapter 2, the evolution of the density operator through the pulse sequence is 90x
Iz ! Iy t1
! Iy cosðt1 Þ þ Ix sinðt1 Þ 90x
! Iz cosðt1 Þ þ Ix sinðt1 Þ t2
! Iz cosðt1 Þ þ Ix sinðt1 Þ cosðt2 Þ þ Iy sinðt1 Þ sinðt2 Þ
½4:1
Thus, the complex signal detected during the acquisition period is proportional to sin(t1) exp(it2) and, as a result, depends parametrically upon the value of t1. As shown in Fig. 4.3, following Fourier transformation of the data recorded during t2, the amplitude of the resulting resonance signal depends on sin(t1). A null signal is obtained if t1 ¼ k, a signal with maximal amplitude is obtained if t1 ¼ (4k þ 1)/2, and an inverted signal is obtained if t1 ¼ (4k þ 3)/2, with k an integer. Formally, a correlation is established between the two time domains. The amplitude of the resonance signal obtained from the Fourier transformation of the data recorded during t2, when displayed as a function of t1, i.e., the data arrayed as a function of t1 at a fixed value of F2, forms an interferogram similar in appearance to the FID. The t1 interferogram is indirectly sampled and differs in this respect
275
4.1 TWO-DIMENSIONAL NMR SPECTROSCOPY
t1
t2
b
t1
a
F2
FIGURE 4.3 Dependence of the NMR signal on the evolution period, t1. (a) The signal detected during the acquisition period, t2, is modulated as a function of the evolution period, t1. The vertical bars represent 908 pulses. The separation between the pulses is equal to t1. (b) Following Fourier transformation of the recorded data with respect to t2, the amplitude of the resonance signal varies periodically as a function of t1 at a fixed value of F2.
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from the FID, which is directly detected by the spectrometer during t2. Fourier transformation of the interferogram with respect to t1 yields the F1 dimension of the two-dimensional spectrum. The correlation between the two time domains in the twodimensional spectrum represented by [4.1] is trivial because the I spin operators precess at the same frequencies during t1 and t2 under the freeprecession Hamiltonian. Consequently, signals are observed only at positions satisfying the relationship F1 ¼ F2, and the conventional one-dimensional spectrum is reproduced along the diagonal of the two-dimensional spectrum. To provide additional information, a twodimensional spectrum must contain resonance signals for which F1 6¼ F2; this condition requires that the components of the density operator that eventually give rise to the observed resonance must evolve under different Hamiltonians during t1 and t2. Fortunately, because the components of the density operator that evolve during t1 (or in fact any time before t2) are never actually recorded, the mixing period can serve to transfer magnetization, or more generally, coherence, among spins prior to acquisition. The presence of a signal in the twodimensional spectrum at one frequency in F1 and at a second frequency in F2 is direct evidence for transfer of coherence during the mixing period. With carefully constructed sequences of rf pulses and delays during the mixing period, correlations are established between the coherences present during t1 and t2 that result in chemically useful information. To illustrate the importance of the mixing period, the example just discussed is extended to include a second spin, S, of the same nuclear species. The Larmor frequencies of the I and S spins are now designated I and S; the two spins are assumed to have a scalar coupling interaction with a coupling constant, J. Focusing on magnetization that originates on the I spin, 90x
Iz ! Iy t1
! Iy cosðI t1 Þ cosðJt1 Þ þ 2Ix Sz cosðI t1 Þ sinðJt1 Þ þ Ix sinðI t1 Þ cosðJt1 Þ þ 2Iy Sz sinðI t1 Þ sinðJt1 Þ 90x
! Iz cosðI t1 Þ cosðJt1 Þ 2Ix Sy cosðI t1 Þ sinðJt1 Þ þ Ix sinðI t1 Þ cosðJt1 Þ 2Iz Sy sinðI t1 Þ sinðJt1 Þ:
½4:2
Only the two terms proportional to Ix and 2IzSy on the last line of [4.2] result in detectable signals during the acquisition period. The first term leads to a detected signal proportional to sin(It1) cos(Jt1)
277
D
ΩS
D
X
F1
X
ΩI
4.1 TWO-DIMENSIONAL NMR SPECTROSCOPY
ΩI
ΩS F2
FIGURE 4.4 Schematic two-dimensional NMR spectrum for two spins, I and S. D represents diagonal peaks that occur at the same frequency in both dimensions and X represents cross-peaks that appear at different frequencies in each dimension. Cross-peaks result from the transfer of coherence from one spin to the other during the mixing period of the experiment. Diagonal peaks result from coherence that is not transferred between spins during the mixing period.
cos(Jt2) exp(iIt2); the second term leads to a signal proportional to sin(It1) sin(Jt1) sin(Jt2) exp(iSt2). A complementary coherence transfer pathway also exists, whereby magnetization originating on the S spin is transferred to the I spin during the mixing period. This complementary pathway leads to detectable signals proportional to sin(St1) cos(Jt1) cos(Jt2) exp(iSt2) and sin(St1) sin(Jt1) sin(Jt2) exp(iIt2). Ignoring multiplet structures and peak shapes for the present discussion (see Section 6.2.1), Fourier transformation with respect to t1 and t2 generates a (schematic) spectrum of the form shown in Fig. 4.4. Two peaks, D, known as diagonal peaks, are observed at frequencies (F1, F2) ¼ (I, I) and (F1, F2) ¼ (S, S). The diagonal peaks result from magnetization that remains on the same spin I or S throughout the experiment and essentially form a onedimensional spectrum. On the other hand, the two peaks, X, known as cross-peaks, result from magnetization that has been transferred from one spin to the other during the mixing period. Inspection of [4.2] shows that the cross-peak at (F1, F2) ¼ (I, S) results from the 2IzSy operator present at the beginning of t2. This operator is
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generated by the second 90x pulse in the sequence acting on the 2IySz operator. The action of this pulse results in the conversion of an antiphase I operator, which evolves with Larmor frequency I during t1, into an antiphase S operator, which evolves with Larmor frequency S during t2; thereby, coherence is transferred from the I spin to the S spin. The cross-peak at (F1, F2) ¼ (S, I) results from the complementary transfer of coherence from the S spin to the I spin. The second pulse in this experiment comprises the entire mixing period. In the present case, the mixing sequence was designed to effect transfer of magnetization via the scalar coupling between two spins; consequently, the appearance of cross-peaks between spins I and S in the spectrum unambiguously indicates that the two spins are scalar coupled and establishes a through-bond correlation between the spins. In fact, this gedanken experiment is the basis of the original two-dimensional NMR technique, the COSY (correlated spectroscopy) experiment, discussed in Section 6.2 (8–10). Detection of a scalar coupling interaction between the two spins does not depend upon observation of changes in the multiplet structure or intensity of the diagonal resonances as in one-dimensional NMR experiments, but rather upon the appearance of cross-peaks in the two-dimensional spectrum. This property gives multidimensional spectroscopy its immense power: not only are important correlations established, but the two independent frequency coordinates effectively increase the resolution in the spectrum. Overlapping signals in the conventional one-dimensional spectrum, which arise, for example, from multiple scalar coupling interactions, are dispersed into the additional frequency dimension in a process called separation of interactions. The frequency in the F2 dimension (recorded during t2) of each peak in the two-dimensional spectrum must correspond to the frequency of a peak in the conventional one-dimensional spectrum. The converse is not true, because certain resonances in the one-dimensional spectrum can be suppressed in the two-dimensional spectrum. Removal of undesirable resonances by using multiple-quantum and isotope filters are discussed in Chapters 6, 7, and 9. The F1 frequency of a peak in the two-dimensional spectrum need not correspond to a frequency in the conventional spectrum; the exact form of the relationship between the F1 and F2 frequencies of a peak in the two-dimensional spectrum will depend on the particular manipulations of the spins before acquisition. Two very useful features arise because the evolution of a coherence during t1 is never actually physically detected. First, the coherence that is present during t1 can be of a type that cannot
4.1 TWO-DIMENSIONAL NMR SPECTROSCOPY
279
normally be recorded, such as multiple-quantum coherence. A typical experiment of this kind would proceed by preparing multiple-quantum coherence, frequency labeling the coherence during t1, and returning the multiple-quantum coherence to single-quantum coherence for detection. Second, the frequencies of peaks in F1 can be manipulated to be different from the actual frequencies with which coherences evolve during t1. Examples of this type of experiment include the removal of chemical shift evolution and heteronuclear scalar couplings by the application of a 1808 pulse at the midpoint of t1. The functions of the different periods of a two-dimensional NMR experiment are summarized as follows: 1. Preparation: The desired nonequilibrium state of the spin system is prepared from the initial (equilibrium) state of the spin system. The preparation period in its simplest form consists of a single pulse that generates transverse magnetization, but more complex sequences of pulses can be used to prepare other coherences, such as multiple quantum coherences, and to perform solvent suppression. 2. Evolution: The off-diagonal components of the density operator prepared in step (1) evolve under the Hamiltonian, He . During the course of the experiment, the incrementable time t1 normally begins at an initial value and increases in discrete steps to a maximum value, t1max. The Hamiltonian, He , may be the free-precession Hamiltonian or may include applied rf fields. The frequencies with which the detected coherence evolves during t1 results in signals appearing at those frequencies in the F1 dimension of the final two-dimensional spectrum. This process is known as F1 frequency labeling of the coherence. 3. Mixing: During the mixing period, coherence is transferred from one spin to another. The mixing period is the key to establishing the type of correlation between the two dimensions and consequently dictates the information content of the spectrum. Depending on the type of experiment, the mixing period consists of one or more pulses and delays. 4. Acquisition: The FID is recorded in the conventional fashion. As discussed in Section 4.3, if more than one coherence transfer pathway is feasible, phase cycling or field gradient pulses are used to determine which coherence transfer processes contribute to the final spectrum. The evolution of the density operator of the spin system during the pulse sequence is described, using the product operator formalism, as a transformation of the density operator from its initial to final value
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during each of the four parts of the experiment. Thus, eq ð0Þ ðt1 Þ ðt1 , 0Þ
Preparation
) ð0Þ,
Evolution Mixing
) ðt1 Þ, ) ðt1 , 0Þ,
Acquisition
) ðt1 , t2 Þ:
½4:3
The modular nature of two-dimensional NMR experiments evident from this discussion facilitates the construction of new experiments from combinations of prefabricated ‘‘building-block’’ pulse sequences that effect particular transformations of the density operator. This approach is even more powerful for the design of three- and four-dimensional experiments, as will be discussed in subsequent chapters.
4.2 CoherenceTransfer and Mixing The key to obtaining useful chemical or structural information from two-dimensional NMR spectroscopy is the transfer of coherence from one spin to another during the mixing period. In the following sections, two mechanisms for transferring coherence between spins in multidimensional NMR spectroscopy are discussed. Coherence transfer in homonuclear spin systems is discussed first; the generalization to heteronuclear spin systems follows directly. This presentation is not meant to represent a comprehensive account of magnetization transfer processes in multidimensional spectroscopy. Rather, the idea that through-bond scalar coupling and through-space dipolar interactions are responsible for transfer of coherence between spins should be appreciated.
4.2.1 THROUGH-BOND COHERENCE TRANSFER A crucial mixing process in many multidimensional experiments is migration of coherence among scalar-coupled nuclei. This process, known as coherence transfer, has already been introduced in Section 2.7.6. Coherence transfer can be produced by evolution of the spin system under a series of rf pulses and free-precession delays (‘‘pulseinterrupted free precession’’ or COSY-type coherence transfer), or by cross-polarization of the spin system by using continuous, timevarying rf fields [TOCSY-type or homonuclear Hartmann–Hahn
4.2 COHERENCE TRANSFER
AND
MIXING
281
(HOHAHA)-type coherence transfer]. Scalar coupling interactions are mediated by covalent bonding interactions; therefore, COSY and TOCSY mixing generate through-bond coherence transfer. 4.2.1.1 COSY-Type Coherence Transfer The density operator must contain antiphase terms in order to transfer coherence from one spin to another by COSY techniques (8, 9). Thus, before coherence transfer can be achieved, antiphase coherence must develop from in-phase coherence by evolution under the scalar coupling interaction between spins. The antiphase coherence with respect to one spin is transferred by the mixing pulse into antiphase coherence with respect to the scalar-coupled partner. This coherence transfer process is illuminated by considering the effect of the pulse sequence, 90x –=2–180y –=2–90y ,
½4:4
on a weakly coupled homonuclear IS spin system, with a mutual scalar coupling constant JIS. Using [2.121], the propagator for the pulse sequence is U ¼ exp i 2 Iy þ Sy exp½i2JIS Iz Sz exp i 2ðIx þ Sx Þ : ½4:5 As discussed in Section 2.7.7.1, the 1808 pulse refocuses chemical shift evolution; in addition, this pulse effectively inverts the phase of the second 908 pulse. The principles of the experiment are deduced by concentrating on the fate of just one of the spins. Starting with equilibrium I spin magnetization, Iz, the analysis proceeds as follows: 90x
Iz ! Iy
! Iy cosðJIS Þ þ 2Ix Sz sinðJIS Þ 90y
! Iy cosðJIS Þ 2Iz Sx sinðJIS Þ:
½4:6
The term proportional to Iy represents coherence of spin I that is not transferred to spin S during the sequence. The term 2IzSx corresponds to antiphase single-quantum coherence of the S spin and represents coherence transferred to the S spin from the I spin by the mixing sequence. Notice that the 2IzSx operator was generated by the application of the 90y pulse to the antiphase operator 2IxSz. Of course, the analogous treatment for the S spin results in transfer of coherence from the S spin to the I spin during the mixing sequence. The coefficients of the operators depend upon the rate at which the antiphase coherence evolves. The amount of coherence passed from one spin to a
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scalar-coupled partner is termed the coherence transfer amplitude. In the present example, the coherence transfer amplitude is given by sin(JIS), and is maximal when ¼ 1/(2JIS). As stated previously, for coherence transfer to proceed at all, the system must have evolved to an antiphase state with respect to the scalar coupling during the period . Now consider the extension to the situation in which spin I is coupled to two other spins, R and S, with scalar coupling constants JIR and JIS. The R and S spins are assumed to lack a scalar coupling interaction (i.e., JRS ¼ 0). The product operator analysis of the mixing sequence yields 90x
Iz ! Iy
! Iy cosðJIR Þ cosðJIS Þ þ 2Ix Rz sinðJIR Þ cosðJIS Þ þ 2Ix Sz cosðJIR Þ sinðJIS Þ þ 4Iy Rz Sz sinðJIR Þ sinðJIS Þ 90y
! Iy cosðJIR Þ cosðJIS Þ 2Iz Rx sinðJIR Þ cosðJIS Þ 2Iz Sx cosðJIR Þ sinðJIS Þ þ 4Iy Rx Sx sinðJIR Þ sinðJIS Þ: ½4:7
Again, the term proportional to Iy represents coherence of spin I that is not transferred to either coupled spin during the sequence. The term 2IzRx corresponds to antiphase single-quantum coherence of the R spin and represents coherence transferred to the R spin from the I spin by the mixing sequence. Similarly, the term 2IzSx corresponds to antiphase single-quantum coherence of the S spin and represents coherence transferred to the S spin from the I spin by the mixing sequence. The term 4IyRxSx represents a linear combination of multiple-quantum coherences and is not of further interest here (but see Section 6.4.2). Analogous treatments for the R and S spins result in transfers of coherence from the R and S spins to the I spin. No coherence is transferred between spins R and S because they are not scalar coupled to each other, even though they are mutually coupled to the I spin. In the present example, the I spin evolves under the influence of two scalar coupling interactions during . In the evolution leading to the term 2IxRz, the scalar coupling to spin R is called the active coupling and the scalar coupling to spin S is called the passive coupling; in the evolution leading to the term 2IxSz, the scalar coupling to spin R is called the passive coupling and the scalar coupling to spin S is called the active coupling. As shown, each active coupling contributes a factor of
4.2 COHERENCE TRANSFER
AND
MIXING
283
sin(Ja), in which Ja is the active scalar coupling constant, and each passive coupling contributes a factor of cos(Jp), in which Jp is the passive scalar coupling constant, to the magnitude of the product operators. If a given spin is scalar coupled to N other spins, the operators that lead to coherence transfer in the COSY-type mixing sequences have a single active coupling and N 1 passive couplings; operators with no active couplings represent operators for which no coherence transfer occurs, and operators with greater than one active coupling represent the creation of multiple-quantum coherences. As before, the coefficients of the operators depend upon the rate at which antiphase coherence is generated. However, to maximize coherence transfer from I to R or S requires knowledge of the values of JIR and JIS. In general, coherence transfers from I to R and from I to S are not maximized for the same value of unless JIR ¼ JIS. In addition, coherence transfer efficiencies cannot be simultaneously maximized for two and three (or more) spin systems because of the different trigonometric expressions encountered in [4.6] and [4.7]. The final operator of interest in [4.6] is an antiphase S operator. In some circumstances, coherence transfer to an in-phase operator is desirable. In the COSY-style mixing sequences, a second delay period must be used to refocus the antiphase operator. The entire mixing sequence is 90x –=2–180y –=2–90y – 2 =2–180y – 2 =2:
½4:8
By the same reasoning as for [4.4] and [4.5], the effects of the pulse sequence are obtained by analysis of the propagator: U ¼ exp½i2JIS 2 Iz Sz exp i 2 Iy þ Sy exp½i2JIS Iz Sz exp i 2 ðIx þ Sx Þ , ½4:9 in which only the scalar coupling Hamiltonian is effective during and 2. The evolution up to the 90y pulse has already been presented in [4.6] and [4.7]; only the analysis of the additional effects of the 2 period must be considered here. For simplicity, only the antiphase term 2IzSx, which results from the coherence transfer step analyzed in [4.6], is treated: 2Iz Sx sinðJIS Þ
2
) 2Iz Sx sinðJIS Þ cosðJIS 2 Þ þ Sy sinðJIS Þ sinðJIS 2 Þ:
½4:10
The second term on the right-hand side of [4.10] represents in-phase transverse magnetization of the S spin. Complete refocusing of the
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
antiphase operator is obtained for 2 ¼ 1/(2JIS). Thus, coherence transfer from an in-phase state on one spin to an in-phase state on a coupled spin requires a total time of 1/JIS when employing pulse-interrupted freeprecession methods. The principal limitation of COSY-type coherence transfer arises from the antiphase multiplet structure of the resulting cross-peaks in the spectrum. If the size of the active coupling is comparable to the linewidth, partial cancellation of the multiplet occurs due to destructive interference between the antiphase components of the peak. To avoid self-cancellation, the antiphase components must be refocused, so that the resulting cross-peak multiplet is composed of peaks entirely of the same algebraic sign. The destructive interference effects are then eliminated, but only at the expense of an additional refocusing period of duration 1/(2JIS). A more detailed discussion of COSY experiments is given in Section 6.2. 4.2.1.2 TOCSY Transfer Through-Bonds Thus far, coherence transfer has been limited to pulse-interrupted free-precession techniques, or COSY-type transfer via evolution under the scalar coupling Hamiltonian in the weak coupling limit. To begin the present discussion, consider the evolution of the density operator under the strong scalar coupling Hamiltonian between two spins I and S (11): H ¼ 2JIS I S ¼ 2JIS Ix Sx þ Iy Sy þ Iz Sz :
½4:11
The evolution of the Iz operator is given by expðiH m ÞIz expðiH m Þ ¼ exp i2JIS m Ix Sx þ Iy Sy þ Iz Sz Iz exp i2JIS m Ix Sx þ Iy Sy þ Iz Sz ¼ expði2Ix Sx Þ exp i2Iy Sy expði2Iz Sz ÞIz expði2Iz Sz Þ exp i2Iy Sy expði2Ix Sx Þ ¼ expði2Ix Sx Þ exp i2Iy Sy Iz exp i2Iy Sy expði2Ix Sx Þ,
½4:12
in which ¼ JIS m, m is the mixing time, and the third line is obtained because the operators IxSx, IySy, and IzSz commute with each other. The evolution is calculated by making use of [2.121] to evaluate the effects of
4.2 COHERENCE TRANSFER
AND
MIXING
285
the propagator U ¼ exp(–iIySy), exp i2Iy Sy Iz exp i2Iy Sy ¼ exp i 2 Ix exp i 2Sx expði2Iz Sz Þ exp i 2 Sx exp i 2 Ix Iz exp i 2 Ix exp i 2Sx expði2Iz Sz Þ exp i 2Sx exp i 2Ix ¼ exp i 2 Ix exp i 2Sx expði2Iz Sz ÞIy expði2Iz Sz Þ exp i 2 Sx exp i 2 Ix ¼ exp i 2 Ix exp i 2Sx Iy cos 2Ix Sz sin exp i 2 Sx exp i 2 Ix ¼ exp i 2 Ix Iy cos þ 2Ix Sy sin exp i 2 Ix ¼ Iz cos þ 2Ix Sy sin , ½4:13 followed by U ¼ exp(–iIxSx), expði2Ix Sx Þ Iz cos þ 2Ix Sy sin expði2Ix Sx Þ ¼ exp i 2Iy exp i 2 Sy expði2Iz Sz Þ exp i 2Sy exp i 2Iy Iz cos þ 2Ix Sy sin exp i 2Iy exp i 2 Sy expði2Iz Sz Þ exp i 2 Sy exp i2 Iy ¼ Iz cos2 þ 2Iy Sx þ 2Ix Sy cos sin þ Sz sin2 , ½4:14 to yield the final result, Iz Sz
H m
)Iz cos2 ðJIS m Þ þ Sz sin2 ðJIS m Þ þ 2 Ix Sy Iy Sx cosðJIS m Þ sinðJIS m Þ,
Hm
)Iz sin2 ðJIS m Þ þ Sz cos2 ðJIS m Þ 2 Ix Sy Iy Sx cosðJIS m Þ sinðJIS m Þ,
½4:15
in which the evolution of the Sz operator is obtained by exchanging the I and S labels. Equation [4.15] predicts that the sum, Iz þ Sz, is a constant and that the difference, Iz – Sz, is given by ðIz Sz Þ
H m
) ðIz Sz Þ cosð2JIS m Þ þ 2ðIx Sy Iy Sx Þ sinð2JIS m Þ: ½4:16
If m ¼ 1/(2JIS), Iz magnetization is transferred completely to Sz magnetization and vice versa. Evolution under the strong coupling Hamiltonian transfers in-phase magnetization between spins in a time of 1/(2JIS), compared with the time of 1/JIS required for in-phase coherence
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transfer in weakly coupled systems via free-precession techniques. In addition, in a three-spin IRS system, magnetization can be transferred from R to S even if JRS ¼ 0 by the two-step transfer Rz ! Iz ! Sz. In real situations, the Hamiltonian for a spin system contains chemical shift and rf terms in addition to the scalar coupling interaction. Magnetization transfer via the strong scalar coupling interaction is efficient only if all chemical shift and rf terms of the Hamiltonian governing the spin system have identical values for each of the two spins I and S. This is the Hartmann–Hahn condition (12). Coherence transfer via Hartmann–Hahn cross-polarization has been used extensively in heteronuclear NMR experiments in the solid state. Braunschweiler and Ernst first demonstrated the feasibility of Hartmann–Hahn crosspolarization in homonuclear solution-phase NMR spectroscopy (11). Hartmann–Hahn matching in the rotating reference frame can be achieved by application of an rf field of sufficient strength that any offset and chemical shift effects are negligible in comparison. Mu¨ller and Ernst (13) demonstrated that transfer of I spin magnetization to the S spin, keeping only the in-phase operators of interest, proceeds as, Iz Sz
H m H m
) Iz aII ð m Þ þ Sz aIS ð m Þ,
½4:17
) Iz aSI ð m Þ þ Sz aSS ð m Þ,
in which aII ð m Þ ¼ 1 sin2 sin2 ðq m Þ, aSS ð m Þ ¼ 1 sin2 sin2 ðq m Þ,
½4:18
aIS ð m Þ ¼ aSI ð m Þ ¼ sin2 sin2 ðq m Þ, and 1=2 , q ¼ ðIðeffÞ SðeffÞ Þ2 þ ðJIS sinI sinS Þ2 1 2JIS sinI sinS tan ¼ , 2 IðeffÞ SðeffÞ 1=2 1=2 and SðeffÞ ¼ 2S þ !21S , IðeffÞ ¼ 2I þ !21I !1I !1S I ¼ tan1 and S ¼ tan1 : I S
½4:19
Here I and S are the tilt angles of the effective field at the I and S spins, respectively, !1I and !1S are the rf field strengths experienced by the I and S spins, respectively; I is the I spin offset, S is the S spin offset, and JIS is the scalar coupling constant between the I and S spins.
4.2 COHERENCE TRANSFER
AND
287
MIXING
In [4.17], aII( m) is the amount of magnetization remaining on spin I, aSS( m) is the amount of magnetization remaining on spin S, and aIS( m) is the amount of magnetization transferred to spin S at time m. The functions aII( m), aSS( m), and aIS( m) frequently are called the mixing coefficients. If I ¼ S, then the Hartmann–Hahn matching condition is satisfied because I(eff) ¼ S(eff). For the special case I ¼ S ¼ 0, [4.17] reduces to [4.15]. If the two scalar-coupled spins have different offsets from the rf carrier frequency, |I| 6¼ |S|, Hartmann–Hahn matching becomes more difficult, and magnetization transfer is reduced drastically. In practice, the power required to accomplish efficient Hartmann–Hahn matching over a significant frequency range I 6¼ |S| by using a continuous rf field would produce disastrous sample and probe heating effects. As shown by [4.17], differences in chemical shift between two coupled spins prevent efficient Hartmann–Hahn matching by application of a continuous rf field. Ideally, an effective Hamiltonian is required such that it eliminates the chemical shift terms over a significant frequency range while the rf field is applied. In effect, the spin Hamiltonian, H, must be reduced from H¼
X
!i Iiz þ 2
i
X
Jij Ii Ij
½4:20
i6¼j
to an effective Hamiltonian, H ¼ 2
X
Jij Ii Ij
½4:21
i6¼j
for a period m, which is of the order of 1/Jij (11). The last equation consists of just the pure scalar coupling term, in which all shift terms or linear operators are removed, leaving only bilinear operators. The absence of chemical shift terms means that the Hartmann–Hahn condition is always satisfied. A pulse sequence that generates an average Hamiltonian given by [4.21] is said to be isotropic. Magnetization transfer under the influence of such a sequence is a continuous mixing process, with the magnetization moving in a periodic fashion among all the spins in the scalar-coupled network. Such pulse sequences shall be referred to as isotropic mixing sequences. An important practical consequence of an isotropic mixing sequence is that the transfer of magnetization occurs equally well for all angular momentum components. That is, coherent exchange of difference magnetization will occur under the isotropic scalar coupling Hamiltonian according to [4.16] with
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similar relations for the x and y components, obtained by cyclic permutation of the indices: ðIx Sx Þ
ðIy Sy Þ
H m
) ðIx Sx Þ cosð2JIS m Þ þ ðIy Sz Iz Sy Þ sinð2JIS m Þ, ½4:22
H m
) ðIy Sy Þ cosð2JIS m Þ þ ðIx Sz Iz Sx Þ sinð2JIS m Þ: ½4:23
The design and performance properties of experimental techniques to achieve isotropic mixing are described in Section 6.5. Until now, the second terms on the right-hand sides of [4.16], [4.22], and [4.23] have not been considered. These terms contain bilinear product operators that are orthogonal to the in-phase terms of interest. The bilinear term in [4.16] is a multiple-quantum coherence term, while bilinear terms in [4.22] and [4.23] represent antiphase single-quantum coherence operators. The significance of the bilinear terms will be discussed for [4.22], although the arguments are applicable equally to the other cases. Simple inspection shows that the antiphase term has y-phase while the in-phase term has x-phase; thus a 908 phase difference exists between the terms. If the signal resulting from the in-phase magnetization is phased to be absorptive, then the signal resulting from the antiphase term automatically becomes dispersive. The dispersive antiphase multiplets occur in the same spectral position in which the in-phase absorptive peaks occur, and can disrupt the lineshapes in the two-dimensional NMR spectrum. Indeed, for small molecules with narrow linewidths, interference from the dispersive antiphase components can be observed. In contrast, the resonance peaks for large biological macromolecules usually appear to be completely absorptive, because the linewidths are notably larger, and the dispersive antiphase components self-cancel very efficiently. Dispersive antiphase resonances can be suppressed further by z-filtration (Section 6.5) (14, 15). To summarize, coherence transfer is obtained when two scalarcoupled spins are subjected simultaneously to an rf field that effectively removes the chemical shifts of the spins. When used for homonuclear coherence transfer under the scalar coupling Hamiltonian, this technique often is referred to as a homonuclear Hartmann–Hahn (16) experiment to indicate the required Hartmann–Hahn matching condition. Throughout the remainder of this text, isotropic mixing sequences that satisfy the Hartmann–Hahn condition will be used to mediate coherence transfer. Pulse sequences utilizing isotropic mixing will be
4.2 COHERENCE TRANSFER
AND
MIXING
289
referred to as TOCSY (total correlation spectroscopy) experiments, as originally suggested by Braunschweiler and Ernst (11). The use of the word ‘‘total’’ in deriving the acronym implies that all spins belonging to a scalar-coupled network are connected by such an experiment.
4.2.2 THROUGH-SPACE COHERENCE TRANSFER As will be discussed in Sections 5.1.2 and 5.5, perturbing the populations of stationary states within a spin system causes timedependent changes in the intensities of dipolar-coupled resonance signals via the NOE (nuclear Overhauser effect) (3, 4). Dipolar cross-relaxation is an extremely useful mixing process in multidimensional NMR spectroscopy, because the efficiency of mixing depends upon the distance between interacting spins. Thus, through-space, rather than through-bond, magnetization transfer generates cross-peaks in the NOE mixing process. Consider the effect of the following pulse sequence on a pair of dipolar coupled spins, I and S, which have no scalar coupling between them: 90x –t1 –90x – m –90x –t2 :
½4:24
This pulse sequence is known as the NOESY (nuclear Overhauser enhancement spectroscopy) (17) experiment and is the most powerful and important technique available for structural investigations of biomolecules by solution-state NMR spectroscopy. A more detailed account of this experiment will be presented in Section 6.6.1. Concentrating solely on the I spin, the following product operators are present after the second 90x pulse: Iz
ð=2Þx t1 ð=2Þx
) Iz cosðI t1 Þ þ Ix sinðI t1 Þ:
½4:25
Experimentally the Ix term is suppressed by phase cycling or by application of a field gradient pulse (Section 4.3), which leaves only the Iz cos(It1) term. A close analogy to the one-dimensional transient NOE experiment (Section 5.1.2) now is apparent. The term Iz cos(It1) represents a perturbation of the I spin from the equilibrium þIz state. The perturbation depends upon the value of t1; for example, whenever It1 ¼ 2, the populations are inverted across the I spin transitions. Consequently an NOE will be induced on the S spin during the fixed delay, m, because the I and S spins have a dipolar coupling. The delay, m, is known as the mixing time and is set to a suitable value (of the
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order of 1/R1) to allow a significant NOE to develop. The effect of cross-relaxation during the mixing time is represented as: Iz cosðI t1 Þ
m
) Iz cosðI t1 ÞaII ð m Þ Sz cosðI t1 ÞaIS ð m Þ, ½4:26
in which aII( m) represents the fraction of the original magnetization remaining on the I spin, and aIS( m) represents the fraction of the original magnetization transferred from the I spin to the S spin during the mixing time by dipolar cross relaxation. The functional forms of aII( m) and aIS( m) are discussed more fully in Section 5.1.2. The final pulse generates an observable term on the S spin of the form Sy cos(It1)aIS( m); upon two-dimensional Fourier transformation, a crosspeak will be generated at frequency (F1, F2) ¼ (I, S) with amplitude proportional to aIS( m). For any magnetization that remains on the I spin, the final pulse will result in an observable term of the form Iy cos(It1)aII( m), which will yield a diagonal peak with frequency coordinates of (F1, F2) ¼ (I, I). Identical pathways also exist for magnetization transfer S to I and corresponding diagonal and crosspeaks result.
4.2.3 HETERONUCLEAR COHERENCE TRANSFER In multidimensional NMR spectroscopy, different spins in a molecule are correlated by separating their interactions into more than one frequency dimension; however, the interacting spins do not necessarily have to be of the same nuclear species. Coherence can be transferred between different nuclear species using techniques analogous to those presented for homonuclear spin systems. The corollaries to the COSY-style homonuclear coherence transfer sequences are the INEPT (18, 19) and distortionless enhancement by polarization transfer (DEPT) (20, 21) family of pulse sequences. Heteronuclear crosspolarization corresponds to the TOCSY-style homonuclear sequences. Heteronuclear NOESY magnetization transfer via the heteronuclear dipolar coupling is analogous to the homonuclear experiment (22, 23). In heteronuclear experiments, rf pulses are applied at more than one frequency (typically differing by hundreds of megahertz) in order to manipulate both the heteronuclear and the 1H spins. Thus, in contrast to homonuclear experiments in which nonselective pulses affect all nuclei, different nuclear species are manipulated independently by rf pulses. As has already been shown in Section 2.7.7.2, the product operator approach can be used to describe manipulations of spin systems that
4.2 COHERENCE TRANSFER
AND
291
MIXING
contain operators corresponding to different nuclear species. As will be seen in Chapter 7, some of the most powerful multidimensional NMR methods rely on heteronuclear coherence transfer between 13C/15N and 1H.
4.2.4 COHERENCE TRANSFER UNDER RESIDUAL DIPOLAR COUPLING HAMILTONIANS As described in Section 2.8, in an anisotropic environment, induced by external fields or physical media such as liquid crystalline materials, the direct dipole–dipole Hamiltonian is not averaged identically to zero and a residual dipole coupling Hamiltonian given by [2.325] is obtained. When the coupling is weak, the Hamiltonian is truncated to [2.328]. This Hamiltonian and the weak scalar coupling Hamiltonian have the same functional form. Therefore, through-space coherence transfer under the residual dipole coupling Hamiltonian is obtained by COSY-type approaches identical to those described in Section 4.2.1.1 for throughbond coherence transfer under the scalar coupling Hamiltonian. Circumstances are different if an isotropic mixing sequence is applied to a system of weakly aligned molecules. The mixing sequence, as for the analogous TOCSY experiment (Section 4.2.1.2), removes the chemical shift Hamiltonian, and evolution occurs under the full residual dipole coupling Hamiltonian given by [2.325]. However, in contrast to TOCSY transfer under the strong scalar coupling Hamiltonian, the residual dipole coupling Hamiltonian [2.325] is not isotropic. In the presence of the mixing sequence, the spin operators in [2.325] are averaged by rotation. For a mixing sequence applied along the x-axis, 2Ix Sx ¼ 2Ix Sx , 2Iy Sy ¼ 2Iy Sy cos2 þ 2Iy Sz þ 2Iz Sy cos sin þ 2Iz Sz sin2 ¼ Iy Sy þ Iz Sz ,
½4:27
2Iz Sz ¼ 2Iz Sz cos2 2Iy Sz þ 2Iz Sy cos sin þ 2Iy Sy sin2 ¼ Iy Sy þ Iz Sz , in which is the time-dependent rotation angle and averaging over the entire mixing sequence is denoted by the overbars. The effective residual dipole–dipole coupling Hamiltonian during the mixing sequence is given by H ¼ 12DIS ð2Ix Sx 2Iy Sy 2Iz Sz Þ:
½4:28
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
As [4.27] makes clear, the isotropic strong scalar coupling Hamiltonian is unchanged by averaging during the mixing sequence. Evolution under the Hamiltonian [4.28] can be analyzed using the same approach used in Section 4.2.1.2. The resulting expressions for evolution of Ix and Iz magnetization are (24) Ix
Iz
H m
) Ix 12ð1 þ cosDIS m Þ þ Sx 12ð1 cosDIS m Þ þ 2Iy Sz 2Iz Sy 12 sinDIS m ,
H m
) Iz
1 2
cos32DIS m þ 12 cos 12DIS m
½4:29
þ Sx 12 cos32DIS m 12 cos 12DIS m þ 2Iy Sx þ 2Ix Sy 12 sin32DIS m þ 2Iy Sx 2Ix Sy 12 sin 12DIS m : ½4:30 Because the Hamiltonian is not isotropic, different transfer functions are obtained for the orthogonal Ix and Iz operators. Evolution of an initial density operator proportional to Ix is formally identical to the transfer obtained in the TOCSY experiment. In contrast, if the initial state of the density operator is proportional to Iz, [4.30] indicates that the diagonal peaks (proportional to Iz) and cross-peaks (proportional to Sz) have opposite signs. This unique property of evolution under the residual dipole coupling Hamiltonian can be used to identify evolution due to three-spin interactions (24). In summary, homonuclear coherence transfer under the residual dipole–dipole Hamiltonian for partially aligned molecules in the presence of a mixing sequence has been termed the DCOSY (dipolar correlation spectroscopy) experiment (24). The resulting Hamiltonian is not isotropic and the coherence transfer amplitudes obtained depend on the initial state of the density operator.
4.3 Coherence Selection, Phase Cycling, and Field Gradients Modern NMR experiments consist of the application of multiple rf pulses to the system under investigation and detection of the resulting resonance signals. These multipulse NMR techniques are described by the pulse sequences used to generate the observed signal and by the evolution of the density operator through the pulse sequence. If an experiment consists of multiple pulses and delays, then more than
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
FIELD GRADIENTS
293
one coherence transfer pathway that leads to observable signals may exist for the spin system of interest. A spectrum derived from many different but simultaneously occurring coherence transfer pathways would be extremely complex and difficult to interpret. Phase cycling or field gradients are used to select a specific pathway and provide an interpretable spectrum. Phase cycling refers to the process of repeating a pulse sequence several times with a systematic variation of the relative phases of the pulses within the sequence. Coherence selection by means of phase cycling normally is implemented during the process of signal averaging. Field gradients are spatially inhomogeneous magnetic fields that are activated for specific periods within a pulse sequence. Coherence selection using pulsed field gradients is achieved during a single repetition of the pulse sequence. In the following sections, the principles of coherence selection with phase cycles and pulsed field gradients are illustrated. The text follows closely the excellent approach presented by Keeler (25) and employs the coherence transfer pathway methods of Bodenhausen (26).
4.3.1 COHERENCE LEVEL DIAGRAMS The concept of coherence developed in Section 2.6 is crucial to an understanding of phase cycling and field gradient techniques. The order of a coherence between two eigenstates of a spin system is defined as p ¼ m, in which m is the difference between the magnetic quantum numbers of the two eigenstates. If an arbitrary operator is expressed in the shift operator basis (Section 2.7.2), then the coherence order is given by the number of raising operators minus the number of lowering operators comprising the representation. Thus, the absolute value of p must be less than or equal to pmax, which is given by the number of spins involved in the coupling network. Longitudinal magnetization, although not strictly a coherence, has properties in common with zero-quantum coherence and is treated as such for phase cycling procedures. At any particular point during a pulse sequence, various coherences may be present simultaneously. Normally, only one, or a small number, of the possible coherences are retained to generate a useful signal. At any time, the coherences present are classified according to their various orders (double, single, zero, etc.), and each coherence order is said to correspond to a different coherence level. For example, double-quantum coherence has a coherence level of 2, and longitudinal magnetization has a coherence level of zero. Formally, the density operator, , is
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
written as the expansion ¼
pmax X
p,
½4:31
p¼pmax
in which p is the component of the density operator associated with a particular coherence level, p. The sum in [4.31] contains 2pmax þ 1 terms and consequently does not represent an expansion of the density operator into a complete set of basis functions, because a complete set consists of 4p elements. Each term in [4.31] would have to be further subdivided to generate a complete basis set. For example, longitudinal magnetization and zero-quantum coherence are part of the same component 0 in [4.31] but are represented by two different orthogonal basis operators. Each term p is given in terms of single transition operators by X cab jaihbj, ½4:32 p ¼ a, b
in which the sum extends over all combinations of eigenstates |ai and hb| for which the magnetic quantum numbers satisfy the relationship ma – mb ¼ p. The effects of a pulse sequence (i.e., pulses and periods of free precession) on coherence order are encapsulated in two rules: first, rf pulses can cause the transfer of coherence from one level to another, and second, periods of free precession conserve the order of coherence. Indeed, an rf pulse potentially can transfer coherence to all coherence levels available to the spin system. The generation of different coherence orders during an NMR experiment is subject to the following three corollaries: (i) the coherence transfer pathway must start at coherence level p ¼ 0, as this corresponds to thermal equilibrium longitudinal magnetization, (ii) only coherence orders p ¼ 1 are created by an rf pulse acting on the thermal equilibrium density operator, and (iii) if the complex signal is observed using quadrature detection then the coherence pathway must end at p ¼ 1. As has been stated several times, most coherences generated during a pulse sequence are suppressed, and only the coherences that will generate the desired NMR spectrum are retained. Coherence is transferred in a specific manner between other coherences by rf pulses during a pulse sequence. The trace of coherence level changes that result in the desired NMR spectrum is known as the coherence transfer pathway. The objective is to use an appropriately designed phase cycle or application of an appropriate set of field gradient pulses to detect only
4.3 COHERENCE SELECTION, PHASE CYCLING, 90°
AND
90°
FIELD GRADIENTS
295
90° t1
coherence level +2 +1 0 –1 –2 coherence transfer pathway
FIGURE 4.5 Coherence transfer pathway for a double-quantum experiment. Double-quantum coherence is selected during the t1 period. The pathway indicated is only one of many pathways that are generated during the pulse sequence; unwanted coherence transfer pathways are rejected by phase cycling.
those signals that follow the chosen coherence transfer pathway. For example, in a two-dimensional double-quantum experiment (Section 6.4.1), the intention is to have double-quantum coherence evolve during t1. The coherences present at each point in the pulse sequence are expressed conveniently using a coherence level diagram, as shown in Fig. 4.5 for a double-quantum experiment. The feasible coherence levels (2, 1, 0, 1, 2 for a two-spin system) are shown as horizontal lines. The heavy solid lines trace the desired coherence transfer pathway by showing the desired coherence levels at every point in the pulse sequence. The indicated trace is only one of many possible coherence pathways that are generated by this particular pulse sequence. Pathways (not shown) that have coherence levels of 1, 0, or þ1 during t1 must be suppressed by the phase cycle or field gradient pulses.
4.3.2 PHASE CYCLES Single-quantum coherence between two nuclear spin angular momentum states, or transverse magnetization, is responsible for the induction of a voltage in the receiver coil. Coherence is an oscillating function of time and is conveniently represented by a vector rotating in a circle (at least for an isolated spin treatable by the Bloch formalism). The angular position of this coherence ‘‘vector’’ at the beginning of the free induction decay determines the phase of the corresponding line in the spectrum. Conventionally, one axis (the reference axis) is chosen such that an absorption mode line is produced when the coherence
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
vector is aligned with this axis at the start of the acquisition period; other orientations of the vector give different phases of the resonance signal. Figure 4.6 illustrates the relationship between the phase of the rf pulse, the initial orientation of the coherence vector, and the phase of the resonance signal. An alternative way to change the phase of a resonance line is to shift the reference axis while keeping the pulse phase fixed. Pulse phase
Type of spectrum
FT + absorption
90x
FT 90y
+ dispersion
FT 90–x
– absorption
FT – dispersion
90–y
= receiver position
FIGURE 4.6 Pulse phase. The phase of a 908 pulse is shifted in increments of 908 while keeping the phase of the receiver constant. The phase of the resulting signal in a one-pulse experiment changes in conjunction with the phase change of the pulse. Co-addition of the four resonance signals results in the cancellation of the signal.
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
FIELD GRADIENTS
297
Figure 4.7 illustratesthe relationship between the receiver phase and the phase of the resonance signal. Comparison of Figs. 4.6 and 4.7 indicates that a given phase difference in the signal is achieved equally well by adjusting the phase of the rf pulses or of the receiver. Referring to the block diagram of an NMR spectrometer shown in Chapter 3 (Fig. 3.1), the receiver phase can be adjusted in two ways: the phase of the rf reference signal in the phase-sensitive detector can be Receiver phase
Type of spectrum
FT 0°
+ absorption
FT 90°
– dispersion
FT 180°
– absorption
FT 270°
+ dispersion
= receiver position
FIGURE 4.7 Receiver phase. The phase of the receiver is shifted in increments of 908 while keeping the phase of the 908 pulse constant. The phase of the resulting signal in a one-pulse experiment changes in conjunction with the phase change of the receiver. Co-addition of the four resonance signals results in the cancellation of the signal.
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
changed (see also Fig. 3.5) and the digitizer phase can be changed. A change in digitizer phase really is a purely digital manipulation of the FID in computer memory. A 1808 digitizer phase shift is obtained by negating the data after they have been digitized in the ADC; a 908 digitizer phase shift is obtained by exchanging the real and imaginary parts of the complex signal after they have been digitized in the ADC. Herein, a distinction between these two approaches to shifting the receiver phase rarely needs to be made; however, the effects of imperfections in components in the phase-sensitive detector, such as electrical dc (nonsinusoidal) baseline offset and gain imbalance in the receiver channels (Section 4.3.2.3), only can be eliminated by phase cycling executed on the digitizer phase. If the pulse is cycled through the four phases, x, y, –x, and –y, on four successive experiments and the transients added together (in either the time or frequency domain), then a null signal is obtained because the two absorptive signals are exactly 1808 out of phase with each other, as are the two dispersive signals. A similar result is obtained for the transients shown in Fig. 4.7. However, if the phase of the pulse is cycled and the receiver phase is moved in concert to track the change in the phase of the coherence, then each transient results in an absorption signal with the same phase, as shown in Fig. 4.8. If the four experiments are combined, the signals add constructively and a final spectrum is obtained that contains a single absorption line. This simple example, which forms the basis of the CYCLOPS technique (Section 4.3.2.3), illustrates the basic principle of phase cycling. The signal of interest is forced to change phase, by shifting the phase of rf pulses, in conjunction with the receiver, so as to cause the signal recorded from different transients to accumulate. In the same manner, unwanted signals are suppressed by ensuring that signals recorded from a series of transients cancel. 4.3.2.1 Selection of a Coherence Transfer Pathway The property used to select a specific coherence transfer pathway by phase cycling is as follows: If a pulse is changed in phase by an amount, , then a coherence undergoing a change in coherence level of p, due to that pulse, acquires a phase shift of p.
For example, consider a coherence at level þ3 being transferred to level þ1 by the action of a pulse. If, during the experiment the pulse changes phase by , then the coherence will acquire phase p, where
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
Pulse phase
FIELD GRADIENTS
299
Type of spectrum
FT 90x
+ absorption
FT 90y
+ absorption
FT 90–x
+ absorption
FT + absorption
90–y
= receiver position
FIGURE 4.8 Pulse and receiver phase. The phase of the 908 pulse is shifted in increments of 908 while simultaneously shifting the phase of the receiver in increments of 908. In this case, the resulting signal retains the same phase in each experiment (shown as absorptive here). Co-addition of the four resonance signals results in the coherent summation of the signal.
p ¼ (þ1) (þ3) ¼ 2. Thus the coherence acquires phase þ2. The coherences are labeled with phase shifts during the pulse sequence. The ultimate phase of the observed magnetization depends on the total phase angle that the coherence acquires during the coherence transfer steps in the sequence. The accumulated phase angles of the desired
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
coherence transfer pathway enable its selection. Coherence selection is accomplished simply by changing the phases of the relevant pulses in the sequence by the appropriate amounts. To prove the preceding statement, the effect on a particular coherence of a nonselective z-rotation (affecting all spins identically) is proved first: X cab expðiFz Þjaihbj expðiFz Þ expðiFz Þp expðiFz Þ ¼ a, b
¼
X
cab expðima Þjaihbj expðimb Þ
a, b
¼
X
cab jaihbj expðipÞ
a, b
¼ p expðipÞ:
½4:33
Next suppose that some propagator U transfers coherence from an element p to an element q: q ¼ U p U1 :
½4:34
If all rf pulses are shifted in phase by an angle , then (note that the Zeeman and scalar coupling interactions are unaffected by z-rotations) expðiFz ÞU expðiFz Þ p expðiFz ÞU1 expðiFz Þ ¼ expðiFz ÞU p expðipÞU1 expðiFz Þ ¼ expðiFz Þ q expðiFz Þ expðipÞ ¼ q expðiqÞ expðipÞ ¼ q expðipÞ,
½4:35
with p ¼ q – p, which provides the desired proof. Phase shifts of the receiver are represented by applying z-rotations to the detection operator Fþ: expðirec Fz ÞFþ expðirec Fz Þ ¼ Fþ expðirec Þ:
½4:36
Clearly, to coherently accumulate signals from a desired coherence transfer pathway, Tr ðtÞFþ ¼ Tr ðtÞFþ expðipÞ expðirec Þ, ½4:37 and rec ¼ p. Thus, as already noted, the receiver phase shifts must follow the phase shifts accrued by the density operator through the series of pulse phase cycles.
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
FIELD GRADIENTS
301
To illustrate the use of phase cycling to retain one coherence transfer pathway while rejecting another, suppose that in some pulse sequence a mixture of double-quantum coherence (p ¼ 2) and zero-quantum coherence (p ¼ 0) has been created. Application of a 908 pulse to the initial zero- and double-quantum states causes the desired and undesired transfers between coherence levels shown in Fig. 4.9. The goal of the phase cycle is to convert double-quantum coherence to observable single-quantum coherence (p ¼ 1) while suppressing any signal from the zero-quantum coherence. The experiment is repeated four times; upon each repetition, the phase of the pulse is incremented by 908 to yield a phase cycle of 08, 908, 1808, 2708 (which conventionally is written as x, y, x, y). The desired
wanted pathways +2 +1 0 –1 –2 ∆p = –3 and +1 unwanted pathway +2 +1 0 –1 –2 ∆p = –1
FIGURE 4.9 Selection of double-quantum coherence. The upper part of the figure illustrates the desired coherence transfer pathways corresponding to coherence-level changes from double-quantum (p ¼ þ2 and –2) to observable single-quantum (p ¼ –1) coherence. These pathways have coherence order changes of p ¼ –3 and þ1, respectively. The lower part of the figure illustrates the coherence transfer pathway to be rejected corresponding to a coherence-level change from zero-quantum (p ¼ 0) to observable singlequantum (p ¼ –1) coherence. This pathway has a coherence order change of p ¼ –1.
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
coherence (p ¼ 2) goes through two pathways, with p ¼ 3 (p ¼ þ2 to p ¼ 1) and p ¼ þ1 (p ¼ 2 to p ¼ 1). The coherence that undergoes the level change p ¼ 3 changes phase by an amount –p ¼ – (–3) ¼ þ3. The acquired phase of the observable coherence for the four transients will be 08, 2708, 5408, 8108, or in terms of 08 to 3608 rotations, 08, 2708, 1808, 908. In a similar way, the coherence undergoing a level change of p ¼ þ1 acquires a phase of –p ¼ (1) ¼ –. The coherence acquires phase 08, –908, –1808, –2708, which in the 08 to 3608 reference frame is equivalent to 08, 2708, 1808, 908. Thus, in the experiment described, the two coherence level changes of p ¼ 3 and p ¼ þ1 result in the observable coherences acquiring phase in an identical fashion. To ensure that the required signal (from both pathways) accumulates constructively on successive transients, then the receiver phase must follow exactly the phase shifts of the wanted coherence. Therefore the receiver phase cycle is 08, 2708, 1808, 908. This information is contained in Table 4.1. The coherence level change from zero-quantum coherence to observable single-quantum coherence has p ¼ 1. Employing the same analysis as previously, the coherence acquires phase –p ¼ (1) ¼ þ, or 08, 908, 1808, 2708. This result, along with the receiver phase cycle determined previously, is tabulated in Table 4.2. While steps 1 and 3 of the receiver phase cycle follow the coherence, steps 2 and 4 are exactly opposite to the coherence phase. Consequently, the same phase cycle, including pulse and receiver phase shifts, that retains the wanted
TABLE 4.1 Selection of Double-Quantum Coherence Pulse phase ()
p
Equivalent cycle
Receiver phase
0 90 180 270
Coherence change p ¼ 3 0 0 270 270 540 180 810 90
0 270 180 90
0 90 180 270
Coherence change p ¼ þ1 0 0 90 270 180 180 270 90
0 270 180 90
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
303
FIELD GRADIENTS
TABLE 4.2 Rejection of Zero-Quantum Coherence Pulse phase () 0 90 180 270
p
Equivalent cycle
Coherence change p ¼ 1 0 0 90 90 180 180 270 270
Receiver phase 0 270 180 90
pathways (p ¼ 3 and p ¼ þ1) also serves to eliminate the unwanted pathway (p ¼ 1), exactly as required. In the preceding example, a four-step phase cycle with increments in phase of 908 was able to discriminate between coherence transfer pathways of p ¼ 3, þ1 and p ¼ 1. The effect of this phase cycle is represented conveniently using the nomenclature of Bodenhausen and co-workers, as: 3 (2) (1) (0) 11 (þ2) (þ3), where the pathways passed by the cycle are set in bold and the pathways blocked by the cycle are set in parentheses. Under the proposed scheme, two pathways are allowed to pass. In general, if a phase cycle uses increments in phase of 360/N degrees, then along with the pathway p selected, pathways m ¼ p nN, where n ¼ 1, 2, 3. . . , will also be selected. Bodenhausen showed that the length and selectivity of a phase cycle are related (26). If a particular value of p is to be chosen from r consecutive values, then N must be at least r. In practical terms, increased selectivity in choosing a specific coherence transfer pathway requires a larger number of smaller steps in the phase cycle. Continuing with the preceding example, now consider discriminating between the two pathways p ¼ 3 and p ¼ þ1, both of which were retained by the original phase cycle. For instance, suppose only the p ¼ þ1 pathway is to be retained and the p ¼ 3 pathway is to be rejected. Table 4.3 shows the effects of extending the phase cycle to six steps rather than four and using a phase increment of 608 rather than 908. The analysis proceeds exactly as before, making sure that the receiver shifts in phase so as to follow the phase acquired by the coherence going through the p ¼ þ1 pathway. Consulting Table 4.3, the signal from the p ¼ þ1 pathway will add on all transients. However, the effect of this receiver phase cycle on the signal arising from the p ¼ 3 pathway is not obvious. In the approach adopted by Keeler (25), the net effect of the phase cycle is represented by
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
TABLE 4.3 Distinguishing p ¼ þ1 from p ¼ 3 Pulse phase ()
Equivalent cycle
p
Receiver phase
0 60 120 180 240 300
Coherence change p ¼ þ1 0 0 60 300 120 240 180 180 240 120 300 60
0 300 240 180 120 60
0 60 120 180 240 300
Coherence change p ¼ 3 0 0 180 180 360 0 540 180 720 0 900 180
0 300 240 180 120 60
subtracting the phase acquired by the coherence from the receiver phase (or vice versa) and representing the differences as a vector diagram. The Equivalent cycle and Receiver phase in Table 4.3 are reproduced in columns A and B of Table 4.4. Subtracting column A from column B gives the results shown in Table 4.4. Figure 4.10 shows that the net effect of the vectors is zero, indicating any signal resulting from the p ¼ –3 pathway will be canceled. As anticipated, a longer phase cycle with smaller phase increments on each step allows greater selectivity. Remembering the rules noted TABLE 4.4 Rejecting p ¼ 3 Coherence phase [A]
Receiver phase [B]
[B] – [A]
Equivalent phase [B] – [A]
0 180 0 180 0 180
0 300 240 180 120 60
0 120 240 0 120 –120
0 120 240 0 120 240
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
FIELD GRADIENTS
305
0°
240°
120°
or
or
–120°
–240°
FIGURE 4.10 Vectorial picture of phase cycling. From the results in Table 4.4, vectors are drawn corresponding to the difference in phase acquired by the coherence and the receiver phase, for the coherence transfer pathway p ¼ þ2 to p ¼ 1 (p ¼ 3). In this case, the goal is to discriminate between the coherence order change p ¼ þ1 and p ¼ 3. The sum of these vectors is zero, and any signal from the p ¼ 3 pathway is eliminated.
earlier, this phase cycle will also retain pathways where p ¼ þ1 6n. All other pathways are rejected. Using the nomenclature of Bodenhausen (26), the effect of this phase cycle is written as (6) –5 (4) (3) (2) (1) (0) 11 (þ2) (þ3) (þ4) (þ5) (þ6) 17. In fact, the six-step phase cycle used in this example is overly selective; as the notation of Bodenhausen makes clear, a five-step phase cycle with 728 increments would also have been satisfactory (although more difficult to visualize). The preceding principles can be restated as follows: 1. A phase cycle (affecting a single rf pulse or a group of pulses) that consists of N steps with ¼ 3608/N increments selects coherence transfer pathways satisfying m ¼ p nN, in which n is an integer. 2. The value of p is selected, from the N consecutive possible values p, p þ 1, p þ 2, . . . , p þ N 1, by shifting the receiver phase by rec ¼ –3608p/N synchronously with the pulse phase cycle. 4.3.2.2 Saving Time To unambiguously select a definite coherence transfer pathway in a pulse sequence, each pulse must have a specific phase cycle and each phase cycle must be executed independently. Unfortunately, strict application of this rule generates extremely long
306
CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY 90°
90°
+2 +1 0 –1 –2
FIGURE 4.11 Two-pulse segment used to generate zero-quantum coherence.
phase cycles, and some mechanisms must be found to reduce the phase cycle to an acceptable length. Rather than considering a coherence transfer step as being mediated by just one pulse, several pulses and intervening delays can be grouped together and regarded as a single unit that causes the desired coherence transfer. The phases of all the pulses in the unit are shifted simultaneously in order to reduce the number of steps in the overall phase cycle. For example, consider the two-pulse experiment shown in Fig. 4.11. The overall aim is to select the overall pathway p ¼ 0 (coherence that starts at level 0 and ends at 0). Cycling each of the pulses independently in 908 increments to select the coherence transfer pathway shown results in a phase cycle of 16 steps. Phase cycling both the pulses simultaneously through the four phases 08, 908, 1808, 2708, while holding the receiver phase constant, selects the pathway p ¼ 0 as required (this assertion can be checked by using the identity previously described, rec¼p). In this way the phase cycle has been reduced from 16 steps to 4 steps. Note that all other pathways that have p ¼ 0 will also be retained, because this approach leaves the coherence level between the pulses undefined. In many circumstances, these other pathways may be disregarded when additional considerations (discussed later) are taken into account. However, if an undesired coherence with p ¼ 0 results in observable signals when the 4-step phase cycle is used, then the full 16step phase cycle must be employed. Normally the first pulse in a sequence is applied to equilibrium magnetization and only coherences with p ¼ 1 are generated. Indeed, a phase cycling scheme will only work if the same initial state exists for successive transients, and consequently suitable delays must be inserted between each transient to enable the system to return to equilibrium. Unless a specific reason exists for distinguishing the coherence levels þ1
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
FIELD GRADIENTS
307
and –1 following the first pulse, then phase cycling of that pulse formally is unnecessary. However, the present theoretical discussion ignores relaxation and other effects that may make phase cycling of the first pulse desirable in practice (Section 4.3.2.3). If an unambiguous coherence transfer pathway has been chosen by the phase cycles of earlier pulses in the sequence, then the last pulse does not need to be phase cycled. Although application of the last pulse to the system may generate many different coherence orders and therefore many coherence pathways, only those pathways that lead to a final coherence level of p ¼ 1 (single-quantum coherence) are observable. The experiment, in essence, chooses the last coherence transfer step itself. Certain coherence transfer pathways are improbable in a given spin system. The maximum coherence available in a system is restricted by the number of nuclear spins in that system. For spin-1/2 nuclei, at least N coupled spins are required to produce N-quantum coherence. In principle, phase cycles that discriminate against coherence orders higher than N are unnecessary. In practice, generating large magnitudes of high-order coherences is difficult, even if theoretically possible, and coherence transfer pathways containing these coherence levels can be ignored. For example, coherence orders greater than four or five in 1H spin systems may not require consideration and the resulting phase cycles can be correspondingly shorter. An alternative approach to designing phase cycling procedures, called cogwheel phase cycling, offers shorter phase cycles in some applications (27, 28). 4.3.2.3 Artifact Suppression Before continuing, three simple phase cycling procedures that are employed to reduce instrumental artifacts are discussed: CYCLOPS, EXORCYCLE, and axial peak suppression. Quadrature detection is obtained during acquisition of the NMR signal by two phase-sensitive detection channels (Section 3.2.2). The two channels, in principle, should be identical except for a relative phase shift of 908. Anomalies arise if differences exist between the two phasesensitive detector channels. If the two quadrature channels have different sensitivities or are not completely orthogonal, then the NMR spectrum contains spurious peaks called quadrature images. The images are located in symmetric positions with respect to the center of the spectrum as genuine peaks (i.e., if a resonance has an offset of , then the quadrature image appears at ). Also, if the electrical dc (nonsinusoidal) baseline offset between the two channels differs, then a spike appears in the middle of the spectrum. This artifact most commonly is referred to as a quadrature glitch.
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
In order to remove these artifacts, a simple phase cycling routine known as CYCLOPS is used (29). For the case of a simple pulse–acquire experiment, CYCLOPS consists of cycling the pulse and receiver through the phases 08, 908, 1808, 2708 synchronously. Any gain difference between the two channels is compensated for by the 908 phase incrementation, while baseline offset errors are eliminated by the 1808 phase inversions. As discussed in Section 4.3.2.2, dc baseline offset differences between receiver channels only can be suppressed if the digitizer phase is inverted. For longer and more complicated pulse sequences, a phase cycle normally is employed that closely mimics the action of CYCLOPS. If this is not the case, CYCLOPS is implemented by adding the phase incrementations 08, 908, 1808, 2708 to all pulses in the sequence along with the receiver. The one drawback to this procedure is that the length of the phase cycle, and therefore the minimal experimental time, is increased by a factor of four. If phase cycling limitations preclude full CYCLOPS phase cycling in a particular pulse sequence, a two-step phase cycle consisting of 08 and 908 phase shifts may be satisfactory in reducing quadrature image artifacts (but not quadrature glitches). As an alternative to CYCLOPS, spectra that are free of quadrature artifacts can be obtained using digital filter technology implemented on modern NMR spectrometers (Section 3.2.3). To illustrate the CYCLOPS technique more fully, an imbalanced detection operator is defined as Fþ ð"Þ ¼ Ix þ ið1 þ "ÞIy ¼ I þ þ "=2ðI þ I Þ ¼ ð1 þ "=2ÞI þ "=2I , ½4:38 in which " 6¼ 0 is the imbalance term. The detected signal is given as usual by the trace of the density operator with the detection operator: pmax X þ þ s ðtÞ ¼ Tr ðtÞF ð"Þ ¼ Tr p ðtÞFþ ð"Þ , ½4:39 pmax þ ¼ ð1 þ "=2Þ Tr 1 ðtÞI ð"=2Þ Tr þ1 ðtÞI : If " 6¼ 0, any components of the density operator that are proportional to I þ, which has p ¼ þ1, will have a nonzero trace with the I operator and generate artifacts in the NMR spectra. Referring back to the concepts of coherence transfer pathways, CYCLOPS phase cycling is equivalent to selecting p ¼ 1 for the entire sequence. Thus, assuming that the initial density operator is given by the thermal equilibrium operator, sþ ðtÞ ¼ Tr 1 ðtÞFþ ¼ ð1 þ "=2Þ Tr 1 ðtÞI þ , ½4:40 and the artifacts in the spectrum are suppressed.
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
FIELD GRADIENTS
309
180°
+2 +1 0 –1 –2
FIGURE 4.12 EXORCYCLE. Coherence transfer pathway for a spin echo sequence. The 1808 pulse serves to change the sign of the coherence order.
The spin echo sequence, –1808–, is a vital component of a large number of NMR experiments. The coherence level diagram for this pulse sequence is shown in Fig. 4.12. Spin echo sequences are sensitive to common imperfections in 1808 pulses, such as miscalibrated pulse lengths and off-resonance effects, that can generate spurious responses. The EXORCYCLE phase cycling scheme is designed to compensate for imperfect 1808 pulses (30). An ideal 1808 refocusing pulse simply has the effect of changing the sign of the coherence level. For example, if the initial density operator has coherence level p ¼ þ1, then following the 1808 pulse the coherence level would be p ¼ 1. The desired coherence transfer pathway has p ¼ 2, so the appropriate phase cycle for the 1808 pulse would be 08, 908, 1808, 2708, while the receiver is cycled 08, 1808, 08, 1808. This is the EXORCYCLE phase cycle. This phase cycle also selects the mirror image pathway p ¼ þ2 so that the EXORCYCLE procedure can be employed if the spin echo segment is part of a more complicated sequence (Section 4.3.4). The undesired pathway of p ¼ 0, which corresponds to unrefocused magnetization, and p ¼ 1, which corresponds to a coherence transfer process, are both suppressed by the EXORCYCLE procedure. If suppression of p ¼ 0 is not important in a particular application, then a two-step phase cycle can be utilized, in which the 1808 pulse is cycled by 08 and 1808 and the receiver phase is not altered. Axial peaks occur in multidimensional NMR experiments because magnetization relaxes toward equilibrium during free-precession evolution periods, such as the t1 interval. This magnetization is not frequency labeled during the t1 period, and is not sensitive to the phase cycling of
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
pulses occurring earlier in the sequence. If the relaxed longitudinal magnetization is converted to observable magnetization prior to the acquisition period, then spurious signals will be generated along the line F1 ¼ 0 in the NMR spectrum (see Section 4.3.4.3). Axial peaks are eliminated by phase cycling a pulse or pulses prior to the t1 period and the receiver by phase angles of 08 and 1808. The most common procedure for obtaining axial peak suppression is by phase cycling the initial pulse in the pulse sequence in conjunction with the receiver. This phase cycle selects p ¼ 1, corresponding to creation of transverse magnetization by the first pulse. Many multidimensional NMR experiments have complex phase cycles that function incidentally to suppress axial peaks. 4.3.2.4 Limitations of Phase Cycling A phase cycle works by requiring signals arising from the desired coherence transfer pathway to add constructively on successive transients, whereas signals from unwanted pathways cancel. Evidently, phase cycling is simply a difference method and consequently will only work if experimental conditions remain constant from transient to transient. Unfortunately, as a practical matter, slight variations occur between transients. For example, amplitude or phase changes in the pulses or field frequency variations in the lock circuitry can contribute to variability from transient to transient and reduce the effectiveness of phase cycling. One of the most common sources of instability is temperature fluctuations that cause resonances (including the lock resonance) to shift slightly. The magnitude of the signals derived from the desired coherence transfer pathway compared to the magnitude of signals from unwanted pathways is an important determinant in the success of phase cycling. If signals from unwanted pathways are expected to be large, errors in the difference procedure may produce artifacts of intensity comparable to the intensity of desired signals. Although the deleterious effects of random fluctuations on coherence selection would be expected to cancel after extended signal averaging, instrumental instabilities are frequently periodic, and even for random effects convergence is generally slow. From a practical point of view, the order in which the individual steps of the phase cycle are employed may result in better or worse suppression of undesirable signals. Unfortunately, the particular order that the steps in a phase cycle should be applied can vary from one spectrometer to another and must be determined empirically. Phase cycles also assume that at the beginning of each transient the system is in thermal equilibrium and only longitudinal magnetization exists. Leaving a long recycle time between successive transients is the optimum way to ensure this condition, but this approach can cause
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
311
FIELD GRADIENTS
lengthy experiments, and recycle times on the order of 1/R1 to 1.5/R1 are commonly employed. In some instances, the phase cycle is designed to suppress artifacts that arise from rapid repetition of a pulse sequence.
4.3.3 PULSED FIELD GRADIENTS An alternative method for coherence transfer pathway selection has been developed that makes use of pulsed field gradients and avoids many of the problems associated with phase cycling procedures (31–41). Dephasing of transverse magnetization by a field gradient pulse is described by [3.131] in Section 3.5.3. For application in coherence selection, this result is generalized to include coherences of arbitrary order. The critical principle, easily obtained by an approach similar to that employed in deriving [3.131], is based on [3.132] and is stated as follows: Coherence dephases in an inhomogeneous magnetic field at a rate proportional to the coherence order and the magnetogyric ratios of the affected nuclei.
4.3.3.1 Selection of a Coherence Transfer Pathway A particular coherence transfer pathway is selected by using field gradient pulses to generate gradient echoes for specific coherences, while leaving unwanted coherences randomized. Consider the case in which field gradient pulses are applied on either side of a mixing period that mediates coherence transfer between coherence levels pi and pf. The first gradient pulse, applied with a shape factor s1, a strength Bg1, and a duration t1, induces a spatially dependent phase of i, and the second gradient pulse, applied with a shape factor s2, a strength Bg2, and a duration t2, induces a phase of f, where i ¼ s1 pi i Bg1 t1 , f ¼ s2 pf f Bg2 t2 ,
½4:41
in which i and f are the magnetogyric ratios of the nuclei comprising the two coherences. Following the second gradient pulse, the net phase accrued by the final coherence is i þ f. Selection of a particular coherence transfer pathway p1 ! p2 occurs by ensuring that the overall phase change is zero; therefore, the durations and amplitudes of the two gradient pulses must be adjusted such that 1 ¼ 2. The second gradient can be thought to ‘‘unwind’’ the effects of the first gradient
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
t
2t
field gradients +2 +1 0 –1 –2
FIGURE 4.13 Selection of the coherence transfer pathway p ¼ þ2 to p ¼ –1 by the use of pulsed field gradients. The double-quantum coherence p ¼ þ2 accrues a phase proportional to þ2t, in which t is the duration of the first gradient pulse. The following rf pulse (solid bar) transfers the coherence to p ¼ –1. To rephase only the final coherence that originated at the p ¼ þ2 coherence level, application of the second gradient, for duration 2t, serves to ‘‘unwind’’ exactly the required coherence.
to form an echo. Coherence orders for which i þ f 6¼ 0 remain dephased and do not contribute to the resulting signal. An illustrative example of coherence selection by gradient pulses is shown in Fig. 4.13. In this example a gradient is applied for a time t prior to the pulse that causes the coherence transfer. The coherence p ¼ þ2 dephases by an amount that is proportional to þ2t. The pulse transfers the coherence to p ¼ 1, then a gradient, in the same sense, is reapplied, but this time for a time 2t. The coherence at this level dephases by an amount proportional to 2t, which is in an opposite sense to that induced by the first gradient pulse. Consequently, after a time 2t, the required coherence is fully rephased. By using strong field gradients, all other coherences involved in other pathways are dephased and coherence selection is achieved. The real advantage of gradient pulses, compared with phase cycling, is that signals arising from unwanted pathways are removed by the gradients in each individual transient rather than relying on subtraction processes after digitization of the signal. Consequently, artifacts from instrumental instabilities may be significantly smaller than in
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
FIELD GRADIENTS
313
experiments using phase cycling for coherence selection. In addition, for some experiments, the number of transients that must be accumulated to achieve a particular signal-to-noise ratio is smaller than the number of transients required to select a particular coherence transfer pathway by phase cycling procedures; the experiment can be executed in less time if gradient pulses are used for coherence selection. This attribute becomes important when recording higher dimensional experiments that require large amounts of spectrometer time. 4.3.3.2 Artifact Suppression Artifact suppression using CYCLOPS and EXORCYCLE phase cycling was discussed in Section 4.3.2.3. Implementation of these schemes requires a minimum number of phase cycle steps, and employing more than one of these schemes can increase the length of a phase cycle enormously (i.e., independent EXORCYCLE phase cycling of n 1808 pulses requires 4n steps). To reduce such prolonged phase cycles, very simple combinations of gradients are used for artifact suppression. In practice, three- and four-dimensional experiments, with correspondingly large numbers of pulses, coherence transfer steps, and refocusing periods, exhibit the most artifacts and are subject to the most severe restrictions on the overall length of the phase cycle. The performance of a number of common components of heteronuclear multidimensional NMR experiments is augmented by the introduction of appropriate pulsed field gradients (42). The 1808 refocusing pulse in a spin echo sequence is notoriously prone to pulse artifacts that historically would be removed by the EXORCYCLE phase cycle. The gradient-enhanced homonuclear spin echo sequence is þz-gradient
180 ðIÞ
þz-gradient:
½4:42
A transverse operator — for example, either Ix or Iy — is dephased by the first gradient, the coherence order is inverted by the 1808(I) pulse, and then the operator is rephased by a gradient of the same sign and the same strength. The gradient pulses eliminate the effect of pulse imperfections that lead to transfer between transverse and longitudinal magnetization and, additionally, any transverse operator of a different spin not affected by the 1808 pulse is effectively removed. This is all accomplished in a single transient. In another common application, a 1808(I) heteronuclear decoupling pulse is used to invert the longitudinal I spin component of antiphase
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
heteronuclear coherence, 2IzSx or 2IzSy, from þz to z (or vice versa). This serves to decouple the two spins during the time period 2: þz-gradient
180 ðIÞ
z-gradient:
½4:43
The z-component inverted by the 1808 pulse is unaffected by the application of either gradient. The þz and –z gradients refocus the transverse S magnetization. Conversion of z-magnetization into transverse magnetization by an imperfect 1808(I) pulse is eliminated because such transverse magnetization will be irreversibly dephased by the second gradient. Again, artifact suppression is accomplished in one transient. In INEPT (Section 2.7.7.2) and other coherence transfer sequences, evolution under the scalar coupling Hamiltonian converts in-phase magnetization into antiphase coherence — for example, Iy ! Iy cosðJIS tÞ 2Ix Sz sinðJIS tÞ. Longitudinal two-spin order, 2IzSz, is created from the antiphase coherence, 2IxSz, by application of a 908(I) pulse with y-phase. Two-spin order has a coherence order of zero and is unperturbed by gradient pulses. Accordingly, the residual Iy magnetization, resulting if t 6¼ 1/(2JIS), for example, is selectively dephased by applying a field gradient pulse after the creation of two-spin order. However, other operators with coherence order zero, such as longitudinal magnetization, also are unperturbed by gradient pulses. In the case of zero-quantum (ZQ) coherence, the efficacy of dephasing by a gradient pulse depends on the difference in resonance frequencies for the spins comprising the ZQ coherence. For example, dephasing of a heteronuclear ZQ coherence depends on ( I S)B0, which is on the order of megahertz, and normally is efficient. In contrast, dephasing of homonuclear ZQ coherence depends on I S, which is on the order of kilohertz (or even zero for identical spins), and is highly inefficient. Special approaches for dephasing ZQ coherences by combinations of adiabatic pulses and gradient pulses have been described (43). 4.3.3.3 Limitations of Pulsed Field Gradients The main limitation of pulsed field gradients for coherence selection is evident from [4.41]: if a coherence pathway pi ! pf is selected by gradient techniques, then the corresponding pathway –pi ! pf cannot be selected simultaneously. As discussed in Section 4.3.4, frequency discrimination in indirect evolution periods requires that the signals be recorded for both the pi ! pf and –pi ! pf pathways (in which pi is the coherence order during t1 and pf ¼ –1 for observable magnetization during t2). In most pulsed
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
FIELD GRADIENTS
315
field gradient experiments, signals from the two pathways must be acquired sequentially (i.e., in two separate experiments), with the result that the sensitivity of the pulsed field gradient experiment is reduced by a factor of 21/2 compared to the corresponding phase-cycled experiment (39, 44, 45). A second limitation is that the use of pulsed field gradients usually requires lengthening of the pulse sequence. Even with actively shielded gradients, delays on the order of 0.1–1.0 ms may be required to permit the spectrometer system to stabilize following a gradient pulse. Additional spin echo sequences also may be necessary to refocus the effects of chemical shift evolution arising from the unperturbed Larmor frequency during a gradient pulse [3.128]. Inevitable relaxation during the inserted delays reduces sensitivity.
4.3.4 FREQUENCY DISCRIMINATION As noted in Section 4.3.3, coherence order is a signed quantity. The sign indicates the sense of precession of the coherence relative to a reference frame rotating at the transmitter frequency. Differentiating between evolution frequencies higher or lower than the transmitter frequency is called frequency discrimination or quadrature detection (Section 3.2.2). In high-resolution multidimensional NMR spectroscopy, spectra are desired in which frequency discrimination is obtained and optimal lineshapes are retained in all dimensions. Methods that have been designed to achieve frequency discrimination during indirect evolution periods are outlined in the following discussion. The two basic techniques for frequency discrimination during evolution periods are termed the hypercomplex (or States) method and the time-proportional phase-incrementation (TPPI) method. The analysis here is based on the work of Bodenhausen (26, 46) and on the seminal paper by Keeler and Neuhaus (47). The following discussion considers two-dimensional spectroscopy; extension to higher dimensions is straightforward. As discussed in Section 3.2.2, frequency discrimination during the acquisition period is obtained by quadrature detection: the sine- and cosine-modulated components of the evolving magnetization are recorded independently by orthogonal detectors (the exact method depends upon the construction of the spectrometer) and treated as a complex signal during subsequent processing. Conventional quadrature detection cannot be used to determine the relative sense of precession of magnetization in the t1 dimension of a two-dimensional experiment, because the signal during t1 is never actually recorded. Nonetheless, the fundamental result that both cosine- and sine-modulated components of
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the appropriate coherences must be recorded is equally valid for the indirect evolution period as for the acquisition period. To continue with the analysis, some important and useful relationships must be developed. Adopting the conventions and notations of Keeler and Neuhaus, the cosine-modulated time-domain (t1) function Sc(t1, t2) is Sc ðt1 , t2 Þ ¼ cosð 1 t1 Þ expði2 t2 Þ 1 ¼ ½expði1 t1 Þ þ expði1 t1 Þ expði2 t2 Þ: 2
½4:44
Here 1 and 2 are the chemical shifts in the first and second dimensions, respectively. The sine-modulated time domain (t1) function, Ss(t1, t2), is Ss ðt1 , t2 Þ ¼ sinð1 t1 Þ expði2 t2 Þ i ¼ ½expði1 t1 Þ expði1 t1 Þ expði2 t1 Þ: 2
½4:45
In both Sc(t1, t2) and Ss(t1, t2), the evolution during the t1 period modulates the amplitude of the signal recorded during t2. Data in which evolution during the t1 period modulates the phase of the signal recorded during t2 are referred to as P-type and N-type signals, respectively: SP ðt1 , t2 Þ ¼ exp½ið2 t2 þ 1 t1 Þ,
½4:46
SN ðt1 , t2 Þ ¼ exp½ið2 t2 1 t1 Þ:
½4:47
In P-type modulation, the sense of the frequency modulation is the same in t1 and t2, whereas in N-type modulation, the sense of the frequency modulation is opposite in t1 and t2 (48, 49). The following relationships are obtained trivially: Sc ðt1 , t2 Þ ¼ ½SP ðt1 , t2 Þ þ SN ðt1 , t2 Þ=2,
½4:48
Ss ðt1 , t2 Þ ¼ i ½SP ðt1 , t2 Þ SN ðt1 , t2 Þ=2:
½4:49
For amplitude-modulated signals, the precession of magnetization during t1 is described by a superposition of two complex signals with opposite frequency, 1 and 1. These signals result from evolution of the shift operators I and I þ, respectively, which in turn correspond to the coherence levels p ¼ 1 and p ¼ þ1 (more generally, for multiplequantum coherences of order pi during t1, p ¼ pi and p ¼ pi). Amplitudemodulated data sets require that both positive and negative coherence levels are selected during t1. In contrast, for phase-modulated signals, precession of magnetization during t1 is described by a complex signal,
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
317
FIELD GRADIENTS
with frequency given by either 1 (for P-type signals) or 1 (for N-type signals). These signals result from selection of only one of the coherence levels p ¼ 1 or p ¼ þ1 during t1 (more generally, for multiple quantum coherences of order pi during t1, p ¼ pi or p ¼ pi). The Fourier transform of the function exp(it – R2t) is given by (Section 3.3.1) Ffexpðit R2 tÞg ¼ A þ iD,
½4:50
in which A¼
R2 ð! Þ2 þ R22
½4:51
is an absorptive Lorentzian line and D¼
! ð! Þ2 þ R22
½4:52
is a dispersive Lorentzian line, and relaxation of the form exp(–R2t) has been assumed. The shorthand notation A2, D2 and A1, D1 will be used to represent the absorption and dispersion parts of the signal in the F2 and F1 dimensions, respectively. Depending on whether the peaks are located þ at þ1 or 1 in the F1 dimension, the signals will be noted as Aþ 1 , D1 or A1 , D1 , respectively. Fourier transformation of [4.44] with respect to t2 yields Sc ðt1 , F2 Þ ¼ ½expði1 t1 Þ þ expði1 t1 Þ½A2 þ iD2 =2:
½4:53
Performing a real (cosine) Fourier transform of [4.53] with respect to t1 yields the two-dimensional spectrum: Sc ðF1 , F2 Þ ¼ ½Aþ 1 A2 þ A1 A2 =2:
½4:54
Alternatively, if the imaginary part of [4.53] is discarded to give S0c , S0c ðt1 , F2 Þ ¼ ½expði1 , t1 Þ þ expði1 t1 ÞA2 =2,
½4:55
and a complex Fourier transform of [4.55] is performed with respect to t1, then the real part of the resulting spectrum is exactly as represented by [4.54]. Equation [4.54] represents two signals, one at þ1, the other at 1, that are absorptive in both dimensions, as shown in Fig. 4.14a. Fourier transformation of [4.45] with respect to t2 yields Ss ðt1 , F2 Þ ¼ i½expði1 t1 Þ expði1 t1 Þ½A2 þ iD2 =2:
½4:56
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
319
FIELD GRADIENTS
Performing a real Fourier transform of [4.56] with respect to t1 yields the two-dimensional spectrum: Ss ðF1 , F2 Þ ¼ i½Aþ 1 A2 A1 A2 =2:
½4:57
Alternatively, if the imaginary part of [4.56] is discarded to give S0s , S0s ðt1 , F2 Þ ¼ i½expði1 t1 Þ expði1 t1 ÞA2 =2,
½4:58
and a complex Fourier transform of [4.58] is performed with respect to t1, then the real part of the resulting spectrum is exactly as represented by [4.57]. Equation [4.57] represents two signals, one at þ1, the other at 1, that are absorptive in both dimensions; however, one peak is positive and the other is negative (Fig. 4.14b). Combining the results of [4.54] and [4.57] as a complex pair generates a frequency-discriminated spectrum of the form SðF1 , F2 Þ ¼ Sc ðF1 , F2 Þ þ iSs ðF1 , F2 Þ ¼ Aþ 1 A2 ,
½4:59
which provides frequency discrimination with retention of a pure double-absorption lineshape. The process is shown schematically in Fig. 4.14. In contrast, two-dimensional Fourier transformations of [4.46] and [4.47] with respect to t1 and t2 yield SP ðF1 , F2 Þ ¼ ½A 1 þ iD1 ½A2 þ iD2 ¼ ½A 1 A2 D1 D2 þ i ½A1 D2 þ D1 A2 ,
½4:60
þ SN ðF1 , F2 Þ ¼ ½Aþ 1 iD1 ½A2 þ iD2 þ þ þ ¼ ½Aþ 1 A2 þ D1 D2 þ i ½A1 D2 D1 A2 :
½4:61
The real parts of these spectra represent frequency-discriminated spectra, as desired; however, the lineshape is a superposition of doubly absorptive and doubly dispersive signals. This lineshape is called phase twisted and is extremely undesirable in high-resolution NMR spectroscopy because the dispersive tails in the lineshape degrade the resolution in the spectrum. The P and N signals are phase twisted in the opposite sense. Absorption and phase-twisted lineshapes are compared in Fig. 4.15. As will be seen in Section 4.3.4.2, [4.48] and [4.49] can be used to generate amplitude-modulated data from separately recorded P-type and N-type signals, and the resulting amplitude-modulated data can be used to generate a phase-sensitive spectrum, as previously described.
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
a
b
FIGURE 4.15 Comparison of (a) absorptive and (b) phase-twisted lineshapes.
4.3.4.1 Frequency Discrimination by Phase Cycling Both positive and negative coherence levels are selected during t1 in a natural fashion in phase-cycled experiments by using the periodicity of phase-cycled coherence filters to select both desired pathways simultaneously. The practical problem is then to separately record the two data sets Sc(t1, t2) and Ss(t1, t2). The key is to recognize that, if a given pulse sequence selects a coherence of order p during t1 and results in a cosine-modulated signal, then the sine-modulated signal is recorded in a second experiment by shifting the phase of an appropriate pulse sequence element (usually preceding the evolution period) by /(2|p|) [4.35]. In the hypercomplex (50) method, cosine- and sine-modulated data sets are recorded using a sampling interval in t1 of 1/SW1, where SW1
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
FIELD GRADIENTS
321
is the required spectral width in the F1 dimension, and are processed exactly as previously described. The combination of the two separate data sets can be performed before the second complex Fourier transform (with respect to t1), because the Fourier transform is linear. An alternative method was designed by Marion and Wu¨thrich (51). This procedure achieves identical results but employs real Fourier transformations. The idea finds its roots in Redfield’s method for quadrature-detected spectra using a single analog-to-digital converter (ADC) (52). Some NMR instruments were originally designed with only one ADC, rather than two, as this was deemed an expensive component. One-dimensional quadrature detection on such spectrometers requires recording data points every 1/(2SW), where SW is the spectral width, twice the normal rate of data acquisition. In addition, the phase of the receiver is incremented by 908 after each data point is recorded. For this reason, the method is known as time-proportional phase incrementation (TPPI). The spectrum is subsequently obtained by application of a real Fourier transform. Overall, the effect of the TPPI procedure is to add a frequency of SW/2 to each data point in the transformed spectrum, thus achieving frequency discrimination. Those peaks in the spectrum below the transmitter frequency (at zero), between SW/2 and zero, are shifted to between zero and SW/2, while those peaks between zero and þSW/2 are shifted to between þSW/2 and SW. Therefore, all resonances in the spectrum appear with the same sign of precession. The same method can be employed in two-dimensional NMR spectroscopy. The incrementable period, t1, is incremented in steps of 1/(2SW1). Between each successive t1 increment, the phase of the coherence during t1 is shifted by 908 (by shifting the phases of the pulses prior to t1 appropriately). Consider the effect of this phase incrementation for the odd and even t1 increments for a sampling interval of t1 ¼ 1/(2SW1). For example, assume the odd-numbered data points sample the cosine-modulated signal cos(1t1) and the even-numbered data points sample the sine-modulated signal sin(1t1), where 1 is the frequency of the coherence of interest evolving during the t1 period. Consequently, the sampled data set is described by the series over the integer index m 1: S½ðm 1Þt1 ¼ 1, sinð1 t1 Þ, cosð1 2t1 Þ, sinð1 3t1 Þ, cosð1 4t1 Þ, . . . :
½4:62
The alternating signs of the cosine and sine terms result from the phase incrementation procedure. This series of data samples is represented in a
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CHAPTER 4 MULTIDIMENSIONAL NMR SPECTROSCOPY
compact form as S½ðm 1Þt1 ¼ cos½ð1 2=4t1 Þðm 1Þt1 ¼ cos½ð1 TPPI Þðm 1Þt1 ,
½4:63
in which TPPI/2 ¼ SW1/2. Real Fourier transformation of this data set produces a spectrum with a signal at the apparent frequency 1 – TPPI. Thus, the TPPI procedure eliminates the need for explicit discrimination of the signs of the coherence frequencies within the unaliased bandwidth by shifting the effective reference frequency to one edge of the spectrum, resulting in all of the frequencies having the same sign. This procedure is analogous to that proposed by States and co-workers; in particular, the TPPI method requires exactly the same total number of transients and total acquisition time as does the States (or hypercomplex) method. In order to avoid baseline distortion, use of the TPPI procedure requires that the coherences be represented in the form cos(1t1 þ ), where is an integral multiple of /2 (in other words, the resulting spectrum needs either 08 or 908 zero-order phase correction (Section 4.3.4.3) (53). In principle, a TPPI procedure can be superimposed on the hypercomplex quadrature scheme to achieve an apparent frequency shift in an indirect dimension of a multidimensional experiment. In most cases, physically shifting the rf transmitter frequency (assuming the shift can be done phase coherently if necessary) is preferable in order to minimize resonance offset effects. Salient features of the States (hypercomplex), TPPI, and hybrid TPPI–States (54) protocols are summarized in Table 4.5. Although the discussions and derivations here have been applied solely to twodimensional NMR spectroscopy, the methods are equally applicable to higher dimensional experiments. 4.3.4.2 Frequency Discrimination by Pulsed Field Gradients Coherence selection by pulsed field gradients naturally results in Ptype or N-type modulation of the observed signal. Either P-type or Ntype data are selected, depending on the relative sense of the initial dephasing gradient pulse and the final, refocusing gradient. To obtain frequency-discriminated phase-sensitive spectra by using the gradient approach, both the P-type and the N-type data are recorded separately (39, 44, 45). Cosine- and sine-modulated data are obtained by combining the P- and N-type data using [4.48] and [4.49]. The resulting amplitude-modulated data are processed as complex data by using the States method of frequency discrimination.
4.3 COHERENCE SELECTION, PHASE CYCLING,
AND
FIELD GRADIENTS
323
TABLE 4.5 Quadrature Detection Methodsa Experiment
Increment
Pulse phase
Receiver phase
(4k þ 1) (4k þ 2) (4k þ 3) (4k þ 4)
t1(0) þ (4k) t1(0) þ (4k þ 1) t1(0) þ (4k þ 2) t1(0) þ (4k þ 3)
x y –x –y
x x x x
(4k þ 1) (4k þ 2) (4k þ 3) (4k þ 4)
States t1(0) þ (4k)2 t1(0) þ (4k)2 t1(0) þ (4k þ 1)2 t1(0) þ (4k þ 1)2
x y x y
x x x x
(4k þ 1) (4k þ 2) (4k þ 3) (4k þ 4)
TPPI–States t1(0) þ (4k)2 t1(0) þ (4k)2 t1(0) þ (4k þ 1)2 t1(0) þ (4k þ 1)2
x y –x –y
x x –x –x
TPPI
a
The index k ¼ 0, 1, . . . , N/4 – 1, in which N is the number of experiments acquired in the t1 dimension, and ¼ 1/(2SW1). The initial sampling delay is t1(0) and is usually set to 0 or . The t1 interferogram consists of N real points for the TPPI method and N/2 complex points for the States and TPPI–States methods. The t1 increment is twice as large for the States and TPPI–States methods as for the TPPI method, but SW1 and t1max are identical for all methods.
Although this procedure produces frequency discrimination in the indirect dimension of the spectrum while keeping absorption lineshapes, the signal-to-noise ratio is reduced by a factor of 21/2 compared to the phase-cycled approaches to frequency discrimination. However, in many experiments, this reduction is avoided by the use of so-called PEP (preservation of equivalent pathways) experiments described in Chapter 7 (Section 7.1.3.3). 4.3.4.3 Aliasing, Folding, and Phasing in Multidimensional NMR Spectroscopy The location of axial peaks in a multidimensional NMR spectrum depends upon the manner in which frequency discrimination is performed in the F1 dimension. If the hypercomplex (States) method is used, then the axial peaks occur at F1 ¼ 0 and result in a ridge of axial peaks across the center of the spectrum, parallel to the F2 axis. If the TPPI method is used, then the axial peaks occur at the edge of the spectrum for the following reason. The TPPI phase cycle adds a
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frequency TPPI to the resonance frequencies in F1; however, the axial peaks are unaffected by the phase cycle and consequently appear in the spectrum at the apparent frequency –TPPI. Generally, even if axial peak suppression phase cycling is employed, placement of axial peaks at the edge of the spectrum minimizes artifacts. The hybrid TPPI–States protocol is identical to the conventional States method except that the axial peaks are shifted to the edge of the spectrum. As described in Section 3.2.1, resonance frequencies outside the spectral width appear artifactually within the spectral width. For complex data (States or TPPI–States frequency discrimination), resonance signals are aliased: signals upfield (downfield) of the edge of the spectral region appear in the downfield (upfield) spectral region. The apparent resonance frequency, a, and the true resonance frequency, 0, are related by a ¼ 0 mSW,
½4:64
in which m is an integer equal to the number of times the signal has been aliased. For real data (TPPI frequency discrimination), resonance signals are folded: signals upfield (downfield) of the edge of the spectral region reflect back across the upfield (downfield) edge of the spectral region. The apparent resonance frequency, a, and the true resonance frequency, 0, are related by a ¼ ð1Þm ð0 mSWÞ:
½4:65
Aliasing and folding (55) are illustrated in Fig. 4.16. Aliasing is useful particularly for minimizing the spectral width in heteronuclear NMR experiments and is discussed further in Section 7.1.2.5. Distortions in the baseline caused by phase correction (Section 3.3.2.3) are particularly serious problems in multidimensional NMR spectroscopy. Fortunately, the receiver reference phase can be easily adjusted on modern NMR spectrometers in order to set the initial signal phase (usually equal to zero). A number of experimental techniques have been developed to ensure that the initial sampling delay in the acquisition dimension t0 ¼ 0 or t0 ¼ 1/(2SW). For example, the Hahn echo sequence can be used to adjust the initial sampling delay (Section 3.6.4.2). Adjustment of the initial sampling delay for indirectly detected evolution periods in multidimensional NMR experiments must account for phase evolution during the preparation and mixing periods. The accrued phase depends upon chemical shift evolution during the evolution period and phase evolution during off-resonance rf pulses within or flanking the evolution period.
4.3 COHERENCE SELECTION, PHASE CYCLING, AND FIELD GRADIENTS
325
FIGURE 4.16 Folding and aliasing in the F) dimension. (a) Schematic spectrum with a full spectral width. (b) The F) spectral width is halved and frequency discrimination is performed using real (TPPJ) acquisition. Resonances outside the spectral width are folded into the spectrum. (c) The F, spectral width is halved and frequency discrimination is performed using hypercomplex (States or TPPJ-States) acquisition. Resonances outside the spectral width are aliased into the spectrum.
Complex (States, States-TPPI, echo/antiecho) frequency discrimination is considered first. The pulse sequence element 90 0-t,-90° is encountered frequently in homonuclear multidimensional pulse sequences. Utilizing the expression for phase evolution during rf pulses given by [3.70], to ensure an initial sampling delay of to= 1/(2SW)), the initial value of t) must be set to t, (0) = 1/(2SW,) - 4L90/n,
[4.66]
in which L90 is the length of a 90° pulse. If an initial sampling delay of zero is desired, then the pulse sequence element 90 0-t,-.6.-1800-.6.-90°, in which .6. is a fixed delay, can be used. The 180° pulse refocuses the effects of phase evolution during the flanking 90° pulses; thus [4.67]
can be chosen as desired. In heteronuclear correlation NMR experiments, the pulse sequence element 90 0(S)-tl/2-1800(J)-t,/2-900(S) commonly is used to decouple I and S spins during the evolution period. In this case, the initial value of t, must be set to [4.68]
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to ensure t0 ¼ 1/(2SW1). Processing details for t0 ¼ 0 or t0 ¼ 1/(2SW1) are discussed in Section 3.3.2.3. For real (TPPI) frequency discrimination, an additional consideration arises. The real signal can be written as the sum of two complex exponential signals. Thus, assuming an initial sampling delay t0 ¼ 0, sðjtÞ ¼ I0 cos½jt þ 0 exp½l0 jt ¼ ð1=2ÞI0 fexp½ði l0 Þjt þ i0 þ exp½ði l0 Þjt i0 g: ½4:69 The transformed spectrum, obtained by analogy to [3.53], is
I0 1 1 I0 þ Sð!k Þ þ cos 0 : 2t ði i!k l0 Þ ði i!k l0 Þ 2
½4:70
The baseline offset term vanishes if 0 ¼ /2. In this case, [4.69] becomes sðjtÞ ¼ I0 sin½jt exp½l0 jt
½4:71
and the signal is said to be sine modulated. If the indirect evolution period has the form 90 –t1 –90x , then sine-modulated data with t0 ¼ 0 is obtained by the following procedure. The first FID is acquired with an initial sampling point set according to [4.66] and ¼ x. The interferogram is sampled using the TPPI method applied to the phase . After acquisition and processing in the acquisition dimension, a zero value is preappended to the beginning of the interferogram (typically by right shifting the data vector) prior to processing the indirect dimension. A 908 zero-order phase correction is applied after Fourier transformation.
4.4 Resolution and Sensitivity The sensitivity of a multidimensional spectrum is close to that of an equivalent one-dimensional experiment recorded in the same total experiment time. Although a particular peak may appear only weakly in each F2 spectrum (recorded for each t1 value), the Fourier transform with respect to t1 concentrates all the signal into a few points in the final multidimensional spectrum. In effect, the signal-to-noise ratio of a peak in the multidimensional spectrum is a function of the time-average signal throughout the entire multidimensional NMR experiment. Three factors lead to a reduction in sensitivity of multidimensional spectra when compared to their one-dimensional counterparts: (i)
4.5 THREE-
AND
FOUR-DIMENSIONAL NMR SPECTROSCOPY
327
relaxation during the incrementable time periods results in a progressive loss of signal as the variable delay increases, (ii) cancellation of antiphase signals due to overlap (Section 6.2.1.5), and (iii) the integrated intensity of a single peak in the one-dimensional spectrum is associated with several peaks in the multidimensional spectrum (i.e., diagonal and crosspeaks). The trade-off between sensitivity and resolution is the same as for one-dimensional spectroscopy: more t1 increments must be recorded to increase resolution (t1max); however, relaxation during the added t1 time periods causes the signal decay and loss of sensitivity. Because the signal nearly always is truncated in the incrementable time periods of multidimensional NMR experiments, suitable apodization functions must be applied to generate spectra containing the required information. Such functions and their effects on truncated data are discussed in Section 3.3.2.2.
4.5 Three- and Four-Dimensional NMR Spectroscopy Two-dimensional NMR spectroscopy has proved to be one of the most important developments in high-resolution NMR. However, for proteins with masses greater than 10 to 12 kDa, even the increased resolution of the two-dimensional spectra is insufficient, and so alternative approaches have been sought. The approach that has now become widely adopted is to increase the number of frequency dimensions possessed by the spectrum. The principles and fundamental ideas already discussed for two-dimensional NMR extend into higher dimensional experiments: the same types of magnetization transfer processes are active, the same principles concerning coherence selection, quadrature detection, resolution, and sensitivity are applicable, and product operator analyses are employed in the same way. Higher dimensional experiments are built from combinations of two-dimensional experiments and can combine the magnetization transfer capabilities of both dipolar and scalar coupling interactions in the same sequence. Three-dimensional pulse sequences are derived from a combination of two two-dimensional pulse sequences (56–58), as shown schematically in Fig. 4.17. The acquisition period of the first pulse sequence and the preparation period of the second pulse sequence are omitted in concatenating the two experiments. In three-dimensional experiments, the signal is recorded conventionally during an acquisition time, denoted t3, as a function of two evolution times, t1 and t2, which are incremented independently from one experiment to another. This procedure generates
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preparation
evolution
mixing
preparation
2D
acquisition
evolution
2D mixing
acquisition
combine
t2
t1 preparation
evolution
mixing
evolution
t3 mixing
acquisition
3D FIGURE 4.17 Schematic generation of a three-dimensional NMR experiment from the combination of two two-dimensional NMR experiments. The mixing period of one two-dimensional experiment and the preparation period of a second two-dimensional experiment are combined. The three-dimensional experiment contains three independent time periods. The FID is recorded during the acquisition time, t3, as a function of two independently incremented evolution times, t1 and t2. A mixing period follows each evolution time, causing a potential two-step magnetization transfer process.
a three-dimensional time-domain data matrix to which a threedimensional Fourier transformation is applied. The corresponding frequency dimensions are denoted F1, F2, and F3. The spectrum can be represented as a three-dimensional cube, but analysis of a threedimensional spectrum is more convenient if two-dimensional slices are taken from the cube as shown in Fig. 4.18. In this case the tiers or planes from the cube are seen as sets of two-dimensional spectra (F3, F1) separated by another interaction along the F2 dimension. Again, as in two-dimensional spectroscopy, either similar or different nuclear types can appear in different dimensions, as required, and correlations between the separate dimensions can be established via NOE effects or through scalar couplings. Those experiments in which all three dimensions contain 1H chemical shifts or couplings are referred to as homonuclear three-dimensional experiments. Those experiments in which one or more dimension is not 1H (usually 13C and/or 15N) are referred to as heteronuclear three-dimensional experiments. Analyses of the most important three-dimensional methods for biological studies are presented in Chapters 6, 7 and 9.
4.5 THREE-
AND
329
FOUR-DIMENSIONAL NMR SPECTROSCOPY
F1 F2
F2
2D
F1 F3
3D
FIGURE 4.18 The development of a three-dimensional data set from a twodimensional data set. The two-dimensional data set depicted here shows a set of resonances that, although resolved in the F1 dimension, are not clearly determined in the F2 dimension. The introduction of an additional evolution period generates a third frequency dimension perpendicular to the first two. The increased resolution afforded by virtue of a second magnetization transfer step can alleviate ambiguities in the two-dimensional spectrum.
In a similar fashion, a four-dimensional experiment consists of a combination of three two-dimensional sequences, omitting the detection periods of the first and second experiments and the preparation stages of the second and third experiments (59). As shown in Fig. 4.19, the general experiment contains three independently incrementable time periods (t1, t2, t3) and the acquisition time period, t4, and consequently the resulting data are a function of these four time periods. Four-dimensional experiments are used in those cases when there is still ambiguity arising due to degeneracy and overlap even in three-dimensional spectra. Up to this point in time, four-dimensional experiments have been exclusively
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preparation
t1
mixing(1)
t2
mixing(2)
t3
mixing(3) acquisition (t4)
FIGURE 4.19 A schematic representation of a four-dimensional NMR experiment. The four-dimensional experiment contains four independent time periods. The FID is recorded during the acquisition time, t4, as a function of three independently incremented evolution times, t1, t2, and t3. A mixing period follows each evolution period, causing a potential three-step magnetization transfer process.
F3
F2
F1 F4
FIGURE 4.20 A four-dimensional experiment is visualized as a series of threedimensional cubes. Each cube represents the three-dimensional F1, F2, F4 subspectrum for a different value of F3.
heteronuclear techniques. Detailed discussions of some useful fourdimensional experiments are presented in Chapters 7 and 9. Visualizing a four-dimensional spectrum can be difficult, although one convenient method is to imagine each two-dimensional slice of a three-dimensional spectrum expanded into an additional dimension by another type of interaction. The progression from two-dimensional spectra to four-dimensional spectra is represented schematically by Fig. 4.20.
4.5 THREE-
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331
References 1. R. Freeman, W. A. Anderson, J. Chem. Phys. 37, 2053–2073 (1962). 2. R. A. Hoffman, S. Forse´n, Prog. NMR Spectrosc. 1, 15–204 (1966). 3. D. Neuhaus, M. Williamson, ‘‘The Nuclear Overhauser Effect in Structural and Conformational Analysis,’’ pp. 1–522. VCH Publishers, New York, 1989. 4. J. H. Noggle, R. E. Shirmer, ‘‘The Nuclear Overhauser Effect: Chemical Applications,’’ pp. 1–259 Academic Press, New York, 1971. 5. R. R. Ernst, J. Chem. Phys. 45, 3845–3861 (1966). 6. W. A. Anderson, R. Freeman, J. Chem. Phys. 37, 85–103 (1962). 7. W. A. Anderson, F. A. Nelson, J. Chem. Phys. 39, 183–189 (1963). 8. J. Jeener, Ampe`re Summer School, Basko Polje, Yugoslavia, 1971. 9. W. P. Aue, E. Bartholdi, R. R. Ernst, J. Chem. Phys. 64, 2229–2246 (1976). 10. J. Jeener, in ‘‘NMR and More, in Honour of Anatole Abragam’’ (M. Goldman, M. Porneuf, eds.), pp. 1–388. Les Editions de Physique, Les Ulis, France, 1994. 11. L. Braunschweiler, R. R. Ernst, J. Magn. Reson. 53, 521–528 (1983). 12. S. R. Hartmann, E. L. Hahn, Phys. Rev. 128, 2042–2053 (1962). 13. L. Mu¨ller, R. R. Ernst, Mol. Phys. 38, 963–992 (1979). 14. R. Bazzo, I. D. Campbell, J. Magn. Reson. 76, 358–361 (1988). 15. M. Rance, J. Magn. Reson. 74, 557–564 (1987). 16. A. Bax, D. G. Davis, J. Magn. Reson. 65, 355–360 (1985). 17. J. Jeener, B. H. Meier, P. Bachmann, R. R. Ernst, J. Chem. Phys. 71, 4546–4553 (1979). 18. G. A. Morris, R. Freeman, J. Am. Chem. Soc. 101, 760–762 (1979). 19. D. P. Burum, R. R. Ernst, J. Magn. Reson. 39, 163–168 (1980). 20. D. M. Doddrell, D. T. Pegg, M. R. Bendall, J. Magn. Reson. 48, 323–327 (1982). 21. M. R. Bendall, D. T. Pegg, J. Magn. Reson. 53, 272–296 (1983). 22. K. E. Kover, G. Batta, Prog. NMR Spectrosc. 19, 223–266 (1987). 23. M. W. F. Fischer, L. Zeng, E. R. P. Zuiderweg, J. Am. Chem. Soc. 118, 12457–12458 (1996). 24. M. R. Hansen, M. Rance, A. Pardi, J. Am. Chem. Soc. 120, 11210–11211 (1998). 25. J. Keeler, in ‘‘Multinuclear Magnetic Resonance in Liquids and Solids — Chemical Applications’’ (P. Granger, R. K. Harris, eds.), pp. 103–129, Vol. 322. NATO ASI Series C. Kluwer Academic Press, Netherlands, 1990. 26. G. Bodenhausen, H. Kogler, R. R. Ernst, J. Magn. Reson. 58, 370–388 (1984). 27. M. H. Levitt, P. K. Maduh, C. E. Hughes, J. Magn. Reson. 155, 300–306 (2002). 28. C. E. Hughes, M. Carravetta, M. H. Levitt, J. Magn. Reson. 167, 259–265 (2004). 29. D. I. Hoult, Proc. Roy. Soc. Lond., Ser. A 344, 311–340 (1975). 30. G. Bodenhausen, R. Freeman, D. L. Turner, J. Magn. Reson. 26, 373–378 (1977). 31. R. E. Hurd, J. Magn. Reson. 87, 422–428 (1990). 32. R. E. Hurd, B. K. John, J. Magn. Reson. 92, 658–668 (1991). 33. R. E. Hurd, B. K. John, J. Magn. Reson. 91, 648–653 (1991). 34. A. Bax, P. G. D. Jong, A. F. Mehlkopf, J. Smidt, Chem. Phys. Lett. 69, 567–570 (1980). 35. P. Barker, R. Freeman, J. Magn. Reson. 64, 334–338 (1985). 36. G. W. Vuister, R. Boelens, R. Kaptein, R. E. Hurd, B. John, P. C. M. van Zijl, J. Am. Chem. Soc. 113, 9688–9690 (1991). 37. G. W. Vuister, R. Boelens, R. Kaptein, M. Burgering, P. C. M. van Zijl, J. Biomol. NMR. 2, 301–305 (1992). 38. B. K. John, D. Plant, P. Webb, R. E. Hurd, J. Magn. Reson 98, 200–206 (1992). 39. A. L. Davis, J. Keeler, E. D. Laue, D. Moskau, J. Magn. Reson. 98, 207–216 (1992).
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40. A. L. Davis, E. D. Laue, J. Keeler, D. Moskau, J. Lohman, J. Magn. Reson. 94, 637–644 (1991). 41. J. Keeler, R. T. Clowes, A. L. Davis, E. D. Laue, Meth. Enzymol. 239, 145–207 (1994). 42. A. Bax, S. S. Pochapsky, J. Magn. Reson. 99, 638–643 (1992). 43. M. J. Thrippleton, J. Keeler, Angew. Chem. Int. Ed. Engl. 42, 3938–3941 (2003). 44. J. R. Tolman, J. Chung, J. P. Prestegard, J. Magn. Reson. 98, 462–467 (1992). 45. J. Boyd, N. Soffe, B. K. John, D. Plant, R. E. Hurd, J. Magn. Reson. 98, 660–664 (1992). 46. G. Bodenhausen, R. Freeman, R. Niedermeyer, D. L. Turner, J. Magn. Reson. 26, 133–164 (1977). 47. J. Keeler, D. Neuhaus, J. Magn. Reson. 63, 454–472 (1985). 48. K. Nagayama, A. Kumar, K. Wu¨thrich, R. R. Ernst, J. Magn. Reson. 40, 321–334 (1980). 49. A. Bax, R. Freeman, J. Magn. Reson. 44, 542–561 (1981). 50. D. J. States, R. A. Haberkorn, D. J. Ruben, J. Magn. Reson. 48, 286–292 (1982). 51. D. Marion, K. Wu¨thrich, Biochem. Biophys. Res. Commun. 113, 967–974 (1983). 52. A. G. Redfield, S. D. Kunz, J. Magn. Reson. 19, 250–254 (1975). 53. D. Marion, A. Bax, J. Magn. Reson. 79, 352–356 (1988). 54. D. Marion, M. Ikura, R. Tschudin, A. Bax, J. Magn. Reson. 85, 393–399 (1989). 55. R. R. Ernst, G. Bodenhausen, A. Wokaun, ‘‘Principles of Nuclear Magnetic Resonance in One and Two Dimensions,’’ pp. 1–610. Clarendon Press, Oxford, 1987. 56. C. Griesinger, O. W. Sørensen, R. R. Ernst, J. Magn. Reson. 73, 574–579 (1987). 57. H. Oschkinat, C. Griesinger, P. J. Kraulis, O. W. Sørenson, R. R. Ernst, A. M. Gronenborn, G. M. Clore, Nature 332, 374–376 (1988). 58. C. Griesinger, O. W. Sørensen, R. R. Ernst, J. Magn. Reson. 84, 14–63 (1989). 59. L. E. Kay, G. M. Clore, A. Bax, A. M. Gronenborn, Science 249, 411–414 (1990).
CHAPTER
5 RELAXATION AND DYNAMIC PROCESSES
Previous chapters have utilized the density matrix and product operator formalisms to describe the evolution of the density operator under the chemical shift, scalar coupling, dipolar coupling, and rf Hamiltonians, which are responsible for the chemical shifts, multiplet structures, and coherence transfer phenomena observed by NMR spectroscopy. In principle, NMR experiments begin from the equilibrium state, in which all coherences (off-diagonal elements of the density operator) are zero and the populations of the energy levels of the system (diagonal elements of the density operator) are described by the Boltzmann distribution. Although multiple pulse and multidimensional NMR techniques permit generation of off-diagonal density matrix elements and observation of complex coherence transfer processes, eventually the equilibrium state is restored. As with similar phenomena in other areas of spectroscopy, the process by which an arbitrary density operator returns to the equilibrium operator is called nuclear magnetic, or spin, relaxation. The present chapter describes the general theoretical framework of spin relaxation. Consequences of spin relaxation processes for particular multidimensional NMR experiments are described in Chapters 6 and 7, and experimental methods for studying spin relaxation and protein dynamics are described in Chapter 8.
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As relaxation is one of the fundamental aspects of magnetic resonance, an extensive literature on theoretical and experimental aspects of relaxation has developed since the earliest days of NMR spectroscopy [see McConnell (1) and references therein]. Relaxation has important consequences for the NMR experiment: (i) relaxation rate constants for single-quantum transverse operators determine the natural linewidths of the resonances detected during the acquisition period, (ii) relaxation rate constants for operators of interest during multidimensional experiments determine the linewidths of resonances in indirectly detected dimensions, unless constant-time or very short evolution periods are utilized, (iii) relaxation rate constants for longitudinal magnetization and coherences generated by the pulse sequence determine the length of the recycle delay needed between acquisitions, and (iv) relaxation rate constants for spin operators created during coherence transfer sequences affect overall sensitivity. Conversely, unique information about the physical processes governing relaxation, including molecular motions and intramolecular distances, is available from NMR experiments. In particular, dipolar cross-relaxation gives rise to the nuclear Overhauser effect (NOE) and makes possible the determination of three-dimensional molecular structures by NMR spectroscopy. Additionally, a variety of chemical kinetic processes can be studied through effects manifested in the NMR spectrum; in many cases, such phenomena are studied while the molecular system remains in chemical equilibrium. Because the theoretical formalism describing relaxation is more complicated mathematically than is the product operator formalism emphasized in this text, the present treatment emphasizes applications of semiclassical relaxation theory. More detailed descriptions of the derivation of the relaxation equations are presented elsewhere (1–4), and numerous review articles are available (5–10).
5.1 Introduction and Survey of Theoretical Approaches Introductory theoretical treatments of optical spectroscopy emphasize the role of spontaneous and stimulated emission in relaxation from excited states back to the ground state of a molecule. The probability per unit time, W, for transition from the upper to lower energy state of an isolated magnetic dipole by spontaneous emission of a photon of energy E ¼ h!0 is given by (2, 11) W¼
0 2 h!30 , 6c3
½5:1
5.1 INTRODUCTION
AND
SURVEY
OF
THEORETICAL APPROACHES
335
in which c is the speed of light. For an 1H spin with a Larmor frequency of 500 MHz, W 10–21 s–1; thus, spontaneous emission is a completely ineffective relaxation mechanism for nuclear magnetic resonance. Stimulated emission also can be shown to have a negligible influence on nuclear spin relaxation, although calculation of transition probabilities is complicated by consideration of the coil in the probe (11). Spontaneous and stimulated emissions are important in optical spectroscopy because the relevant photon frequencies are orders of magnitude larger than the rf frequencies relevant in NMR spectroscopy. Instead, nuclear spin relaxation is a consequence of coupling of the spin system to the surroundings. The surroundings have historically been termed the lattice following the early studies of NMR relaxation in solids where the surroundings genuinely were a solid lattice. The lattice includes other degrees of freedom of the molecule containing the spins of interest (such as rotational degrees of freedom) as well as other molecules comprising the system. The energy levels of the lattice are assumed to be quasi-continuous with populations that are described by a Boltzmann distribution. Furthermore, the lattice is assumed to have an infinite heat capacity and consequently to be in thermal equilibrium at all times. The lattice modifies the local magnetic fields at the locations of the nuclei. As a consequence, the local fields weakly couple the lattice and the spin system. Stochastic Brownian rotational motions of molecules in liquids render local magnetic fields time dependent. More precisely, the local fields are composed of a rotationally invariant, and consequently time-independent, component and a rotationally variant, time-dependent component. The time-dependent local magnetic fields can be resolved into components perpendicular and parallel to the main static field. In addition, the fields can be decomposed by Fourier analysis into a superposition of harmonically varying magnetic fields with different frequencies. Thus, the Hamiltonian acting on the spins is given by H ¼ Hz þ Hlocal ðtÞ þ Hanisotropic ðtÞ ¼ Hz þ Hisotropic local local anisotropic þ Hanisotropic ¼ Hz þ Hisotropic local longitudinal ðtÞ þ Htransverse ðtÞ,
½5:2
in which Hz is the Zeeman Hamiltonian, Hisotropic contains the isotropic local chemical shift and scalar coupling interactions, and Hanisotropic ðtÞ, local anisotropic Hanisotropic ðtÞ, H ðtÞ are the Hamiltonians for stochastic anisotransverse longitudinal tropic interactions. The anisotropic Hamiltonians have an ensemble average of zero by construction owing to the rotational invariance
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of isotropic solution. Additionally, the correlations between stochastic fluctuations tend to zero for t c, in which c is defined as the correlation time of the stochastic process (vide infra). In isotropic solution, c is approximately the rotational correlation time of the molecular species. Relaxation processes that require exchange of energy with the lattice are termed nonadiabatic. Transverse components of the stochastic local field are responsible for nonadiabatic contributions to relaxation. If the Fourier spectrum of the fluctuating transverse magnetic fields at the location of a nucleus contains components with frequencies corresponding to the energy differences between eigenstates of the spin system, then transitions between eigenstates can occur. In this case, transition of the spin system from a higher (lower) energy state to a lower (higher) energy state is accompanied by an energy-conserving transition of the lattice from a lower (higher) to higher (lower) energy state. A transition of the spin system from higher energy to lower energy is more probable because the lattice is always in thermal equilibrium and has a larger population in the lower energy state. Thus, exchange of energy between the spin system and the lattice brings the spin system into a state of thermal equilibrium in which the populations of the stationary states have the Boltzmann distribution. Furthermore, transitions between stationary states caused by nonadiabatic processes decrease the lifetimes of these states and thereby broaden the energies of the stationary states through a Heisenberg uncertainty relationship. As a result, resonance frequencies vary randomly and phase coherence between spins is reduced over time. Consequently, nonadiabatic fluctuations that cause transitions between states result in both thermal equilibration of the spin state populations and decay of off-diagonal coherences. Relaxation processes that do not require exchange of energy with the lattice are termed adiabatic. Fluctuating fields parallel to the main static field are responsible for adiabatic contributions to relaxation. These fluctuating fields generate variations in the total magnetic field in the z-direction and, consequently, in the energies in the nuclear spin energy levels. Thus, adiabatic processes cause resonance frequencies of affected spins to vary randomly. Over time, the spins lose phase coherence and off-diagonal elements of the density matrix decay to zero. The populations of the states are not altered and no energy is exchanged between the spin system and the lattice, because transitions between stationary states do not occur. For macromolecules with ! c 1, the adiabatic contributions to the relaxation of off-diagonal elements of the density matrix are much more important compared to nonadiabatic lifetime effects.
5.1 INTRODUCTION
AND
SURVEY
5.1.1 RELAXATION
OF
THEORETICAL APPROACHES
IN THE
337
BLOCH EQUATIONS
In the simplest theoretical approach to spin relaxation, the relaxation of isolated spins is characterized in the Bloch equations [1.28] by two phenomenological first-order rate constants: the spin– lattice, or longitudinal, relaxation rate constant, R1, and the spin–spin, or transverse, relaxation rate constant, R2 (12). In the following discussion, rate constants rather than time constants are utilized; the two quantities are reciprocals of each other (T1 ¼ 1/R1 and T2 ¼ 1/R2). The spin–lattice relaxation rate constant describes the recovery of the longitudinal magnetization to thermal equilibrium, or, equivalently, return of the populations of the energy levels of the spin system (diagonal elements of the density operator) to the equilibrium Boltzmann distribution. The spin–spin relaxation rate constant describes the decay of the transverse magnetization to zero, or, equivalently, the decay of transverse single-quantum coherences (off-diagonal elements of the density matrix). Nonadiabatic processes contribute to both spin–lattice and spin–spin relaxation. Adiabatic processes contribute only to spin–spin relaxation; spin–lattice relaxation is not affected because adiabatic processes do not change the populations of stationary states. The Bloch formulation provides qualitative insights into the effects of relaxation on the NMR experiment, and the phenomenological rate constants can be measured experimentally. For example, the Bloch equations predict that the FID is the sum of exponentially damped sinusoidal functions and that, following perturbation of a spin system from equilibrium, R2 governs the length of time that the FID can be observed and R1 governs the minimum time required for equilibrium to be restored. The Bloch formulation does not provide a microscopic explanation of the origin or magnitude of the relaxation rate constants, nor is it extendible to more complex, coupled spin systems. For example, in dipolar-coupled two-spin systems, multiple spin operators, such as zero-quantum coherence, have relaxation rate constants that differ from both R1 and R2. In the spirit of the Bloch equations, the results for a product operator analysis of the evolution of a spin system under a particular pulse sequence can be corrected approximately for relaxation effects simply by adding an exponential damping factor for each temporal period post hoc. If product operator analysis of a two-dimensional pulse sequence yields a propagator U ¼ Ua(t2)UmUe(t1)Up, in which Up is the propagator for the preparation period, etc., then relaxation effects
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approximately are included by writing ðt1 , t2 Þ ¼ U ð0ÞU1 exp½Rp tp Re t1 Rm tm Ra t2 ,
½5:3
in which Rp is the relaxation rate constant for the operators of interest during the preparation time, tp, etc. Cross-correlation and cross-relaxation effects are assumed to be negligible. For example, the signal recorded in an 1H–15N HSQC spectrum is found by product operator analysis to be proportional to cosð!N t1 Þ cosð!H t2 Þ cosðJHN H t2 Þ, in which !N and !H are the Larmor frequencies of the 15N and 1HN spins, respectively and JHN H is the scalar coupling constant between the amide and 1H spins. The phenomenological approach modifies this expression to cosð!N t1 Þ cosð!H t2 Þ cosðJHN H t2 Þ exp ½R2N t1 R2H t2 , in which R2N and R2H are the transverse relaxation rate constants for the 15N and 1H operators present during t1 and t2, respectively, and relaxation during the INEPT sequences has been ignored. Relaxation effects on HSQC spectra are discussed in additional detail in Section 7.1.2.4. As a second example, product operator analysis of the INEPT pulse sequence [2.285], in the absence of relaxation, yields a density operator term proportional to 2IzSy sin(2JISt). Coherence transfer is maximized for 2t ¼ 1/(2JIS) [2.287]. If relaxation is considered, the result is modified to –2IzSy sin(2JISt) exp(2R2It), in which R2I is the relaxation rate of the I spin operators present during the period 2t. Maximum coherence transfer is obtained for 2t ¼ ðJIS Þ1 tan1 ðJIS =R2I Þ 1=ð2JIS Þ:
½5:4
5.1.2 THE SOLOMON EQUATIONS Spin–lattice relaxation for interacting spins can be treated theoretically by considering the rates of transitions of the spins between energy levels, as was demonstrated first by Bloembergen, Pound, and Purcell (13). Figure 5.1 shows the energy levels for a two-spin system with transition frequencies labeled. The four energy levels are labeled in the normal way as |mImSi. The rate constants for transitions between the energy levels are denoted by W0, WI, WS, and W2, and are distinguished according to which spins change spin state during the transition. Thus, WI denotes a relaxation process involving an I spin flip, WS denotes a relaxation process involving an S spin flip, W0 is a relaxation process in
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bb WS ba WI W2 W0 WI
ab WS aa
FIGURE 5.1 Transitions and associated rate constants for a two-spin system.
which both spins are flipped in opposite senses (flip-flop transition), and W2 is a relaxation process in which both spins are flipped in the same sense (flip-flip transition). A differential equation governing the population of the state |i is written by inspection: dP ¼ ðWI þ WS þ W2 ÞP þ WI P þ WS P þ W2 P þ K, ½5:5 dt in which P is the population of the state |i and K is a constant chosen to ensure that the population P returns to the equilibrium value P0 . At equilibrium, dP/dt ¼ 0; therefore, the value of K is found by setting the left-hand side of [5.5] equal to zero: K ¼ ðWI þ WS þ W2 ÞP0 WI P0 WS P0 W2 P0 :
½5:6
Writing P ¼ P P0 yields an equation for the deviation of the population of the |i state from the equilibrium population, dP ¼ ðWI þ WS þ W2 ÞP þ WI P þ WS P þ W2 P : dt ½5:7
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Similar equations are written for the other three states: dP ¼ ðW0 þ WI þ WS ÞP þ W0 P þ WI P þ WS P , dt dP ¼ ðW0 þ WI þ WS ÞP þ W0 P þ WI P þ WS P , dt dP ¼ ðWI þ WS þ W2 ÞP þ WI P þ WS P þ W2 P : dt ½5:8 Now, recalling that hIzi(t) ¼ Tr{(t)Iz} ¼ 11 þ 22 33 44 ¼ P þ P P P and hSzi(t) ¼ Tr{(t)Sz} ¼ 11 22 þ 33 44 ¼ P P þ P P leads to dIz ðtÞ ¼ ðW0 þ 2WI þ W2 ÞIz ðtÞ ðW2 W0 ÞSz ðtÞ, dt dSz ðtÞ ¼ ðW0 þ 2WS þ W2 ÞSz ðtÞ ðW2 W0 ÞIz ðtÞ, dt
½5:9
in which Iz ðtÞ ¼ hIz iðtÞ hI 0z i and hI 0z i is the equilibrium magnitude of the Iz operator. Corresponding relationships hold for Sz. Making the identifications I ¼ W0 þ 2WI þ W2 , S ¼ W0 þ 2WS þ W2 ,
½5:10
IS ¼ W2 W0 leads to the Solomon equations for a two-spin system (14): dIz ðtÞ ¼ I Iz ðtÞ IS Sz ðtÞ, dt dSz ðtÞ ¼ S Sz ðtÞ IS Iz ðtÞ: dt
½5:11
The Solomon equations reveal that the temporal evolution of the I spin longitudinal magnetization depends not only on its own departure from equilibrium (autorelaxation), but also on the state of the S spin longitudinal magnetization and vice versa. The time dependence of the two magnetizations are linked: this connection is called cross-relaxation. The rate constants I and S are the autorelaxation rate constants (or the spin–lattice relaxation rate constants, R1I and R1S, in the Bloch terminology) for the I and S spins, respectively, and IS is the cross-relaxation rate constant for exchange of magnetization between the two spins.
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The Solomon equations are extended to N interacting spins as X dIkz ðtÞ ¼ k Ikz ðtÞ kj Ijz ðtÞ, ½5:12 dt j6¼k in which k ¼
X
kj
½5:13
k6¼j
reflects the relaxation of the kth spin by all other spins (in the absence of interference effects; see Section 5.2.1). Equation [5.12] is written in matrix nomenclature as dMz ðtÞ ¼ RMz ðtÞ, ½5:14 dt in which R is an N N matrix with elements Rkk ¼ k and Rkj ¼ kj, and Mz(t) is an N 1 column vector with entries Mk(t) ¼ Ikz(t). The Solomon equations in matrix form have the formal solution Mz ðtÞ ¼ eRt Mz ð0Þ ¼ U1 eDt UMz ð0Þ,
½5:15
in which D is a diagonal matrix of the eigenvalues of R, U is a unitary matrix, and D ¼ URU1
½5:16
is the similarity transformation that diagonalizes R. These differential equations show that if the populations of the energy levels of the spin system are perturbed from equilibrium, then relaxation of a particular spin is in general a multiexponential process. For a two-spin system, I IS R¼ , IS S
þ 0 , D¼ 0 ½5:17 1=2 o 1n
¼ ð I þ S Þ ð I S Þ2 þ 4 2IS , 2 2 3 IS IS 1=2 1=2 ½ð I þ Þ2 þ 2IS ½ð I Þ2 þ 2IS 5, U¼4 I þ I 1=2 1=2 ½ð I þ Þ2 þ 2IS ½ð I Þ2 þ 2IS
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and upon substituting into [5.15], the result obtained is MI ðtÞ aII ðtÞ aIS ðtÞ MI ð0Þ ¼ , MS ðtÞ aSI ðtÞ aSS ðtÞ MS ð0Þ
½5:18
in which the matrix elements, aij(t), are given by 1 S S 1 I expð tÞ þ 1 þ I expð þ tÞ , aII ðtÞ ¼ 2 ð þ Þ ð þ Þ 1 I S I S 1þ expð tÞ þ 1 expð þ tÞ , aSS ðtÞ ¼ 2 ð þ Þ ð þ Þ IS aIS ðtÞ ¼ aSI ðtÞ ¼ ½expð tÞ expð þ tÞ: ð þ Þ ½5:19 These equations frequently are written in the form 1 S S 1 I þ 1þ I expðRC tÞ expðRL tÞ, aII ðtÞ ¼ 2 RC RC 1 I S I S 1þ aSS ðtÞ ¼ þ 1 expðRC tÞ expðRL tÞ, 2 RC RC IS aIS ðtÞ ¼ aSI ðtÞ ¼ ½1 expðRC tÞ expðRL tÞ RC ½5:20 by defining the cross-rate constant, RC and a leakage rate constant, RL: RC ¼ þ ¼ ½ð I S Þ2 þ 4 2IS 1=2 , R L ¼ :
½5:21
The leakage rate constant results in irreversible relaxation toward Boltzmann equilibrium for both spins, while the cross-rate constant determines the rate of magnetization transfer between spins. If 1 ¼ 2 ¼ , and IS ¼ , [5.19] simplifies to aII ðtÞ ¼ aSS ðtÞ ¼ 12½1 þ expð2tÞ exp ð Þt , ½5:22 aIS ðtÞ ¼ aSI ðtÞ ¼ 12½1 expð2tÞ exp ð Þt : The time dependence of the matrix elements aII(t) and aIS(t) are illustrated in Fig. 5.2. To illustrate aspects of longitudinal relaxation as exemplified by the Solomon equations, four different experiments are analyzed. For
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1.0 0.8
aij(t)
0.6 0.4 0.2
0
5
10
15
20
t (sec)
FIGURE 5.2 Time dependence of (—) aII(t) and (- - -) aIS(t) calculated using [5.22] with ¼ 0.30 s–1 and ¼ –0.15 s–1.
simplicity, a homonuclear spin system with I ¼ S, I ¼ S ¼ , and IS ¼ are assumed. The experiments use the generalized pulse sequence: 180 –t–90 –acquire:
½5:23
The initial state of the longitudinal magnetization is prepared by application of the 1808 pulse to equilibrium magnetization. The longitudinal magnetization relaxes according to the Solomon equations during the delay t. The final state of the longitudinal magnetization is converted into transverse magnetization by the 908 pulse and recorded during the acquisition period. In the selective inversion recovery experiment, the 1808 pulse is applied selectively to the I spin. The initial conditions are Iz ð0Þ ¼ hIz ið0Þ hI 0z i ¼ 2hI 0z i, and Sz ð0Þ ¼ hSz ið0Þ hS0z i ¼ 0. The time decay of the I spin magnetization is given by
hIz iðtÞ= I 0z ¼ 1 exp ð Þt ½1 þ expð2tÞ ½5:24 and is generally biexponential. In the initial rate regime, the slope of the recovery curve is given by d hIz iðtÞ= I 0z
½5:25
¼ 2 : dt t!0
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In the nonselective inversion recovery experiment, the 1808 pulse affects both the I and S spins equally. The initial conditions are Iz ð0Þ ¼ hIz ið0Þ hI 0z i ¼ 2hI 0z i and Sz ð0Þ ¼ hSz ið0Þ hS0z i ¼ 2hS0z i. The time course of the I spin magnetization is given by
hIz iðtÞ= I 0z ¼ 1 exp ð Þt ½1 þ expð2tÞ þ Sz0 = I 0z exp ð Þt ½1 expð2tÞ ¼ 1 2 exp ð þ Þt , ½5:26 in which the second line is obtained by using hI 0z i=hSz0 i ¼ S =I ¼ 1. The recovery curve is monoexponential with rate constant þ . In the initial rate regime, d hIz iðtÞ= I 0z
½5:27
¼ 2ð þ Þ, dt t!0 in which þ ¼ 2ðWI þ W2 Þ does not depend on W0. For macromolecules (Section 5.3),W0 WI W2 ; consequently, recovery of equilibrium magnetization following a nonselective 1808 pulse is inefficient. In the transient NOE experiment, the S spin longitudinal magnetization is inverted with a selective 1808 pulse to produce initial conditions Iz ð0Þ ¼ hIz ið0Þ hI 0z i ¼ 0 and Sz ð0Þ ¼ hSz ið0Þ hSz0 i ¼ 2hSz0 i. The time course of the I spin magnetization is given by
hIz iðtÞ= I z0 ¼ 1 þ Sz0 = Iz0 exp ð Þt ½1 expð2tÞ ¼ 1 þ exp ð Þt ½1 expð2tÞ, ½5:28 and is biexponential. In the initial rate regime, d hIz iðtÞ= Iz0
¼ 2: dt t!0
½5:29
Thus, the initial rate of change of the I spin intensity in the transient NOE experiment is proportional to the cross-relaxation rate, . In this initial rate approximation, in which hIz ið0Þ ¼ hIz0 i, solving [5.28] to first order in the mixing time, m, the time during which cross-relaxation occurs, gives hIz ið m Þ ¼ hI 0z i þ 2 m hSz0 i ¼ hI 0z ið1 þ 2 m Þ:
½5:30
Therefore, for m ¼ 0, the I spin magnetization is equal to its equilibrium value, but as m increases, the I spin magnetization has an
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additional contribution that is proportional to the mixing time and the cross-relaxation rate constant. This change in the magnitude of the I spin magnetization due to perturbation of the S spin is the NOE enhancement. In the decoupled inversion recovery experiment, the S spin is irradiated by a weak selective rf field (so as not to perturb the I spin) throughout the experiment in order to equalize the populations across the S spin transitions. In this situation, hSzi(t) ¼ 0 for all t, and the S spins are said to be saturated. Equation [5.11] reduces to
dhIz iðtÞ ¼ hIz iðtÞ Iz0 þ Sz0 dt
¼ hIz iðtÞ Iz0 ð1 þ = Þ :
½5:31
Following the 1808 pulse, Iz ð0Þ ¼ hIz ið0Þ hIz0 i ¼ 2hIz0 i and the time course of the I spin magnetization is given by
hIz iðtÞ= Iz0 ¼ 1 þ = ð2 þ = Þ expð tÞ: In the initial rate regime, d hIz iðtÞ= Iz0
¼ 2 þ : dt t!0
½5:32
½5:33
In this case, the recovery curve is monoexponential with rate constant . These analyses indicate that, even for an isolated two-spin system, the time dependence of the longitudinal magnetization usually is biexponential. The actual time course observed depends upon the initial condition of the spin system prepared by the NMR pulse sequence. Examples of the time courses of the I spin magnetization for these experiments are given in Fig. 5.3. The preceding derivation does not provide theoretical expressions that relate the transition rate constants, W0, WI, WS, and W2 to particular stochastic Hamiltonians. The semiclassical relaxation theory as described in Section 5.2 will be used to obtain such expressions, rather than pursuing further the original approach of Bloembergen et al. (13). As will be shown, the transition rate constants depend upon the different frequency components of the stochastic magnetic fields [5.118]. Thus, the transition characterized by WI is induced by molecular motions that produce fields oscillating at the Larmor frequency of the I spin, and the transition characterized by WS is induced by molecular motions that produce fields oscillating at the Larmor frequency of the S spin. The W0
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/
1
0
–1 0
5
10
15
20
t (sec)
FIGURE 5.3 Magnetization decays for inversion recovery experiments. (—) Selective inversion recovery calculated using [5.24]; ( ) nonselective inversion recovery calculated using [5.26]; (— - —) transient NOE recovery calculated using [5.28]; and (- - -) decoupled inversion recovery calculated using [5.32]. Calculations were performed for a homonuclear IS spin system with I ¼ S, ¼ 0.30 s–1, and ¼ –0.15 s–1.
pathway is induced by fields oscillating at the difference of the Larmor frequencies of the I and S spins, and the W2 pathway is induced by fields oscillating at the sum of the Larmor frequencies of the two spins. Most importantly, the cross-relaxation rate constant is nonzero only if W2 W0 6¼ 0; therefore, the relaxation mechanism must generate nonzero rate constants for the flip-flip (double-quantum) and/or flipflop (zero-quantum) transitions. For biological macromolecules, dipolar coupling between nuclear spins is the main interaction for which W2 and W0 are nonzero. The Solomon equations are central to the study of the NOE and will be discussed in additional detail in Section 5.5.
5.1.3 RANDOM-PHASE MODEL
FOR
TRANSVERSE RELAXATION
A very simple model for the effect of longitudinal stochastic fluctuations on the transverse relaxation of nuclear spins will now be derived. The instantaneous longitudinal component of the Hamiltonian experienced by a spin is Hlongitudinal ðtÞ ¼ !0 þ !ðtÞ Iz : ½5:34
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The local, stochastic component to the precession frequency of the spin, !(t), varies due to molecular motion. The average value of !(t) ¼ 0 by construction, because any nonzero average value is incorporated into !0. The time dependence of the complex magnetization is given by dMþ ðtÞ ¼ i !0 þ !ðtÞ Mþ ðtÞ: dt
½5:35
This equation will be solved in a slightly unusual manner, in order to provide insights into later derivations. First, the effect of !0 is removed by changing to the variable MTþ(t) ¼ exp(–i!0t)Mþ(t); this is identical to a transformation to a rotating frame resonant with !0. The new variable satisfies the differential equation: þ
dMT ðtÞ dMþ ðtÞ ¼ i!0 expði!0 tÞMþ ðtÞ þ expði!0 tÞ dt dt ¼ i!0 expði!0 tÞMþ ðtÞ þ i expði!0 tÞ !0 þ !ðtÞ Mþ ðtÞ ¼ i!ðtÞMTþ ðtÞ, ½5:36 which is integrated to give the solution for a particular !(t): Zt MTþ ðtÞ ¼ exp i !ðt0 Þ dt0 Mþ ð0Þ,
½5:37
0
where MTþ(0) ¼ Mþ(0). The observed signal is obtained by averaging over all possible instances of !(t), which will be indicated by overbars: Zt Tþ 0 0 M ðtÞ ¼ exp i !ðt Þ dt Mþ ð0Þ 0 ( ) Zt Zt Zt 1 !ðt0 Þ dt0 !ðt0 Þ dt0 !ðt00 Þ dt00 þ Mþ ð0Þ ¼ 1þi 2 0 0 0 Z Z 1 t t 0 00 0 00 ¼ 1 !ðt Þ!ðt Þ dt dt þ Mþ ð0Þ: 2 0 0 ½5:38 Between lines two and three, the order of averaging and integration has been reversed and the condition !ðtÞ ¼ 0 has been used. This equation provides an expression for the ensemble average magnetization. The assumption has been made that the variation in !(t) is uncorrelated with the variation in Mþ(0) so that the ensemble averaging can be performed
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independently. A differential form of this equation is obtained by differentiating [5.38]: Zt dMTþ ðtÞ ¼ !ðtÞ!ðt0 Þ dt0 Mþ ð0Þ: ½5:39 dt 0 Equation [5.39] is converted into a differential equation for MTþ ðtÞ by making the following two assumptions: (i) Mþ ð0Þ can be replaced with MTþ ðtÞ on the right-hand side of [5.39] and (ii) the limit of the integral can be extended from t to infinity. The result is Z1 dMTþ ðtÞ ¼ !ðtÞ!ðt0 Þ dt0 MTþ ðtÞ: ½5:40 dt 0 The function !ðtÞ!ðt0 Þ is called the autocorrelation function of the stochastic process. Autocorrelation functions play a central role in the analysis of stochastic processes. If t ¼ t0 , the value of the autocorrelation function is simply the mean square amplitude of the process (i.e., the variance of the fluctuations). For stationary random processes, the autocorrelation function depends only on the time difference, ¼ t t0 . The autocorrelation function will be denoted C(). The relaxation rate constant, R2, is then identified as Z1 Z1 R2 ¼ !ðtÞ!ðt Þ d ¼ CðÞ d: ½5:41 0
0
With this expression, [5.40] is transformed back to an equation for Mþ ðtÞ: dMþ ðtÞ ¼ fi!0 R2 gMþ ðtÞ, dt
½5:42
which is simply the Bloch equation for evolution of Mþ ðtÞ. To proceed, the assumption will be made that C() ¼ C(0) exp(/ c), in which Cð0Þ ¼ !2 ðtÞ is the variance of the fluctuating fields and c is the characteristic time over which the fields vary. The final result after performing the integral is R2 ¼ !2 ðtÞ c :
½5:43
This expression shows that the relaxation rate constant depends on the mean square magnitude of the fluctuating fields and the time scale of the fluctuations. This result, although derived for a very simple model,
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contains much of the qualitative physics of the more complete treatment to be presented in Section 5.2. Now, the two assumptions made in deriving [5.40] are justified post facto. To first order, the fractional change in MTþ ðtÞ is given by ½MTþ ðtÞ MTþ ð0Þ=Mþ ð0Þ ¼ R2 t. For a time t 1/R2, MTþ ðtÞ and Mþ ð0Þ differ negligibly and MTþ ðtÞ can be substituted for Mþ ð0Þ in [5.39]. The correlation time, c, typically is related to the time scale for molecular rotational diffusion in solution; consequently, the time scales of interest for spin relaxation satisfy t c. In this case, the value of the integrand in [5.39] is zero for times greater than t and the upper limits of the integral can be extended to infinity. As a result of these two assumptions, the present theory is valid on a so-called coarse-grained time scale c t 1/R2. The restrictions on t 1/R2 would appear to constitute a fatal weakness because relaxation in NMR experiments frequently must be considered for times T 1/R2. To rectify this, T conceptually is defined as T ¼ nt, in which n is an integer and t satisfies the coarse-grained temporal restrictions. Relaxation over the period T is calculated by piecewise evaluation of [5.42] for each of the n intervals in succession. In the limit of n ! 1 while t ! 0, the expected exponential relaxation behavior is obtained even for T 4 1/R2. To finish the analysis, the variance in !(t) must be determined. The variance will depend, of course, on the particular relaxation mechanism of interest. As an illustration, relaxation due to the CSA interaction will be considered. For the CSA Hamiltonian with an axially symmetric CSA tensor (Section 5.4.4), qffiffi qffiffi anisotopic þ 2 2 0 1 1 1 1 ðtÞ ¼ 3B0 3Y 2 ½ðtÞIz þ 2Y2 ½ðtÞI 2Y2 ½ðtÞI , Hlocal ½5:44 in which is the CSA, Y q2 ½ are modified spherical harmonic functions given in Table 5.1, and (t) ¼ { (t), (t)} are the timedependent angles defining the orientation of the symmetry axis of the CSA principal axis system in the laboratory frame. The term proportional to Iz represents the fluctuating longitudinal interactions (giving rise to adiabatic relaxation) and the terms proportional to I þ and I – represent the fluctuating transverse interactions (giving rise to nonadiabatic relaxation). An expression for !(t) is obtained from the longitudinal component of the CSA Hamiltonian in [5.44]: 0 2 Hanisotropic longitudinal ðtÞ ¼ !ðtÞIz ¼ 3B0 Y2 ½ðtÞIz :
½5:45
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TABLE 5.1 Modified Second-Order Spherical Harmonicsa q
Y q2
q q Y q 2 ¼ ð1Þ Y 2
0
(3 cos2 – 1)/2 pffiffiffiffiffiffiffiffi – 3=2 sin cos ei pffiffiffiffiffiffiffiffi 3=8 sin2 ei2
(3 cos2 – 1)/2 pffiffiffiffiffiffiffiffi 3=2 sin cos ei pffiffiffiffiffiffiffiffi 2 i2 3=8 sin e
1 2 a
The modified spherical harmonic functions are normalizedp(to give ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi the conventional spherical harmonic functions) by multiplying by 5=ð4Þ.
For the present, the assumption will be made that the molecule reorients isotropically; therefore the probability distribution for the orientation of the principal axis system is p( , ) ¼ 1/(4). The form of !2 ðtÞ is: 2 Z 2 Z 2 1 4 2 B0 ! ðtÞ ¼ Y02 ½ 2 sin d d ¼ ðB0 Þ2 : ½5:46 3 4 0 45 0 For B0 ¼ 11.7 T, the predicted transverse relaxation rate constant for a 13C spin with ¼ 200 ppm in a molecule with c ¼ 10 ns is 22 s–1. An exact calculation (using expressions presented in Table 5.8; see Section 5.4.4) yields R2 ¼ 22.5 s–1. The difference between the two results arises through neglect in the present derivation of the lifetime broadening (nonadiabatic) effects of fluctuating transverse fields.
5.1.4 BLOCH, WANGSNESS,
AND
REDFIELD THEORY
A microscopic semiclassical theory of spin relaxation was formulated by Bloch, Wangsness, and Redfield (BWR) and has proved to be the most useful approach for practical applications (15, 16). In the semiclassical approach, the spin system is treated quantum mechanically and the surroundings (the heat bath or lattice) are treated classically. This treatment suffers primarily from the defect that the spin system evolves toward a final state in which energy levels of the spin system are populated equally. Equivalently, the semiclassical theory is formally correct only for an infinite Boltzmann spin temperature; at finite temperatures, an ad hoc correction to the theory is required to ensure that the spin system relaxes toward an equilibrium state in which the populations are properly described by a Boltzmann distribution. A fully quantum mechanical treatment of spin relaxation overcomes this defect
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and predicts the proper approach to equilibrium; however, the computational details of the quantum mechanical relaxation theory are outside the scope of this text (2, 16).
5.2 The Master Equation In the semiclassical theory of spin relaxation, the Hamiltonian for the system is written as the sum of a deterministic quantum mechanical Hamiltonian that acts only on the spin system, H0 , and a stochastic Hamiltonian, H1 ðtÞ, that couples the spin system to the lattice: HðtÞ ¼ H0 þ H1 ðtÞ:
½5:47
The Hamiltonian H1 ðtÞ is regarded as a time-dependent perturbation acting on the main time-independent Hamiltonian, H0 . This Hamiltonian is written in the absence of an applied rf field (see Section 5.2.3 for the effects of rf fields). The Liouville equation of motion of the density operator is (Section 2.2.3) n o dðtÞ ¼ i½H0 þH1 ðtÞ, ðtÞ ¼ i L^ 0 þ L^ 1 ðtÞ ðtÞ, ½5:48 dt ^ ¼ ½HðtÞ, is the commutation superoperator or in which LðtÞ Liouvillian. By formally identifying L^ 0 and L^ 1 ðtÞ with !0 and !(t), respectively, the ideas used in the random-phase model are used to solve [5.48]. First, the explicit influence of H0 is removed by transforming to the new variable, T ðtÞ ¼ expðiL^ 0 tÞðtÞ ¼ expðiH0 tÞðtÞ expðiH0 tÞ,
½5:49
in which the second equality is established by expanding the exponential factors in Taylor series. The change of variables transforms the Liouville equation into a new reference frame, which is called the interaction frame: d T ðtÞ dðtÞ ¼ iL^ 0 expðiL^ 0 tÞðtÞ þ expðiL^ 0 tÞ dt ndt o ^ ^ ^ ¼ i expðiL0 tÞL0 ðtÞ i expðiL0 tÞ L^ 0 þ L^ 1 ðtÞ ðtÞ n o ¼ i expðiL^ 0 tÞ L^ 1 ðtÞðtÞ ¼ iL^ T1 ðtÞ T ðtÞ ¼ i½HT1 ðtÞ, T ðtÞ,
½5:50
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in which L^ T1 ðtÞ ¼ HT1 ðtÞ, and HT1 ðtÞ ¼ L^ 0 H1 ðtÞ ¼ exp fiH0 tgH1 ðtÞ exp fiH0 tg:
½5:51
The transformation into the interaction frame is isomorphous to the rotating-frame transformation; however, important differences exist between the two. The rotating-frame transformation removes the explicit time dependence of the rf Hamiltonian and renders the Hamiltonian time independent in the rotating frame. The Hamiltonian H0 is active in the rotating frame. The interaction-frame transformation removes the explicit dependence on H0 ; however, HT1 ðtÞ remains time dependent. As discussed in Section 5.2.3, the rotating-frame and interaction-frame transformations are performed sequentially in some circumstances. Equation [5.50] is solved by the same formal approach used in the random-phase model for relaxation to yield an analog to [5.40], d T ðtÞ ¼ dt
Z
1 0
d½HT1 ðtÞ, ½HT1 ðt Þ, T ðtÞ 0 ,
½5:52
in which the overbar indicates ensemble averaging over the stochastic Hamiltonians and T(t) now designates the ensemble average of the density matrix (the overbar is omitted for convenience). This equation is subject to the following assumptions: 1. The ensemble average of HT1 ðtÞ is zero. Any components of HT1 ðtÞ that do not vanish upon ensemble averaging are incorporated into H0 . 2. HT1 ðtÞ and T(t) are uncorrelated so that the ensemble average is taken independently for each quantity. 3. The times of interest satisfy c t 1/R, in which c is the characteristic correlation time for HT1 ðtÞ and R is the relevant relaxation rate constant. In liquids, c is on the order of the rotational diffusion correlation time for the molecule, 1012107 s. 4. T(t) can be replaced by T(t) 0, in which 0 is the equilibrium density operator. By construction, T0 ¼ 0 . Assumptions 1, 2, and 3 are similar to assumptions made for the random-phase model derived in Section 5.1.3. Assumption 4 ensures that the spin system relaxes toward thermal equilibrium, a concern that did not arise in the random-phase model for transverse relaxation. The term 0 naturally enters the differential equation in a full quantum
353
5.2 THE MASTER EQUATION
mechanical derivation. More detailed discussions of the range of validity of these assumptions are found elsewhere (2, 3). In order to transform [5.52] back into the laboratory frame, the transformation properties of H1 (t) must be established. The approach to be utilized follows the derivation of the nuclear spin Hamiltonian in Section 2.8. The stochastic Hamiltonian is decomposed as H1 ðtÞ ¼
k X
q ð1Þq F q k ðtÞAk ,
½5:53
q¼k
in which F qk ðtÞ is a random function of spatial variables, Aqk is a tensor spin operator, and k is the rank of the tensor (2, 17, 18). Additionally, q q qy q q Aq k ð1Þ Ak and F k ðtÞ ð1Þ F k ðtÞ. For the Hamiltonians of interest in NMR spectroscopy, k is one or two, and the decomposition is always possible. The Aqk are chosen to be spherical tensor operators because these operators have simple transformation properties under rotations. To proceed, the operators Aqk are expanded in terms of basis operators X q Aqk ¼ Akp ½5:54 p
that satisfy the relationship n o h i L^ 0 Aqkp H0 , Aqkp ¼ !qp Aqkp :
½5:55
Here Aqkp and !qp are called the eigenfunctions and eigenfrequencies of the Hamiltonian commutation superoperator. The index p serves to distinguish spin operators with the same order q but distinct eigenfrequencies. This additional label is important particularly for the dipolar Hamiltonian, because the interacting spins will have different eigenfrequencies in the absence of magnetic equivalence. Equation [5.55] implies the additional property exp iL^ 0 t Aqkp ¼ expðiH0 tÞAqkp expðiH0 tÞ ¼ exp i!qp t Aqkp , ½5:56 which is proved as usual by expanding the exponential factors in the Taylor series. For example, if H0 ¼ !IIz þ !SSz, then the single-element operator 2IzSþ ¼ I Sþ – ISþ ¼ |ih| |ih| (see [2.215]) is an
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CHAPTER 5 RELAXATION
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DYNAMIC PROCESSES
eigenoperator with eigenfrequency !S: H0 , I S þ I S þ
¼ ð!I Iz þ !S Sz Þ ji ji ð!I Iz þ !S Sz Þ
¼ !I Iz ji Iz ji Iz þ Iz
þ !S Sz ji Sz ji Sz þ Sz ¼ 12!I ji þ ji þ 12!S ji þ ji ¼ !S ji ¼ !S I S þ I S þ :
½5:57
Applying [5.56], in the interaction frame, X q expfiH0 tgAqkp expfiH0 tg AqT k ¼ expfiH0 tgAk expfiH0 tg ¼ p
¼
X
Aqkp
n
o exp i!qp t ,
½5:58
p
AqT ¼ expfiH0 tgAq k k expfiH0 tg ¼
X
n o q Aq exp i! t p , kp
½5:59
p q in which !q p ¼ !p . Substituting [5.53] and [5.58] into [5.52] yields n o 0 XX 0 0 d T ðtÞ ¼ ð1Þqþq exp ið!qp þ !qp0 Þt ½Aqkp0 , ½Aqkp , T ðtÞ 0 dt q,q0 p,p0
Z
1
0
0 q q F q k ðtÞF k ðt Þ exp i!p d: ½5:60 0
The random processes F qk ðtÞ and F qk ðtÞ are assumed to be statistically independent unless q0 ¼ q; therefore, the ensemble average in [5.60]
355
5.2 THE MASTER EQUATION
vanishes if q0 6¼ q. Thus, k X n o X d T ðtÞ q T ¼ exp ið!qp !qp0 Þt ½Aq kp0 , ½Akp , ðtÞ 0 dt q¼k p,p0 Z1 q F qk ðtÞF q k ðt Þ exp i!p d:
½5:61
0
A second simplification of this equation is commonly utilized. Terms q q in [5.61] in which j!qp þ !q p0 j ¼ j!p !p0 j 0 are nonsecular in the sense of perturbation theory (Section 5.2.2), and do not affect the long-time behavior of T(t) because the rapidly oscillating factors expfið!qp !qp0 Þtg average to zero much more rapidly than relaxation occurs. Furthermore, if none of the eigenfrequencies are degenerate, terms in [5.61] are secular and nonzero only if p ¼ p0 . Thus, Z1 k X n o X d T ðtÞ q q T q ¼ ½Akp , ½Akp , ðtÞ 0 F kq ðtÞF q ðtÞ exp i! d: p k dt 0 q¼k p ½5:62 The correlation functions F qk ðtÞF q k ðt Þ are real, even-valued, functions of for diffusive stochastic processes of interest in the theory of spin relaxation in macromolecules. The real part of the integral in [5.62] is called the power spectral density function, j q(!): Z 1 F qk ðtÞF q ðt Þ exp f i! g d j q ð!Þ ¼ 2 Re k Z 10 q q F k ðtÞF k ðt Þ expfi! g d ¼ Re Z1 1 q q F k ðtÞF k ðt þ Þ expfi! g d : ½5:63 ¼ Re 1
The factor of two is introduced in the first line of [5.63] for convenience in extending the lower limit of the integral. Thus, the power spectral q q q q density is an even function of ! and jq ð!q p Þ ¼ j ð!p Þ ¼ j ð!p Þ ¼ q q j ð!p Þ. The imaginary part of the integral, Z 1 q q q F k ðtÞF k ðt Þ expfi! g d k ð!Þ ¼ Im Z0 1 q q F k ðtÞF k ðt þ Þ expfi! g d ½5:64 ¼ Im 0
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q q is an odd function of !. Consequently, kq ð!q p Þ ¼ k ð!p Þ ¼ q q q q q k ð!p Þ ¼ k ð!p Þ, k (0) ¼ 0, and k X X
q T q q ½Aq kp , ½Akp , ðtÞ 0 k ð!p Þ
q¼k p
¼
k Xn X q¼0
¼
p
k X X q¼0
o q q q T T q q ½Aq kp , ½Akp , ðtÞ 0 ½Akp , ½Akp , ðtÞ 0 k ð!p Þ
q T q q ½½Aq kp , Akp , ðtÞ 0 k ð!p Þ:
½5:65
p
Further, in the high-temperature limit, 0 / H0 ; consequently, using q [5.55] yields ½½Aq kp , Akp , 0 ¼ 0. With this notation (19), k X d T ðtÞ 1X q T q q ¼ ½Aq kp , ½Akp , ðtÞ 0 j ð!p Þ dt 2 q¼k p
þi
k X X q¼0
q T q q ½½Aq kp , Akp , ðtÞk ð!p Þ:
p
½5:66 Equation [5.66] is transformed to the laboratory frame to yield the Liouville–von Neumann differential equation for the density operator: dðtÞ ^ ¼ i½H0 , ðtÞ i½, ðtÞ ððtÞ 0 Þ, dt
½5:67
in which the relaxation superoperator is k X 1X q ^ ¼ j q ð!qp Þ½Aq kp , ½Akp , 2 q¼k p
½5:68
and ¼
k X X q¼0
q kq ð!qp Þ½Aq kp , Akp :
½5:69
p
The second term in [5.67] represents second-order frequency shifts of the resonance lines, which are called dynamic frequency shifts, and is called the dynamic frequency shift operator. This term has the same
357
5.2 THE MASTER EQUATION
form as the first term and thus can be incorporated into H0 , by redefining H0 þ ! H0 , to give the final result: dðtÞ ^ ¼ i½H0 , ðtÞ ððtÞ 0 Þ: dt
½5:70
The functions j q(!) and kq(!) obey the Kramers–Kro¨nig relation and form a Hilbert transform pair. Dynamic frequency shifts are not considered further herein, but are discussed extensively elsewhere (2, 20). Equation [5.70] is converted into an equation for product, or other basis operators, by expanding the density operator in terms of the basis operators to yield the matrix form of the master equation, X dbr ðtÞ=dt ¼ firs bs ðtÞ rs ½bs ðtÞ bs0 g, ½5:71 s
in which
rs ¼ Br j½H0 , Bs =hBr jBr i
½5:72
is a characteristic frequency, D E ^ s =hBr jBr i rs ¼ Br jB ¼
k X nD o E 1 X q h i jq ð!qp Þ Br j½Aq , ½A , B = B jB s r r kp kp 2 q¼k p
½5:73
is the rate constant for relaxation between the operators Br and Bs, and
bj ðtÞ ¼ Bj jðtÞ = Bj jBj : ½5:74 For normalized basis operators with Tr{Br2} ¼ Tr{Bs2}, rs ¼ sr. Equations [5.71]–[5.74] are the main results of this section for relaxation in the laboratory reference frame. As shown by [5.71], the evolution of the base operators for a spin system is described by a set of coupled differential equations. Diagonal elements rr are the rate constants for auto- or self-relaxation of Br; off-diagonal elements rs are the rate constants for cross-relaxation between Br and Bs. Crossrelaxation between operators with different coherence orders is precluded as a consequence of restricting [5.60] (and hence [5.71]) to terms satisfying q0 ¼ q. For example, cross-relaxation does not occur between zero- and single-quantum coherence. Furthermore, if none of the transitions in the spin system is degenerate (to within approximately a linewidth), then cross-relaxation rate constants between off-diagonal elements of the density operator in the laboratory reference frame are
358
CHAPTER 5 RELAXATION Populations
ZQT
1QT
AND
DYNAMIC PROCESSES 2QT Populations ZQT 1QT 2QT
FIGURE 5.4 Redfield kite. Solid blocks indicate nonzero relaxation rate constants between operators in the absence of degenerate transitions. Populations have nonzero cross-relaxation rate constants, but all other coherences relax independently. If transitions are degenerate, the dashed-outline blocks indicate the additional nonzero cross-relaxation rate constants observed between coherences with the same coherence level.
also zero through the secular approximation. Consequently, the matrix of relaxation rate constants between operators has a characteristic block diagonal form, known as the Redfield kite, illustrated in Fig. 5.4. Two critical requirements for a stochastic Hamiltonian to be effective in causing relaxation are encapsulated in [5.70] and [5.71]: q (i) the double commutator ½Aq kp , ½Akp , ðtÞ 0 must not vanish and (ii) the spectral density function for the random process that modulates the spin interactions must have significant components at the characteristic frequencies of the spin system, !qp . The former requirement can be regarded as a kind of selection rule for whether the term in the stochastic Hamiltonian that depends upon the operator Aqkp is effective in causing relaxation of the density operator. In most cases, the stochastic random process is a consequence of molecular reorientational motions. This observation is central to the dramatic differences in spin relaxation and, thus, in NMR spectroscopy, of rapidly rotating small
359
5.2 THE MASTER EQUATION
molecules and slowly rotating macromolecules. Calculation of relaxation rate constants involves two steps: (i) calculation of the double commutator and trace formation over the spin variables and (ii) calculation of the spectral density function. These two calculations are pursued in the following sections.
5.2.1 INTERFERENCE EFFECTS In many instances, more than one stochastic Hamiltonian capable of causing relaxation of a given spin may be operative. In this circumstance, [5.53] is generalized to H1 ðtÞ ¼
k XX
q ð1Þq F q mk ðtÞAmk ,
½5:75
m q¼k
in which the summation over the index m refers to the different relaxation interactions or stochastic Hamiltonians. Using [5.75] rather than [5.53] in this derivation leads once more to [5.71], with rs given by a generalization of [5.73] o E 1 X X X nD q h i j q ð!qp Þ rs ¼ Br j½Aq , ½A , B = B jB s r r mkp mkp 2 m q p o E 1 X X X nD q h i j qmn ð!qp Þ Br j½Aq , ½A , B = B jB þ s r r mkp nkp ½5:76 2 m,n q p ¼
X m
m6¼n
m rs þ
X
mn rs ,
m,n m6¼n
in which the cross-spectral density is Z 1 F qmk ðtÞF q ðt þ Þ exp f i! gd : j qmn ð!Þ ¼ Re nk
½5:77
1
Here m rs is the autorelaxation rate constant due to the mth relaxation mechanism and mn is the relaxation rate constant arising from rs interference or cross-correlation between the mth and nth relaxation mechanisms. Clearly, j qmn ð!Þ ¼ 0 unless the random processes F qmk ðtÞ and F qnk ðtÞ are correlated. In the absence of correlation between the different relaxation mechanisms, mn rs ¼ 0 for all m and n and each mechanism contributes additively to relaxation of the spin system.
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DYNAMIC PROCESSES
The two most frequently encountered interference or cross-correlation effects in biological macromolecules arise from interference between dipolar and anisotropic chemical shift interactions and interference between the dipolar interactions of different pairs of spins. The prototypical example of the former is the interference between the dipolar and CSA interactions for 15N (21). The prototypical example of the latter is the interference between the dipolar interactions in an I2S or I3S spin system such as a methylene (I2 represents the two methylene 1H spins; S represents either a remote 1H or the methylene 13C) or methyl group (I3 represents the three methyl 1H spins; S represents either a remote 1H or the methyl 13C) (10). Interference effects can result in cross-relaxation between pairs of operators for which cross-relaxation would not be observed otherwise. The observation of otherwise ‘‘forbidden’’ cross-relaxation pathways is one of the hallmarks of interference effects (22). Relaxation interference also forms the basis for Transverse Relaxation Optimized Spectroscopy (TROSY), in which interference between relaxation mechanisms is used to obtain narrower resonance linewidths (Chapter 7).
5.2.2 LIKE SPINS, UNLIKE SPINS, APPROXIMATION
AND THE
SECULAR
A distinction frequently is made between like and unlike spins and relaxation rate constants are derived independently for each case (2). Like spins are defined as spins with identical Larmor frequencies and unlike spins are defined as spins with widely different Larmor frequencies. Such distinctions can obscure the generality of the theory embodied in [5.71]. In actuality, the presence of spins with degenerate Larmor frequencies has straightforward consequences for relaxation. First, particular operators Aqkp in [5.53] may become degenerate (i.e., have the same eigenfrequency, !qp ) and are therefore secular with respect to each other. Thus, prior to applying the secular condition, the set of Aqkp must be redefined as Aqkp ¼
X
Aqkm ,
½5:78
m
in which the summation extends over the operators for Aqkm for which !qp ¼ !qm . For example, operators with eigenfrequencies of 0 and !I !S belong to different orders p for unlike spins; the eigenfrequencies are degenerate for like spins and the corresponding operators would be summed to yield a single operator with eigenfrequency of zero. Second,
361
5.2 THE MASTER EQUATION
for spins that are magnetically equivalent, such as the three 1H spins in a methyl group, basis operators that exhibit the maximum symmetry of the chemical moiety are derived using group theory (18, 23). Although such basis operators simplify the resulting calculations, the group theoretical treatment of relaxation of magnetically equivalent spins is beyond the scope of the present text; the interested reader is referred to the original literature (18, 23). The following discussions focus on spin systems without degenerate transitions. Results of practical interest that arise as a consequence of degeneracy are presented as necessary. The concepts of like and unlike spins arise as limiting cases because the secular approximation has been imposed in deriving [5.62] from [5.61]. This assumption, although widely applied, is not necessary to relaxation theory (6). If the secular approximation is not applied, then [5.61] is transformed directly to the laboratory frame to yield dðtÞ ¼ i½H0 , ðtÞ dt Z k X X q q ½Akp0 , ½Akp , ðtÞ 0 q¼k p,p0
1 0
n o q F qk ðtÞF q ðt Þ exp i! d: p k ½5:79
Ignoring dynamic frequency shifts for simplicity, this equation is identical to [5.70] except that the relaxation superoperator is redefined as 1 ^ ¼ 2
k X X q¼k
q j q ð!qp Þ½Aq kp0 , ½Akp , :
½5:80
p,p0
Thus, if the secular approximation is not invoked, then the relaxation superoperator contains additional terms. To illustrate the main consequences of not utilizing the secular approximation, [5.61] in the interaction frame will be analyzed again. The density operator is expanded in the set of eigenoperators of the Hamiltonian commutation superoperator, X ðtÞ ¼ bs ðtÞBs , ½5:81 s
in which, for simplicity, Bs is used to represent the eigenoperators, rather than Aqkp , and the index s runs over all combinations of q and p. Using [5.56], X bs ðtÞei!s t Bs : ½5:82 T ðtÞ ¼ s
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CHAPTER 5 RELAXATION
AND
DYNAMIC PROCESSES
Equation [5.61] is written as n o X XX dbr ðtÞ ¼ i!r br ðtÞ bs ðtÞ bs0 ei!s t exp i !s !r þ !qp !qp0 t dt s q p,p0 nD oZ 1 E q q Br j½Aq Fkq ðtÞF q kp0 ,½Akp ,Bs =hBr jBr i k ðt Þ exp i!p d: 0
½5:83 The rate constant for relaxation between the operators Br and Bs is given by: o E 1 X X nD q h i rs ¼ j q ð!qp Þ, Br j½Aq , ½A , B = B jB ½5:84 0 s r r kp kp 2 q p,p0 provided that !s !r þ !qp !qp0 ¼ 0. This condition can be regarded as a generalization of the secular requirement that !qp !qp0 ¼ 0. For autorelaxation, r ¼ s and the usual secular condition is obtained; thus, the autorelaxation rate constants are not affected by whether or not the secular condition is applied. If r 6¼ s, then [5.84] predicts that additional cross-relaxation rate constants will be nonzero compared to results obtained from [5.73] when the secular approximation is utilized. If only two operators are considered, then d br ðtÞ br ðtÞ br0 i!r r , ½5:85 ¼ i!s s bs ðtÞ bs0 dt bs ðtÞ in which k ¼ kk and ¼ rs. The eigenvalues of this equation are 1=2
¼ i! 12 !2 þ 2 þ 4 2 2i! , ½5:86 in which ! ¼ ð!r þ !s Þ=2, ¼ ð r þ s Þ=2, ! ¼ (!r !s), and ¼ ( r s). Whenever |!2 2| 4 2, can be neglected in [5.85] and each operator evolves independently. Essentially, crossrelaxation between two basis operators is negligible if the difference between eigenfrequencies is greater than the linewidth. The additional cross-relaxation pathway predicted by neglect of the secular approximation is suppressed because the two operators evolve relative to each other. Over a time period t ¼ 2/!, the instantaneous effect of crossrelaxation is averaged to zero. As !s ! !r, ! ! 0, and cross-relaxation gradually becomes effective. When the secular approximation is not imposed, distinctions between like and unlike spins do not arise because the relative precession frequencies of pairs of operators naturally control
363
5.2 THE MASTER EQUATION
which cross-relaxation terms will be effective. The drawback to the neglect of the secular approximation is that many more terms must be evaluated in the summations in [5.84] compared with [5.73]. Dipolar relaxation in a scalar-coupled spin system, discussed in Section 5.4.2, is a practical example of the concepts discussed herein.
5.2.3 RELAXATION
IN THE
ROTATING FRAME
In the presence of an applied rf field [for example, in a RotatingFrame Overhauser Effect Spectroscopy (ROESY) or TOCSY experiment], the transformation into the interaction frame involves, first, a transformation into a rotating frame to remove the time dependence of Hrf (t), followed by transformation into the interaction frame of the resulting time-independent Hamiltonian. If H0 Hz — that is if the Zeeman Hamiltonian is dominant (i.e., ignoring the scalar coupling Hamiltonian) — then the interaction frame is equivalent to a doubly rotating tilted frame. As a consequence of the initial rotating-frame transformation, the eigenfrequencies !qp used as arguments of the spectral density function jq ð!qp Þ are modified to !qp þ !pqðrf Þ , in which Þ !qðrf is defined by: p K h i X Þ q !rf,i Izi , Aqkp ¼ !qðrf Akp , ½5:87 p i¼1
in which !rf,i is the frequency of the rotating frame for the ith spin and K is the number of irradiated spins in the spin system. Homonuclear spin operators transform identically under the rotating-frame transformation; therefore, the relaxation superoperator in the rotating frame is calculated as for like spins (Section 5.2.2). For macromolecules with !1 c 1, in which !1 is the strength of the applied rf field and c is the Þ rotational correlation time of the molecule, jq ð!qp þ !qðrf Þ jq ð!qp Þ p (Section 5.3). In this case, approximate values for the relaxation rate constants in the rotating frame are calculated using [5.73]0 by first 0 transforming the operators in the tilted frame, Br and Bs to the laboratory frame. Thus (24), D E
0rs ¼ U1 B0r Uj^ U1 B0s U = B0r jB0r : ½5:88 For an rf field applied with x-phase, the transformation U is defined as a y-rotation, ( ) K X U ¼ exp i i Iyi , ½5:89 i¼1
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CHAPTER 5 RELAXATION
AND
DYNAMIC PROCESSES
Iz
–Iy = –I'y
I'z q Ix
I'x FIGURE 5.5 Relative orientations of the laboratory and tilted reference frames used to determine the transformation [5.89].
0
0
in which i is defined by [1.21] (if Br and Bs refer to different spins, then i may differ for each spin). The relative orientations of the tilted and untilted reference frames are illustrated in Fig. 5.5 for a single spin. If ¼ 0, either because !1 ¼ 0 or because !1 |!0 !rf |, then [5.88] reduces to [5.73]. In general, operators that do not commute with the Hamiltonian in the rotating frame decay rapidly as a consequence of rf inhomogeneity (Section 3.5.3). Thus, if a cw rf field is applied, as in a basic ROESY experiment, only operators with effective frequencies in the rotating frame equal to zero must be considered: such operators are usually limited to longitudinal operators and homonuclear zero-quantum operators. If the rf field is phase modulated to compensate for resonance offset and rf inhomogeneity, e.g., by applying a suitably constructed coherent decoupling scheme, such as DIPSI-2 (decoupling in the presence of scalar interaction), single- and multiple-quantum operators also must be considered (25). For operators containing transverse components in the rotating frame, the relaxation rate constant given by [5.88] is an instantaneous rate constant; the effective average rate constant is obtained by averaging the rate constant over the trajectory followed by the operator under the influence of the Hamiltonian in the rotating frame (26).
5.3 SPECTRAL DENSITY FUNCTIONS
365
5.3 Spectral Density Functions A general expression for the spectral density function is given by [5.63]. As discussed elsewhere, for relaxation in isotropic liquids in the high-temperature limit (27), jq ð!Þ ¼ ð1Þq j 0 ð!Þ ð1Þq jð!Þ,
½5:90
therefore, only one spectral density function need be calculated. The relaxation mechanisms of interest in the present context arise from tensorial operators of rank k ¼ 2. The random functions F 02 ðtÞ are factored to give F 02 ðtÞ ¼ c0 ðtÞY 02 ½ðtÞ and, consequently, Z 1 0 0 jð!Þ ¼ Re c0 ðtÞc0 ðt þ ÞY 2 ½ðtÞY 2 ½ðt þ Þ expði!Þ d Z1 1 ¼ Re Cð Þ expði! Þ d ,
½5:91
½5:92
1
in which the stochastic correlation function is given by Cð Þ ¼ c0 ðtÞc0 ðt þ ÞY02 ½ðtÞY02 ½ðt þ Þ,
½5:93
where c0(t) is a function of physical constants and spatial variables, Y 02 ½ðtÞ is a modified second-order spherical harmonic function, and (t) ¼ { (t), (t)} are polar angles in the laboratory reference frame. The polar angles define the orientation of a unit vector that points in the principal direction for the interaction. For the dipolar interaction, the unit vector points along the line between the two nuclei (or between the nucleus and the electron for paramagnetic relaxation). For CSA interaction with an axially symmetric chemical shift tensor, the unit vector is collinear with the symmetry axis of the tensor. For the quadrupolar interaction, the unit vector is collinear with the symmetry axis of the electric field gradient tensor. The modified spherical harmonics are given in Table 5.1 (28). The functions c0(t) for dipolar, CSA, and quadrupolar interactions are given in Table 5.2. As a molecule rotates stochastically in solution due to Brownian motion, the oscillating magnetic fields produced are not distributed uniformly over all frequencies. A small organic molecule tumbles at a greater rate as compared to a biological macromolecule in the same solvent, and the
366
CHAPTER 5 RELAXATION
AND
DYNAMIC PROCESSES
TABLE 5.2 Spatial Functions for Relaxation Mechanisms Interaction
c(t) pffiffiffi h I S rIS ðtÞ3 6ð0 =4Þ pffiffiffi I B0 = 3
Dipolar CSAa Quadrupolarb
e2 qQ=½4 hIð2I 1Þ
a
The chemical shift tensor is assumed to be axially symmetric with principal values zz ¼ k , xx ¼ yy ¼ ?, and ¼ k – ?. b Q is the nuclear quadrupole moment and e is the charge of the electron. The electric field gradient tensor is assumed to be axially symmetric with the principal value of the field gradient defined by Vzz ¼ eq, and Vxx ¼ Vyy.
distribution of oscillating magnetic fields resulting from rotational diffusion of the two molecules will be different. The power spectral density function measures the contribution to orientational (rotational) dynamics of the molecule from motions with frequency components in the range ! to ! þ d!. For a rigid spherical molecule undergoing rotational Brownian motion, c0(t) ¼ c0 is a constant and the autospectral density function is jð!Þ ¼ d00 Jð!Þ, in which the orientational spectral density function is Z 1 C200 ðÞ expfi!g d , Jð!Þ ¼ Re
½5:94
½5:95
1
the orientational correlation function is C200 ðÞ ¼ Y 02 ½ðtÞY 02 ½ðt þ Þ,
½5:96
and d00 ¼ c20 . For isotropic rotational diffusion of a rigid rotor or spherical top, the correlation function is given by (10) C200 ðÞ ¼ 15 exp½= c ,
½5:97
in which the correlation time, c, is approximately the average time for the molecule to rotate by 1 radian. The correlation time varies due to molecular size, solvent viscosity, and temperature, but generally c is of the order of picoseconds for small molecules and of the order of
367
5.3 SPECTRAL DENSITY FUNCTIONS 5
J(w) × 10–9 (s)
4 3 2 1 0 100
102
104
106
108
1010
w (s–1)
FIGURE 5.6 Spectral density functions for an isotropic rotor. Calculations were performed using [5.98] with (—) c ¼ 2 ns and ( ) c ¼ 10 ns.
nanoseconds for biological macromolecules in aqueous solution (Section 1.4). The corresponding spectral density function is Jð!Þ ¼
2 c : 5 ð1 þ !2 2c Þ
½5:98
The functional form of the spectral density function for a rigid rotor is Lorentzian; a graph of J(!) versus ! is shown in Fig. 5.6. The logarithmic plot of J(!) is relatively constant for !2 2c 1 and then begins to decrease rapidly at !2 c2 1. If molecular motion is sufficiently rapid 2 q q to satisfy !q2 p c 1 for !p 6¼ 0, then Jð!p Þ Jð0Þ. This limit is called the extreme narrowing regime. For sufficiently slow molecular motion, q 2 q2 !q2 p c 1, then Jð!p Þ / !p . This limit is called the slow tumbling, or spin diffusion regime. Local fields are modulated stochastically by relative motions of nuclei in a molecular reference frame as well as by overall rotational Brownian motion. Rigorously for isotropic rotational diffusion and approximately for anisotropic rotational diffusion, the total correlation function is factored as (29) CðÞ ¼ CO ðÞCI ðÞ:
½5:99
The correlation function for overall motion, CO(), is given by [5.96] or [5.97]. The correlation function for internal motions, CI(), is given
368
CHAPTER 5 RELAXATION
AND
DYNAMIC PROCESSES
by [5.93], in which the orientational variables are defined in a fixed molecular reference frame, rather than the laboratory reference frame. Calculations of CI() have been performed for a number of diffusion and lattice jump models for internal motions. Rather than describing in detail calculations of spectral density functions for diffusion and jump models of intramolecular motions, two useful limiting cases of N-site models are given without proof [see Bru¨schweiler (17) for a more extensive review].The N-site lattice jump models assume that the nuclei of the relevant spins jump instantaneously between N allowed conformations. Therefore, the transition rates reflect the lifetimes of each conformation. The spectral density function depends upon the time scale of the variation in the spatial variables, c0(t). If the transition rates between sites approaches zero, then jð!Þ ¼ Jð!Þ
N X
pk c20k ¼ Jð!Þc20 ,
½5:100
k¼1
in which pk is the population and c0k is the value of the spatial function for site k. If the transition rates between sites approaches infinity, then
2
q X N 2
2 X X
q q jð!Þ ¼ Jð!Þ pk c0k Y 2 ðk Þ ¼ Jð!Þ
c0 Y 2 ðÞ , ½5:101
q¼2 k¼1 q¼2 in which k are the polar angles for site k. An extremely useful treatment that incorporates intramolecular motions in addition to overall rotational motion is provided by the Lipari–Szabo model free formalism (29, 30). In this treatment, the spectral density function is given by 2 S2 c ð1 S2 Þ jð!Þ ¼ c20 þ , ½5:102 5 1 þ ð! c Þ2 1 þ ð!Þ2 2 1 in which 1 ¼ 1 c þ e , S is the square of the generalized order parameter that characterizes the amplitude of intramolecular motions in a molecular reference frame, and e is the effective correlation time for internal motions. The order parameter is defined by
2
2 h i1 X
q S2 ¼ c20
c0 Y 2 ðÞ ,
½5:103
q¼2
in which the overbar indicates an ensemble average performed over the equilibrium distribution of orientations in the molecular reference
5.3 SPECTRAL DENSITY FUNCTIONS
369
frame. The order parameter satisfies the inequality, 0 S2 1, in which lower values indicate larger amplitudes of internal motions. A significant advantage of the Lipari–Szabo formalism is that specification of the microscopic motional model is not required. If e approaches infinity, [5.102] reduces to the same form as [5.100]; if e approaches zero, [5.102] reduces to the same form as [5.101]. Equation [5.102] has been used extensively to analyze spin relaxation in proteins (31, 32). The expressions given in [5.100], [5.101], and [5.102] are commonly encountered in discussions of dipolar relaxation between two spins, I and S. Using c0(t) from Table 5.2 gives jð!Þ ¼ Jð!Þr6 IS ,
½5:104
2 2 q
X
Y 2 ðk Þ jð!Þ ¼ Jð!Þ
3 ,
rIS q¼2
½5:105
2 6 S2 c ð1 S2 Þ þ jð!Þ ¼ rIS , 5 1 þ ð! c Þ2 1 þ ð!Þ2
½5:106
2 2 q
i1 X Y ð Þ
2 S ¼ r6
3 , IS
r IS q¼2
½5:107
2
h
in which ¼ 6½ð0 =4Þh I S 2 . Equation [5.104] (slow internal motion) is called ‘‘r–6 averaging’’ and [5.105] (fast internal motion) is called ‘‘r–3 averaging’’ with respect to the conformations of the molecule. The former equation is appropriate for treating the effects of aromatic ring flips and the latter equation is appropriate for treating methyl group rotations (33, 34). The spectral density function [5.100] can be modified to include cross-correlation between relaxation interactions with fixed relative orientations (35). The cross-spectral density function is given by n jmn ð!Þ ¼ cm 0 c0 P2 ðcos mn ÞJð!Þ,
½5:108
in which P2(x) ¼ (3x2 1)/2, and mn is the angle between the principal axes of the two interactions. The cross-spectral density function for the Lipari–Szabo model free formalism is given by P2 ðcos mn Þ S2mn 2 m n S2mn c jmn ð!Þ ¼ c0 c0 þ , ½5:109 5 1 þ ð! c Þ2 1 þ ð!Þ2
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CHAPTER 5 RELAXATION
AND
DYNAMIC PROCESSES
in which 2
n 1 X
m q
q S2mn ¼ cm c
c0 Y 2 ðm Þ
cn0 Y 2 ðn Þ : 0 0
½5:110
q¼2
Other expressions for j(!) have been derived for molecules that exhibit anisotropic rotational diffusion or specific internal motional models (10). Spin relaxation measurements have proved to be a powerful approach for experimental investigation of the rotational diffusion anisotropy of macromolecules (36–38).
5.4 Relaxation Mechanisms A very large number of physical interactions give rise to stochastic Hamiltonians capable of mediating spin relaxation. In the present context, only the intramolecular magnetic dipolar, anisotropic chemical shift, quadrupolar, and scalar coupling interactions will be discussed. Intramolecular paramagnetic relaxation has the same Hamiltonian as for nuclear dipolar relaxation, except that the interaction occurs between a nucleus and an unpaired electron. Other relaxation mechanisms are of minor importance for macromolecules or are only of interest in very specialized cases. For spin-1/2 nuclei in diamagnetic biological macromolecules, the dominant relaxation mechanisms are the magnetic dipolar and anisotropic chemical shift mechanisms. For nuclei with spin 4 1/2, notably 14N and 2H in proteins, the dominant relaxation mechanism is the quadrupolar interaction. Relaxation rate constants for nuclei in proteins depend upon a large number of factors, including overall rotational correlation times, internal motions, the geometrical arrangement of nuclei, and the relative strengths of the applicable relaxation mechanisms. If the overall correlation time and the three-dimensional structural coordinates of the protein are known, relaxation rate constants are calculated in a relatively straightforward manner using expressions derived in the following sections. In general, 1H relaxation in proteins is dominated by dipolar interactions with other 1H spins (within approximately 5 A˚) and by interactions with directly bonded heteronuclei. The latter arise from dipolar interactions with 13C and 15N in labeled proteins or from scalar relaxation of the second kind between the quadrupolar 14N nuclei and amide 1H nuclei. Relaxation of protonated 13C and 15N heteronuclei is dominated by dipolar interactions with the directly bonded 1H spins, and secondarily by CSA (for 15N spins and aromatic 13C spins). Relaxation
371
5.4 RELAXATION MECHANISMS
of unprotonated heteronuclei is dominated by CSA interactions and dipolar interactions with remote 1H spins.
5.4.1 INTRAMOLECULAR DIPOLAR RELAXATION SYSTEM
FOR
IS SPIN
Any nucleus with a nonzero spin angular momentum generates an instantaneous magnetic dipolar field that is proportional to the magnetic moment of the nucleus. As the molecule tumbles in solution, this field fluctuates and constitutes a mechanism for relaxation of nearby spins. Most importantly for structure elucidation, the efficacy of dipolar relaxation depends on the nuclear moments and on the inverse sixth power of the distance between the interacting nuclei. As a result, nuclear spin relaxation can be used to determine distances between nuclei. Hydrogen nuclei have a large magnetogyric ratio; therefore, dipole– dipole interactions cause the most efficient relaxation of 1H spins and constitute a sensitive probe for internuclear distances. Initially, a two-spin system, IS, will be considered with !I !S and scalar coupling constant JIS ¼ 0. The energy levels of the spin system and the associated transition frequencies are shown in Fig. 5.7. The terms Aq2p are given in Table 5.3. The spatial functions for the different interactions are given in Tables 5.1 and 5.2.
bb wS ba wI w I + wS w I – wS wI
ab wS aa
FIGURE 5.7 Transitions and associated eigenfrequencies for a two-spin system.
372
CHAPTER 5 RELAXATION
DYNAMIC PROCESSES
AND
TABLE 5.3 Tensor Operators for the Dipolar Interaction q
p
0
0
0
1
0
1
Aq2p pffiffiffi ð2= 6ÞIz Sz pffiffiffi 1=ð2 6ÞI S þ pffiffiffi 1=ð2 6ÞI þ S þ
1
0
(1/2) IzS
1
1
(1/2) IþSz
2
þ þ
0
(1/2) I S
q qy Aq 2p ¼ ð1Þ A2p pffiffiffi ð2= 6ÞIz Sz pffiffiffi 1=ð2 6ÞI þ S pffiffiffi 1=ð2 6ÞI S þ
!qp 0 !S – !I !I – !S
!S
(1/2) I Sz
!I
(1/2) IzS
(1/2) I S
!I þ !S
TABLE 5.4 Commutator Relationshipsa [Ix, Iy] ¼ iIz [I, 2IS ] ¼ 2[I, I]S [2IS , 2ISe] ¼ [I, I]" a
I ¼ Ix, Iy, or Iz; S ¼ Sx, Sy, or Sz. Equivalent expressions for S operators are obtained by exchanging I and S labels; " is the Kronecker delta.
The relaxation rate constants are calculated using [5.73]. To aid in the calculation of the double commutators, the commutation relations given in Table 5.4 are useful. To begin, the identity operator is disregarded because it has no effect on the relaxation equations. Next, the zero-order block consists of the operators with total coherence order equal to zero, Iz, Sz, 2IzSz, I þS , and I S þ, and has dimension 5 5. The operators with coherence order equal to 1 form a series of blocks of dimension 2 2: {I þ, 2I þSz}, {I , 2I Sz}, {Sþ, 2IzS þ}, and {S , 2IzS}. The operators with coherence order equal to 2 form a series of blocks of dimension 1 1: {I S } and {I þSþ}. Because of the secular approximation, the longitudinal operators Iz, Sz, and 2IzSz do not crossrelax with the zero-quantum operators I þS and I S þ. Dipolar crossrelaxation between the operators 2IzSz and Iz or between 2IzSz and Sz does not occur; therefore, the 2IzSz operator relaxes independently of the Iz and Sz operators. Similarly, dipolar cross-relaxation between
5.4 RELAXATION MECHANISMS
373
in-phase and antiphase operators, such as I þ, 2I þSz, does not occur. These results are obtained by direct calculation of the cross-relaxation rate constants or are anticipated using the symmetry of the dipole Hamiltonian and group theory arguments beyond the scope of this text (10, 18, 23). Cross-relaxation between these operators does arise due to interference between dipolar and CSA relaxation mechanisms (Section 5.5.4) (21). The relaxation matrix for the zero-order block for longitudinal magnetization has dimensions 2 2, with individual elements, rs, giving the rate constant for relaxation between operators Br and Bs for r, s ¼ {1, 2}, B1 ¼ Iz, and B2 ¼ Sz. The double commutators ½Aq 2p , ½Aq2p , Iz are calculated as follows for each combination of p and q in Table 5.3: h i 0 A20 , ½A020 , Iz ¼ ð2=3Þ Iz Sz , ½Iz Sz , Iz ¼ 0, h i 0 A21 , ½A021 , Iz ¼ ð1=24Þ I S þ , ½I þ S , Iz ¼ ð1=24Þ½I S þ , I þ S ¼ ð1=24ÞfIz Sz g, h i 0 A21 , ½A021 , Iz ¼ ð1=24Þ I þ S , ½I S þ , Iz ¼ ð1=24Þ½I þ S , I S þ ¼ ð1=24ÞfIz Sz g, 1 1 A20 , ½A20 , Iz ¼ ð1=4Þ Iz S , ½Iz S þ , Iz ¼ 0, 1 þ ½5:111 A20 , ½A1 20 , Iz ¼ ð1=4Þ Iz S , ½Iz S , Iz ¼ 0, 1 1 A21 , ½A21 , Iz ¼ ð1=4Þ I Sz , ½I þ Sz , Iz ¼ ð1=4ÞS2z I , I þ ¼ ð1=8ÞIz , 1 þ 2 þ A21 , ½A1 21 , Iz ¼ ð1=4Þ I Sz , ½I Sz , Iz ¼ ð1=24ÞSz I , I ¼ ð1=8ÞIz , 2 2 A20 , ½A20 , Iz ¼ ð1=4Þ I S , ½I þ S þ , Iz ¼ ð1=4Þ I S , I þ S þ ¼ ð1=4ÞfSz þ Iz g, þ þ 2 þ þ A20 , ½A2 20 , Iz ¼ ð1=4Þ I S , ½I S , Iz ¼ ð1=4Þ I S , I S ¼ ð1=4ÞfSz þ Iz g:
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CHAPTER 5 RELAXATION
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DYNAMIC PROCESSES
For autorelaxation of the Iz operator, the preceding operators are premultiplied by Iz and the trace operation is performed:
ð1=24ÞhIz jfIz Sz gi ¼ ð1=24Þ I2z Iz Sz
¼ ð1=24Þ jI2z Iz Sz j þ jI2z Iz Sz j
þ jI2z Iz Sz j þ jI2z Iz Sz j ¼ 1=24,
ð1=8ÞhIz jIz i ¼ ð1=8Þ I2z
¼ ð1=8Þ jI2z j þ jI2z j
þ jI2z j þ jI2z j
½5:112
¼ 1=8,
ð1=4ÞhIz jfSz þ Iz gi ¼ ð1=4Þ I2z þ Iz Sz
¼ ð1=4Þ jI2z þ Iz Sz j þ jI2z þ Iz Sz j
þ jI2z þ Iz Sz j þ jI2z þ Iz Sz j ¼ 1=4: For cross-relaxation between the Sz and the Iz operator, the operators of [5.112] are premultiplied by Sz and the trace operation is performed:
ð1=24ÞhSz jfIz Sz gi ¼ ð1=24Þ Iz Sz S2z
¼ ð1=24Þ jIz Sz S2z j þ jIz Sz S2z j
þ jIz Sz S2z j þ jIz Sz S2z j ¼ 1=24, ð1=8ÞhSz jIz i ¼ ð1=8ÞhIz Sz i
½5:113 ¼ ð1=8Þ hjIz Sz ji þ jIz Sz j
þ jIz Sz j þ jIz Sz j ¼ 0,
ð1=4ÞhSz jfSz þ Iz gi ¼ ð1=4Þ S2z þ Iz Sz
¼ ð1=4Þ jS2z þ Iz Sz j þ jS2z þ Iz Sz j
þ jS2z þ Iz Sz j þ jS2z þ Iz Sz j ¼ 1=4:
5.4 RELAXATION MECHANISMS
375
Auto- and cross-relaxation rate constants of the Sz operator are obtained by exchanging I and S operators in the preceding expressions. Substituting the values of these trace operations into [5.73] (and using hIz|Izi ¼ 1) yields 11 ¼ ð1=24Þ jð!I !S Þ þ 3jð!I Þ þ 6jð!I þ !S Þ , 22 ¼ ð1=24Þ jð!I !S Þ þ 3jð!S Þ þ 6jð!I þ !S Þ , ½5:114 12 ¼ ð1=24Þ jð!I !S Þ þ 6jð!I þ !S Þ : If the I and S spins are separated by a constant distance, rIS, then, 11 ¼ ðd00 =4Þ Jð!I !S Þ þ 3Jð!I Þ þ 6Jð!I þ !S Þ , 22 ¼ ðd00 =4Þ Jð!I !S Þ þ 3Jð!S Þ þ 6Jð!I þ !S Þ , ½5:115 12 ¼ ðd00 =4Þ Jð!I !S Þ þ 6Jð!I þ !S Þ , in which h2 2I 2S r6 d00 ¼ ð0 =4Þ2 IS :
½5:116
Thus, the evolution of the longitudinal operators, Iz and Sz, is governed by d hIz iðtÞ I 0z =dt ¼ 11 hIz iðtÞ I 0z 12 hSz iðtÞ S 0z , d hSz iðtÞ S 0z =dt ¼ 22 hSz iðtÞ S 0z 12 hIz iðtÞ I 0z : ½5:117 Making the identification 11 ¼ I (¼ R1I), 22 ¼ S (¼ R1S), and 12 ¼ IS puts [5.117] into the form of the Solomon equations [5.11], in which I and S are the autorelaxation rate constants and IS is the cross-relaxation rate constant. The Solomon transition rate constants (Section 5.1.2) are W0 ¼ jð!I !S Þ=24, WI ¼ jð!I Þ=16, WS ¼ jð!S Þ=16,
½5:118
W2 ¼ jð!I þ !S Þ=4: Next, relaxation of the transverse I þ operator is considered. The q þ double commutators ½Aq 2p , ½A2p , I are calculated as follows for each
376
CHAPTER 5 RELAXATION
AND
DYNAMIC PROCESSES
combination of p and q in Table 5.3:
A020 , A020 ,I þ ¼ ð2=3Þ Iz Sz , Iz Sz ,I þ ¼ ð1=6ÞI þ , 0 0 þ A21 , A21 ,I ¼ ð1=24Þ I S þ , I þ S ,I þ ¼ 0, 0 0 A21 , A21 ,I þ ¼ ð1=24Þ I þ S , I S þ ,I þ ¼ ð1=12Þ I þ S , Iz S þ ¼ ð1=24ÞI þ , 1 1 þ ¼ ð1=4Þ Iz S , Iz S þ ,I þ ¼ ð1=8Þ Iz S , I þ S þ A20 , A20 ,I ¼ ð1=8ÞI þ , ½5:119 1 1 þ þ þ þ þ A20 , A20 ,I ¼ ð1=4Þ Iz S , Iz S ,I ¼ ð1=8Þ Iz S , I S ¼ ð1=8ÞI þ , 1 1 þ A21 , A21 ,I ¼ ð1=4Þ I Sz , I þ Sz ,I þ ¼ 0, 1 1 þ ¼ ð1=4Þ I þ Sz , I Sz ,Iþ ¼ ð1=4ÞS2z I þ , Iz ¼ ð1=8ÞI þ , A21 , A21 ,I 2 2 þ A20 , A20 ,I ¼ ð1=4Þ I S , I þ S þ ,Iþ ¼ 0, 2 2 þ A20 , A20 ,I ¼ ð1=4Þ I þ S þ , I S ,Iþ ¼ ð1=4Þ I þ S þ , Iz S ¼ ð1=4ÞI þ : All nonzero results are proportional to I þ; therefore, because the operator basis is orthogonal, no operator cross-relaxes with I þ. This result is a consequence of the secular approximation. For autorelaxation of the I þ operator, the operators in [5.119] are premultiplied by I þ and the trace operation is performed:
þ þ þ I jI ¼ I I ¼ jI I þ j þ jI I þ j
½5:120 þ jI I þ j þ jI I þ j ¼ 2: This same factor is the normalization in the denominator of [5.73]. Thus, R2I ¼ ð1=48Þ 4jð0Þ þ jð!I !S Þ þ 3jð!I Þ þ 6jð!S Þ þ 6jð!I þ !S Þ ½5:121
377
5.4 RELAXATION MECHANISMS
TABLE 5.5 Relaxation Rate Constants for IS Dipolar Interaction Coherence level
Operator
Populations Iz 0
Relaxation rate constanta
Iz
(d00/4) {J(!I !S) þ 3J(!I) þ 6J(!I þ !S)}
Sz
(d00/4) {J(!I !S) þ 3J(!S) þ 6J(!I þ !S)}
! Szb
(d00/4) {J(!I !S) þ 6J(!I þ !S)}
2IzSz
(3d00/4) {J(!I) þ J(!S)}
ZQx, ZQy (d00/8) {2J(!I !S) þ 3J(!I) þ 3J(!S)}
1
I þ, I
(d00/8) {4J(0) þ J(!I !S) þ 3J(!I) þ 6J(!S) þ 6J(!I þ !S)}
S þ, S
(d00/8) {4J(0) þ J(!I !S) þ 3J(!S) þ 6J(!I) þ 6J(!I þ !S)}
2I þSz, 2 I Sz (d00/8) {4J(0) þ J(!I !S) þ 3J(!I) þ 6J(!I þ !S)} 2IzS þ, 2IzS (d00/8) {4J(0) þ J(!I !S) þ 3J(!S) þ 6J(!I þ !S)} 2
DQx, DQy (d00/8) {3J(!I) þ 3J(!S) þ 12J(!I þ !S)}
d00 ¼ ð0 =4Þ2 h2 I2 S2 r6 IS : Cross-relaxation only occurs between Iz and Sz.
a b
and
d I þ =dt ¼ ði!I R2I Þ I þ :
½5:122
If rIS is constant, R2I ¼ ðd00 =8Þ 4Jð0Þ þ Jð!I !S Þ þ 3Jð!I Þ þ 6Jð!S Þ þ 6Jð!I þ !S Þ : ½5:123 Analogous equations are written by inspection for the I , S þ, and S operators. The complete set of dipolar relaxation rate constants for the basis operators for the two spin system is given in Table 5.5. The dependence of R1 and R2 on c for a rigid molecule is illustrated in Fig. 5.8. Notice that R1 has a maximum for !0 c ¼ 1 while R2 increases monotonically with c.
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CHAPTER 5 RELAXATION
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DYNAMIC PROCESSES
10 8
Ri (s–1)
6 4 2
0
1
2
3
|wNtc|
FIGURE 5.8 Relaxation rate constants for an 1H–15N dipolar spin system. (—) 15 N R1 spin–lattice rate constants; ( ) 15N R2 spin–spin rate constants. Calculations were performed using expressions given in Table 5.5 together with [5.116] and [5.98]. Parameters used were B0 ¼ 11.74 T, I ¼ 2.675 108 T–1 s–1 (1H), S ¼ –2.712 107 T1 s1 (15N), and rIS ¼ 1.02 A˚.
5.4.2 INTRAMOLECULAR DIPOLAR RELAXATION COUPLED IS SPIN SYSTEM
FOR
SCALAR-
The Iz and Sz operators both commute with the scalar coupling Hamiltonian; consequently, dipolar spin–lattice relaxation is unaffected by the scalar coupling interaction and the expressions given in [5.114] and [5.117] remain valid. The in-phase and antiphase transverse operators, I þ and 2I þSz, are coupled together by the scalar coupling Hamiltonian. Applying [5.71] yields the following equations (assuming the I spin is on-resonance):
d I þ ðtÞ=dt ¼ iJIS 2I þ Sz ðtÞ R2I I þ ðtÞ,
½5:124 d 2I þ Sz ðtÞ=dt ¼ iJIS I þ ðtÞ R2IS 2I þ Sz ðtÞ, in which R2I and R2IS are given in Table 5.5 and !I ¼ 0 is assumed for simplicity. These equations are written in matrix form as " #" þ
" þ
# # I ðtÞ I ðtÞ R2I iJIS d ¼ ½5:125
þ
dt 2I þ Sz ðtÞ 2I Sz ðtÞ iJIS R2IS
5.4 RELAXATION MECHANISMS
379
and are solved by analogy to [5.15] to yield þ
1 R2I R2IS 1 expð tÞ I ðtÞ ¼ 2 ð þ Þ
R2I R2IS þ 1þ expð þ tÞ I þ ð0Þ ð þ Þ
iJIS ½expð tÞ expð þ tÞ 2I þ Sz ð0Þ, ð Þ þ ½5:126 þ
1 R2I R2IS 1þ expð tÞ 2I Sz ðtÞ ¼ 2 ð þ Þ
R2I R2IS þ 1 expð þ tÞ 2I þ Sz ð0Þ ð þ Þ
iJIS ½expð tÞ expð þ tÞ I þ ð0Þ, ð þ Þ in which n 1=2 o :
¼ ðR2I þ R2IS Þ=2 ððR2I R2IS Þ=2Þ2 ðJIS Þ2 If (2JIS)2 (R2I R2IS)2, then þ
I ðtÞ ¼ 12 exp iJIS þ R2 t
þ exp iJIS þ R2 t I þ ð0Þ exp iJIS þ R2 t
exp iJIS þ R2 t 2I þ Sz ð0Þ, þ
2I Sz ðtÞ ¼ 12 exp iJIS þ R2 t
þ exp iJIS þ R2 t 2I þ Sz ð0Þ exp iJIS þ R2 t
exp iJIS þ R2 t I þ ð0Þ,
½5:127
½5:128
with R2 ¼ ðR2I þ R2IS Þ=2 ¼ ðd00 =8Þ 4Jð0Þ þ Jð!I !S Þ þ 3Jð!S Þ þ 3Jð!I Þ þ 6Jð!I þ !S Þ : ½5:129
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CHAPTER 5 RELAXATION
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DYNAMIC PROCESSES
Equation [5.128] predicts that the signal arising from I þ has the expected form of a doublet with linewidth R2 =. The doublet is in-phase if h2I þSzi(0) ¼ 0 and antiphase if hI þi(0) ¼ 0. Evolution of the scalar coupling interaction on a faster time scale compared to the relaxation processes averages the two relaxation rate constants, R2I and R2IS, because coherence is rapidly exchanged between the I þ and 2I þSz operators. An alternative viewpoint, consistent with the presentation of Section 5.2.2, is obtained by transforming to a new operator basis using the following transformation matrix, 1 1 1 U ¼ pffiffiffi , ½5:130 2 1 1 to give " þ
# " þ
# " # I ðtÞ I ðtÞ R2I iJIS d 1 ¼ U , U U U
dt 2I þ Sz ðtÞ iJIS R2IS 2Iþ Sz ðtÞ þ # " þ
1 d I ðtÞ þ 2I Sz ðtÞ pffiffiffi ½5:131
2 dt Iþ ðtÞ 2Iþ Sz ðtÞ
# " #" þ
I ðtÞ þ 2Iþ Sz ðtÞ ðR2I R2IS Þ=2 iJIS þ R2 1 , ¼ pffiffiffi þ
2 ðR2I R2IS Þ=2 iJIS þ R2 I ðtÞ 2Iþ Sz ðtÞ " þ # " #" þ # I S ðtÞ ðR2I R2IS Þ=2 iJIS þ R2 d I S ðtÞ ¼ : þ
þ
dt I S ðtÞ I S ðtÞ ðR2I R2IS Þ=2 iJIS þ R2 In the new basis, which is the eigenbasis of the scalar coupling Hamiltonian, all precession terms are diagonal elements of the Hamiltonian. The term h(I þS i(t) ¼ h(I þ þ 2I þSz)/2i(t) represents the multiplet component of the scalar-coupled doublet with frequency J/2 Hz and the term h(I þSi(t) ¼ h(I þ 2I þSz)/2i(t) represents the multiplet component of the scalar-coupled doublet with frequency J/2 Hz. When (2JIS)2 (R2I R2IS)2, the off-diagonal elements can be neglected and the two multiplet components relax independently with identical relaxation rate constants. In this limit, the doublet components are nonsecular with respect to each other and cross-relaxation is quenched. The off-diagonal terms in [5.131] illustrate the effect of the breakdown of the secular approximation as JIS and (R2I R2IS)/2 become comparable.
381
5.4 RELAXATION MECHANISMS
For the purely dipolar IS interaction in the spin diffusion limit, R2I R2IS ¼ 3d00 Jð!S Þ=4 ¼
320 h2 2I 1602 B20 r6IS c
½5:132
normally is quite small. For example, if I ¼ 15N, S ¼ 1HN, and c ¼ 5 ns, then R2I R2IS ¼ 0.031 s–1, compared with JIS ¼ 92 Hz. However, the Sz operator may have relaxation pathways other than the IS dipolar interaction. In the cited example, the Sz operator would be dipolar coupled to other 1H spins, and the relaxation rate constant for the 2IþSz operator contains a contribution, Rext, from 1H dipolar longitudinal relaxation. Ignoring cross-correlation and cross-relaxation effects, Rext is simply additive to R2IS. The additional contribution from Rext has two important effects. First, R2 is increased by Rext/2. Practical consequences of the increased linewidth in heteronuclear NMR spectra are discussed in Section 7.1.2.4. Second, if Rext is sufficiently large, then ðR2I R2IS Rext Þ2 R2ext ð2JIS Þ2 , þ ¼ R2I, – ¼ R2IS þ Rext, and [5.126] reduces to þ
I ðtÞ ¼ Iþ ð0Þ exp½R2I t, þ
½5:133 2I Sz ðtÞ ¼ 2Iþ Sz ð0Þ exp½ðR2IS þ Rext Þt: The expected doublet has been reduced to a singlet resonance in a process called self-decoupling, which is similar both to scalar relaxation of the second kind (Section 5.4.5) and to chemical exchange (Section 5.6.2). For (R2I R2IS Rext)2 (2JIS)2, the doublet is partially decoupled and broadened, as for intermediate chemical exchange (Section 5.6.1). Self-decoupling can complicate the measurement of scalar coupling constants (Section 6.2.1.5, 6.3.3, and 7.5) (39). A similar set of equations is obtained for the Sþ and 2SþIz coherences by interchanging the I and S labels. For an uncoupled IS spin system, R2I 6¼ R2S, but for a scalar-coupled spin system undergoing free precession, R2 is identical for the I and S spins.
5.4.3 INTRAMOLECULAR DIPOLAR RELAXATION SYSTEM IN THE ROTATING FRAME
FOR
IS SPIN
An IS homonuclear spin system, in which the two spins interact through the dipolar interaction, but are not scalar coupled, is treated here. The spin lock field is assumed to be applied with x-phase. The autorelaxation rate constant of the I z0 operator and the cross-relaxation rate constant between the I z0 and S z0 operators are calculated in the
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tilted rotating frame. As discussed in Section 5.2.3, in the presence of the spin lock field, the I and S spins are treated as like spins (alternatively, the relaxation rate constants are calculated using [5.84] without invoking the secular hypothesis); thus, the components of the dipolar interaction listed in Table 5.3 are redefined according to [5.78] as A02 ¼ A020 þ A021 þ A021 , 1 1 A1 2 ¼ A20 þ A21 ,
2 A2 2 ¼ A20 :
½5:134
From [5.89], I 0z ¼ sin I Ix þ cos I Iz , S0z ¼ sin S Sx þ cos S Sz :
½5:135
q 0 Applying [5.88], the double commutators ½Aq 2 , ½A2 , Iz are calculated first. Straightforward, but tedious, calculations yield
½A02 ,½A02 , Iz0 ¼ sin I ð5Ix þ 4Sx Þ=24 þ cos I ðIz Sz Þ=6, 1 1 0 1 0 ½A1 2 , ½A2 , Iz ¼ A2 A2 , Iz ¼ sin I ð2Ix þ 2Sx þ 2I Þ=8 cos I Iz =8, 2 2 0 0 ½A2 , ½A2 , Iz ¼ A22 , A2 2 , Iz ¼ sin I I =8 cos I ðIz þ Sz Þ=8:
½5:136
The autorelaxation rate constant is determined by premultiplying the preceding expressions by I z0 and forming the trace:
sin I Ix þ cos I Iz j sin I ð5Ix þ 4Sx Þ=24 þ cos I ðIz Sz Þ=6
¼ ð5=24Þ sin2 I þ ð1=12Þ cos2 I ,
sin I Ix þ cos I Iz j sin I ð2Ix þ 2Sx þ 2I Þ=8 cos I Iz =8 ¼ ð3=16Þ sin2 I ð1=8Þ cos2 I ,
½5:137
sin I Ix þ cos I Iz j sin I I =8 cos I ðIz þ Sz Þ=8 ¼ ð1=8Þ sin2 I þ ð1=4Þ cos2 I :
Thus, the autorelaxation rate, R1( I) (which commonly is called R1 ) is given by RI ð I Þ ¼ ð1=48Þ 2 cos2 I þ 5 sin2 I jð0Þ þ 6 cos2 I þ 9 sin2 I jð!0 Þ þ 12 cos2 I þ 6 sin2 I jð2!0 Þg ¼ R1I cos2 I þ R2I sin2 I : ½5:138
383
5.4 RELAXATION MECHANISMS
Similarly, the cross-relaxation rate constant is found by premultiplying the expressions in [5.136] by Sz0 and forming the trace:
sin S Sx þ cos S Sz j sin I ð5Ix þ 4Sx Þ=24 þ cos I ðIz Sz Þ=6 ¼ ð1=6Þ sin S sin I ð1=12Þ cos S cos I ,
sin S Sx þ cos S Sz j sin I ð2Ix þ 2Sx þ 2I Þ=8 cos I Iz =8 ½5:139 ¼ ð1=8Þ sin S sin I ,
sin S Sx þ cos S Sz j sin I I =8 cos I ðIz þ Sz Þ=8 ¼ ð1=4Þ cos S cos I : Thus, the cross-relaxation rate, RIS( I, S) is given by RIS ð I , S Þ ¼ ð1=24Þfð cos S cos I þ 2 sin S sin I Þjð0Þ þ 3 sin S sin I jð!0 Þ þ 6 cos S cos I jð2!0 Þg þ sin I sin S ROE ¼ cos I cos S NOE IS IS ,
½5:140
NOE in which the pure laboratory-frame cross-relaxation rate constant, IS , is given in [5.114] and the pure rotating-frame cross-relaxation rate constant is given by (40)
ROE ¼ ð1=24Þf2jð0Þ þ 3jð!0 Þg: IS
½5:141
For both autorelaxation and cross-relaxation, the effect of the tilted field is to average the relaxation rate constants of the laboratory (longitudinal) and rotating frames (transverse) by the projection of the spin operators onto the tilted reference frame.
5.4.4 CHEMICAL SHIFT ANISOTROPY RELAXATION
AND
QUADRUPOLAR
Chemical shifts are reflections of the electronic environments that modify the local magnetic fields experienced by different nuclei (Section 1.5). These local fields are anisotropic; consequently, the components of the local fields in the laboratory reference frame vary as the molecule reorients due to molecular motion. These varying magnetic fields are a source of relaxation. Very approximately, the maximum CSA for a particular nucleus is of the order of the chemical shift range for the nucleus. Consequently, CSA is most important as a relaxation mechanism for nuclei with large chemical shift ranges. In the NMR spectroscopy of biological molecules, carbon, nitrogen, and phosphorous have significant CSA contributions to relaxation. CSA is generally a
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small effect for 1H relaxation, except at very large static magnetic field strengths. CSA relaxation rate constants have a quadratic dependence on the applied magnetic field strength. Thus, use of higher magnetic field strengths does not always increase the achievable signal-to-noise ratio as much as anticipated, because increased CSA relaxation broadens resonance linewidths. This effect is particularly significant for 13C spins in carbonyl groups and for 31P. Nuclei with I 4 1/2 also possess nuclear electric quadrupole moments. The quadrupole moment is a characteristic of the particular nucleus and represents a departure of the nuclear charge distribution from spherical symmetry. The interactions of the quadrupole moment with local oscillating electric field gradients (due to electrons) provide a relaxation mechanism. Quadrupolar interactions can be very large and efficient for promoting relaxation. Quadrupolar nuclei display broad resonance lines in NMR spectra, unless the nuclei are in a highly symmetric electronic environment (which reduces the magnitudes of the electric field gradients at the locations of the nuclei). As discussed in more detail elsewhere, Bloch spin–lattice and spin–spin relaxation rate constants can only be defined for quadrupolar nuclei under extreme narrowing conditions or for quadrupolar nuclei with I ¼ 1 (2). The terms Aq2p for the CSA and quadrupolar interactions are given in Tables 5.6 and 5.7, respectively. The spherical harmonic and spatial functions for the different interactions are given in Tables 5.1 and 5.2. Relaxation rate constants for the CSA and quadrupolar interactions are calculated by the same procedure as for the dipolar interactions and are given in Tables 5.8 and 5.9, respectively. The results are calculated for axially symmetric chemical shift and electric field gradient tensors (i.e., xx ¼ yy 6¼ zz and Vxx ¼ Vyy 6¼ Vzz). Extensions to these results for anisotropic tensors are given elsewhere (2).
TABLE 5.6 Tensor Operators for the CSA Interaction q
p
0
0
1
0
2
0
Aq2p pffiffiffi ð2= 6ÞIz (1/2) I —
þ
q qy Aq 2p ¼ ð1Þ A2p pffiffiffi ð2= 6ÞIz
(1/2) I —
!qp 0 !I 2!I
385
5.4 RELAXATION MECHANISMS
TABLE 5.7 Tensor Operators for the Spin-1 Quadrupolar Interaction
0
0
Aq2p pffiffiffi 2 1= 6 3I z 2
1
0
(1/2) (IzI þ þ I þIz)
(1/2) (IzI þ I Iz)
!I
2
0
(1/2) I þI þ
(1/2) I I
2!I
q
p
q qy Aq 2p ¼ ð1Þ A2p pffiffiffi 2 1= 6 3I z 2
!qp 0
TABLE 5.8 CSA Relaxation Rate Constants Coherence level
Relaxation rate constanta
Operator
Populations 1
d00 J(!I) (d00/6) {4J(0) þ 3J(!I)}
Iz I þ, I
d00 ¼ ðgI B0 Þ2 =3 ¼ 2 !2I =3:
a
TABLE 5.9 Relaxation Rate Constants for the Spin-1 Quadrupolar Interaction Coherence level
Relaxation rate constanta
Operator
Populations
Iz
0
2
3d00 {J(!I) þ 4J(2!I)}
3Iz – 2 þ
þ
I ,I
1 þ
I Iz þ 2IzI , I Iz þ 2IzI
9d00 J(!I) (3d00/2) {3J(0) þ 5J(!I) þ 2J(2!I)}
I þI þ, I I
2
(3d00/2) {3J(0) þ J(!I) þ 2J(2!I)} 3d00 {J(!I) þ 2J(2!I)}
2 d00 ¼ e2 qQ=ð4 hÞ :
a
5.4.5 RELAXATION INTERFERENCE As discussed in Section 5.2.1, correlations between two stochastic Hamiltonians results in cross-correlated relaxation or relaxation interference. The principal cause of the correlation between the Hamiltonians is that the same molecular motions affect each Hamiltonian. The interference between the dipolar and chemical shift anisotropy relaxation
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mechanisms has been recognized for many years [see Goldman (21) and references therein]. The hallmark of this phenomenon is that the two lines in a scalar-coupled doublet have different linewidths. Dipole–CSA relaxation interference has long been regarded as a curiosity or a complication to be suppressed experimentally (41–43). In the past few years, a number of new applications have emerged that make use of relaxation interference for line narrowing in heteronuclear correlation spectroscopy (44), for measuring chemical shift anisotropies in solution (45, 46), and for investigating dynamic properties of macromolecules (45). The present discussion is based on the treatment of Goldman (21). A two-spin system is considered. The I spin has a dipolar relaxation interaction with the S spin and has a significant chemical shift anisotropy. The evolution of longitudinal components of the density operator is given by 0
0
dhIz iðtÞ CSA ¼ ðRDD 1I þ R1I ÞðhIz iðtÞ Iz Þ IS ðhSz iðtÞ Sz Þ dt z h2Iz Sz iðtÞ, 0
0
dhSz iðtÞ ¼ RDD 1S ðhSz iðtÞ Sz Þ IS ðhIz iðtÞ Iz Þ, dt 0
dh2Iz Sz iðtÞ CSA ¼ ðRDD 1IS þ R1I Þh2Iz Sz iðtÞ z ðhIz iðtÞ Iz Þ: dt ½5:142 The evolution of the transverse components of the density operator is given by
þ
d Iþ ðtÞ CSA þ ¼ iJIS 2Iþ Sz ðtÞ ðRDD 2I þ R2I Þ I ðtÞ xy 2I Sz ðtÞ dt
þ
d 2Iþ Sz ðtÞ CSA ¼ iJIS Iþ ðtÞ ðRDD ½5:143 2IS þ R2I Þ 2I Sz ðtÞ dt þ
xy I ðtÞ, in which the superscripts refer to the dipole–dipole (DD) and CSA relaxation mechanisms, respectively. All terms other than z and xy are relaxation rate constants determined earlier. If z ¼ 0, the first two equations in [5.142] are simply the Solomon equations. For an axially symmetric chemical shift tensor and a rigid spherical molecule, the interference rate constants are given by pffiffiffi
z ¼ 3cdP2 ðcos ÞJð!I Þ, ½5:144
387
5.4 RELAXATION MECHANISMS
xy
pffiffiffi 3 ¼ cdP2 ðcos Þ 4Jð0Þ þ 3Jð!I Þ , 6
½5:145
pffiffiffi in which d ¼ ð0 h I S Þ=ð4r3IS Þ and c ¼ I ð jj ? ÞB0 = 3 are obtained from Table 5.2, and is the angle between the symmetry axis of the CSA principal axis system and the vector connecting the I and S spins. These results are derived using [5.76]. The meaning of [5.143] is seen more easily by transforming using [5.130] to yield DD " #" þ # " þ # R2I RDD iJIS þ R2 þ xy I S ðtÞ 2IS =2 d I S ðtÞ ¼ , þ
þ
DD DD dt I S ðtÞ I S ðtÞ R2I R2IS =2 iJIS þ R2 xy ½5:146 DD CSA in which R2 ¼ ðRDD 2I þ R2IS Þ=2 þ R2I . As discussed in Section 5.4.2, the off-diagonal terms are unimportant provided that ð2JIS 2 Þ DD 2 ðRDD 2I R2IS Þ . These terms represent cross-relaxation between the two doublet components and can become important in multipulse experiments that suppress the effects of the scalar coupling interaction. The effect of relaxation interference is that the interference term adds to the relaxation rate (and hence the linewidth) of the doublet component with frequency JIS/2 Hz and subtracts from the relaxation rate (and hence the linewidth) of the doublet component with frequency –JIS/2 Hz. This result is the basis of the TROSY technique for line narrowing at high static magnetic field strengths, for which the magnitude of xy can approach R2 due to the field dependence of the chemical shift anisotropy (44).
5.4.6 SCALAR RELAXATION As discussed in Sections 1.6 and 2.5.2, the isotropic scalar coupling Hamiltonian, HJ ¼ 2JIS I S, slightly perturbs the Zeeman energy levels of the coupled spins; the resonances thereby are split into characteristic multiplet patterns. Spin I experiences a local magnetic field that depends on the value of the coupling constant and the state of spin S. The local magnetic field becomes time dependent if the value of JIS is time dependent or if state of the S spin varies rapidly. The former relaxation mechanism is termed scalar relaxation of the first kind; the latter mechanism is termed scalar relaxation of the second kind. Scalar relaxation of the first kind results from transitions of the spin system between environments with different values of JIS. For example, the
388
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AND
DYNAMIC PROCESSES
three-bond scalar coupling constant for a pair of 1H spins depends upon the intervening dihedral angle according to the Karplus relationship (see Chapter 9, [9.2]). If the dihedral angle is time dependent, the consequent time dependence of JIS can lead to scalar relaxation. Scalar relaxation of the second kind results if the S spin relaxes rapidly (e.g., S is a quadrupolar nucleus) or is involved in rapid chemical exchange. Scalar relaxation of the second kind also can be significant for 1H S spins in macromolecules, in which case the homonuclear relaxation rate constants (reflecting the dipolar interaction of the S spins with 1H spins other than the I spin) are large. Normally, field fluctuations produced by this mechanism are not fast enough for effective longitudinal relaxation, but transverse relaxation may be induced (Section 5.4.2). sc For relaxation of the I spin, expressions for Rsc 1 and R2 are given by (2) 2A2 2 SðS þ 1Þ , Rsc 1 ¼ 3 1 þ ð!I !S Þ2 22 " # ½5:147 A2 2 sc SðS þ 1Þ þ 1 , R2 ¼ 3 1 þ ð!I !S Þ2 22 assuming that the scalar relaxation mechanism is fast enough to result in a single, broadened resonance, rather than a resolved multiplet. For scalar relaxation of the first kind, A ¼ 2( p1p2)1/2(J1J2), in which J1 and J2 are the scalar coupling constants in the two environments; p1 and p2 are the site populations, 1 ¼ 2 ¼ e, and e is the exchange time constant for transitions between the two environments. For scalar relaxation of the second kind, A ¼ 2JIS, and 1 and 2 are the spin– lattice and spin–spin relaxation time constants for the S spin, respectively. If the S spin is a quadrupolar nucleus, then the relaxation time constants are calculated using the expressions given in Table 5.9. The secular contribution to Rsc 2 can be calculated using the random-phase model for transverse relaxation (Section 5.1.3). A more general treatment of scalar relaxation applicable to all time scales has been given by London (47).
5.5 Nuclear Overhauser Effect The nuclear Overhauser effect (NOE) is a manifestation of the prediction [5.117] that dipolar-coupled spins do not relax independently. The NOE is without doubt one of the most important effects in NMR spectroscopy and more detailed discussions are found in monographs
389
5.5 NUCLEAR OVERHAUSER EFFECT
devoted to the subject (48, 49). The Solomon equations ([5.11]) are extremely useful for explication of NOE experiments. The NOE is characterized by the cross-relaxation rate constant, NOE IS , defined by [5.114], or the steady-state NOE enhancement, IS, which is defined in the following discussion. These two quantities naturally arise in transient or steady-state NOE experiments, respectively. The steady-state NOE experiment is illustrated here by using a dipole-coupled two-spin system as an example. If the S spin is irradiated by a weak rf field (so as not to perturb the I spin) for a period of time t 1/ S, 1/ I, then the average populations across the S spin transitions are equalized and the I spin magnetization evolves to a steady-state value, hIzss i. In this situation, the S spins are said to be saturated. Setting 0 dIz ðtÞ=dt ¼ 0 and hSzi(t) ¼ 0 in [5.11] and solving for hI ss z i=hI z i yields
0
d I ss z NOE 0 ¼ I I ss Sz ¼ 0, z I z þ IS ½5:148 dt ss 0
0 0 I z = I z ¼ 1 þ NOE = : S I I z IS z Using hSz0 i=hIz0 i ¼ S = I yields
0
NOE IS S I ss ¼ 1 þ IS , z = Iz ¼ 1 þ I I
½5:149
in which
IS
NOE IS S : I I
½5:150
As shown, the value of the longitudinal magnetization (or population difference) for the I spin is altered by saturating (equalizing the population difference) the S spin. If IS is positive, then the population differences across the I spin transitions are increased by reducing the population differences across the S spin transitions. Because the equilibrium population differences are inversely proportional to temperature, this result appears to indicate that heating the S spins (reducing the population difference) has the effect of cooling the I spins (increasing the population difference). This conclusion would appear to violate the Second Law of Thermodynamics; however, if coupling between the spin system and the lattice is properly taken into account, then no inconsistency with thermodynamics exists. The value of the NOE enhancement, IS, is measured using the steady-state NOE difference experiment. In this experiment, two spectra are recorded. In the first spectrum, the S spin is saturated for a period of
390
CHAPTER 5 RELAXATION
AND
DYNAMIC PROCESSES
time sufficient to establish the NOE enhancement of the I spin, a 908 pulse is applied to the system, and the FID is recorded. The intensity of
. In the the I spin resonance in the spectrum is proportional to I ss z second experiment, the S spin is not saturated. Instead a 908 pulse is applied to the system at equilibrium and the FID is recorded. The
intensity of the I spin resonance in this spectrum is proportional to I 0z . The value of IS is calculated from [5.150]. In practice, the steady-state NOE difference experiment is performed somewhat differently than described in order to maximize the accuracy of the results; such issues are not relevant to the present discussion and are discussed elsewhere (49). are made by use of the one-dimensional Measurements of NOE IS transient NOE experiment, discussed in Section 5.1.2, or the twodimensional NOESY experiment (Section, 6.6.1). These laboratoryframe relaxation transient NOE experiments have rotating-frame analogs: the transient ROE experiment and the two-dimensional ROESY experiment (Section 6.6.2) in which the rotating-frame crossrelaxation rate constant, ROE IS , is given by [5.141]. Using the isotropic rotor spectral density function [5.98], the crossrelaxation rate constants for a homonuclear spin system ( I ¼ S ¼ ) are given by h2 20 4 c 6 NOE 1 þ IS ¼ , 1 þ 4!20 2c 1602 r6IS ½5:151 2 2 4 h 3 c 0 ¼ 2þ ROE , IS 1 þ !20 2c 1602 r6IS and the NOE enhancement is given by . 6 3 6 þ 1 þ :
IS ¼ 1 þ 1 þ 4!20 2c 1 þ !20 2c 1 þ 4!20 2c
½5:152
The cross-relaxation rate constants are proportional to the inverse sixth power of the distance between the two dipolar interacting spins, rIS, but the enhancement IS does not depend upon rIS. Thus, a measurement of
IS indicates whether two spins are close enough in space to experience dipolar cross-relaxation, but a quantitative estimate of the distance separating the spins cannot be obtained. To estimate the distance or ROE must be measured directly (or IS between two nuclei, NOE IS IS measured in one experiment and I in a second experiment).
5.6 CHEMICAL EXCHANGE EFFECTS
IN
NMR SPECTROSCOPY
391
In the extreme narrowing limit (!0 c 1), [5.151] and [5.152] reduce to ¼ ROE ¼ NOE IS IS
h2 20 4 c , 322 r6IS
1
IS ¼ , 2
½5:153
and in the spin diffusion limit (!0 c 1), ¼ NOE IS
h2 20 4 c , 1602 r6IS
ROE ¼ IS
h2 20 4 c , 802 r6IS
IS ¼ 1:
½5:154
In the slow tumbling regime, the laboratory- and rotating-frame crossrelaxation rate constants are related by ROE ¼ 2 NOE :
½5:155
This relationship has been used to compensate approximately for crossrelaxation effects in NMR spectra (50, 51). The values of NOE and IS IS are zero if ! c ¼ 1.12, whereas, ROE > 0 for all . c IS The principal use of the NOE in biological NMR spectroscopy is the determination of distances between pairs of 1H spins (52). The NOE enhancements of interest arise from slowly tumbling biological macromolecules in the spin diffusion limit. For such molecules, relatively large transient homonuclear 1H NOE (or ROE) enhancements build up quickly and are detected most effectively by transient NOE and NOESY (or transient ROE and ROESY) methods (Section 6.6).
5.6 Chemical Exchange Effects in NMR Spectroscopy NMR spectroscopy provides an extremely powerful and convenient method for monitoring the exchange of a nucleus between environments due to chemical reactions or conformational transitions. In the first instance, the nucleus exchanges intermolecularly between sites in different molecules; in the second, the nucleus exchanges intramolecularly between conformations. The exchange process can be monitored by NMR spectroscopy even if the sites are chemically equivalent, provided that the sites are magnetically distinct. Nuclear spins can be manipulated during the NMR experiment without affecting the chemical states of the system, because of the weak coupling between the spin system and the lattice. Thus, chemical reactions and conformational exchange processes can be studied by NMR spectroscopy while the system remains in chemical equilibrium.
392
CHAPTER 5 RELAXATION
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To establish a qualitative picture of the effects of exchange on an NMR spectrum, suppose that a given nucleus exchanges with rate constant k between two magnetically distinct sites with resonance frequencies that differ by !. On average, the resonance frequency of the spin in each site can only be observed for a time of the order of 1/k before the spin jumps to the other site and begins to precess with a different frequency. The finite observation time places a lower limit on the magnitude of ! required to distinguish the two sites. If the exchange rate is slow (k !), then distinct signals are observed from the nuclei in the two sites; in contrast, if the exchange rate is fast (k !), then a single resonance is observed at the population-weighted average chemical shift of the nuclei in the two sites. The NMR chemical shift time scale is defined by the difference between the frequencies of spins in the two sites.
5.6.1 CHEMICAL EXCHANGE
FOR
ISOLATED SPINS
For simplicity, only the case of chemical exchange in spin systems without scalar coupling interactions is treated here. In this situation, the exchange process is treated using an extension of the Bloch equations (Section 1.2). The results obtained using the Bloch equations are applicable to scalar-coupled spin systems if relaxation processes other than exchange are in the macromolecular limit and scalar coupling constants are not modified by the exchange process (53, 54). If scalar coupling constants are modified by the exchange process, as might be observed for 1H 3JHH scalar coupling interactions, then a complete treatment based on the density operator formalism is required (55). A first-order chemical reaction (or two-site chemical exchange) between two chemical species, A1 and A2, is described by the reaction k1
A1 ! A2 ,
½5:156
k1
in which k1 is the reaction rate constant for the forward reaction and k–1 is the reaction rate constant for the reverse reaction. The chemical kinetic rate laws for this system are written in matrix form as d ½A1 ðtÞ k1 ¼ ½A ðtÞ k1 dt 2
k1 k1
½A1 ðtÞ : ½A2 ðtÞ
½5:157
5.6 CHEMICAL EXCHANGE EFFECTS
IN
NMR SPECTROSCOPY
393
For a coupled set of N first-order chemical reactions between N chemical species, this equation is generalized to dAðtÞ ¼ KAðtÞ, dt
½5:158
in which the matrix elements of the rate matrix, K, are given by Kij ¼ kji Kii ¼
ði 6¼ jÞ, N X
½5:159
kij ,
j¼1 j6¼i
and the chemical reaction between the ith and jth species is kij
Ai ! Aj :
½5:160
kji
The modified Bloch equations are written in matrix form for the jth chemical species as N X dMjx ðtÞ ¼ ð1 j Þ½Mj ðtÞ BðtÞx R2j Mjx ðtÞ þ Kjk Mkx ðtÞ, dt k¼1 N X dMjy ðtÞ Kjk Mky ðtÞ, ¼ ð1 j Þ½Mj ðtÞ BðtÞy R2j Mjy ðtÞ þ dt k¼1
½5:161
N X dMjz ðtÞ ¼ ð1 j Þ½Mj ðtÞ BðtÞz R1j Mjz ðtÞ Mj0 ðtÞ þ Kjk Mkz ðtÞ, dt k¼1
with Mj0 ðtÞ ¼ M0 ½Aj ðtÞ
N .X
½Aj ðtÞ:
½5:162
j¼1
The Bloch equations modified for chemical reactions are called the McConnell equations (56). The index j in [5.161] and [5.162] refers to the same spin in different chemical environments, not to different nuclear spins (cf. Section 1.2). If the system is in chemical equilibrium, then [Aj](t) ¼ [Aj] and N X k¼1
Kjk Mk0 ðtÞ ¼ 0:
½5:163
394
CHAPTER 5 RELAXATION
AND
DYNAMIC PROCESSES
Using this result, and defining Mk0 ¼ Mk0(t), the expression for the evolution of longitudinal magnetization in [5.161] is expressed as dMjz ðtÞ ¼ ð1 j Þ½Mj ðtÞ BðtÞz R1j Mjz ðtÞ Mj0 dt N X Kjk ½Mkz ðtÞ Mk0 : þ
½5:164
k¼1
The preceding equations are generalized to the case of higher order chemical reactions by defining the pseudo-first-order rate constants: kij ¼
_ij ðtÞ , ½Ai ðtÞ
½5:165
in which _ij ðtÞ is the rate law for conversion of the ith species containing the nuclear spin of interest into the jth species containing the nuclear spin of interest. The effect of the chemical reactions is to shift the spin of interest between molecular environments. For example, consider the elementary reaction k1
A1 þ B! A2 þ C,
½5:166
k1
in which a spin in species A1 is transferred to species A2 as a result of the chemical reaction. The chemical kinetic rate laws for this system are " # " #" # k1 ½BðtÞ k1 ½CðtÞ ½A1 ðtÞ d ½A1 ðtÞ ¼ , ½5:167 dt ½A2 ðtÞ k1 ½BðtÞ k1 ½CðtÞ ½A2 ðtÞ which has the same form as [5.158] in which the elements of K are defined using [5.159] and [5.165]. Notice that the rate expressions for [B](t) and [C](t) are not included in [5.167] because the spin of interest is not contained in either species. In the absence of applied rf fields, the equation governing the evolution of longitudinal magnetization becomes N X dMjz ðtÞ ¼ R1j Mjz ðtÞ Mj0 ðtÞ þ Kjk Mkz ðtÞ: dt k¼1
½5:168
5.6 CHEMICAL EXCHANGE EFFECTS
IN
NMR SPECTROSCOPY
395
Defining 2
3 M1z ðtÞ 6 7 Mz ðtÞ ¼ 4 ... 5
½5:169
MNz ðtÞ yields the compact expression, dMz ðtÞ ¼ ðR þ KÞ Mz ðtÞ M0 ðtÞ þ KM0 ðtÞ, dt
½5:170
in which the elements of R are given by Rij ¼ ijR1j. For simplicity, the possibility of simultaneous dipolar cross-relaxation and chemical exchange is not considered. If the system is in chemical equilibrium, KM0(t) ¼ KM0 ¼ 0 and defining Mz(t) ¼ Mz(t) M0 dMz ðtÞ ¼ ðR þ KÞMz ðtÞ: dt
½5:171
By similar reasoning, the equation of motion for the transverse magnetization is written in the rotating frame as dMþ ðtÞ ¼ ði: R þ KÞMþ ðtÞ, dt
½5:172
in which the elements of : are given by ij ¼ ij j, and the elements of R are given by Rij ¼ ijR2j. Equations [5.171] and [5.172] have the same functional form as [5.14] and are solved by the same methods ([5.15]). For example, the rate matrix for longitudinal relaxation in a system undergoing two-site exchange is given by k1 1 þ k1 ½5:173 R K¼ k1 2 þ k1 with eigenvalues
¼
1=2 o 1n ð 1 þ 2 þ k1 þ k1 Þ ð 1 2 þ k1 k1 Þ2 þ 4k1 k1 : 2 ½5:174
396
CHAPTER 5 RELAXATION
AND
DYNAMIC PROCESSES
The time course of the magnetization is given by "
M1z ðtÞ M2z ðtÞ
#
" ¼
a11 ðtÞ
a12 ðtÞ
a21 ðtÞ
a22 ðtÞ
#"
M1z ð0Þ
#
M2z ð0Þ
,
½5:175
in which 1 1 2 þ k1 k1 1 expð tÞ a11 ðtÞ ¼ 2 ð þ Þ 1 2 þ k1 k1 expð þ tÞ , þ 1þ ð þ Þ 1 1 2 þ k1 k1 1þ a22 ðtÞ ¼ expð tÞ 2 ð þ Þ 2 þ k1 k1 expð þ tÞ , þ 1 1 ð þ Þ a12 ðtÞ ¼
k1 ½expð tÞ expð þ tÞ, ð þ Þ
a21 ðtÞ ¼
k1 ½expð tÞ expð þ tÞ: ð þ Þ
½5:176
If the initial perturbation is nonselective, so that M1z(0) / p1 and M2z(0) / p2, in which p1 and p2 are the equilibrium fractional site populations for sites A1 and A2, and if 1 ¼ 2, then longitudinal relaxation is not affected by the exchange process. In addition, if these conditions are not met, but exchange is fast on the chemical shift time scale and |kex(p1 p2)| | 1 2|, in which kex ¼ k1 þ k–1, then longitudinal relaxation similarly is unaffected and the population-averaged longitudinal relaxation rate constant is observed. If these conditions are not satisfied, that is, if exchange is slow on the chemical shift time scale and either a selective perturbation is applied to the spins in the two sites or 1 6¼ 2, then the exchange process transfers longitudinal magnetization between sites. To obtain some insight into the form of these equations, assume that 1 ¼ 2 ¼ . Under these conditions, the time dependence of the
5.6 CHEMICAL EXCHANGE EFFECTS
IN
397
NMR SPECTROSCOPY
1
aij(t) exp(r t)
0.8 p1 0.6 0.4 p2 0.2
0
1
2
3
4
5
kext
FIGURE 5.9 Population transfer due to chemical exchange. The transfer function amplitudes (—) a11(t), (- - -) a22(t), ( ) a12(t), and (- -) a21(t) calculated using [5.177].
longitudinal magnetization is given by a11 ðtÞ ¼ ½p1 þ p2 expð2kex tÞ expð tÞ, a22 ðtÞ ¼ ½p2 þ p1 expð2kex tÞ expð tÞ,
½5:177
a12 ðtÞ ¼ p1 ½1 expð2kex tÞ expð tÞ, a21 ðtÞ ¼ p2 ½1 expð2kex tÞ expð tÞ: The time dependence of the transfer amplitudes, aij(t), given in [5.177] is shown in Fig. 5.9. The homology between [5.22] and [5.177] illustrates the similarity between the effects of cross-relaxation and chemical exchange on longitudinal magnetization. Indeed, similar experimental techniques are utilized to study both phenomena (such as NOESY and ROESY experiments, Section 6.6). The rate matrix for transverse relaxation in a system undergoing two-site exchange is given by " i: þ R K ¼
i1 þ 1 þ k1
k1
k1
i2 þ 2 þ k1
# ,
½5:178
398
CHAPTER 5 RELAXATION
AND
DYNAMIC PROCESSES
with eigenvalues 1n
¼ ði1 i2 þ 1 þ 2 þ k1 þ k1 Þ 2 1=2 o ði1 þ i2 þ 1 2 þ k1 k1 Þ2 þ4k1 k1 : The time course of the magnetization is given by " # # " a11 ðtÞ a12 ðtÞ Mþ Mþ 1 ðtÞ 1 ð0Þ ¼ , a21 ðtÞ a22 ðtÞ Mþ Mþ 2 ðtÞ 2 ð0Þ
½5:179
½5:180
in which
1 i1 þ i2 þ 1 2 þ k1 k1 1 expð tÞ 2 ð þ Þ i1 þ i2 þ 1 2 þ k1 k1 expð þ tÞ , þ 1þ ð þ Þ 1 i1 þ i2 þ 1 2 þ k1 k1 1þ expð tÞ a22 ðtÞ ¼ 2 ð þ Þ ½5:181 i1 þ i2 þ 1 2 þ k1 k1 expð þ tÞ , þ 1 ð þ Þ
a11 ðtÞ ¼
a12 ðtÞ ¼
k1 ½expð tÞ expð þ tÞ, ð þ Þ
a21 ðtÞ ¼
k1 ½expð tÞ expð þ tÞ: ð þ Þ
The NMR spectrum is given by the Fourier transformation of þ Mþ 1 ðtÞ þ M2 ðtÞ. The eigenvalues are resolved into real and imaginary parts, corresponding to the relaxation rates and precession frequencies, by using the identity 1=2 1=2 1 i ða þ ibÞ1=2 ¼ pffiffiffi a þ ða2 þ b2 Þ1=2 þ pffiffiffi a þ ða2 þ b2 Þ1=2 : ½5:182 2 2 If 1 ¼ 2 ¼ is assumed for simplicity, then ¼ i R2 , in which (57, 58) h i1=2 1=2 kex 1 2 2 2 2 2 2 2 p ffiffi ffi R2 ¼ þ , kex ! þ kex þ ! 16p1 p2 ! kex 2 8 ½5:183
5.6 CHEMICAL EXCHANGE EFFECTS
¼
IN
NMR SPECTROSCOPY
399
h i1=2 1=2 2 1 þ 2 1 pffiffiffi !2 k2ex þ k2ex þ !2 16p1 p2 !2 k2ex , 2 8 ½5:184
and ! ¼ 2 – 1. When exchange is slow, kex 5 |!|, the magnetization components for the two sites are nonsecular with respect to each other and the offdiagonal terms in [5.178] can be neglected. In this case, the magnetization components in the two sites evolve independently with þ Mþ 1 ðtÞ ¼ M1 ð0Þ exp ði1 þ 1 þ k1 Þt , ½5:185 þ Mþ 2 ðtÞ ¼ M2 ð0Þ exp ði2 þ 2 þ k1 Þt : When exchange is fast, kex 4 !, only the averaged magnetization þ Mþ ðtÞ ¼ Mþ 1 ðtÞ þ M2 ðtÞ is observable. The evolution of the averaged magnetization is derived simply by using the random-phase model for transverse relaxation (Section 5.1.3) together with the identification c ¼ 1/kex. The averaged resonance evolves as ½5:186 Mþ ðtÞ ¼ Mþ ð0Þ exp i þ þ p1 p2 !2 =kex , in which the average resonance offset is ¼ p1 1 þ p2 2 and ¼ p1 1 þ p2 2 . Simulated spectra are shown in Fig. 5.10 for two situations: the first is symmetric unimolecular exchange in which p1 ¼ p2 ¼ 0.5; the second is skewed unimolecular exchange in which p1 ¼ 3p2. In the absence of exchange, spectra f and l in Fig. 5.10 show that resolved lines are observed for the two sites with resonance frequencies 1 and 2, and relaxation decay constants 1 and 2. As the exchange rate increases, the resonance lines broaden, as shown in spectra e and k, and the evolution is described by [5.185]. When the exchange rate is of the order of the chemical shift separation between the two sites, the lines become very broad and begin to coalesce when kex ! (spectra c and i). This is known as the intermediate exchange regime, or coalescence. Intermediate exchange processes can cause peaks to disappear in spectra because the broadening becomes so great that the resonance line becomes indistinguishable from the baseline noise. Increasing the exchange rate for the system above the coalescence point pushes the system into fast exchange, kex 4 !. A single averaged resonance line is observed at the population-weighted average shift and evolution is described by [5.186], as shown in spectra b and h in Fig. 5.10. As the
400
CHAPTER 5 RELAXATION
a
g
b
h
c
i
d
j
e
k
f
–100
AND
DYNAMIC PROCESSES
l
–50
0
50
100
–100
–50
0
50
100
Frequency (Hz)
FIGURE 5.10 Chemical exchange for a two-site system. Shown are the Fourier transformations of FIDs calculated by using [5.181]. The calculations used !/2 ¼ 180 Hz and 1 ¼ 2 ¼ 10 s–1. In (a–f ), p1 ¼ p2 ¼ 0.5; in g–l, p1 ¼ 3p2. Calculations were performed for values of the exchange rate, kex, equal to (a, g) 10000 s–1, (b, h) 2000 s–1, (c, i) 900 s–1, (d, j) 200 s–1, (e, k) 20 s–1, and (f, l) 0 s–1.
exchange rate continues to increase, the resonance lineshape becomes increasingly narrow until, in the limit kex ! 1, the relaxation decay constant is given by . Equations [5.185] and [5.186] confirm the qualitative conclusions about the slow and fast exchange regimes stated herein.
5.6 CHEMICAL EXCHANGE EFFECTS
IN
401
NMR SPECTROSCOPY
5.6.2 QUALITATIVE EFFECTS OF CHEMICAL EXCHANGE SCALAR-COUPLED SYSTEMS
IN
Multiplet structure due to scalar couplings is affected by chemical exchange. Detailed theoretical treatment using the density matrix formalism is beyond the subject matter of this text (59); instead, the discussion here presents qualitatively the most important effects. Formally, scalar relaxation (Section 5.4.6) and chemical exchange in scalar-coupled systems are homologous. Two different cases must be considered: intermolecular (homologous to scalar relaxation of the second kind) and intramolecular exchange (homologous to scalar relaxation of the first kind). Intermolecular chemical exchange in scalar-coupled systems is encountered frequently in biological NMR applications. For example, exchange between labile amide and solvent protons perturbs the 1HN to 1 H scalar coupling interaction. In an IS spin system, the I spin resonance is a doublet, with the lines separated by JIS. One line of the doublet is associated with the S spin in the state, and the other line is associated with the S spin in the state. Suppose that a given I spin is coupled to an S spin in the state. If the S spin exchanges with another S spin originating from the solvent (intermolecular exchange), then after the exchange, the I spin has equal probability of being coupled to an S spin in the and states because the incoming spin has a 50% chance of either being in its state or in its state. Similar considerations hold for an I spin initially coupled to an S spin in the state. Consequently, the I spin sees the S spin state constantly changing due to exchange and thus the frequency of the I spin resonance constantly changes between the frequencies of the two lines of the doublet. This phenomenon constitutes a two-site exchange process and exhibits properties of slow, intermediate, and fast exchange. If the exchange is fast compared to the difference in frequency between the two lines (i.e., compared to the scalar coupling constant), a single line is observed at the mean frequency (the Larmor frequency of the I spin). Because homonuclear scalar coupling constants tend to be small, relatively slow exchange processes, which would minimally perturb the chemical shifts of the exchanging spins, can result in collapse of multiplet structure. Indeed, the broadening of multiplets and the disappearance of multiplet structure are the first clues to the existence of exchange phenomena in NMR spectra. Intramolecular exchange constitutes a slightly different situation. Consider a system in which spin I is scalar coupled to spin S, but due to the presence of multiple conformers, spin S experiences n environments,
402
CHAPTER 5 RELAXATION
AND
DYNAMIC PROCESSES
S1, S2, . . . , Sn, with different scalar coupling constants. For simplicity, the chemical shift of the I spin is assumed to be identical in all conformers. If the conformers interconvert on a time scale that is long compared to the scalar coupling constants, the I spin multiplet is a superposition of n doublets arising from the IS1, IS2, . . . , ISn scalar coupling interactions. On the other hand, if the conformers interconvert at a rate much larger than the scalar coupling constants, the I spin resonance is a doublet with an effective scalar coupling constant that is a population-weighted average of the n scalar coupling constants. An example of this effect arises for the scalar coupling between 1H and 1H spins in amino acids. If the conformations of the H nuclei are fixed relative to the H nuclei, then the H multiplet is split by two coupling constants, one from each of the 1H spins to the 1H spin (e.g., 12 and 3 Hz, respectively, for a trans and gauche conformation). On the other hand, if the H nuclei exchange between trans, gaucheþ, and gauche– rotomeric sites due to free rotation about the C–C bond, then the H multiplet is split by a single average coupling constant, with a value (12 þ 3 þ 3)/3 ¼ 6 Hz, due to the 1H spins.
References 1. J. McConnell, ‘‘The Theory of Nuclear Magnetic Relaxation in Liquids,’’ pp. 1–196. Cambridge University Press, New York, 1987. 2. A. Abragam, ‘‘Principles of Nuclear Magnetism,’’ pp. 1–599. Clarendon Press, Oxford, 1961. 3. M. Goldman, ‘‘Quantum Description of High-Resolution NMR in Liquids,’’ pp. 1–268. Clarendon Press, New York, 1988. 4. B. Cowan, ‘‘Nuclear Magnetic Resonance and Relaxation,’’ pp. 1–434. Cambridge University Press, Cambridge, 1997. 5. P. Luginbu¨hl, K. Wu¨thrich, Prog. NMR Spectrosc. 40, 199–247 (2002). 6. M. Goldman, J. Magn. Reson. 149, 160–187 (2001). 7. D. M. Korzhnev, M. Billeter, A. S. Arseniev, V. Y. Orekhov, Prog. NMR Spectrosc. 38, 197–266 (2001). 8. V. A. Daragan, K. H. Mayo, Prog. NMR Spectrosc. 31, 63–105 (1997). 9. M. W. F. Fischer, A. Majumdar, E. R. P. Zuiderweg, Prog. NMR Spectrosc. 33, 207–272 (1998). 10. L. G. Werbelow, D. M. Grant, Adv. Magn. Reson. 9, 189–299 (1977). 11. D. I. Hoult, N. S. Ginsberg, J. Magn. Reson. 148, 182–199 (2001). 12. F. Bloch, Phys. Rev. 70, 460–474 (1946). 13. N. Bloembergen, E. M. Purcell, R. V. Pound, Phys. Rev. 73, 679–712 (1948). 14. I. Solomon, Phys. Rev. 99, 559–565 (1955). 15. R. K. Wangsness, F. Bloch, Phys. Rev. 89, 728–739 (1953). 16. A. G. Redfield, Adv. Magn. Reson. 1, 1–32 (1965). 17. R. Bru¨schweiler, D. A. Case, Prog. NMR Spectrosc. 26, 27–58 (1994).
5.6 CHEMICAL EXCHANGE EFFECTS 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54.
IN
NMR SPECTROSCOPY
403
N. C. Pyper, Mol. Phys. 22, 433–458 (1971). R. Bru¨schweiler, J. Chem. Phys. 105, 6164–6167 (1996). L. G. Werbelow, J. Chem. Phys. 70, 5381–5383 (1979). M. Goldman, J. Magn. Reson. 60, 437–452 (1984). N. Mu¨ller, G. Bodenhausen, K. Wu¨thrich, R. R. Ernst, J. Magn. Reson. 65, 531–534 (1985). N. C. Pyper, Mol. Phys. 21, 133 (1971). D. Abergel, A. G. Palmer, Concepts Magn. Reson. 19A, 134–148 (2003). M. H. Levitt, G. Bodenhausen, R. R. Ernst, J. Magn. Reson. 53, 443–461 (1983). C. Griesinger, R. R. Ernst, Chem. Phys. Lett. 152, 239–247 (1988). P. S. Hubbard, Phys. Rev. 180, 319–326 (1969). D. M. Brink, G. R. Satchler, ‘‘Angular Momentum,’’ pp. 1–170. Clarendon Press, Oxford, 1993. G. Lipari, A. Szabo, J. Am. Chem. Soc. 104, 4546–4559 (1982). G. Lipari, A. Szabo, J. Am. Chem. Soc. 104, 4559–4570 (1982). A. G. Palmer, Curr. Opin. Biotechnol. 4, 385–391 (1993). G. Wagner, Curr. Opin. Struc. Biol. 3, 748–753 (1993). H. Liu, P. D. Thomas, T. L. James, J. Mag. Reson. 98, 163–175 (1992). T. M. G. Koning, R. Boelens, R. Kaptein, J. Magn. Reson. 90, 111–123 (1990). L. E. Kay, D. A. Torchia, J. Magn. Reson. 95, 536–547 (1991). R. Bru¨schweiler, X. Liao, P. E. Wright, Science 268, 886–889 (1995). L. K. Lee, M. Rance, W. J. Chazin, A. G. Palmer, J. Biomol. NMR. 9, 287–298 (1997). N. Tjandra, S. E. Feller, R. W. Pastor, A. Bax, J. Am. Chem. Soc. 117, 12562–12566 (1995). G. S. Harbison, J. Am. Chem. Soc. 115, 3026–3027 (1993). A. A. Bothner-By, R. L. Stephens, J. Lee, J. Am. Chem. Soc. 106, 811–813 (1984). J. Boyd, U. Hommel, I. D. Campbell, Chem. Phys. Lett. 175, 477–482 (1990). L. E. Kay, L. K. Nicholson, F. Delagio, A. Bax, D. A. Torchia, J. Magn. Reson. 97, 359–375 (1992). A. G. Palmer, N. J. Skelton, W. J. Chazin, P. E. Wright, M. Rance, Mol. Phys. 75, 699–711 (1992). K. Pervushin, R. Riek, G. Wider, K. Wu¨thrich, Proc. Natl. Acad. Sci. U.S.A. 94, 12366–12371 (1997). N. Tjandra, A. Szabo, A. Bax, J. Am. Chem. Soc. 118, 6986–6991 (1996). D. Fushman, N. Tjandra, D. Cowburn, J. Am. Chem. Soc. 120, 10947–10952 (1998). R. E. London, J. Magn. Reson. 86, 410–415 (1990). J. H. Noggle, R. E. Shirmer, ‘‘The Nuclear Overhauser Effect: Chemical Applications,’’ pp. 1–259. Academic Press, New York, 1971. D. Neuhaus, M. Williamson, ‘‘The Nuclear Overhauser Effect in Structural and Conformational Analysis,’’ 2nd edn., pp. 1–656. Wiley-VCH, New York, 2000. C. Griesinger, G. Otting, K. Wu¨thrich, R. R. Ernst, J. Am. Chem. Soc. 110, 7870–7872 (1988). J. Cavanagh, M. Rance, J. Magn. Reson. 96, 670–678 (1992). K. Wu¨thrich, ‘‘NMR of Proteins and Nucleic Acids,’’ pp. 1–292. Wiley, New York, 1986. F. A. A. Mulder, N. R. Skrynnikov, B. Hon, F. W. Dahlquist, L. E. Kay, J. Am. Chem. Soc. 123, 967–975 (2001). N. R. Skrynnikov, F. A. A. Mulder, B. Hon, F. W. Dahlquist, L. E. Kay, J. Am. Chem. Soc. 123, 4556–4566 (2001).
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55. J. I. Kaplan, G. Fraenkel, ‘‘NMR of Chemically Exchanging Systems,’’ pp. 1–165. Academic Press, New York, 1980. 56. H. M. McConnell, J. Chem. Phys. 28, 430–431 (1958). 57. J. S. Leigh, J. Magn. Reson. 4, 308–311 (1971). 58. D. E. Woessner, J. Chem. Phys. 35, 41–48 (1961). 59. R. R. Ernst, G. Bodenhausen, A. Wokaun, ‘‘Principles of Nuclear Magnetic Resonance in One and Two Dimensions,’’ pp. 1–610. Clarendon Press, Oxford, 1987.
CHAPTER
6 EXPERIMENTAL 1 H NMR METHODS
Because of the high natural abundance and large magnetogyric ratio of the 1H nucleus, protein NMR studies traditionally have utilized predominantly homonuclear 1H spectroscopic techniques. This chapter describes the homonuclear 1H NMR experiments required to obtain complete 1H resonance assignments and to ascertain structural and dynamical features of proteins with molecular masses of up to 10 to 12 kDa, provided that the proteins are well-behaved in solution and display reasonable chemical shift dispersion (as a consequence of differences in chemical shift dispersion, relatively larger -sheet proteins, compared to -helical proteins, are amenable to investigation). Hundreds of 2D and 3D 1H NMR experiments have been described in the literature; however, many of these are not generally applicable, or have been superseded by superior techniques. This chapter provides a concise compendium of useful NMR experiments from which resonance assignments and subsequent structural and dynamical investigations can be performed by using a minimum of spectrometer time. Heteronuclear NMR spectroscopy, utilizing 13C and 15N spins as well as 1H spins is described in Chapters 7 and 9. Throughout this chapter, phase-sensitive, rather than magnitudemode, spectra have been presented, because resolution of the resonances is superior and analysis of the cross-peak fine structure is facilitated.
405
406
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
Sections of correlation spectra with antiphase lineshapes are shown with multiple contours for the positive peak components, and a single contour for negative components. Generally, spectra with in-phase lineshapes are depicted with only positive or negative levels displayed; spectra in which both sets of peaks are displayed are discussed in the appropriate figure captions.
6.1 Assessment of the1D 1H Spectrum Although the majority of spectroscopic analyses will depend on 2D and 3D NMR spectroscopy, important preparatory work can be performed by using 1D 1H NMR spectroscopy. To begin, a 1D 1H NMR spectrum of the protein in H2O solution (containing 5–10% D2O as a lock reference) is acquired using the Hahn echo (Section 3.6.4.2), excitation sculpting (Section 3.5.3), jump–return (Section 3.5.2), or other one-dimensional NMR experiments. The first spectral feature of interest is the signal-to-noise (S/N) ratio. This parameter obviously depends upon the concentration of protein, but also is affected by the linewidth and multiplet structure of the resonance signals. If more than the most basic correlation experiments (i.e., COSY for homonuclear spectroscopy, HSQC/HMQC for heteronuclear spectroscopy) are to be feasible, then the 1D 1H spectrum must contain a reasonable amount of signal after co-adding 16 or 32 transients. The standard sample used for 1H experiments in this text is a 2 mM ubiquitin solution (see Preface). The 1H NMR spectrum shown in Fig. 6.1a was collected with 32 transients and has S/N ratios of 243, for a resolved upfield-shifted methyl group, and 46, for a resolved downfield-shifted amide group. S/N ratios are measured as the resonance peak height divided by the root-mean-square baseline noise in the spectrum. Rapid rotation narrows the resonance linewidths for methyl groups (relative to 1H resonances); in contrast, amide proton linewidths are broadened by amide proton solvent exchange, scalar relaxation to the 14N nucleus, and partially resolved scalar coupling to the 1H spin. In addition, the intensity of the amide resonance is reduced by the fraction of D2O present in the sample. Thus, the value of the S/N ratio for the amide is less than one-third of the value for the methyl group. The second parameter of interest is the resonance linewidth, given by the full-width at half-height (FWHH) or the transverse relaxation rate constant (R2). Beyond the contribution to the S/N ratio, FWHH and R2 affect the efficacy of coherence transfer by evolution under the scalar
6.1 ASSESSMENT
OF THE
1D 1H SPECTRUM
407
c
b a
10
8
6
4 1H
2
0
(ppm)
FIGURE 6.1 (a) 1D spectrum of a 2 mM ubiquitin sample in H2O acquired at 500 MHz and 278C. The 908 acquisition pulse was preceded by 1.5 s of solvent presaturation and followed by a Hahn echo (Section 3.6.4.2). The spectrum is the result of 32 transients collected with a spectral width of 12.5 kHz and a 30-kHz filter width. The region around the upfield-shifted methyl group of Leu50 (arrow) is enlarged in the inset (b). The RMS noise at the edges of the spectrum is 0.0023 (arbitrary units), while the Leu50 methyl 1H signal has a height of 0.56, giving an S/N ratio of 243. An S/N ratio of 46 is obtained for the resolved downfield-shifted 1HN resonance of Ile13 (filled circle). (c) The 1D spectrum of a 2 mM ubiquitin sample in 8 M urea/H2O solution. The reduced chemical shift dispersion is characteristic of unfolded or denatured proteins.
408
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
coupling Hamiltonian. Furthermore, in indirectly detected 1H dimensions of multidimensional NMR experiments, the transverse relaxation rate constant, R2, of the coherence of interest is important in deciding an appropriate value for t1max. Extending t1 beyond 2/R2 or 3/R2 is undesirable because the later increments of t1 only contribute noise to the processed spectrum. Values of FWHH and R2 are related to the rotational correlation time of the protein, which in turn depends on the solution viscosity and temperature (Section 1.4). Larger proteins have larger linewidths and lower apparent S/N ratios even at similar concentrations. Aggregation causes line broadening, and, in some unfortunate cases, proteins may aggregate to such an extent that no peaks are observed in the 1D 1H spectrum (the linewidth becomes so large that the peaks merge into the baseline), even though other methods of detection indicate that the actual protein concentration in solution is in the millimolar range. The phenomenological 1H single-quantum relaxation rate constant, R2 ¼ R2 þ Rinhom , in which Rinhom represents the contribution from magnetic field inhomogeneity, can be estimated from the full-width-at-half-height linewidth (FWHH) of resolved peaks as R2 ðFWHH Þ. For ubiquitin (Mr ¼ 8565), the full-width-at-half-height linewidth (FWHH) of a resolved upfield methyl group and downfield amide proton are 6.4 and 8.6 Hz, respectively, at 300 K. Linewidths were obtained by curve-fitting to an in-phase Lorentzian doublet because the observed lineshape of both resonances has a contribution from a partially resolved scalar coupling. Transverse relaxation rate constants for the amide 1H spins also can be estimated from the one-dimensional jump–return Hahn echo experiment (Section 3.6.2.6). The third spectral feature of interest is the resonance dispersion in the 1D spectrum. The degree of dispersion indicates the integrity of the protein under the particular experimental conditions chosen. Denatured proteins have chemical shifts close to those found in short linear peptides (the so-called random coil shifts; Section 1.5), whereas folded proteins will exhibit a range of chemical shifts due to the anisotropic magnetic fields of proximal aromatic or carbonyl groups. Thus, if very little chemical shift dispersion is observed, then the protein may be unfolded, or may have very little stable structure. As an example, a spectrum of ubiquitin denatured in 8 M urea is shown in Fig. 6.1c. The key resonances to examine arise from the amide protons (random coil shifts, 8.5–8.0 ppm), -protons (random coil shifts, 4.4–4.1 ppm), and methyl groups of valine, isoleucine, and leucine (random coil shifts, 1.1–0.8 ppm). Examination of the chemical shift dispersion also indicates the ease with which resonance assignments can be made by the
6.2 COSY-TYPE EXPERIMENTS
409
sequential spin system method described in Section 9.1.1. If overlap in the important 1HN region is significant (and to a lesser extent in the 1H and methyl 1H regions), sequential assignments will be hard won, and the number of NOEs that can be assigned unambiguously will be low (leading to poor structural definition). Schemes have been proposed that use the observed dependence of 1HN and 1H chemical shift on secondary structure to estimate the number of residues in different types of regular secondary structure (1, 2). Finally, the purity of the sample can be gauged from the 1D spectrum. Low molecular weight impurities are apparent as sharp peaks amid the broader envelope of protein resonances. Of course, a pure protein sample also can exhibit linewidth variations due to differential internal mobility, particularly at side chain termini, or in flexible loop regions, so sharp lines are not necessarily proof of contamination. Observation of variations in the relative peak heights of resolved resonances may indicate inhomogeneous protein preparations, although such information usually is better gauged from COSY (Section 6.2.1) or HSQC/HMQC spectra (Section 7.1.1). Low molecular weight impurities also can be identified from a TOCSY experiment (Section 6.5) recorded with a long (200-ms) mixing time because protein resonances are attenuated preferentially by relaxation. If any of the basic attributes of the 1D 1H NMR spectrum (S/N ratio, linewidth, chemical shift dispersion) are less than ideal, the 1D spectrum provides an efficient way to probe the dependence of these aspects on sample conditions, including concentration, temperature, pH, and ionic strength. As a word of caution, care should be taken when performing such studies, as extremes of temperature and pH may lead to irreversible denaturation or loss of protein integrity. Furthermore, unless spectra are recorded without presaturation of the solvent resonance (Section 3.5.3), spectra may suffer from a loss of amide proton signal intensity from saturation transfer via exchange with solvent at elevated temperature or pH.
6.2 COSY-Type Experiments COSY and related experiments are based on coherence transfer through evolution under the scalar coupling Hamiltonian during pulse-interrupted free-precession pulse sequence elements. The basic aspects of coherence transfer are described in Section 4.2.2.1. The following sections describe in detail theoretical and practical aspects of these experiments.
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
410
6.2.1 COSY COSY, or correlated spectroscopy, was the first 2D NMR experiment to be devised (3, 4), and remains useful for NMR studies of small proteins. COSY cross-peaks arise through coherence transfer between coupled spins; in practice for protein studies, this limits transfer to protons separated by two or three bonds. The pulse sequence simply consists of two pulses separated by an incrementable delay (t1). The recycle delay precedes the first pulse, and the acquisition period (t2) follows the second pulse, as shown in Fig. 6.2. The basic phase cycle consists of eight steps: the phases of both pulses and the receiver are cycled together using the CYCLOPS scheme, and the phases of the first pulse and receiver are inverted to reduce axial peak intensity. 6.2.1.1 Product Operator Analysis Most of the homonuclear NMR experiments discussed in this chapter begin with a 90x –t1–90x pulse sequence element (multiple-quantum experiments are the principal exceptions). In the following discussion, spins will be designated Ik for k ¼ 1, 2, . . . , K (for a K-spin system). The chemical shift of the kth spin is k, and the scalar coupling constant between the jth and kth spins is Jjk f2
f1 t1
t2
p +1 0 –1
FIGURE 6.2 Pulse sequence and coherence level diagram for the COSY experiment. Narrow bars represent 908 pulses. The basic phase cycle is 1 ¼ x, 2 ¼ x, and receiver ¼ x. Axial peak suppression and CYCLOPS phase cycling are performed to obtain an eight-step phase cycle. Frequency discrimination in F1 is obtained by shifting the phase of 1 and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4).
6.2 COSY-TYPE EXPERIMENTS
411
(assumed to represent a three-bond scalar interaction). For a two-spin system, initial I1z magnetization evolves through the pulse sequence element as 2
x
t1 2
x
I1z ! I1z cosð1 t1 Þ cosðJ12 t1 Þ 2I1x I2y cosð1 t1 Þ sinðJ12 t1 Þ þ I1x sinð1 t1 Þ cosðJ12 t1 Þ 2I1z I2y sinð1 t1 Þ sinðJ12 t1 Þ, ½6:1
in which 2 x represents a nonselective rf pulse with x-phase applied to the I spins. Parallel evolution beginning with I2z magnetization is exhibited by exchanging I1 and I2 labels. A product operator analysis of the COSY experiment reveals important features that must be considered while acquiring, processing, and analyzing COSY spectra. The essence of the product operator analysis for a two-spin system has already been presented (Section 4.2.1). In summary, the first two terms of [6.1] do not lead to observable magnetization and can be ignored, provided that the spin system achieves thermal equilibrium during the recycle delay (Section 6.3). The third term gives rise to a diagonal peak and the fourth term leads to a cross-peak modulated by 1 in t1 and 2 in t2. Manipulation of the trigonometric terms of [6.1] leads to a clearer understanding of the multiplet fine structure: sinð1 t1 Þ cosðJ12 t1 Þ ¼ 12½ sinð1 t1 J12 t1 Þ þ sinð1 t1 þ J12 t1 Þ, ½6:2 sinð1 t1 Þ sinðJ12 t1 Þ ¼ 12½ cosð1 t1 J12 t1 Þ cosð1 t1 þ J12 t1 Þ: ½6:3 As indicated by [6.2], the diagonal peak has an in-phase lineshape in F1 with the two multiplet components centered at 1 and separated by 2J12; in contrast, [6.3] indicates that the cross-peak has an antiphase lineshape with the two components centered at 1 and separated by 2J12 (if is given in units of angular frequency). The sinusoidal modulation of [6.2] and cosinusoidal modulation [6.3] mean that the diagonal and cross-peaks differ in phase by 908 and cannot both be phased to absorption simultaneously. Consideration of the evolution of I1x and 2I1zI2y during t2 indicates that the F2 lineshapes of the diagonal and cross-peaks are the same as in the F1 dimension. The antiphase lineshapes of COSY cross-peaks have important implications for the way in which these data are collected and processed (Sections 6.2.1.2 and 6.2.1.3). In addition, the differences in the relative phase of diagonal and cross-peaks are one of the main shortcomings of
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
412
the COSY experiment: namely, when the cross-peaks are phased to absorption, the in-phase dispersive tails of the diagonal peaks obscure cross-peaks near the diagonal (arising from scalar-coupled protons close in chemical shift). COSY cross-peaks between spins in more complex spin systems have fine structure in addition to the simple antiphase splitting of [6.3] (5). The product operator formalism can be used to elucidate the nature of this fine structure, although the calculations quickly become tedious to perform by hand if more than a few coupled spins are involved. For a cross-peak between spins I1 and I2, the component of the time-domain signal at the chemical shift of spin I1 is proportional to (6) sinð1 tÞ sinðJ12 tÞ
K Y
cosðJ1k tÞ
k¼3 K Y 1 cosðJ1k tÞ, ¼ ½cosð1 t J12 tÞ cosð1 t þ J12 tÞ 2 k¼3
½6:4
in which t represents either the t1 or t2 evolution period (depending upon whether the cross-peak represents I1 ! I2 or I2 ! I1 coherence transfer), J12 is the active scalar coupling between spin I1 and spin I2, and J1k is the value of the passive scalar coupling between spin I1 and spin Ik (k 4 2). A similar equation would represent the signal at the chemical shift of the I2 spin, except that the product would extend over the passive scalarcoupled partners of the I2 spin. The product of the two sine terms on the left-hand side of [6.4] gives rise to the antiphase splitting by the active coupling J12. Because the product of two cosine terms can be decomposed into a sum of cosine terms, after Fourier transformation, [6.4] yields in-phase absorption components for the passive scalar coupling interactions. The appearance of cross-peaks for amino acid spin systems for a variety of coupling constants and linewidths has been described (7). In the special case of scalar coupling between spin I1 and an In group, in which n is the number of magnetically equivalent spins, the cross-peak is considered to have one active coupling and (n 1) passive couplings; as a result, the relative intensities of the fine-structure components are described by the antiphase Pascal triangle (6). 6.2.1.2 Experimental Protocol Aside from the details of experimental protocols common to all 2D experiments (e.g., setting the rf transmitter frequency, calibrating the 908 pulse length, choosing spectral widths, and determining the recycle delay), the nature of coherence transfer and of lineshapes in a COSY experiment requires additional
413
6.2 COSY-TYPE EXPERIMENTS 1.0
Intensity
0.5
0.0
–0.5
–1.0 –30
–20
–10
0
10
20
30
n (Hz) FIGURE 6.3 Variation in the peak height of an antiphase absorptive doublet as the digital resolution is decreased to approach the size of the peak splitting. Both curves represent data for an antiphase pair of Lorentzian lines with half-height width 5 Hz and a separation of 5 Hz. The solid curve has a digital resolution of 0.2 Hz/point and accurately traces the lineshape. The broken curve has a digital resolution of 5 Hz/point and clearly has a smaller vertical separation of the positive and negative extrema. Note that the exact decrease in peak height will depend on where the low digital resolution points fall on the curve.
consideration. Digital resolution in the frequency domain has a profound influence on the relative cross-peak intensity of antiphase lineshapes. If the digital resolution in F1 or F2 is too low, the positive and negative lobes of the cross-peak will cancel partially, and the intensity of the cross-peak will be reduced as indicated in Fig. 6.3. Equivalently, in the time domain the cross-peak product operators contain sin(JISt1) and sin(JISt2) trigonometric terms arising from the active scalar coupling interaction [6.1] that are superposed upon the terms reflecting chemical shift evolution; in this respect, the COSY experiment is said to generate sine-modulated data. Consequently, evolution in t1 and t2 must occur for times comparable to or greater than 1/(4J12) to 1/(2J12) (approximately 62 to 125 ms for a 4-Hz coupling constant) if observable cross-peaks are to be obtained. The parameter t ¼ 1/(2J12) occurs many times in discussions of scalar correlated experiments, because coherence transfer occurs via antiphase terms with magnitudes proportional to sin(J12t). Acquisition for a sufficient length of time in t2 is rarely
414
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
a problem, as a greater proportion of the recycle delay can be spent recording each FID without lengthening the total acquisition time of the experiment. Commonly, each FID is collected for t2 max 1=ð2J12 Þ and is truncated at the processing stage to achieve the desired amount of sensitivity or resolution enhancement. In contrast, increasing t1max by increasing the number of data points in each interferogram adds linearly to the total acquisition time because a new FID must be collected for each additional t1 value. For a protein in the 8- to 10-kDa range, COSY is one of the most sensitive proton 2D experiments. A reasonable S/N ratio can be obtained in a few hours using a 500-MHz NMR spectrometer equipped with a conventional probe for a protein sample concentration of 1 mM; even less sample is required if a cryogenic probe is available. For example, assuming a 10-ppm spectral width in F1, and a recycle delay of 2.0 s (t2 acquisition for 0.3 s and equilibrium recovery and/or solvent presaturation for 1.7 s), a COSY experiment with eight transients for each of 512 t1 increments could be recorded in just over 2 hours and would have a t1max of 50 ms. If cross-peaks arising from small coupling constants are to be observed (e.g., 3 JHN H couplings in residues adopting helical conformations), then additional t1 increments can be recorded so as to bring t1max into the 80-ms range. Alternatively, the number of t1 increments can be kept constant and t1max increased by decreasing the F1 spectral width; however, care must be taken to avoid folding diagonal peaks on top of cross-peaks. Increasing t1max beyond 100 ms is unlikely to bring significant improvements in S/N because of extensive relaxation during the longer values of t1. For more dilute samples, the number of transients collected for each t1 increment must be increased. Except as noted in the figure captions in this chapter, the COSY spectra of ubiquitin were acquired with 16 transients for each of 800 t1 values with an F1 spectral width of 5800 Hz; TPPI was used for frequency discrimination in t1 and t1max ¼ 69 ms. The acquisition time for each FID was 330 ms (spectral width, 6250 Hz over 2048 complex points), although t2max was usually reduced at the processing stage (see later). The water resonance was suppressed by presaturation for 1.5 s. The total acquisition time was just over 7 hr. The effect of t1max on COSY spectra of ubiquitin is illustrated in Fig. 6.4. As t1max decreases from 69 to 17.2 ms, the cross-peaks become less intense because of self-cancellation of the antiphase multiplet components. At the shortest values of t1max, the cross-peak of Asp21 is not observed and the characteristic glycine fine structure of the Gly47 cross-peak disappears.
a
b
c
4.0
415
6.2 COSY-TYPE EXPERIMENTS
A28
L46
F1 (ppm)
G47
D58 D32 D52 + D60
4.5
N25 K48 D21
8.1
8.0
8.1
8.0
8.1
8.0
F2 (ppm)
FIGURE 6.4 Sections of the 1HN–1H region of COSY spectra acquired with the same total acquisition time (8 hr) but with different values of t1max: (a) 16 transients per increment, t1max ¼ 69 ms; (b) 32 transients per increment, t1max ¼ 34.5 ms; (c) 64 transients per increment, t1max ¼ 17.2 ms. All spectra were processed with an unshifted sine bell in t2 and a sine bell shifted 208 over the available data and zero-filled to 1024 points in t1. As t1max decreases, many of the peaks become weaker and that of Asp21 is lost completely. Also note that the characteristic glycine fine structure is not present at the shorter values of t1max (see cross-peak of Gly47), making identification of these residues difficult.
6.2.1.3 Processing As surmised from [6.1], the cross- and diagonal peaks are 908 out of phase in COSY spectra. In the normal mode of display, the cross-peaks are phased to have antiphase absorptive lineshapes in both dimensions, and the diagonal peaks are phased to have in-phase dispersive lineshapes in both dimensions. Because the tails of the diagonal peaks can obscure information-containing cross-peaks, processing procedures for COSY spectra are designed to maximize cross-peak intensity and minimize diagonal peak intensity. Because cross-peak lineshapes present in a COSY spectrum are antiphase, apodization functions emphasizing the initial parts of each FID do not improve the S/N ratio; instead, window functions must include data points up to 1/(2JIS), where JIS is the magnitude of the
416
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
active coupling constant leading to the cross-peak. Strongly resolutionenhancing window functions such as unshifted or slightly phase shifted sine bells are used for COSY spectra. As an added bonus, the use of an unshifted sine bell in t2 severely attenuates any residual H2O peak, leading to spectra with very flat baselines and no significant ridges emanating from the F2 water stripe. The t1 window function is a compromise between reduction of the diagonal tails (unshifted or slightly shifted sine bell) and sensitivity (increasingly shifted sine bell). An unshifted sine bell applied over 150 ms of the acquired data in t2 and a 158–308 phase-shifted sine bell applied over all data points in t1 will usually give adequate results for cross-peaks far from the diagonal, such as those arising from 1HN–1H correlations. Enhanced resolution and a concomitant reduction in diagonal peak intensity can be achieved by increasing t2max to 200–300 ms, and by shifting the F1 sine bell by 08–108. Such processing is used for the observation of cross-peaks close to the diagonal, such as in those involving leucine, valine, or isoleucine methyl resonances. The exact processing parameters used for the example ubiquitin spectra are described in the figure captions in this chapter. Figure 6.5 illustrates the use of strong resolution enhancement to aid in the observation of the correlations involving the methyl groups of the leucine, isoleucine, and valine. The choice of correct phase parameters in F2 is not readily apparent from the Fourier transform of the row of data collected with the smallest value of t1. In order to determine the phase parameters, a 1D pulse– acquire spectrum is acquired with the same pulse length, carrier position, and spectral width as for the 2D COSY experiment. The phase corrections determined for the 1D spectrum can then be applied during the F2 processing of the COSY, possibly with the zero-order phase correction adjusted by 908, depending on the exact phase cycles used in the two experiments. With the high speed of modern computer workstations, an alternative is to process the spectrum in both dimensions, determine the required phase parameters by examining several rows of the data, and then reprocess the entire spectrum with phase corrections added. The phase correction required in F1 will result from precession of the spins during the finite pulse lengths of the pulses on either side of t1, and during the initial value of t1. The amount of precession can be calculated, thus the F1 phase parameters may be calculated and applied during processing (Section 3.3.2.3). During the extraction of coupling constants from the cross-peak fine structure (Section 6.2.1.5), analysis should be performed on F2 sections of spectra that have been substantially zero-filled in this dimension. Obtaining a digital resolution of about 0.5 Hz/point is usually adequate.
417
2.0
b
2.0
d
1.0
c
2.0
1.0
F1 (ppm)
1.0
a
2.0
1.0
6.2 COSY-TYPE EXPERIMENTS
1.0
0.5
F2 (ppm)
FIGURE 6.5 The effect of t1 window functions on COSY spectra. The same data set was reprocessed with the sine bells applied in t1 shifted by the following amounts: (a) 308, (b) 208, (c) 108, (d) 58. As the window function becomes more resolution enhancing, the peaks close to the diagonal are less obscured by the dispersive tails from the diagonal.
418
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
In order to reduce the data storage requirements and processing time of such a large spectrum, F2 regions that do not contain the peaks of interest may be discarded after the F2 Fourier transformation. 6.2.1.4 Information Content The power of the COSY experiment lies in the ability to provide correlations between pairs of protons separated by two or three bonds. However, assigning entire spin systems in the COSY spectrum is rarely possible because of chemical shift degeneracy in the upfield region of the spectrum. Instead, the COSY spectrum is best used to identify correlations in the so-called fingerprint regions. These regions are well separated from each other and usually contain well-resolved cross-peaks, the number of which reflect the amino acid composition of the protein. The usual regions of interest are the 1 N 1 H – H cross-peaks (the backbone fingerprint), the 1H–1H crosspeaks (this region can get very crowded and is therefore of limited usefulness), cross-peaks between the aromatic resonances of phenylalanine, tyrosine, tryptophan, and histidine side chains, cross-peaks involving the isoleucine, valine, and leucine methyl groups, and the cross-peaks involving alanine and threonine methyl groups. Other correlation experiments such as TOCSY (Section 6.5) or relayed COSY (Section 6.2.2) can then be used to connect these fragments together to form complete spin system assignments. Figure 6.6 indicates these regions of the COSY spectrum while Figs. 6.7 and 6.8 show details of the two methyl fingerprint regions of ubiquitin. COSY is a relatively sensitive experiment (with the provisos given later for resonances with large linewidths or small coupling constants), and can be used to check sample purity or homogeneity. While the absence of some expected correlations in the fingerprint regions usually is indicative of experimental shortcomings (e.g., resonance overlap), the presence of extra resonances indicates deficiencies in the purity or conformational homogeneity of the sample. The COSY spectrum of a protein of N residues, containing P proline and G glycine residues, should display N P þ G 1 correlations in the 1HN–1H fingerprint region (the N-terminal residue usually is not observed). At low pH, arginine 1H–1H" cross peaks can be observed in the 1HN–1H fingerprint region. The number of observed correlations is often less than expected due to rapid amide proton exchange with solvent, coincidence of the 1H and water resonances (and therefore attenuation by the solvent presaturation process), or degeneracy of both 1H protons of a glycine residue. Double-quantum (2Q) spectroscopy (Section 6.4.1) can provide a useful means of observing correlations absent for the two latter reasons, while the pre-TOCSY COSY
419
6.2 COSY-TYPE EXPERIMENTS
0
Leu Val Ile CH3 Ha-Hb
5
Thr Ala CH3
F1 (ppm)
HN-Ha
10
Aromatic
10
5
0
F2 (ppm)
FIGURE 6.6 The five regions of the COSY spectrum containing the fingerprint cross-peaks.
(Sections 6.2.1.6 and 6.5.5) and gradient-enhanced 2Q filtered (2QF)COSY (Section 6.3) experiments circumvent problems of 1H coincidence with the water resonance. Ubiquitin contains 76 residues, including three prolines and six glycines; therefore, 78 1HN–1H resonances are expected. The 70 peaks are plainly visible in Fig. 6.9. A more detailed analysis of the COSY and other spectra indicates that the other eight cross-peaks are not observed for the following reasons: (i) the 1H resonances of Val5, Leu15, and Arg54 are coincident with the H2O signal, (ii) the amide protons of Glu24 and Gly53 are very broad, (iii) 1HN and 1H resonances of Asp52 0 and Asp60 have identical shifts, and (iv) 1H and 1H of Gly75 are degenerate. Thus, the backbone resonances of all residues can be identified.
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS 0.5
420 I30d
I3 δ
1.0
I44d
L69
I23d
I36g L73 L43
L56 I44g
I13g
L67
I61d
L50 I3g
2.0
L71
L15
I61g
1.5
I13d
V5
L8
F1 (ppm)
I36d
V70
1.0
I23g
0.8
I30g
V17 2.5
V26
0.6
0.4
F2 (ppm)
FIGURE 6.7 Section of the H2O COSY spectrum showing the cross-peaks in the leucine, valine, and isoleucine methyl fingerprint region. The cross-peaks are labeled with the corresponding resonance assignment. This spectrum is the same as that in Fig. 6.5d.
6.2.1.5 Quantitation of Scalar Coupling Constants in COSY Spectra The value of the 3J scalar coupling constant can be determined from the fine structure in a COSY cross-peak for spins without passive coupling partners, such as the 3 JHN H of residues other than glycine in H2O solution, and the 3 JH H for residues with a single 1H in D2O solution. The presence of a passive coupling in glycine 1HN–1H crosspeaks and 1H–1H cross-peaks within residues with two 1H protons complicates the analysis and the scalar coupling constants are more conveniently obtained from exclusive correlation spectroscopy (E.COSY) spectra (Section 6.3.3). Although the product operator analysis of COSY indicates that the lobes of a cross-peak in a two-spin system are separated by J12, the effect of the linewidth on the antiphase lineshape has not been considered. Qualitatively, as the linewidth approaches or exceeds the size of the coupling constant, cancellation of the positive and negative multiplet components reduces the intensity and increases the apparent separation
421 4.5
6.2 COSY-TYPE EXPERIMENTS
T12
A46 4.0
T14
F1 (ppm)
T66 A28
T55 4.5
T9 T7
T22
1.2
1.0
0.8
F2 (ppm)
FIGURE 6.8 Section of the COSY spectrum showing the cross-peaks in the alanine and threonine methyl fingerprint region. The cross-peaks are labeled by the one-letter amino acid code and the residue number. The arrow indicates a cross-peak falling outside the spectral region shown. For Thr7 and Thr22, the 1 H protons are coincident with the water resonance and are attenuated by presaturation. These correlations are observed in D2O solution, in a pre-TOCSY COSY (Section 6.2.1.5 and Fig. 6.15), and in a gradient-enhanced 2QF-COSY (Section 6.3.1 and Fig. 6.26).
of the multiplet as shown in Fig. 6.10. Quantitatively, the antiphase absorptive lineshape is given by [3.26] 1 1 1 þ ðJ12 2Þ2 =2FWHH 1 þ ðJ12 þ 2Þ2 =2FWHH 1 1 ¼ , 2 2 2 1 þ p ð1 2=J12 Þ 1 þ p ð1 þ 2=J12 Þ2
AðÞ ¼
½6:5
in which the individual multiplet components are assumed to have Lorentzian lineshapes with linewidth FWHH, p ¼ J12/FWHH, and the lineshape is normalized to unit amplitude. The extrema are obtained by setting the derivative of [6.5] with respect to equal to zero, and solving the resulting equation for : i1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J12 h ¼ pffiffiffi p2 þ 2 p4 þ p2 þ 1 1 : ½6:6 2 3p
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
422
V26
I30
E64
G76
Q31 G75 K63
L56
L50 L8
A28
R72 D39
I13
I3 S57 R74 L73
D52 D60
V70 V17
E34
V5
Q49
R42 Q51 K27
T7 H68
F45 K6
K11 T9
D58 E40
N25 K48
D21
L15
I44
Q41 K29
G47 D32
4.0
G53
K33 E62 S65
F1 (ppm)
G35
S20 Y59 I36
E16 R54
T14
T22
T12 T55 E18
L71
5.0
E24
L67
I61
G10
I23 A46
L69 Q2
T66 L43 F4
9.0
F2 (ppm)
8.0
FIGURE 6.9 1HN–1H fingerprint region of ubiquitin. The cross-peaks are labeled by the one-letter amino acid code and the residue number. The arrows indicate cross-peaks falling outside the spectral region shown. The boxes indicate residues whose H protons are coincident with the H2O resonance, hence the cross-peak is suppressed by presaturation. These correlations are observed in a pre-TOCSY COSY (Section 6.2.1.5 and Fig. 6.15) or gradient-enhanced 2QF-COSY (Section 6.3.1 and Fig. 6.26). See text for a more complete account of the peaks present in this spectrum.
Because this equation provides the values of at the maximum and minimum amplitude of the lineshape, substitution of [6.6] back into [6.5] allows a determination of the apparent peak heights of the multiplet lines. The resulting equation is shown graphically in Fig. 6.11. This confirms the qualitative view of Fig. 6.10: as J12 is reduced with respect to FWHH, the intensity of a COSY cross-peak decreases. Thus, in the case of equivalent linewidth and coupling constant (p ¼ 1), the resultant COSY peak intensity will be 80% of the value of a single Lorentzian line; linewidths in the range 6–8 Hz are expected for ubiquitin, and the intensity of many cross-peaks will be reduced by this amount or more. The decrease in cross-peak intensity caused by broad lines can severely limit the information content of a COSY spectrum, but more insidious is the effect on the separation of the peak maxima. Because the
423
6.2 COSY-TYPE EXPERIMENTS 1.0
Intensity
0.5
0.0
–0.5
–1.0 –30
–20
–10
0
10
20
30
n (Hz)
FIGURE 6.10 Changes observed for a Lorentzian antiphase absorptive doublet as the apparent peak separation varies as a function of linewidth. The Lorentzian lines are separated by 5 Hz and have a half-height width of 3 (solid curve), 5, 10, or 20 Hz. All curves are plotted such that a single Lorentzian line would have a height of unity.
1.0
Intensity
0.8
0.6
0.4
0.2
0
1
2
3
4
(J12/∆n FWHH)
FIGURE 6.11 Peak height of a Lorentzian antiphase absorptive doublet varies as the ratio of the peak separation to the linewidth, J12/FWHH.
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
424
% increase in separation
100
80
60
40
20
0
1
2
3
4
(J12/∆n FWHH)
FIGURE 6.12 Apparent separation of peak maxima relative to the actual separation of a pair of absorptive antiphase Lorentzian lines as a function of linewidth. As J12/FWHH approaches infinity, the separation approaches J12. As J12/FWHH approaches zero, the separation tends to a limit of FWHH/31/2.
values of given by [6.6] represent the frequency at which the amplitude is a maximum or a minimum (given by the ‘‘þ’’ or ‘‘’’ solutions, respectively), the total separation of the peak maxima is given by i1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J12 h : a ¼ pffiffiffi p2 þ 2 p4 þ p2 þ 1 1 3p
½6:7
This equation was first described by Neuhaus et al. (6) and is most useful in the form shown in Fig. 6.12. The graph indicates that the observed separation is always larger than the actual value of J12. Moreover, Fig. 6.12 indicates that the smaller the value of p (the larger the linewidth with respect to the multiplet separation), the larger the difference between the actual separation of the initial Lorentzian lines (J12) and the observed separation of the peak maxima. Thus, to continue the example just described, when the linewidth is equal to the J12 coupling (p ¼ 1), the observed separation of peak maxima will be 7% higher than J12. The error in J12 increases rapidly as p decreases below 1. Self-cancellation is especially problematic for residues in a helical environment; the 3 JHN H coupling constant calculated for an ideal helix (with ¼ 608 and ¼ | 608|) using the Karplus equation [9.2] is 4.2 Hz.
425
6.2 COSY-TYPE EXPERIMENTS
a
b
J
relative intensity
0.5
vd
0
–0.5 va –20
0
20
–20
0
20
n (Hz)
FIGURE 6.13 Measurement of the maxima separation for a pair of antiphase Lorentzian lines in absorption (a) and dispersion (d). In both panels, real separation is 5.0 Hz (denoted by J in the left-hand panel) and the half-height linewidth is 10 Hz (p ¼ 0.5). Measurement of d is difficult when p 4 1.0 because of the low intensity of the broad positive lobes.
A method of determining scalar coupling constants that takes into account the natural properties of a Lorentzian line to overcome the problems of self-cancellation has been proposed by Kim and Prestegard (8). The method takes advantage of the differences in lineshapes of absorptive and dispersive antiphase Lorentzian multiplets. Analogously to [6.7], a pair of simultaneous equations describing the coupling constant and the linewidth, J12 and FWHH, in terms of the absorptive and dispersive peak separations, a and d, can be solved to give (Fig. 6.13): 6 4 642a JIS 642a 2d JIS þ 1446a þ 964a 2d þ 362a 4d J2IS ½6:8 þ 818a 366a 2d 424a 4d 42a 6d þ 8d ¼ 0: The real root of this cubic equation in J212 provides the coupling constant. The method is performed most easily by processing the COSY with antiphase absorptive cross-peaks (i.e., in the normal way, although
426
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
with high digital resolution). The absorptive peak separation, a, is measured from a row (F2 cross-section) through a given cross-peak. Adding a zero-order phase correction of 908 to the row allows measurement of the dispersive peak separation, d, for the same crosspeak. This method assumes that the lineshape is Lorentzian and is not applicable if window functions such as shifted sine bells or Lorentzian– Gaussian transformations have been employed in t2. The method works reasonably well for cross-peaks with high signal-to-noise ratio; however, for noisy data, accurate measurement of d is difficult because the tops of the peaks are broad. Alternatively, [6.5] can be fit directly to the F2 cross-section through a cross-peak (or several rows co-added to improve the S/N ratio) by a nonlinear least-squares algorithm to determine the values of the scalar coupling constant and linewidth most consistent with the lineshape (9, 10). Once again, the spectrum should be processed to maintain Lorentzian lineshapes in F2. The accuracy of this approach also is compromised as p decreases, owing to the lower S/N ratios. Several methods of analysis of 1H spectra have been reported to be of use in situations where the linewidth is greater then J12 (11–13). Practically, scalar coupling constants that are less than about half of the linewidth are difficult to measure by analysis of antiphase splittings in the COSY experiment. Even the ‘‘best’’ method of direct line-fitting is rarely useful for the larger proteins studied by NMR (415 kDa) and scalar coupling constants are better obtained using heteronuclear experiments (Section 7.5). 6.2.1.6 Experimental Variants The basic COSY pulse sequence has been extensively modified since its initial conception. Four examples are briefly mentioned here: COSY-, purged COSY (P.COSY), pre-TOCSY COSY, and COSY with SCUBA. The first of these provides a means to simplify cross-peak fine structure while the latter three address deficiencies in the basic experiment. In the COSY- experiment, the rotation angle of the second pulse is 5 908. Reduction of the length of the final pulse in the COSY- experiment has two effects on the resulting spectra. First, the diagonal peaks are more intense relative to the cross-peaks because coherence transfer scales as sin2(). Second, the fine structure of the cross-peaks changes because connected and unconnected transitions have different intensities. In spin systems with three or more mutually coupled spins, certain components of a given cross-peak arising from passive coupling will be reduced in intensity. Acquisition with ¼ 358 provides reasonable sensitivity while suppressing cross-peak components arising from
427
6.2 COSY-TYPE EXPERIMENTS COSY-35
3.5
COSY
HN-Hα"
G10
4.0
G10
K29
HN-Hα'
F1 (ppm)
HN-Hα"
HN-Hα'
4.5
N25 7.9
7.8
7.9
7.8
F2 (ppm)
FIGURE 6.14 Cross-peaks of Gly10 in the COSY-35 experiment. Compared to the normal COSY, contributions from the passive coupling 2 JH0 H00 are removed from the 1HN–1H cross-peaks in F1 while the passive coupling of 1HN to the other 1H is removed in F2. The fine structure of cross-peaks to nonglycine residues is not affected by the reduced flip angle.
unconnected transitions by tenfold, compared to those of connected transitions (4). An example of the simplification obtained is shown in Fig. 6.14 for the cross-peaks arising from Gly10. The multiplet fine structure in COSY- is similar to that observed in E.COSY (14), and will be discussed in more detail later (Section 6.3.3). Although the removal of some elements of fine structure can facilitate identification of peaks in crowded regions of the spectrum, the COSY- experiment offers little benefit for spin system identification and principally permits scalar coupling constants to be measured without errors of the type described in Fig. 6.12. The P.COSY experiment provides a way to remove the dispersive diagonal tails of the COSY experiment without a significant loss in sensitivity (15). The central idea in the technique is similar to that of the primitive E.COSY (P.E.COSY; Section 6.3.3). In the P.E.COSY, a COSY-35 and a COSY-0 are subtracted, with the result that the diagonal is severely attenuated (16). This experiment suffers from a lack of sensitivity because the cross-peaks have low intensity in the COSY-35
428
CHAPTER 6 EXPERIMENTAL I H NMR METHODS
spectrum and are not present at all in the COSY-O spectrum. These issues are rectified in the P.COSY by generating cross-peaks in a normal COSY experiment and preparing a "diagonal-only" spectrum by repeated left shifting of a 10 spectrum acquired with high SIN (15). The resulting spectrum is comparable in quality to a 2QF-COSY (Section 6.3.1) but with twofold greater sensitivity. As already described, cross-peaks in the IHN_IHO' fingerprint region of COSY spectra are attenuated by presaturation of the solvent resonance. The simple expedient of including a short isotropic mixing period (Section 4.2.2) between the presaturation period and the first 90° pulse restores intensity to these cross-peaks. The mixing period transfers magnetization from IHN to IHO' via the 3JI-INW scalar coupling interaction; therefore, I HO' magnetization is present at the start of the COSY sequence even for the protons coincident with the water resonance. This variant is referred to as the pre-TOCSY COSY (17). An example of such a spectrum is shown in Fig. 6.15; the cross-peaks
"
..t
0"
~
~? .,0
§§
V5
L15
eo
o
0.,
·S~"0
~.
D
<0
~t-~ 6
Tt
..t
E Cl.
ro ..t
.s u::
T22 0
to
GO C»
~
9.2
9.0
8.8
N
to
1.2
F2 (ppm)
FIGURE 6.15 Two sections of the pre-TOCSY COSY spectrum of ubiquitin. The experiment was recorded with 1.5 s of presaturation followed by 27 ms of DIPSI-2rc isotropic mixing (70); 16 transients were collected for each of 800 increments of II (Ilmax = 69 ms). The labeled cross-peaks were not present in the standard COSY experiment (Figs. 6.9 and 6.8) because of presaturation.
6.2 COSY-TYPE EXPERIMENTS
429
from Val5, Leu15, and Arg54 are plainly visible in this spectrum, but are missing from Fig. 6.9. Further discussion of pre-TOCSY sequences can be found in Section 6.5.5. Cross-relaxation between spins whose resonances have been unavoidably saturated along with the water and spatially nearby unsaturated spins can be used as an alternative to the pre-TOCSY technique. This simple strategy, referred to as SCUBA (Stimulated Cross-peaks Under Bleached Alphas) (18), involves inserting a short period after the water presaturation sequence has ended and just prior to the initial 908 pulse of the COSY sequence. During this time period, which should be kept short relative to the 1H T1 values, equilibration occurs as a result of dipolar cross-relaxation between the saturated and unsaturated protein resonances. Thus, some magnetization will be restored to the saturated resonances prior to the start of the COSY sequence, allowing signals to be observed from the normally bleached region of the spectrum. This technique can be employed in any experiment that uses presaturation to suppress the water signal. Caution needs to be taken in using the SCUBA and pre-TOCSY techniques if peak intensities or volumes are to be quantified.
6.2.2 RELAYED COSY Conceptually, the relayed coherence transfer experiments (RCTCOSY, relayed COSY, or R.COSY) are simple extensions of the COSY experiment (19). Instead of acquiring the spectrum after a single coherence transfer step from spin I1 to spin I2, a delay is introduced prior to acquisition to allow antiphase magnetization to develop for a second time. If spin I2 has coupling partners other than I1 (e.g., spin I3), antiphase coherence will develop between I2 and I3 and a third 908 pulse will transfer coherence from I2 to I3. Because chemical shifts are only monitored during t1 (prior to the first coherence transfer step) and t2 (after the second coherence transfer step), the result will be a cross-peak between spin I1 and spin I3 even if I1 and I3 are not directly coupled. For simple molecules containing only a few resonances, such spin gymnastics are redundant, because in the COSY experiment both the I1 ! I2 and I2 ! I3 cross-peaks are readily observed. However, in the complex spectra of proteins, chemical shift degeneracy of spin I2 with resonances of other spin systems can hinder the assignment of I1 and I3 to the same spin system. In such cases, the cross-peak between I1 and I3 in the R.COSY experiment places I1 and I3 unequivocally in the same amino acid spin system. The pulse sequence used to observe relayed coherence transfer is shown in Fig. 6.16.
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
430 f1
f2 t1
f3 t 2
f4 t 2
t2
p +1 0 –1
FIGURE 6.16 The pulse sequence and coherence level diagram for the R.COSY experiment. The phases are cycled as follows: 1 ¼ 4(x, x); 2 ¼ 4(x), 4(x); 3 ¼ 8(x); 4 ¼ 2(x, x, x, x); and receiver ¼ 4(x, x). CYCLOPS is performed on all pulses and the receiver to produce a 32 step cycle. The 1808 pulse during the mixing period can be a composite 902 902þ=2 902 pulse. Frequency discrimination in F1 is obtained by shifting the phase of 1 and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4).
6.2.2.1 Product Operator Analysis For this analysis, the linear spin system I1–I2–I3 is considered, in which J12 and J23 are resolved and J13 ¼ 0. Considering initial I1 magnetization, the operator terms present after the 90x – t1 – 90x sequence are given by [6.1]. The 2I1zI2y term of this expression represents transfer of coherence from I1 to I2 and is the first step in the generation of an I1 ! I3 cross-peak. Appropriate phase cycling preserves this term (and also the I1x single-quantum coherence) and suppresses other coherence orders. During the spin echo =2 – 180x – =2 sequence, chemical shifts are refocused and evolution under the J12 and J23 scalar coupling Hamiltonians yields the following terms: =2x =2
I1x sinð1 t1 Þ cosðJ12 t1 Þ 2I1z I2y sinð1 t1 Þ sinðJ12 t1 Þ ! I1x cosðJ12 Þ þ 2I1y I2z sinðJ12 Þ sinð1 t1 Þ cosðJ12 t1 Þ þ 2I1z I2y cosðJ23 Þ þ 4I1z I2x I3z sinðJ23 Þ sinð1 t1 Þ sinðJ12 t1 Þ cosðJ12 Þ þ ½I2x cosðJ23 Þ þ 2I2y I3z sinðJ23 Þ sinð1 t1 Þ sinðJ12 t1 Þ sinðJ12 Þ:
½6:9
6.2 COSY-TYPE EXPERIMENTS
431
Application of the final 90x pulse creates the following terms immediately prior to t2: I1x cosðJ12 Þ 2I1z I2y sinðJ12 Þ sinð1 t1 Þ cosðJ12 t1 Þ þ 2I1y I2z cosðJ23 Þ þ 4I1y I2x I3y sinðJ23 Þ sinð1 t1 Þ sinðJ12 t1 Þ cosðJ12 Þ þ I2x cosðJ23 Þ 2I2z I3y sinðJ23 Þ sinð1 t1 Þ sinðJ12 t1 Þ sinðJ12 Þ:
½6:10
Of the six components in [6.10], only the three-spin coherence 4I1yI2xI3y is unobservable. The most interesting peak arises by relay from I1 via I2 to I3, and is described by the 2I2zI3y term. This peak appears at the I3 chemical shift and is antiphase with respect to spin I2 in F2. It appears at the chemical shift of I1 in F1, and the sin(1t1) sin(J12t1) modulation indicates antiphase lineshape with respect to I2. In the normal mode of display, relay peaks are phased to be absorptive in both dimensions and this will be assumed in the following discussion of the phases of other peaks. The I1x and 2I1yI2z terms contribute to the I1 diagonal peak. The former is analogous to the diagonal peak in a normal COSY experiment and has a dispersive in-phase lineshape in both dimensions. The second diagonal component is an absorptive antiphase signal in both dimensions. Thus, as in the regular COSY experiment, the diagonal is dominated by dispersive in-phase terms that must be attenuated by severe window functions. Finally, the R.COSY spectrum contains cross-peaks between spins that are directly coupled (‘‘COSY’’-type peaks). The 2I1zI2y and I2x terms of [6.10] contribute to these peaks. The former possesses absorptive antiphase character in F2 and dispersive in-phase character in F1, while the latter possesses dispersive in-phase character in F2 and absorptive antiphase character in F1. The contribution of each component to the overall lineshape of these peaks is dependent on the length of the mixing period as described by the sin(J12) and cos(J23) sin(J12) trigonometric functions, respectively. These peaks can never be phased to pure absorption lineshapes, and dispersive tails emanate from the COSY peaks in the R.COSY spectrum. Although the results observed in R.COSY spectra of the larger spin systems found in real amino acids are more complex than the equations just described, the general trends are similar. A more complete description of the transfer functions in particular amino acids has been presented by Bax and Drobny (20).
432
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
6.2.2.2 Experimental Protocol From the preceding discussion, cancellation of antiphase cross-peaks also must be avoided in R.COSY experiments. In addition to the standard COSY acquisition parameters, a value must be chosen for the mixing delay, , during which the second antiphase state develops. In H2O solution, the most useful peaks appear at the chemical shift of 1H in F1 and at the chemical shift of 1HN in F2 (Section 9.1.1). For such cross-peaks, coherence will be transferred from 1 H to 1H during t1, and from 1H to 1HN during the relay period, i.e., I1 ¼ 1H, I2 ¼ 1H, and I3 ¼ 1HN. Given the preceding discussion, t1max must be long enough to allow transfer via the 1H–1H coupling i:e:, t1 max 1=ð23JH H Þ . For many residues in folded proteins, at least one 3JH H coupling will be in the 8- to 12-Hz range. Thus, R.COSY experiments utilize t1max in the range of 40 to 60 ms. For an F1 spectral width of 5 kHz, 400 to 600 t1 increments are recorded (assuming quadrature detection with TPPI). During the mixing time, , the antiphase coherence transferred from 1 H to 1H during t1 must refocus with respect to 1H and defocus with respect to 1HN. Thus, the magnitude of the transfer from 1H to 1HN depends on both 3 JHN H and 3 JH H . Analytical functions describing the transfer have been calculated for a variety of spin system types using values of coupling constants commonly found in proteins (20, 21). Maxima in the transfer functions occur for ¼ 40–60 ms; however, relaxation during reduces the intensity of all peaks. Therefore, compromise values of ¼ 20–40 ms provide adequate coherence transfer without extensive relaxation losses. Usually, with a 2–4 mM sample, adequate 1HN–1H cross-peak intensities are obtained by acquiring 32 to 64 transients per t1 increment. The R.COSY spectrum analyzed below was acquired from a 2 mM ubiquitin solution in H2O. Thirty-two transients were collected for each of 800 increments (t1max ¼ 69 ms). A mixing time of 22 ms was used during the relay portion of the experiment (). A composite 1808 pulse was used to refocus chemical shifts in the middle of (Section 3.4.2). The total acquisition time was 14 hr. 6.2.2.3 Processing Given the results of the product operator analysis, processing of R.COSY data is very similar to that used for COSY data. Harsh window functions (unshifted sine bells in F2 and slightly shifted sine bells in F1) are required to minimize the dispersive diagonal peaks and COSY-type cross-peaks. 6.2.2.4 Information Content The most interesting cross-peaks in the R.COSY experiment result from two coherence transfer steps.
6.2 COSY-TYPE EXPERIMENTS
433
As a rule, all possible 1H–1HN cross-peaks in the R.COSY spectrum are not observed, because of the dependence on two coupling constants (either or both of which could be small). However, in side chains containing CH2 groups, at least one of the 1H–1H couplings is commonly greater than 8 Hz and provides an efficient means of generating one 1H–1HN cross-peak. These generalizations are exemplified by the peaks observed in the R.COSY spectrum of ubiquitin. Of the 45 side chains that have an observable backbone amide proton and a -methylene group, six residues have no 1HN–1H correlations, 30 (67%) have a single 1HN–1H correlation, and in only 9 cases (20%) are both 1HN–1H correlations observed. Also, 11 of the 18 side chains with an amide proton and a single -proton exhibit an 1HN–1H cross-peak. Although acquisition of R.COSY spectra with longer mixing times can permit identification of additional 1HN–1H correlations arising from small 3 JHN H coupling constants, double-relayed COSY (Section 6.2.3) and TOCSY (Section 6.5) experiments are better suited to detecting such correlations. The section of the R.COSY spectrum shown in Fig. 6.17a depicts several of the relayed 1HN–1H cross-peaks. The section of the R.COSY spectrum shown in Fig. 6.17b depicts the 1HN–1H correlations to emphasize the dispersive nature of the COSY-type peaks in the R.COSY spectrum.
6.2.3 DOUBLE-RELAYED COSY Additional coherence transfer steps can be obtained by inserting a second /2–1808–/2–908 sequence prior to t2 in the R.COSY experiment (22). This variant is termed the double-relayed COSY (DR.COSY), and the pulse sequence is shown in Fig. 6.18. The DR.COSY correlates protons through a network of three scalar couplings. In principle, the COSY experiment could be extended indefinitely until correlations are observed between all spins in a particular spin system. However, inefficient coherence transfer and relaxation during the periods of free precession reduce the sensitivity to such an extent that the DR.COSY represents the practical limit of such experiments. Set up for DR.COSY contains essentially the same considerations as for the R.COSY, except that values must be selected for both 2 and 1. The most useful peaks will be transferred through three couplings to 1 N H . Instead of optimizing the transfer from 1H to 1HN, the most 1 0 1 00 effective results are00 obtained by transfer from H to H during t1 (via 0 00 the large 1H –1H coupling), 1H to 1H during 1 (via the largest of the two possible 1H–1H couplings), and from 1H to 1HN during 2. In this way, relatively intense cross-peaks are obtained to both 1H protons,
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
434
a E18 E34
1.5
Q2
E40 E51 2.0
Q49
2.5
F1 (ppm)
E16
4.5
b
9.0
8.5 F2 (ppm)
8.0
FIGURE 6.17 Two sections of an R.COSY spectrum ( ¼ 22 ms) of ubiquitin in H2O solution showing (a) 1HN–1H and (b) 1HN–1H correlations. The glutamate and glutamine resonances in the depicted region are surrounded by rectangles. Note that only one of the possible 1HN–1H correlations has appreciable intensity and that many of the 1HN–1H COSY-type peaks have distinct dispersive tails.
and DR.COSY represents a significant improvement over R.COSY, in which correlations usually are observed to only one 1H. Although a product operator analysis is not explicitly presented here for the DR.COSY, important results are obtained by analogy with the R.COSY sequence. Double-relay cross-peaks are absorptive antiphase in both dimensions, whereas COSY and single-relay peaks contain some inphase dispersive contributions. Choices of mixing time to optimize the intensity of various correlations have been determined by calculating the cross-peak intensities as a function of 2 and 1 for a variety of possible coupling constants and spin systems (21). Setting the delays 1 ¼ 28 ms
435
6.2 COSY-TYPE EXPERIMENTS f1
f2 t1
f3 t1 2
f4 t1 2
f5 t2 2
f6 t2 2
t2
p +1 0 –1
FIGURE 6.18 Pulse sequence and coherence level diagram for the DR.COSY experiment. The 1808 pulses can be replaced with composite pulses as described in Fig. 6.16. The phases are cycled as follows: 1 ¼ 8(x, x); 2 ¼ 8(x), 8(x); 3 ¼ 16(x); 4 ¼ 4(x), 4(x), 4(x), 4(x); 5 ¼ 16(x); 6 ¼ 4(x, x, x, x); and receiver ¼ 8(x, x). CYCLOPS is performed on all pulses to yield a 64-step phase cycle. Frequency discrimination in F1 is obtained by shifting the phase of 1 and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4).
0
00
and 2 ¼ 35 ms maximizes the 1H –10H –100H–1HN transfer. Transfer during t1 is via the large 15-Hz 1H –1H coupling, so t1max can be slightly shorter than in the R.COSY; however, the cross-peaks are still antiphase, and subject to self-cancellation if t1max is too short. Alternatively, setting the delays 1 ¼ 20 ms and 2 ¼ 31 ms will maximize transfer from 1H to 1HN (21, 23). Total acquisition times usually need to be longer than for R.COSY because the most interesting correlations result from three transfer steps. Forty-eight-hour acquisitions are not uncommon in the literature for this experiment. Processing is analogous to that used in COSY and R.COSY experiments (Sections 6.2.1.3 and 6.3.1.3). The spectrum depicted in Fig. 6.19 was acquired from an H2O solution of ubiquitin. Sixty-four transients were collected for each of 576 increments in t1, with a t1max ¼ 50 ms and a total acquisition time of 21 hr. The mixing times 1 and 2 were set to 28 and 35 ms, respectively, to emphasize H–HN correlations, and composite 1808 pulses were used in the middle of both delays (Section 3.4.2).
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
1.5
436
2.0
Q49
F1 (ppm)
E40
E16
E34 E18 2.5
Q2
E51 9.0
8.5
8.0
F2 (ppm)
FIGURE 6.19 Section of a DR.COSY spectrum ( 1 ¼ 28 ms, 2 ¼ 35 ms) of ubiquitin acquired from H2O solution. The same spectral region as in the top of Fig. 6.17 is shown, with glutamine and glutamate 1H and 1H resonance positions denoted by the rectangles and ellipses, respectively. The particular delays used during the relay periods have optimized transfer of both 1H to 1HN at the expense of 1H –1HN correlations.
Figure 6.19 depicts a region of the DR.COSY in which several correlations from both protons of -methylene groups are observed to the backbone amide proton. In the equivalent region of the R.COSY, only one of the correlations is observed (compare with Fig. 6.17). Correlations from 1H to 1HN are absent for the glutamine and glutamate side chains identified in Fig. 6.19. In the spectrum as a whole, with mixing times 1 ¼ 28 ms and 2 ¼ 35 ms, correlations from 0 1 00 H and 1H to 1HN are observed in 39 (87%) of the 45 residues with a -methylene group. Of these side chains, 31 also contained -protons, but in only three cases were 1H –1HN correlations observed (both 1 1 N H – H correlations were also observed for these three cases). For side chains containing a single -proton, more 1HN–1H correlations were observed in the DR.COSY than in the R.COSY experiment (14 out of 18, compared to 11 out of 18), and 13 correlations from 1HN to at least one 1H were observed.
437
6.3 MULTIPLE-QUANTUM FILTERED COSY
6.3 Multiple-Quantum Filtered COSY The double-quantum filtered (2QF)-COSY experiment was developed by the Ernst and Freeman laboratories as an alternative to COSY (24–26). The 2QF-COSY is actually the simplest version of a family of experiments based on filtration through a p-quantum state. Not all amino acid spin systems are capable of achieving some of the higher quantum states; hence, experiments with p 4 2 provide useful spectral simplification (27). All pQF-COSY experiments that use phase cycling for coherence selection have the same basic pulse sequence consisting of three 908 pulses, with the first two separated by t1 and the last two separated by a short delay, as shown in Fig. 6.20. As described in Section 4.3.2.1, phase cycling is used to cancel all but the particular coherence level of interest.
f2 f3
f1 t1
t2
p +2 +1 0 –1 –2
FIGURE 6.20 Pulse sequence and coherence level diagram for a 2QF-COSY experiment. The basic four-step phase cycle for 2QF-COSY is 1 ¼ 2 ¼ x, y, x, y; 3 ¼ x; and receiver ¼ 2(x, x). The same pulse sequence is used for 3QF-COSY with the following basic six-step phase cycle: 1 ¼ 2 ¼ 08, 608, 1208, 1808, 2408, 3008; 3 ¼ 08; and receiver ¼ 3(08, 1808). Axial peak suppression and CYCLOPS phase cycling yields a 32-step phase cycle for 2QF-COSY and 48step phase cycle for 3QF-COSY. An improved phase cycle for the 2QF-COSY that suppresses rapid pulsing artifacts is 1 ¼ 4(y), 4(y); 2 ¼ 4(x), 4(x); 3 ¼ (y, x, –y, x, x, –y, x, y); and receiver ¼ (x, –y, x, y, –y, x, y, x) (32, 33). Frequency discrimination in F1 is obtained by shifting the phase of 1 and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4).
438
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
Selection of a desired order of multiple-quantum filtration is achieved by incrementing the phases of the pulses as follows: 1 and 3: receiver:
2:
n=p 0 n
½6:11
In [6.11], the integer n is incremented from 0 to 2p 1; thus, the basic phase cycle consists of 2p steps. Note that for p 4 2, appropriate selection involves phase shift increments of less than 908, hence the x, y, x, y notation is no longer appropriate to describe the phases. Such shifts are readily accomplished with the digital phase-shifting hardware present on modern spectrometers. Alternative cycles involving phase shifting of 3 also can be used to obtain the desired coherence selection. Formally, these phase cycles retain coherence orders kp, where k ¼ 1, 3, 5, . . . , but the higher order coherences have low intensity (see later). In addition to the basic 2p cycle, artifacts in the spectra are reduced by performing CYCLOPS on all pulses and axial peak suppression on the first pulse (Section 4.3.2.3), leading to 16p steps in the cycle. A number of approaches have been developed for incorporating coherence selection using pulsed field gradients into the pQF-COSY experiment. In the approach shown in Fig. 6.21, the basic pulse sequence incorporates short spin echo periods to allow the application of gradient pulses during the multiple-quantum filter and prior to acquisition (28, 29). As discussed in Section 4.3.3.1, filtration for signed coherence order p is obtained by requiring that the coherence transfer p ! 1 is selected by the gradient pair. Assuming that the gradient pulses have the same duration and amplitudes G1 and G2, respectively, the filter requires that G2 ¼ pG1:
½6:12
Thus, for example, a þ2 quantum filter is obtained by setting G2 ¼ 2G1. Phase cycling is not necessary for coherence selection and the only phase cycling utilized is for artifact suppression using CYCLOPS or EXORCYCLE (Section 4.3.2.3). The gradient-selected pQF-COSY experiment has the principal disadvantage of being a factor of two less sensitive than the phase-cycled experiment because only the coherence order p1 and not p, can be transferred to detectable
439
6.3 MULTIPLE-QUANTUM FILTERED COSY 1
H
f1
f2 f2 f3 f3 t1
t2
Grad G1
G2
p +2 +1 0 –1 –2
FIGURE 6.21 Pulse sequence and coherence level diagram for a gradientenhanced 2QF-COSY experiment. The gradient pair satisfy the relationship G2 ¼ pG1 to select the signed coherence level p ¼ þ2 The basic four-step phase cycle for 2QF-COSY is 1 ¼ 4(y), 4(x); 2 ¼ 4(x), 4(y); 3 ¼ y, x, y, x, x, y, x, y; and receiver ¼ x, y, x, y, y, x, y, x. Frequency discrimination in F1 is obtained by shifting the phase of 1 and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4). A bipolar pair of gradients can be inserted into the t1 period to prevent radiation damping (31).
magnetization. In addition, the gradient-selected experiment is slightly longer, allowing additional relaxation losses and unwanted evolution of scalar coupling interactions. These disadvantages are outweighed by the high degree of filtration that is obtained by gradient selection. In particular, when magic-angle field gradient pulses are utilized (Section 4.3.3), very high-quality water suppression can be obtained without presaturation of the solvent resonance (28, 30). If the water signal is not suppressed prior to the t1 evolution period, then the full water 1H magnetization is in the transverse plane at the
440
CHAPTER
6 EXPERIMENTAL IH NMR METHODS
start of tl. Difficulties will arise because the consequent radiation damping of the water 1H magnetization will have the effect of a selective rf pulse coincident with the water resonance frequency. Thereby, radiation damping will result in undesirable perturbations of protein resonances that are (nearly) degenerate with the water resonance. These perturbations can have the form of phase distortion and/or signal attenuation of the affected resonances. These difficulties are avoided if significant radiation damping is prevented from occurring during t I. An effective solution with general applicability was proposed by Sklenar- and consists of.the application of a pair of bipolar gradient pulses during the tl period (31). The first gradient pulse, applied at the beginning of the tl period, defocuses the magnetization and therefore severely attenuates the effect of radiation damping. A second gradient pulse, of equal strength but opposite polarity, is then applied at the end of the tl period in order to refocus the 1H magnetization. In the event that the evolution period contains rf pulses for some purpose, the polarity and exact positioning of the defocusing/refocusing gradient pulses may need to be modified. Careful attention to the trajectory of the water magnetization is essential for the execution of any NMR experiment performed on a sample in H 20 solution.
6.3.1 2QF-COSY As noted previously, one of the major shortcomings of the COSY experiment is the 90° phase difference between the diagonal and crosspeaks, leading to a dispersive diagonal in the usual mode of presentation. The intense, sprawling nature of the diagonal, particularly for uncoupled singlet resonances, distorts and obscures cross-peaks close to the diagonal. The double-quantum filtered COSY experiment overcomes some of the drawbacks of COSY (24, 25). The multiplet structures of cross-peaks in the COSY and 2QF-COSY are identical. The benefits of the 2QF-COSY are the absorptive antiphase lineshapes of the diagonal peaks of coupled spins and the severe attenuation of diagonal peaks from uncoupled resonances. The drawbacks of 2QF-COSY are a twofold decrease in sensitivity and a longer phase cycle.
6.3.1.1 Product Operator Analysis To begin, a scalar-coupled twospin system is treated. The product operators present following the pulse sequence segment, 90~ - tl - 90~ are given by [6.1]. The phase cycle or gradient pair selects only double-quantum coherence present after the second pulse; therefore, all terms can be disregarded except the multiplequantum term -2hJ2y cOS(Q1tl) sin(n'l12tl)' This product operator is a
6.3 MULTIPLE-QUANTUM FILTERED COSY
441
superposition of double-quantum and zero-quantum coherence and can be expanded accordingly as (see Section 2.7.5), ð1=2Þ½2I1x I2y þ 2I1y I2x þ 2I1x I2y 2I1y I2x cosð1 t1 Þ sinðJ12 t1 Þ ¼ ð1=2iÞ½I1þ I2þ I1 I2 I1þ I2 þ I1 I2þ cosð1 t1 Þ sinðJ12 t1 Þ ¼ ½DQy ZQy cosð1 t1 Þ sinðJ12 t1 Þ:
½6:13
In the phase-cycled 2QF-COSY, the double-quantum coherence DQy ¼ (1/2)(2I1xI2y þ 2I1yI2x) is selected by the phase cycle. The zeroquantum coherence ZQy ¼ (1/2)(2I1yI2x – 2I1xI2y) is rejected by the phase cycle. The final 90x pulse generates the following observable singlequantum coherence terms: ð12Þ½2I1x I2y þ 2I1y I2x cosð1 t1 Þ sinðJ12 t1 Þ ð2 Þx ! ð12Þ½2I1x I2z þ 2I1z I2x cosð1 t1 Þ sinðJ12 t1 Þ:
½6:14
In the gradient-selected 2QF-COSY, only the double-quantum coherence I1þ I2þ ¼ (1/2)[2I1xI2x þ 2I1yI2y þ i(2I1xI2y þ 2I1yI2x)] is selected by the gradient pair G1 and G2. Only the terms 2I1xI2y and 2I1yI2x are converted to observable single-quantum coherences by the final 90x pulse: ð14Þ½2I1x I2y þ 2I1y I2x cosð1 t1 Þ sinðJ12 t1 Þ ð2Þx ! ð14Þ½2I1x I2z þ 2I1z I2x cosð1 t1 Þ sinðJ12 t1 Þ:
½6:15
The terms 2I1xI2x and 2I1yI2y result in undetectable multiple-quantum and longitudinal two-spin order after the final 90x pulse. As indicated by the prefactors in [6.14] and [6.15], the sensitivity of the gradient-selected experiment is reduced by a factor of two compared to the phase-cycled experiment, but otherwise, the signal obtained is identical (neglecting evolution and relaxation during the spin echo periods of the gradientselected experiment). Simple inspection shows that the 2I1xI2z product operator evolves at the chemical shift of the I spin and is antiphase with respect to the scalar coupling during t2. The 2I1zI2x product operator evolves at the chemical shift of the I2 spin and is antiphase with respect to the scalar coupling during t2. Because both terms evolve with a frequency of 1 during t1, the former represents an I1 spin diagonal peak, and the latter represents
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
442
an I1 ! I2 cross-peak. The diagonal and cross-peak terms have the same phase in F2 (both terms contain x-operators) and are modulated by the same t1 trigonometric terms; expansion of the trigonometric coefficients indicates that both the diagonal and the cross-peaks are antiphase in the F1 dimension. Consequently, the diagonal and cross-peaks in a 2QFCOSY spectrum can be phased to pure absorption in both dimensions. Thereby, the 2QF-COSY experiment eliminates problems associated with dispersive tails emanating from the diagonal resonances in COSY spectra. This attribute influences the processing and extraction of information from 2QF-COSY spectra. Consideration of the evolution of a three-spin system during a 2QFCOSY is useful at this point to reveal further features of the spectrum. The system now contains three spins, I1, I2, and I3, with couplings J12, J13, and J23. Evolution of initial I1y magnetization under the chemical shift, J12 scalar coupling, and J13 scalar coupling Hamiltonians lead to the following antiphase terms: ð2 Þx Iz ! I1y t1
! 2I1x I2z cosð1 t1 Þ sinðJ12 t1 Þ cosðJ13 t1 Þ þ 2I1y I2z sinð1 t1 Þ sinðJ12 t1 Þ cosðJ13 t1 Þ þ 2I1x I3z cosð1 t1 Þ cosðJ12 t1 Þ sinðJ13 t1 Þ þ 2I1y I3z sinð1 t1 Þ cosðJ12 t1 Þ sinðJ13 t1 Þ þ 4I1y I2z I3z cosð1 t1 Þ sinðJ12 t1 Þ sinðJ13 t1 Þ 4I1x I2z I3z sinð1 t1 Þ sinðJ12 t1 Þ sinðJ13 t1 Þ
½6:16
ð2 Þx
! 2I1x I2y cosð1 t1 Þ sinðJ12 t1 Þ cosðJ13 t1 Þ: 2I1z I2y sinð1 t1 Þ sinðJ12 t1 Þ cosðJ13 t1 Þ 2I1x I3y cosð1 t1 Þ cosðJ12 t1 Þ sinðJ13 t1 Þ 2I1z I3y sinð1 t1 Þ cosðJ12 t1 Þ sinðJ13 t1 Þ þ 4I1z I2y I3y cosð1 t1 Þ sinðJ12 t1 Þ sinðJ13 t1 Þ 4I1x I2y I3y sinð1 t1 Þ sinðJ12 t1 Þ sinðJ13 t1 Þ: The resulting 2I1zI2y and 2I1zI3y terms of [6.16] describe antiphase singlequantum coherence and will be rejected by the phase cycling. Similarly, the 4I1xI2yI3y term describes a mixture of 3Q and three-spin singlequantum coherences and will also be rejected (this term is discussed in more detail with respect to the 3QF-COSY in Section 6.3.2.1). The other
6.3 MULTIPLE-QUANTUM FILTERED COSY
443
resulting terms of [6.16] contain mixtures of ZQ and DQ coherences (Section 2.7.5) and can be rewritten as 2I1x I2y ¼ 12 2I1x I2y þ 2I1y I2x 2I1y I2x þ 2I1x I2y ¼ 2i1 I1þ I2þ I1 I2 I1þ I2 þ I1 I2þ h i 12 ½6:17 ¼ DQ12 y ZQy , 2I1x I3y ¼ 12 2I1x I3y þ 2I1y I3x 2I1y I3x þ 2I1x I3y ¼ 2i1 I1þ I3þ I1 I3 I1þ I3 þ I1 I3þ h i 13 ¼ DQ13 y ZQy ,
½6:18
4I1z I2y I3y ¼ I1z 2I2y I3y 2I2x I3x þ 2I2x I3x þ 2I2y I3y ¼ I1z I1þ I2þ þ I1 I2 I1þ I2 I1 I2þ 23 ¼ 2I1z DQ23 x ZQx ,
½6:19
in which ZQijp and DQijp indicate zero-quantum coherence and doublequantum coherence, respectively, of phase p (x or y) between spins Ii and Ij. The DQ23 x operator is antiphase with respect to I1. In the phase-cycled 2QF-COSY, only the DQijp coherences in [6.17]–[6.19] are retained by the phase cycling. The final 90x pulse will transfer the DQ terms back to observable magnetization to yield 12 cosð1 t1 Þ sinðJ12 t1 Þ cosðJ13 t1 Þ½2I1x I2z þ 2I1z I2x , 12 cosð1 t1 Þ cosðJ12 t1 Þ sinðJ13 t1 Þ½2I1x I3z þ 2I1z I3x , þ 12 cosð1 t1 Þ sinðJ12 t1 Þ sinðJ13 t1 Þ 4I1y I2x I3x 4I1y I2z I3z :
½6:20
In the gradient-selected 2QF-COSY, only the double-quantum coherences Iiþ Ijþ and 2Ikz Iiþ Ijþ in [6.17]–[6.19] are selected by the gradient pair G1 and G2. After the final 90x pulse, the observable coherences are given by [6.20] scaled, as noted previously, by a factor of two due to the gradient coherence selection. The two terms in the first line of [6.20] describe an I1 diagonal peak and an I1 ! I2 cross-peak; both have antiphase character and are of the same relative phase, as discussed for [6.16]. Similarly, the terms on line 2 describe an I1 diagonal peak, and an I1 ! I3 cross-peak. The cos(Jt1) trigonometric terms in the first two lines indicate additional in-phase splittings in F1 arising from passive coupling (see also Section 6.2.1.1); similar splittings will also be present in F2. The first term on line 3 is
444
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
unobservable, while the second term is a third I1 diagonal component doubly antiphase with respect to I2 and I3. When the cross-peaks are phased to absorption, the doubly antiphase term will be in dispersion. Fortunately, the antiphase nature of this term leads to pronounced self-cancellation, hence the dispersive tails will not extend far from the diagonal. In conclusion, this analysis has demonstrated that, even in more complex spin systems, the diagonal resonances of the 2QF-COSY experiment have predominantly in-phase absorption lineshapes and do not obscure cross-peaks between protons close in chemical shift. 6.3.1.2 Experimental Protocol The cross-peaks in the COSY and 2QF-COSY experiments have similar lineshape properties. Thus, many experimental requirements for the 2QF-COSY, notably high digital resolution in F1 and F2, are similar to those already described for the COSY experiment. The only extra parameter required in the experimental protocol for the phase-cycled experiments is the delay between the two final 908 pulses. This delay should be on the order of a few microseconds, i.e., as short as possible, while still allowing the rf hardware to make accurate phase shifts of the pulses. In the gradientselected experiments, the gradient durations and strengths are chosen to obtain satisfactory coherence selection and water suppression. The spin echo delays are chosen to be long enough to encompass the gradients and gradient recovery periods. If three-axis pulsed field gradients are available, then the gradient strengths are adjusted to yield a magic-angle gradient pulse (Section 4.3.3). When 2QF-COSY experiments are acquired from D2O solution, the digital resolution in F1 is improved by reducing the spectral width to span only the aliphatic resonances. Because correlations between the aromatic and aliphatic portions of the spectrum are not observed, folding of the diagonal and cross-peaks of the aromatic resonances does not confuse analysis of the F1 chemical shifts (Section 4.3.4.3). By reducing the spectral width, a high t1max is obtained with fewer increments of t1. One artifact common in 2QF-COSY spectra results from incomplete recovery of the spins during the recycle delay and gives rise to extra diagonal peaks at F1 ¼ 2F2 (in this case F1 and F2 refer to absolute frequencies from the carrier position, not chemical shifts) (32, 33). Figure 6.22 depicts a region of a spectrum acquired with a short recycle delay (1.33 s) where such artifacts are particularly prominent. Due to the particular spectral widths and carrier position in this spectrum, the artifact peaks occur at F1 (ppm) ¼ 2 F2 (ppm) 5.6 for F2 between 5.6 and 2.6 ppm; due to folding, above F2 (ppm) ¼ 2.6 they occur at
445
5.0
4.0
2.0 3.0 F1 (ppm)
1.0
0.0
6.3 MULTIPLE-QUANTUM FILTERED COSY
5.0
4.0
3.0
2.0 F2 (ppm)
1.0
0.0
FIGURE 6.22 A 2QF-COSY spectrum acquired with a short recycle delay. The ‘‘double diagonal’’ artifacts lie along the dotted line. Quadrature detection in F1 was achieved with TPPI; consequently, above 2.6 ppm, the artifact peaks fold to F1 ¼ 4.82F2. The spectrum was acquired using the pulse sequence of Fig. 6.20 (16-step phase cycle) with 32 scans for each of 512 increments and with a total recycle delay of 1.33 s (0.33-s acquisition time and 1-s weak presaturation).
F1 (ppm) ¼ 4.8 2 F2 (ppm). In particular, the artifacts obscure weak peaks in the 1H–1H fingerprint region (Fig. 6.23, panel a). Lengthening the recycle delay (panel b) and the incorporation of axial peak suppression in the phase cycle (panel c) reduces the size of these artifacts and allows several weak proline 1H–1H cross-peaks to be identified. Derome and Williamson (32) and Turner (33) have provided detailed analyses of the origins of the rapid-pulsing artifacts in 2QF-COSY spectra such as that shown in Fig. 6.22, and have derived modified phase-cycling schemes that can be very effective in suppressing these artifacts. One such modified, eight-step phase cycle is given in the caption to Fig. 6.20. 6.3.1.3 Processing Although the cross-peaks of the 2QF-COSY are similar to those of the COSY experiment, processing requirements are usually quite different. Strongly resolution-enhancing window functions
446
6
CHAPTER
a
b
I
I
EXPERIMENTAL
b ~~ ~
0
'H
'~
c
'~ DCV~
~~:,~
.~~~~,1 II"
NMR METHODS
~
P38y"·o
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f
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3.9
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n
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.
'D ~'IJ C
I
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~.
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a-f3': '
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,.
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P38y'-o
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3.7
F2 (ppm) FIGURE 6.23 Cross-peaks obscured by the "double diagonal" artifacts. (a) Enlargement of the spectrum shown in Fig. 6.22. The spectra in band c were obtained with a longer total recycle delay (2.3 s) and the phase cycle for panel c included axial peak suppression. Other acquisition and processing parameters were the same for all three spectra. Assignments for two of the obscured crosspeaks are included in panel c.
are not required to reduce the intensity of the predominantly antiphase absorptive diagonal peaks. Nonetheless, the appearance of the spectrum is improved by using mild resolution enhancement to minimize overlap of neighboring cross-peaks. Sine bells shifted by 90° to 30° in both tl and t2 are appropriate window functions. The sine bells should span between 150 and 300 ms of each FID (depending on the degree of resolution required and the SIN ratio of the data) and over the entire tl interferogram. Phase parameters are determined by the methods already described for the COSY experiment. All of the 0 20 2QF-COSY spectra depicted in this chapter were processed with a sine bell shifted 60° in t2 with a t2max of 256 ms, and a sine bell shifted 45° over all 512 tl points.
6.3.1.4 Information Content The 2QF-COSY contains essentially the same information as does the normal COSY experiment. The regions of interest in the 2QF-COSY experiment are still the fingerprint regions described in Fig. 6.6.
447
6.3 MULTIPLE-QUANTUM FILTERED COSY
A disadvantage of using the double-quantum filter is a decrease in signal-to-noise ratio by a theoretical factor of two for the phase-cycled 2QF-COSY and a factor of four for the gradient-selected 2QF-COSY relative to the COSY experiment (25). The origin of the twofold reduction in sensitivity between the phase-cycled and gradient-selected 2QF-COSY experiments is evident from [6.14] and [6.15]. The origin of the decreased sensitivity for the phase-cycled 2QF-COSY compared to the COSY experiment can be appreciated by considering the first four steps of the phase cycle: 1: 2: 3: receiver:
x, x, x, x,
y, y, x, x,
x, x, x, x,
y y x x
Because the final two pulses of the experiment are contiguous, the first and third transients have a 1808 pulse, or 08 pulse, respectively, after t1; these transients suppress the in-phase dispersive diagonal components and add noise to the spectrum but do not contribute any intensity to the cross-peaks (15). Thus, only half of the transients in the phase-cycled 2QF-COSY experiment contribute to cross-peak intensity, leading to a twofold decrease in sensitivity relative to a COSY experiment acquired in the same amount of time. The difference in sensitivity is demonstrated in Fig. 6.24, which shows cross-sections through H2O COSY and phasecycled 2QF-COSY spectra at the F1 chemical shift of the 1H resonance of Ile13. The absolute magnitudes of the cross-peaks are similar, as the H2O COSY was acquired with 16 scans per t1 increment while the 2QFCOSY spectra was acquired with 32 scans per t1 increment; all other acquisition and processing parameters are identical, with a sine bell shifted by 208 being necessary to prevent the tails of diagonal peaks from interfering with the COSY cross-peaks. Although doubling the number of transients in the 2QF-COSY experiment equalizes the signal height, the root-mean-square (RMS) noise also increases by a factor of 21/2, hence the actual S/N ratio is lower than for the COSY experiment. The difference in sensitivity for 2QF-COSY and COSY experiments is observed if both spectra are recorded and processed in exactly the same way. In practice, however, the more severe window functions used for processing COSY spectra reduce the sensitivity of the experiment. Thus, in Fig. 6.24 the top trace depicts a row of data from the 2QFCOSY spectrum reprocessed with sine bells shifted 608 and 908 in t2 and t1, respectively. The S/N ratio is now 1.3-fold higher than in the COSY spectrum. As a caveat, the severe window functions used with COSY
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
448
high S/N 2QF-COSY rms noise = 0.82
2QF-COSY rms noise = 0.85
COSY rms noise = 0.58
vertical scale x10
10.6
10.2
9.8
9.6
9.4
chemical shift (ppm)
FIGURE 6.24 Comparison of the S/N ratio in COSY and 2QF-COSY spectra for the 1HN–1H cross-peak of Ile13. Except for the number of transients (COSY ¼ 16, 2QF-COSY ¼ 32), both spectra were acquired under identical conditions. The two lower traces were also processed identically (sine bell shifted 208 in t1 and an unshifted sine bell in t2), whereas the top trace was processed with less resolution-enhancing window functions (a cosine bell in t1 and a sine bell shifted 608 in t2). See text for a more detailed description.
spectra also are resolution enhancing, so although the final COSY and 2QF-COSY experiments can have similar signal-to-noise ratios, the resolution in the COSY will be higher. A useful strategy for the study of proteins employs the COSY experiment in H2O solution to observe 1HN–1H cross-peaks that are far from the diagonal, and uses the 2QF-COSY in D2O solution to identify cross-peaks in the vicinity of the diagonal. Even though harsh window functions can be used to observe peaks close to the diagonal in COSY spectra, distortions arise and the peaks are more readily observed in the 2QF experiment (Fig. 6.25). The gradient-selected 2QF-COSY experiment has the advantage over phase-cycled COSY and 2QF-COSY experiments that presaturation of the solvent signal is not necessary. As a consequence, cross-peaks near the water resonance can be observed. Figure 6.26 shows an identical region as shown for the pre-TOCSY COSY spectrum in Fig. 6.15.
449
6.3 MULTIPLE-QUANTUM FILTERED COSY COSY
2QF-COSY d-e
7.5
F1 (ppm)
7.3
d-e
e- z
7.5
e- d
e-z
7.4
7.5
e- d
7.4
F2 (ppm)
FIGURE 6.25 Comparison of cross-peaks close to the diagonal in COSY and 2QF-COSY. The COSY spectrum was acquired with 16 transients for each of 800 t1 increments and processed with a sine bell shifted 58 in t1 and an unshifted sine bell in t2. The 2QF-COSY was acquired with 32 transients for 800 t1 increments and processed with a cosine bell in t1 and a sine bell shifted 608 in t2. The severe window functions required to reduce the streaks emanating from the diagonal of the COSY spectrum distort the aromatic cross-peaks of Phe45, especially those between 1H" and 1H .
Thus, if a probe equipped with three-axis gradients is available, a single gradient-enhanced 2QF-COSY spectrum may substitute for the COSY, pre-TOCSY COSY, and 2QF-COSY spectra.
6.3.2 3QF-COSY As described in Section 6.3, the 2QF-COSY is just one member of a family of experiments involving filtration through a p-quantum state; experiments with p 4 2 offer spectral simplification because some resonances cannot participate in a p-quantum coherence (see later, discussion of selection rules). As a practical matter, pQF-COSY experiments with p 4 3 are applied rarely to proteins. 6.3.2.1 Product Operator Analysis A product operator analysis of the 3QF-COSY experiment initially proceeds as described for the 2QF-COSY, and [6.16] describes the terms present after the second 908
450
CHAPTER 6 EXPERIMENTAL IH NMR METHODS 60
~
Oe
~
(jo ~ V5
-
- D-
&i
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-
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,...
D
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~ \:J'
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l!l
,...
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1-C\1 l!l I
9.2
9.0
8.8
1.2
F2 (ppm)
FIGURE 6.26 Two sections of the gradient-enhanced 2QF-COSY spectrum of ubiquitin. The illustrated spectral regions are identical to those shown in Fig. 6.15. Sixteen transients were collected for each of 800 increments of 11 (tlmax = 69 ms). The labeled cross-peaks were not present in the standard COSY experiment (Figs. 6.9 and 6.8) because of presaturation.
pulse in the sequence. Only the three-spin term 4hJ2 yh y can contribute to the 3QF-COSY spectrum; all other terms in [6.16] are suppressed by the phase cycle or gradient selection. Just as the product operator 2hJ2y term in the 2QF-COSY experiment was expressed as the combination of DQ and ZQ coherences [6.17], the 4hJ2.J3y operator is expanded as the combination of 3Q coherences and three-spin singlequantum coherences. Using the single-element basis set of Section 2.7.2, and referring to the eigenstates for a three-spin system (Fig. 2.2), a 3Qx coherence is: 3Qx
= HI~ Ii It + 1,12- I:;] = ![(h~ + iIly)(hx + ihy)(hx + ihy)
+ (h~ -
iI1y)(hx - ihy)(hx - iI3y )]
= H4I 1x hxhx - 4I1yh yhx - 4I1x h yh y - 4I 1y hxh),].
[6.21]
6.3 MULTIPLE-QUANTUM FILTERED COSY
451
Three-spin single-quantum coherences connect eigenstates for which two spins change quantum number by þ1 and the third changes by 1, or for which two spins change quantum number by 1 and the third changes by þ1; formally, the net coherence order is þ1 or 1. For example, 1Q123 ¼ 12½I1þ I2þ I3 þ I1 I2 I3þ ¼ 12½ðI1x þ iI1y ÞðI2x þ iI2y ÞðI3x iI3y Þ þ ðI1x iI1y ÞðI2x iI2y ÞðI3x þ iI3y Þ ¼ 14½4I1x I2x I3x 4I1y I2y I3x þ 4I1x I2y I3y þ 4I1y I2x I3y ,
½6:22
in which the overbar in the subscript on Q indicates the spin changing quantum number in the opposite sense to the other two spins. Taking the appropriate combinations of the three-spin single-quantum terms and the 3Qx term yields the following result: 4I1x I2y I3y ¼ 3Qx þ 1Q123 þ 1Q123 : 1Q123
½6:23
In the phase-cycled 3QF-COSY experiment, the phase cycle selects only the 3Qx component of [6.23]. Thus, from [6.21], the action of the final 908 pulse acting on the 3Qx term is 3Qx sinð1 t1 Þ sinðJ12 t1 Þ sinðJ13 t1 Þ ð2 Þx ! 14½4I1x I2x I3x þ 4I1z I2z I3x þ 4I1x I2z I3z þ 4I1z I2x I3z sinð1 t1 Þ sinðJ12 t1 Þ sinðJ13 t1 Þ:
½6:24
In the gradient-selected 3QF-COSY experiment, the gradient pair selects only the I1þ I2þ I3þ component of [6.23]. The resulting product operators after the final 908 pulse are described by [6.24] except that the amplitude is reduced by a factor of two. The first term of [6.24] is unobservable, while the last three correspond to an I1 ! I3 cross-peak, an I1 diagonal peak, and an I1 ! I2 cross-peak, respectively. The three observable terms are modulated by the same trigonometric functions; hence, the resonances have the same relative phase in F1. The combination of three sine terms indicates that all three peaks are antiphase with respect to J12 and J13. Likewise in F2, all three peaks have the same relative phase (x) and are in antiphase with respect to both couplings. The appearance of a double antiphase Lorentzian lineshape is discussed in more detail in Section 6.4.1.1 and Fig. 6.36. 6.3.2.2 Experimental Protocol and Processing The sensitivity of a cross-peak arising from a p-quantum coherence in a pQF-COSY
452
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
decreases by a factor of 2p1 (cf. the factors of 1/2 in [6.20] and 1/4 in [6.24]) for a phase-cycled experiment (27) and 2p for a gradientselected experiment; consequently, more transients must be co-added to obtain a suitable S/N ratio. For this reason, the increased number of steps in the phase cycle to achieve p-quantum selection is rarely a problem. Other practical considerations in implementing 3QF-COSY experiments parallel those described above for the 2QF case (Section 6.3.1.2). The sections of 3QF-COSY spectra shown below were acquired from D2O solution with 48 transients for each of 512 increments; the spectral width was 3100 Hz and a t1max was 82.6 ms. Phase cycling was used for coherence selection. The spectrum was folded in F1 as described above. The complete phase cycle used in this experiment is listed in the caption to Fig. 6.20. Considerations for processing pQF-COSY spectra are similar to those for the 2QF-COSY described in Section 6.3.1.3. The cross-peak fine structure is antisymmetric with respect to the chemical shift axes for even p, and symmetric for odd p (for example, see Fig. 6.27 or 6.30). This simple observation aids in determining appropriate phase corrections for higher order pQF-COSY spectra. 6.3.2.3 Information Content The main reason for filtration via higher coherence orders is the spectral simplification achieved. The following selection rules provide the basis for this simplification: (i) diagonal peaks will be observed when the active spin has resolved scalar couplings to p 1 spins; (ii) a cross-peak between two spins will be observed when the two active spins have p 2 mutual resolved coupling partners (27). A resolved scalar coupling is a scalar interaction that would give rise to an observable multiplet splitting in the 1D spectrum (assuming no overlap with resonances of other spin systems); thus, degenerate protons can never have a mutual resolved coupling. The cross-peaks expected from these definitions are strictly valid only in the weak coupling limit and are summarized in Fig. 2 of Mu¨ller et al. (27). Experiments with p 4 3 have been reported in the literature (27, 34), but the lack of sensitivity and modest gains in spectral simplification preclude their common application to proteins. Some of the specific advantages of D2O 3QF-COSY over the 2QF counterpart are (i) a removal of 1H diagonal peaks and 1H–1H crosspeaks of threonine, alanine, valine, and isoleucine residues, (ii) complete removal of all glycine peaks, (iii) removal of all diagonal and cross-peaks involving methyl groups except 1H –1H of isoleucine, and (iv) removal of tyrosine aromatic ring resonances (unless the four bond 1H1–1H2 or
453
6.3 MULTIPLE-QUANTUM FILTERED COSY ,;
..
. .~~" '13 . '113
'o3~~ ~
~ ~
,~,~~ 136'
160
E a.
144
S
u::
0:'0; ~
."
~ ~ I"
'~'
'-00
~
"''? 123
130
0
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; :~
3QF-COSY
1.0
0.5
1.0
0.5
F2 (ppm)
FIGURE 6.27 Comparison of the methyl fingerprint region in 2QF- and 3QFCOSY experiments. Assignments for the isoleucine IHyl_IH& cross-peaks are labeled in the 3QF-COSY. The single resolved couplings to the methyl groups of leucine and valine and the Cy 2 H 3 group of isoleucine are insufficient to allow cross-peaks in the 3QF spectrum.
IHEI_IHE2 couplings are resolved) and removal of IHQ'_IHtJ cross-peaks of spin systems with degenerate .a-methylene protons. The effect on the methyl fingerprint region is shown in Fig. 6.27. The methyl groups of alanine, threonine, valine, and leucine residues have a single resolved coupling, and formally cannot produce diagonal or crosspeaks in pQF-COSY experiments with p > 2. This simplification allows ready identification of isoleucine I HY_I HO cross-peaks. Multiexponential relaxation of methyl group coherences results from cross-correlation (Section 5.2.1) and causes violation of the selection rules. Weak forbidden cross-peaks and diagonal peaks are sometimes observed from these sRin systems in 3QF- and 4QF-COSY experiments (35, 36). In the HQ'_IH.B fingerprint region, serine IHQ'_IHtJ and proline IHy_IHo cross-peaks are often more readily observed because of the removal of glycine and threonine cross-peaks in this region. In opposition to the general trend in sensitivity for pQF-COSY experiments, the double antiphase absorptive fine structure in the 3QF-COSY
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
454
3QF
5.3
5.2
5.3
F1 (ppm)
1.4 1.8
Q2
1.6
K6
L69
L43
1.2
2QF
5.2
F2 (ppm)
FIGURE 6.28 Comparison of several 1H–1H cross-peaks in 2QF-COSY and 3QF-COSY spectra. For the upfield 1H of Leu43 and Leu69 (boxed crosspeaks), the active coupling is small and the cancellation of the antiphase components severely attenuates the cross-peaks in the 2QF-COSY. The components of the 3QF-COSY cross-peaks are not separated by the sum of the individual 3 JH H , hence the cancellation is not as pronounced. The arrows indicate cancellation (2QF-COSY) and reinforcement (3QF-COSY) of overlapping parts of the downfield Lys6 and the upfield Gln2 1H–1H cross-peaks as shown schematically in Fig. 6.30.
leads to less self-cancellation than occurs in the absorptive single antiphase peaks in the 2QF-COSY, and a higher apparent S/N ratio sometimes is observed for particular resonances in 3QF-COSY spectra. (For an example of an ideal absorptive double antiphase Lorentzian line, see Fig. 6.36). An experimental example is illustrated in 0 00 Fig. 6.28. Although 1H–1H –1H fragments with identical 1H shifts cannot participate in a triple-quantum coherence, even slight deviations from degeneracy can lead to peaks in the 3QF-COSY. In the case of Glu24, other correlation spectra0 were unable to resolve the slight 00 difference in chemical shift of 1H and 1H , but a peak was observed in
6.3 MULTIPLE-QUANTUM FILTERED COSY
455 ~
,,
,~O~
~
~P'
Ly s63 9
6
Glu24
9
~O~
'd#fJP d
0
Ec.
N .3
u:
oO~ IX
IX
)
0°
~ f)\)
4.0
0
Q
d
Q
b(
C\J
N 3.9
3.8
F2 (ppm)
FIGURE 6.29 IH"'_I H.B' and IH"'_IH.B" cross-peaks of Glu24 in the 3QF-COSY spectrum. Although the ,B-protons appear degenerate in other spectra (COSY, 2QF-COSY, and 2Q), the fine structure in the 3QF-COSY clearly shows that the chemical shifts are slightly different (approximately 0.02 ppm).
the 3QF-COSY (Fig. 6.29). The fine structure clearly indicates that I H B' and IH,B" are separated by approximately 0.02 ppm. A detailed catalogue of the fine structure observed in cross-peaks in pQF-COSY has been presented elsewhere (27). In a system of n mutually coupled spins, cross-peaks in a pQF-COSY (where n > p) will contain contributions from all possible p-quantum states (including combination bands if n - p = 2, 4, ...). Symmetry properties of the cross-peak multiplets indicate that partially overlapping cross-peaks will be more subject to cancellation in spectra filtered with an even value of p than with an odd value of p (Fig. 6.30) (37).
6.3.3 E.COSY Values of 3J can provide dihedral angle information that can be used as experimental restraints during protein structure determination
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
456 2QF-COSY
3QF-COSY
FIGURE 6.30 Schematic view of partially overlapped cross-peaks in 2QF- and 3QF-COSY spectra. Due to the symmetry of the cross-peaks, those in the 2QFCOSY cancel while those in the 3QF-COSY reinforce. A real example of this is observed in Fig. 6.28.
(Section 9.2.1). For spins that possess a single active coupling, the scalar coupling constant can be extracted from the fine structure of a peak in the COSY spectrum (Section 6.2.1.5). If both actively coupled spins have additional passive coupling partners, then the COSY crosspeak fine structure becomes more complicated, and in general the individual multiplet component positions (and hence the scalar coupling constants) are not well resolved for the broad resonance lines observed in protein spectra. A number of methods have been developed that excite only connected transitions (Section 2.6; see Fig. 2.2) — i.e., the passive spins remain unperturbed and the crosspeaks contain only active coupling components. These methods have been given the generic name of exclusive COSY or (E.COSY) (38). In 1 H NMR of proteins, E.COSY spectra provide a useful means to measure 3 JH H in side chains containing -methylene groups; limitations of sensitivity and linewidth have precluded widespread use in measuring other scalar coupling interactions. However, the small linewidths and high concentrations available in peptide studies have allowed the measurement of virtually all 1H–1H couplings in some peptide systems using E.COSY spectra (39).
457
6.3 MULTIPLE-QUANTUM FILTERED COSY
6.3.3.1 Product Operator Analysis The most common approach to obtaining E.COSY data is construction of a linear combination of pQF-COSY spectra where the level of filtration includes the orders p ¼ 0, 1, 2 . . . , K (38, 40). The weighting factor given to a particular COSY of multiple-quantum order p in this combination is given by Wk ¼ 14 p2 þ W0 Wk ¼
1 2 4 ðp
1Þ þ W1
ðp evenÞ, ðp oddÞ,
½6:25
where the first and second lines of [6.25] are applicable to even and odd k, respectively. Because spectra acquired with p ¼ 0 and p ¼ 1 contribute only to the diagonal, W0 and W1 are customarily set to zero to minimize interference from intense diagonal peaks. Thus, the appropriate weights become W2 ¼ 1, W3 ¼ 2, W4 ¼ 4 etc. For a given cross-peak, pQF-COSY is included in the combination if p 2 passive spins are common to the two active spins (38). Given the pQF-COSY selection rules (Section 6.3.2.3), if a cross-peak appears in the pQF-COSY, then the spectrum should be included in the linear combination comprising the E.COSY experiment. For a spin system containing a -methylene group, a single common passive spin 1H contributes to both 1H–1H cross-peaks; therefore, the maximum level of multiple-quantum coherence required in the linear combination to remove the passive spin components is 3. The results of the product operator analyses performed for the 2QF-COSY (Section 6.3.1.1) and 3QF-COSY (Section 6.3.2.1) demonstrate the appearance of cross-peaks in the E.COSY of a three-spin system. From [6.20], the observable term describing the I1 ! I2 crosspeak in the 2QF-COSY spectrum of the I1, I2, I3 spin system is proportional to ð2QFÞ12 ¼ 12 cosð1 t1 Þ sinðJ12 t1 Þ cosðJ13 t1 Þ2I1z I2x :
½6:26
Similarly from [6.24], the equivalent operator in the 3QF-COSY is ð3QFÞ12 ¼ þ14 sinð1 t1 Þ sinðJ12 t1 Þ sinðJ13 t1 Þ4I1z I2x I3z :
½6:27
Considering quadrature detection in t1 and evolution and detection during t2 gives the following signals for the 2QF-COSY and 3QF-COSY
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
458 experiments, respectively:
sð2QFÞ12 ¼ 4i expði1 t1 Þ sinðJ12 t1 Þ cosðJ13 t1 Þ expði2 t2 Þ sinðJ12 t2 Þ cosðJ23 t2 Þ ¼ 16i expði1 t1 Þ sinðJ12 t1 Þ expði2 t2 Þ sinðJ12 t2 Þ expðiJ13 t1 Þ expðiJ23 t2 Þ þ expðiJ13 t1 Þ expðiJ23 t2 Þ þexpðiJ13 t1 Þ expðiJ23 t2 Þ þ expðiJ13 t1 Þ expðiJ23 t2 Þ , ½6:28
sð3QFÞ12 ¼ 8i expði1 t1 Þ sinðJ12 t1 Þ sinðJ13 t1 Þ expði2 t2 Þ sinðJ12 t2 Þ sinðJ23 t2 Þ ¼ 32i expði1 t1 Þ sinðJ12 t1 Þ expði2 t2 Þ sinðJ12 t2 Þ expðiJ13 t1 Þ expðiJ23 t2 Þ expðiJ13 t1 Þ expðiJ23 t2 Þ expðiJ13 t1 Þ expðiJ23 t2 Þ þ expðiJ13 t1 Þ expðiJ23 t2 Þ : ½6:29 Combining [6.28] (2QF) and [6.29] (3QF) according to [6.25] yields sð2QFÞ12 þ 2sð3QFÞ12 ¼ 8i expði1 t1 Þ sinðJ12 t1 Þ expði2 t2 Þ sinðJ12 t2 Þ expðiJ13 t1 Þ expðiJ23 t2 Þ þ ½6:30 expðiJ13 t1 Þ expðiJ23 t2 Þ : Thus, two components arising from the passive J13 and J23 couplings in [6.28] and [6.29] have been removed in [6.30]. The overall effect on the I1 ! I2 cross-peak is shown schematically in Fig. 6.31. The E.COSY cross-peak contains a superposition of two twospin antiphase multiplets, with the displacement being equal to the size of the passive coupling in each dimension. The active coupling can be measured as the antiphase separation (as in Section 6.2.1.5). More importantly, the passive coupling can be measured from the displacements of the two multiplets as a simple peak-to-peak separation, as this distance will not be subject to systematic errors associated with finite
459
6.3 MULTIPLE-QUANTUM FILTERED COSY 2QF-COSY
3QF-COSY
J12
J23
J12
E.COSY
J13 (12) J3
J13 f3
J23
FIGURE 6.31 Schematic view of the I1–I2 cross-peak for a three-spin system during 2QF-COSY, 3QF-COSY, and E.COSY experiments. Filled and open circles indicate positive and negative components, respectively. Traces parallel to F2 through each cross-peak are shown on the right, and were calculated for Lorentzian lines of 4-Hz half-height width, J12 ¼ 6 Hz, J13 ¼ 10 Hz, and J23 ¼ 15 Hz. The vector construction at the bottom is described in the text.
linewidth (Section 6.2.1.5). This feature allows the E.COSY method to determine scalar couplings much smaller than the linewidth, provided the appropriate cross-peaks can be generated. A displacement vector, J3ð12Þ , connecting peaks of like sign (constructed as shown at the bottom of Fig. 6.31) is associated with a particular passive spin (I3 in this example). The angle 3 between the F2 axis and the displacement vector reflects the sign of the product J13 J23, and can distinguish the relative signs of these two coupling constants (14). Provided spectra are processed to have the diagonal running from top right to bottom left, and 3 is constrained to be between 08 and 1808, positive and negative products of J13 J23 are indicated by 08 5 3 5 908 and 908 5 3 5 1808, respectively.
460
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
Although this discussion has been limited to 1H NMR, the E.COSY concept may be extended to include heteronuclear couplings. Small passive couplings (usually 1H–1H) may then be measured in F2 between two components separated by a large heteronuclear coupling (one bond 1 H–X or X–X, where X is 15N or 13C). These applications of the E.COSY principle are discussed in more detail in Section 7.5. A more thorough analysis extensible to spin systems of greater complexity involves consideration of single-transition basis operators within the energy level diagram of the spin system of interest. Pairs of transitions give rise to each element of fine structure in a crosspeak and are described in the terms discussed in Section 2.6: regressive or progressive, connected or anticonnected. Depending on these qualities, the transition pairs display particular characteristics in the final spectrum, and the appearance of each multiple-quantum filtered (MQF) spectrum, or combination of such spectra, can be deduced (40). 6.3.3.2 Experimental Protocol The majority of experimental details for acquisition of 2QF- and 3QF-COSY experiments have been covered in Sections 6.3.1.2 and 6.3.2.2, however, a few aspects are specific to the E.COSY experiment. The weights indicated by [6.25] indicate that the 3QF-COSY data should be scaled by a factor of two compared to the 2QF-COSY if both spectra have been acquired identically (i.e., spectral widths, number of transients, and number of t1 increments). However, noise introduced by the addition (or subtraction) of the two data sets is minimized if no scaling is performed after acquisition; consequently, the 3QF-COSY spectrum is acquired with twice the number of F2 transients as for the 2QF-COSY, and the two spectra simply are added. In order to complete the basic phase cycles of both experiments (16 and 24 transients) and acquire twice the number of transients in the 3QF-COSY, 48 and 96 transients must be collected for the 2QF- and 3QF-COSY, respectively. The spectra should be acquired in an interleaved fashion to avoid subtraction artifacts, and combined at the processing stage. In this mode of acquisition, the individual 2QF-COSY and 3QF-COSY experiments also can be analyzed. However, 144 transients must be collected for each t1 increment in the E.COSY experiment, which requires considerable total acquisition time. An alternative mode of acquisition uses phase cycling within a single experiment to perform the appropriate data combination between transients (14). Many of the steps in the individual phase cycles of the 2QF- and 3QF-COSY experiments are identical, and only need be
461
6.3 MULTIPLE-QUANTUM FILTERED COSY
TABLE 6.1 Phase Cycles for the E.COSY Experimenta Pulse phase Receiver phase Transients
08 08 4
608 1808 3
1208 08 1
2408 08 1
3008 1808 3
a
The relative number of transients acquired for each phase increment in the combined E.COSY experiment. The phase shifts are applied to 1 and 2, while keeping the phase of 3 fixed (08) in Fig. 6.20.
performed once. Table 6.1 indicates the number of transients that must be collected for each of the six phase increments required for an E.COSY experiment suitable for analysis of three-spin systems. The basic phase cycle contains 12 steps, and can be extended to 48 steps by performing CYCLOPS (Section 4.3.2.3). Although only 48 transients have to be acquired to complete the phase cycle in this method, separate analysis of the 3QF-COSY and 2QF-COSY is not possible. Finally, the two antiphase multiplets must be resolved in both dimensions to permit accurate measurements of the coupling constants. Normally, the cross-peaks at the F2 frequency of 1H0 are analyzed. These 00 cross-peaks are separated in F1 by the passive 1H –1H coupling, and given the large size of this interaction (15 Hz), sufficient t1 increments to properly resolve the components are easily acquired. The spectra are acquired from D2O solution to avoid interference from passive couplings to 1HN, and may be folded in F1 to improve the digital resolution (Section 6.2.1.2). The sections of spectra shown in Fig. 6.32 were acquired as separate 2QF- and 3QF-COSY spectra in an interleaved fashion with 48 and 96 transients, respectively. The spectra were folded in F1, and contained 512 increments of t1 (t1max ¼ 82 ms). The total acquisition time was 42 hr. 6.3.3.3 Processing Processing parameters for E.COSY data are identical for both methods of acquisition and were generally outlined in Section 6.3.1.3. The 1H–1H cross-peaks are of primary interest for measuring 3 JH H , and are best resolved at the F2 frequency of 1H. In order to improve the accuracy of the measurement, the spectra should be extensively zero-filled in F2 (to less than 0.5 Hz per point). The spectrum shown in Fig. 6.32 was processed with moderately
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS 1.0
462
b"
3J
ab" =6
Hz
Hz
b'
1.5
ab' =10
F1 (ppm)
3J
a 2QF 5.3
5.2
3QF 5.3
5.2
E.COSY 5.3
5.2
F2 (ppm)
FIGURE 6.32 Section of the E.COSY spectrum of ubiquitin obtained by the co-addition of a 2QF-COSY (48 transients per t1 increment) and a 3QF-COSY (96 transients per t1 increment). Measurements of 3 JH H from the multiplet patterns are illustrated.
shifted cosine bells and zero-filled from 2048 to 16,384 points to give a final digital resolution of 0.38 Hz/point in F2. Line-fitting procedures can be used to improve the accuracy of the coupling constant measurement, although coupling constant data are rarely interpreted in such detail that errors of 0.5 Hz are significant (Section 9.2.1). 6.3.3.4 Information Content The principal information provided from the E.COSY is 3 JH H . The analysis of the product operators has shown that co-addition of 2QF-COSY and 3QF-COSY is sufficient to provide an E.COSY pattern for the cross-peaks of a three-spin system. Longer spin systems have additional scalar coupling interactions with 1 H spins and passive coupling components remain in an E.COSY spectrum constructed from 2QF-COSY and 3QF-COSY; higher orders of pQF-COSY must be included to remove these additional passive interactions. However, the passive components are only present in the F1 (1H) dimension and 3 JH H can still be measured from the displacements in the F2 (1H) dimension. A section of the E.COSY of ubiquitin is shown in Fig. 6.32. Measurements are most accurate if taken from the displacement by the passive coupling because the artifacts described in Section 6.2.1.5 are
463
6.4 MULTIPLE-QUANTUM SPECTROSCOPY 00
not present. Thus, 3 JH H0 is best measured from the 1H–1H cross-peak and vice versa. In order to00 measure both 1H–1H coupling constants, 1 1 0 the H – H and 1H–1H cross-peaks must be resolved, which is not always the case in this crowded region of the spectrum. 6.3.3.5 Experimental Variants Linear combinations of COSY spectra acquired with mixing pulses less than 908 also produce E.COSY-type cross-peak fine structure. In the COSY- spectrum of an I1, I2, I3 spin system, the following operators give rise to a cross-peak representing coherence transfer I1 ! I2 (the t1 trigonometric factors have been ignored for simplicity): 2I1z I2y sin2 ðÞ 4I1z I2y I3z sin2 ðÞ cosðÞ:
½6:31
The second term of [6.31] would not be observed in a normal COSY experiment because the term at the end of t1 from which it originates (4I1yI2zI3z) would be converted entirely into unobservable magnetization (4I1zI2yI3y) by the 90x mixing pulse (for example, see [6.16]). The combination of single and double antiphase lineshapes, represented by the first and second terms in [6.31], respectively, is responsible for generating E.COSY fine structure (Section 6.3.3.1). Thus, combining COSY- spectra with different rotation angles suppresses the passive coupling components. Two main disadvantages of this approach are that (1) the cross-peak intensity is low due to the dependence on sin2() and (2) the diagonal is very intense and also contains many dispersive components. As is evident from [6.31], a single COSY- experiment does have a degree of E.COSY character. The ratio of connected to unconnected transition intensities is maximal when 358. The undesired passive components are suppressed by more than 10-fold, which is adequate for many applications (4). Thus, COSY-35 provides a simple method to obtain 3 JH H . An example of a COSY-35 spectrum has already been shown in Fig. 6.14 for the H2O case of glycine 1HN–1H peaks, and Fig. 6.33 compares a section of the 1H–1H region in D2O with the equivalent region of the E.COSY.
6.4 Multiple-Quantum Spectroscopy The concept of multiple-quantum (MQ) coherence was introduced in Chapter 2, and multiple-quantum filtration of COSY spectra was
464
CHAPTER
6
EXPERIMENTAL IH NMR METHODS
o
E0S
,.... u..
o cri
COSY-35
5.2
E.COSY
5.1
5.2
5.1
F2 (ppm) FIGURE
6.33 Comparison of the COSY-35 and E.COSY spectra of ubiquitin
in D 2 0.
described in Section 6.3. In this section, another useful application of multiple-quantlll11 coherence is discussed. Although multiple-quantum states cannot induce directly an observable signal in the receiver coil, the pQ family of experiments indirectly observes multiple-quantum states with coherence level p> 1 during the t, period of a 2D experiment (41). The resulting spectra possess unique characteristics that complement the information obtained from COSY or pQF-COSY experiments (37). The pulse sequence for a pQ experiment is shown in Fig. 6.34. Following the initial 90° pulse, antiphase coherence develops during a fixed spin echo sequence (in contrast to COSY experiments in which antiphase coherence develops during tl)' The 180° pulse serves to refocus chemical shift evolution.· Multiple-quantum coherence is generated by the second 90° pulse, and the precession of the desired pQ state is monitored during the tl period. The final 90° pulse creates observable single-quantum magnetization, which is recorded during t2. The phases of the pulses in a pQ experiment are described by the
465
6.4 MULTIPLE-QUANTUM SPECTROSCOPY f2
f1 t 2
f3 t 2
f4 t1
t2
p +2 +1 0 –1 –2
FIGURE 6.34 Pulse sequence for nQ and coherence level diagram for 2Q experiments. The basic 2Q phase cycle is as follows: 1 ¼ 2 ¼ 3 ¼ x, y, x, y; 4 ¼ x; receiver ¼ 2(x, x). The basic 3Q phase cycle is as follows: 1 ¼ 2 ¼ 08, 608, 1208, 1808, 2408, 3008; 3 ¼ 908, 1508, 2108, 2708, 3308, 308; 4 ¼ 908; and receiver ¼ 3(08, 1808). Axial peak suppression and CYCLOPS phase cycling yields a 32-step phase cycle for 2Q and a 48-step phase cycle for 3Q. Frequency discrimination in F1 is obtained by shifting the phase of 1, 2, 3, and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4).
following formula: 1 and 2: 3: 4: receiver:
n=p 1 þ
½6:32
n þ 2
where is 0 for even p and /2 for odd p. In these expressions, n takes integer values from 0 to 2p 1, thus the basic phase cycle consists of 2p steps. This can be expanded by performing CYCLOPS to yield a total of 8p steps (note the close similarity between this experiment and the pQF-COSY experiments (Section 6.3).
6.4.1 2Q SPECTROSCOPY Although pQ experiments can be adjusted to observe a range of multiple-quantum states, 2Q and 3Q experiments usually are adequate
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
466
for studies of proteins. The 2Q experiment, discussed in the present section, is the most commonly performed, and the theoretical analysis of the 2Q experiment presented here illustrates the main differences and advantages compared to COSY and MQF-COSY techniques. The 3Q experiment is described in Section 6.4.2. 6.4.1.1 Product Operator Analysis Initially, evolution of a two-spin system during the 2Q pulse sequence is treated. In this example, the simplest results are obtained if the fates of both I1 and I2 initial z-magnetization are calculated. At the end of the 90x –=2–180x –=2– 90x sequence, the following operators are present at the start of the t1 period: 2
x
2x 2 2
x
I1z þ I2z ! ½I1z þ I2z cosðJ12 Þ þ 2I1x I2y þ 2I1y I2x sinðJ12 Þ:
½6:33
If the delay ¼ 1/(2J12), the longitudinal terms, I1z þ I2z, will vanish, leaving only the pure DQy coherence, 2I1xI2y þ 2I1yI2x. However, in a real sample, a range of values of J12 will be encountered, and not all magnetization can be converted into two-quantum coherence; however, the phase cycling suppresses the residual longitudinal magnetization and retains only the two-quantum coherence. In the rest of this analysis the longitudinal components will be ignored. During t1, the two-quantum coherence evolves at the sum of the I1 and I2 chemical shifts, 12 ¼ 1 þ 2, but does not evolve under the influence of J12 (Section 2.7). The final 90x pulse creates the product operators that evolve during t2: t1 2I1x I2y þ 2I1y I2x sinðJ12 Þ ! 2I1x I2y þ 2I1y I2x cosð12 t1 Þ sinðJ12 Þ 2I1x I2x þ 2I1y I2y sinð12 t1 Þ sinðJ12 Þ, 2
x
! ½2I1x I2z þ 2I1z I2x cosð12 t1 Þ sinðJ12 Þ ½2I1x I2x þ 2I1z I2z sinð12 t1 Þ sinðJ12 Þ: ½6:34 The final line of [6.34] contains a combination of longitudinal twospin order magnetization and a mixture of zero- and two-quantum
467
6.4 MULTIPLE-QUANTUM SPECTROSCOPY Phe45
F1 = δ1 + ε2
F1 = δ + ε
δ2
δ δ
14.5 F1 (ppm)
ζ
ε
Phe4 F1 = δ + ε Tyr59 F1 = δ + ε
δ
7.5
ε
14.0
ε
15.0
F1 = δ + ζ
His68
7.0
F2 (ppm)
FIGURE 6.35 Example of the appearance of 2Q cross-peaks arising from twospin systems. The peaks arising from aromatic ring protons of Tyr59 are present at the bottom of the figure, and represent a two-spin system, because 4 JH1 H2 and 4 JH"1 H"2 are not resolved. Although not formally two-spin systems, the crosspeaks from the aromatic ring protons of Phe4, and Phe45 are also labeled. The small 4 JH2 H"1 of His68 also gives rise to a weak cross-peak. The spectrum was acquired from D2O solution with an excitation delay, ¼ 22 ms.
coherence; neither of these operators leads to observable magnetization during t2 and can be ignored. By analogy with the previous results obtained for the COSY experiment, the 2I1xI2z and 2I1zI2x terms in the penultimate line represent antiphase magnetization with peaks centered at 1 J12 and 2 J12 in F2 (again with in units of angular frequency). Both of the product operators are modulated by cos[(1 þ 2)t1] and give rise to resonances at 1 þ 2 in F1. The lineshapes have no fine structure in this dimension. The appearance of a two-spin system in the 2Q spectrum is exemplified by the peaks from Tyr59 of ubiquitin Fig. 6.35. More interesting results are obtained if an equivalent analysis is performed on a system containing the three spins I1, I2, and I3 with coupling constants of J12, J13, and J23. During the initial spin echo period, evolution of all three couplings will take place but chemical shifts
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
468
will be refocused. Concentrating on the I1 magnetization only, 2
x
2x 2 2
x
I1z ! I1z cosðJ12 Þ cosðJ13 Þ þ 2I1x I2y sinðJ12 Þ cosðJ13 Þ þ 2I1x I3y cosðJ12 Þ sinðJ13 Þ 4I1z I2y I3y sinðJ12 Þ sinðJ13 Þ:
½6:35
The resulting longitudinal operator is suppressed by the phase cycle. The remaining three terms are mixtures of double- and zero-quantum coherence (Section 2.7.5) described by [6.17]–[6.19]. Once again, the phase cycling retains only the two-quantum operators. Allowing for evolution as described in Section 2.7.5, the following terms are present at the end of t1: 13 DQ12 y sinðJ12 Þ cosðJ13 Þ þ DQy cosðJ12 Þ sinðJ13 Þ
þ 2I1z DQ23 x sinðJ12 Þ sinðJ13 Þ h i t1 12 cosðK t Þ 2DQ I sinðK t Þ ! DQ12 12 1 12 1 y x 3z cosð12 t1 Þ sinðJ12 Þ cosðJ13 Þ h i 12 cosðK t Þ þ 2DQ I sinðK t Þ DQ12 12 1 12 1 x y 3z sinð12 t1 Þ sinðJ12 Þ cosðJ13 Þ h i 13 cosðK t Þ 2DQ I sinðK t Þ þ DQ13 13 1 13 1 y x 2z cosð13 t1 Þ cosðJ12 Þ sinðJ13 Þ h i 13 DQ13 cosðK t Þ þ 2DQ I sinðK t Þ 13 1 13 1 x y 2z sinð13 t1 Þ cosðJ12 Þ sinðJ13 Þ h i 23 cosðK t Þ DQ sinðK t Þ þ 2I1z DQ23 23 1 23 1 x y cosð23 t1 Þ sinðJ12 Þ sinðJ13 Þ h i 23 cosðK t Þ þ DQ sinðK t Þ þ 2I1z DQ23 23 1 23 1 y x sinð23 t1 Þ sinðJ12 Þ sinðJ13 Þ:
½6:36
In this expression, Kij ¼ Jik þ Jjk is the double-quantum splitting and is equal to the sum of the coupling constants between the active spins (Ii and Ij) and the mutual passive spin (Ik).
6.4 MULTIPLE-QUANTUM SPECTROSCOPY
469
Expansion of the right-hand side of [6.36] into single Cartesian operators indicates that, following the final 90x pulse, none of the resulting terms 2, 4, or 6 leads to observable magnetization, and the terms 1, 3, and 5 yield 1 2½2I1x I2z þ 2I1z I2x cosð12 t1 Þ cosðK12 t1 Þ sinðJ12 Þ cosðJ13 Þ þ 12 4I1x I2x I3y 4I1z I2z I3y cosð12 t1 Þ sinðK12 t1 Þ sinðJ12 Þ cosðJ13 Þ þ 12½2I1x I3z þ 2I1z I3x cosð13 t1 Þ cosðK13 t1 Þ cosðJ12 Þ sinðJ13 Þ þ 12 4I1x I2y I3x 4I1z I2y I3z cosð13 t1 Þ sinðK13 t1 Þ cosðJ12 Þ sinðJ13 Þ þ 12 4I1y I2x I3x þ 4I1y I2z I3z cosð23 t1 Þ cosðK23 t1 Þ sinðJ12 Þ sinðJ13 Þ þ 12½2I2x I3z þ 2I2z I3x cosð23 t1 Þ sinðK23 t1 Þ sinðJ12 Þ sinðJ13 Þ:
½6:37 The terms proportional to 2I1xI2z þ 2I1zI2x on line 1 of [6.37] are observable during t2 and generate cross-peaks at frequencies 1 and 2 in F2 with an antiphase splitting of J12. Both of these peaks have an F1 shift of 1 þ 2 and are referred to as direct peaks. Because cosð12 t1 Þ cosðK12 t1 Þ ¼ 12½cosð12 t1 K12 t1 Þ þ cosð12 t1 þ K12 t1 Þ, ½6:38 the direct peaks have in-phase lineshapes in F1. The direct peaks are phased to absorption in both dimensions, and the phases of other peaks in the spectrum are described relative to these direct peaks. Direct peaks between spins I1 and I3 are represented by line 3 and the direct peaks between spins I2 and I3 are on line 6 of [6.37]. In all cases, the F2 lineshapes are absorptive and antiphase with respect to the active coupling. The F1 lineshape of the I1–I3 peaks is equivalent to that described above for the I1–I2 peaks, while the I2–I3 peaks are dispersive antiphase in this dimension. The 2I1xI2xI3y operator term on line 2 of [6.37] is a three-spin coherence that is not observable. The 2I1zI2zI3y term describes an I3 coherence that is antiphase with respect to I1 and I2 in F2. Because both J12 and J13 are resolved, this is an observable term and will have a double antiphase dispersive lineshape in F2; this lineshape and several others observed in 2Q spectra are displayed in Fig. 6.36. The cross-peak has an F2 chemical shift of 3 and is modulated at a frequency 1 þ 2 during F1. This resonance is described as a remote peak. Because of the cos(12t1) sin(K12t1) modulation, this peak has a dispersive antiphase lineshape in F1 (Fig. 6.36). The first terms on lines 4 and 5 of [6.37] are not observable, while the second terms describe
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
470 1.0
a
b
c
d
relative intensity
0.0
–1.0 1.0
0.0
–1.0 –20
0
20 –20
0
20
n (Hz)
FIGURE 6.36 Lineshapes of single (a, b) and double (c, d) antiphase Lorentzian lines phased to absorption (a, c) and dispersion (b, d). These lineshapes are commonly found in cross-peaks in 2Q spectra. In all panels, the linewidths of the individual Lorentzian lines are 4.0 Hz and a single line has an intensity of 1. The scalar coupling constants correspond to a 10-Hz coupling (top) or a 10- and a 6-Hz coupling (bottom). Note that in real spectra (poor digital resolution, linewidths comparable to the splittings, and the presence of passive couplings), distinguishing between absorptive antiphase and dispersive double antiphase lineshapes or between dispersive antiphase and absorptive double antiphase lineshapes can be difficult.
the character of the I1–I3 and I2–I3 remote peaks, respectively. The F2 lineshape of both peaks is double antiphase dispersive, while in F1 the former is dispersive antiphase and the latter is absorptive in-phase. The positions of all of the peaks described by [6.37] are depicted schematically in Fig. 6.37 and the lineshapes of these peaks are summarized in Table 6.2. As a result of this analysis, several interesting points emerge for a spin system in which all three coupling constants are resolved: (i) the initial I1 magnetization contributes to all nine cross-peaks in the spectrum, (ii) peaks at a given F1 and F2 frequency can arise by three different pathways starting initially from I1, I2 or I3 magnetization, (iii) the three pathways contributing to a given peak all have the same
471
6.4 MULTIPLE-QUANTUM SPECTROSCOPY J13= 0 g
h
i
d
f
e
d
a
b
c
a
Ω1
Ω2
Ω3
Ω1
Ω2
Ω3
a
b
Ω1
Ω2
Ω1+2
b
e
Ω1+3
F1
i
Ω2+3
J23= 0
Ω3
F2
FIGURE 6.37 Contributions of initial I1 magnetization to the peaks observed in the 2Q spectrum of a three-spin system. The filled circles and open squares represent direct and remote peaks, respectively, while the lines indicate the pseudo-diagonal (F1 ¼ 2F2). The left panel assumes that J12, J13, and J23 are resolved. If J13 (center) or J23 (right) is not resolved, then I1 contributes to fewer peaks. The F1 and F2 lineshapes of the cross-peaks, labeled a–i, are described in Table 6.2.
F2 lineshape (absorptive antiphase or dispersive double antiphase for direct or remote peaks, respectively), and (iv) one of the pathways leading to a direct peak has dispersive antiphase character in F1, whereas two of the pathways have this lineshape for the remote peaks. Consequently, all peaks will have dispersive tails in the F1 dimension. Generally, 2Q spectra are run with short mixing times [ 5 (2Jjk)–1] to avoid unnecessary loss of magnetization via relaxation, and sin(Jjk) and cos(Jjk) will both be positive. Due to the similarity of a negative dispersive antiphase peak and a positive absorptive in-phase peak (Fig. 6.36), the three contributions to a remote connectivity will add constructively in such experiments. Furthermore, the remote and direct peaks can be differentiated on the basis of the F2 lineshape: if the spectrum is phased so that the positive lobe of the absorptive antiphase direct peak is downfield of the peak center, then the major positive lobe
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
472
TABLE 6.2 Lineshapes in a 2Q Spectruma Chemical shift Peak a b c d e f g h i
Type Direct Direct Remote Direct Direct Remote Direct Direct Remote
F1 1 þ 2 1 þ 2 1 þ 2 1 þ 3 1 þ 3 1 þ 3 2 þ 3 2 þ 3 2 þ 3
Lineshape
F2 1 2 3 1 3 2 2 3 1
F1 abs abs disp abs abs disp disp disp abs
F2 in in anti in in anti anti anti in
abs abs disp abs abs disp abs abs disp
anti anti anti 2 anti anti anti 2 anti anti anti 2
a
Shown are the lineshapes of the cross-peaks in a 2Q spectrum of a three-spin system that result from initial I1 magnetization. The peak labels (a–i) refer to the schematic spectrum shown in Fig. 6.37. In the table, 1, 2, and 3 are the resonance frequencies of spins I1, I2, and I3, respectively; ‘‘abs’’ and ‘‘disp’’ indicate absorption and dispersion lineshapes, respectively; ‘‘in,’’ ‘‘anti,’’ and ‘‘anti 2’’ refer to in-phase, antiphase, and doubly antiphase multiplet structures, respectively.
of the dispersive double antiphase remote peak will be upfield of the peak center. Due to changes in sign of the trigonometric terms for long , these generalizations are not always valid, although the relative signs of the peaks do change in a predictable manner if the coupling constants are known [e.g., see Dalvit et al. (42)]. Finally, a linear three-spin system is examined by considering the effect of setting one of the couplings to zero; I1 may be considered a terminal spin in a linear system if J13 ¼ 0. In this case, sin(J13) ¼ 0 and terms such as 4I1zI2zI3y are no longer observable because the antiphase state between I1 and I3 can never evolve into observable magnetization during t2. Thus, of the nine terms (peaks a–i) described in Table 6.2, only lines 1 and 2 (peaks a and b) correspond to observable peaks; i.e., initial I1 magnetization can only contribute to the direct peaks between I1 and I2. Alternatively, I1 may be considered the middle spin of a linear system if J23 ¼ 0. In this case, initial I1 magnetization contributes to the I1–I2 direct peaks (peaks a and b), the I1–I3 direct peaks (peaks d and e), and the I2–I3 remote peak (peak i). Consideration of these results indicates that a remote peak is only observed when both of the spins contributing to the doublequantum frequency have resolved couplings to the third passive spin
6.4 MULTIPLE-QUANTUM SPECTROSCOPY
473
whose frequency is measured in F2. Results such as these form the basis of the selection rules that are discussed in more detail in Section 6.4.2.3. 6.4.1.2 Experimental Protocol As already described, the spin echo sequence in Fig. 6.34 serves to generate antiphase magnetization. Thus, the choice of delay, , will depend upon the magnitudes of the active scalar coupling constants. For 2Q spectra acquired for proteins in H2O and D2O solution, ¼ 30 ms compromises between coherent evolution and incoherent relaxation. In order to emphasize particular correlations, spectra with as low as 20 ms and as large as 80 ms have been reported in the literature (23, 42). The 1808 pulse in the middle of the delay can be a source of several types of artifact in the final spectrum, many of which can be alleviated by the use of a composite 1808 pulse of the form 901 1801þ=2 901 (Section 3.4.2) (43) or by insertion of a pair of gradient pulses flanking the 1808 pulse (Section 3.6). In addition, spectra of high quality can usually be obtained without presaturation of the solvent resonance during the 2Q excitation period. This often leads to the observation of more intense correlations involving protons resonating close to the solvent signal (see later). As discussed in Section 4.3.4.1, quadrature detection in the t1 dimension of two-dimensional NMR spectra is achieved by shifting the phase of pulses prior to t1 according to the TPPI, States, or TPPI– States protocols in order to shift the phase of the indirectly detected coherences by 908. Multiple-quantum coherences are p-fold more sensitive to rf phase shifts than are single-quantum coherences. Thus, in a pQ experiment, the phases 1, 2, and 3 must be incremented by /(2|p|) — that is, by /4 for a 2Q experiment. The choice of spectral width in F1 of the 2Q experiment is not straightforward because resonance peaks appear at sums of the chemical shifts of coupled resonances, and some prior knowledge of the system is helpful. For work in H2O solution, the spectrum must extend downfield of the largest sum of two coupled spins; most usually, this will result from the most downfield 1HN or 1H resonance, or from aromatic ring protons. This limit can be calculated if COSY or other correlation spectra have already been acquired, or else estimated from the sum of the most downfield 1HN and 1H resonances observed in a 1D spectrum. The upfield spectral limit can be calculated from the cross-peaks observed in the upfield region of the COSY (usually involving methyl resonances) or else estimated as twice the frequency of the most upfield resonance in the 1D spectrum.
474
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
The carrier usually is positioned on resonance with the water signal. The resulting large spectral width suggests that many t1 points will have to be acquired to achieve the required resolution of resonances in the F1 dimension. Alternatively, because the focus of 1H NMR spectroscopy in H2O is primarily the 1HN–1H correlations, the F1 spectral width can be reduced and the carrier position shifted to span the spectral region expected for the 1HN–1H correlations only. In similar fashion, the digital resolution for 2Q spectra acquired from D2O solution can be increased by setting the carrier position and spectral width to span resonances arising from the aliphatic resonances only. The lineshapes of all peaks in the F1 dimension of the 2Q spectrum are either absorptive in-phase or dispersive antiphase (Table 6.2 and Fig. 6.36); consequently, large values of t1max are not required. Given the preceding discussion of product operators, the acquisition parameters chosen for F2 are similar to those used for COSY spectra. The 2Q spectra obtained from H2O solution discussed in the next section were collected with 32 transients for each of 800 t1 increments and a mixing period, , of 22 or 32 ms. The F1 spectral width was 5600 Hz and t1max ¼ 71.4 ms; this was sufficient to span all of the 1 N 1 H – H correlations, but resulted in folding some of the upfield aliphatic side chain correlations. An equivalent spectrum was recorded in D2O solution with 600 t1 increments, ¼ 30 ms, an F1 spectral width of 5000 Hz (t1max ¼ 60 ms), and with the transmitter in the center of the aliphatic region; aromatic resonances were folded. 6.4.1.3 Processing As for COSY spectra, unshifted sine-bell window functions are applied in F2 of 2Q spectra acquired in H2O solution to attenuate the residual solvent resonance; in D2O solution, the window function can be phase shifted to increase sensitivity. Applying the sine bell over more points will decrease the S/N ratio but increase the resolution. The 2Q spectrum does not contain dispersive in-phase peaks in F1; therefore, window functions need not be as strongly resolution enhancing as for COSY. A simple cosine bell to prevent truncation artifacts is usually sufficient. The center of the 2Q spectrum in F1 is referenced to be double the carrier frequency. Phase parameters are derived in a fashion similar to derivation of those of the COSY experiment. 6.4.1.4 Information Content Like COSY spectra, 2Q spectra contain information about scalar coupling networks; however, 2Q spectra have unique features that circumvent some of the inherent
475
6.4 MULTIPLE-QUANTUM SPECTROSCOPY
Hb ' + Hb "
Ha + Hb "
F1
Ha + Hb '
HN + Hb " HN + Hb ' HN + Ha HN
Ha
Hb '
Hb "
F2
FIGURE 6.38 Schematic representation of the peaks expected in the 2Q spectrum of a 1HN–1H–1H2 spin system (4 JHN H is assumed to be zero). The filled circles and open squares represent direct and remote peaks, respectively, while the line indicates the pseudo-diagonal (F1 ¼ 2F2).
problems of COSY experiments and provide additional information through both direct and remote correlations. Direct peaks occur at the sum of two chemical shifts of two coupled spins in F1 and at the chemical shift of each spin in F2. From the product operator analysis (Section 6.4.1.1), remote peaks also occur in systems of three or more coupled spins. Remote peaks occur at the sum of the frequencies of two actively coupled spins in F1, and at the frequency of the third passively coupled spin in F2. The peaks expected in the 2Q spectrum of an 1 HN – 1 H – 1 H2 spin system are presented schematically in Fig. 6.38. The presence of these peaks are formally described by selection rules (41) that are discussed more fully in Section 6.4.2.3. The problems of COSY spectra that can be partially alleviated in the 2Q spectroscopy fall into three categories: diagonal peaks, selfcancellation, and attenuation by solvent presaturation. The dispersive tails of diagonal peaks in COSY spectra can curtail observation of correlations between resonances with similar chemical shifts. The 2Q spectrum does not contain diagonal peaks and observation of cross-peaks is facilitated. In this respect, 2Q spectra are comparable to 2QF-COSY spectra, provided that the latter have been acquired with
476
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
sufficient t1 points to allow resolution of the diagonal and cross-peaks at the same F2 shift. Self-cancellation of COSY cross-peaks has been described in detail in Section 6.2.1.5. The attenuation that this causes is most severe in F1, where the linewidth is determined by t1max. In 2Q spectroscopy, the peaks have absorptive in-phase and dispersive antiphase character in F1, and are not subject to cancellation. Thus, for interactions with small scalar coupling constants or for proteins with large linewidths, 2Q spectra can display correlations that are unobserved in COSY spectra. Finally, presaturation of the solvent peak in the COSY experiment also attenuates correlations involving protons (usually 1H) that resonate close to, or coincident with, the solvent resonance. In a 2Q experiment, even if presaturation is employed prior to the initial 908 pulse (Fig. 6.34), antiphase magnetization between 1HN and 1H still develops, from the initial 1HN magnetization, provided that the solvent resonance is not irradiated during the delay (44). Thus, 2Q coherence can still be generated for these resonances. In the COSY spectrum of ubiquitin, three 1HN–1H cross-peaks were absent because of 1H presaturation (Fig. 6.9). The equivalent region of the 2Q spectrum clearly reveals these correlations (Fig. 6.39). Perhaps the biggest advantage of 2Q over COSY experiments is the presence of remote peaks. Usually, remote peaks occur in addition to direct peaks, and for short values of are easily recognized by the 1808 phase difference in F2 relative to the direct peaks (see product operator analysis in Section 6.4.1.1). Because the remote F1 frequency only gives information about the sum of two of the chemical shifts, at least one direct peak must also be observed at the same F2 shift to determine all three chemical shifts of the spin system. However, the remote peaks are observed even if two of the coupled spins have degenerate chemical shifts. This feature makes 2Q spectroscopy one of the few reliable methods to identify cases of chemical shift degeneracy. In addition to the important direct peaks with 1H (Fig. 6.39), amide protons are involved in two sets of remote connectivities involving glycine residues and all residues containing an 1H proton (45). Glycine residues are unique in that their amide protons are directly coupled to two -protons. The coupling within this group gives rise to three peaks 0 00 at F2 ¼ 1HN: the two direct peaks at F1 ¼ 1HN þ 1H and 1HN þ 1H and 0 00 also a remote peak at F1 ¼ 1H þ1H . In COSY spectra, one of the two cross-peaks expected for the glycine 1HN–1H residue is often missing, due to either a small active coupling (leading to self-cancellation), an overlap with other 1HN–1H correlations, or a degeneracy of the 1H and 1 00 H shifts, hence the observation of the remote peak in the 2Q spectrum
6.4
477
MULTIPLE-QUANTUM SPECTROSCOPY
o
C'i .....
E 0-
.e,
o
-.q:
u::
Ii}
-.q:
.....
I
9.5
I
9.0
8.5
F2 (ppm)
F'GURE 6.39 Section of the H 2 0 2Q spectrum of ubiquitin (r = 22 ms) showing part of the IHN_'HQ' region. The diagonal line represents the possible 2Q frequencies of 'H N coupled to IHa spins on resonance with the H 20 signal. The cross-peaks arising from Val5 and Leul5 were not observed in the COSY spectrum because of saturation of the 'HQ' resonance (compare Fig. 6.9). The cross-peak arising from Arg54 is observed in the 2Q spectrum, but does not appear in the illustrated region. Due to partial saturation in the COSY, the cross-peak due to Vall? is much more intense in the 2Q spectrum.
provides an unequivocal assignment of these amino acid spin systems (Fig. 6.40). The lobes surrounding the central positive and negative components of the glycine 2Q remote peaks are not resolution enhancement artifacts, but are actually the result of the natural dispersive double antiphase lineshape in F2 and dispersive antiphase lineshape in F, (see Fig. 6.36). In all residues except glycine and proline, the 'H a spin is coupled to both 'H N and at least one 'HIJ, hence remote peaks at F, = 'H N + IHIJ are expected at F2 = 'H a . Such peaks provide a useful means of correlating an I H N spin directly with an I HIJ spin in the same spin system. Unfortunately, the usefulness is compromised by the water stripe present at F2 = 8(H 2 0), which will obscure many 'H a resonances; as a result, IHN_'HIJ correlations are more readily established in TOCSY (Section 6.5) or relayed COSY (Section 6.2.2) spectra.
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS G47
K63
G75
G76 4.0
+
7.5
3.8
478
4.2
F1 (ppm)
G35
8.0
G75
G10
G35
8.4
8.0 F2 (ppm)
FIGURE 6.40 Section of the H2O 2Q spectrum of ubiquitin ( ¼ 22 ms) showing 0 00 the remote glycine peaks at F1 ¼ 1H þ 1H . The peaks from five of the six glycine residues are clearly present and labeled. The amide proton of the remaining glycine (Gly53) is unusually broad, and no cross-peaks involving it are observed in any of the correlation spectra. The inset shows the cross-peaks of 0 00 Gly35 and Gly75 in the COSY spectrum. The degeneracy of 1H and 1H of Gly75 and the overlap with Lys63 1HN–1H in the COSY are confirmed by the presence of the 2Q remote peak.
The 1H–1H peaks form the main fingerprint region of interest in the upfield region of the 2Q spectrum and are most easily observed in D2O solution at the F2 shift of the 1H resonances. This region of the COSY spectrum is normally crowded, making the observation of all 1 1 H – H cross-peaks problematic. In addition, for -methylene-containing side chains, one of the 1H–1H coupling constants frequently is small; thus, only one of the two expected peaks may be observed in COSY spectra. Although the direct 1H–1H peaks in the 2Q spectrum are equally crowded and possibly weak, the remote peaks are not; hence, observation of the remote peak and one direct peak allows assignment of all three resonance positions (Fig. 6.41). Commonly, resonance positions within a side chain will be inferred from correlations relayed to the amide proton in TOCSY experiments
479
6.4 MULTIPLE-QUANTUM SPECTROSCOPY
7.5
b'+b "
8.0
a +b "
F1 (ppm)
b '+b "
a+b"
S20a
S65a
S20b '
4.5
S65b'
4.0
S20b "
8.5
a+b'
a+b'
S65b " 3.5
F2 (ppm)
FIGURE 6.41 Example of remote and direct peaks for CH–CH2 fragments in the D2O 2Q spectrum ( ¼ 30 ms). Squares and ellipses indicate direct and remote correlations, respectively. The cross-peaks arising from Ser65 and Ser20 are connected by solid and dashed lines, respectively. The F1 and F2 frequencies are indicated adjacent to the lines connecting the cross-peaks. For this value of , the direct and remote peaks are distinguished by virtue of the opposite phase in F2 (see text).
acquired in H2O. Although chemical shift arguments allow assignment of the cross-peaks to a particular side chain resonance, this process is fallible. Discriminating 1H and 1H resonances within five-spin residues is one case where such a simple analysis commonly leads to incorrect side-chain assignments. Thus, the unambiguous determination of 1H resonance positions in the 2Q spectrum provides a useful complement to other relay-based techniques (e.g., see Fig. 6.42). Indeed, during the analysis of spectra for this book, several 1H resonances that had not been assigned previously were observed in MQ spectra (46, 47). Loss of signal intensity due to chemical exchange broadening (Section 5.6) frequently causes difficulties when analyzing standard 2D spectra such as COSY or 2QF-COSY. Under appropriate conditions, the 2Q experiment provides valuable assistance in such cases. One example is provided by the following situation (48). Doublequantum coherence between two spins, A and B, evolves at the sum,
480
CHAPTER 6 EXPER'MENTAL 'H NMR METHODS 2QF-COSY
TOCSY
()l
a
C\J
Y+Y"
f3"
~aD OD
f3'
G
()l
i.n
O:>a+ f3" a+ f3' <0
0
N
o
E 0S
f3"
u::
0
1'1
[;>
3.3
3.3 F2 (ppm)
3.3
F'GURE 6.42 Cross-peaks between IHQ' of GluM and its side chain resonances. The 'HQ'-'H.B' cross-peak in the 2QF-COSY and the 'HQ'-'H.B' direct peak in the 2Q spectrum are extremely weak due to the small coupling between IHQ' and IH.B'. All side chain resonance positions are identified in the TOCSY (r m = 48 ms); however, the 'HQ'-'H.B' and 'HQ'_IHY resonances cannot be unambiguously assigned. Observation of the remote peak in the 2Q spectrum (r = 30 ms) unambiguously identifies the chemical shift of 'H.B'.
Q A + QB, of their respective single-quantum frequencies. This sum frequency is unaffected by mutual exchange of the A and B spins; therefore, double-quantum coherence is insensitive to the exchange process. The direct peaks involving spins A and B will still be exchange . broadened in the observed dimension F 2 . However, if the doublequantum coherence is transferred to a third spin C that is coupled to both A and B, and if this third spin is not involved in the exchange process, then a remote peak will be observable at the frequencies Q A + Q B in F, and Q c in F2 . This scenario occurs for phenylalanine aromatic ring spin systems. The two IH e spins interchange as a result of 1800 flips of the ring, but the 'H~ proton is unaffect~d. Consequently, the remote peak at the sum of the two 'He frequencies in F, and the 'H~ frequency in F 2 is insensitive to the exchange process. The 2Q experiment has an additional advantage for exchangebroadened resonances because double-quantum peaks mostly have an
481
6.4 MULTIPLE-QUANTUM SPECTROSCOPY
in-phase multiplet structure. Thus the double-quantum peaks are less sensitive to self-cancellation as the linewidth increases, compared to cross-peaks with antiphase multiplet structure in COSY or 2QF-COSY experiments. Loss of signal intensity due to the effects of chemical exchange during the preparation period can be reduced by employing multiple refocusing pulses (49) or by using a Hartmann–Hahn mixing sequence (50).
6.4.2 3Q SPECTROSCOPY As with 2QF- and 3QF-COSY, the 2Q experiment is just one of many experiments based on observation of multiple-quantum states. Multiple-quantum states higher than 3Q, although attainable theoretically, are rarely applied to protein systems. Thus, the 3Q experiment is the focus of this section. The full phase cycle for the 3Q experiment was described in the caption to Fig. 6.34. 6.4.2.1 Product Operator Analysis Due to the similarity of the initial parts of the 2Q and 3Q pulse sequences, evolution of the product operators is the same up to the end of the 90x – =2 – 180x – =2 sequence. At this point, a 90y pulse must be applied in order to generate odd orders of multiple-quantum coherence. Thus, the following operators are obtained for a three-spin system (I1, I2, and I3): 2
x
2x 2 2
y
I1z !I1y cosðJ12 Þ cosðJ13 Þ þ 2I1z I2x sinðJ12 Þ cosðJ13 Þ þ 2I1z I3x cosðJ12 Þ sinðJ13 Þ 4I1y I2x I3x sinðJ12 Þ sinðJ13 Þ:
½6:39
The 3Q phase cycle suppresses the first three terms of [6.39]. The fourth term is a mixture of 3Qy and three-spin single-quantum coherence; a similar operator was encountered in the analysis of the 3QF-COSY experiment (Section 6.3.2.1). Once again, phase cycling retains only the pure three-quantum term and only its evolution needs to be considered during t1: t1
3Qy sinðJ12 Þ sinðJ13 Þ ! 3Qy cosð123 t1 Þ sinðJ12 Þ sinðJ13 Þ 3Qx sinð123 t1 Þ sinðJ12 Þ sinðJ13 Þ, ½6:40
482
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
where 123 ¼ 1 þ 2 þ 3. Application of the final 90y pulse to the Cartesian single operator components of [6.40] produces the following operators prior to t2: 4I1z I2y I3z 4I1y I2z I3z 4I1z I2z I3y þ 4I1y I2y I3y cosð123 t1 Þ sinðJ12 Þ sinðJ13 Þ þ 4I1z I2z I3z 4I1y I2y I3z 4I1z I2y I3y 4I1y I2z I3y sinð123 t1 Þ sinðJ12 Þ sinðJ13 Þ:
½6:41
The fourth term on line 1 and all four terms on line 3 of [6.41] are unobservable operators. The first three terms on line 1 give rise to observable peaks at the chemical shifts of each of the three spins in F2 and at the sum of their chemical shifts in F1. In F2, each peak is doubly antiphase with respect to the other two spins, whereas in F1, all three peaks are singlets without fine structure and can be phased to absorption. The preceding product operator analysis has been extended to include additional passive spins to give results applicable to other spin systems (51). Thus, the contribution of spin I1 to the 3Q coherence (I1 þ I2 þ I3) is described by
A1 ¼ sinðJ12 Þ sinðJ13 Þ
K Y
cosðJ1k Þ,
½6:42
k¼4
in which I1, I2, and I3 are active spins and all spins, Ik, for k 4 3, are passive spins. If all three active spins are passively coupled to a fourth spin, I4, then I4 also contributes to the 3Q coherence (I1 þ I2 þ I3) in the following manner: A4 ¼ sinðJ14 Þ sinðJ24 Þ sinðJ34 Þ
K Y
cosðJ4k Þ:
½6:43
k¼5
These expressions have been used to generate excitation profiles as a function of for the spin systems and coupling constants commonly found in amino acids (51). By considering evolution of the density matrix during a multiplequantum experiment, Braunschweiler et al. (41) have described two selection rules for the peaks observed in such spectra: (i) multiplequantum coherence involving a set of q spins can only be transferred to
6.4 MULTIPLE-QUANTUM SPECTROSCOPY
483
the single quantum transitions of a spin I when the scalar couplings between I and all q spins are resolved (if I belongs to the set of q spins, only the couplings to the remaining q 1 spins need be resolved), and (ii) multiple-quantum coherence involving two or more equivalent nuclei cannot be transferred to the single-quantum coherence of the equivalent nuclei because couplings between magnetically equivalent nuclei are ineffective in isotropic solution phase. As in the case of multiplequantum filtered COSY experiments, violations of the second selection rule by methyl groups in proteins are manifestations of multiexponential relaxation (35). 6.4.2.2 Experimental Protocol and Processing Carrier positions and spectral widths are chosen in a manner analogous to the methods described in the preceding section for the 2Q experiment. The principal difference is that now F1 peak positions occur at the sum of three chemical shifts. Considering only the aliphatic resonances (i.e., folding the aromatic signals00 in F1), the downfield spectral limit will be at the sum 0 of H, H , and H chemical shifts. The largest sum usually arises from serine residues. As 3Q coherences are now evolving during t1, the phases of all pulses prior to t1 must be incremented by /(2|p|) ¼ /6 in order to achieve quadrature detection (Section 4.3.4.1). Generally, ¼ 30 to 36 ms; additional detail is given in Section 6.4.2.3. Other details of the experimental protocol are similar to those already described above for the 2Q experiment (Section 6.4.1.2). The sections of spectra shown in Fig. 6.43 were obtained in about 16 hours on a 2 mM ubiquitin sample in D2O solution using a preparation period of ¼ 36 ms; 48 transients were collected for each of 600 t1 increments using a spectral width of 7000 Hz in F1 (t1max ¼ 43 ms). The carrier was shifted from the water resonance to 2.0 ppm at the start of the pulse sequence and the aromatic resonances were folded in F1. The experiment was acquired with TPPI quadrature detection in F1. Window functions applied in F2 are chosen in a fashion similar to those chosen for the 2Q experiment (Section 6.4.1.3). Phase parameters are chosen to give a symmetric three-lobed profile for direct peaks in this dimension (formally an absorptive double antiphase lineshape; Fig. 6.36). Given the predominantly absorptive lineshape in F1, processing in this dimension should be sufficient to prevent truncation, and possibly to provide slight resolution enhancement (e.g., sine bells shifted by 608 to 908). The center of the spectrum in F1 is referenced to three times the carrier frequency.
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
Ser65
a
b"
b'
a
b'
F1 = a+b '+b"
12.2
F1 (ppm)
F1 = a+b '+b "
12.0
11.8
484
b"
12.4
Ser20
4.5
4.0
F2 (ppm)
FIGURE 6.43 Demonstration of the spectral simplification achieved for a spin system containing three coupled spins in a 3Q spectrum; only three direct peaks are expected for such systems. Spin systems of Ser20 and Ser65 are shown (for comparison, the 2Q cross-peaks arising from these residues are depicted in Fig. 6.41).
6.4.2.3 Information Content The justifications for using 3Q experiments are twofold. First, moving to a higher quantum state results in spectral simplification because some spin systems cannot attain the requisite level of coherence (much as the 3QF-COSY offers advantages over the 2QF-COSY). Second, the remote peaks present in the 3Q spectrum offer unique assignment information. Even for spin systems that are present in pQF-COSY or 2Q spectra, the 3Q experiment may offer considerable simplification. For example, Fig. 6.43 depicts the correlations arising from the spin systems of Ser20 and Ser64; each spin system leads to three direct peaks only. Comparable regions of other spectra contain larger numbers of peaks: the COSY spectra contains six cross-peaks and three diagonal peaks and the 2Q contains six direct and three remote peaks. The simplification makes observation of the spin systems of Ser20 and Ser64 very straightforward (e.g., compare with Fig. 6.41).
6.4 MULTIPLE-QUANTUM SPECTROSCOPY
485
The three main uses for the 3Q spectrum are (i) verification of 1H and 1H resonance assignments made in other spectra, (ii) identification of resonance positions for phenylalanine and tryptophan ring protons, and (iii) identifying cases of resonance degeneracy at the end of longer spin systems (five-spin, lysine, arginine, and proline residues). Amide protons of glycine residues contribute0 to a direct 3Q 0 00 00 coherence (F1 ¼ 1HN þ 1H þ 1H ; F2 ¼ 1HN, 1H , or 1H ) that allows determination of all three chemical shifts; this information is obtained equally well from the remote peaks in a 2Q spectrum (Section 6.2.3.4). In the case of residues other than glycine or proline, an amide proton contributes to a direct (F1 ¼ 1HN þ 1H þ 1H) peak, and if the amino acid contains a -methylene group, a remote peak 0 00 (F1 ¼1HN þ 1H þ 1H ). The former correlations have excitation maxima in the range ¼ 30–40 ms, while the latter are maximized for ¼ 40–50 ms (51). From the multiple-quantum selection rules, the 3Q coherence is observable only at F2 ¼ 1H in both of these cases. Not all of these correlations are observed in spectra acquired from H2O solution because of interference from the water resonance. The main peaks of interest in a 3Q spectrum acquired from D2O solution are (i) coherences involving 1H and 1H protons, (ii) coherences arising from the aromatic resonances of tryptophan and phenylalanine residues, and 0 (iii) coherences involving the terminal 00 1 0 1 00 1 " protons of lysine (1H0 þ 1H" 00þ 1H" and H þ H þ H ), arginine, 0 00 and proline (1H þ 1H þ 1H and 1H þ 1H þ 1H) residues. Using the analytical expressions [6.42] and [6.43], values of to provide optimal intensity of these correlations can be deduced (51). Regardless of the size of 3 JH H and extensions of the spin system beyond 1H, the intensities of 1 1 0 1 00 the H þ H þ H coherences are maximal at 0 ¼ 30 ms except for 00 serine residues, for which the smaller geminal 1H –1H coupling leads to maximal intensity at ¼ 40 ms. The linear and aromatic spin systems have a broad maximum centered at ¼ 65 ms. Thus, all types of correlations should be detectable for in the 30- to 40-ms range. The efficiency of excitation for the terminal protons of longer side chains is harder to categorize because of the influence of contributions from passive spins. The various active and passive contributions have very different F1 lineshapes that can lead to severe cancellation for some values of . This complex behavior also is very dependent on the exact linewidths of the resonances involved because the dispersive terms are also prone to self-cancellation. In summary, a value of ¼ 36 ms is appropriate for most aliphatic side chain correlations (23). Acquisition of experiments with longer or shorter values of may be necessary to observe all correlations involving the terminal groups of arginine,
486
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
proline, and lysine. The work of Chazin should be consulted when optimizing for a particular correlation (51). The 3Q correlations among resonances at the termini of longer side chains provide one of the few ways to positively identify resonance degeneracy (51). However, the number of possible 3Q frequencies can be large, and the ability to obtain useful assignment information will depend on the particular distributions of chemical shifts. The aromatic regions of 3Q spectra are usually quite simple, as they only contain peaks from phenylalanine or tryptophan residues. For phenylalanine residues, direct peaks arising from three-bond scalar coupling interactions are observed at (F1, F2) frequencies of ( þ " þ , ") and ( þ " þ ", ). Additional direct and remote peaks are observed if long-range four-bond or five-bond scalar couplings are resolved. Resonances from the aromatic ring of tyrosine are absent unless the long-range 4 JH1 H2 or 4 JH"1 H"2 couplings are resolved. This region of the ubiquitin 3Q spectrum is further simplified by the degeneracy of H" and H of one of the two phenylalanine residues (Fig. 6.44).
6.5 TOCSY Soon after the introduction of the R.COSY and related experiments (Sections 6.2.2 and 6.2.3), a new method of obtaining relayed connectivities was introduced, namely, Total Correlation Spectroscopy (TOCSY) (52). This technique is also known by the acronym HOHAHA (homonuclear Hartmann–Hahn) spectroscopy (53). Instead of relying on pulse-interrupted free-precession sequences to transfer coherence between antiphase states (Section 4.2.1.1), TOCSY utilizes isotropic mixing to transfer in-phase magnetization between spins via the strong scalar coupling Hamiltonian (Section 4.2.1.2), with the result that magnetization can be transferred through several couplings during the course of the mixing period. In the absence of relaxation, cross-peaks potentially are generated between all resonances within a spin system.
6.5.1 PRODUCT OPERATOR ANALYSIS The TOCSY experiment can be performed in numerous ways. In the method described here, after frequency labeling during t1, magnetization is returned to the z-axis for isotropic mixing using the relaxationcompensated DIPSI-2 (DIPSI-2rc) or other mixing sequence, and then returned to the transverse plane for detection (54). In addition to several
487 22.4
6.5 TOCSY Phe45 F1 = e+z+e
z
d e +z
d
F1 (ppm)
e
22.0
F1 = d+e+z
F1 = e+z+e 21.6
F1 = d+e+z Phe4 7.5
7.0 F2 (ppm)
FIGURE 6.44 Section of the 3Q spectrum of ubiquitin in D2O solution showing the aromatic side chain resonances. Direct peaks arising from three-bond scalar coupling interactions are indicated by rectangles; direct peaks that result from at least one resolved four-bond scalar coupling are indicated by ellipses; a remote peak arising from other long-range scalar couplings is indicated by a diamond. Peaks from Phe45 are readily apparent, whereas the degeneracy of 1H" and 1H
for Phe4 prevents the generation of 3Q coherence, hence the direct peaks are not observed. In the absence of resolved four-bond scalar couplings, no correlations are observed for Tyr59.
advantages described here, this method also enables the addition of a Hahn echo at the end of the sequence to provide a flatter baseline (Section 3.6.4.2). The pulse sequence, coherence level diagram, and phase cycle for such an experiment are shown in Fig. 6.45. If a probe equipped with pulsed field gradients is available, then pulsed field gradients can be used for artifact and solvent suppression. The pulse sequence, coherence level diagram, and phase cycle for an experiment that incorporates the excitation sculpting water suppression technique are shown in Fig. 6.46. The phase cycling used in these versions of the TOCSY experiment is identical to that developed for the NOESY experiment, and is discussed in more detail in Section 6.6. Following the 90x –t1 –90x sequence, the density operator is described by [6.1]. The I1x and 2I1zI2y terms in [6.1] are eliminated by the TOCSY phase cycle or the gradient pulse G1. The DQ component of
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
488 f2
f1 t1
f3 ∆
Isotropic Mixing
∆
f4 ∆1
∆2
t2
p +1 0 –1
FIGURE 6.45 Pulse sequence and coherence level diagram for the TOCSY experiment. A Hahn echo is included prior to detection. The basic eight steps of the phase cycle are 1 ¼ 2(x, x, x, x); 2 ¼ 2(x, x, x, x); 3 ¼ 8(x); 4 ¼ 4(y), 4(y); and receiver ¼ 4(x, x); the full 32-step phase cycle is completed by performing CYCLOPS on all pulses and the receiver. The delay allows the transmitter power to be changed and is short enough (20 s) to prevent development of NOE cross-peaks. Frequency discrimination in F1 is obtained by shifting the phase of 1 and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4).
the term 2I1xI2y is dephased by the rf inhomogeneity present during the isotropic mixing pulse sequence of duration m (Section 3.5.3). During the isotropic mixing sequence, magnetization proportional to I1z is transferred throughout the spin system via the strong coupling Hamiltonian as described in Section 4.2.1.2: K X m Ikz a1k ðm Þ cosðI t1 Þ cosðJ12 t1 Þ, I1z cosðI t1 Þ cosðJ12 t1 Þ ! k¼1
½6:44 in which a1k( m) are mixing coefficients for transfer of magnetization through the spin system from spin I1 to spin Ik, and zero-quantum terms (Section 4.2.1.2) have been ignored. Following the final 90x pulse and the Hahn echo or excitation sculpting sequence, the density operator prior to t2 is given by K X Iky a1k ðm Þ cosðI t1 Þ cos ðJ12 t1 Þ: ½6:45 k¼1
489
6.5 TOCSY 1H
f1
f2
f3 f4
t1
f5
f5 f4
t2
isotropic mixing ∆1
∆1+∆2
∆2
Grad G1
G2 G3
G3 G4
G4 p +1 0 –1
FIGURE 6.46 Pulse sequence and coherence level diagram for a TOCSY experiment using excitation sculpting to avoid presaturation of the solvent resonance. The curved pulse shapes indicate selective 1808 pulses at the water frequency. The basic eight steps of the phase cycle are 1 ¼ 2(x, x, x, x); 2 ¼ 2(x, x, x, x); 3 ¼ 8(x); 4 ¼ 4(y), 4(y), 5 ¼ 4(y), 4(y); and receiver ¼ 4(x, x); the full 32-step phase cycle is completed by performing CYCLOPS on all pulses and the receiver. Delays before and after mixing allow the transmitter power to be changed and are short enough (20 s) to prevent development of NOE cross-peaks. Frequency discrimination in F1 is obtained by shifting the phase of 1 and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4). A bipolar pair of gradients can be inserted into the t1 period to prevent radiation damping (31).
The term proportional to I1y represents the diagonal peak, and the terms proportional to Iky for k 6¼ 1 represent cross-peaks. The observation of a cross-peak between two spins in the TOCSY experiment does not indicate that the spins are directly coupled; rather, the cross-peak indicates that magnetization can be transferred between the two spins by a series of steps through two- and three-bond scalar couplings between mutually coupled spins. The magnitudes of the cross-peaks [governed by aij( m)] depend upon the topology of the spin system, the coupling constants between pairs of spins, the efficiency of the isotropic mixing sequence employed,
490
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
and the rate of relaxation during the isotropic mixing pulse. The mixing coefficients usually satisfy aij( m) 4 0, but aij( m) 5 0 can arise for certain scalar coupling topologies (55). Numerical calculations based on density matrix theory have been used to calculate the transfer efficiencies for the coupling networks expected in naturally occurring amino acids using scalar coupling constants representative of a variety of conformations present in proteins (56). Such data are of much value in choosing an appropriate mixing time, as will be discussed in the next section. Inspection of [4.16] indicates that the mixing process also leads to the transfer of zero-quantum coherence that gives rise to antiphase components in both the cross-peaks and the diagonal peaks. The 908 phase difference between the in-phase and antiphase components indicates that when the former are phased to absorption, the latter will produce dispersive tails spreading out from peaks in the TOCSY spectrum. Fortunately, the resonance linewidths present in most proteins are sufficiently broad that these antiphase components of the peak are reduced by self-cancellation. One advantage of performing the experiment in the manner indicated in Fig. 6.45 is that the size of the antiphase components may be further reduced by varying the delays on either side of the isotropic mixing period over the course of the experiment. This process is known as ‘‘z-filtration’’ (54, 57), and is analogous to techniques for suppression of zero-quantum peaks in NOESY spectra. A more complete discussion is given in Section 6.6.1. The choice of appropriate isotropic mixing sequence has been the subject of much research (58). The first successful mixing sequences were derived from phase-modulated irradiation schemes used for spin decoupling, such as MLEV-17 (53) and WALTZ-16 (59, 60) (Section 3.4.3). Recently, many pulse sequences have been developed that are isotropic over wide frequency ranges with minimal power requirements. The most popular of these, at present, are the DIPSI (61, 62) and flipflop spectroscopy (FLOPSY) (63) family of pulse sequences developed by Shaka and co-workers. The DIPSI-2 sequence is given by the pulse sequence element R ¼ 320 410 290 285 30 245 375 265 370,
½6:46
in which pulse lengths are given in degrees and overbars indicate 1808 phase shifts. As is the case for broadband excitation (Section 3.4.4) and spin decoupling (Section 3.4.5), adiabatic pulses also have been incorporated into isotropic mixing schemes (64–67). Adiabatic mixing schemes are less sensitive to mis-setting of the B1 field than are conventional mixing sequences. Just as in the case for spin decoupling
6.5 TOCSY
491
(Section 3.4.5), the performance of isotropic mixing sequences is improved by combining the basic pulse sequence element into a The performance of mixing sequences is difficult to supercycle RR RR. establish by product operator analysis. Instead, the efficiency of coherence transfer is calculated numerically as a function of resonance offsets using the full density matrix formalism. Panels b–f in Fig. 6.47 illustrate the efficiency of coherence transfer for the MLEV17, WALTZ-16, DIPSI-2, FLOPSY-16, and WURST-2 adiabatic isotropic mixing sequences. MLEV-17 and WALTZ-16 provide significantly less efficient coherence transfer than do more modern sequences. The initial orientation of the proton magnetization must be considered when applying a given TOCSY mixing sequence. If the zfiltered version of the TOCSY experiment, Fig. 6.45, is used, then the initial magnetization is aligned along the þz or z axes, whereas if zfilters are not employed, the initial magnetization normally is oriented in the transverse plane. Some sequences, such as WALTZ-16 or DIPSI-2, can work equally well in these two situations. However, in the second case, the initial magnetization must be aligned orthogonally to the mixing sequence, i.e., along the y axes if the rf pulses of the mixing scheme are applied along the x axes. In contrast, sequences such as FLOPSY and adiababtic mixing require that the initial magnetization is aligned along the z axes, and MLEV-17 requires that the initial magnetization is aligned parallel to, or antiparallel to, the axis along which the 17th pulse is applied. Magnetization should never be spin-locked along a transverse axis, because magnetization then decays subject to (rapid) spin–spin relaxation processes; when properly aligned, the net relaxation behavior of the magnetization is determined by a combination of (rapid) spin–spin and (slow) spin–lattice relaxation processes (68). The most recent isotropic mixing schemes not only have desirable magnetization transfer properties, but also minimize transfer via dipolar coupling during the mixing period (via the rotating-frame Overhauser effect, or ROE; Section 5.4.3) (69). Direct ROE peaks and the TOCSY peaks are of opposite sign, and the ROE peaks may attenuate or completely cancel TOCSY cross-peaks between proximal spins within the same spin system. The dipolar effects are particularly problematic during the long mixing times required to see all correlations in extended spin systems and are also more acute for large proteins because dipolar relaxation is more efficient. The ‘‘clean’’ isotropic mixing sequences take advantage of the difference in sign of the NOE (laboratory-frame Overhauser effect) and the ROE [5.155]. By placing suitable delays in the pulse train, NOEs develop that offset the ROE contributions. The two
493
6.5 TOCSY
most widely used sequences in this respect are the DIPSI-2rc sequence (70) and the clean CITY sequence (71). Schemes also have been designed that achieve compensation between ROE and NOE effects without introducing delays during the mixing sequence (72). Clean adiabatic mixing sequences based on the WURST-2 pulse have been described (64, 67). The supercycle sequence of the ‘‘clean’’ DIPSI-2rc sequence is in which RR RR, R ¼ 1801403209027020200 853012512030075255 10190180,
½6:47
and should be compared with the original DIPSI-2 sequence [6.46]. The delays occur when (for ideal rotations) magnetization is aligned along the z-axis to permit compensatory NOE cross-relaxation. The delay has a nominal duration equal to the length of a 143.878 pulse. None of these schemes is capable of suppressing transfer via chemical exchange (Section 5.6). The effect of such a process on the peaks observed in a TOCSY spectrum has been described by Feeney and co-workers (73). Discrimination of chemical exchange peaks is discussed in more detail in Section 6.6.2.4.
6.5.2 EXPERIMENTAL PROTOCOL In all of the experiments described so far in this chapter, cross-peaks have had antiphase lineshapes in F2. This fact, coupled with the severely resolution-enhancing window functions used during processing, means little attention was paid to the quality of the baseline in the resulting spectra. Provided the spectrometer is reasonably well shimmed to minimize the residual water resonance, the baselines will be flat and the cross-peaks will be easily visible. In contrast, after the t2 Fourier transformation of a 2D TOCSY spectrum, the peaks in each F2 slice are in-phase and absorptive. Thus, they have many similarities with 1D spectra, including many of the same problems of baseline distortion discussed in Section 3.3. Upon Fourier transformation in t1, the baseline distortions, which tend to vary between t1 increments, are manifested as alternating positive and negative ridges running parallel to the F2 axis. In severe cases, the cross-peaks are obscured. In order to obtain the flattest baseline possible in absorptive homonuclear experiments such as TOCSY, the following points should be considered: (i) shim the sample with the specific aim of reducing the residual water signal
494
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
remaining after presaturation (Section 3.6.2.2), (ii) replace the final 908 pulse with a Hahn echo (Section 3.6.4.2) or pulsed field gradient watersuppression technique (Section 3.7.3), (iii) adjust the preacquisition delay to remove the need for frequency-dependent phase corrections in F2 (Section 3.3.2.3), (iv) digitally oversample in t2 (74), and (v) set the initial sampling delay in t1 to obtain either 08 or 1808 first-order phase correction in F1 (Section 3.4.1). More details on these topics can be found elsewhere in this book. Because the diagonal and cross-peaks in the TOCSY spectrum are predominantly in-phase and absorptive, the choice of t1max is limited by the desired resolution in F1. Sampling t1 beyond 1.5/R2 to 2/R2 is unnecessary, and values of t1max in the range of 40 to 60 ms are usually adequate. Generally, isotropic mixing is used during a pulse sequence to achieve one of two goals: (i) efficient transfer of magnetization through a single three-bond scalar coupling only and (ii) maximum transfer of magnetization between resonances at extreme ends of a spin system. The former aim is required to identify all possible 1HN–1H correlations, or if the isotropic mixing is part of a longer pulse sequence (Sections 6.5.5 and 6.7). The latter objective intends to provide observable crosspeaks between all spins within a spin system and is critical for successful completion of the spin system assignment stage of the sequential assignment process (Section 9.1.1). Both of these aims cannot be achieved simultaneously with a single mixing time because of the oscillatory nature of the magnetization transfer process (56): as the mixing time is increased, cross-peaks between directly coupled spins tend to increase in intensity quickly and then decrease before cross-peaks to more distant protons have even started to gain intensity, as illustrated for an isoleucine spin system in Fig. 6.48. During the assignment process, several TOCSY spectra are commonly acquired with different mixing times to avoid the possibility of missing correlations that have a minimum in the transfer function at the mixing time used in a single experiment. During the mixing period, the supercycle comprising the mixing sequence must be executed an integral number of times. Given the rf field strengths commonly used in TOCSY experiments (10– 12 kHz), the duration of a supercycle is 2–4 ms (depending on the sequence used). Therefore, the mixing time can only be varied in increments of this amount. Simulations indicate that for residues with large 3 JHN H , transfer from 1HN to 1H will be maximal at 30–50 ms, whereas transfer from 1 N H to other side chain protons increases approximately monotonically for mixing times up to 100 ms (56). In cases with smaller values of 3 JHN H , transfer from 1HN to all other protons is approximately
495
6.5 TOCSY 0.6
a
0.4
a1k(t m)
0.2
0.0 0.6
b
0.4
0.2
0.0 0
50
100
150
200
250
tm (ms)
FIGURE 6.48 Variation of cross-peak intensity as a function of isotropic mixing time for an isoleucine spin system. (a) Cross-peak intensity for transfer from the 1 H spin with the 1HN spin removed and (b) cross-peak intensity for magnetization transfer from the 1HN spin. The curves for the destination spins are (—) 1H, (– – –) 1H, (– –) H 1, (– –) 1H 2, and ( ) 1H. The transfer functions were calculated using the following coupling constants: 3 JHN H ¼ 10:0 Hz, 3 JH H ¼ 12 Hz, vicinal couplings to methyl groups were 6.7 Hz, all geminal couplings were 15 Hz, and all other vicinal couplings were 7 Hz. The effect of relaxation during the mixing was not considered.
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
496
monotonic up to 100 ms. Thus, maximizing transfer from 1HN to 1H will most likely be obtained with an isotropic mixing period 35–45 ms in duration. The simulations indicate that in order to maximize transfer to protons distant from 1HN, longer mixing times should be employed. However, relaxation during the mixing sequence, which was not incorporated into the simulations, reduces the intensity of all crosspeaks at longer mixing times. For a protein the size of ubiquitin, the maximal useful length of isotropic mixing is of the order of 100 to 120 ms; for larger proteins, this limit is shorter. A usual course of action would be to acquire one TOCSY with a mixing time of 40–60 ms to obtain a high number of 1HN–1H correlations, and then to acquire a second TOCSY with as long a mixing time as possible (i.e., 75 ms or longer) without severe loss of signal intensity. Although transfer through a single coupling is as efficient as in the COSY experiment (Sections 4.1.1.1 and 4.2.1.2), the peaks resulting from relay through several couplings are often of very low intensity. Thus, total acquisition times will generally be longer than for COSY. One or two passes through the phase cycle described in Fig. 6.45 or 6.46 (32 or 64 scans, respectively) will give acceptable sensitivity for a protein in the 1–4 mM range. Assuming a t1max of 50 ms and a reasonable F1 spectral width, TOCSY experiments require total acquisition times of 10–20 hr. The spectra shown in Fig. 6.49 were acquired using the DIPSI-2rc sequence with a pulse field strength of 12.5 kHz applied for 48, 83, or 102 ms. Thirty-two transients were collected for each of 576 t1 increments, yielding a t1max of 50 ms and total acquisition times of about 10.5 hr. Quadrature detection in F1 was achieved with TPPI. The delays on either side of the isotropic mixing time only facilitated a change in transmitter power; z-filtration (54) was not performed.
6.5.3 PROCESSING Appropriate window functions for TOCSY spectra are far less resolution enhancing than are those used for COSY-type spectra. Sine bells shifted by 608 to 908 in F1 and F2 provide some degree of resolution enhancement while reducing truncation effects. The trade-off between sensitivity and resolution can be fine tuned by varying the width of the sine bell from 80 to 300 ms in t2. Alternatively, matched exponential apodization or weak (i.e., not very resolution enhancing) Lorentzianto-Gaussian transformations can be performed. The Lorentzian-toGaussian transformations are beneficial when observing cross-peaks close to the diagonal; the intensity of the diagonal is such that the
497
g" g'
g" g'
b
b"
g" g'
d" g"
g" g'
d" g"
d b
g'
d b
g' d' b"
b" b'
b'
e K48 8.0
K29 7.9
K48
e
e K29
8.0
7.9
K48 8.0
3.0
F1 (ppm)
b'
2.0
6.5 TOCSY
e K29 7.9
F2 (ppm)
FIGURE 6.49 Sections of H2O TOCSY spectra acquired with mixing times of 48 (left), 83 (center), and 102 ms (right). The cross-peaks observed to the amide protons of Lys29 and Lys48 are assigned at the different mixing times. Crosspeaks to the spin-system termini are observed only at the longer mixing times.
Lorentzian tails are very obtrusive. Commonly, spectra are processed with at least two different window functions to maximize sensitivity (for analyzing most of the spectrum) and enhance resolution (for analyzing heavily overlapped regions of the spectrum). In addition to the precautions taken while setting up the experiment, other processing ‘‘tricks’’ (some of which are discussed in Chapter 3) can be beneficial in 2D spectra with absorptive lineshapes. In brief, these include (i) deconvolution to remove the residual water resonance in each FID (75), (ii) linear prediction of the first (or first few) points of the FID, and (iii) applying a baseline correction of a functional form sufficient to remove dc offsets or correct tilting, bowing, or rolling of the baseline. In most cases, if all other acquisition and processing precautions are taken, baseline corrections will not be needed. The H2O TOCSY spectra presented in this section were processed with weak Lorentzian-to-Gaussian transformations in t1 and t2. A cosine bell was also applied in t1 to eliminate truncation artifacts. The solvent resonance was removed by deconvoluting the FID with a sine lineshape
498
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
averaged over 32 points. Any dc offset was removed by considering the intensity of the final 5% of each FID. The first point of each FID was linear predicted using the first 128 points (using the HSVD algorithm) and then scaled by 0.5 prior to Fourier transformation.
6.5.4 INFORMATION CONTENT COSY-type experiments are very good at identifying sections of spin systems within the so-called fingerprint regions (Section 6.2.1.4). Connecting fragments from the different fingerprint regions is difficult in such spectra because of overlap and self-cancellation (due to complex fine structure) of the intervening methylene signals. TOCSY, and other relayed experiments discussed later, circumvent these problems by producing cross-peaks between the fingerprint resonances at opposite ends of extended spin systems. Because of the large chemical shift dispersion of the amide resonances, relayed connectivities involving these protons play a vital role in identification and assignment of spin system type (Section 9.1.1). Thus, in cases where two spin systems have identical 1H chemical shifts, the complete spin systems can be assigned by observing correlations to the associated 1HN spin. In TOCSY spectra acquired from H2O solution, the most useful crosspeaks include correlations from 1HN to the methyl resonances of alanine, threonine, valine, leucine, or isoleucine; the 1H or 1H" resonances of arginine or lysine residues, respectively; and the 1H of three-spin side chains, or 1H and 1H of five-spin side chains. Because of the length of leucine, isoleucine, arginine, and lysine side chains, the interesting endto-end cross-peaks are only observed for longer mixing times. Some examples of the correlations observed in these long side-chains as the mixing time is increased are shown in Fig. 6.49. Commonly, not all of these correlations will be observed because small values of 3 JHN H or small values for 3 JH H0 and 3 JH H00 will limit transfer from 1HN or 1H. Provided some 1H resonances are resolved, TOCSY spectra acquired from D2O solution permit observation of cross-peaks between 1H and the spin system termini. Analysis of TOCSY spectra is prone to two possible pitfalls. First, even if all side chain resonances are observed, definitive assignment of a resonance to a particular position within the side chain may not be possible. This is particularly true for the methylene groups of five-spin side chains, arginine, and lysine. Estimates of the assignment from chemical shift arguments, or by following TOCSY cross-peak intensity as a function of mixing time, are not infallible. Other correlation techniques that transfer coherence over a specific number of couplings
6.5 TOCSY
499
are required to eliminate ambiguity (for example, see Fig. 6.42). Second, the variation of cross-peak intensity with mixing time, including initial growth and then decay, may prevent observation of cross-peaks at certain mixing times. In the 48-ms mixing-time TOCSY experiment of ubiquitin, correlations between 1HN and the terminal protons of many shorter side chains are observed (i.e., transfer as far as 1H ). However, very few correlations are observed from 1HN to 1H" of lysine or 1H of leucine (zero out of seven lysine residues and four out of nine leucine residues); many more such correlations are observed at 84 ms (five out of seven lysine residues and all nine leucine residues). Extending the mixing period to 102 ms leads to the observation of weak 1HN–1H" correlations for the two other lysine residues. For the seven isoleucine residues, cross-peaks from 1HN to both terminal methyl groups are readily apparent after 48 or 84 ms of isotropic mixing. In two cases, the correlations from 1HN to the intervening H 1 are only observed at 102 ms, and in a third case one of the 1HN–1H 1 cross-peaks was not observed at all.
6.5.5 EXPERIMENTAL VARIANTS The use of isotropic mixing has become a mainstay of 1H NMR analysis of proteins. Four less common applications involving isotropic mixing are discussed in this section: pre-TOCSY experiments, TOCSY with very short acquisition times, sensitivity-enhanced TOCSY, and TOCSY with short mixing times. The concept of using isotropic mixing prior to a pulse sequence has already been encountered in the pre-TOCSY COSY experiment (Section 6.2.1.6). The idea behind this and all pre-TOCSY experiments is to use a short TOCSY mixing period to transfer magnetization to protons that have been attenuated by the solvent presaturation pulse (17). Thus, cross-peaks can be observed at the F1 frequency of the water resonance. The process is potentially applicable to all homonuclear 2D experiments utilizing solvent presaturation, but is most commonly used in the COSY experiment. The introduction of the isotropic mixing period alters all cross-peak intensities in the final spectrum. This is particularly pertinent for NOESY spectra because cross-peak intensities are interpreted in a quantitative fashion to generate distance restraints for protein structure calculations. In scalar correlation spectra, peak positions, and not intensities, are generally of interest, and perturbations from the pre-TOCSY sequence are unimportant. Investigations of amide proton exchange with solvent play an important role in understanding protein structure and internal dynamics
500
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
(76). Exchange rates can be obtained by rapidly transferring protein from H2O to D2O and repeatedly acquiring NMR spectra to follow the decay of amide 1H signal as protons exchange with solvent deuterons. One-dimensional NMR spectra permit high temporal resolution (of the order of a few minutes); however, overlap of the 1HN resonances precludes detailed, site-specific analyses of the exchange rates. Twodimensional spectra permit the cross-peaks from individual amide protons to be resolved; however, each spectrum must be acquired quickly (on the order of 30 min or less) and have a high sensitivity for amide protons. TOCSY spectra provide a useful compromise between these two factors [HSQC spectra (Section 7.1.1.1) of 15N-labeled proteins are even more useful]. A short t1max (and short total acquisition time) can be used because cross-peaks in TOCSY spectra have in-phase absorptive lineshapes. The mixing time is kept short (no more than 40 ms) to maximize the intensity of 1HN–1H correlations, and the F1 resolution is maximized by reducing the spectral width in F1 to cover the 1 N 1 H – H fingerprint region only (care should be taken to avoid folding or aliasing diagonal peaks into the region of interest). Finally, total acquisition times are kept short by only performing 4 or 8 transients per increment. Bax and colleagues have discussed some of the experimental aspects of short 2D experiments (77). Figure 6.50 shows a section of a TOCSY spectrum acquired from a 2 mM H2O sample of ubiquitin in 15 min. The data consisted of four transients co-added for each of 160 t1 experiments. The pulse sequence used was the gradient-enhanced TOCSY experiment of Fig. 6.46. The optimum four-step phase cycle (see legend, Fig. 6.50) was deduced empirically to determine which artifacts were dominant (and therefore required attenuation by the limited phase cycle) on the NMR spectrometer utilized. Although the acquisition time was short, most of the 1HN–1H correlations are visible. In the version of the TOCSY experiment just described, after the initial 90x –t1 –90x period, a mixture of frequency-labeled I1z and I1x magnetization is created [6.1]. If an isotropic mixing sequence is applied orthogonally (i.e., all pulses in the sequence have phases y or –y), then magnetization transfer occurs independently by the pathways I1z ! Ikz [4.16] and I1x ! Ikx [4.22]. Thus, the TOCSY experiment is unusual in that equivalent coherence transfer is obtained simultaneously along two orthogonal axes. In the conventional experiment, phase cycling (54, 57) or rf inhomogeneity effects (52, 53) are used to remove one or other of the components and obtain amplitude-modulated phase-sensitive data. In Fig. 6.45, inversion of the phases 1 and 2 every second transient suppresses the signal from the I1x ! Ikx pathway. Cavanagh and Rance
6.5 TOCSY
501
~
0°0 ~.
o
o:&'Q) o~
C>
<>
co
9.0
-o
8.0 F2 (ppm)
0
EC-
<0
~o
o l!i
<>
S u::
7.0
FIGURE 6.50 Section of a TOCSY experiment acquired in 15 min on a 2 mM ubiquitin sample in H 2 0 with a mixing time of 36 ms. Only a single contour is shown for the negative (folded) diagonal and cross-peaks. The pulse sequence of Fig. 6.46 was used. The entire F I spectral width is shown. Four transients were collected for each of 160 II experiments (tlmax=29ms). The phase cycle was c/>3 = receiver = x, y, -x, - y. In spite of the short phase cycle, the excitation sculpting gradient sequence helps reduce the level of artifacts in the spectrum. Even though the acquisition time is short, virtually all of the expected IHN_IH'" correlations are observed. The negative peaks near the top of the spectrum result from correlations to side chain protons that are folded in Fl.
have shown that an improvement in signal-to-noise ratio of .j2 is obtained by altering the phase cycle to retain both components entering the isotropic mixing period (78). Absorptive lineshapes are achieved by recording two independent data sets in which the phase of the 90° pulse prior to the mixing period differs by 180°. For example, in Fig. 6.45, the first two steps of the phase cycle (
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
502
over a conventional spectrum acquired with the same total acquisition time (78). Similar principles have been used in several heteronuclear pulse schemes to improve sensitivity (Section 7.1.3.2). Because the rate of buildup of a TOCSY peak depends on the size of the scalar coupling constant, quantitative analysis of cross-peak intensities can be used to estimate coupling constants (79). The size of a TOCSY cross-peak depends on all of the couplings within a spin system; therefore, extraction of single coupling constants must be performed with a degree of caution. Short mixing times (5–20 ms) must be used to ensure that magnetization transfers primarily through a single scalar coupling interaction; however, zero-quantum artifacts (see Section 6.6.1.1) can distort cross-peaks in such spectra. The rather crude estimates of scalar coupling constants obtained by this approach can be used to aid in the stereospecific assignments of -methylene protons (80).
6.6 Cross-Relaxation NMR Experiments All of the experiments described so far in this chapter have relied upon magnetization or coherence transfer via scalar couplings and only provide correlations between protons within the same amino acid residue. The sequential assignment process in an unlabeled protein sample is completed using the dipole–dipole cross-relaxation to correlate 1 H spins that are close in space. Additionally, distance restraints for structure determination of proteins are derived primarily from 1H–1H dipole–dipole cross-relaxation. The two experimental approaches in existence are based on longitudinal cross-relaxation (the NOE) and transverse cross-relaxation (the ROE). Theoretical origins of these crossrelaxation mechanisms are described in Section 5.5.
6.6.1 NOESY The pulse sequence for the NOESY (Nuclear Overhauser Effect Spectroscopy) experiment is shown in Fig. 6.51. Initially, a 908–t1–908 element is used to frequency label the spins and return the magnetization to the z-axis. Magnetization transfer occurs via dipolar coupling for a period m before observable transverse magnetization is created by the final 908 pulse. The final pulse can be replaced by a Hahn echo sequence with a concomitant improvement in the flatness of the baseline (Sections 3.6.4.2 and 6.5). A coherence level diagram for this pulse sequence is presented in Fig. 6.51. If a probe equipped with
503
6.6 CROSS-RELAXATION NMR EXPERIMENTS f1
f2 t1
f3
f4 ∆1
tm
∆2
t2
p +1 0 –1
FIGURE 6.51 Pulse sequence and coherence level diagram for the NOESY experiment. A Hahn echo sequence is included prior to detection. Appropriate values for the delays are discussed in the text. The same 32-step phase cycle as for the TOCSY is employed (Fig. 6.45).
pulsed field gradients is available, then pulsed field gradients can be used for artifact and solvent suppression. The pulse sequence, coherence level diagram, and phase cycle for an experiment that incorporates the excitation-sculpting water-suppression technique is shown in Fig. 6.52. 6.6.1.1 Product Operator Analysis The theory of magnetization transfer by dipolar coupling (the NOE) has been discussed in Sections 4.2.2 and 5.5. For a spin system containing two scalar-coupled spins, evolution through the 90x –t1 –90x sequence is described by [6.1]. Evolution of the I1z term in [6.1] during m is governed by the Solomon equations, in which the initial condition is I1z cos(1t1) cos(J12t1) and the equilibrium magnetization, I 01z , is rejected by phase cycling for axial peak suppression. If K 1 spins (Ik for k ¼ 2, . . . , K) are close in space to spin I1 (this notation allows for the possibility that the scalar-coupled spin, I2, is dipolar coupled to I1 as well), then the resulting evolution during m is m
I1z cosðI t1 Þ cosðJt1 Þ !
K X
Ikz a1k ðm Þ cosð1 t1 Þ cosðJ12 t1 Þ,
k¼1
½6:48
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
504 1H f1
f2 t1
f3 tm
f5 f4
∆1
f5 f4
∆1+∆2
t2
∆2
Grad G1 G2
G2 G3
G3
p +1 0 –1
FIGURE 6.52 Pulse sequence and coherence level diagram for the NOESY experiment using excitation sculpting to avoid presaturation of the solvent resonance. The curved pulse shapes indicate selective 1808 pulses at the water frequency. Appropriate values for the delays are discussed in the text. The same 32-step phase cycle as for the TOCSY is employed (Fig. 6.46). Frequency discrimination in F1 is obtained by shifting the phase of 1 and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4). A bipolar pair of gradients can be inserted into the t1 and m periods to prevent radiation damping (31).
in which a1k( m) ¼ [exp(R m)]1k is the (1, k)th element of the matrix exponential and R is the matrix of rate constants i and ij (Section 5.1.2). After the final 908 pulse and Hahn echo, the density operator terms that result from the longitudinal magnetization are given by K X
Iky a1k ðm Þ cosð1 t1 Þ cosðJ12 t1 Þ:
½6:49
k¼1
The final spectrum contains diagonal peaks (the k ¼ 1 term in [6.49]) and I1 ! Ik, for k 4 1, NOE cross-peaks. All of the peaks are in-phase with respect to homonuclear scalar coupling in F1 and F2, and also can be phased to absorption in both dimensions.
6.6 CROSS-RELAXATION NMR EXPERIMENTS
505
The magnetization that will give rise to the NOE cross-peaks stems from longitudinal magnetization (coherence level p ¼ 0) during m, and the phase cycling rejects other coherence levels during this period, including the single-spin terms I1x and 2I1xI2z in [6.1]. The 12 second term of [6.1] is a mixture of ZQ12 y (p ¼ 0) and DQy (p ¼ 2) coherences (Section 2.7.5); while the double-quantum operator is suppressed by the phase cycling, the zero-quantum term survives. During m the zero-quantum term will precess according to the difference in chemical shift of I1 and I2. The following terms will be generated by the final 908 pulse: ZQ12 y cosð1 t1 Þ sinðJ12 t1 Þ m 2
x
! þ 12½2I1x I2x þ 2I1z I2z cosð1 t1 Þ sinðJ12 t1 Þ sin½ð1 2 Þm 12½2I1z I2x 2I1x I2z cosð1 t1 Þ sinðJ12 t1 Þ cos½ð1 2 Þm : ½6:50 The last line of [6.50] contains observable terms and therefore must be considered in an analysis of the NOESY spectrum. Such artifacts arise via a zero-quantum pathway and are referred to as zero-quantum peaks. These peaks are in antiphase in both dimensions, and are also in dispersion when the normal NOE peaks ([6.49]) are phased to absorption. During analysis of a NOESY spectrum, the integrated intensity of a given cross-peak is interpreted in terms of the distance between the two protons giving rise to the peak (Sections 6.2.5.4 and 10.2.1). Clearly, from [6.50] the real NOE and zero-quantum peaks between two coupled spins appear at identical chemical shifts in F1 and F2. Although the net integrated intensity of the dispersive zero-quantum component is zero, accurately integrating the contributions from the dispersive tails of this component may not be possible, and errors in the measurement of the NOE cross-peak volume result; in addition, the antiphase dispersive tails can interfere with the integration of other cross-peaks in crowded regions of the spectrum. The magnitude of the zero-quantum component varies as cos[(1 2) m], which depends on the chemical shifts of the spins involved and the mixing time. In addition, because the zero-quantum terms have transverse components during m, relaxation is faster than for longitudinal magnetization, and the zero-quantum component is reduced in intensity relative to the true NOE peak when a long mixing time is
506
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
employed. The use of a z-filter to suppress the zero-quantum terms is discussed in Section 6.6.1.2. 6.6.1.2 Experimental Protocol The peaks of interest in the NOESY spectrum are absorptive and in-phase in both dimensions ([6.49]). Thus, all of the precautions adopted in Section 6.5.2 to ensure flat baselines in TOCSY spectra should also be employed in the NOESY experiment. Obtaining a flat baseline is even more imperative in the NOESY spectrum; at the least, the size of the cross-peaks will be interpreted in a semiquantitative fashion, hence any offset in the baseline will lead to systematic errors in the NOE cross-peak volume. The number of observable NOEs will have a direct bearing on the quality of structures produced from the data, hence obtaining a high S/N ratio is also important. Acquisition times of 24 to 48 hr are not uncommon, even with relatively concentrated samples. Accurate quantitative analysis also requires minimization of any other effect that may systematically alter the intensity of peaks, especially when a detailed quantitative analysis is to be performed (e.g., see Sections 6.6.1.4 and 10.2). The recycle delay should be 3/R1, in which R1 is the smallest longitudinal relaxation rate in the protein, to avoid steady-state effects that perturb the intensity of cross-peaks. Also, solvent presaturation should be avoided as a means of solvent suppression in these applications because the intensity of cross-peaks involving protons that resonate close to the water, or labile protons that are exchanging with the solvent, will be reduced. Methods that attempt to combat these effects are discussed in Section 6.6.1.5. The theoretical time dependence of the NOE cross-peaks in the NOESY experiment (Section 5.1.2) suggests that the mixing time should be on the order of 1/R1 to maximize the intensities of NOE cross-peaks. A long mixing time also has the advantage that zero-quantum artifacts will be of low intensity. However, long mixing times will also allow multiple magnetization transfers, or spin diffusion, to contribute substantially to the cross-peak intensity. The origins and consequences of spin diffusion are illustrated for a three-spin system with the following relaxation rate matrix: 2 3
1 12 0 ½6:51 R ¼ 4 12 2 23 5: 0 23 3 By construction, spins I1 and I3 are too far apart to have an appreciable dipolar coupling ( 13 ¼ 0), thus direct magnetization transfer between I1
6.6 CROSS-RELAXATION NMR EXPERIMENTS
507
and I3 is not possible. The time dependence of the I1 magnetization is given to third-order in time by hI1z iðm Þ ¼
3 X
½expðRm Þ1k hIkz ið0Þ
k¼1
3 X 2 3 hIkz ið0Þ E1k R1k m þ 12R21k m 16R31k m k¼1
2 1 3 3 2 2 2 m 6 1 þ 2 1 12 m þ 2 12 ¼ hI1z ið0Þ 1 1 m þ 12 21 þ 12 2 þ hI2z ið0Þ 12 m þ 12ð 1 þ 2 Þ 12 m 3 2 2 12 þ ð 1 þ 2 Þ 2 12 þ 12 23 m 16 21 þ 12 1 2 3 1 þ hI3z ið0Þ 2 12 23 m 6ð 1 þ 2 þ 3 Þ 12 23 m : ½6:52 Each of the terms in [6.52] can be assigned a physical interpretation; however, only three terms will be discussed in detail. The first-order term 12 m hI2z ið0Þ represents direct transfer of magnetization from spin I2 to spin I1 and gives rise to a cross-peak in the NOESY spectrum. In the initial rate regime, only this term contributes to the cross-peak intensity, and the cross-peak intensity is proportional to the cross-relaxation rate 2 hI3z ið0Þ exemplifies constant, 12. The second-order term ð1=2Þ 12 23 m spin diffusion. This term gives rise to a cross-peak between spins I1 and I3 by an indirect two-step transfer from I3 ! I2 ! I1. In the quadratic time regime, the intensity of the spin diffusion cross-peak depends on the product of the individual cross-relaxation rate constants. Finally, the 2 3 third-order term 2 12 m hI1z ið0Þ represents a back transfer pathway I1 ! I2 ! I1. The back transfer has the effect of reducing the intensity of the cross-peak that would otherwise result from cross-relaxation between I1 and I2. Therefore, even for a two-spin system, outside of the initial rate regime, NOE cross-peak intensities are not proportional to the crossrelaxation rate constants. The assumed linearity between the NOE crosspeak intensities and cross-relaxation rate constants sometimes is called ‘‘the isolated two-spin approximation’’; as the present discussion shows, this phrase is a misnomer. As a consequence of spin diffusion, cross-peaks between pairs of protons that are far apart will gain intensity from magnetization that has been transferred via intervening spins, while cross-peak between pairs of protons that are close together will be decreased by the loss of magnetization to other nearby protons. Failure to adequately account for spin diffusion results in the derivation of inaccurate distance
508
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
constraints between pairs of protons; overly tight constraints derived from NOE cross-peaks dominated by spin diffusion lead to overly constrained and incorrect protein structures (Sections 6.6.1.4 and 10.2.1). Spin diffusion effects can be minimized by using a short mixing time, but in these experiments all cross-peak intensities will be low, and zero-quantum artifacts will be emphasized. A compromise is usually struck, with mixing times of 50–150 ms providing reasonable cross-peak intensities that are not overly influenced by spin diffusion or zero-quantum contributions. Dipolar relaxation is more efficient in systems with long rotational correlation times, hence a shorter mixing time is required to limit spin diffusion in large proteins. The dispersive antiphase components observed in NOE cross-peaks between scalar-coupled protons can be suppressed by several methods. Early strategies involved randomly changing the phase of the zeroquantum coherence by randomly varying the mixing time or by applying a 1808 pulse at random positions within a fixed mixing time from one t1 increment to the next (81). A disadvantage of these methods is that they effectively transfer the problem of zero-quantum coherence into additional t1 noise in the NOESY spectra. More effective procedures have been proposed for the elimination of zero-quantum coherences. For example, the mixing time can be kept fixed, and a 1808 pulse applied at a selected position within the mixing period; this pulse causes a partial refocusing of the zero-quantum coherence. By co-adding several data sets with the 1808 pulse varied in its position, suppression of zeroquantum coherence can be achieved (82, 83). More recently, a new method has been developed that involves the simultaneous application of a swept-frequency 1808 pulse and a magnetic field gradient (84, 85). In this scheme, the presence of the magnetic field gradient causes the 1H spins at different positions in the sample to be inverted at different time points; this leads to a position-dependent variation in refocusing of the zero-quantum coherence. Integrated over the sample volume, the variable refocusing will lead to destructive interference of the zeroquantum coherences and will therefore provide an effective level of suppression. An alternative scheme for dealing with zero-quantum coherences has also been proposed that separates such zero-quantum peaks from the NOESY peaks, rather than attempting to suppress the zero-quantum resonances. In this procedure, a 2D NOESY experiment is run instead as a 3D experiment in which the second evolution period is created by incrementing the position of a 1808 pulse in the NOE mixing period (86). In this case, the NOE peaks will appear in the zero frequency plane in the second dimension, whereas the zero-quantum peaks will appear in planes at the appropriate zero-quantum frequencies. In all of
509
6.6 CROSS-RELAXATION NMR EXPERIMENTS
a
G75
b
R74
R74 R72
R72
L73
L73
8.6
8.4
8.6
4.5
F1 (ppm)
4.0
G75
8.4
F2 (ppm)
FIGURE 6.53 Comparison of sections of the (a) m ¼ 40 and (b) m ¼ 100 ms NOESY spectra of ubiquitin in H2O solution. Several intraresidue and sequential 1HN–1H cross-peaks are denoted by the rectangles and ellipses, respectively, and allow sequential assignment of several residues near the Cterminus. Due to the presence of zero-quantum artifacts, several of the intraresidue peaks contain dispersive antiphase components in the 40-ms mixing time experiment.
these suppression schemes, elimination of zero-quantum coherence arising from spins closely spaced in resonance frequency is difficult. The phase cycling used in the NOESY experiment must incorporate axial peak-suppression phase cycling (of the first 908 pulse) and selection of p ¼ 1 and p ¼ þ1 by phase cycling of the second 908 pulse. The basic phase cycle is four steps (1 ¼ x, x, x, x; 2 ¼ x, x, x, x; and receiver ¼ x, x, x, x). Alternatively, both pulses can be phase cycled in synchrony to select p ¼ 0 (Section 4.3.2.2). EXORCYCLE phase cycling is used for the Hahn echo and CYCLOPS is applied to all pulses. Bodenhausen and co-workers have discussed phase cycles for NOESY experiments (87). The NOESY spectra depicted in Fig. 6.53 were acquired from H2O solution using the Hahn echo sequence of Fig. 6.51 with the sample delay set to avoid phase correction-associated baseline distortions in F1 (Section 3.3.2.3). The solvent was saturated during the recycle delay and during the mixing time. Thirty-two transients were collected for each of 576 increments of t1 (t1max ¼ 50 ms), a mixing time of 40 or 100 ms was
510
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
used, and total acquisition time was 10.0 or 10.5 hr. No attempt was made to suppress zero-quantum contributions. 6.6.1.3 Processing The similarity of TOCSY and NOESY lineshapes dictates that processing of these two spectra is similar. All of the details discussed in Section 6.2.4.3 are relevant to the present discussion. Accurate determination of cross-peak volumes is important for quantitative analysis of NOESY spectra. A variety of methods have been proposed to improve the quality of cross-peak volume extraction from NOESY spectra (88–91). The ubiquitin spectra shown in Fig. 6.53 were processed in a fashion identical to that for the TOCSY spectra of Section 6.5.3.3, except that deconvolution of the residual water resonance was not necessary. 6.6.1.4 Information Content NOESY spectra provide a powerful means of elucidating conformational details of molecules in solution. The requirement that two protons be separated by less than 5 A˚ (or so) in order to give rise to an NOE immediately allows a loose restraint to be placed on their separation. Furthermore, the size of the NOE depends inversely on the distance, hence the restraint can be tighter than 5 A˚ if the NOE is intense. In order to calculate the structure of a protein, many such restraints must be identified in an unambiguous fashion (Section 9.2.1). In most applications, NOE cross-peaks simply are placed into one of several size categories associated with an upper bound for the proton separation. More accurate calibration is difficult because of the complex relationship between NOE buildup, local correlation time, and the distribution of neighboring protons. Analysis of NOESY spectra with different mixing times (called a buildup or m series) allows the initial slope of the NOE buildup to be estimated and facilitates calibration. Methods for interpreting NOESY spectra are discussed in more detail in Section 10.2.1. Figure 6.53 shows regions of the 40- and 100-ms NOESY spectra of ubiquitin in H2O solution, and demonstrates that sequential 1H–1HN NOEs can be used to obtain sequence-specific assignments of the residues near the C-terminus. As expected for a protein of this size, the cross-peaks are all more intense at the longer mixing time. The residues at the C-terminus are conformationally mobile, have narrower linewidths, and therefore have very prominent dispersive contributions in the shorter mixing-time spectrum (Section 6.6.1.1); self-cancellation reduces the impact of these components for other, broader resonances. Increasing the mixing time also leads to a decrease in the relative intensity of zero-quantum effects.
6.6 CROSS-RELAXATION NMR EXPERIMENTS
511
Besides cross-relaxation, chemical exchange can also lead to crosspeaks in NOESY spectra. In cases of slow exchange between two species (Section 5.6), a cross-peak is observed at the frequencies of a particular nucleus in the two different sites if the exchange rate between the species is comparable to 1/ m. For proteins, the chemical exchange peaks will be of the same sign as NOE cross-peaks (the same sign as the diagonal peaks, formally negative), hence discrimination of the two can be difficult. Very complicated spectra can result due to peaks arising from combinations of exchange and cross-relaxation; in effect, these are spin diffusion-type peaks, as they involve two transfer steps. Identification of exchange effects is discussed in more detail in Section 6.6.2.4. 6.6.1.5 Experimental Variants Suppressing the solvent resonance by presaturation unavoidably results in transfer of saturation to labile protons undergoing exchange with solvent, and attenuation of crosspeaks involving protons that resonate close to the solvent resonance (Section 3.5.1). Several alternatives to presaturation in 1D 1H NMR have already been discussed (Section 3.5) and these methods are valuable for NOESY experiments. The final pulse of the NOESY experiment rotates longitudinal magnetization into the transverse plane much the same as in a 1D experiment; hence, this read pulse can be replaced with a sequence used for 1D selective excitation. The simplest method of acquiring NOESY spectra in H2O solution without presaturation employs a jump-and-return observe sequence in place of the final 908 pulse (Section 3.5.2) (92). This experiment is usually referred to as a jump–return NOESY (JR.NOESY), and a pulse sequence is depicted in Fig. 6.54. Note that a Hahn echo cannot be incorporated into this experiment. Experimental protocol for the JR.NOESY is similar to a NOESY experiment acquired with presaturation. A variety of more complicated schemes can also be incorporated into the NOESY experiment with some advantages over the jump–return version (93, 94). The F1 phase discrimination in the experiment depicted in Fig. 6.54 will result in longitudinal water magnetization during m for some FIDs and in transverse water magnetization during m for other FIDs, depending on the relative phase of 1 and 2 (this is true for both the TPPI and hypercomplex methods; Section 4.3.4). As a consequence, the water resonance will be suppressed to different degrees for the different FIDs, unless m is sufficiently long to allow radiation damping of the water resonance back to the z-axis (relaxation is far too slow to be useful in this regard). The difference between the state of the water magnetization for different FIDs can be reduced somewhat by shifting
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
512 f1
f2 t1
f4
f3 tm
∆
t2
p +1 0 –1
FIGURE 6.54 Pulse sequence and coherence level diagram for the NOESY experiment with a jump–return observe pulse (JR.NOESY). The basic eightstep phase cycle is as follows: 1 ¼ x, x, y, y, x, x, y, y; 2 ¼ x, x, y, y, x, x, y, y; 3 ¼ 8(x); 4 ¼ 8(x); and receiver ¼ 4(x, x). Cyclops is performed on all pulses and the receiver to yield a 32-step cycle. Adjusting the rotation angle and phase of pulse 4 a few degrees can improve the degree of solvent suppression. Frequency discrimination in F1 is obtained by shifting the phase of 1 and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4).
the phase 1 by 458. If large values of m are used to allow efficient radiation damping, then JR.NOESY spectra may include contributions from spin diffusion. The JR.NOESY spectrum depicted in Fig. 6.55 was acquired from H2O solution, with 32 transients for each of 576 t1 increments (t1max ¼ 50 ms). A mixing time of 150 ms was used, and the total acquisition time was 11 hr. Sections of the JR.NOESY and NOESY with presaturation are depicted in Fig. 6.55. Many cross-peaks close to the water resonance are observed only in the JR.NOESY spectrum and other peaks up to 0.2 ppm. from the water line are noticeably more intense in the jump–return experiment (even allowing for the longer mixing time of this experiment). In Fig. 6.55 (and elsewhere in this spectrum), a large number of peaks occur at the exact F1 frequency of the water resonance. These peaks can arise from a variety of mechanisms other than cross-relaxation with nonlabile protein protons coincident with the solvent resonance, including chemical exchange between water and labile 1HN or 1HO groups, NOEs from bound water to protein
513
5.0 presaturation
4.5
49
F1 (ppm)
4.5
6.6 CROSS-RELAXATION NMR EXPERIMENTS
65 66 48-49 17-18
75 74
73 15-16
15 22 b 23N 16
5.0
18
22-23 jump-return
8.7
8.1
F2 (ppm)
FIGURE 6.55 Comparison of NOESY spectra acquired from H2O solution in which solvent was suppressed by presaturation (top) or selective excitation with a jump–return sequence (bottom). The spectra were collected under identical conditions except for the mixing times, which were 100 and 150 ms in presaturation and jump–return spectra, respectively. Intraresidue and sequential NOEs are denoted by rectangles and ellipses, respectively, with the peaks arising between 1HN and 1H unless otherwise noted. The three peaks outlined by broken ellipses probably arise from exchange of amide protons with the solvent, as residues 73 to 75 are close to the C-terminus and are flexible.
protons, or NOEs from protein hydroxyl protons on resonance with the water resonance. Modified NOESY sequences using pulsed field gradients to help suppress the water have been described (95). As for the TOCSY
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
514
experiment, presaturation of the solvent resonance is avoided if the Hahn echo sequence element in Fig. 6.51 is replaced by a gradient watersuppression technique, such as the excitation-sculpting sequence element shown in Fig. 6.52. In contrast to the jump–return pulse sequence of Fig. 6.54, which returns the water magnetization to the þz-axis prior to detection, the excitation-sculpting approach dephases the water magnetization prior to detection. Thus, depending on the recycle delay and sample pH, some saturation transfer from water 1H spins to the amide 1 N H spins can occur, although at a reduced level compared to the NOESY experiment using presaturation (Fig. 6.51). As for the JR.NOESY, the difference between the state of the water magnetization during m for different FIDs can be reduced by shifting the phase 1 by 458. Gradient pairs applied during the t1 period and during m minimize radiation damping and reduce perturbations of signals close to resonance with the water signal (Section 6.3) (31). A section of the NOESY spectrum acquired using the pulse sequence of Fig. 6.52 is shown in Fig. 6.56. The region shown is identical to that shown in Fig. 6.55 and shows similar improvements as obtained using the jump– return experiment relative to the spectrum acquired using presaturation.
4.4
4.8
F1 (ppm)
4.6
5.0
8.8
8.6
8.4
8.2
8.0
F2 (ppm)
FIGURE 6.56 NOESY spectra acquired from H2O solution in which solvent was suppressed by the excitation-sculpting approach of Fig. 6.52. The spectra were collected under the same conditions as for Fig. 6.55 (bottom) and the same region of the spectrum is shown.
515
6.6 CROSS-RELAXATION NMR EXPERIMENTS f1
f2 t1
f3 Isotropic Mixing
tm
f4 ∆1
∆2
t2
p +1 0 –1
FIGURE 6.57 Pulse sequence and coherence level diagram for the relayed NOESY experiment. The phase cycle is identical to that used in Figs. 6.51 and 6.45.
In another variant of the basic NOESY experiment, the mixing period includes both isotropic mixing and cross-relaxation periods (Fig. 6.57), and is known as a relayed NOESY. Experimental setup is identical to that for a normal NOESY experiment. The aim is to use the isotropic mixing to transfer magnetization via a single scalar coupling, usually between 1HN and 1H, prior to NOESY cross-relaxation. The isotropic mixing period normally is 20–30 ms in duration (Section 6.5.2). The choice of NOESY mixing time is based on a trade-off between crosspeak intensity and contributions from spin diffusion (Section 6.6.1.2). The experiment discussed in the following analysis was acquired with 32 transients for each of 576 t1 increments (total acquisition time 10.5 hr); 27 ms of DIPSI-2rc isotropic mixing (70) was used prior to 100 ms of cross-relaxation. Analysis of a relayed NOESY experiment may help in the sequential assignment process in cases in which the 1H protons of adjacent residues are degenerate. The sequential assignment process relies on the observation of NOEs from 1HN of residue i to 1HN, 1H, and 1H of residue i 1. For peptide sections in an extended conformation, the sequential 1HN–1HN and 1HN–1H NOEs are of zero or low intensity; degeneracy of the 1H resonances of adjacent residues will stymie the assignment procedure (the important sequential NOE will be overlapped with the intraresidue 1HN–1H NOE of residue i 1). In the relayed NOESY experiment, 1HN and 1H resonance positions of residue i 1 are recorded in t1, then this magnetization is passed on to 1H of
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
516
residue i 1 during the isotropic mixing and then to 1HN of residue i during the NOE mixing period. Intense sequential 1H–1HN and 1 N 1 N H – H NOE cross-peaks result from this process, allowing the sequential assignment process to continue. In the relayed NOESY spectrum, cross-peaks no longer have a direct dependence on the distance between the protons at the F2 and F1 frequencies of the crosspeak, and such data should be used for assignment purposes only. An example is shown in Fig. 6.58, where many sequential 1HN–1HN NOEs
9.0
49 48 71 72 52
3
F1 (ppm)
51
4
72 73 56 55
3
8.0
2 71
69
70 68 NOESY 8.5
Relayed NOESY 8.0
8.5
8.0
F2 (ppm)
FIGURE 6.58 Comparison of sections of NOESY (left) and relayed NOESY (right) spectra. Both experiments were performed under identical conditions except for the mixing period, which included 27 ms of DIPSI-2rc isotropic mixing in the relayed NOESY. Weak coherent irradiation was used to suppress the solvent before the experiment and during the 100-ms NOE mixing period. Relayed NOESY peaks outlined by boxes indicate sequential 1HN–1HN NOEs between residues in the -sheet of ubiquitin that are weak or not observed in the conventional NOESY experiment. The greater intensity allows sequential assignments to be made even if sequential 1H resonances are degenerate (as is the case for His68 and Leu69). The labels denote residue numbers of the amide protons contributing to each cross-peak.
6.6 CROSS-RELAXATION NMR EXPERIMENTS
517
readily are observable in the relayed NOESY experiment but are of negligible intensity in the normal NOESY experiment.
6.6.2 ROESY Rotating-frame Overhauser Effect Spectroscopy (ROESY) was first developed by Bothner-By and co-workers and was initially known by the acronym CAMELSPIN (cross-relaxation appropriate for minimolecules emulated by locked spins) (96). As both names suggest, the experiment monitors cross-relaxation between spins that are spin-locked by the application of rf pulses (96, 97). ROESY has the advantage that the rotating-frame Overhauser effect (ROE) cross-relaxation rate constant is positive for all rotational correlation times: the maximum size of the ROE varies from 0.38 for !0 c 1 to 0.68 for !0 c 1. Therefore, ROESY cross-peaks are observable even if !0 c 1; in contrast, cross-peaks vanish in laboratory-frame NOESY experiments if !0 c 1. ROESY is very useful in studies of peptides in which laboratory frame NOEs are weak, but the experiment also has merits appropriate for the study of proteins. ROESY, NOESY (Section 6.6.1), and TOCSY (Section 6.5) are experimentally very similar; consequently, comparisons to NOESY and TOCSY will be made throughout this discussion. A more detailed discussion of relaxation in the rotating frame is given in Section 5.4.3. 6.6.2.1 Product Operator Analysis The original version of the ROESY experiment simply consisted of a 908–t1– m–t2, sequence in which the spin-locking field during m was provided by continuous lowpower irradiation (2–4 kHz), as illustrated in Fig. 6.59a. For a scalarcoupled two-spin system, the evolution up to the mixing period is given by 2
x
t1
I1z ! I1y cosð1 t1 Þ cosðJ12 t1 Þ þ 2I1x I2z cosð1 t1 Þ sinðJ12 t1 Þ þ I1x sinð1 t1 Þ cosðJ12 t1 Þ þ 2I1y I2z sinð1 t1 Þ sinðJ12 t1 Þ: ½6:53 During the subsequent spin-locking period (with a y-phase rf field), any operators orthogonal to the rf field in a tilted rotating frame are dephased by rf inhomogeneity (Section 3.4.3). The x-axes of the rotating and tilted reference frames are coincident; thus, all terms containing x-operators are dephased. The transformation of z- and y-operators into
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
518
the tilted frame is performed using [5.89]: I1y ) I01z sin1 þ I01y cos1 , 2I1y I2z ) 2 I01z sin1 þ I01y cos1 I02z cos2 I02y sin2 ¼ 2I01z I02z sin1 cos2 2I01z I02y sin1 sin2 þ 2I01y I02z cos1 cos2 2I01y I02y cos1 sin2 ,
½6:54
in which 1 and 2 are the tilt angles for spins I1 and I2. The only terms that commute with the spin lock Hamiltonian are proportional to I01z and 2I01z I02z . If K 1 spins (Ik for k ¼ 2, . . . , K) are close in space to spin I1, the resulting evolution of the longitudinal magnetization is I01z sin1 cosð1 t1 Þ cosðJ12 t1 Þ m
!
K X
I0kz a1k ðm Þ sin1 cosð1 t1 Þ cosðJ12 t1 Þ,
½6:55
k¼1
in which a1k( m) ¼ [exp(–R m)]1k is the (1, k)th element of the matrix exponential and R is the matrix of rotating-frame relaxation rate constants Rkk(i) and jk(i, j) (Section 5.4.3). Transforming back from the tilted frame to the rotating frame yields the observable operators: K X
Iky a1k ðm Þ sin1 sink cosð1 t1 Þ cosðJ12 t1 Þ:
½6:56
k¼1
The first term represents a diagonal peak and the remaining K 1 terms represent cross-peaks. Diagonal peaks and cross-peaks have in-phase absorptive lineshapes in F1 and F2. In the usual methods of acquisition, the cross-peaks are of phase opposite to the diagonal because j and jk are both positive (96). The two-spin term 2I 01z I 02z does not cross-relax with other I1 or I2 spin operators during m; however the amplitude of the operator is reduced by relaxation (with relaxation rate constant designated Rzz). Transformation back into the rotating frame yields the observable operators: 2I1y I2z sin1 cos2 þ 2I1z I2y cos1 sin2 sin1 cos2 sinð1 t1 Þ sinðJ12 t1 Þ expðRzz m Þ:
½6:57
The limitations of the simple ROESY experiment are now evident: (i) amplitude of cross-peaks are reduced by a factor of sin 1 sin 2 and
6.6 CROSS-RELAXATION NMR EXPERIMENTS
519
(ii) two-spin order generates cross-peaks with antiphase lineshapes in both dimensions that distort the in-phase multiplet patterns expected for ROESY cross-peaks. Griesinger and Ernst developed a simple and clever modification to the ROESY pulse sequence that overcomes these limitations. In this sequence (Fig. 6.59b), evolution through the 90x –t1 –90x block proceeds as described in [6.1]. 2
x
t1 2
x
I1z ! I1z cosð1 t1 Þ cosðJ12 t1 Þ 2I1x I2y cosð1 t1 Þ sinðJ12 t1 Þ þI1x sinð1 t1 Þ cosðJ12 t1 Þ 2I1z I2y sinð1 t1 Þ sinðJ12 t1 Þ: ½6:58 Again, y-operators are dephased by the spin lock rf field. Transformation of the z- and x-operators into the tilted frame yields I1z cosð1 t1 Þ cosðJ12 t1 Þ þ I1x sinð1 t1 Þ cosðJ12 t1 Þ ) I 01x sin 1 þ I 01z cos 1 cosð1 t1 Þ cosðJ12 t1 Þ þ I 01x cos 1 þ I 01z sin 1 sinð1 t1 Þ cosðJ12 t1 Þ:
½6:59
The only term that commutes with the spin lock Hamiltonian is I 01z ðcos 1 cosð1 t1 Þ sin 1 sinð1 t1 ÞÞ cosðJ12 t1 Þ ¼ I 01z cosð1 t1 þ 1 Þ cosðJ12 t1 Þ:
½6:60
Cross-relaxation during m yields I 01z cosð1 t1 þ 1 Þ cosðJ12 t1 Þ m
!
K X
I 0kz a1k ðm Þ cosð1 t1 þ 1 Þ cosðJ12 t1 Þ:
½6:61
k¼1
Transforming back from the tilted frame to the rotating frame and applying the last 908 pulse yields the observable operators: K X
Iky cos k Ikx sin k a1k ðm Þ cosð1 t1 þ 1 Þ cosðJ12 t1 Þ:
½6:62
k¼1
The offset dependence of the ROESY cross-peaks appears in [6.62] as a phase error of 1 in t1 and k in t2. Because k is approximately linear for 0 k B1 (Section 3.4.1), the resonance offset effects are compensated by phase correction during processing. No two-spin operators that commute with the spin lock Hamiltonian are created; therefore,
520
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
the cross-peak multiplet structure is undistorted (minor contributions from evolution of zero-quantum coherences in the tilted frame have been ignored). Although the Griesinger and Ernst approach eliminates the offset dependence that arises from the projection of the spin operators between tilted and untilted frames, the magnitudes of cross-relaxation rate constants in a ROESY experiment also depend upon resonance offset as shown by [5.140]. As a result, relaxation for off-resonance spins will contain a laboratory-frame component (i.e., an NOE) as well as a rotating-frame component. Interestingly (and somewhat counterintuitively) for large biomolecules, the apparent offset-dependent crossrelaxation rate between two spins is actually most efficient for cross-peaks along the antidiagonal and least efficient for cross-peaks close to the diagonal away from the center of the spectrum (98). Any quantitative analysis of ROESY cross-peak intensities must consider the offset dependence of the rate constants. A practical problem encountered in the ROESY experiment is that the spin lock pulse is capable of inducing coherence transfer (52). The TOCSY (or J) cross-peaks are of the same sign as the diagonal (Section 6.5); consequently, TOCSY transfer within a scalar-coupled system tends to cancel the cross-relaxation components and render quantitation of the ROE (and hence the interproton separation) difficult. More insidiously, cross-peaks that arise through consecutive TOCSY and ROE magnetization transfers have the same sign as do the actual ROE peaks (99) and can be misinterpreted. Fortunately, a long, weak pulse is not efficient at achieving a Hartmann–Hahn match between two protons unless they are close in chemical shift or symmetrically disposed about the carrier position (Fig. 6.47a). Unambiguous ROE cross-peaks can be identified by recording two ROESY spectra with very different rf carrier offsets (99). Development of pulse sequences that eliminate TOCSY transfer and generate pure ROE cross-peaks is an area of active research (100, 101). 6.6.2.2 Experimental Protocol and Processing In order to achieve baselines flat enough to allow accurate quantitation of cross-peak intensities, the points discussed for the TOCSY and NOESY spectra (Sections 6.5.2 and 6.6.1.2) are also pertinent in the ROESY experiment. In addition, parameters must be chosen for the spin lock mixing period. Sample heating and minimization of J transfer are jointly accommodated by using weak spin lock field strengths (2–5 kHz), although the offset dependence of the ROE may be nontrivial in such cases. The ROE builds up at a rate twice that of the laboratory-frame NOE (Section
521
6.6 CROSS-RELAXATION NMR EXPERIMENTS
5.4.3); therefore, shorter mixing times are required to obtain ROE peaks comparable in size to their NOESY counterparts. For proteins, mixing times are usually kept short (40–200 ms). The absorptive inphase lineshape expected for diagonal and cross-peaks in the ROESY spectrum indicates that processing will be very similar to the procedures already described for TOCSY and NOESY (Sections 6.5.3 and 6.6.1.3). ROESY spectra were recorded with the pulse sequences shown in a and b of Fig. 6.59. Thirty-two transients were collected for each of 512 t1 increments (t1max ¼ 44 ms). The spins were locked by a continuous
a f1 t2
t1 spin lock (q)
b
f2
f1
f2 t2
t1 spin lock (q)
p +1 0 –1
FIGURE 6.59 Pulse sequences and coherence level diagram for the ROESY experiments. (a) The basic phase cycle is 1 ¼ x, x and receiver ¼ x, x. The spin lock phase ¼ y. (b) The basic phase cycle is 1 ¼ x, x; 2 ¼ x, x; and receiver ¼ x, x. The spin lock phase ¼ x. The full phase cycle is completed by performing CYCLOPS on all pulses and the receiver. Frequency discrimination in F1 is obtained by shifting the phase of 1 and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4).
522
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
low-power pulse (2.5 kHz field strength) of 40-ms duration. Sections of the spectra are shown in a and b of Fig. 6.60. The increased cross-peak amplitude obtained with the pulse sequence of Fig. 6.59b is illustrated in Fig. 6.60c. 6.6.2.3 Information Content Although resonance offset effects hinder quantitation of ROESY spectra, the ROESY experiment has several redeeming qualities for studies of proteins. Foremost, as discussed previously, the ROE is always positive and cross-peaks can be observed in ROESY spectra even if the peaks cannot be observed in NOESY spectra because !0 c 1 (Section 5.4.3). A further advantage of ROESY over NOESY is that spin diffusion (or three-spin effects) produces contributions to cross-peaks that are of sign opposite to the sign of direct ROE peaks. Conceptually, the rotating-frame cross-relaxation rate constant is positive, and magnetization transfer between two spins occurs with inversion of sign. Thus, a diagonal peak and a cross-peak arising by a direct ROE between two spins have opposite signs. Transfer of the cross-peak magnetization to a third spin involves another change of sign. As a result, cross-peaks dominated by spin diffusion will be of the same sign as the diagonal has. If a small three-spin interaction contributes to a ROESY cross-peak, the measured intensity is reduced and may be interpreted as a longer interproton separation. Consequently, the upper bound restraint applied in structure calculations will not be overly restrictive. The influence of spin diffusion in NOESY spectra is particularly pronounced for NOEs involving geminal methylene groups. Efficient 00 0 spin diffusion between the 1H and 1H tends to equalize the intensity 00 0 of NOEs to other protons even if the distances to 1H and 1H are not equal. Stereospecific assignment of -methylene protons plays an important role in defining side chain conformation and depends heavily on estimating the 0 relative sizes of intraresidue and sequential 00 distances to 1H and 1H (102, 103). The use of ROESY spectra for this process significantly reduces the chance of incorrectly making such assignments. Another important facet of the ROESY experiment is that chemical exchange peaks are of the same sign as the diagonal, i.e., opposite in sign to peaks arising from direct cross-relaxation. Thus, rotating-frame experiments are invaluable in the study of dynamic processes involving slow exchange between two or more states. Protein–protein or peptide– protein interactions are one area where discrimination of crossrelaxation and chemical exchange is not possible from NOESY, but is apparent from ROESY data. As with chemical exchange in TOCSY
6.6
523
CRoss-RELAXATION NMR EXPERIMENTS
,a
A46a
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Il (j
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00'
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(@
Q
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E 0.
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M
u::
((;) 0
0
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b
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A
0
-.
c
9.0
8.5 F2 (ppm)
8.0
FIGURE 6.60 ROESY spectra of ubiquitin acquired from H 2 0 solution with a mixing time of 40 ms and a spin lock field of 2.5 kHz. (a) Section of the spectrum acquired with the sequence of Fig. 6.59a. (b) Section of the spectrum acquired with the sequence of Fig. 6.59b. (c) Traces through the cross-peaks for the IH'" of Ala46 for (...) the spectrum shown in panel a and (-) the spectrum shown in panel b.
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
524
spectra (73), complex situations can arise where peaks result from both cross-relaxation and exchange. 6.6.2.4 Experimental Variants Just as in the JR.NOESY, ROESY experiments frequently are acquired without presaturation of the solvent resonance. A variety of suppression techniques have been devised that accomplish this (the jump–return sequence cannot be used effectively following the spin lock pulse) (93), some of which have been discussed in Section 3.5.3. One possible pulse sequence is shown in Fig. 6.61. This sequence is derived from that shown in Fig. 6.59b by adding the excitation-sculpting segment for water suppression.
1H f1
f2
f4
f2 f3
t1
f4 f3
t2
spin lock (q) ∆1
∆1+∆2
∆2
Grad G1
G1 G2
G2 p +1 0 –1
FIGURE 6.61 Pulse sequence and coherence level diagram for a ROESY experiment acquired from H2O solution using excitation sculpting to avoid presaturation of the water resonance. The curved pulse shapes indicate selective 1808 pulses at the water frequency. The basic phase cycle is 1 ¼ 2(x, x); 2 ¼ 4(x); 3 ¼ y, y, y, y; 4 ¼ y, y, y, y; and receiver ¼ 2(x, x). The spin lock phase ¼ x. The full phase cycle is completed by performing CYCLOPS on all pulses and the receiver. Frequency discrimination in F1 is obtained by shifting the phase of 1 and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4). A bipolar pair of gradients can be inserted into the t1 period to prevent radiation damping (31).
6.7 1H 3D EXPERIMENTS
525
6.7 1H 3D Experiments Given the vast improvement in effective resolution between 1D and 2D NMR spectra, 3D NMR spectroscopy is a logical approach to increasing the effective resolution still further. The increase in dimensionality from 2D to 3D is achieved by inserting a second incrementable delay and mixing period immediately before the acquisition period of a normal 2D experiment, as was discussed in Section 4.5 and shown schematically in Fig. 4.17. The first example of a 3D NMR experiment useful in the study of proteins was reported in 1988 (104), and combined NOESY and TOCSY mixing with the measurement of proton frequencies in all three dimensions. The acquisition of 3D homonuclear NMR spectra introduces many technical challenges. The digital resolution in the indirect dimensions needs to be reasonably high because of the large number of protons and their relatively poor chemical shift dispersion. Simultaneously, the total acquisition time must be minimized because two evolution delays have to be incremented independently. As a consequence, such spectra are acquired with minimal digital resolution in t1 and t2 (usually less than 128 complex data points are collected) and with relatively few transients (never more than 16). The phase cycle must be chosen with great care so as to achieve the maximum amount of artifact suppression in the fewest steps. Even with optimization of conditions, homonuclear 3D experiments commonly require four to eight days to acquire. Improvements in digital resolution (or a reduction in the total acquisition time) may be obtained by the use of selective pulses to limit the frequencies observed in one or both of the indirectly detected dimensions (105–107). Aside from the technical issues, homonuclear 3D spectra are much more complicated than 2D spectra. For a protein the size of ubiquitin, 2000 to 3000 individual cross-peaks are observable in the 2D NOESY spectrum; in the corresponding homonuclear 3D NOESY–TOCSY spectrum, approximately five times as many cross-peaks might be observed because the magnetization transferred between spins by a particular NOE interaction (generating one cross-peak in the 2D spectrum) is transferred to all of the spins in the same spin system by the isotropic mixing. In contrast, heteronuclear 3D and 4D spectra have similar numbers of cross-peaks as 2D experiments (Chapter 7). Thus, although homonuclear 3D experiments were developed first, heteronuclear multidimensional spectroscopy experiments, because of the greater simplicity and superior resolving power, are preferable.
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
526 1H
f3
f1 f2 t1
tm
f5
f4 t2
y
y
–y
–y
∆1
∆1+∆2
t2
isotropic mixing
∆2
Grad G1
G2
G3 G4
G4 G5
G5
p +1 0 –1
FIGURE 6.62 Pulse program and coherence level diagram for a homonuclear 3D NOESY–TOCSY experiment. An excitation-sculpting segment is included prior to detection to avoid presaturation of the water resonance. The DIPSI mixing pulses are of y phase, and the phases of the pulses are cycled as follows: 1 ¼ 2(x, x, y, y); 2 ¼ 2(x, x, y, y); 3 ¼ x, x, y, y, x, x, y, y; 4 ¼ 2(x, x, y, y); 5 ¼ 2(x, x, y, y); 6 ¼ 8(y); 7 ¼ 8(y); and receiver ¼ x, x, y, y, x, x, y, y. Frequency discrimination in F1 (F2) is obtained by shifting the phase of 1 (3) and the receiver according to the TPPI, States, or TPPI–States protocols (Section 4.3.4).
Nevertheless, homonuclear 3D spectra can offer additional information in cases where isotopic labeling is not possible. The main implementation of homonuclear 3D spectroscopy in the study of proteins combines NOESY and TOCSY mixing because both mechanisms transfer in-phase magnetization and produce lineshapes that do not suffer from self-cancellation under conditions of limiting digital resolution. The pulse sequence for a 3D NOESY–TOCSY experiment is shown in Fig. 6.62.
6.7.1 EXPERIMENTAL PROTOCOL The precautions already described to obtain flat baselines in 2D TOCSY (Section 6.5.2) and NOESY (Section 6.6.1.2) spectra are also relevant for 3D spectroscopy, with the additional limitation that the phase cycle must be kept to a bare minimum. The eight-step phase cycle described in Fig. 6.62 should be considered a starting point for
6.7 1H 3D EXPERIMENTS
527
optimization of this experiment on a given spectrometer; the dominant artifacts will vary from one spectrometer to another and some optimization may be necessary. Assuming that the artifact level is acceptable from eight transients, then 256 increments can be collected for both t1 and t2 in just over a week of spectrometer time.
6.7.2 PROCESSING Not only is the acquisition of a homonuclear 3D experiment time consuming, but processing also provides challenges. In general, processing strategies discussed for TOCSY (Section 6.5.3) and NOESY (Section 6.6.1.3) experiments are applied to 3D spectra as well. However, even modest zero-filling can lead to excessively large data matrices, and using linear prediction or maximum entropy reconstruction to improve the resolution in F1 and F2 is a considerable computational burden. During the initial stages of resonance assignment, the peaks of most interest involve amide protons in F3; therefore, processing can be simplified by saving only this region after the t3 Fourier transformation.
6.7.3 INFORMATION CONTENT Homonuclear 3D spectra are analyzed with the aim of obtaining resonance assignments or to obtain interproton distance restraints from the unambiguous assignment of NOE peaks. However, the spectra contain a vast number of redundant cross-peaks, and complete analysis (as might be performed for a 2D spectrum) is not an attractive proposition unless the process is assisted by computer automation. Alternatively, certain regions of the 3D spectrum may be analyzed in detail to resolve specific ambiguities arising from the 2D spectra. The most intense peaks lie along the body-diagonal at F1 ¼ F2 ¼ F3 and arise from magnetization that has not been transferred during either mixing period. The next most intense peaks arise from transfer by either NOE or TOCSY, but not both. The former occur in the F2 ¼ F3, or NOE, plane, while the latter occur in the F1 ¼ F2, or TOCSY, plane. The weakest cross-peaks arise from NOE and TOCSY transfer and are sometimes referred to as real or true 3D cross-peaks because they have no equivalent in 2D spectra. Further, the 3D cross-peaks in the F1 ¼ F3 plane are referred to as back-transfer peaks, as they arise from transfer from one proton to a second during the NOE mixing, and then back to the first proton during the isotropic mixing. Analysis of the 3D spectrum is most readily accomplished from 2D plots corresponding to F1–F2 or F1–F3 planes; the intersection of these planes with the NOE,
528
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
TOCSY, and back-transfer planes gives rise to the NOE, TOCSY, and back-transfer lines, respectively, which play an important role in analysis of the spectrum (108, 109). Information additional to that present in 2D spectra is contained in the peaks with F1 6¼ F2 6¼ F3. Note that the intensity of the peaks will depend on the efficiency of transfer via both dipolar relaxation and isotropic mixing. Because cross-peak intensity is not related directly to the NOE between the spins, calibration of interproton distances cannot be performed in a precise manner. The dependence of cross-peak intensity on the rate of TOCSY transfer can be especially problematic for the amide protons of helical residues; the small value of 3 JHN H will reduce the size of all correlations, even if the interproton separation is relatively short. Finally, although the 3D spectra may be able to resolve correlations that are overlapped in 2D spectra, correlations between two protons with identical chemical shifts still cannot be observed. Either four proton dimensions or a single proton and two heteronuclear dimensions (e.g., the HMQC–NOESY–HMQC; Section 7.2.4) are required to observe such cross-peaks.
6.7.4 EXPERIMENTAL VARIANTS Several variants of homonuclear 3D experiments have been described that differ from the example described here in the choice of mixing schemes. Thus, the 3D TOCSY–NOESY has been proposed as a complementary method of performing the 3D NOESY–TOCSY experiment. Because both experiments have one TOCSY and one NOESY mixing period, the information content is identical, although cross-peaks containing equivalent information will be present in different regions of the spectra. Thus, if a particular cross-peak is obscured by some artifact in one type of spectrum, equivalent information may be present in an artifact-free region of the other experiment. In addition, a jump–return read pulse (Section 3.5.2) can be incorporated into the TOCSY–NOESY experiment, improving solvent suppression and allowing the observation of peaks close to the water resonance (110, 111). Homonuclear 3D spectra have also been used to study spin diffusion pathways by utilizing NOESY transfer during both mixing periods in the 3D NOESY–NOESY experiment (112). In such a spectrum, the intensity of a true 3D cross-peak between spins I1, I2, and I3 at F1 ¼ 1, F2 ¼ 2 and F3 ¼ 3 depends upon the product of the cross-relaxation rates, 12 and 23, and thus has intensity comparable to the spin diffusion
6.7 1H 3D EXPERIMENTS
529
contribution to a peak between I1 and I3 in a 2D NOESY spectrum [6.52]. This is one of the few direct experimental methods to investigate spin diffusion pathways. The 3D NOESY–NOESY spectra have been used for complete resonance assignments and identification of the secondary structure present in a protein (113, 114). Finally, 3D TOCSY– TOCSY spectra have been used as an aid to automated resonance assignment routines (115). This experiment has the advantage that spectra of high quality can be obtained with very little phase cycling, and total acquisition times can be kept short.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29.
D. S. Wishart, F. M. Richards, FEBS Lett. 293, 72–80 (1991). D. S. Wishart, B. D. Sykes, F. M. Richards, J. Mol. Biol. 222, 311–333 (1991). J. Jeener, Ampe`re Summer School, Basko Polje, Yugoslavia, 1971. W. P. Aue, E. Bartholdi, R. R. Ernst, J. Chem. Phys. 64, 2229–2246 (1976). D. Marion, K. Wu¨thrich, Biochem. Biophys. Res. Commun. 113, 967–974 (1983). D. Neuhaus, G. Wagner, M. Vasak, J. H. R. Ka¨gi, K. Wu¨thrich, Eur. J. Biochem. 151, 257–273 (1985). H. Widmer, K. Wu¨thrich, J. Magn. Reson. 74, 316–336 (1987). Y. Kim, J. H. Prestegard, J. Magn. Reson. 84, 9–13 (1990). L. J. Smith, M. J. Sutcliffe, C. Redfield, C. M. Dobson, Biochemistry 30, 986–996 (1991). S. Ludvigsen, K. V. Andersen, F. M. Poulsen, J. Mol. Biol. 217, 731–736 (1991). J. J. Titman, J. Keeler, J. Magn. Reson. 89, 640–646 (1990). L. McIntyre, R. Freeman, J. Magn. Reson. 96, 425–431 (1992). T. Szperski, P. Gu¨ntert, G. Otting, K. Wu¨thrich, J. Magn. Reson. 99, 552–560 (1992). C. Griesinger, O. W. Sørensen, R. R. Ernst, J. Magn. Reson. 75, 747–492 (1987). D. Marion, B. Bax, J. Magn. Reson. 80, 528–533 (1988). L. Mueller, J. Magn. Reson. 72, 191–196 (1987). G. Otting, K. Wu¨thrich, J. Magn. Reson. 75, 546–549 (1987). S. C. Brown, P. L. Weber, L. Mueller, J. Magn. Reson. 77, 166–169 (1988). G. Eich, G. Bodenhausen, R. R. Ernst, J. Am. Chem. Soc. 104, 3731–3732 (1982). A. Bax, G. Drobny, J. Magn. Reson. 61, 306–320 (1985). W. J. Chazin, K. Wu¨thrich, J. Magn. Reson. 72, 358–363 (1987). G. Wagner, J. Magn. Reson. 55, 151–156 (1983). W. J. Chazin, M. Rance, P. E. Wright, J. Mol. Biol. 202, 603–622 (1988). M. Rance, O. W. Sørensen, G. Bodenhausen, G. Wagner, R. R. Ernst, K. Wu¨thrich, Biochem. Biophys. Res. Commun. 117, 479–485 (1983). U. Piantini, O. W. Sørensen, R. R. Ernst, J. Am. Chem. Soc. 104, 6800–6801 (1982). A. J. Shaka, R. Freeman, J. Magn. Reson. 51, 169–173 (1983). N. Mu¨ller, R. R. Ernst, K. Wu¨thrich, J. Am. Chem. Soc. 108, 6482–6492 (1986). P. C. M. van Zijl, M. O‘Neil Johnson, S. Mori, R. E. Hurd, J. Magn. Reson., Ser A 113, 265–270 (1995). A. L. Davis, E. D. Laue, J. Keeler, D. Moskau, J. J. Lohman, J. Magn. Reson. 94, 637–644 (1991).
530
CHAPTER 6 EXPERIMENTAL 1H NMR METHODS
30. D. L. Mattiello, W. S. Warren, L. Mueller, B. T. Farmer, J. Am. Chem. Soc. 118, 3253–3261 (1996). 31. V. Sklena´rˇ , J. Magn. Reson., Ser A 114, 132–135 (1995). 32. A. E. Derome, M. P. Williamson, J. Magn. Reson. 88, 177–185 (1990). 33. C. J. Turner, J. Magn. Reson. 96, 551–562 (1992). 34. M. Rance, C. Dalvit, P. E. Wright, Biochem. Biophys. Res. Commun. 131, 1094–1102 (1985). 35. N. Mu¨ller, G. Bodenhausen, K. Wu¨thrich, R. R. Ernst, J. Magn. Reson. 65, 531–534 (1985). 36. M. Rance, P. E. Wright, Chem. Phys. Lett. 124, 572–575 (1986). 37. M. Rance, W. J. Chazin, C. Dalvit, P. E. Wright, Meth. Enzymol. 176, 114–134 (1989). 38. C. Griesinger, O. W. Sørensen, R. R. Ernst, J. Magn. Reson. 75, 474–492 (1987). 39. Z. L. Madi, C. Griesinger, R. R. Ernst, J. Am. Chem. Soc. 112, 2908–2914 (1990). 40. C. Griesinger, O. W. Sørensen, R. R. Ernst, J. Chem. Phys. 85, 6837–6852 (1986). 41. L. Braunschweiler, G. Bodenhausen, R. R. Ernst, Mol. Phys. 48, 535–560 (1983). 42. C. Dalvit, M. Rance, P. E. Wright, J. Magn. Reson. 69, 356–361 (1986). 43. M. Levitt, R. Freeman, J. Magn. Reson. 33, 473–476 (1979). 44. G. Otting, K. Wu¨thrich, J. Magn. Reson. 66, 359–363 (1986). 45. G. Wagner, E. R. P. Zuiderweg, Biochem. Biophys. Res. Commun. 113, 854–860 (1983). 46. D. L. Di Stefano, A. J. Wand, Biochemistry 26, 7272–7281 (1987). 47. P. L. Weber, S. C. Brown, L. Mueller, Biochemistry 26, 7282–7290 (1987). 48. M. Rance, J. Am. Chem. Soc. 110, 1973–1974 (1988). 49. L. Mueller, P. Legault, A. Pardi, J. Am. Chem. Soc. 117, 11043–11048 (1995). 50. V. V. Krishnan and M. Rance, J. Magn. Reson., Ser A 116, 97–106 (1995). 51. W. J. Chazin, J. Magn. Reson. 91, 517–526 (1991). 52. L. Braunschweiler, R. R. Ernst, J. Magn. Reson. 53, 521–528 (1983). 53. A. Bax, D. G. Davis, J. Magn. Reson. 65, 355–360 (1985). 54. M. Rance, J. Magn. Reson. 74, 557–564 (1987). 55. M. Rance, Chem. Phys. Lett. 154, 242–247 (1989). 56. J. Cavanagh, W. J. Chazin, M. Rance, J. Magn. Reson. 87, 110–131 (1990). 57. R. Bazzo, I. D. Campbell, J. Magn. Reson. 76, 358–361 (1988). 58. S. J. Glaser, G. P. Drobny, Adv. Magn. Reson. 14, 35–58 (1990). 59. A. J. Shaka, J. Keeler, T. Frenkiel, R. Freeman, J. Magn. Reson. 52, 335–338 (1983). 60. A. J. Shaka, J. Keeler, R. Freeman, J. Magn. Reson. 53, 313–340 (1983). 61. A. J. Shaka, C. J. Lee, A. Pines, J. Magn. Reson. 77, 274–293 (1988). 62. S. P. Rucker, A. J. Shaka, Mol. Phys. 68, 509–517 (1989). 63. M. Kadkhodael, O. Rivas, M. Tan, A. Mohebbi, A. J. Shaka, J. Magn. Reson. 91, 437–443 (1991). Kupcˇe, P. Schmidt, M. Rance, G. Wagner, J. Magn. Reson. 135, 361–367 (1998). 64. E. 65. A. E. Bennett, J. D. Gross, G. Wagner, J. Magn. Reson. 165, 59–79 (2003). 66. W. Peti, C. Griesinger, W. Bermel, J. Biomol. NMR 18, 195–205 (2000). Kupcˇe, W. Hiller, Magn. Reson. Chem. 39, 231–235 (2001). 67. E. 68. C. Griesinger, R. R. Ernst, Chem. Phys. Lett. 152, 239–247 (1988). 69. C. Griesinger, G. Otting, K. Wu¨thrich, R. R. Ernst, J. Am. Chem. Soc. 110, 7870–7872 (1988). 70. J. Cavanagh, M. Rance, J. Magn. Reson. 96, 670–678 (1992). 71. J. Briand, R. R. Ernst, Chem. Phys. Lett. 185, 276–285 (1991). 72. M. Kadkhodael, T.-L. Hwang, J. Tang, A. J. Shaka, J. Magn. Reson., Ser. A 105, 104–107 (1993).
6.7 1H 3D EXPERIMENTS
531
73. J. Feeney, C. J. Bauer, T. A. Frenkiel, B. Birdsall, M. D. Carr, G. C. K. Roberts, J. R. P. Arnold, J. Magn. Reson. 91, 607–613 (1991). 74. M. A. Delsuc, J. Y. Lallemand, J. Magn. Reson. 69, 504–507 (1986). 75. D. Marion, M. Ikura, A. Bax, J. Magn. Reson. 84, 425–430 (1989). 76. G. Wagner, Quart. Rev. Biophys. 16, 1–57 (1983). 77. D. Marion, M. Ikura, R. Tschudin, A. Bax, J. Magn. Reson. 85, 393–399 (1989). 78. J. Cavanagh, M. Rance, J. Magn. Reson. 88, 72–85 (1990). 79. F. Fogolari, G. Esposito, P. Viglino, J. Magn. Reson., Ser. A 102, 49–57 (1993). 80. G. M. Clore, A. Bax, A. M. Gronenborn, J. Biomol. NMR 1, 13–22 (1991). 81. S. Macura, Y. Huang, D. Suter, R. R. Ernst, J. Magn. Reson 43, 259–281 (1981). 82. G. Otting, J. Magn. Reson. 86, 496–508 (1990). 83. M. Rance, G. Bodenhausen, G. Wagner, K. Wu¨thrich, R. R. Ernst, J. Magn. Reson. 62, 497–510 (1985). 84. K. E. Cano, M. J. Thrippleton, J. Keeler, A. J. Shaka, J. Magn. Reson. 167, 291–297 (2004). 85. M. J. Thrippleton, J. Keeler, Angew. Chem. Int. Ed. 42, 3938–3941 (2003). 86. H. Wang, G. D. Glick, E. R. P. Zuiderweg, J. Magn. Reson., Ser. A 102, 116–121 (1993). 87. G. Bodenhausen, H. Kogler, R. R. Ernst, J. Magn. Reson. 58, 370–388 (1984). 88. W. Denk, R. Baumann, G. Wagner, J. Magn. Reson. 67, 386–390 (1986). 89. G. H. Weiss, J. E. Kiefer, J. A. Ferretti, J. Magn. Reson. 97, 227–234 (1992). 90. V. Stoven, A. Mikou, D. Piveteau, E. Guittet, J.-Y. Lallemand, J. Magn. Reson. 82, 163–168 (1989). 91. J. Fejzo, Z. Zolnai, S. Macura, J. L. Markley, J. Magn. Reson. 88, 93–110 (1990). 92. P. Plateau, M. Gue´ron, J. Am. Chem. Soc. 104, 7310–7311 (1982). 93. G. Otting, K. Wu¨thrich, J. Am. Chem. Soc. 111, 1871–1875 (1989). 94. V. Sklena´rˇ , A. Bax, J. Mag. Reson. 75, 378–383 (1987). 95. J. Stonehouse, G. L. Shaw, J. Keeler, J. Biomol. NMR 4, 799–805 (1994). 96. A. A. Bothner-By, R. L. Stephens, J.-M. Lee, C. D. Warren, R. W. Jeanloz, J. Am. Chem. Soc. 106, 811–813 (1984). 97. A. Bax, D. G. Davis, J. Magn. Reson. 63, 207–213 (1985). 98. C. Griesinger, R. R. Ernst, J. Magn. Reson. 75, 261–271 (1987). 99. D. Neuhaus, J. Keeler, J. Magn. Reson. 68, 568–574 (1986). 100. T.-L. Hwang, A. J. Shaka, J. Magn. Reson. 135, 280–287 (1998). 101. J. Schleucher, J. Quant, S. J. Glaser, C. Griesinger, in ‘‘Encyclopedia of Nuclear Magnetic Resonance’’ (D. M. Grant, R. K. Harris, eds.), vol. 8, pp. 4789–4804. John Wiley & Sons, Ltd., Chichester, 1996. 102. G. Wagner, W. Braun, T. F. Havel, T. Schaumann, N. Go, K. Wu¨thrich, J. Mol. Biol. 196, 611–639 (1987). 103. P. Gu¨ntert, W. Braun, M. Billeter, K. Wu¨thrich, J. Am. Chem. Soc. 111, 3997–4004 (1989). 104. H. Oschkinat, C. Grieinger, P. J. Kraulis, O. W. Sørensen, R. R. Ernst, A. M. Gronenborn, G. M. Clore, Nature 332, 374–376 (1988). 105. C. Griesinger, O. W. Sørensen, R. R. Ernst, J. Magn. Reson. 73, 574–579 (1987). 106. C. Griesinger, O. W. Sørensen, R. R. Ernst, J. Am. Chem. Soc. 109, 7227–7228 (1987). 107. H. Oschkinat, C. Ciesler, T. A. Holak, G. M. Clore, A. M. Gronenborn, J. Magn. Reson. 83, 450–472 (1989). 108. G. W. Vuister, R. Boelens, R. Kaptein, J. Magn. Reson. 80, 176–185 (1988). 109. G. W. Vuister, R. Boelens, A. Padilla, G. J. Kleywegt, R. Kaptein, Biochemistry 29, 1829–1839 (1990).
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110. H. Oschkinat, C. Cieslar, A. M. Gronenborn, G. M. Clore, J. Magn. Reson. 81, 212–216 (1989). 111. J. P. Simorre, D. Marion, J. Magn. Reson. 94, 426–432 (1991). 112. R. Boelens, G. W. Vuister, T. M. G. Koning, R. Kaptein, J. Am. Chem. Soc. 111, 8525–8526 (1989). 113. G. W. Vuister, R. Boelens, A. Padilla, R. Kaptein, J. Biomol. NMR 1, 421–438 (1991). 114. R. Bernstein, A. Ross, C. Cieslar, T. A. Holak, J. Magn. Reson., Ser. B 101, 185–188 (1993). 115. C. Cieslar, T. A. Holak, H. Oschkinat, J. Magn. Reson. 89, 184–190 (1990).
CHAPTER
7 HETERONUCLEAR NMR EXPERIMENTS
The 2D and 3D 1H NMR methods discussed in Chapter 6 are ineffective for proteins with molecular masses greater than approximately 10–12 kDa. The number of hydrogen atoms present in proteins scales approximately linearly with molecular mass. The rotational correlation times of globular proteins, and thus the linewidths of the NMR resonances, also increase linearly with molecular mass. The increased number and linewidth of the resonances in homonuclear 1H NMR spectra result in extensive chemical shift overlap and degeneracy. Conventional assignment procedures, based upon observation of sequential NOE correlations (Chapter 10), become difficult or impossible. In addition, larger linewidths (larger spin–spin relaxation rate constants) result in decreased sensitivity for 1H correlation experiments that rely on small (510Hz) homonuclear 3J scalar couplings for coherence transfer (e.g., COSY, multiple-quantum, and TOCSY experiments). Heteronuclear NMR spectroscopy (1–3) effectively circumvents these problems for proteins of at least up to molecular masses of 25–30 kDa, provided that the proteins can be uniformly labeled with the NMR active isotopes 13C and 15N (4); as discussed in Chapter 9, even larger proteins are accessible by combining 15N and 13C labeling with fractional or complete deuteration, in which nonexchangeable 1H atoms
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are replaced with 2H atoms (5, 6). Spectral resolution is improved by increasing the dimensionality of the NMR spectrum so that the highly overlapped 1H resonances, present in 1H 2D spectra, are separated in 3D and 4D spectra according to the more highly resolved heteronuclear resonances. At the same time, the efficiency of coherence transfer is increased by utilizing relatively large one-bond and two-bond (1J and 2 J) scalar coupling interactions between pairs of heteronuclei and between heteronuclei and their directly attached hydrogen nuclei, rather than the relatively small 1H homonuclear three-bond scalar coupling interactions. The general principles of heteronuclear NMR spectroscopy are discussed in this Chapter. A selection of experiments is described, focusing on those that are necessary for resonance assignment and structure determination of isotopically labeled proteins. The selection of experiments presented in this chapter is made for pedagogical purposes and is by no means complete; indeed, a given resonance correlation may be obtained using several different methods, and development of new experimental techniques continues apace. Furthermore, except for certain illustrative cases, the pulse sequences presented do not include details concerning pulsed field gradient techniques for artifact suppression and coherence-selection or water-suppression techniques. Pulsed field gradients and water-suppression can be incorporated in multiple ways using the methods described in Chapters 3 and 4. A selection of pulse sequences containing such elaborations is presented in Chapter 9. A comprehensive survey of modern techniques has been published (7). All multidimensional heteronuclear NMR experiments correlate heteronuclear resonances, typically 13C or 15N in proteins, with 1H resonances by transfer of coherence (or polarization) between the heteronuclear (S) and 1H (I) spins. Regardless of the specific protocol utilized to effect coherence transfer, the NMR experiment can start with excitation of either I or S spin polarization and must end with detection of either I or S spin magnetization. Ignoring the relative efficiency of different coherence or polarization transfer schemes, the overall sensitivity of heteronuclear correlation NMR experiments is proportional to (see [3.69] and [3.188]) S=N / ex 3=2 det ½1 expðR1ex TÞ,
½7:1
in which ex and det are the magnetogyric ratios of each nucleus excited at the beginning of the sequence and detected at the end of the sequence, respectively, T is the recycle time of the experiment, and R1ex is the spin–lattice relaxation rate constant of the excited nucleus (assuming
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
535
multiple transients are accumulated) (8). Therefore, indirect, or proton, detection is used whenever possible in order to maximize sensitivity. In these techniques, 1H spin polarization initially is transferred to the heteronucleus, the desired heteronuclear spin manipulations are performed, and the heteronuclear coherence finally is transferred back to 1H magnetization for detection. The gain in sensitivity compared to a correlation experiment in which 1H magnetization is transferred to the S nucleus for detection is thus nð I = S Þ3=2 , in which I and S are the magnetogyric ratios of the I and S nuclei, respectively, and n is the number of hydrogen atoms attached to the S nucleus. For 1H–13C correlations, the gain in sensitivity is approximately 24 for methyl 1H spins, 16 for methylene 1H spins, and 8 for methine 1H spins, while for 1 H–15N correlations of backbone amides the gain is approximately 30. The sensitivity gain relative to an experiment that starts with S-nucleus polarization and detects 1H magnetization is simply the ratio of the magnetogyric ratios, I/ S; for 1H–13C correlations this ratio is about 4, and for 1H–15N correlations it is about 10. The larger spin–lattice relaxation rate constants of hydrogen nuclei compared to heteronuclei (R1I 4 R1S) give an additional sensitivity advantage to the experiments starting with 1H magnetization (i.e., the schemes involving I ! S and I ! S ! I transfers) because of the [1 – exp(–R1exT)] factor in [7.1]. In practice, several factors can reduce the empirical gain in sensitivity, including splitting of the resonance signal into multiplets due to homonuclear scalar couplings, relaxation during coherence transfer steps, incomplete coherence transfer in multispin systems (e.g., I2S and I3S systems), and resonance offset effects arising from the increased heteronuclear, relative to 1H, chemical shift range. Direct 13C-detected experiments also have been introduced for application to biomacromolecules (9).
7.1 Heteronuclear Correlation NMR Spectroscopy Two-dimensional proton-detected heteronuclear correlation experiments most commonly use HMQC, HSQC, or TROSY coherence transfer mechanisms. The techniques are distinguished by whether the transferred coherence evolves during an indirect evolution period as heteronuclear multiple-quantum coherence (HMQC), heteronuclear single-quantum coherence (HSQC), or heteronuclear coherence, usually but not always spin state selective, chosen for its favorable relaxation properties (TROSY) (10–14). As will be seen subsequently, the two-dimensional heteronuclear correlation experiments are integral
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components of all heteronuclear three- and four-dimensional NMR experiments.
7.1.1 BASIC HMQC
AND
HSQC EXPERIMENTS
Pulse sequences for the HMQC and HSQC heteronuclear correlation experiments are illustrated in Fig. 7.1. A heteronuclear IS spin system (I ¼ 1H, S ¼ 15N or 13C), in which the I and S spins are directly covalently bonded and the 1H spin I is scalar coupled to a remote 1H spin, K, forms the basis for the following discussion. The homonuclear 1 H scalar coupling constant (JIK) is assumed to be much smaller than JIS. 7.1.1.1 The HMQC Experiment Evolution during the HMQC scheme of Fig. 7.1a is described using the product operator formalism as follows: 2 ðIx þKx Þ2 2 Sx
Iz ! 2Ix Sy t1 =2ðIx þKx Þt1 =2
! 2Ix Sy cosðS t1 Þ cosðJIK t1 Þ 4Iy Kz Sy cosðS t1 Þ sinðJIK t1 Þ 2 Sx 2
! Iy cosðS t1 Þ cosðJIK t1 Þ þ 2Ix Kz cosðS t1 Þ sinðJIK t1 Þ, ½7:2 in which the delay 2 is set to 1/(2JIS) (approximately 5.4 ms for one-bond 1H–15N JNH ¼ 92 Hz, and 3.6 ms for one-bond 1H–13C JCH ¼ 140 Hz). If desired, all delays of nominal length 1/(2JIS) can be shortened slightly to account for relaxation (Section 5.1.1). Only operators leading to observable terms have been propagated through the pulse sequence. Evolution of scalar coupling interactions other than JIS during the periods 2 has been ignored. Chemical shift evolution of the I spin during the 2 periods and during t1 is refocused by the 1808(I) pulse. Heteronuclear multiple-quantum (MQ) coherence, represented by the 2IxSy operator, does not evolve under the influence of the active scalar coupling, JIS, during the t1 period (Section 2.7.5). Evolution under the homonuclear JIK scalar coupling Hamiltonian is not refocused, because both 1H spins I and K experience the effect of the nonselective 1808(I) pulse. The resulting correlation spectrum exhibits homonuclear J coupling multiplet structure in the F1 dimension. In addition, the F2 lineshapes consist of the superposition of in-phase absorptive and
537
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY f3
a
t2 I f1 S
f2 t1
t1
2
2
2t
b
2t
decouple
f3
y
t2 I f1
S
t
f1 t
c
f2 t1
t1
2
2
t
t
decouple
f4
y
t2 I f1
S
t
f3
f1 t
f2
t1
T
T
2
2
2
–
t1 2
t
t
decouple
T
FIGURE 7.1 Pulse sequences for basic (a) HMQC, (b) HSQC, and (c) constanttime HSQC heteronuclear correlation experiments. Thin bars represent 908 pulses and thick bars represent 1808 pulses. Pulses are applied with x-phase unless the phase is indicated above the bar. The nominal value for 2 ¼ 1/(2JIS). Decoupling during t2 is achieved by using GARP-1, WALTZ-16, or other decoupling sequences. (a) Phase cycling for the HMQC experiment is 1 ¼ x, x; 2 ¼ 8(x), 8(x); 3 ¼ 2(x), 2(y), 2(x), 2(y); and receiver ¼ 2(x, x, x, x), 2(x, x, x, x). (b) Phase cycling for the HSQC experiment is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ 4(y), 4(y); and receiver ¼ x, x, x, x. (c) Phase cycling for the constant-time HSQC experiment is 1 ¼ x, x; 2 ¼ 8(x), 8(x); 3 ¼ 2(x), 2(y), 2(x), 2(y); 4 ¼ 16(y), 16(y); and receiver ¼ 2(x, x, x, x), 2(x, x, x, x). If desired, this 32-step phase cycle can be reduced to 8 steps by eliminating the cycling of 2 and using only the first 4 steps of the phase cycle of 3 (with appropriate changes to 4 and the receiver); an additional reduction by a factor of two can be obtained by eliminating cycling of 4. Frequency discrimination is obtained by TPPI, States, or TPPI–States phase cycling of 1.
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CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
antiphase dispersive components, represented by Iy and 2IxKz operators in the last line of [7.2], respectively. The antiphase dispersive component of the signal can be purged by inserting a 90y (I) pulse prior to the acquisition period. The purge pulse transforms the final operators in [7.2] to Iy cosðS t1 Þ cosðJIK t1 Þ þ 2Ix Kz cosðS t1 Þ sinðJIK t1 Þ 2 ðIy þKy Þ ! Iy cosðS t1 Þ cosðJIK t1 Þ 2Iz Kx cosðS t1 Þ sinðJIK t1 Þ:
½7:3 The antiphase I spin operator is transformed into an antiphase K spin operator. If the K spin resonance frequency occurs in an unimportant region of the spectrum (if it is not coupled to an S spin of interest), then the IS correlation spectrum has pure in-phase absorptive lineshapes in both frequency dimensions. Inclusion of homonuclear scalar coupling evolution during the periods 2 is facilitated by using [2.121] to simplify the propagator for the pulse sequence prior to product operator analysis. The propagator is U ¼ exp½i2H exp i2 Sx exp½iHt1 =2 exp½iðIx þ Kx Þ ½7:4 exp½iHt1 =2 exp i2 Sx exp½i2H exp i2 ðIx þ Kx Þ , in which the free-precession Hamiltonian for weak scalar coupling is given by H ¼ I Iz þ S Sz þ 2JIS Iz Sz þ 2JIK Iz Kz :
½7:5
Inserting E ¼ exp[i(Ix þ Kx)] exp[–i(Ix þ Kx)] and applying [2.121] yields U ¼ exp½i2H exp i2 Sx exp½iHt1 =2 exp½iðIx þ Kx Þ exp½iHt1 =2 exp½iðIx þ Kx Þ exp½iðIx þ Kx Þ exp i2 Sx exp½i2H exp i2ðIx þ Kx Þ ¼ exp½i2H exp i2 Sx exp½iHt1 =2 exp½iðI Iz þ S Sz 2JIS Iz Sz þ 2JIK Iz Kz Þt1 =2 exp i2Sx exp½iðI Iz þ S Sz 2JIS Iz Sz þ 2JIK Iz Kz Þ2 exp i2ðIx þ Kx Þ ¼ exp½i2JIK Iz Kz ðt1 þ 4 Þ exp½iðS Sz þ 2JIS Iz Sz Þ2 exp i2Sx exp½iS Sz t1 exp i2 Sx exp½iðS Sz 2JIS Iz Sz Þ2 exp i2 ðIx þ Kx Þ : ½7:6
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
539
Evolution through the pulse sequence is represented as 2 ðIx þKx ÞðS Sz 2JIS Iz Sz Þ2 2 Sx
Iz ! 2Ix Sy S Sz t1
2 Sx ðS Sz þ2JIS Iz Sz Þ2
! 2Ix Sy cosðS t1 Þ ! Iy cosðS t1 Þ 2JIK Iz Kz ðt1 þ4 Þ
! Iy cosðS t1 Þ cos½JIK ðt1 þ 4 Þ þ 2Ix Kz cosðS t1 Þ sin½JIK ðt1 þ 4 Þ:
½7:7
Evolution of the homonuclear scalar coupling during the 2 periods introduces a phase error in the F1 dimension of the 2D HMQC experiment. The magnitude of the phase error depends upon JIK and thus will vary nonlinearly throughout the spectrum. For a 10 Hz scalar coupling constant and 2 ¼ 1/(2JNH) ¼ 5.4 ms, the phase error is 19.48. The linewidth in the F1 dimension of an HMQC spectrum is determined by the relaxation rate constant of heteronuclear MQ coherence plus the contributions from inhomogeneous broadening (10, 13). In this and the following discussions of other heteronuclear correlation experiments, inhomogeneous broadening is ignored and relaxation rate constants are calculated using the methods outlined in Chapter 5. The model for relaxation assumes that (i) the S spins relax by dipole–dipole interactions with the directly attached I spins and by chemical shift anisotropy, (ii) the I spins relax by dipole–dipole interactions with the S spins, by dipole–dipole interactions with k additional remote 1H spins and by chemical shift anisotropy, (iii) in the limit of slow overall tumbling, which typically applies for proteins, !I c !S c 1, and J(0) J(!S) J(!I) J(!I !S), (iv) !I !K, and (v) rotational diffusion is isotropic, with J(!) given by [5.98]. Individual relaxation rate constants are obtained from Tables 5.5 and 5.8. in Chapter 5. In the resulting expressions, dIS reflects the heteronuclear dipolar coupling between the scalar-coupled I and S spins; dCSA(I) and dCSA(S) arise from chemical shift anisotropy of the I and S spin, respectively, and the terms containing dIk reflect the homonuclear dipolar coupling between 1H spins (Sections 5.4.1 and 5.4.4). In 13C- or 13C/15N-labeled samples, the S spin (either 13C or 15N) has additional dipolar interactions with nearby, predominantly directly bonded, 13C spins, designated as R spins. These interactions are smaller than the dipolar IS interaction by a factor of dRS =dIS ¼ 2R r6IS = 2I r6RS
½7:8
540
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
and are neglected in the present discussion. The effects of relaxation interference are ignored until the TROSY experiment is discussed in Section 7.1.3.5. The 1808(I) pulse during the HMQC scheme of Fig. 7.1a interconverts heteronuclear zero- and double-quantum coherences. Consequently, the relaxation rate constant for MQ coherence is the average of the relaxation rate constants for zero- and double-quantum coherences. Using the preceding model, the relaxation rate constant for MQ coherence is IS IK CSA ðIÞ þ RCSA ðSÞ R2MQ ¼ 12 RIS 2 ðZQÞ þ R2 ðDQÞ þ R2 ðIÞ þ R2 2 dIS ¼ Jð!I !S Þ þ 3Jð!I Þ þ 3Jð!S Þ þ 6Jð!I þ !S Þ 8 dCSAðSÞ dCSAðIÞ 4Jð0Þ þ 3Jð!I Þ þ 4Jð0Þ þ 3Jð!S Þ þ 6 6 1X þ dIk 5Jð0Þ þ 9Jð!I Þ þ 6Jð2!I Þ , ½7:9 8 k in which the summation, , includes all the homonuclear k 6¼ I spins. In the limit of slow overall tumbling, the relaxation rate constant is approximated by " # c 4 4 5X dCSAðIÞ þ dCSAðSÞ þ dIk : ½7:10 R2MQ ¼ 3 4 k 5 3 7.1.1.2 The HSQC Experiment In the HSQC experiment, illustrated in Fig. 7.1b, the INEPT sequence introduced in Section 2.7.7.2 is used to transfer I spin polarization (Iz) into antiphase heteronuclear single-quantum (SQ) coherence (2IzSy). The antiphase heteronuclear SQ coherence evolves during the subsequent t1 evolution period. A second INEPT sequence is used to transfer the frequency-labeled heteronuclear SQ coherence back to 1H magnetization for detection. For a heteronuclear IS spin system, evolution through the pulse sequence is described as follows: 2 ðIx þKx ÞðIx þKx Þ, Sx 2 ðIy þKy Þ, 2 Sx Iz ! 2Iz Sy t1 =2ðIx þKx Þt1 =2
! 2Iz Sy cosðS t1 Þ 2Iz Sx sinðS t1 Þ 2 ðIx þKx Þ, 2 Sx ðIx þKx Þ, Sx
! Ix cosðS t1 Þ 2Iy Sx sinðS t1 Þ,
½7:11
541
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
in which the delay 2 ¼ 1/(2JIS) and evolution of the homonuclear 1H scalar coupling interaction during the INEPT sequences has been ignored. The resultant term proportional to IySx is unobservable multiple-quantum coherence. The 1808(I) pulse in the middle of the evolution period refocuses evolution of the 1H heteronuclear JIS scalar coupling interaction. The 2IzSy operator present during t1 commutes with the homonuclear 1H scalar coupling Hamiltonian and the F1 lineshape does not contain contributions from 1H scalar coupling interactions. Evolution of the homonuclear scalar coupling interaction during the INEPT sequences is analyzed most easily by using [2.121] to simplify the propagator for the pulse sequence: U ¼ exp½iH exp½iðIx þ Kx Þ exp½iSx exp½iH exp i 2 ðIx þ Kx Þ exp i 2 Sx exp½iHt1 =2 exp½iðIx þ Kx Þ exp½iHt1 =2 exp i 2 Sx exp i 2 ðIy þ Ky Þ exp½iH exp½iðIx þ Kx Þ exp½iSx exp½iH exp i 2ðIx þ Kx Þ ¼ exp½ið2JIS Iz Sz þ 2JIK Iz Kz Þ2 exp i 2 ðIx þ Kx Þ exp i 2 Sx exp½iðS Sz þ 2JIK Iz Kz Þt1 exp i 2Sx exp i 2 Iy þ Ky exp½ið2JIS Iz Sz þ 2JIK Iz Kz Þ2 exp i 2 ðIx þ Kx Þ ½7:12 prior to evaluating evolution through the pulse sequence as
2 ðIx þKx Þð2JIS Iz Sz þ2JIK Iz Kz Þ2 2 ðIy þKy Þ, 2 Sx
Iz ! 2Iz Sy cosð2JIK Þ ðS Sz þ2JIK Iz Kz Þt1
! 2Iz Sy cosðS t1 Þ cosð2JIK Þ 2 ðIx þKx Þ, 2 Sx ð2JIS Iz Sz þ2JIK Iz Kz Þ2
! Ix cosðS t1 Þ cos2 ð2JIK tÞ Iy Kz cosðS t1 Þ sinð4JIK Þ, ½7:13 in which only observable terms have been included. Evolution of the 1H scalar coupling during the INEPT periods modulates the amplitude (but not the phase) of the in-phase absorptive resonance and introduces an antiphase dispersive contribution to the F2 lineshape. As in the case of the HMQC experiment, the antiphase dispersive component can be purged by applying a 90y (I) pulse prior to acquisition.
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CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
In Fig. 7.1b, the 908(I) pulse following the t1 period is phase shifted by 908 relative to the 908(I) pulse preceding the t1 period. If the two pulses have the same y-phase, then evolution through the pulse sequence becomes
2 ðIx þKx Þ–ð2JIS Iz Sz þ2JIK Iz Kz Þ2 2
ðIy þKy Þ, 2Sx Iz ! 2Iz Sy cosð2JIK Þ þ 4Iy Sy Kx sinð2JIK Þ ðS Sz þ2JIK Iz Kz Þt1
! 2Iz Sy cosðS t1 Þ cosð2JIK Þ þ 4Iy Sy Kx cosðS t1 Þ sinð2JIK Þ 2 Sx 2 ðIy þKy Þð2JIS Iz Sz þ2JIK Iz Kz Þ2 ! Iy cosðS t1 Þ
cosð4JIK Þ þ 2Ix
Kz cosðS t1 Þ sinð4JIK Þ:
½7:14
Comparison of [7.13] and [7.14] demonstrates that the phase shift of the final 908(I) pulse reduces contributions from evolution of the homonuclear 1H scalar coupling interaction during the initial INEPT period because cos2(2JIK) 4 cos(4JIK) and sin(4JIK) 5 2sin(4JIK). The F1 linewidth of the HSQC spectrum is determined by the relaxation rate of the heteronuclear SQ coherence under conditions of free precession during t1. As discussed in Section 5.4.2, evolution under the scalar coupling Hamiltonian averages the relaxation rate constants for in-phase and antiphase SQ coherence to give 1 IS IK CSA ðIÞ þ RCSA ðSÞ R 2S ¼ RIS 2 ð2Iz S Þ þ R2 ðSÞ þ R1 ðIÞ þ R1 2 2 dIS ¼ 4Jð0Þ þ Jð!I !S Þ þ 3Jð!S Þ þ 3Jð!I Þ þ 6Jð!I þ !S Þ 8 dCSAðIÞ dCSA þ Jð!I Þ þ 4Jð0Þ þ 3Jð!S Þ 2 6 1X þ dIk Jð0Þ þ 3Jð!I Þ þ 6Jð2!I Þ : 8 k
½7:15
In the limit of slow overall tumbling, the relaxation rate constant is approximated by
R2S
" # c 4 1X dIS þ dCSAðSÞ þ ¼ dIk : 3 4 k 5
½7:16
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
543
7.1.1.3 The Constant-Time HSQC Experiment Constant-time evolution periods were originally used in 1H NMR spectroscopy to produce F1-decoupled homonuclear NMR spectra (15–17), but subsequently have been employed in a number of heteronuclear 3D and 4D experiments. The constant-time HSQC (CT-HSQC) experiment (18, 19), illustrated in Fig. 7.1c, differs from the simple HSQC experiment (Fig. 7.1b) only in the way in which the heteronuclear SQ coherence evolves during the constant-time period, T, between the two INEPT sequences. The sequence fragment constituting the constant-time evolution period is t1 =2 – 180 ðI, KÞ – T=2 – 180 ðSÞ – ðTt1 Þ=2
½7:17
and has the propagator U ¼ exp½iHðT t1 Þ=2 exp½iSx exp½iHT=2 exp½iðIx þ Kx Þ exp½iHt1 =2 ¼ exp½i2JIK Iz Kz T exp½iI Iz ðT t1 Þ exp½iS Sz t1 exp½iðIx þ Kx þ Sx Þ:
½7:18
Evolution of the heteronuclear SQ coherence present following the initial INEPT sequence is given by U2Iz Sy U1 ¼ expðiS t1 Sz Þ2Iz Sy expðiS t1 Sz Þ ¼ 2Iz Sy cosðS t1 Þ þ 2Iz Sx sinðS t1 Þ:
½7:19
As shown by [7.18], the heteronuclear scalar coupling interaction, JIS, is active for a total time period (T/2 – t1/2) – T/2 þ t1/2 ¼ 0; consequently, the S spin coherence remains antiphase with respect to the I spins. The homonuclear scalar coupling, JIK, is active for the entire constant-time period, T, because both spins are affected equally by a nonselective 1808 pulse. Consequently, evolution during the t1 period is not modulated by homonuclear scalar coupling interactions and the F1 lineshape does not contain homonuclear multiplet structure. In the present example, this property is not particularly useful because the 2IzSy operator present during t1 commutes with the 1H homonuclear scalar coupling Hamiltonian. However, the same property exists for any homonuclear scalar coupling interaction and will be used to great advantage in 1H–13C HSQC spectra of fully 13C-enriched proteins (Section 7.1.5). Dephasing of the S spin coherence by magnetic field inhomogeneity during the constant-time period is refocused for a time T – t1 by the 1808(S) pulse; consequently, relaxation of the heteronuclear
544
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
single-quantum coherence is proportional to exp(R2T) exp(R2inhomt1), in which R2 is the homogeneous transverse relaxation rate of the SQ coherence and R2inhom is the inhomogeneous contribution to the total relaxation rate R2 ¼ R2 þR2inhom . In the constant-time HSQC experiment, exp(R2T) is a multiplicative factor reducing the intensity of the resonance signals, and exp(Rinhomt1) determines the F1 linewidth (in practice, Rinhomt1max 1 and the F1 linewidth is determined principally by the apodization function employed). Clearly, t1max cannot exceed T, and constant-time HSQC experiments invariably require compromise between resolution in F1 (large values for T to maximize t1max) and sensitivity [small values of T to minimize exp(R2T)].
b
c
124
120
116 F1 (ppm)
112
a
108
7.1.1.4 Comparison of HMQC and HSQC Spectra Selected regions of the 1H–15N HMQC, HSQC, and constant-time HSQC spectra of
8
7
8
7
8
7
F2 (ppm)
FIGURE 7.2 Comparison of selected regions from 1H–15N heteronuclear correlation spectra of 15N-labeled ubiquitin, recorded using the schemes of Fig. 7.1a–c, respectively. Each spectrum was recorded in approximately the same total time with identical t1 and t2 acquisition times (348 and 164 ms, respectively). Each spectrum was processed similarly. No apodization function was applied in the F1 dimension for the spectra shown in panels a and b; a cosine-bell apodization function was applied in the F1 dimension of the spectrum shown in panel c. NH2 correlations are indicated by lines connecting the two nonequivalent proton resonances.
545
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
100
200
a
5.9 Hz
b
2.9 Hz
c
0.9 Hz
300
t1 (ms)
FIGURE 7.3 The t1 interferograms, and their resulting Fourier transforms, taken through the amide proton resonance of Ile36 in the same 1H–15N heteronuclear correlation spectra as illustrated in Fig. 7.2a–c, respectively. The F1 linewidth at half-height is indicated beside each peak. Linewidths were measured by curve fitting the decay of the t1 interferograms. For peak c, the indicated linewidth represents inhomogeneous broadening.
ubiquitin are compared in Fig. 7.2. The multiplet structure and dispersive contribution to the F1 lineshapes associated with the homonuclear J coupling is clearly visible in the HMQC spectrum (Fig. 7.2a). The F1 resolution in the HSQC spectrum (Fig. 7.2b) is clearly superior to the F1 resolution in the HMQC spectrum. The different relaxation properties of the HMQC and HSQC experiments are emphasized in Fig. 7.3, which shows the t1 interferograms through a selected amide 1H resonance, and their corresponding Fourier transforms, for the spectra shown in Fig. 7.2. For backbone amide moieties in proteins, dCSAðSÞ /dIS ¼ 0.055 (assuming ¼ 160 ppm for 15N and B0 ¼ 11.74 T) and dIS =dIk 0:3. In this ˚ –6 calculation, dIk was estimated using a typical value of r6 Ik ¼ 0.027 A for the distances between a backbone hydrogen atom and the other hydrogen atoms in proteins (20). Therefore, R2MQ R2S , and linewidths are narrower in the F1 dimension of an 1H–15N HSQC spectrum than in that of an HMQC spectrum. The observed linewidths in spectra a and b in Fig. 7.3 are consistent with values of 4.9 and 3.0 Hz, respectively, calculated from [7.10] and [7.16] by using c ¼ 4.1 ns.
546
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
The dispersive contribution associated with the homonuclear J coupling is clearly visible in the HMQC spectrum (Fig. 7.3a). As expected, the interferogram for the constant-time HSQC experiment (Fig. 7.3c) exhibits very little decay, and the linewidth in the transformed spectrum is dominated by the apodization applied during processing. For 1H–13C methine moieties, dCSAðSÞ /dIS ¼ 0.002 (assuming ¼ 25 ppm for 13C and B0 ¼ 11.74 T) and dIS =dIk 1:4. Therefore, R2MQ R2S , and linewidth differences in 1H–13C HSQC and 1H–13C HMQC spectra are not as pronounced as for 1H–15N correlations.
7.1.2 ADDITIONAL CONSIDERATIONS CORRELATION EXPERIMENTS
IN
HETERONUCLEAR
Aspects of phase cycling, quadrature detection, multiplet structure, folding or aliasing, and processing schemes that have not been discussed in detail for the basic HMQC and HSQC experiments are presented in the following sections. 7.1.2.1 Phase Cycling and Artifact Suppression The minimum phase cycling required for HMQC and HSQC experiments comprises two steps necessary for spectral editing or isotope filtration and two steps necessary for frequency discrimination, or quadrature detection, in the indirectly detected dimension (Section 4.3.4). Frequency discrimination normally is obtained by TPPI, States, or TPPI–States phase cycling (Section 4.3.4.1) of the first 901 ðSÞ pulse in the HMQC and HSQC experiments (Fig. 7.1). In the HSQC experiments (Fig. 7.1b–c), the phase of the 1808(S) pulses preceding the t1 period should be phase cycled in concert with the 901 ðSÞ pulse for optimal results (21). The initial t1 sampling delay is adjusted to 1/(2SW1) as described in Section 3.3.2.3 by adjusting t1(0) such that t1 ð0Þ ¼ 1=ð2SW1 Þ – 4 90 ðSÞ =
½7:20
for conventional evolution periods (e.g., Fig. 7.1a–b). For the constanttime evolution period of Fig. 7.1c, t1 ð0Þ ¼ 1=ð2SW1 Þ. The isotope filtration phase cycle is critically important for most heteronuclear experiments, because only signals from 1H spins directly attached to the heteronucleus of interest (15N or 13C) are selected. Normally, changes in the coherence order of the S spin between pS ¼ 0 (corresponding to a product operators containing an Sz component) and pS ¼ 1 (corresponding to product operators containing a Sx or Sy
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
547
component) are selected in a two-step phase cycle by simultaneously inverting the phase of a 908(S) pulse and the receiver. Signals from I spins that are not scalar coupled to S spins are unaffected by the phase shift of the 908(S) pulse and are canceled by inversion of the receiver phase. Straightforward product operator analysis demonstrates that an isotope filter can be implemented in the HMQC and HSQC (Fig. 7.1) by use of a two-step (x, x) phase cycle on the 901 ðSÞ pulse or the 902 ðSÞ together with an (x, x) receiver phase cycle. Normally, the first pulse is chosen for the isotope filter phase cycling because a degree of axial peak suppression is obtained as well. Time permitting, both pulses can be cycled independently to give a four-step ‘‘double-difference’’ isotope filter (22). Additional phase cycling is utilized to eliminate artifacts resulting from pulse imperfections. The 1808(I) pulse in the middle of the t1 evolution period of the HMQC experiment (Fig. 7.1a) typically benefits from EXORCYCLE phase cycling (Section 4.3.2.3) to suppress artifacts from imperfect 1808 pulses. If a 1808 pulse is used simply to invert a longitudinal operator [e.g., the 1808(I) pulse applied to IzSy in the t1 period of HSQC experiments], then artifacts caused by pulse imperfections are suppressed by inverting the phase of the 1808 pulse in a two-step (x, x) phase cycle without changing the receiver phase. Time permitting, CYCLOPS phase cycling (Section 4.3.2.3) can be applied to the entire pulse sequence to eliminate quadrature artifacts. The two-step (x, x) phase cycle applied to the 1808(I) pulse in the t1 period of the HSQC experiment provides an important example of artifact suppression in heteronuclear correlation experiments. If the pulse has a nominal rotation angle of þ , rather than the ideal value of , then evolution through the t1 period is given by (for simplicity, S is assumed to be zero) t1 2
2Iz Sy ! 2Iz Sy cosðJIS t1 =2Þ þ Sx sinðJIS t1 =2Þ ðþÞIx
! 2Iz Sy cosðJIS t1 =2Þ cosð þ Þ 2Iy Sy cosðJIS t1 =2Þ sinð þ Þ þ Sx sinðJIS t1 =2Þ t
1 2 ! 2Iz Sy cos2 ðJIS t1 =2Þ cosð þ Þ þ sin2 ðJIS t1 =2Þ þ Sx 1 þ cosð þ Þ cosðJIS t1 =2Þ sinðJIS t1 =2Þ 2Iy Sy cosðI t1 =2Þ cosðJIS t1 =2Þ sinð þ Þ
2Ix Sy sinðI t1 =2Þ cosðJIS t1 =2Þ sinð þ Þ:
½7:21
548
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
The first and last terms of the result in [7.21] are converted to observable 1 H magnetization by the reverse INEPT sequence: 2Iz Sy cos2 ðJIS t1 =2Þ cosð þ Þ þ sin2 ðJIS t1 =2Þ 2Ix Sy sinðI t1 =2Þ cosðJIS t1 =2Þ sinð þ Þ INEPT ! Ix cos2 ðJIS t1 =2Þ cosð þ Þ þ sin2 ðJIS t1 =2Þ Iy sinðI t1 =2Þ cosðJIS t1 =2Þ sinð þ Þ:
½7:22
This result can be simplified to Ix cos2 ðJIS t1 =2Þ cosð þ Þ þ sin2 ðJIS t1 =2Þ Iy sinðI t1 =2Þ cosðJIS t1 =2Þ sinð þ Þ ¼ Ix 1 þ cosðÞ =2 þ Ix cosðJIS t1 Þ 1 cosðÞ =2 Iy sinðI t1 =2Þ cosðJIS t1 =2Þ sinðÞ ¼ Ix 1 2 =4 þ Ix cosðJIS t1 Þ2 =4 Iy sinðI t1 =2Þ cosðJIS t1 =2Þ ½7:23 by applying standard trigonometric identities. The last line is obtained by assuming 1. The first term represents the desired heteronuclear correlation resonance peak, with slightly reduced intensity. The second term represents an in-phase doublet split by the heteronuclear scalar coupling constant. The third term generates dispersive doublets at frequencies þ I/2 and I/2 (i.e., symmetrically positioned with respect to the main heteronuclear correlation resonance peak), with apparent splitting equal to JIS/2 and amplitudes /4. For ¼ 208 (0.35 radians), the relative amplitudes of the resonance peaks are 0.97, 0.015, and 0.09. The third term in [7.23], which represents the largest artifact, is eliminated by the two-step phase cycle. The second term cannot be eliminated by phase cycling (or field gradient pulses); however, this term normally is small if the pulse lengths are measured carefully. Artifacts associated with 1808 pulses can also be eliminated by using field gradient pulses (Section 4.3.3.2), composite pulses (Section 3.4.3), and adiabatic pulses (Section 3.4.6). Composite and adiabatic pulses are beneficial particularly for the 1808(S) pulses during the INEPT sequences of the HSQC experiment (Fig. 7.1b), because optimal inversion of the Sz operator component (over a typically wide spectral width) directly improves the sensitivity of the experiment. 7.1.2.2 13C Scalar Coupling and Multiplet Structure The preceding analyses of heteronuclear correlation experiments have indicated the
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
549
multiplet structure arising due to heteronuclear IS and homonuclear IK scalar coupling interactions. For proteins enriched in 13C, additional scalar coupling interactions must be considered. In most of the other experiments designed for application to 13 C-labeled proteins, aliphatic 13C and 13CO spins are treated theoretically as different nuclear species because of the large difference in resonance frequencies ( 100 ppm). The 13C or 13CO spins can be decoupled by applying selective 1808 pulses or semi-selective decoupling schemes to the 13CO or 13C spins during t1. However, rf fields applied at aliphatic 13C frequencies can have significant effects at the carbonyl frequencies (and vice versa) that must be considered in practice. First, the carbonyl (aliphatic) pulses should minimally excite the aliphatic (carbonyl) spins, either by adjusting the power and duration of rectangular pulses as described in Section 3.4.1 or by using selective pulses (Section 3.4.4). Second, phase errors and frequency shifts caused by off-resonance effects of carbonyl (aliphatic) rf fields applied during aliphatic (carbonyl) 13C evolution periods should be compensated in the pulse sequence whenever possible (Section 3.4.1). In 1H–15N heteronuclear correlation NMR spectroscopy of 15N/13C double-labeled proteins, the 13C–15N scalar coupling interaction must be decoupled. In most instances, decoupling of 13CO–15N and aliphatic 13 C–15N interactions is performed independently by using selective 1808 pulses or semiselective composite pulse decoupling sequences. In 1H–13C heteronuclear correlation NMR spectroscopy of 15N/13C double-labeled proteins, an additional 1808(15N) pulse is applied in the middle of the 13C evolution period, or a broadband 15N decoupling scheme is applied throughout t1, in order to decouple the 15N and 13C spins. These methods clearly will not work in the case of homonuclear 13 C–13C couplings between aliphatic 13C spins because all aliphatic 13C spins will experience the effect of the inversion pulses. Therefore, the F1 lineshapes in 1H–13C HSQC and HMQC spectra typically are composed of 13C–13C scalar coupled multiplets. The multiplet structure can be collapsed by using a constant-time 13C evolution period as described in Section 7.1.5 or by homonuclear decoupling techniques (23). 7.1.2.3 Folding and Aliasing For a fixed number of t1 increments, the digital resolution of the F1 dimension of an HSQC (or other) experiment generally can be increased by reduction of the spectral width to less than the actual maximum frequency range of the resonances of interest. However, if the maximum number of t1 increments is being utilized in a constant-time experiment (t1max ¼ T), then the digital resolution cannot
550
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
be improved because reducing the spectral width requires that the number of t1 increments be reduced to maintain t1max T. Whenever the spectral width is reduced, resonance signals from outside the acquired spectral window will be folded or aliased in the NMR spectrum (Sections 3.2.1, and 4.3.4.3). Data acquired with the hypercomplex methods (States or TPPI–States) are aliased, in which resonances upfield (downfield) of the acquired spectral window appear at the low- (high)-field side of the spectrum. Folding, in which resonances upfield (downfield) of the acquired spectral window fold back into the spectrum at the upfield (downfield) edge, occurs for real acquisition (TPPI). If the initial sampling delay in the aliased dimension is set to 1/(2SW), then peaks that have been folded or aliased an odd number of times have opposite phase to those that have not been aliased or have been aliased an even number of times. The effect of reducing the 13C spectral width is shown schematically in Fig. 7.4 for hypercomplex and real acquisition in t1. Clearly, the States /TPPI-States
F1
F2
F1 TPPI
F1
F2 F2
FIGURE 7.4 A schematic illustration of the effect of halving the F1 spectral width in an 1H–13C heteronuclear correlation experiment, for both hypercomplex (States or TPPI–States) and real (TPPI) acquisition in t1.
551
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
empirical relationship between 1H and 13C chemical shifts in proteins allows extensive aliasing via hypercomplex acquisition in the 13C dimensions of 2D, 3D, and 4D NMR spectra. The 13C spectral width can be readily set to as low as 20 ppm without adversely complicating the interpretation of the spectrum. Each apparent 13C frequency in the final spectrum corresponds to several 13C chemical shifts, separated by intervals equal to the acquired 13C spectral width. Identification of the true 13C chemical shift of a given cross-peak is determined from the associated aliphatic 1H chemical shift together with knowledge of the sequential resonance assignments. Aliasing also can be used advantageously in 15N correlation experiments (particularly to alias upfield Arg 15 " N resonances). When beginning investigations of a new protein, a trial spectrum should be recorded with a wide enough spectral width to encompass the actual resonance positions and enable optimal parameters for folding to be determined. The use of aliasing is illustrated in practice for the 1H–13C HSQC spectra of ubiquitin in Fig. 7.5. Both experiments were acquired using a simple HSQC pulse sequence (Fig. 7.1b) and the TPPI–States hypercomplex method of quadrature detection in t1. The improvement in digital resolution can be seen clearly in the aliased spectrum acquired
20
a
60
40
40 F1 (ppm)
b
4
4
2 F2 (ppm)
2 F2 (ppm)
FIGURE 7.5 1H–13C HSQC spectra of uniformly 15N/13C-labeled ubiquitin, recorded with F1 spectral widths of (a) 9090.9 Hz (72.3 ppm) and (b) 3971.4 Hz (31.6 ppm). Negative cross-peaks in panel b, which correspond to resonances that have been aliased in the F1 dimension, are plotted with a single level only.
552
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
with the reduced F1 spectral width, as more of the homonuclear 13C–13C couplings are resolved in this spectrum. One of the main advantages of using a narrower 13C spectral width is that a given digital resolution (set by t1max) can be achieved by using fewer t1 increments. Consequently, for a fixed total acquisition time, more transients can be acquired per increment, more extensive phase cycling can be employed, and data storage requirements are reduced. For given values of t1max and the total acquisition time, reducing the spectral width and increasing the number of transients per increment do not affect sensitivity of the NMR experiment, because proportionally fewer increments are acquired. These features are particularly important when recording 3D and 4D spectra. 7.1.2.4 Processing Heteronuclear Correlation Experiments HMQC, HSQC, and other heteronuclear correlation experiments produce lineshapes that are predominately in-phase and absorptive in both the F1 and F2 dimensions. Accordingly, processing these spectra is relatively straightforward and does not require careful optimization of t1max and window functions to avoid self-cancellation of antiphase lineshapes (Section 6.2.1.3). In F2 (the acquisition dimension), the FID will rarely be truncated, and exponential matched filtering or Lorentzian-to-Gaussian transformation are satisfactory. In the F1 dimension, similar window functions are satisfactory if the interferograms are not truncated. More commonly, the interferograms will be truncated and Kaiser, Hamming, or cosine bell window functions provide satisfactory apodization (Section 3.3.2.2). Severely truncated data frequently are extended by linear prediction prior to Fourier transformation or are analyzed by maximum entropy reconstruction (Section 3.3.4). Interferograms in constant-time HSQC experiments are ideal for mirror-image linear prediction (Section 3.3.4.1).
7.1.3 DECOUPLED HSQC, SENSITIVITY-ENHANCED HSQC, TROSY EXPERIMENTS
AND
Pulse sequences for decoupled HSQC, sensitivity-enhanced HSQC, and TROSY experiments are shown in Fig. 7.6. Each of these pulse sequences contains a reverse polarization scheme that consists of two pulse-interrupted free-precession periods appended to each other (i.e., two INEPT-like sequences separated by mixing pulses). Additional insights into the design of these experiments are obtained by considering them as members of a family of related pulse sequences.
553
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
7.1.3.1 The Decoupled HSQC Experiment The basic HSQC experiment transfers coherence from I to S spins in the form of antiphase SQ coherences. In some experiments, net magnetization transfer from I to S spins in the form of in-phase SQ coherence is desirable, either because of the improved linewidth or because of the requirements of subsequent coherence transfer steps in an extended pulse sequence (see Section 7.4.3 for an example) (10). A decoupled HSQC experiment that achieves net magnetization transfer is illustrated in Fig. 7.6a. A refocused INEPT sequence (Section 2.7.7.3) is used to transfer I spin polarization to in-phase S magnetization (24). The S magnetization evolves during the t1 evolution period. Continuous 1H decoupling is applied to suppress evolution under the IS scalar coupling Hamiltonian. A reverse refocused INEPT sequence transfers S spin magnetization back to I spin magnetization for detection. A more elegant implementation of this experiment, which does not alter the following discussion, is described by Ottiger and Bax (25). In the modified experiment, the pair of 1808 pulses in the first 2 1 period are shifted in a semi-constant-time manner and the pair of 1808 pulses in the second 2 1 period are eliminated. The modified experiment provides increased resolution and requires fewer 1808 pulses. Evolution through the decoupled HSQC pulse sequence for a scalar coupled heteronuclear InS spin system can be described as follows (for simplicity, homonuclear scalar coupling to remote 1H spins is ignored): n X
P P P Ikx Ikx , Sx 2 Iky , 2 Sx 2
Ikz !
k¼1 1
P
n X
2Ikz Sy
k¼1 Ikx , Sx 1
! nSx sinð2JIS 1 Þ cosn1 ð2JIS 1 Þ t1
! nSx cosðS t1 Þ sinð2JIS 1 Þ cosn1 ð2JIS 1 Þ P P n Iky 1 Ikx , Sx 1 X 2 n2Ikz Sy cosðS t1 Þ ! k¼1
sin2 ð2JIS 1 Þ cos2n2 ð2JIS 1 Þ P P n Ikx , 2 Sx Ikx , Sx X 2 nIkx cosðS t1 Þ2n ð2 1 Þ; !
½7:24
k¼1
in which 2 ¼ 1=ð2JIS Þ and only observable terms arising from in-phase Sx magnetization have been shown. The coherence transfer function,
554
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS y
a
y t2
I
decouplex f1
S
t
f1
τ1
t1
t
f2
f1
f4
y
b
t1
t1
f5
f5
t1
f5
t
t
decouple
f6 t2
I
f1 S
t
f2
f1 t
t1
t1
2
2
t
f3 t
f3 t
t
decouple
f3
y
c
f2
t2 I
f1 S
t
f2
f1 t
t1
t
t
t
t
FIGURE 7.6 Pulse sequences for (a) decoupled HSQC, (b) sensitivity-enhanced PEP-HSQC, and (c) TROSY heteronuclear correlation experiments. Thin bars represent 908 pulses and thick bars represent 1808 pulses. Pulses are applied with x-phase unless the phase is indicated above the bar. The nominal value for 2 ¼ 1/(2JIS). Decoupling during t1 in panel a is achieved by using WALTZ-16. Decoupling during t2 in panels a and b is achieved by using GARP-1, WALTZ16, or other decoupling sequences. (a) Phase cycling for the decoupled HSQC experiment is 1 ¼ x, x; 2 ¼ 2(x), 2(x); and receiver ¼ x, x, x, x. Frequency discrimination is obtained by TPPI, States, or TPPI–States phase cycling of 1. (b) Phase cycling for the PEP-HSQC experiment can be performed in two ways. In the original version, the phase cycling is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ 2(y), 2(y); 4 ¼ x; 5 ¼ y; 6 ¼ x; and receiver ¼ x, x, x, x for the first FID acquired. The second FID is aquired with inversion of 2. Quadrature detection in the F1 dimension is performed using the TPPI–States method applied to 1. In the modified version of the experiment, the phase
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
555
n(t), is n ðtÞ ¼ sinðJIS tÞ cosn1 ðJIS tÞ
½7:25
and is graphed in Fig. 7.7. Complete refocusing cannot be obtained for InS spin systems with n 4 1 because during the delays, 2 1, coherences evolve under one active JIS scalar coupling interaction and n 1 passive JIS scalar coupling interactions. The relative intensity of the I spin resonance in an IS spin system has a maximum value of 1.0 for 2 1 ¼ 1/ (2JIS). The relative intensity of each I spin resonance in an I2S spin system has a maximum value of 0.50 for 2 1 ¼ 1/(4JIS) and is nulled for 2 1 ¼ 1/(2JIS). The relative intensity for each I spin resonance in an I3S spin system has a maximum value equal to approximately 0.45 for 2 1 ¼ 1/(5JIS) and is nulled for 2 1 ¼ 1/(2JIS). If the I spins are degenerate, as in a methyl moiety, the total signal intensity is obtained by summing the contributions from each I spin. The properties of the coherence transfer function can be used to edit a refocused HSQC experiment according to the I spin multiplicity. A compromise value of 2 1 ¼ 1/(3JIS) (2.4 ms for 1JCH ¼ 140 Hz) yields n(2 1) ¼ 0.87, 0.86, and 0.63 for n ¼ 1, 2, and 3. 1 H decoupling during t1 is achieved by the use of a composite pulse decoupling scheme (WALTZ-16 or DIPSI-2, for instance). Decoupling must be applied synchronously: the decoupling sequence must begin at the same position within the decoupling supercycle for each recorded transient, because evolution of remote 1H spins not scalar coupled to the S spin must be identical during consecutive transients to obtain effective isotope filtration (Section 7.1.2.1). Evolution under the passive scalar coupling interactions during the delays, 2 1, generates 2n 1 antiphase product operators in addition to the desired in-phase magnetization. For example, in an I2S spin system, FIGURE 7.6—Continued cycling is 1 ¼ x, x; 2 ¼ x, x, y, y; 3 ¼ y, y, x, x; 4 ¼ y, y, x, x; 5 ¼ x, x, y, y; 6 ¼ y, y, x, x; and receiver ¼ x, x, x, x for the first FID acquired. The second FID is aquired with inversion of 2. For each t1 increment, 1 and the receiver are inverted; no other quadrature detection scheme is required. (c) Four (three) step phase cycling for the TROSY experiment is 1 ¼ x, y, x, y (2708, 308, 1508); 2 ¼ y; 3 ¼ y; and receiver ¼ x, y, x, y (08, 2408, 1288) for the first FID acquired. The second FID is acquired with 1 ¼ x, y, x, y (2708, 308, 1508); 2 ¼ y; 3 ¼ y; and receiver ¼ x, y, x, y (08, 1208, 2408). Depending on the spectrometer, y and y phases may need to be interchanged for the TROSY experiment. Processing for PEP-HSQC and TROSY experiments are described in the text.
556
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS 1.5 I3S
IS 1
nΓn(t)
0.5
0
-0.5
-1 I S 2 -1.5 0
2
4
6
8
10
t (ms)
FIGURE 7.7 Plots of the refocused INEPT coherence transfer functions, nn(t), for InS spin systems ([7.25]). Results are shown for methine IS (___), methylene I2S (- - -), and methyl I3S (- -) groups with JCH coupling constants of 140 Hz. The dashed vertical line at 2.2 ms indicates the optimal value of t to maximize n(t) simultaneously for methine, methylene, and methyl carbons.
evolution of 2I1z Sy during the 2 1 delay of the forward refocused INEPT sequence gives P 1
Ikx , Sx 1
2I1z Sy ! 2I1z Sy cosð2JI1 S 1 Þ cosð2JI2 S 1 Þ þ 4I1z I2z Sx cosð2JI1 S 1 Þ sinð2JI2 S 1 Þ þ 2I2z Sy sinð2JI1 S 1 Þ sinð2JI2 S 1 Þ þ Sx sinð2JI1 S 1 Þ cosð2JI2 S 1 Þ:
½7:26
The antiphase terms of the type I1zSy, I2zSy, and I1zI2zSx present in [7.26] are not efficiently dephased by a highly rf inhomogeneity-compensated composite pulse decoupling scheme (as discussed in Section 3.5, terms such as 4I1zI2zSx are particularly insensitive to rf inhomogeneity). No net rotation of the I1z and I2z operators is expected for an integral number of cycles of an ideal composite pulse decoupling scheme applied to the I
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
557
spins (ignoring homonuclear TOCSY transfer); however, the length of the evolution period, t1, generally will not be equivalent to an integral number of supercycles. If all the pulses of the composite decoupling sequence are applied along the x-axis, then the I spin operators experience a net rotation about the x-axis by an angle, , given by the sum of the flip angles of the pulses in the applied fraction of the final supercycle. Transfer of antiphase S magnetization back to the I spin magnetization by the reverse INEPT sequence depends upon and therefore is a function of t1. Subsequent Fourier transformation results in a pattern resembling t1 noise for the I2S and I3S signals. The 90y ðIÞ pulse immediately following the t1 period, depicted as an open bar in Fig. 7.6a, suppresses this spurious magnetization transfer by converting the antiphase operators into multiple-quantum operators that are not refocused into observable 1H magnetization by the reverse INEPT sequence. The F1 linewidth of a decoupled HSQC spectrum is determined by the relaxation rate of in-phase SQ coherence: CSA ðSÞ R2S ¼ RIS 2 ðSÞ þ R2 dIS ¼ 4Jð0Þ þ Jð!I !S Þ þ 3Jð!S Þ þ 6Jð!I Þ þ 6Jð!I þ !S Þ 8 dCSA 4Jð0Þ þ 3Jð!S Þ , þ 6
½7:27
in which the individual relaxation rate constants are obtained from Tables 5.5 and 5.8 (Chapter 5) and other assumptions are identical to those used in the discussion of the HMQC and HSQC experiments. In the limit of slow overall tumbling, the relaxation rate constant is approximated by c 4 dIS þ dCSA : ½7:28 R2S ¼ 3 5 For backbone amide moieties in proteins, R2S R2S , because decoupling of the I and S spins during t1 eliminates broadening due to longitudinal relaxation of Iz. Consequently, narrower linewidths are obtained in decoupled 1H–15N HSQC spectra, compared with conventional HSQC spectra. For example, a linewidth of 2.0 Hz is calculated from [7.28], compared with 3.0 Hz calculated from [7.16], by using c ¼ 4.1 ns. The linewidth differences in conventional and decoupled 1H–13C HSQC spectra are not as pronounced because the C–H dipole–dipole contribution to the transverse relaxation
558
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
rate constant is large compared to the H–H dipole–dipole contribution. To illustrate an analysis (26) that will prove very useful in analyzing the sensitivity-enhanced HSQC and TROSY experiments, the following discussion specializes to an IS spin system. In this case, 2 1 ¼ 2 ¼ 1/(2JIS) and the 90y ðIÞ pulse, shown as an open bar in Fig. 7.6a, following the t1 period can be eliminated. The propagator for the reverse polarization transfer scheme is written as U ¼ exp½iH exp½iðIx þ Sx Þ exp½iH exp i 2 ðIx þ Sx Þ ¼ ¼ ¼ ¼
exp½iH exp½iðIx þ Sx Þ exp½iH exp i 2 2Iz Sz exp ið2 ÞðIx þ Sx Þ exp i 2 2Iz Sz exp i 2 Sx exp i 2 2Iz Sy exp i 2 2Iy Sz exp i 2 Ix exp i 2 Sx exp i 2 Sy exp i 2 Sy exp i 2 2Iz Sy exp ið2Þ2Iy Sz exp i 2 Iy exp i 2 Iy exp i 2 Ix exp i 2 Sy exp i 2 Sz exp iI Sy exp iIy S exp i 2 Iz exp i 2 Iy :
½7:29
The two z-rotations in the final result introduce only phase shifts and can be ignored for the present discussion. The 90y ðIÞ and 90y ðSÞ rotations do not affect the experiment if the S spin coherence is purely in-phase at the end of the t1 period and if the detected I spin coherence also is in-phase; the composite pulse decoupling sequences applied during t1 and t2 ensure these conditions. Consequently, the part of the propagator responsible for coherence transfer can be written simply as U ¼ exp iI Sy exp iIy S :
½7:30
This propagator corresponds to a selective inversion of coherences (and populations) between the ji and ji eigenstates, followed by a selective inversion between the ji and ji eigenstates. The effects of these rotations are depicted in Fig. 7.8a. The propagator causes the transfers: Iy S
I Sy
Iy S
I Sy
I S ! I S ! I S , I S ! I þ S ! I þ S :
½7:31
559
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY a
ββ βα
IαS–
I+S−
πIySα
αβ
I−Sβ
I−S−
IβS–
αα
I+Sα
πIαSy
b
ββ βα IβS– αβ IαS–
I−Sβ I−Sα
πZQ αα
FIGURE 7.8 Energy level representation for reverse polarization transfers for (a) decoupled HSQC and TROSY experiments and (b) a PEP-HSQC experiment. Coherences are represented by wavy lines connecting energy eigenstates. Selective 1808 rotations are represented by curved lines and are labeled with the appropriate rotation operator.
The corresponding transfers for the S þ coherences are found by interchanging the and þ signs: Iy S
I Sy
Iy S
I Sy
I Sþ ! I þ S þ ! I þ S , I Sþ ! I Sþ ! I S :
½7:32
Only the operators containing an I factor are potentially detectable during t2; thus, including the effects of chemical shift evolution during t1 gives I S eiS t1 þ I Sþ eiS t1 ! I S eiS t1 þ I S eiS t1 ¼ I cosðS t1 Þ i2I Sz sinðS t1 Þ:
½7:33
The antiphase term is not detectable during t2, because composite pulse decoupling is utilized. Consequently, in agreement with the preceding product operator analysis, an in-phase, amplitude-modulated signal is recorded. On average, one-half of the signal (the antiphase term) is not detectable in this experiment, which is typical of
560
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
amplitude-modulation schemes for frequency discrimination during t1 of 2D NMR experiments. 7.1.3.2 Sensitivity-Enhanced HSQC In the conventional HSQC experiment (Fig. 7.1b), heteronuclear SQ coherence evolves under the influence of the S spin chemical shift Hamiltonian during the t1 evolution period to yield two orthogonal terms proportional to 2IzSy and 2IzSx [7.11]. The second term is not refocused by the reverse INEPT sequence and does not contribute to the final observed magnetization [7.11] (this is essential if amplitude-modulated, pure-phase spectra are to be recorded). Therefore, on average, one-half of the initial I spin polarization does not contribute to the detected signal. Modification of the HSQC experiment to permit refocusing and detection of both orthogonal transverse magnetization components forms the basis of a class of heteronuclear correlation experiments, developed originally by Rance and co-workers,pffiffithat can provide ffi sensitivity improvements by factors of up to 2 relative to the conventional experiments (20, 27, 28). This technology has been denoted ‘‘preservation of equivalent pathways’’ (PEP) (29) or ‘‘coherence order selective’’ (COS) (30). The principle of the PEP technique for sensitivity improvement will be demonstrated for the PEP-HSQC experiment illustrated in Fig. 7.6b. The evolution of the density operator for the enhanced PEP-HSQC sequence proceeds exactly as for the conventional HSQC experiment up to the end of the t1 evolution period in Fig. 7.6b. For an InS spin system, the evolution through the remainder of the pulse sequence is given by, for the operators of interest, 2Iz Sy cosðS t1 Þ 2Iz Sx sinðS t1 Þ 2 Ix , 2 Sx
! 2Iy Sz cosðS t1 Þ þ 2Iy Sx sinðS t1 Þ Iy , Sx
! Ix cosðS t1 Þ þ 1,n 2Iy Sx sinðS t1 Þ 2 Iy , 2 Sy
! Iz cosðS t1 Þ 1,n 2Iy Sz sinðS t1 Þ Iy , Sy
! Iz cosðS t1 Þ 1,n Ix sinðS t1 Þ 2 Ix
! Iy cosðS t1 Þ 1,n Ix sinðS t1 Þ,
½7:34
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
561
in which 1,n ¼ cosn–1(2JIS) for 2 ¼ 1/(2JIS) results from evolution of the MQ coherence due to passive scalar coupling interactions. The sine-modulated component is stored as multiple-quantum coherence while the cosine-modulated component is refocused to I spin magnetization; subsequently, the cosine-modulated I spin coherence is stored as longitudinal magnetization while the sine-modulated component is refocused to I spin coherence. Refocusing of both orthogonal signal components following t1 is possible only for IS (n ¼ 1) spin systems. For example, evolution through the pulse sequence for an I2S spin system yields 2I1z Sy cosðS t1 Þ 2I1z Sx sinðS t1 Þ 2 Ix , 2 Sx
! 2I1y Sz cosðS t1 Þ þ 2I1y Sx sinðS t1 Þ Iy , Sx
! I1x cosðS t1 Þ þ 4I1y I2z Sy sinðS t1 Þ 2 Iy , 2 Sy
! I1z cosðS t1 Þ þ 4I1y I2x Sy sinðS t1 Þ Iy , Sy
! I1z cosðS t1 Þ 4I1y I2x Sy sinðS t1 Þ 2 Ix
! Iy cosðS t1 Þ þ 4I1z I2x Sy sinðS t1 Þ:
½7:35
The first term on the last line of [7.35] is observable magnetization; in accordance with [7.34], the second term is unobservable multiple-quantum coherence. The resultant –Iy cos(St1) and – 1,nIx sin(St1) terms in [7.34] describe orthogonal in-phase I spin magnetization components that have evolved at the frequency of the S spin during t1. The final two terms in [7.34] represent superposed observable signals with a 908 phase difference in both dimensions of a 2D spectrum (resulting in phase-twisted lineshapes). In order to separate the two orthogonal terms and obtain purely absorptive lineshapes, a second experiment is performed in which the phase 2 is inverted. The relevant operator terms are (again beginning with the operators present at time a, given at the
562
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
end of [7.11]) as follows: 2Iz Sy cosðS t1 Þ 2Iz Sx sinðS t1 Þ 2 Ix , 2 Sx
! 2Iy Sz cosðS t1 Þ þ 2Iy Sx sinðS t1 Þ Iy , Sx
! Ix cosðS t1 Þ þ 1,n 2Iy Sx sinðS t1 Þ 2 Iy , 2 Sy
! Iz cosðS t1 Þ 1,n 2Iy Sz sinðS t1 Þ Iy , Sy
! Iz cosðS t1 Þ 1,n Ix sinðS t1 Þ 2 Ix
! Iy cosðS t1 Þ 1,n Ix sinðS t1 Þ:
½7:36
Addition of the two data sets [7.34] and [7.36] gives the single observable term 1,n 2Ix sinðS t1 Þ"MQ ,
½7:37
while subtraction of the two data sets yields the single observable term 2Iy cosðS t1 Þ"I :
½7:38
The data set represented by [7.37] contains only signals from IS spin systems, because the extended reverse polarization transfer sequence implements a spin multiplicity filter. The coefficients of 2 in [7.37] and [7.38] arise because two acquisitions have been performed; in practice, each experiment would be recorded with one-half of the total number of transients desired to maintain the same total acquisition time as the conventional experiment. The factors eMQ and eI have been introduced in [7.37] and [7.38] to account for the different relaxation rates of the sine- and cosine-modulated components. These factors are given by "MQ ¼ expð2R2MQ Þ
½7:39
"I ¼ expð2R1I Þ,
½7:40
and
in which R2MQ is transverse relaxation rate constant for heteronuclear multiple-quantum coherence, and R1I is the longitudinal relaxation rate
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
563
constant of the I spin (28): IK CSA ðIÞ R1I ¼ RIS 1 ðIÞ þ R1 ðIÞ þ R1 dIS ¼ Jð!I !S Þ þ 3Jð!I Þ þ 6Jð!I þ !S Þ þ dCSAðIÞ Jð!I Þ 4 1X dIk Jð0Þ þ 3Jð!I Þ þ 6Jð2!I Þ : þ 4 k
½7:41
In the limit of slow overall tumbling, the relaxation rate constant is approximated by R1I ¼
c X dIk : 10 k
½7:42
The data sets described by [7.37] and [7.38] differ in phase by 908 in both dimensions. The phase difference in the acquisition dimension can be corrected by applying a 908 zero-order phase correction to the data set represented by [7.38] after processing in F2. Alternatively, and now more commonly, the phase correction is performed in the time domain by swapping the real and imaginary components of the FID and negating the new imaginary component. The procedure to first add and subtract the data sets followed by time-domain phase correction of one of the resulting data sets is called ‘‘Rance–Kay’’ processing in most NMR data processing programs. In the original implementation of the PEP-HSQC experiment, either real or hypercomplex acquisition in t1, by appropriate phase cycling of 1, is used to obtain two phase-sensitive data sets, corresponding to [7.37] and [7.38]. The data sets are transformed and phased to obtain two 2D heteronuclear correlation spectra with purely absorptive peak shapes in both dimensions. The final sensitivity-enhanced spectrum is obtained by adding the two pure-absorption spectra. The random noise in the two spectra described by [7.37] and [7.38] is uncorrelated (31) and increases pffiffiffi only by 2 in the sensitivity-enhanced spectrum. The achievable sensitivity improvement over a conventional HSQC experiment recorded in the same total time is pffiffiffi ½7:43 " ¼ 2ð"I þ 1,n "MQ Þ=2, pffiffiffi which has a maximum value of 2 for an IS spin system. The F1 lineshape of the sensitivity-enhanced HSQC experiment is identical to that of the conventional HSQC experiment because identical spin operators are present during t1.
564
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
In most current implementations of the PEP approach, the two data sets obtained from Rance–Kay processing are treated as a hypercomplex pair for subsequent Fourier transformation in t1. No additional data acquisitions are utilized for quadrature detection in the indirect dimension. However, processing the hypercomplex data defined by [7.37] and [7.38] yields quadrature artifacts in the F1 dimension of the spectrum due to the amplitude differences, "I and 1,n"MQ, between the two data sets. The PEP method can be modified, by appropriate phase cycling (see the caption to Fig. 7.6) (32), to equalize the relaxation differences between the coherence transfer pathways used to refocus the orthogonal magnetization components. The sine- and cosine-modulated terms that result from addition and subtraction of the data sets acquired using the modified phase cycle are given by Ix sinðS t1 Þ "I þ 1,n "MQ ,
½7:44
Iy cosðS t1 Þ "I þ 1,n "MQ ,
½7:45
respectively. The two components have identical magnitudes and are treated as a hypercomplex quadrature pair in the t1 dimension (following a 908 zero-order phase correction in the acquisition dimension of the data described by [7.45]). Following the same formal discussion presented for the decoupled HSQC experiment, the evolution of an IS spin system in the sensitivity-enhanced reverse polarization transfer is considered again. The propagator for the reverse polarization transfer scheme can be written as U ¼ exp i ð2ÞIx exp½iH exp iðIy þ Sy Þ exp½iH exp i ð2ÞðIy þ Sy Þ exp½iH exp iðIy þ Sy Þ exp½iH exp i ð2ÞðIx þ Sx Þ ¼ exp ið2ÞIx exp ið2 Þ2Iz Sz exp ið2 ÞðIy þ Sy Þ exp ið2 Þ2Iz Sz exp ið2ÞðIx þ Sx Þ ¼ exp ið2ÞIx exp ið2 ÞðIy þ Sy Þ exp ið2ÞðIx þ Sx Þ exp ið2 Þ2Ix Sx exp ið2 Þ2Iy Sy ¼ exp ið2ÞSy exp ið2ÞIz exp ið2ÞSz exp ið2 Þ2Ix Sx exp ið2 Þ2Iy Sy ¼ exp ið2ÞSy exp ið2ÞIz exp ið2ÞSz exp iZQy : ½7:46
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
565
Again, the two z-rotations introduce only phase shifts and can be ignored for the present discussion. In contrast to [7.29], in-phase S spin coherence is not required following t1. The detected I spin coherence is in-phase because composite pulse decoupling is applied during t2. Thus, the 908 Sy rotation does not affect evolution of the detected signal. Consequently, the propagator can be written simply as U ¼ exp iZQy :
½7:47
This propagator corresponds to a selective inversion of coherences (and populations) between the ji and ji eigenstates. The effects of these rotations are depicted in Fig. 7.8b. The propagator (ignoring any phase shifts) causes the transfers: ZQy
I S ! iI S ; ZQy
I S ! iI S ; ZQy
½7:48
I Sþ ! iI þ S ; ZQy
I Sþ ! iI þ S : The phase factors i and –i are not obtainable from the pictorial description of the ZQy rotation and must instead be determined by direct calculation of the product operators. The sign inversion for two of the transfers in [7.48] means that the pulse sequence transfers antiphase S spin coherence to in-phase I spin coherence. Only the operators containing an I factor are potentially detectable during t2; thus, including the effects of chemical shift evolution during t1 gives I S eiS t1 I S eiS t1 ! I S eiS t1 þ I S eiS t1 ¼ I eiS t1 :
½7:49
In agreement with this product operator analysis, an in-phase, phasemodulated signal is recorded. On average, the detected signal has twice the amplitude, compared to the signal recorded in the amplitudemodulated decoupled HSQC; however, as previously discussed, deconvoluting p the ffiffiffi phase-modulated signals results in an increase pffiffiffi pinffiffiffi the noise level by 2 and results in a net sensitivity gain of 2/ 2 ¼ 2: The requirement for decoupling during acquisition to ensure that the detected signal is in-phase can be eliminated by inserting a 90y ðSÞ pulse immediately prior to acquisition. This additional pulse cancels the effect of the 908 Sy rotation in the last line of [7.46]. In fact, this modified
566
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
sequence element has been utilized in TROSY experiments by Kay and co-workers (33). 7.1.3.3 TROSY Experiment The basic pulse sequence for the original 1H–15N TROSY (transverse relaxation optimized spectroscopy) experiment is shown in Fig. 7.6c. Rather than performing a detailed product operator analysis of this sequence, the insights obtained in the analyses of the decoupled HSQC and PEP-HSQC experiments will be utilized. The initial part of the TROSY experiment is identical to the HSQC or sensitivity-enhanced HSQC experiments, except that decoupling is not applied during the t1 period. Consequently, evolution during t1 is described by t1
2Iz Sy ! 2Iz Sy cosðS t1 Þ cosðJIS t1 Þ þ 2Iz Sx sinðS t1 Þ cosðJIS t1 Þ þ Sx cosðS t1 Þ sinðJIS t1 Þ þ Sy sinðS t1 Þ sinðJIS t1 Þ: ½7:50 The propagator for the reverse polarization transfer scheme can be written as U ¼ exp i ð2ÞSy Þ exp½iH exp½iðIx þ Sx Þ exp½iH exp i ð2 ÞðIx þ Sx Þ exp½i1H exp½iðIx þ Sx Þ exp½iH exp i ð2 ÞIy ¼ exp i 2 Sy exp i 2 2Iz Sz exp i 2 ðIx þ Sx Þ exp i 2 2Iz Sz exp i 2 Iy ¼ exp i 2 Sy exp i 2 Sx exp i 2 2Iz Sy exp i 2 2Iy Sz exp i 2 Ix exp i 2 Iy ¼ exp i 2 Sz exp i 2 Sy exp i 2 2Iz Sy exp i 2 2Iy Sz exp i 2 Iy exp i 2 Iz ¼ exp i 2 Sz exp iI Sy exp iIy S exp i 2 Iz : ½7:51 The two z-rotations introduce only phase shifts and can be ignored for the present discussion. In contrast to the decoupled HSQC or sensitivityenhanced HSQC experiments, no unnecessary 908 rotations are present in the propagator; therefore, decoupling is not required during t1 or t2. Consequently, the propagator can be written simply as U ¼ exp iI Sy exp iIy S :
½7:52
567
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
This propagator is exactly that considered previously for the decoupled HSQC experiment and causes the transfers depicted in Fig. 7.8a and [7.31]–[7.32]. Only the operators containing an I factor are potentially detectable during t2; thus, including the effects of chemical shift evolution during t1 gives I S eiðS þJIS Þt1 þ I Sþ eiðS JIS Þt1 ! I S eiðS þJIS Þt1 þ I S eiðS JIS Þt1 : ½7:53 The transfer from I S þ to I S is suppressed either by phase cycling or by gradient coherence selection (Section 7.1.4.2). Phase cycling of 1 selects for the coherence order change of the S spin from pS ¼ 0 to 1, while suppressing the coherence order change from pS ¼ 0 to þ1. In principle, this selection could be obtained using a three-step phase cycle; however, a four-step cycle is normally used for the convenience provided by 908 phase shifts. Both phase cycles are given in the caption to Fig. 7.6c. The net transfer, including the phase cycle, is I S eiðS þJIS Þt1 ! I S eiðS þJIS Þt1 :
½7:54
The full intensity of the I S operator is recorded during acquisition because decoupling is not employed. However, on average, one-half of the signal is recorded in this experiment compared to the sensitivityenhanced HSQC experiment, resulting in a twofold decrease in overall sensitivity, in the absence of relaxation effects. The resonance correlation peak has frequency coordinates (S þ JIS, I þ JIS), compared with HSQC spectra, in which the resonance coordinates are (S, I). The final signal in [7.53] is phase modulated in t1 and is deconvoluted by procedures similar to that used for the PEP-HSQC experiment. To achieve the desired transfer for the second data set required for Rance–Kay processing, two changes are made to the phase cycle. First, the the phase cycling of 1 is reversed to select for the coherence order change of the S spin from pS ¼ 0 to þ1, while suppressing the coherence order change from pS ¼ 0 to 1. Second, the phases 2 and 3 are shifted by 1808; this is tantamount to inserting a 180x pulse on the I spins immediately after t1 and a 180y pulse on the S spins immediately prior to acquisition. The propagator in [7.52] is modified to U ¼ exp iSy exp iI Sy exp iIy S exp½iIx : ½7:55 This propagator results in the desired coherence transfer pathway: Ix
Iy S
I Sy
Sy
I Sþ ! I S þ ! I S þ ! I S ! I S ,
½7:56
568
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
with the final result, including chemical shift evolution, I Sþ eiðS þJIS Þt1 ! I S eiðS þJIS Þt1 :
½7:57
Equations [7.54] and [7.57] represent the pair of spectra for Rance–Kay processing. The TROSY experiment is unusual in that the S spin natural abundance magnetization contributes constructively to the signal (in other heteronuclear experiments, the natural spin polarization of the S spin usually is destroyed, prior to the first 1H 908 pulse, by a 908 pulse on the S spin followed by a gradient pulse). Prior to the t1 period, evolution of the S spin magnetization is described by Sz ! Sy ¼ 2i ðSþ S ÞðI þ I Þ:
½7:58
Referring to [7.50], the component of the density operator arising from the 1H spin polarization, prior to the t1 period, is described by 2Iz Sy ¼ 2i ðSþ S ÞðI I Þ:
½7:59
Comparison of [7.58] and [7.59] shows that the desired operator SI adds constructively from the initial 1H and 15N polarizations. As in the PEP-HSQC experiment, differential relaxation of operators present through the reverse polarization transfer affect the final detected resonance in the TROSY spectrum. Examining the effects of relaxation is more straightforward in the Cartesian basis. The initial operator of interest after the t1 period is SI ¼ (E/2 Iz)(Sx iSy). Evolution through the reverse polarization transfer, including relaxation, is given by 2 Iy
ðE=2Iz ÞðSx iSy Þ ! ðE=2þIx ÞðSx iSy Þ 2
!Iz Sy exp½R2S 2þIx Sx exp½R2MQ 2 þ iIz Sx exp½R2S 2iIx Sy exp½R2MQ 2 2 Ix ; 2 Sx
! Iy Sx exp½R2S 2þIx Sx exp½R2MQ 2 iIy Sx exp½R2S 2iIx Sz exp½R2MQ 2 2 ! 12Ix
exp½ðR2S þR2I Þ2þIx Sx exp½2R2MQ 2
iIy Sx exp½ðR2S þR2MQ Þ2 2i Iy exp½ðR2MQ þR2I Þ2 2 Sy
! 12Ix exp½ðR2S þR2I Þ2Ix Sz exp½2R2MQ 2 þ iIy Sz exp½ðR2S þR2MQ Þ2 2i Iy exp½ðR2MQ þR2I Þ2, ½7:60
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
in which IS CSA CSA R2I ¼ 12 RIS ðSÞ þ RIK ðIÞ 2 ð2I Sz Þ þ R2 ðIÞ þ R1 2 ðIÞ þ R2 dIS ¼ 4Jð0Þ þ Jð!I !S Þ þ 3Jð!S Þ þ 3Jð!I Þ þ 6Jð!I þ !S Þ 8 dCSAðSÞ dCSAðIÞ Jð!S Þ þ 4Jð0Þ þ 3Jð!I Þ þ 2 6 1X þ dIk 5Jð0Þ þ 9Jð!I Þ þ 6Jð2!I Þ 8 k
569
½7:61
is the relaxation rate constant of the I spin transverse magnetization. In the limit of slow tumbling, this rate constant is approximated by ( ) c 4 5X dIS þ dCSAðIÞ þ R2I ¼ dIk ½7:62 3 4 k 5 The operators on the final line in [7.60] can be expressed in the spherical basis as I S fexp½ðR2S þ R2I Þ2 þ exp½2R2MQ 2 þ exp½ðR2S þ R2MQ Þ2 þ exp½ðR2MQ þ R2I Þ2g=4 þ I S fexp½ðR2S þ R2I Þ2 exp½2R2MQ 2 exp½ðR2S þ R2MQ Þ2 þ exp½ðR2MQ þ R2I Þ2g=4 I S f1 ðR2S þ R2I þ 2R2MQ Þg þ I S fðR2MQ R2I Þg:
½7:63
The last line of this equation is obtained by expanding the exponentials to first order, assuming that 2R2 1, in which ¼ {I, S, MQ}. This result indicates that when the relaxation rate constants are both large and different, then the desired ‘‘TROSY’’ resonance with frequency coordinates (S þ JIS, I þ JIS), associated with I S coherence, has reduced intensity, while an undesired resonance associated with I S coherence appears in the spectrum with frequency coordinates (S þ JIS, I JIS). A similar analysis shows that the spin operator S I present after the t1 period yields an observable operator, I S ! I S fðR2S R2MQ Þg,
½7:64
due to differential relaxation. This transfer yields a correlation with frequency coordinates (S JIS, I þ JIS). Even in the presence of differential relaxation, the resonance correlation at (S JIS, I JIS) is efficiently suppressed by the TROSY sequence. Techniques for
570
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
additional suppression of these artifactual peaks have been described (34, 35). 7.1.3.4 Comparison of Decoupled HSQC, PEP-HSQC, and TROSY Experiments Selected regions of the 1H–15N decoupled HSQC, sensitivity-enhanced PEP-HSQC, and TROSY spectra of ubiquitin are presented in Fig. 7.9. As predicted by [7.25], the NH2 resonances are suppressed in the decoupled HSQC pulse sequence; a similar product operator analysis predicts that the NH2 resonances are suppressed in the TROSY spectrum as well. The relative sensitivity of the three experiments is illustrated by the 1D traces through a selected 1H–15N correlation peak.
b
c
124
120
116 F1 (ppm)
112
a
108
7.1.3.5 Relaxation Interference and TROSY Spectra of Larger Proteins TROSY is both a general principle of pulse sequence design and a particular experiment. The concept behind TROSY is
8
7
8
7
8
7
F2 (ppm)
FIGURE 7.9 Comparison of selected regions from 1H–15N heteronuclear correlation spectra of 15N-labeled ubiquitin, recorded using the schemes of Fig. 7.6a–c, respectively. Each spectrum was recorded in approximately the same total time with identical t1 and t2 acquisition times and processed similarly. NH2 correlations are indicated by lines connecting the two nonequivalent proton resonances. A slice parallel to the F2 axis through a selected resonance is shown in the lower right corner of each spectrum to illustrate the relative sensitivity of the experiments.
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
571
that certain coherences (usually specific multiplet components, but also multiple-quantum operators) have smaller transverse relaxation rate constants than do single-quantum operators representing total spin magnetization. Of course, recording only the signal from a subset of multiplet components, rather than the total magnetization, incurs an initial loss of sensitivity. For example, the TROSY experiment for IS spin systems, such as 1HN–15N, has one-half the sensitivity of the PEP-HSQC experiment if relaxation is neglected. However, if the selected spin operators have small enough transverse relaxation rate constants, and if t1max or free-precession delays are long enough, then improved resolution and sensitivity can be obtained, particularly for large macromolecules in which relaxation losses are particularly severe. The application of line narrowing through relaxation interference was anticipated by Griffey and Redfield (36). The first TROSY experiment was described by Wu¨thrich and co-workers and utilized relaxation interference between CSA and dipole–dipole (DD) interactions within the amide 1HN–15N IS spin system. Since the original description, TROSY concepts have been developed for other IS spin systems (37), as well as for I2S (CH2) (38) and I3S (CH3) spin systems (39). Relaxation interference between DD/CSA interactions is discussed in Section 5.4.5. The effect of differential relaxation on the 1H–15N multiplet components in large proteins at high static magnetic field, where the DD/CSA relaxation interference is most prominent, is shown in Fig. 7.10 for the protein calbindin D28k, with a molecular mass of 30 kDa. The figure shows expansions from a series of heterocorrelation spectra recorded at 800 MHz; in each spectrum, only resonances arising from a single resolved 1H–15N amide moiety are shown. In Fig. 7.10a, an HSQC spectrum in which decoupling was omitted in both t1 and t2 evolution periods is shown. The effects of relaxation interference are manifest in the different intensities of the four multiplet components. The relaxation rate constants for the spin operators contributing to the observed cross-peaks are R2 ðI S Þ ¼ R2I þ I,IS , R2 ðI S Þ ¼ R2I I,IS , R2 ðI S Þ ¼ R2S þ S,IS , R2 ðI S Þ ¼ R2S S,IS ,
½7:65
572
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS 115.7
I−Sb
116.7
IaS−
a
b 9.9
9.8
c 9.9 1HN
9.8
9.9
9.8
117.7
15
IbS−
N (ppm)
I−Sa
(ppm)
FIGURE 7.10 1H–15N multiplet structure. A resonance is shown for calbindin D28k recorded at 800 MHz. (a) HSQC spectrum recorded without decoupling in t1 or t2 periods to illustrate the differential linewidths of multiplet components. (b) Conventional HSQC spectrum, in which the multiplet components are collapsed to a singlet. (c) TROSY spectrum that selects the narrowest multiplet component.
in which I,IS ( S,IS) is the transverse CSA/DD relaxation interference rate constant for cross-correlation between the CSA interaction of the I (S) spin and the DD interaction of the IS spin pair [5.145], and R2I ðR2S Þ is the averaged relaxation rate constant for single-quantum coherence of the I (S) spin during free precession. In Fig. 7.10b, a conventional HSQC spectrum is shown. Decoupling in the t1 and t2 dimensions suppresses relaxation interference and essentially averages the relaxation rate constants, and consequent intensities, of the multiplet components. In Fig. 7.10c, the TROSY experiment selects the most intense multiplet component, resulting from destructive interference between DD and CSA relaxation during both t1 and t2 evolution periods. An important aspect of TROSY experiments arises from remote 1 H–1H dipole–dipole interactions. The narrow TROSY component arises from evolution of IS during t1 and I S during t2. Both of these operators contain 1H spin components and consequently both R2I and R2S contain contributions from remote 1H–1H dipole–dipole relaxation. These additional relaxation pathways reduce the line narrowing achievable in the TROSY experiment. As a result, optimal use of TROSY techniques commonly necessitates replacement of remote 1H spins with 2H in order to reduce these dipole–dipole interactions.
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
7.1.4 WATER SUPPRESSION AND GRADIENT ENHANCEMENT HETERONUCLEAR CORRELATION SPECTRA
573 OF
The previous sections have outlined the basic coherence transfer and relaxation effects that are important for the most widely utilized two-dimensional heteronuclear correlation experiments. Practical implementations of these techniques, either as two-dimensional methods or for incorporation into higher dimensionality NMR experiments, frequently utilize a number of enhancements to obtain high-quality water suppression and reduced artifacts, particularly by incorporating field gradient pulses. Aspects of water-suppression techniques and gradient coherence selection in heteronuclear correlation spectroscopy are described in the following sections. 7.1.4.1 Solvent Suppression 1H–13C heteronuclear correlation spectra of 13C-labeled proteins normally are recorded using proteins in D2O solution, and presaturation of the residual HDO solvent resonance usually is satisfactory. The 1H–13C heteronuclear spectra of 13 C/15N-labeled proteins in H2O solution can be recorded by using presaturation of the solvent resonance; however, pulsed field gradient techniques for solvent suppression are preferable to avoid obscuring 1 13 H – C correlations. For 1H–15N heteronuclear spectra of 15N- and 13 C/15N-labeled proteins acquired in H2O solution, presaturation of the solvent signal usually is avoided to minimize solvent saturation transfer to the amide 1H spins that reduces signal intensities (40–42). Although jump–return versions of the HMQC experiment have been designed, the most effective solvent-suppression schemes for 1H–15N heteronuclear NMR spectroscopy incorporate spin lock purge pulses (43) or field gradient pulses (40, 42, 44) (Section 3.7.3) into the HSQC experiment. Use of these techniques is facilitated because evolution under the 1H–15N scalar coupling Hamiltonian provides an efficient mechanism for independently manipulating the protein and solvent resonances. For example, evolution through the initial INEPT sequence of the HSQC experiment yields Iy , Sx
Iy ! 2Ix Sz , Wy
½7:66
Wy ! Wy , in which I ¼ 1HN and S ¼ 15N are the solute spins, and W designates the solvent 1H spins. The resulting solute and solvent 1H operators are orthogonal. In these pulse sequence elements, the phase of the 1808 pulse
574
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
on the 1H spins has been changed to þy compared with the sequences of Fig. 7.1. At this point, a purge pulse applied with x-phase spin-locks the 2IxSz coherence and dephases the Wy coherence. Alternatively, a 90y pulse applied to the 1H spins converts the 2IxSz coherence into longitudinal two-spin order, 2IzSz, without affecting the Wy operator. A subsequent z-axis field gradient or homospoil pulse selectively dephases the solvent magnetization without affecting the solute coherence. Finally, if a selective 90y pulse is applied to the solvent spins prior to the first nonselective 90x pulse, then evolution through the initial INEPT sequence gives 2 Ix Iy , Sx 2 Iy
Iz ! 2Iz Sz , 2 Wy Wy 2 Wy
½7:67
Wz ! Wz : This pulse sequence fragment is the basis for ‘‘water flip-back’’ techniques, because the solvent magnetization is returned to the z-axis by the INEPT sequence; similar concepts are used during other portions of pulse sequences to ensure that the solvent signal remains aligned along the z-axis during the acquisition period. Examples of the ‘‘fast’’ HSQC pulse sequence and a water flip-back TROSY pulse sequence are shown in Fig. 7.11. In both of these sequences, the water magnetization is transverse during the t1 period. The gradients labeled g2 dephase and rephase the water magnetization at the beginning and end of the t1 period to avoid deleterious effects of radiation damping. The water magnetization is returned to the þz-axis by the final 908 pulse applied to the I spins. The final 1808 pulse applied to the I spins is part of a WATERGATE element, along with the flanking field gradient pulses. Gradient pulses during periods of free precession, such as the gradient pulses labeled g1 or g3 in sequences a and b in Fig. 7.11 or the gradients g2 in Fig. 7.11b are applied in such manner as to keep transverse water magnetization dephased as much as possible to minimize deleterious radiation damping. Thus, in the sequences shown, the first gradient of the g1 pair is applied immediately after the preceding 908 pulses and the second gradient of the pair is applied immediately before the subsequent 908 pulse. 7.1.4.2 Gradient-Enhanced HSQC and TROSY NMR Spectroscopy Although pulsed field gradients (PFGs) can be used to generate frequency-discriminated pure-absorption spectra by gradient coherence
575
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
a
y
f3
y
t2 I f1
S
t
f1 t
g1
g1
f2 t1
t1
2
2
g2
g2
t
t
g3
g3
decouple
Grad
f3
y
y
b
t2 I
f1 S
f1
t
t
g1
g1
f2 t1
g2
t
–g2 g3
t
t
t
g3
g4
g4
Grad
FIGURE 7.11 (a) Fast HSQC and (b) TROSY pulse sequences incorporating water flip-back solvent-suppression schemes. Thin bars represent 908 pulses and thick bars represent 1808 pulses. The 1808 pulse represented as an open bar is a crafted pulse that leaves the water magnetization unperturbed; 3–9–19, or soft–hard–soft, pulse schemes commonly used for this pulse are described in Section 3.7. Pulses are applied with x-phase unless the phase is indicated above the bar. The nominal value for 2 ¼ 1/(2JIS). (a) Decoupling during t2 is achieved by using GARP-1, WALTZ-16, or other decoupling sequences. Phase cycling for the fast HSQC experiment is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ 4(y), 4(y); and receiver ¼ x, x, x, x. (b) Phase cycling for the TROSY experiment is 1 ¼ x, y, x, y; 2 ¼ y; 3 ¼ y; and receiver ¼ x, y, x, y for the first FID acquired. The second FID is acquired with 1 ¼ x, y, x, y; 2 ¼ y; 3 ¼ y; and receiver ¼ x, y, x, y. The two data sets are stored separately and then combined as described in the text. Depending on the spectrometer, y and y phases may need to be interchanged for the TROSY experiment.
576
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
selection, a signal-to-noise loss of 21/2 normally is incurred relative to techniques employing phase cycling (TPPI or hypercomplex methods) (Section 4.3.4.2). Kay and co-workers showed that pulsed field gradients can be used for coherence selection in a PEP-HSQC experiment without sacrificing sensitivity (45). A PFG-PEP-HSQC experiment is shown in Fig. 7.12a. Coherence selection is obtained using gradient pulses g3 and g6; the other gradient pulses are used to suppress artifacts associated with the 1808 pulses (Section 4.3.3.2). The sequence also incorporates water flip-back techniques for solvent suppression. The evolution of the density operator for the PFG-PEP-HSQC sequence proceeds exactly as for the conventional HSQC experiment up to the end of the t1 period in Fig. 7.12a. Evolution through the first spin echo period yields 2Iz Sy cosðS t1 Þ 2Iz Sx sinðS t1 Þ 1 Sx 1
! 2Iz Sy cos½S t1 þ S ðzÞ 2Iz Sx sin½S t1 þ S ðzÞ, ½7:68 in which S(z) is the spatially dependent phase acquired by the S spin coherence during the gradient pulse g3. The PEP reverse INEPT sequence yields 2Iz Sy cos½S t1 þ S ðzÞ 2Iz Sx sin½S t1 þ S ðzÞ PEP
! Iy cos½S t1 þ S ðzÞ"I 1,n Ix sin½S t1 þ S ðzÞ"MQ :
½7:69
Following the second spin echo period and gradient pulse g6, the I spin coherences acquire a spatially dependent phase I(z). If the gradient pulses are adjusted such that S(z) ¼ I(z), the resulting observable magnetization is Iy cosðS t1 Þ Ix sinðS t1 Þ "I þ 1,n "MQ =2: ½7:70 Imbalance between the two coherence transfer pathways is suppressed by the N/P selection of the second field gradient pulse. To deconvolute the orthogonal magnetization components, a second acquisition is performed in which the phase of the 902 ðSÞ pulse immediately following the t1 evolution period is inverted and the sign of gradient g6 is reversed so that S(z) ¼ –I(z). The resulting observable magnetization is given by ½7:71 Iy cosðS t1 Þ Ix sinðS t1 Þ "I þ 1,n "MQ =2: Addition and subtraction of these two data sets yield results equivalent to [7.44] and [7.45]. The resulting phase-sensitive PFG-PEP-HSQC
577
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY y
a
f3
y
y
–y
d2
I f1
f1 t
t
g1
g1 g2
y
y
S
f2 t1
t1
2
2
d1
d1
g3 –g3
d2
t2
y t
t
t
t
g4
g4
g5
g5
decouple g6
Grad
b
f3 d2
I f1
f1 t
t
g1
g1 g2
d2
t2
f2 t1
d1
S
–g2
d1
g3 –g3
t
t
t
t
g4
g4
g5
g5
g6
Grad
FIGURE 7.12 Pulse sequence for the (a) PFG-PEP-HSQC and (b) PFG-TROSY experiments. Thin bars represent 908 pulses and thick bars represent 1808 pulses. Pulses are applied with x-phase unless the phase is indicated above the bar. The short thick bar represents a water-selective soft 908 pulse. The nominal value for 2 ¼ 1/(2JIS); delays 1 and 2 are long enough to encompass the gradients. Gradient coherence selection is performed using g3 and g6 (black); other gradients (white) are for artifact suppression. Gradients g3 are applied as a pair in order to minimize the delay 1. (a) Decoupling during t2 is achieved by using GARP-1, WALTZ-16, or other decoupling sequences. The phase cycling for the PFG-PEP-HSQC experiment is 1 ¼ x, x; 2 ¼ x; and receiver ¼ x, x for the first FID acquired. The second FID is acquired with inversion of 2 and gradient g6. For each t1 increment, 1 and the receiver are inverted; no other quadrature detection scheme is required. (b) Phase cycling for the TROSY experiment is 1 ¼ x, x; 2 ¼ y; 3 ¼ y; and receiver ¼ x, x for the first FID acquired. The second FID is acquired with 1 ¼ x, x; 2 ¼ y; 3 ¼ y; and receiver ¼ x, x. The gradient g6 also is inverted for the second FID. Depending on the spectrometer, y and y phases may need to be interchanged for the TROSY experiment. For both PFG-PEP-HSQC and PFG-TROSY, the two data sets are stored separately and then combined as described in the text.
578
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
spectrum has the same nominal sensitivity as does the phase-cycled PEP-HSQC spectrum, or double the sensitivity of a conventional PFG-HSQC spectrum. The 21/2 sensitivity loss associated with coherence selection by pulsed field gradients is recovered and the nominal 21/2 PEP sensitivity enhancement is obtained independently in the PFG-PEPHSQC experiment. The gradient-enhanced pulse sequence is necessarily longer than the PEP-HSQC experiment, because the gradient pulses required for coherence selection are placed within spin echoes in order to refocus chemical shift evolution. The sensitivity of the PFG-PEP-HSQC experiment is reduced relative to the PEP-HSQC experiment by a factor of exp½2 1 R2S 2 2 R2I , in which R2S and R2I are the relaxation rate constants for the S and I spin coherence present during the spin echo periods (46). The same considerations can be applied to the TROSY experiment. A PFG-TROSY experiment is shown in Fig. 7.12b. In addition to gradient coherence selection, this sequence also incorporates water flip-back solvent suppression.
7.1.5 THE CONSTANT-TIME 1H–13C HSQC EXPERIMENT As noted in Section 7.1.2.2, multiplet structure in the F1 dimension of conventional 1H–13C HSQC or HMQC spectra arises from aliphatic homonuclear 13C–13C scalar coupling interactions. The multiplet structure can be eliminated by using a constant-time 13C evolution period (18, 19). The constant-time 1H–13C HSQC (CT-HSQC) experiment illustrated in Fig. 7.13 differs from the pulse sequence of Fig. 7.1c by addition of 15N coherent decoupling throughout the constant-time period T and addition of two carbonyl 1808 pulses. Evolution under the 13CO–13C scalar coupling Hamiltonian is refocused during the initial t1 fraction of the constant-time evolution period by the 1808 pulse applied to the carbonyl spins, and during the remaining fraction of the constant-time period, T t1, by the 1808 pulse applied to the aliphatic carbon spins. The 13C–13C scalar coupling interaction between the S spin and n other aliphatic 13C spins, R, with coupling constant JCC, is active during the entire period T. The net evolution of 2IzSy coherence during T is obtained by generalizing [7.19]: T
2Iz Sy ! 2Iz Sy cosðS t1 Þ cosn ðJCC TÞ þ 2Iz Sx sinðS t1 Þ cosn ðJCC TÞ þ antiphase terms: ½7:72
579
7.1 HETERONUCLEAR CORRELATION NMR SPECTROSCOPY
The antiphase terms (of the form 4IzSxR1z and 8IzSyR1zR2z, etc.) are not converted back to observable 1H magnetization by the reverse INEPT sequence following the constant-time evolution period, and can be ignored. The 13C magnetization present at the end of the constant-time evolution period is modulated only by its chemical shift during t1; therefore, cross-peaks in the 2D correlation spectrum appear as singlets in the F1 dimension. To achieve effective decoupling of the 13C–13CO scalar coupling interaction, the rectangular 1808(13C) pulse pffiffiffi applied during T of Fig. 7.13 has a field strength given by B1 ¼ = 3, where is the offset between the 13CO and aliphatic 13C carrier frequencies, and selective shaped 1808(13CO) pulses are used (Section 7.1.2.2). The second 1808(13CO) f4
y
t2 1H
f1
f1 13C
t
t
f3
f2
t1
T
T
2
2
2
–
t1 2
t
t
decouple
T
13CO
15N
decouple
FIGURE 7.13 Pulse sequences for the 1H–13C constant-time HSQC experiment. Thin bars represent 908 pulses and thick bars represent 1808 pulses. Pulses are applied with x-phase unless the phase is indicated above the bar. The nominal value for 2 ¼ 1/(2JIS), T ¼ n=JCC , and n ¼ 1 or 2. The first and last 1808(13C) pulses are best applied as composite pulses of the type 90x 180y 90x , or other broadband inversion pulse, in order to minimize resonance offset and rf inhomogeneity effects. The 15N decoupling during the constant-time evolution period is accomplished using a WALTZ-16 or DIPSI-2 decoupling scheme, and 13 C decoupling during the t2 1H data acquisition period is accomplished with a GARP-1 or other decoupling scheme. The phase cycle is 1 ¼ x, x; 2 ¼ 8(x), 8(x); 3 ¼ 2(x), 2(y), 2(x), 2(y); 4 ¼ 16(y), 16(y); and receiver ¼ 2(x, x, x, x), 2(x, x, x, x); if required, this 32-step phase cycle can be reduced by a factor of two by eliminating cycling of 4. Frequency discrimination is obtained by TPPI, States, or TPPI–States phase cycling of 1.
580
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
pulse has no effect on the product operator analysis of the pulse sequence and serves to refocus aliphatic 13C evolution caused by the off-resonance effects of the first 1808(13CO) pulse. If the second selective 1808(13CO) pulse were omitted, the resulting spectrum would have a frequency-dependent phase error in the F1 dimension (Section 3.4.1). The 13C–13CO scalar coupling interaction is not active during the 2 INEPT delays, so the aliphatic 13C pulses may be applied at full power to maximize excitation of the aliphatic spins. The 1808(13CO) pulses can be applied in an alternative sequence: ðTt1 Þ=4 – 180 ð13 COÞ – ðTt1 Þ=4 – 180 ð13 CÞ – ðT þ t1 Þ=4 – 180 ð13 COÞ – ðT þ t1 Þ=4:
½7:73
In this case, the 13C operators of interest are in-phase with respect to the 13C –13CO scalar coupling when the 1808(13C) pulse is applied. Consequently, this pulse can be applied at full power to obtain optimal inversion of aliphatic 1808(13C) spins. The uniformity of one-bond aliphatic 13C–13C couplings (JCC ranges from 32 to 40 Hz) facilitates optimization of the length of T to maximize the cosn(JCCT) factor in [7.72]. The experiment also can provide information on the number of aliphatic carbons attached to a given 13 C nucleus. As can be seen from [7.72], if T ¼ 1=JCC , the sign of the 13 C magnetization will be opposite for carbons coupled to an odd, relative to an even, number of other aliphatic carbons. If, on the other hand, T ¼ 2=JCC , all cross-peaks will have the same sign. The resolution enhancement and spectral editing features of the CT-HSQC experiment are illustrated in Fig. 7.14, which compares 1 H–13C HSQC spectra of uniformly 15N/13C-labeled ubiquitin acquired using the conventional HSQC sequence of Fig. 7.1b with spectra acquired using the CT-HSQC scheme of Fig. 7.13 and constant-time evolution periods, T, of 27 and 54 ms. Expansions of the 1H–13C regions of the spectra are shown in Fig. 7.15. The conventional HSQC spectrum (Fig. 7.14a) and the CT-HSQC spectrum acquired with T ¼ 54 ms (Fig. 7.14c) were recorded with identical digital resolution in the t1 dimension (480 complex t1 data points were acquired with t1max ¼ 52.8 ms), while the CT-HSQC spectrum acquired using T ¼ 27 ms (Fig. 7.14b) was recorded with one-half the digital resolution of the other two experiments (240 complex t1 data points were acquired with t1max ¼ 26.4 ms). Each spectrum was zero-filled to give a final F1 digital resolution of 8.9 Hz. The resolution obtained using T ¼ 2=JCC 54 ms is clearly greater than that obtained using T ¼ 1=JCC 27 ms. However, 13C transverse
581
7.2 HETERONUCLEAR-ED'TED NMR SPECTROSCOPY
a
c
b
... ':"t :
.
.;;~: ., .:.. . . .. ,. .
0
C\J
$
.
. ....
t~\'I"" ~, ' I " t.:
'''.1'
,t.
~"::~" DP........
,'''.
. 0 ~.
. ·t;.:~:
I
. , ,'I "'~"I
,1,1
,".
,'. 't 6
4
...
:..:.;.:~/
I
I 1. ;"1'• ~,:11
Itl
. o
<0
:0 ,..
2
o
6
4
2
o
0
4
2
o
F2 (ppm)
F'GURE 7.14 Comparison of the IH_ 13 C HSQC spectra of uniformly 'SN/13C-labeled ubiquitin recorded using the conventional HSQC experiment (a), and the CT-HSQC experiment with T= 27 ms (b) and T= 54 ms (c). The CT-HSQC spectra were acquired with the initial sampling delay, 1,(0), equal to zero, so that no phase correction was required in the F, dimension. Negative cross-peaks in panel b, corresponding to 13C nuclei that are coupled to zero or two other aliphatic carbons, are plotted with a single level only. Expansions of the 'H"_ 13 C" regions of these spectra are shown in Fig. 7.15.
relaxation during the constant-time period attenuates the observable signal by a factor of exp( -R2T). For proteins larger than ubiquitin, this attenuation can be significant, even for T= 27 ms. To obtain the maximum 13C resolution, the maximum number of t, increments must be acquired. In 3D or 4D experiments, the number of increments in the indirectly detected dimensions is limited by the time available for total acquisition of the spectrum and 13C constant-time evolution periods are limited practically to 27 ms.
7.2 Heteronuclear-Edited NMR Spectroscopy Heteronuclear-edited NMR experiments represent the simplest use of heteronuclear spins to facilitate NMR spectroscopy of larger proteins (1, 2). Three- and four-dimensional heteronuclear-edited NMR experiments resolve cross-peaks between 'H spins according to the chemical shift of the heteronuclei honded directly to the IH spins.
582
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
a
C
o~oo
e• • 0
8 0
,
0
0
.l@iiO~,*
o '. , ~ fl.
, ,
II "
0
i' ~ ,0
.
... ..... 0.- •• 0 0:.
I
,p
I
..
0 0 nO
, •
-: -. - --...;:-:.-
o~.
~
.~
=
•
..
.0: .0. •
,
·0
.-
• •0
0
_0
.-.--
-:- .-.. -: .-...
. --
•
...
...
:
• 0
<0 <0
o
I'
:
o o
C\l
r--
6
5
4
6
5
4
6
5
4
F2 (ppm)
FIGURE 7.15 Expansions of the IH"'_13C'" regions of the three spectra shown in Fig. 7.14.
A 3D heteronuclear-edited experiment consists of a homonuclear pulse sequence, usually a NOESY (Section 6.6.1) or TOCSY (Section 6.5) experiment, and an HSQC (or HMQC) pulse sequence (Section 7.1) catenated as discussed in Section 4.5 (47). A 4D heteronuclear-edited NOESY experiment consists of the catenation of a homo nuclear NOESY pulse sequence and two HSQC (or HMQC) building blocks.
7.2.1 3D NOESY-HSQC SPECTROSCOPY The basic I-S (1= IH, and S= 13C or 15N) NOESY-HSQC experiment is illustrated in Fig. 7.16 (48-50). Up until time a, the sequence is a homonuclear NOESY experiment (Section 6.6.1) with S spin decoupling during the tl evolution period. Decoupling of the lIS coupling interaction is achieved by application of a 180 (S) pulse at the midpoint of tl (illustrated), or by application of a composite decoupling pulse scheme throughout ti. The 90°(1) pulse immediately preceding time a is equivalent to the first 90°(1) pulse in the HSQC experiment, and the remainder of the pulse sequence following time a is identical to the HSQC experiment (Fig. 7.1 b). The product operator treatment of the NOESY-HSQC experiment is obtained by catenating the results for the NOESY and HSQC experiments. For example, for two 'H spins I and K and a heteronuclear S spin 0
583
7.2 HETERONUCLEAR-EDITED NMR SPECTROSCOPY
that is scalar coupled only to the I spin, evolution through the pulse sequence is given by 2 ðIx þKx Þt1 =2Sx t1 =2 2 ðIx þKx Þ
Iz þ Kz ! Iz cosðI t1 Þ
Y
þ Kz cosðK t1 Þ m
cos JI t1 Y
cosðJK t1 Þ
(
! Iz aII ð m Þ cosðI t1 Þ
Y
cos JI t1
þ aIK ð m Þ cosðK t1 Þ
Y
) cosðJK t1 Þ
2 ðIx þKx ÞðIx þKx Þ, Sx 2 ðIy þKy Þ, 2 Sx ! (
2Iz Sy aII ð m Þ cosðI t1 Þ
Y
cos JI t1
þ aIK ð m Þ cosðK t1 Þ
Y
cosðJK t1 Þ
(
t2 =2ðIx þKx Þt2 =2
)
! 2Iz Sy cosðS t2 Þ aII ð m Þ cosðI t1 Þ
Y
þ aIK ð m Þ cosðK t1 Þ 2 ðIx þKx Þ, 2 Sx ðIx þKx Þ,Sx
Y
(
cos JI t1 )
cosðJK t1 Þ
! Ix cosðS t2 Þ aII ð m Þ cosðI t1 Þ
Y
) Y cos JI t1 þ aIK ð m Þ cosðK t1 Þ cosðJK t1 Þ ,
½7:74 in which 2 ¼ 1/(2JIS), aII( m) and aIK( m) are transfer functions for dipolar cross-relaxation (Sections 5.1.2 and 5.4.1), homonuclear scalar coupling interactions of the I and K spins with other 1H spins are represented by terms containing JI and JK , and only terms leading to observable signals are derived. Magnetization originating on the K spin that is not transferred to the I spin during the NOESY mixing
584
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
period is suppressed by the HSQC isotope filter. If the K spins were coupled to S spins, then analogous detectable Kx coherence terms would appear in [7.74]. The complete magnetization transfer pathway can be abbreviated as follows: 1
NOE
1
JIS
JIS
Ki ! Ij ! Sj ! Ij : ðt1 Þ ðt2 Þ ðt3 Þ
½7:75
Coherence transfer steps are indicated above the arrows, and the locations of the independent t1, t2, and t3 evolution periods are indicated on the second line. In order to obtain sufficient digital resolution in the 3D experiment, a large number of (t1, t2) experiments are required (typically 128 32 complex data points are acquired in t1 and t2, respectively), therefore only a limited number of transients can be recorded for every (t1, t2) value if the entire experiment is to be completed in a reasonable time period. As a consequence, a short phase cycle must be utilized and incorporation of pulsed field gradients for artifact suppression is a powerful approach for reducing the length of the requisite phase cycle (Section 7.2.4.3). Using the eight-step phase cycle illustrated in Fig. 7.16, a 3D spectrum can be acquired in 3–4 days. A shorter phase cycle can be obtained by eliminating the axial peak suppression phase cycle, 1.
f2
f1 I
t1
t1
2
2
y tm
t
t
t
t
t3
a f3 S
f3 t2
t2
2
2
decouple
FIGURE 7.16 Pulse sequence for the 3D 1H–15N NOESY–HSQC experiment. Thin bars represent 908 pulses and thick bars represent 1808 pulses. Pulses are applied with x-phase unless the phase is indicated above the bar. The nominal value for 2 ¼ 1/(2JIS) and m is the mixing time. The phase cycling is 1 ¼ 2(x), 2(x); 2 ¼ 4(x), 4(x); 3 ¼ x, x; and receiver ¼ x, x, x, x, x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1 and 3, respectively, and the receiver phase, in a TPPI, States, or TPPI–States manner.
7.2 HETERONUCLEAR-EDITED NMR SPECTROSCOPY
585
Quadrature detection in the t1 and t2 dimensions is achieved by shifting the phases of 1 and 3 independently in a TPPI or TPPI–States manner. Processing NOESY–HSQC spectra follows the general methods outlined in Section 6.6.1.3 for processing NOESY spectra and Section 7.1.2.4 for processing HSQC spectra, apart from the more limited resolution of the F1 and F2 dimensions. In the final 3D spectrum, the F1(1H)–F3(1H) projection corresponds to the F1(1H)–F2(1H) region of a conventional 2D 1H–1H NOESY spectrum, and the F2(S)–F3(1H) projection corresponds to the F1(S)–F2(1H) dimensions of a 2D HSQC (or HMQC) correlation spectrum, as illustrated schematically in Fig. 7.17. The NOESY–HSQC pulse sequence illustrated in Fig. 7.16 can be simplified by replacing the HSQC sequence with the HMQC sequence. From the previous discussion of the HSQC and HMQC experiments, such simplification may appear to be undesirable due to the superior resolution and relaxation properties of the HSQC experiment. However, the resolution in the heteronuclear dimension of a 3D experiment is typically limited by the digital resolution, so the detrimental effect of using the HMQC sequence is not as great in the 3D as in the 2D case. In addition, the simple HSQC sequence can be replaced with any of the variant heteronuclear correlation pulse sequences (decoupled HSQC, constant-time HSQC, PEP-HSQC, TROSY) as desired. 7.2.1.1 3D 1H–15N NOESY–HSQC Presaturation of the solvent resonance normally is avoided in 1H–15N NOESY–HSQC NMR spectroscopy of 15N-labeled proteins. Instead, water suppression is achieved by incorporating spin lock purge pulses or field gradient pulses into the pulse sequence, as discussed in Section 7.1.4. The use of these techniques avoids saturation of the 1H spins and allows observation of important 1HN–1H cross-peaks. The advantage of not presaturating the water resonance in the 1 H–15N NOESY–HSQC experiment is illustrated clearly by Fig. 7.18, which shows selected regions of F2(15N) slices of a 3D NOESY–HSQC spectrum of ubiquitin compared with the equivalent regions from a 2D homonuclear NOESY spectrum. Several 1H–1HN cross-peaks can be seen close to the water resonance position in the NOESY–HSQC spectrum (Fig. 7.18, a and b), but are not observable in the 2D homonuclear NOESY spectrum acquired using presaturation of the water resonance (Fig. 7.18c). These cross-peaks would be difficult to observe if presaturation of the water resonance had been utilized in the 3D NOESY–HSQC experiment.
586
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
3D
F2 (S)
2D
F1 (1H) F1 (S) F3 (1HS)
F1 (1H)
F2 (1HS)
F2 (1H)
2D
FIGURE 7.17 A schematic illustration showing the relationship between a 3D heteronuclear-edited spectrum and 2D homonuclear and heteronuclear correlation spectra. Cross-peaks representing three different spin systems, with degenerate 1HS chemical shifts, but differing S spin chemical shifts, are indicated. The 3D spectrum is represented as a series of F1(1H)–F3(1HS) slices edited by the chemical shift of the directly attached S spins along the F2 axis.
The phase cycle of 1 necessary for coherence selection and for frequency discrimination in F1 results in a different state of the water magnetization for different values of 1. When 1 ¼ x, the water magnetization is given by Wz at the start of the mixing period. When 1 ¼ x, the state of the water magnetization is Wz. When the phase of 1 is shifted to y for quadrature detection, the state of the water magnetization is described by Wx . The extent of radiation damping during m is very different for these different operators; thus, the quality
587
7.2 HETERONUCLEAR-EDITED NMR SPECTROSCOPY
a
H68
V26 t
)o
V170
161
E24
G53~
L67 E180
1
1123
b
• E24
T'' l R5,,~60 V51 I,d'30
K27
t.I." N25
'130
Kll@
)122
22
~'TI40
.
H68
'155
F2
= 118.9 ppm
d
.(p4J o
K27
E a.
F"O F2=121.1 ppm
e
.e, u:::
, o'
A28:'ih V26(f) •
N60@ 16
H68
t~~
F2
9.0
8.5
= 118.9 ppm
8.0
7.5
F2 = 121.1 ppm
9.0
8.5
8.0
7.5
9.0
8.5
B.O
7.5
F3 (ppm)
FIGURE 7.18 Selected F 1CH")-F3CH N) regions from a 3D IH_ 15 N NOESYHSQC spectrum of 15N-labeled ubiquitin at Fl 5N) chemical shifts of 118.9 ppm (a) and 121.1 ppm (b), and the corresponding region from a 2D homonuclear NOESY spectrum (c) acquired using presaturation of the H 2 0 signal. Plots d and e correspond to F1CHN)-FlH N ) regions at the same F 2( 15 N) chemical shifts as plots a and b, respectively, and plot f corresponds to the equivalent region from the 2D homonuclear NOESY spectrum. Intraresidue NOEs are indicated by a box while interresidue NOEs are indicated by ellipses.
of the water suppression will be very different for the different transients. This undesirable situation is alleviated by shifting the phases of ¢I by 45° (51). In this case, the water magnetization at the start of the mixing period is oriented at either 45° or 135° with respect to the z-axis and radiation damping effects are more similar for different transients. By the same token, any pulsed field gradients applied during T m should be placed at the end of mixing period to allow maximal radiation damping to occur. While all IH spins are frequency labeled during t), only magnetization from those spins directly attached to 15N is retained for detection during t3, and therefore the only cross-peaks observed in the spectrum involve cross-relaxation to amide IH spins. Thus, the F 1(IH)-F3 eH)
588
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
projection of the 3D spectrum corresponds to the F1(1H)–F2(1HN) region of a conventional 2D 1H–1H NOESY spectrum. 7.2.1.2 3D 1H–13C NOESY–HSQC The 1H–13C NOESY–HSQC experiment, which typically is acquired in D2O solution, provides NOE correlations between aliphatic 1H spins (and between aliphatic and aromatic 1H spins). In a 13C-edited NOESY–HSQC spectrum the F1(1H)–F3(1H) projection corresponds to the F1(1H)–F2(1Haliphatic) region of a conventional 2D 1H–1H NOESY spectrum acquired in D2O solution, and the F2(13C)–F3(1H) projection corresponds to the 2D HSQC (or HMQC) correlation spectrum [F1(13C)–F2(1H)] (52). For uniformly 13C-labeled proteins, the NOESY–HSQC experiment illustrated in Fig. 7.16 is modified to include 13CO decoupling during the t2 evolution period by using a selective composite pulse decoupling scheme such as SEDUCE-1 (52). Aliphatic 13C decoupling during the 1H t1 evolution period can be accomplished by using a composite pulse decoupling scheme or, more simply, a composite 1808(13C) pulse. If the 13C-edited NOESY–HSQC experiment utilizes a conventional HSQC sequence (Figs. 7.1b and 7.16), then the maximum t2 acquisition time must be kept shorter than 1/(2JCC) in order to avoid sensitivity losses due to resolution of the JCC couplings in the F2 dimension, unless homonuclear decoupling is used (23). This constraint usually is not limiting in a 3D experiment. For example, on an 800-MHz spectrometer, if 48 complex points are to be acquired in the t2 dimension with an F2 spectral width of 30–35 ppm, then the value of t2max will range from 8.0 to 6.9 ms, compared with 1/(2JCC) ¼ 14 ms. Of course, the CT-HSQC sequence (Fig. 7.13) also can be incorporated into the 1H–13C NOESY–HSQC experiment in order to completely eliminate line broadening due to JCC scalar coupling (Section 7.1.5). Assignment of NOE cross-peaks in a 3D 1H–13C NOESY–HSQC spectrum is greatly aided by the symmetry present in the spectrum. For two proximal spins, H1 and H2, NOE cross-peaks are expected at [F1(1H1), F2(13C2), F3(1H2)] and at [F1(1H2), F2(13C1), F3(1H1)]. By searching the 3D spectrum for such symmetry-related peaks, the 13C chemical shifts associated with both 1H spins involved in the NOE interaction can therefore be identified. Knowledge of all four chemical shifts can potentially lead to an unambiguous NOE assignment. An example in which identification of the symmetry-related peak aids NOE assignment is included in Fig. 7.19, which shows selected regions from F1(1H)–F3(1H) slices of a 3D 1H–13C NOESY–HSQC spectrum of 13C-labeled ubiquitin, together with the equivalent region from
7.2 HETERONUCLEAR-EDITED NMR SPECTROSCOPY
589
b
T
F4jJ' F4jJ"
P388
jJ
I
0
.~.
F2 = 62.3 ppm
5
4
F3 (ppm)
4
F3 (ppm)
5
4 F2 (ppm)
FIGURE 7.19 Selected F,eH)-FlH a ) regions from a 3D 'H_ 13 C NOESYHSQC spectrum of 13C-labeled ubiquitin in D 2 0 solution at F2C'3C) chemical shifts of 54.3 ppm (a) and 30.2/62.3 ppm (b), and the corresponding region from a 2D homonuclear NOESY spectrum (c) acquired using an unlabeled sample of ubiquitin in D 2 0 solution. Intraresidue NOEs are indicated by a box while interresidue NOEs are indicated by ellipses. The Lys6eH a )Thr l2(IHa ) cross-peak discussed in the text is located in the lower left region of each spectrum. a homonuclear 'H-IH NOESY spectrum of ubiquitin. The cross-peak between Lys6(lH") and Thrl2('H") is ambiguous in the homonuclear NOESY spectrum due to degeneracy of the IH" chemical shifts of Lys6 and Thr66; the presence of this cross-peak in Fig. 7.l9a at the F3 H) chemical shift of Lys6, and its absence in Fig. 7.19b at the FiH) chemical shift of Thr66, clearly support the assignment indicated.
e
7.2.2 3D TOCSY-HSQC SPECTROSCOPY The description of the IH_ 15N NOESY-HSQC experiment given in the preceding section also applies to the I H _15N TOCSY-HSQC experiment, with the exception that the NOESY mixing period is replaced by a TOCSY isotropic mixing sequence (1, 49, 50). The information obtained from a 3D IH_ 15 N TOCSY-HSQC spectrum is the same as that obtained from the F 1('H)-F2('H N ) region of a conventional 2D IH- IH TOCSY spectrum, but, as illustrated schematically in Fig. 7.17, it is edited in a third dimension accotding to the 15N chemical shift associated with the amide IH N . In addition to providing intraresidue
590
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS f2
f1
I
t1 2
t1 2
y
tm isotropic mixing
t
t
t
t
t3
a f3
f3 t2 2
S
t2 2
decouple
FIGURE 7.20 Pulse sequence for the 3D 1H–15N TOCSY–HSQC experiment. DIPSI-2rc isotropic mixing is used during m. Other experimental details are as given in Fig. 7.16.
correlations that are important for the sequential assignment process, the 1H–15N TOCSY–HSQC experiment may be used to obtain qualitative estimates of 3 JH H coupling constants from the relative intensities of well-resolved 1HN–1H cross-peaks in spectra acquired with short mixing times (535 ms) (53). A pulse sequence for an 1H–15N TOCSY–HSQC experiment is shown in Fig. 7.20. Following the initial 1H t1 evolution period, a 908(1H) pulse returns the frequency-labeled magnetization to the z-axis for the isotropic mixing period. The DIPSI-2rc isotropic mixing sequence (54) transfers 1H magnetization from aliphatic spins to the corresponding intraresidue amide 1H spins, while minimizing rotatingframe NOE effects. The 908(1H) pulse following the mixing sequence rotates the resulting z-magnetization back into the transverse plane, and is therefore analogous to the first pulse in an HSQC experiment, as already discussed for the 1H–15N NOESY–HSQC experiment. The remainder of the sequence is equivalent to an 1H–15N HSQC experiment. The complete magnetization transfer pathway is TOCSY
Ii ! ðt1 Þ
1J IS
1J IS
Ij ! Sj ! Ij : ðt2 Þ ðt3 Þ
½7:76
Sample F1(1H)–F3(1H) planes from a TOCSY–HSQC spectrum of ubiquitin are compared with the equivalent region of a homonuclear TOCSY spectrum in Fig. 7.21. An analogous 1H–13C TOCSY–HSQC experiment, for use with protein D2O solutions, might also be designed. However, a significantly more sensitive experiment, which relies on coherence transfer via the large and uniform 13C–13C J couplings, rather than the smaller 1H–1H J couplings, gives the same information.
591
7,2 HETERONUCLEAR-EDITED NMR SPECTROSCOPY o y'
a
y' y"O
•r2~
~.~..
Oy
y' •
P't
•P 'P'
y"l
IP
P" y' y"
P', P"
II
I P" ~t
8 •
,I
OJa
oa .a 130
I a L8
•a
a
K27
a,
R54
aCl E18
~
•
•
.
T14
F2 = 118.9 ppm
8
F3 (ppm)
7
9
0
,:I,t.
o '.
I
I
8
F3 (ppm)
7
r>l~0
0 0 Qo
9
E a. a.
,
". I
~~D9"
I 0
•
F2 =121.1 ppm
,'"
e
~
~
,
IIel
.
"'~
u:
• I
~. Qo
K11
0, 0
I
r 'I a
,a
•
• '0
• a.
C
0 0:0
•
N25
V5
'0'
'I
D
til'
IP
9
• D
0
123
oa
.'
o , 1I.0 ~ I D ;!,
.p op
161
H68
~
IJ·
Pi IP"
.
. ::"1';
b
y'
I
0
•
•• • 8
~ • [
.,.
. l{)
7
F2 (ppm)
FIGURE 7.21 Ubiquitin TOCSY-HSQC spectrum. Selected F,CH)-FJC'HN ) regions from a 3D 'H_ 15 N TOCSY-HSQC spectrum of 15N-labeled ubiquitin at F2( 15N) chemical shifts of 118,9ppm (a) and l21.1ppm (b), and the corresponding region from a 2D homo nuclear TOCSY spectrum (c) acquired using presaturation of the H 2 0 signal. Isotropic mixing times of 64 and 48 ms were used for the 3D TOCSY-HSQC and 2D TOCSY spectra, respectively,
These experiments, the HCCH-TOCSY, and the related HCCH-COSY, are discussed in Section 7.3.
7.2.3 3D HSQC-NOESY AND HSQC-TOCSY EXPERIMENTS NOESY and HSQC (or HMQC) pulse sequences conceivably could be combined in the reverse order to yield an HSQC-NOESY (or HMQC-NOESY) experiment, illustrated in Fig. 7.22 (55). In this experiment, the F, and F3 dimensions are exchanged relative to the NOESY-HSQC experiment; thus, the F,('H)-F2(S) projection corresponds to the 2D HSQC (or HMQC) correlation spectrum, and the F,('H)-FiH) projection corresponds to the F,('H)-F2 ('H) region of a conventional 2D 'H-'H NOESY spectrum. Similarly, in an HSQC-TOCSY (or HMQC-TOCSY) experiment, the F,('H)-F2(S) projection corresponds to the 2D HSQC (or HMQC) correlation spectrum, and the F, (' H)-Fi H) projection corresponds to the F,('H)-F2 ('H) region of a conventional 2D 'H-1H TOCSY spectrum.
592
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS f1
I
f1 t
t1 2
t f3
f2
y
f3
t1 2
tm
t3
f3 t2 2
S
t2 2
t
t
decouple
FIGURE 7.22 HSQC–NOESY pulse sequence. Thin bars represent 908 pulses and thick bars represent 1808 pulses. Pulses are applied with x-phase unless the phase is indicated above the bar. The nominal value for 2 ¼ 1/(2JIS) and m is the mixing time. The phase cycling is 1 ¼ x, x; 2 ¼ x, x, x, x; 3 ¼ 4(x), 4(x); and receiver ¼ x, x, x, x, x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1 and 3, respectively, and the receiver phase, in a TPPI, States, or TPPI–States manner.
The coherence transfer pathway is given by 1
JIS
Ii ! ðt1 Þ
1
JIS
Si ! ðt2 Þ
NOESY=TOCSY
Ii ! Ij : ðt3 Þ
½7:77
For 15N-labeled proteins, these experiments have two advantages compared to the 1H–15N NOESY–HSQC and 1H–15N TOCSY–HSQC experiments (the differences are much less pronounced for 1H–13C heteronuclear-edited experiments). First, the 1H–1H planes in HSQC–NOESY and HSQC–TOCSY experiments have higher digital resolutions than do the corresponding planes in NOESY–HSQC and TOCSY–HSQC experiments for a given experimental time, because the full 1H spectral width is digitized during t3 rather than during t1 and the narrower HN spectral region is digitized in t1 rather than in t3. Second, narrower F1 linewidths result from evolution of heteronuclear multiple-quantum coherence during t1 rather than of 1H single quantum coherence (Section 7.1.1). However, the HSQC–NOESY and HSQC–TOCSY experiments have the disadvantage that a number of 1 N H (F1)–1H(F3) cross-peaks may be obscured by the intense residual water peak, unless very efficient water suppression can be obtained using field gradient pulses (Section 3.7.3). PEP sensitivity enhancement cannot be incorporated into the HSQC–NOESY experiment because the orthogonal magnetization
7.2 HETERONUCLEAR-EDITED NMR SPECTROSCOPY
593
components following t2 cannot be converted simultaneously to longitudinal magnetization during the NOESY mixing period. In contrast, PEP sensitivity enhancement can be incorporated independently into the HSQC and TOCSY portions of the HSQC–TOCSY pulse sequence, which yields a theoretical sensitivity improvement of 2 compared to the conventional HSQC–TOCSY or TOCSY–HSQC experiment (56, 57).
7.2.4 HMQC–NOESY–HMQC EXPERIMENTS Although the 3D heteronuclear-edited NOESY spectra previously discussed offer a vast improvement in resolution relative to 2D homonuclear NOESY spectra, the possibility of ambiguity still remains. One limitation of the 3D heteronuclear-edited NOESY experiments is that NOE correlations cannot be observed between 1H spins with degenerate chemical shifts, because these cross-peaks are coincident with the more intense autocorrelation or ‘‘diagonal’’ peak. Observation of such NOEs, which occur between both aromatic and aliphatic 1H spins in 3D 1H–13C NOESY–HSQC spectra and between amide 1H spins (particularly in proteins with a high helical content) in 1H–15N NOESY–HSQC spectra, is important for both resonance assignment and protein structure determination. Indeed, NOEs between aliphatic 1H spins make up the majority of NOEs observed for proteins, and identification of as many of these NOEs as possible is essential. Additionally, in an 1H–15N NOESY–HSQC experiment, ambiguities related to 1HN chemical shift degeneracy are removed provided that either the 1HN or 15N chemical shifts of a given amide group can be resolved; however, ambiguities associated with overlap in the aliphatic 1 H region remain. Therefore, even if a given amide–aliphatic NOE cross-peak is fully resolved in the 3D 1H–15N NOESY–HSQC spectrum, unambiguous assignment on the basis of the aliphatic 1H chemical shift alone may be impossible. HMQC–NOESY–HMQC NMR spectroscopy provides a solution to these problems (analogous HSQC–NOESY–HSQC experiments are discussed in Section 9.1.8) (58). These experiments are derived conceptually by overlapping HMQC–NOESY and NOESY–HMQC experiments or by catenating two HMQC experiments and a NOESY experiment. As the experiment names indicate, a 15N/15N HMQC–NOESY–HMQC combines 1H–15N HMQC, 1H–1H NOESY, and 1H–15N HMQC sequences; a 13C/15N HMQC–NOESY–HMQC combines 1H–13C HMQC, 1H–1H NOESY, and 1H–15N HMQC sequences; and a 13C/13C HMQC–NOESY–HMQC combines 1H–13C HMQC, 1H–1H NOESY, and 1H–13C HMQC sequences. For the case
594
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
of two 1H spins, I1 and I2, covalently bonded to two heteronuclei, S1 and S2, the first HMQC experiment correlates spins I1 and S1, magnetization is exchanged between I1 and I2 during the NOESY mixing period, and the second HMQC experiment correlates I2 and S2. Thus, each NOE cross-peak can be identified by up to four (two 1H and two heteronuclear) chemical shifts in four independent frequency dimensions in 4D versions of these experiments. In some instances, 3D versions of the HMQC–NOESY–HMQC experiments that correlate S1, S2, and I2 in the three frequency dimensions are satisfactory, provided that the S1 resonances are well-resolved (see Section 7.2.5). 7.2.4.1 3D 15N/15N HMQC–NOESY–HMQC The 3D 1H–15N HMQC–NOESY–HMQC experiment illustrated in Fig. 7.23 is used to detect NOEs between amide 1H spins with degenerate chemical shifts. In this experiment, the heteronuclear chemical shifts are labeled in the F1 and F2 dimensions, and the 1H chemical shift is detected in the F3 dimension (59, 60). The initial part of this experiment, between points a and b, is equivalent to an HMQC sequence (less the t2 acquisition period) (Fig. 7.1a), and generates, at time b, transverse 1HN magnetization that is modulated by the chemical shift of its attached 15N nucleus as a function of t1 [7.2]. The following 908(1H) pulse regenerates 1HN z-magnetization, which is transferred by cross-relaxation to proximal 1H spins during the NOESY mixing time, m. The second HMQC sequence converts any 1HN magnetization present following m into heteronuclear multiple-quantum coherence for indirect detection of the associated 15N f4
t3
tm
1H a
15N
f5
f2
b
f1 2t
t1 2
t1 2
2t
c
f3 2t
t2 2
t2 2
2t
decouple
FIGURE 7.23 Pulse sequence for the 3D 1H–15N HMQC–NOESY–HMQC experiment. Experimental details are similar to those given in Fig. 7.16. The phase cycling is 1 ¼ x, x; 2 ¼ 4(x), 4(x); 3 ¼ 2(x), 2(x); 4 ¼ 8(y), 8(y); 5 ¼ 16(y), 16(y); and receiver ¼ x, x, x, x, x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1, and 3, respectively, and the receiver phase, in a TPPI, States, or TPPI–States manner.
595
7.2 HETERONUCLEAR-EDITED NMR SPECTROSCOPY
chemical shifts during t2 and direct detection of the 1HN frequencies during the acquisition period, t3. The delays 2 ¼ 1/(21JNH). The magnetization transfer pathway is 1
1
JNH
HN i !
15
1
JNH
Ni ! ðt1 Þ
1
NOE
HN i !
1
1
JNH
HN j !
1
15
JNH
HN j : ðt3 Þ
1
Nj ! ðt2 Þ
½7:78
The 32-step phase cycle given in Fig. 7.23 is rather long by 3D standards, but because the digital resolution in the heteronuclear dimensions need not be as great as in 1H dimensions, the total acquisition time can still be limited to 4 days. Typically, the acquired 3D matrix comprises 64(t1) 32(t2) 512(t3) complex data points. The first four steps of the phase cycle select signals that have arisen via 1HN multiple-quantum coherence, while the third step (the first and second set of four transients) eliminates artifacts that arise from single-quantum 1 H magnetization present during m. The phase cycling of the two 1808(1H) pulses simply serves to eliminate artifacts that result from imperfections in these pulses. If desired, phase cycling of these two pulses can be eliminated in order to reduce the overall experiment time, or to increase the digital resolution in the t2 dimension by acquiring more increments with fewer transients per increment. 7.2.4.2 4D 13C/15N HMQC–NOESY–HMQC The pulse sequence for the 4D 13C/15N HMQC–NOESY–HMQC experiment (61) is illustrated in Fig. 7.24. Conceptually, the experiment simply comprises a NOESY mixing period between two HMQC sequences; the first HMQC sequence is tuned to 1H–13C couplings and the second is tuned to 1H–15N couplings. Magnetization is therefore transferred from 13 C-attached aliphatic 1H spins to 15N–attached amide 1H spins via the following pathway: 1
1
JCH
Hi !
13
1
JCH
Ci ! ðt1 Þ
1
NOE
Hi ! ðt2 Þ
1
1
JNH
HN j !
15
1
JNH
Nj ! ðt3 Þ
HN j : ðt4 Þ
1
½7:79
Between times a and b in Fig. 7.24, the basic pulse sequence is similar to a 2D 1H–13C HMQC experiment (Section 7.1.1.1), except that the t2 acquisition time of the 2D experiment has been substituted with an incremental t2 evolution period for indirect detection of the aliphatic 1H chemical shifts. The 1808(13C) decoupling pulse applied in the middle of the t2 evolution period should be applied as a broadband inversion composite pulse, such as 90x 180y 90x , in order to minimize resonance
596
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS f1 t2 2
1H a
13C
b
f2 2t1
t2 2
t1 2
t1 2
t4
tm c
2t1 f3 2t2
15N
13CO
decouple
t3 2
t3 2
2t2
decouple
decouple
FIGURE 7.24 Pulse sequence for the 4D 13C/15N HMQC–NOESY–HMQC experiment. The nominal values for 2 1 ¼ 1/(2JCH) and 2 2 ¼ 1/(2JNH). The 1808(13C) decoupling pulse in the middle of the t2 evolution period is applied as a broadband inversion composite pulse, such as 90x 180y 90x , in order to minimize resonance offset effects. Other experimental details are similar to those given in Fig. 7.16. The phase cycling is 1 ¼ x; 2 ¼ x, x; 3 ¼ 2(x), 2(x); receiver ¼ x, x, x, x. Quadrature detection in the t1, t2, and t3 dimensions is achieved by incrementing independently the phases 2, 1, and 3, respectively, and the receiver phase, in a TPPI, States, or TPPI–States manner.
offset effects. The delays 2 1 ¼ 1/(21JCH). The 908(1H) pulse immediately following the initial 1H–13C HMQC sequence rotates the transverse 1H magnetization to the z-axis. During the subsequent NOESY mixing time, m, magnetization can be transferred to proximal 1H spins via dipolar couplings. The remainder of the pulse sequence, following time c, represents an 1H–15N HMQC sequence, with indirect detection of the 15 N chemical shift during t3, and, finally, detection of 1HN during the acquisition time t4. The delays 2 2 ¼ 1/(21JNH). Decoupling of the 13CO spins during t1 and t3 can be achieved using a suitable low-power composite pulse decoupling scheme, such as SEDUCE-1, as indicated in Fig. 7.24, or by application of selective 1808(13CO) pulses at the midpoint of the evolution periods. Alternatively, 13CO decoupling may be omitted all together, because t1max and t3max always are much less than 1=ð2JC CO Þ and1=ð2JNCO Þ, respectively, and therefore evolution of these couplings do not greatly reduce sensitivity. In a 4D experiment, every effort must be made to maximize the digital resolution, but at the same time keep the total acquisition time
597
7.2 HETERONUCLEAR-EDITED NMR SPECTROSCOPY
within reasonable bounds. The minimum phase cycling requires two steps for each heteronuclear filter, therefore the four-step phase cycle given in Fig. 7.24 is used. The double heteronuclear filtering in this experiment is very efficient at removing artifacts, including the intense ‘‘diagonal peaks’’ corresponding to magnetization that has not been transferred from a 13C-attached 1H spin to an 15N-attached 1H spin during the NOESY mixing time. The initial t1, t2, and t3 sampling delays are adjusted to 1/(2SW) as described in Sections 3.3.2.3 and 7.1.2.1. Alternatively, the initial value of t2 can be set to zero if a compensatory delay equal in duration to the 1808(13C) composite pulse is inserted prior to the first 908(13C) pulse (61). In this case, the 1808(1H) pulse in the middle of t1 refocuses the evolution during the delay and the 1808(13C) composite pulse. Quadrature detection in the t1, t2, and t3 dimensions is achieved by shifting the phases of 1, 2, and 3 independently in a TPPI–States manner. A typical acquisition comprises 8–16(t1) 64(t2) 8–16(t3) 128–256(t4) complex data points. Processing of four-dimensional NMR experiments is discussed in Section 7.2.4.4. Pulsed field gradients may be applied with particular advantage in the 4D 13C/15N HMQC–NOESY–HSQC experiment to suppress artifacts, eliminate the H2O signal, and select for the coherence transfer pathway involving 15N magnetization (62). Gradient coherence selection is coupled with the PEP sensitivity improvement technology discussed earlier (Section 7.1.4.2) to decrease the number of phase cycle steps by a factor of two relative to the nongradient experiment. The shorter phase cycle allows spectra to be recorded with increased resolution for a given total acquisition time. 7.2.4.3 4D 13C/13C HMQC–NOESY–HMQC The pulse sequence for the 4D 13C/13C HMQC–NOESY–HMQC experiment (63, 64) illustrated in Fig. 7.25 consists of a NOESY mixing period between two 1H–13C HMQC sequences. The delays 2 ¼ 1/(21JCH). Magnetization is transferred via the following pathway: 1
1
JCH
Hi !
13
1
JCH
Ci ! ðt1 Þ
1
NOE
Hi ! ðt2 Þ
1
1
JCH
Hj !
13
1
JCH
Cj ! 1 Hj : ðt3 Þ ðt4 Þ
½7:80
Suppression of undesired coherence transfer pathways is more difficult in the 4D 13C/13C HMQC–NOESY–HMQC experiment than in the in the 13C/15N HMQC–NOESY–HMQC experiment (63). In particular, artifact peaks observed along the pseudo-diagonal with F2(1H) ¼ F4(1H) and F1(13C) 6¼ F3(13C) would otherwise render identification of genuine
598
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS f1
f1
t2 2
1H
t2 2
t4
tm f3
f2 13C
2t
13CO
Grad
t1 2
t1 2
2t
2t
decouple g1
t3 2
t3 2
2t
decouple
decouple g1 g2
g3 g4
g4
FIGURE 7.25 Pulse sequence for the 4D 13C/13C HMQC–NOESY–HMQC experiment. The 1808(13C) decoupling pulse in the middle of the t2 evolution period is applied as a broadband inversion composite pulse, such as 90x 180y 90x , in order to minimize resonance offset effects. Gradients are used for artifact suppression (Section 3.6). Other experimental details are similar to those given in Fig. 7.16. The phase cycling is 1 ¼ x; 2 ¼ x, x; 3 ¼ 2(x), 2(x); receiver ¼ x, x, x, x. Quadrature detection in the t1, t2, and t3 dimensions is achieved by incrementing independently the phases 2, 1, and 3, respectively, and the receiver phase, in a TPPI, States, or TPPI–States manner.
NOEs between 1H spins with degenerate chemical shifts extremely difficult, if not impossible, in the 4D 13C/13C HMQC–NOESY–HMQC experiment. The pulse sequence shown in Fig. 7.25 incorporates field gradient pulses designed to eliminate spurious magnetization transfer pathways without requiring lengthy pulse phase cycles (64). The gradient pulses are used only for artifact suppression and are positioned within existing delays in the pulse sequence; consequently, sensitivity of the experiment is not compromised by the use of gradient pulses. The gradient g2 eliminates any transverse coherences present during m and the 908(13C)–g3 sequence element eliminates any heteronuclear two-spin order (2IzSz). Unlike the 4D 13C/15N HMQC–NOESY–HSQC experiment (62), the PEP sensitivity improvement scheme (Section 7.1.3.2) does not compensate completely for the sensitivity loss associated with coherence selection by pulsed field gradients because, in addition to CH groups, CH2 and CH3 moieties must be detected (i.e., 13C spins with n 4 1 attached 1H spins) (30).
7.2 HETERONUCLEAR-EDITED NMR SPECTROSCOPY
599
The four steps of the phase cycle correspond to independent cycling of the 908(13C) pulses at the beginning of the HMQC periods, in a fashion identical to that of the 13C/15N HMQC–NOESY–HMQC experiment already discussed; phase cycling of these pulses provides isotope filtration and eliminates axial peaks in the F1 and F3 dimensions. The phase cycle can be shortened to two steps by eliminating phase cycling of the second 908(13C) pulse. In this case, no isotope filtering is used after the NOESY mixing period; therefore, axial peaks occur at the edges of the spectrum in the F3 dimension. The initial t1, t2, and t3 sampling delays are adjusted to 1/(2SW) as described in Sections 3.3.2.3 and 7.1.2.1. Alternatively, the initial value of t2 can be set to zero if a compensatory delay equal in duration to the 1808(13C) pulse is inserted prior to the first 908(13C) pulse (61). In this case, the 1808(1H) pulse in the middle of t1 refocuses the evolution during the delay and the 1808(13C) pulse. Quadrature detection in the t1, t2, and t3 dimensions is achieved by shifting the phases of 1, 2, and 3 independently in a TPPI–States manner. A typical acquisition is limited to 16(t1) 64(t2) 16(t3) 128–256(t4) complex data points, resulting in an acquisition time of 3.5 days using the two-step phase cycle. Processing of four-dimensional NMR experiments is discussed in Section 7.2.4.4. 7.2.4.4 Processing 4D HMQC–NOESY–HMQC Spectra The acquired digital resolution in the indirectly detected dimensions of a 4D HMQC–NOESY–HMQC spectrum is necessarily low in order to keep the overall measuring time within reasonable limits (less than 7–8 days). In particular, the heteronuclear dimensions are limited to only 8–16 complex points (or slightly more if gradient-enhanced pulse sequences are used). Resolution enhancement of the severely truncated heteronuclear signals (t1 and t3) by either linear prediction or maximum entropy reconstruction (Section 3.3.4) is essential. In maximum-entropy reconstruction, the time-domain data for the 1 H dimensions, t2 and t4, are completely processed first, including apodization, zero-filling, Fourier transformation, and phasing. The imaginary parts of F2 and F4 are discarded following these initial steps. The heteronuclear (t1, t3) planes [for each (t2, t4) pair] are processed by using a two-dimensional maximum-entropy algorithm to directly yield the final 4D spectrum (Section 3.3.4.2). Analogously to maximum-entropy processing, a two-dimensional linear prediction algorithm ideally would be used to increase the resolution in the (t1, t3) planes (65). However, as pointed out by Zhu and Bax, 2D linear prediction requires enormous amounts of computing
600
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
TABLE 7.1 Summary of Steps Used in Processing 4D 13C/15N and Data Setsa
13
C/13C-Edited NOESY
Step
Computation performed
1 2
Fourier transform in t3(13C or 15N) dimension Apodization, zero-filling, Fourier transformation, and phasing in t2(1H) and t4(1H) dimensions Linear prediction of t1(13C) time-domain data Apodization, zero-filling, Fourier transformation, and phasing in t1(13C) dimension Inverse Fourier transformation in t3(13C or 15N) dimension Linear prediction of t3(13C or 15N) time-domain data Apodization, zero-filling, Fourier transformation, and phasing in t3(13C or 15N) dimension
3 4 5 6 7 a
Adapted from Clore et al. (63).
time and is therefore (presently at least) impractical for 4D data sets. Instead, one-dimensional linear prediction routines are used to extend the time-domain data independently in both the t1 and t3 dimensions (63, 66) by using the protocol presented in Table 7.1. A data set containing 8(t1) 64(t2) 8(t3) 128(t4) complex data points typically is processed by the maximum-entropy reconstruction or linear prediction protocols to give a final spectrum comprising 32 128 32 256 real data points.
7.2.5 RELATIVE MERITS OF 3D AND 4D HETERONUCLEAR-EDITED NOESY SPECTROSCOPY The number of cross-peaks observable in these 3D and 4D heteronuclear-edited NOESY spectra is the same as is present in the 2D homonuclear NOESY spectra. Each 1H–1H NOE cross-peak in a 2D NOESY spectrum is separated into a third dimension by the chemical shift of the heteronucleus attached to one 1H spin and, for 4D spectroscopy, into a fourth dimension by the chemical shift of the heteronucleus directly attached to the other 1H spin. Therefore, the increased resolution associated with the extension to three or four dimensions is not accompanied by any increase in the complexity of the spectrum (unlike homonuclear 3D NMR spectroscopy; Section 6.7). In addition, the sensitivity of the 3D and 4D NOESY experiments is relatively high, even
7.3 THE HCCH–COSY
AND
HCCH–TOCSY EXPERIMENTS
601
for larger proteins, because the through-bond coherence transfer steps are highly efficient (the heteronuclear couplings involved, 1 JCH and 1 JNH , are significantly larger than the linewidths). Generally, equivalent information can be obtained from a set of complementary three-dimensional NMR experiments or a single fourdimensional experiment. For example, the information content of the two 3D 1H–15N NOESY–HSQC and 3D 13C/15N HMQC–NOESY– HMQC experiments theoretically is equivalent to that of a single 4D 13 C/15N-edited HMQC–NOESY–HMQC experiment. However, no direct correlation can made between the aliphatic 1H and 13C chemical shifts using the two 3D experiments, so the possibility of ambiguity remains, particularly as the two 3D experiments would be acquired at different times (with possible slight variations in conditions). On the other hand, the two three-dimensional NMR experiments can be acquired with much greater resolutions in the indirect dimensions than can the four-dimensional experiment, which facilitates more accurate determination of resonance frequencies. Assuming complete 1H, 15N, and 13C assignments are available, the 4D 13C/15N-edited HMQC– NOESY–HMQC and 4D 13C/13C-edited HMQC–NOESY–HMQC experiments allow assignment of virtually all observable NOEs, because they eliminate most of the problems associated with resonance overlap.
7.3
13
C^13C Correlations: The HCCH^COSYand HCCH^TOCSY Experiments
The HCCH–COSY (67–69) (1H–13C–13C–1H correlation spectroscopy) and HCCH–TOCSY (70, 71) (1H–13C–13C–1H total correlation spectroscopy) experiments are used in the assignment of aliphatic 1H and 13C resonances of 13C-labeled proteins. These experiments allow dispersion of the 2D 1H–1H COSY or TOCSY spectra into a third (or fourth) 13C frequency dimension by utilizing three magnetization transfer steps: first from an 1H to its directly attached 13C nucleus via the 1JCH coupling ( 140 Hz), then from the 13C to neighboring 13C nuclei via the 1JCC couplings (32–40 Hz), and, finally, from 13C back to the directly attached 1H spins via the 1JCH coupling. For larger proteins the three-step magnetization transfer is significantly more efficient than is transferring 1H magnetization in a single step using the unresolved 1 H–1H J couplings. In the HCCH–COSY experiment, 13C magnetization transfer is achieved by using a 908 13C COSY mixing pulse (in analogous fashion to the 908 1H COSY mixing pulse in the 1H–1H COSY
602
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
experiment discussed in Section 6.2.1.1), so that magnetization is transferred only from a 13C nucleus to its directly bound neighbors; in the HCCH–TOCSY experiment, transfer is achieved by isotropic mixing of the 13C spins, resulting in both direct and multiple-relayed magnetization transfers along the carbon side chain. The amino acid spin system considered in the following sections consists of K (noncarbonyl) carbon spins, Ck for k ¼ 1, . . . , K, and a carbonyl spin, C0 . Carbon spin C1 has M directly bonded 1H spins, H1m for m ¼ 1 to M, and carbon spin Ck has Nk directly bonded 1H spins, Hkn for n ¼ 1 to Nk. The 1H and 13C Larmor frequencies are Hk and Ck , respectively. The one-bond 1H–13C scalar coupling constants are designated as JCH. The 13C–13C scalar coupling constants are designated as JC j C k , and JCk CO ; these interactions can be one bond or multiple bond, depending on the context. Homonuclear 1H–1H scalar coupling interactions are unresolved in larger proteins and are not considered explicitly. For simplicity, the 15N–13C scalar coupling Hamiltonian is not considered. If desired, 15N decoupling can be achieved by applying a composite pulse decoupling sequence during 13C evolution periods. The free-precession Hamiltonian for the C1 spin is given by H1 ¼ C1 C1z þ
M X
2JCH H1mz C1z þ
m¼1
K X
2JC1 Ck C1z Ckz þ 2JC1 CO C1z C0z :
k¼2
½7:81 A similar Hamiltonian can be written for any other Ck. The net evolution through the HCCH–COSY and HCCH–TOCSY experiments is summarized as M X
1 JCH H1m !
1
JCC
1 k JCH
C ! C !
m¼1
ðt1 Þ
1
Nk X n¼1
ðt2 Þ
Hkn :
½7:82
ðt3 Þ
Carbon nuclei C1 and Ck (k ¼ 2) are directly covalently bonded in the HCCH–COSY experiment (this restriction does not apply to the HCCH–TOCSY experiment). As a concrete example, if C1 is the 13 C spin, and C2 is the 13C spin of isoleucine, then K ¼ 5, M ¼1, N2 ¼ 1, C3 is the 13C1 spin, C4 is the 13C2 spin, C5 is the 13C spin, JC1 C2 is a one-bond scalar coupling constant, JC1 Ck for k 4 2 are (negligible) two- or three-bond scalar coupling constants, JC2 C3 and JC2 C4 are one-bond scalar coupling constants, JC2 C5 is a (negligible) two-bond
7.3 THE HCCH–COSY
AND
603
HCCH–TOCSY EXPERIMENTS
scalar coupling constant, JC1 CO is a one-bond scalar coupling constant, and JC 2 CO is a (negligible) two-bond scalar coupling constant.
7.3.1 HCCH–COSY Figure 7.26a illustrates a simple HCCH–COSY pulse sequence (67–69). The basic principles behind this, and other, HCCH-type
a
ψ1
f2
1H
t1
a
f1
f3
f2
f1 t3
ψ2
f6
t
13C
t
f4 t2 2
b
t2 2
d
e
f5
∆+d
d c
f1
∆
d
f5 f
t
t
decouple
d
13CO
T 2
b
ψ1 1H a 13C
f2
t
t1 f1
+
t2 2 f2
f3
f1 t3
t ψ2
f4 t2 4
t2
d1 d2 ∆ + 4
b
e
f5 T t2 – 2 2 f1
c
∆
∆
∆
d
d1 d2
f5
t
f
t decouple
f1
13CO
T
T
FIGURE 7.26 Pulse sequences for the 3D HCCH–COSY (a) and constant-time HCCH–COSY (b) experiments. Thin and thick rectangular bars represent 908 and 1808 pulses, respectively. Rounded bars represent selective 1808 pulses applied to 13CO spins. Pulses are applied with x-phase unless the phase is indicated above the bar. Values for delays are discussed in the text. The phase cycling for both experiments is 1 ¼ x; 2 ¼ x; 1 ¼ 8(x), 8(x); 2 ¼ 4(x), 4(y), 4(x), 4(y); 3 ¼ y, y; 4 ¼ 2(x), 2(y), 2(x), 2(y); 5 ¼ 4(x), 4(x); 6 ¼ 2(x), 2(x); and receiver ¼ x, x, x, x, 2(x, x, x, x), x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1 and 2, respectively, and the receiver phase, in a TPPI–States manner.
604
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
experiments are described here using the product operator formalism. Several shortcomings of this particular pulse sequence are alleviated in a constant-time version of the experiment (Fig. 7.26b). The experiment begins at time a in Fig. 7.26a with longitudinal magnetization of the H1mz spins. At the end of the 1H evolution period, t1, this magnetization is transferred to the attached carbon via an INEPT sequence. The 1808(13C) decoupling pulse in the middle of t1 ensures that the 1H spins effectively are decoupled from 13C spins during this evolution period. A composite pulse (90x –180y –90x ) is used to minimize resonance offset and rf inhomogeneity effects. For 2 ¼ 1=ð2JCH Þ the magnetization at time b is given by b ¼
M X
2H1mz C1y cosðH1 t1 Þ:
½7:83
m¼1
Between points b and c, only the Hamiltonian for the C1 spin need be considered because b commutes with the Hamiltonian for spin C2. The propagator for the pulse sequence is given by ! ! K M X X Ckx expðiH1 t2 =2Þ exp i H1mx U ¼ expðiH1 Þ exp i expðiH1 Þ exp ¼ exp i
M X
k¼1 iC0x
m¼1
!
expðiH1 t2 =2Þ
4JCH H1mz Ckz exp i
m¼1
expðiC1 t2 Þ exp i
K X k¼1
! Cxk
K X
!
2JC1 Ck ð2 þ t2 ÞC1z Ckz
k¼2
exp i
M X
! H1mx
exp iC0x ,
m¼1
½7:84 in which the last line is obtained by applying [2.121]. The positioning of the 1H and 13C 1808 pulses between points b and c results in proton decoupling during t2. The 13C spins are decoupled from 13CO spins during the entire t2 þ 2 delay. The 1808(13C) pulse refocuses chemical shift evolution during the two delays. The magnetization at time c is described by n o c ¼ C1x cos½JC1 C2 ðt2 þ 2 Þ 2C1y C2z sin½JC1 C2 ðt2 þ 2 Þ cosðH1 t1 Þ cosðC1 t2 Þ1 ðt2 þ 2 ÞMM ð2 Þ;
½7:85
7.3 THE HCCH–COSY
AND
605
HCCH–TOCSY EXPERIMENTS
in which n(t) is given by [7.25] (for JIS ¼ JCH), and
j ðtÞ ¼
K Y
cosðJC j C k tÞ
½7:86
k¼3
encapsulates the effect of passive 13C–13C couplings to spin Cj. Only terms that result in observable magnetization have been included in [7.86]. The 908(13C) pulse following time point c transfers the antiphase C1 magnetization into antiphase C2 magnetization in a COSY-like manner, giving, at time d, n o d ¼ C1x cos½JC1 C2 ðt2 þ 2 Þ þ 2C1z C2y sin½JC1 C2 ðt2 þ 2 Þ cosðH1 t1 Þ cosðC1 t2 Þ1 ðt2 þ 2 ÞMM ð2 Þ:
½7:87
During the subsequent time interval, 2 þ 2 , between points d and e, the propagator is
U ¼ expði ½H1 þ H2 Þ exp i
M X
! H1mx
exp i
! H2nx
n¼1
m¼1
expði ½H1 þ H2 Þ exp i
N2 X
K X
! Ckx expði ½H1 þ H2 ½ þ Þ
k¼1
¼ exp i
M X
! 4JCH H1mz C1z
exp i
m¼1
exp i
K X
N2 X
! 4JCH H2nz C2z
n¼1
! 4JC1 Ck ð þ ÞC1z Ckz
k¼3
exp i
K X
! 4JC2 Ck ð þ ÞC2z Ckz
k¼3
exp i4JC1 C2 ð þ ÞC1z C2z ! ! ! N2 K M X X X k 1 2 Cx exp i Hmx exp i Hnx : exp i k¼1
m¼1
n¼1
½7:88
606
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
The magnetization at time e is described by ( M X e ¼ 2H1mz C1y cos½JC1 C2 ðt2 þ 2 Þ m¼1
cos½2JC1 C2 ð þ Þ1 ð2 þ 2ÞM ð2 Þ
N2 X n¼1
2H2nz C2y sin½JC1 C2 ðt2 þ 2 Þ )
sin½2JC1 C2 ð þ Þ2 ð2 þ 2ÞN2 ð2 Þ cosðH1 t1 Þ cosðC1 t2 Þ1 ðt2 þ 2 ÞMM ð2 Þ:
½7:89
The remainder of the experiment is a reverse INEPT sequence. At the start of the detection period, f, the magnetization is described by ( M X 2H1mx cos½JC1 C2 ðt2 þ 2 Þ f ¼ m¼1
cos½2JC1 C2 ð þ Þ1 ð2 þ 2ÞM ð2 Þ N2 X þ 2H2nx sin½JC1 C2 ðt2 þ 2 Þ n¼1
)
sin½2JC1 C2 ð þ Þ2 ð2 þ 2ÞN2 ð2 Þ cosðH1 t1 Þ cosðC1 t2 Þ1 ðt2 þ 2 ÞMM ð2 Þ:
½7:90
For carbons with at least one passive coupling partner, 13C–13C coherence transfer is optimized by setting 2 þ 2 ¼ 1=ð41 JCC Þ. To maximize both M(2 ) and N2 ð2 Þ simultaneously for methine, methylene, and methyl carbons, 2 2.2 ms (Fig. 7.7). The first term in [7.90] represents the autocorrelation or ‘‘diagonal peak’’ and the second term represents the cross-peak resulting from coherence transfer from the H1m protons to the H2n protons by the pathway [7.82]. The principal disadvantage to this pulse sequence is that the efficiency of 13C–13C magnetization transfer between J-coupled carbons depends on t2 via sin½JC1 C2 ðt2 þ 2 Þ1 ðt2 þ 2 Þ. This has three consequences: (i) magnetization transfer is not maximal, (ii) the lineshape in the 13C dimension (F2) is not purely absorptive because the term 2JC1 C2 0:24 radians (148) represents a phase shift, and (iii) the lineshape in F2 is a multiplet with the active JC1 C2 coupling antiphase and passive JC 2 Ck
7.3 THE HCCH–COSY
AND
607
HCCH–TOCSY EXPERIMENTS
couplings in-phase. The antiphase, partially dispersive, character of the lineshape reduces sensitivity and resolution of the spectrum.
7.3.2 CONSTANT-TIME HCCH–COSY The 13C–13C magnetization transfer can be optimized independently of t2, and the multiplet structure in the F2 dimension can be collapsed, by using the constant-time HCCH–COSY experiment shown in Fig. 7.26b (69). The same spin system is considered. The modified 1H evolution period and INEPT sequence between time points a and b in the constant-time HCCH–COSY pulse sequence reduce the number of 1808(13C) pulses from two to one; reducing the number of 1808(13C) pulses reduces artifacts from pulse imperfections (72). Ignoring artifacts and relaxation, the magnetization at time b is identical to the magnetization present at time b of the original HCCH–COSY experiment [7.83]. This modification also could be incorporated into the original pulse sequence (Fig. 7.26a). Between points b and c, the propagator for the pulse sequence is given by ! M X H1mx U ¼ expðiH1 ðT t2 Þ=4Þ exp i m¼1
exp iC0x expðiH1 ðT t2 Þ=4Þ ! K X exp i Ckx expðiH1 ðt2 =4 þ ÞÞ expðiC0x Þ expðiH1 2 Þ k¼1
exp i
M X
! H1mx
expðiH1 ðt2 =4 þ 1 ÞÞ
m¼1
¼ exp i
M X
!
4JCH 2 H1mz C1z
exp i
m¼1
expðiC1 t2 Þ exp i
K X
! Ckx
K X
! 2JC1 Ck TC1z Ckz
k¼2
½7:91
k¼1
for ¼ T/4 ¼ 1 þ 2 and 2 2 ¼ 2.2 ms. Evolution due to 13C–13C scalar coupling interactions occurs during the entire constant-time evolution period, while C1 chemical shift evolution proceeds during t2 only. The selective 1808(CO) pulses remove the effects of one-bond 13C–13CO scalar coupling interaction (selective pulses also can be applied
608
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
simultaneously to the aromatic 13C spins to remove the effects of scalar coupling between the 13C and 13C spins of aromatic residues). The duration of the selective pulses must be minimized in order to maximize the attainable resolution, for a given value of T. The magnetization present at time c is described by n o c ¼ C1x cosðJC1 C2 TÞ þ 2C1y C2z sinðJC1 C2 TÞ cosðH1 t1 Þ cosðC1 t2 Þ1 ðTÞMM ð2 2 Þ:
½7:92
Magnetization transfer through the remainder of the sequence is essentially the same as for the non-constant-time version of the experiment. At the start of the final acquisition period, f, the magnetization is described by ( M X f ¼ H1mx cos2 ðJC1 C2 TÞ1 ðTÞM ð2 2 Þ m¼1
N2 X
)
H2nx sin2 ðJC1 C2 TÞ2 ðTÞN ð2 2 Þ
n¼1
cosðH1 t1 Þ cosðC1 t2 Þ1 ðTÞMM ð2 2 Þ:
½7:93
Magnetization transfer between scalar-coupled carbons is independent of t2 and can be optimized by setting the duration of T. The maximum t2 acquisition time, and therefore digital resolution, is limited to be less than T, while values of T significantly longer than 1=ð4JCC Þ reduce sensitivity due to the passive carbon couplings. In practice, a value of T 7.8 ms gives close to optimal transfer and sufficient digital resolution in the 13C dimension (if the spectral width is chosen to give appropriate aliasing of resonances in this dimension; Section 7.1.2.3). At the same time, purely absorptive, singlet Lorenztian lineshapes are obtained in the F2 dimension. Figure 7.27 shows an example F1(1H)–F3(1H) slice from a constanttime HCCH-COSY spectrum of 13C-labeled ubiquitin. The F2 spectral width is only 32 ppm; therefore, aliasing has occurred and the displayed slice corresponds to two 13C chemical shifts (30.2 and 62.3 ppm). Those resonances that have been aliased in the F2(13C) dimension have phase opposite to the phase of those that have not been aliased.
7.3.3 HCCH–TOCSY The HCCH–TOCSY experiment (70, 71) is similar to the HCCH– COSY experiment, except that the 908(13C) mixing pulse is replaced
7.3 THE HCCH–COSY
Q2
AND
609
HCCH–TOCSY EXPERIMENTS
2
R74 E18 V26
F1 (ppm)
3
M1
I61
I23
4
S57
5
T14 T12 T66
F2 = 30.2, 62.3 ppm 5
4
3
2
1
F3 (ppm)
FIGURE 7.27 A selected F1(1H)–F3(1H) slice from a constant-time HCCH– COSY spectrum of 13C-labeled ubiquitin in D2O solution, acquired using the pulse sequence illustrated in Fig. 7.26b. Negative cross-peaks, which correspond to resonances that have been aliased in the F2(13C) dimension, are plotted with a single level only; these peaks have F2(13C) chemical shifts of 62.3 ppm. The labels indicate the assignment in the F1(1H) and F2(13C) dimensions.
by an isotropic mixing scheme that results in both direct and relayed magnetization transfer along the carbon side chain. The HCCH– TOCSY pulse sequence illustrated in Fig. 7.28 combines features from the two HCCH–COSY pulse sequences (a and b, Fig. 7.26). A version of the HCCH–TOCSY experiment incorporating field gradient pulses for artifact and solvent suppression has been described (73). Up to time b, the HCCH–TOCSY sequence is equivalent to that in Fig. 7.26b, and the magnetization present at time b is described by [7.83]. The following sequence, up to time c, is equivalent to that in Fig. 7.26a, and the relevant magnetization present at time c is given by [7.85]. The short ( 2 ms) spin lock (SL) ‘‘trim-pulse’’ applied along the x-axis
610
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS ψ1
1H
f2
t
a
t1 f1
f2
f3
t3
t ψ2
f4 t2 2
13C
f1
b
d
t2 2
f1
f5 d
f5 SL
d
(isotropic mixing)y
d
d
t
t decouple
c
13CO
FIGURE 7.28 Pulse sequence for the HCCH–TOCSY experiment. Thin and thick rectangular bars represent 908 and 1808 pulses, respectively. Rounded bars represent selective 1808 pulses applied to 13CO spins. Pulses are applied with x-phase unless the phase is indicated above the bar. Values for delays are discussed in the text. The phase cycling for this experiment is 1 ¼ x; 2 ¼ x; 1 ¼ 8(x), 8(x); 2 ¼ 4(x), 4(y), 4(x), 4(y); 3 ¼ y, y; 4 ¼ 2(x), 2(y), 2(x), 2(y); 5 ¼ 4(x), 4(x); and receiver ¼ x, x, x, x, 2(x, x, x, x), x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1 and 2, respectively, and the receiver phase, in a TPPI–States manner.
following time c defocuses the 2C1y Ckz antiphase coherence that is not parallel to the effective field. The subsequent isotropic mixing sequence (Section 4.2.1.2) transfers the in-phase C1x magnetization to its neighbors within the carbon spin system, via the 13C–13C scalar coupling interactions. The remainder of the experiment comprises a refocused reverse INEPT sequence that transfers the in-phase 13C magnetization back to the attached protons for detection. The final magnetization prior to acquisition is given by ( ) M K X N X X 1 k Hmx 1 ð2 ÞM ð2 Þa11 ðm Þ þ Hnx 2 ð2 ÞN ð2 Þa1k ðm Þ d ¼ m¼1
k¼2 n¼1
cosðH1 t1 Þ cosðC1 t2 Þ cos½JC1 C2 ðt2 þ 2 Þ cosð2JC1 C2 Þ 1 ðt2 þ 2 ÞMM ð2 Þ, ½7:94 in which a11( m) and a1k( m) are the isotropic mixing coefficients and m is the mixing time. An efficient broadband isotropic mixing scheme, such as DIPSI or FLOPSY sequences (74–77), must be used for the HCCH–TOCSY
7.3 THE HCCH–COSY
AND
611
HCCH–TOCSY EXPERIMENTS
experiment, because the 13C chemical shifts are dispersed over a wide frequency range; use of FLOPSY requires addition of a z-filter to the pulse sequence (78). The rate of coherence transfer from one carbon to its neighbor depends on the magnitude of the effective scalar coupling during the mixing time. For coupled 13C spins with significantly different chemical shifts (e.g., Thr 13C and 13C ; Ala 13C and 13C), the magnitude of the effective coupling can be reduced substantially and the rate of 13C magnetization transfer during the isotropic mixing period is reduced correspondingly. To determine the optimum 13C isotropic mixing time to be used in an HCCH–TOCSY experiment, rates of the carbon–carbon magnetization transfer must be known for the amino acid spin systems of interest. Calculations for the DIPSI-3 sequence (71, 79) indicate that an isotropic mixing time of 20–35 ms is optimal for the observation of 13 C relayed connectivities. Figure 7.29 shows the calculated net 0.6
a
b
c
d
0.5 0.3
a1k (tm )
0.2 0.0 0.6 0.5 0.3 0.2 0.0 0
10
20
30
40
50
0
10
20
30
40
50
tm (ms)
FIGURE 7.29 Isotropic mixing for 13C spins in isoleucine. Transfer functions for ideal isotropic mixing under the strong coupling Hamiltonian are shown for magnetization transfer for magnetization originating on the (a) 13C, (b) 13C, (c) 13C2, and (d) 13C1 spins. The curves for the destination spins are (—) C, ) C . The one-bond scalar coupling constants were (- - -) C2, (- -) C1, and ( assumed to be 35 Hz. Relaxation and resonance offset effects have been neglected.
612
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
intraresidue 13C–13C magnetization transfer for an isoleucine amino acid spin system as a function of the mixing time during an ideal isotropic sequence. Selected regions from an HCCH–TOCSY spectrum of 13C-labeled ubiquitin, illustrating the assignment of the side chain resonances of Ile23, are shown in Fig. 7.30. Cross-peaks corresponding to 13C, 13C2, and 13C in the F2 dimension have been aliased, and have phase opposite to the cross-peaks corresponding to 13C and 13C1. The low intensity of the 1H–1H1 and 1H2–1H1 correlations in this 22.5-ms mixing time HCCH–TOCSY spectrum is to be expected from Fig. 7.29.
g1''
a
b
g1'
g1''
d
g2
d
b
g1'
g1''
g2
d
a
b
g1'
g1''
g2
d
a
b
g2
d
3.5
3.0
2.5
2.0
1.5
1.0
18.2 F2 (ppm)
a
b
a
g2
9.2
g1'
27.7
g1
b
34.2
g2
a
62.3
d
0.5
F3 (ppm)
FIGURE 7.30 Selected regions from F2(13C) slices of a 22.5-ms mixing-time 3D HCCH–TOCSY spectrum of 13C-labeled ubiquitin, showing correlations originating from the 1H, 1H, 1H1, 1H2, and 1H of Ile23. Isotropic mixing was achieved using a DIPSI-2 sequence with a 7.7-kHz rf field strength.
613
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
7.4 3D Triple-Resonance Experiments Three- and four-dimensional heteronuclear triple-resonance experiments correlate backbone 1HN, 15N, 1H, 13C, and 13CO (and side chain 1H and 13C) spins using one-bond and two-bond scalar coupling interactions. These experiments constitute an alternative to the classical sequential resonance assignment strategy based on observation of characteristic short-range NOEs (Chapter 10). A large number of triple-resonance pulse schemes (7) have been published since the original description of this approach in application to calmodulin (16.7 kDa), by Bax and co-workers in 1990 (80, 81). Using these methods, assignments of 13C/15N-labeled proteins up to 30 kDa can be achieved routinely. Using extensions of these techniques for 2 H/13C/15N-labeled proteins (Section 9.1), backbone, 13C, and side chain methyl (Ile, Leu, Val) assignments have been achieved for malate synthase G (81.4 kDa) (82, 83). The nomenclature established for triple-resonance experiments is more or less systematic. The spins that are frequency labeled during the indirect evolution periods or the acquisition period are listed using HN, N, HA, CA, CO, HB, and CB to represent the 1HN, 15N, 1H, 13C, 13 CO, 1H, and 13C spins. Spins through which coherence is transferred, but not frequency labeled, are given in parentheses. For example, a triple-resonance experiment utilizing the following coherence transfers, 1
HN ! 15 N ! 13 CO ! 13 C ! 13 CO ! 15 N ! 1 HN , ðt1 Þ
ðt2 Þ
ðt3 Þ
½7:95
might be called an (HN)N(CO)CA(CO)(N)HN experiment. However, this unwieldy moniker can be shortened using the following conventions. First, the experiment is a so-called out-and-back pulse sequence in which the initially excited proton spin and the detected proton spin are identical. Omitting the back-transfer steps from the name yields the shorter form, HNN(CO)CA, without introducing ambiguity, because the 13C usually is not the detected spin and the presence of a back-transfer pathway to the 1HN spin thereby is implied. Second, the designation of the 1HN spin is redundant, because the transfer from 1 N H $ 15N is the only available step. Thus, 1HN can be abbreviated as H without confusion to yield the final name HN(CO)CA for this experiment. This abbreviated name describes equally well an experiment
614
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
that rearranges the labeling periods as 1
HN ! 15 N ! 13 CO ! 13 C ! 13 CO ! 15 N ! 1 HN : ðt1 Þ
ðt2 Þ
ðt3 Þ
½7:96
The order in which the frequency labeling is performed is easily determined from the pulse sequence. A four-dimensional version of this experiment that includes a CO evolution period would be designated HNCOCA. Table 7.2 summarizes the correlations that are observed, and the scalar couplings utilized for coherence transfer, in several useful triple-resonance experiments. These experiments offer alternative ways of establishing sequential backbone connectivities, and at least two and often more independent pathways can be found to support a given sequential assignment, without any knowledge of the spin system type. Knowledge of some of the spin systems involved is required to ‘‘align’’ the assignments with the protein amino acid sequence; this information is obtained from the HCCH–COSY and HCCH–TOCSY experiments (Section 7.3), from an 1H–15N TOCSY–HSQC experiment (Section 7.2.2), or from knowledge of 13C and 13C chemical shifts. Assignment strategies using triple resonance experiments are summarized in Chapter 10. The experiments listed in Table 7.2 are discussed in more detail herein to demonstrate the basic principles of triple-resonance NMR spectroscopy. The set of experiments presented is certainly not complete, and new and improved triple-resonance pulse sequences, based on the principles described herein, continue to be published. In the following discussion, the nuclear spins of the ith amino acid residue are designated 1 15 HN N spin, H i and C i for i for the amide H spin, Ni for the amide 1 13 aliphatic H and C spins ( ¼ , , etc.), and C0i for the carbonyl 13C spin. One-bond scalar coupling constants are designated JCH, JNH, JC C ( , ¼ , , , etc.), JC CO , and JNCO (interresidue 15N–13CO scalar coupling). Intraresidue and interresidue 13C–15N scalar coupling constants are designated 1 JC N and 2 JC N , respectively. Aspects of data acquisition and processing common to all triple-resonance NMR experiments are discussed in Section 7.4.6.
7.4.1 A PROTOTYPE TRIPLE-RESONANCE EXPERIMENT: HNCA The HNCA experiment correlates the amide 1HN and 15N chemical shifts with the intraresidue 13C shift, by making use of the relatively small one-bond 15N–13C scalar coupling (7–11 Hz) to establish
Experiment
Correlations observed
HNCA
1 1
HN(CO)CA
1
15 13 HN i Ni Ci 15 13 HN i Ni Ci1
H
1
Niþ1
1
HN iþ1
C C
C
H
H
O
H
H
O
H
C
C
C H
H
C
H
H
O
N
H
C
H
H
C
C
JNH
1
JNC
2
JNC
1
JNH
1
JNCO
1
JC CO
C
H
H
1 H
N
Hi 15 Ni 1 HN i 15
H C
15 13 HN i Ni Ci1
Hi
H
C
H
1
C
N
N
H(CA)NH
J couplingsb
Magnetization transfer
O
H
1
JC H
1
JNC
N
C
C
N
C
C
2
JNC
H
H
O
H
H
O
1
JNH
Ref. 81, 84, 86
84, 89
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
TABLE 7.2 Triple-Resonance Experiments Used for Sequential Resonance Assignmenta
72
(continued)
615
TABLE 7.2 continued Experiment
Correlations observed
HNCO
1
15 13 HN i Ni COi1
Magnetization transfer
J couplingsb
Ref.
13 13
HNCACB
13 13
Ci =13 Ci 15 Ni 1 HN i Ci =13 Ci
15
Niþ1
1
H
HN iþ1
H
H
C
JCH
1
JC C
1
JNC
2
JNC
1
JNH
1
JC C
1
JNC
C
2
JNC
O
1
JNH
H
N
C
C
N
C
C
H
H
O
H
H
O
Ci =13 Ci 15 Ni 1 HN i Ci1 =13 Ci1 15 Ni 1 HN i
C
1
H N H
C C H
H
H C O
N H
C C H
H
98
105
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
CBCANH
a
Only the experiments analyzed in the text are listed. A more extensive tabulation of triple-resonance (and other) NMR experiments is presented elsewhere (7, 157). b1 JNH 91 Hz, 1 JNC 7–11 Hz, 2 JNC 4–9 Hz, 1 JNCO 15 Hz, 1 JC CO 55 Hz, 1 JCH ð1 JC H ,1 JC H Þ 140 Hz, 1 JC C 35 Hz.
617
618
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
correlations between the 15N and 13C spins (81, 84). In addition, this experiment also provides sequential connectivities by transferring coherence from the 15N spins to the 13C of the preceding residue via the interresidue two-bond 15N–13C scalar coupling, which can be as large as 9 Hz. The HNCA experiment utilizes ‘‘out-and-back’’ coherence transfer, in which the 1HN magnetization excited initially in the pulse sequence also is detected during t3. A simple HNCA pulse sequence (81) is illustrated in Fig. 7.31a. Analysis of this pulse sequence using the product operator formalism reveals the basic principles of tripleresonance NMR spectroscopy. Figure 7.31 also includes more sophisticated versions of the same experiment that overcome several shortcomings of the original. The more advanced pulse sequences utilize many of the concepts introduced elsewhere in this text and serve to illustrate the iterative process by which NMR experiments are refined. 7.4.1.1 A Simple HNCA Experiment The design of the HNCA pulse sequence shown in Fig. 7.31a is particularly straightforward and evolution through the sequence can be evaluated by inspection. Magnetization originating on amide 1HN spins is transferred to the directly attached 15N spins via an INEPT sequence. For 2 ¼ 1=ð2JNH Þ, the resulting antiphase 15N magnetization at time a in Fig. 7.31a is represented by a ¼ 2HN iz Niy :
½7:97
The 15N chemical shift evolution proceeds during the subsequent t1 evolution period. Evolution due to scalar coupling interactions between the 15N spin and 1HN, 13C, and 13CO spins is eliminated by 1808 refocusing pulses applied in the middle of the t1 period. The magnetization present at time b is described by b ¼ 2HN iz Niy cosðNi t1 Þ:
½7:98
Omitting the 1808(13CO) pulse from the experiment introduces an additional cos(JNCOt1) factor in [7.98]; fortunately, this does not greatly attenuate the signal because the maximum acquisition time in the 15N dimension (t1max) is typically chosen to be less than 1/(2JNCO). Indeed, the original description of this experiment did not include 13CO decoupling in t1 (81). Following t1 evolution, the 15N magnetization becomes antiphase with respect to the coupled 13C spins during the delay . Evolution due to 1H–15N couplings also proceeds during this delay; therefore,
619
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
a
y 1H
t
t3
t
t
f 1 f1
f
f2 t1 2
15N
a
t1 2 f3
d
d
b
c
t
decouple
f3
13Ca
t2 2
t2 2
d
e
13CO
b
y 1H
t
t3
t
t
f1 f1 15N
f2 t1 2 a
13Ca
T 2 f3
y
y T 2
t1 2 b
T 2 f3
t2 2 c
T 2
t e
decouple
t2 2 d
13CO
FIGURE 7.31 Pulse sequences for the HNCA experiment. Thin and thick rectangular bars represent 908 and 1808 pulses, respectively. Pulses are applied with x-phase unless the phase is indicated above the bar. Short, wide rectangles are selective soft pulses applied at the frequency of the water resonance. Field gradient pulses shown as open bars are used for artifact and water suppression; field gradient pulses shown as filled bars are used for coherence selection. Values for delays are discussed in the text. (a) A non-constant-time HNCA experiment, with phase cycling 1 ¼ x, x; 2 ¼ 4(x), 4(y), 4(x), 4(y); 3 ¼ 2(x), 2(x); and receiver ¼ x, x, x, x, x, x, x, x. (b) A constant-time HNCA experiment with phase cycling 1 ¼ x, x; 2 ¼ 4(x), 4(y), 4(x), 4(y); 3 ¼ 2(x), 2(x); and receiver ¼ x, x, x, x, x, x, x, x. (c) An 1Hdecoupled constant-time HNCA experiment with phase cycling 1 ¼ x, x; 2 ¼ 4(x), 4(y), 4(x), 4(y); 3 ¼ 2(x), 2(x); and receiver ¼ x, x, x, x, x, x, x, x. In sequences ac, quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1 and 3, respectively, and the receiver phase, in a TPPIStates manner. (d) A PFG-PEP-HNCA experiment with phase cycling 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ 4(x), 4(x); 4 ¼ x plus the correction for off-resonance phase shift caused by the 1808(13CO) pulse during t2; 5 ¼ x; and receiver ¼ x, x, x, x for the first FID acquired. The phase 5 ¼ x and the gradient g2 is inverted for the second FID. The 908 pulses flanking decoupling periods are applied with the same field strength as the decoupling sequence. The delay 2 is longer than g5. The data are acquired and processed as described for the PFG-PEP-HSQC experiment (Section 7.1.4.2).
620
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
c
y t
1H
t3 d1
t f1 f1
15N
d1
decouple T 2
t1 2
a
f2
t
t e
T 2
t1 2 b
f3
T 2 f3
13Ca
t2 2
t2 2
c
d
T 2
decouple
13CO
d
–y y
y t
1H
t
y
f1 f1 15N
y
–y
d1
d1
decouplex T 2
f3
T 2 f2
f2 13Cα
t1 2 t2 2
t2 2
t1 2
T 2
T 2
t
t
f5
t3 t
t d2 d2
y decouple
f4
13CO g2 g1
Grad
e t
f1 f1
g3 g4
g4
f5
t
15N
g5 g3
y –y
–y 1H
g1
t f3
y T 2
T 2 f2
f2
13Cα
t2 2
t3 t
t +e
t e f6
y
t2 2
∆
t1 2
t1
t1
∆+
t1 2
z
z
f4 t2
t2
13CO g3
Grad
g1
g1 g2
g2
g6 g4
g4 g5
g5
FIGURE 7.31—Continued (e) A PFG-TROSY-HNCA experiment with phase cycling 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ 4(x), 4(x); 5 ¼ y; 6 ¼ y; and receiver ¼ x, x, x, x for the first FID acquired. The second FID is acquired with 1 ¼ x, x; 5 ¼ y; 6 ¼ y; and receiver ¼ x, x. The gradient g3 also is inverted for the second FID. Depending on the spectrometer, y and y phases may need to be interchanged. The 1808 pulse represented as an open bar leaves the water magnetization unperturbed; 3–9–19, or soft–hard–soft, pulse schemes commonly used for this pulse are described in Section 3.7. The delays ¼ T=2 , " is longer than g6, and ¼ þ "=2. The data are acquired and processed as described for the PFG-TROSY experiment (Section 7.1.4.2).
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
621
the duration of is restricted to be an integral multiple of 1/JNH, so that the 15N magnetization is antiphase with respect to its coupled proton at the end of the delay. Scalar coupling evolution involving the 13CO spins is refocused by the 1808(13CO) pulse in the middle of the t2 evolution period. Ignoring evolution of the 13CO scalar coupling interactions, the relevant components of the density operator present at time c are given by n o N N C ð Þ þ 4H N C ð Þ cosðNi t1 Þ cosðJNH Þ, c ¼ 4HN ix 1 ix 2 iz iz iz ði1Þz ½7:99 in which 1(t) and 2(t) are coherence transfer functions for intraand interresidue scalar connectivities between the 15N and 13C spins, respectively: 1 ðtÞ ¼ sinð1JC N tÞ cosð2JC N tÞ, 2 ðtÞ ¼ cosð1JC N tÞ sinð2JC N tÞ:
½7:100
As shown in Fig. 7.32, these functions have two relatively broad maxima centered around 28 and 35 ms for -helical and -sheet structures, respectively (85). As shown in Fig. 7.33, the total amplitudes of the operators in [7.99], 1( ) cos(JNH ) and 2( ) cos(JNH ), have maxima for equal to 22 ms and 33 ms. The positions of these maxima correspond to 2/JNH and 3/JNH, respectively, and are not significantly affected by the small variations of the 13C–15N coupling constants with local secondary structure. In order to minimize relaxation losses, the value of is set to 22 ms, for which cos(JNH ) 1. The 908 pulses applied to both 1H and 13C spins immediately following point c create multiple-quantum three-spin 1HN–15N–13C coherence represented by n o N d ¼ 4HN N C ð Þ þ 4H N C ð Þ cosðNi t1 Þ cosðJNH Þ: iy ix iy 1 iy ix ði1Þy 2 ½7:101 During the subsequent t2 evolution period, both 1H and 15N chemical shifts are refocused by the 1808 pulses applied to these spins; effectively, the three-spin coherence evolution depends only on the 13C chemical shift. The three-spin coherence does not evolve under the influence of the active scalar couplings between the spins (Section 2.7.5). Application of the 1808(13CO) pulse at the midpoint of the t2 evolution period, however, ensures that evolution due to 13CO scalar coupling is refocused; the
622
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS 0.8 0.6 0.4
Γk (t)
0.2 0 –0.2 –0.4 –0.6 0
10
20
30
40
50
60
t (ms)
FIGURE 7.32 Plots of the HNCA coherence transfer functions, k(t), for (—) k ¼ 1 or (- - -) k ¼ 2 [7.100]. Thin and thick lines represent the results for -helical and -sheet conformations, respectively. Nominal values of 1 JC N and 2 JC N for an -helical conformation are assumed to be 9.6 and 6.4 Hz, respectively, and in a -sheet conformation are assumed to be 10.9 and 8.3 Hz, respectively (85).
small F2 phase errors resulting from the off-resonance effects of the 1808(13CO) pulse are readily corrected when processing the data (Section 3.4.1). The scalar coupling interactions between 13C and 13C spins do, however, remain active during the t2 evolution period. At the end of the t2 period, e, the magnetization is described by n o N t2 Þ1 ð Þ þ 4H Nix C t2 Þ2 ð Þ N C cosð cosð e ¼ 4HN ix C C iy iy iy ði1Þy i i1 cosðNi t1 Þ cosðJC C t2 Þ cosðJNH Þ: ½7:102 Alternatively, the antiphase 15N magnetization present at time c can be transferred into antiphase 13C magnetization by application of 908 pulses to the 15N and 13C spins, rather than to the 1H and 13C spins as described. This alternative method, which is illustrated in the next
623
7.4 3D TRIPLE-RESONANCE EXPERIMENTS 0.8 0.6 0.4
Γk(d)cos(pJNHd)
0.2 0 –0.2 –0.4 –0.6 –0.8 0
10
20
30
40
50
60
d (ms)
FIGURE 7.33 Plots of the HNCA coherence transfer functions, k( ) cos(JNH ) for (—) k ¼ 1 or (- - -) k ¼ 2. The plots were calculated for 1 JC N and 2 JC N , coupling constants of 10 and 7 Hz, respectively, and a JNH coupling constant of 91 Hz.
section for a constant-time version of the HNCA experiment, might be thought to offer a distinct advantage because the signal decay in the t2 dimension is determined by the transverse relaxation rate of singlequantum 13C coherence rather than the three-spin 1HN–15N–13C multiple-quantum coherence. However, as discussed by Grzesiek and Bax (84), the 13C spins remain coupled to the 13C spins during the t2 evolution period [7.102], and the acquisition time in the t2 dimension is therefore kept shorter than 1=ð2JC C Þ, about 8–10 ms in practice. Over this time period, the difference in the intrinsic relaxation rates of transverse 13C magnetization and three-spin 1HN–15N–13C coherence is of little consequence, and the two methods of transferring coherence from 15N to 13C are equivalent. The magnetization present following the t2 evolution period [7.102] is transferred back to observable 1HN magnetization by the pathway reverse of that described previously, with the exception that the t1 evolution period is omitted. The 908 pulses immediately following point e convert the three-spin 1HN–15N–13C coherence back into antiphase 15 N magnetization, which rephases with respect to its coupled 13C spin
624
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
during the delay . The 15N remains antiphase with respect to its coupled amide 1H because, as already discussed, the duration of is set to be an integral multiple of 1=JNH . The final series of pulses represent a reverse INEPT sequence and result in observable 1HN magnetization described by n o 2 2 t2 Þ ð Þ þ cosðC t2 Þ ð Þ f ¼ H N cosð C ix 1 2 i i1 ½7:103 cosðN t1 Þ cosðJC C t2 Þ cosðJNH Þ: A representative F2(13C)–F3(1HN) slice from an HNCA spectrum of N/13C-labeled ubiquitin is shown in Fig. 7.34. The observed 1HN–13C correlations are labeled according to the 13C resonance; in this example, the stronger correlations correspond to the intraresidue connectivities while the weaker correlations correspond to sequential interresidue connectivities. Empirically, average values of 1 JC N 4 2 JC N ; however, the ranges of intra- and interresidue scalar coupling constants observed in proteins overlap, and intensities of the resonance peaks can be affected by differences in relaxation rates. Consequently, identification of the intra- and interresidue connectivities on the basis of resonance intensities in the HNCA experiment is not infallible. 15
G47
G10
50
F1 = 121.6 ppm
R74
55
K48
L73 N25
K11
I44
60
I13
F2 (ppm)
L43
65
T14
V26 9.0
8.5
8.0
7.5
7.0
F3 (ppm)
FIGURE 7.34 A selected F2(13C)–F3(1HN) slice, at an F1(15N) chemical shift of 121.6 ppm, from an HNCA spectrum of 15N/13C-labeled ubiquitin.
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
625
7.4.1.2 The CT-HNCA Experiment The modified version of the HNCA experiment illustrated in Fig. 7.31b includes a ‘‘constant-time’’ period during which evolution of the 15N chemical shift and evolution of the 15N–13C scalar coupling interaction occur simultaneously. In the original HNCA experiment (Fig. 7.31a), the 15N chemical shift, and 15 N–13C scalar coupling evolution periods are independent. The main advantage of the constant-time evolution scheme, as discussed in detail here, is a reduction in relaxation losses and a concomitant improvement in sensitivity. During the first part of the CT-HNCA experiment, magnetization originating on amide 1H spins is transferred to their directly attached 15 N spins via an INEPT sequence, in a fashion identical to that of the original experiment discussed previously. The magnetization present at time a in Fig. 7.31b is therefore represented by [7.97]. Between points a and b, the pulse sequence is similar to the constant-time periods used in the constant-time HSQC (Section 7.1.1.3) and constant-time 1H–13C HSQC (Section 7.1.5) experiments. By similar reasoning, evolution due to the 15N–13C scalar coupling interaction (1 JC N ,2 JC N ) occurs during the entire constant-time evolution period, T, while 15N chemical shift evolution proceeds only during t1. The 15N spins are effectively decoupled from both the 1H and 13CO spins by the application of 1808 pulses to these nuclei; consequently, the 15N magnetization remains antiphase with respect to its coupled 1H spin. The magnetization at time b is described by n o N b ¼ 4HN N C ðT Þ þ 4H N C ðT Þ cosðNi t1 Þ, ½7:104 ix 1 ix 2 iz iz iz ði1Þz in which 1(T) and 2(T) are the coherence transfer functions [7.100]. If relaxation during T is ignored, then intraresidue coherence transfer is optimized by selecting T to maximize 1(T) (Fig. 7.32). The amplitude of the signal is further reduced, however, by a factor of exp(–RT), in which R is the average relaxation rate constant for the HN iz Nix Ciz operator. In practice, T is typically chosen to be between 22 and 28 ms. The magnetization present at the end of the constant-time delay, T, is transferred by the simultaneous 15N and 13C 908 pulses into antiphase 13C magnetization at time c, n o N N C ðT Þ þ 4H N C ðT Þ cosðNi t1 Þ: ½7:105 c ¼ 4HN iz 1 iz 2 iz iy iz ði1Þy The three 1808 pulses applied at the midpoint of the t2 evolution period serve to refocus 1H, 15N, and 13CO scalar coupling evolution. The 13C and 13C spins are scalar coupled during t2; thus, the acquisition time
626
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
in the t2 dimension should be shorter than 1=ð2JC C Þ, or about 8–10 ms in practice. At the end of the t2 evolution period, point d, the relevant operators are n o N d ¼ 4HN iz Niz Ciy cosðCi t2 Þ1 ðT Þ þ 4Hiz Niz Cði1Þy cosðCi1 t2 Þ2 ðT Þ cosðNi t1 Þ cosðJC C t2 Þ:
½7:106
This magnetization is then transferred back into observable amide H magnetization by the pathway reverse of that described previously. At the beginning of the acquisition period, the observable magnetization is described by 1
n o 2 2 t2 Þ ðT Þ þ cosðC t2 Þ ðT Þ e ¼ H N cosð cosðNi t1 Þ cosðJC C t2 Þ: C ix 1 2 i i1 ½7:107 For T ¼ , this expression is identical to that previously derived, [7.103], for the original non-constant-time version of the HNCA experiment. As already mentioned, however, the advantage of the constant-time evolution scheme is found when relaxation effects are considered. In the original HNCA experiment (Fig. 7.31a), signal decays as t1 increases; in the CT-HNCA experiment (Fig. 7.31b), signal does not decay as t1 increases. For T ¼ , the gain in sensitivity that can be obtained from using the constant-time evolution scheme is approximately S ¼ R t1 max 0
t1 max t1 max R2N , ¼ expðt1 R2N Þ dt1 1 exp t1 max R2N
½7:108
in which R2N is the transverse relaxation rate of the antiphase 15N magnetization (2HN iz Niy ) present during the t1 evolution period of the non-constant-time experiment [7.16], and t1max is the maximum duration of the t1 evolution period. As with any constant-time evolution scheme, t1max is limited to be less than (or equal to) the constant-time period, T. For ubiquitin, R2N 9.9 s–1 (calculated as described in Section 7.1.1.2 for c ¼ 4.1 ns), and for t1max ¼ 18.6 ms, [7.108] predicts a sensitivity gain of 10% in the constant-time experiment. Larger gains would be expected for larger proteins with larger relaxation rate constants. 7.4.1.3 The Decoupled CT-HNCA Experiment The sensitivity of the above CT-HNCA experiment (and related pulse sequences) can be
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
627
further increased by the introduction of synchronous broadband 1H decoupling (84, 86), as illustrated in Fig. 7.31c. The magnetization present at time a in Fig. 7.31c is represented by [7.97]. At the end of the first delay, 1 ¼ 1/(2JNH), the 15N magnetization is refocused to be in-phase with respect to its attached proton. Application of synchronous broadband 1H decoupling (WALTZ-16 or DIPSI-2, for instance) prevents the creation of any 15N or 13C quantum states that are antiphase with respect to the attached 1H spin. Contributions to the 15N and 13 C linewidths from 1H longitudinal relaxation are therefore eliminated, because both 15N and 13C transverse magnetization remains in-phase (Section 7.1.3.1). The resulting decrease in the apparent R2N and R2C relaxation rate constants reduces the signal loss caused by 15N and 13C transverse relaxation during delays between the two 1H–15N INEPT sequences. The magnetization at time c is therefore described by c ¼ f2Niz Ciy 1 ðT Þ 2Niz Cði1Þy 2 ðT Þg cosðNi t1 Þ:
½7:109
This expression can be compared directly with [7.101] and [7.105] from the HNCA and CT-HNCA experiments, respectively, which contain N multiple-quantum (4HN iy Nix Ciy ) or antiphase (4Hiz Niz Ciy ) states with 1 respect to H spins. At the end of the t2 evolution period, point d, the relevant operators are d ¼ f2Niz Ciy cosðCi t2 Þ1 ðT Þ 2Niz Cði1Þy cosðCi1 t2 Þ2 ðT Þg cosðNi t1 Þ cosðJC C t2 Þ:
½7:110
This magnetization is then transferred back into observable amide 1H magnetization by the reverse pathway. The magnetization prior to acquisition, point e, is described by [7.107]. 7.4.1.4 The Gradient-Enhanced HNCA Experiment The HNCA and many other triple resonance experiments include an indirect evolution period for amide 15N spins and detect amide proton spins during the acquisition period. These experiments are particularly easy to modify for coherence selection using pulsed field gradients and PEP sensitivity enhancement (Section 7.1.4.2). A pulse sequence for a decoupled PFG-PEP-HNCA experiment is shown in Fig. 7.31d (a nongradient PEP-HNCA could be designed as well). This experiment is very similar to the experiment shown in Fig. 7.31c, with the following exceptions: (i ) 15N frequency labeling is performed during the second period, T, rather than during the first; (ii) the reverse INEPT sequence of Fig. 7.31c
628
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
is replaced by a PEP reverse INEPT sequence; (iii) field gradient pulses are used for coherence selection and quadrature detection in the 15N evolution period; (iv) pulsed field gradient pulses are used to suppress artifacts associated with 1808 pulses; (v) water flip-back solvent suppression is incorporated; and (vi) the off-resonance phase shift caused by the 1808(13CO) pulse during t2 is corrected by adjusting the phase 4. The 90y pulse prior the 1H decoupling sequence rotates the water magnetization from the þz-axis to the þx-axis so that the water magnetization is spin-locked by the decoupling rf field. The 90y pulse returns the water magnetization to the þz-axis at the end of the decoupling period. Acquisition and processing of the PFG-PEP-HNCA experiment are performed as described for the PFG-PEP-HSQC experiment (Section 7.1.4.2). 7.4.1.5 The Gradient-Enhanced TROSY–HNCA Experiment A PFG-TROSY–HNCA experiment is shown in Fig. 7.31e (a nongradient TROSY–HNCA could be designed as well). This experiment is derived from the PFG-PEP-HNCA by omitting the 1H decoupling field and replacing the PEP reverse polarization transfer scheme with the TROSY pulse sequence element. Decoupling of the 1H–13C scalar coupling interaction during t2 is performed with 1808(1H) pulses applied at t2/4 and 3t2/4; the net 08 rotation avoids interconversion of the 1H I and I spin operators, which would destroy the TROSY effect. For 2H-labeled proteins, the 1H decoupling pulses can be omitted, and composite pulse decoupling applied to the 2H spins during t2 when the 13C magnetization is transverse. The sequence incorporates two enhancements compared to the PFG-PEP-HNCA and the originally proposed TROSY– HNCA (87, 88). First, the TROSY element begins with a 908(1H) pulse, but no 15N pulse. Consequently, the refocusing period for the 15N–13C scalar coupling interaction can be extended into the first 2 period of the TROSY element by adding a 1808(13C) pulse. This shortens the overall length of the pulse sequence by 2, with consequent reduction in relaxation losses. Second, the TROSY element ends with a 908(15N) pulse, but no 1H pulse. Consequently, the refocusing gradient g6 can be accommodated by shifting the final 1808(1H) pulse as shown in Fig. 7.31e, rather than by incorporating an additional spin echo element as in Fig. 7.31d. This change reduces the number of 1808 pulses. More importantly, the final 1808 pulse is crafted to leave the water magnetization unperturbed (Section 3.7.3) so that the water magnetization is never inverted during the reverse polarization transfer. Acquisition and processing of the PFG-TROSY–HNCA experiment is performed as described for the PFG-TROSY experiment (Section 7.1.4.2).
7.4.2 A COMPLEMENTARY APPROACH: THE HN(CO)CA EXPERIMENT The HN(CO)CA experiment provides sequential correlations between the amide 1H and 15N chemical shifts of one amino acid residue and the 13C chemical shift of the preceding residue by transferring coherence via the intervening 13CO spin (84, 89). The same sequential information is obtained from the HNCA experiment (Section 7.4.1); however, the HNCA experiment does not always distinguish intraresidue and interresidue connectivities because the 1 JC N and 2 JC N coupling constants can be of comparable magnitude or the intraresidue and interresidue 13C chemical shifts may be coincidentally degenerate. The HN(CO)CA experiment circumvents these problems by providing sequential correlations exclusively. In addition, the sensitivity of the HN(CO)CA experiment is greater than that of the HNCA for larger proteins, because the relay of magnetization via the one-bond JNCO and JC CO scalar coupling interactions is more efficient compared to transfer via the relatively small 2 JC N scalar coupling interaction. The salient features of the CT-HN(CO)CA experiment illustrated in Fig. 7.35 are discussed here. Following the initial INEPT-type transfer of magnetization from the amide 1HN spins to their directly attached 15N spins, the magnetization present at time a in Fig. 7.35 is represented by a ¼ 2Hiz Niy
½7:111
for 2 ¼ 1/(2JNH). During the subsequent constant-time evolution period, 2 1 ¼ 2( 2 þ 3), the propagator is
0 U ¼ exp i 2 C0ði1Þx exp i 2 Nix exp½iHð 1 t1 =2Þ exp iC ði1Þx
X k expðiNix Þ exp½iH 2 exp i Hkx exp½iH 3
X k exp i Ckx exp½iHt1 =2
¼ exp i 2 C 0ði1Þx exp i 2 Nix exp i4
630
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS y
1H
t
t f1
15N
t
t
t1 d 3 2
d2
d1
t1 2
d1
t3
d2
d3
decouple
b
d4 c
13Ca
t e
f2
f1
a
13CO
t
d4 d
f3 t2 2
t2 2
FIGURE 7.35 Pulse sequence for a constant-time HN(CO)CA experiment. Values of delays are described in the text. Other experimental details are similar to those given in Fig. 7.31. The phase cycle is 1 ¼ x, x; 2 ¼ 4(x), 4(y), 4(x), 4(y); 3 ¼ 2(x), 2(x); and receiver ¼ x, x, x, x, x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1 and 3, respectively, and the receiver phase, in a TPPI–States manner.
P k in which exp P i Hkx represents a nonselective proton 1808 pulse k represents a nonselective carbon 1808 pulse (with and exp i Ckx summations over k and implied), only chemical shift and scalar coupling interactions affecting the 15N spin have been included in the Hamiltonian, and the last line is obtained by using [2.121]. Evolution due to 1H–15N coupling occurs only during 2 3, while evolution due to 15 N–13CO coupling occurs during the entire period, 2 1. For 2 3 ¼ 1/(2JNH), the antiphase 15N operator is refocused to become in-phase with respect to the amide 1HN spin at the end of the constant-time period. Chemical shift evolution of the 15N coherence occurs only during t1. The duration of the constant-time evolution period, 2 1, can be adjusted from 1=ð3JNCO Þ to 1=ð2JNCO Þ, independently of the 1H–15N coupling evolution period, 2 3, as required to minimize relaxation losses. Typically, 2 1 22 ms [1/(3JNCO)]. The magnetization present at time b is described by b ¼ 2Niy C0ði1Þz cosðNi t1 Þ sinð2JNCO 1 Þ sinð2JNH 3 Þ:
½7:113
The pair of simultaneous 908 pulses transfer [7.113] into antiphase carbonyl magnetization, given at time c, c ¼ 2Niz C0ði1Þy cosðNi t1 Þ sinð2JNCO 1 Þ sinð2JNH 3 Þ:
½7:114
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
631
The product operator in [7.114] is in-phase with respect to the amide 1 N H spin. If this 13CO magnetization were antiphase, then its transverse relaxation rate (and the relaxation rates of the product operators during t2) would be increased due to contributions from proton longitudinal relaxation, and the signal observed would be correspondingly decreased. This would be the case if the amide 1HN spins were decoupled during the constant-time evolution period, as in the CT-HNCA experiment [7.104]. During the subsequent 4–t2– 4 period, the 13CO spin is correlated with its attached 13C spin in an HMQC-like manner. Ignoring evolution due to couplings between 13C and 15N, which are relatively small in magnitude, and between 13CO and 1H spins, which are not refocused by the 1808 pulses at the midpoint of t2, the coherence present at time d is d ¼ 2Niz C0ði1Þy cosðNi t1 Þ cosðCi1 t2 Þ cosðJC C t2 Þ sin2 ðJC CO 4 Þ sinð2JNCO 1 Þ sinð2JNH 3 Þ:
½7:115
The delay 4 is set from 1=ð3JC CO Þ to 1=ð2JC CO Þ. The 13C and 13C spins remain coupled during t2; therefore, in order to avoid sensitivity losses due to resolved JC C scalar couplings in the F2 dimension, the t2 acquisition time must be shorter than 1=ð2JC C Þ, or about 8–10 ms in practice. The off-resonance effect of the 1808(13CO) pulse in the middle of the t2 evolution period results in a small frequency-dependent phase shift in this dimension, which can be corrected approximately during data processing (Section 3.4.1). Magnetization is transferred back along the same pathway to yield the observable magnetization at time e: e ¼ H N ix cosðNi t1 Þ cosðCi1 t2 Þ cosðJC C t2 Þ
sin2 ðJC CO 4 Þ sin2 ð2JNCO 1 Þ sin2 ð2JNH 3 Þ:
½7:116
As in the HNCA experiment, the sensitivity of the HN(CO)CA experiment can be improved by the use of synchronous broadband proton decoupling after the antiphase term at time a [7.111] has been allowed to refocus and by incorporating PEP sensitivity enhancement. A representative F2(13C)–F3(1HN) slice from an HN(CO)CA spectrum of 15N/13C-labeled ubiquitin is shown in Fig. 7.36. This slice was taken at the same F1(15N) chemical shift as for the F2(13C)–F3(1HN) slice from the HNCA spectrum of ubiquitin shown in Fig. 7.34. By comparison of these spectra, the interresidue correlations in the HNCA
632
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS G10
G47
50
F1 = 121.6 ppm
55
L73 N25
65
60
I13
F2 (ppm)
L43
9.0
8.5
8.0
7.5
7.0
F3 (ppm)
FIGURE 7.36 A selected F2(13C)–F3(1HN) slice, at an F1(15N) chemical shift of 121.6 ppm, from an HN(CO)CA spectrum of 15N/13C-labeled ubiquitin. This slice may be compared directly with the equivalent region from the HNCA spectrum of ubiquitin illustrated in Fig. 7.34.
spectrum can be unambiguously distinguished from the intraresidue correlations, because only the interresidue connectivities are observed in the HN(CO)CA experiment.
7.4.3 A STRAIGHT-THROUGH TRIPLE-RESONANCE EXPERIMENT: H(CA)NH The H(CA)NH experiment (72) correlates 1H chemical shifts with intraresidue amide 1H and 15N chemical shifts, by making use of the one-bond 15N–13C J coupling (7–11 Hz). This experiment also provides sequential connectivities by transferring coherence from the 13C spins to the 15N of the following residue via the interresidue two-bond 15 N–13C J coupling (4–9 Hz). The H(CA)NH experiment differs fundamentally from the HNCA experiment because one-way ‘‘straightthrough’’ or ‘‘out-and-stay’’ transfer of magnetization from 1H to 15N spins is obtained via successive through-bond transfer between the directly coupled 1H–13C, 13C–15N, and 15N–1HN pairs. At each
633
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
transfer step, net magnetization transfer (rather than coherence transfer to antiphase spin states) must be obtained using refocused INEPT sequences. A product operator description of the H(CA)NH experiment illustrated in Fig. 7.37 is given in the following discussion. Beginning with longitudinal 1H magnetization at point a (a ¼ Hiz ), transverse 1H magnetization is frequency labeled during t1 and transferred in an INEPT-like manner to its directly attached 13C, to yield at point b b ¼ 2Hiz Ciy cosðHi t1 Þ
½7:117
for 2 1 ¼ 1/(2JCH) ( 3.5 ms). The identical concatenated t1 evolution period and INEPT magnetization transfer scheme were used in the CT-HCCH–COSY experiment (Fig. 7.26b). Following the simultaneous 908 1H and 13C pulses, the antiphase 13 C magnetization, [7.117], is refocused with respect to the 1H spins during the period 2 1, while dephasing due to 15N–13C one-bond and two-bond scalar couplings, and 13C–13C one-bond scalar coupling occurs during 2 2. For 13C directly coupled to a single proton (i.e., all
f1
a
13Ca
f2
t t1+ 1 2
1H
t1 2 f3
y t2
t1
d1 d2–d1
d2 f4
decouple c
f4
f5 t t d3+ 2 d4–d3 2 2 2 d
13CO
t3
f
f3
b
15N
t2
d4 e
decouple
FIGURE 7.37 Pulse sequence for a constant-time H(CA)NH experiment. Values of delays are described in the text. Other experimental details are similar to those given in Fig. 7.31. The phase cycle is 1 ¼ x, x; 2 ¼ 16(x), 16(x); 3 ¼ 2(x), 2(x); 4 ¼ 4(x), 4(x); 5 ¼ 8(x), 8(y), 8(x), 8(y); and receiver ¼ x, x, x, x, 2(x, x, x, x), x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1 and 4, respectively, and the receiver phase, in a TPPI–States manner.
634
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
amino acids except glycine), the relevant components of the density operator at point c are n o c ¼ 2Ciy Niz 1 ð2 2 Þ 2Ciy Nðiþ1Þz 2 ð2 2 Þ cosðHi t1 Þ sinð2JCH 1 Þ cosð2JC C 2 Þ,
½7:118
in which the coherence transfer functions, 1(t) and 2(t), are given by [7.100]. The optimal value of 2 1 ¼ 1/(2JCH). Intraresidue coherence transfer is optimized by selecting 2 to maximize
ð2 2 Þ ¼ 1 ð2 2 Þ cos 2JC C 2 exp 2R2Ci 2 , ½7:119 in whichR2Ci is the 13C transverse relaxation rate. This function, which is plotted in Fig. 7.38, has a maximum value for 2 ¼ 12 to 13 ms. The 908(13C) and 908(15N) pulses immediately following point c transfer the antiphase 13C coherence into antiphase 15N coherence at point d: d ¼ 2Ciz Niy 1 ð2 2 Þ þ 2Ciz Nðiþ1Þy 2 ð2 2 Þ cosðHi t1 Þ sinð2JCH 1 Þ cosð2JC C 2 Þ:
½7:120
During the subsequent evolution period between points d and e, the propagator is
X X U ¼ exp½iH 4 exp i Nkx exp½iHt2 =2 exp i Ckx
X exp½iHð 4 3 Þ exp i Hkkx exp½iHð 3 þ t2 =2Þ Xn 1 ¼ exp i4JNH 3 HN kz Nkz :exp i4 JC N Ckz Nkz
o Nkz exp iNk t2 Nkz exp i42JC N Cðk1Þz
X X X k exp i Nkx exp i Ckx exp i Hkx , ½7:121 in which only chemical shift and scalar coupling interactions affecting the 15N spins have been included in the Hamiltonian, and the last line is obtained by using [2.121]. The antiphase 15N coherences refocus with respect to the active 15N–13C scalar coupling and dephase due to passive 15 N–13C scalar coupling during 2 4. Thus, 2Ciz Niy refocuses under the intraresidue 1 JC N scalar coupling interaction and dephases under the 2 JC N scalar coupling to the 13C spin of the preceding residue, while
635
7.4 3D TRIPLE-RESONANCE EXPERIMENTS 0.4
0.2
Γa (2d 2)
0
–0.2
–0.4
–0.6
–0.8 0
20
30
40
50
100
2d 2(ms)
FIGURE 7.38 Plots of the H(CA)NH coherence transfer function, (2 2) [7.119], for 1 JC1 N , 2 JC2 N , and 1 JC C coupling constants of 10, 7, and 35 Hz, respectively. The solid line corresponds to the transfer function in the absence of relaxation, while the dashed line includes the relaxation term expð2R2C 2 Þ for 13 C linewidths of 1=ðT2C Þ ¼ 10 Hz.
2Ciz Nðiþ1Þy refocuses under the interresidue scalar coupling interaction and dephases due to the intraresidue coupling. The 15N magnetization also defocuses with respect to the amide proton during the time period 2 3. The relevant coherences present at point e are described by n e ¼ 2HN iz Niy cosðNi t2 Þ1 ð2 2 Þ1 ð2 4 Þ: o 2HN ðiþ1Þz Nðiþ1Þy cosðNiþ1 t2 Þ2 ð2 2 Þ2 ð2 4 Þ cosðHi t1 Þ sinð2JNH 3 Þ sinð2JCH 1 Þ cosð2JC C 2 Þ:
½7:122
Intraresidue coherence transfer through this segment of the experiment is maximized by setting 2 3 ¼ 1/(2JNH) ( 5.4 ms), and by adjusting 2 4 to maximize 1 ð2 4 Þ. This transfer function is identical to that plotted in Fig. 7.32 for the HNCA experiment [7.100]. When 15N transverse relaxation, R2N, is considered, a value of 2 4 of 23 ms is close to optimal.
636
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
The final INEPT step in the pulse sequence transfers the antiphase N magnetization to in-phase 1HN magnetization to yield, at time f, n o N cosð t Þ ð2 Þ ð2 Þ þ H cosð t Þ ð2 Þ ð2 Þ f ¼ H N N 2 1 2 1 4 N 2 2 2 2 4 i iþ1 ix ðiþ1Þx
15
cosðHi t1 Þ sinð2JCH 1 Þ cosð2JC C 2 Þ, ½7:123 for 2 2 ¼ 1/(2JNH). Thus, magnetization that originated as 1H magnetization has been transferred to and is detected as amide 1HN magnetization. A representative F1(1H)–F3(1HN) slice from an H(CA)NH spectrum 15 of N/13C-labeled ubiquitin is shown in Fig. 7.39. The observed 1 N 1 H – H correlations are labeled according to the 1H resonance. In this example, the stronger correlation corresponds to the intraresidue connectivities while the weaker correlation corresponds to sequential interresidue connectivities. Identification of the intra- and interresidue connectivities on the basis of resonance intensities in the H(CA)NH experiment has the same caveats as given for the HNCA experiment (Section 7.4.1).
3
F2 = 121.6 ppm
4 I44
R74 L73
K11 N25
K48
T14
5
I13
F1 (ppm)
V26
L43
9.0
8.5
8.0
7.5
7.0
F3 (ppm) 1
FIGURE 7.39 A selected F1( H)–F3(1HN) slice, at an F2(15N) chemical shift of 121.6 ppm, from an H(CA)NH spectrum of 15N/13C-labeled ubiquitin. Resonances from Gly10 and Gly47 are suppressed for 2 1 ¼ 1/(2JCH).
637
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
7.4.4 BACKBONE CORRELATIONS
WITH THE
13
CO SPINS
The HNCA, HN(CO)CA, and H(CA)NH experiments illustrate many of the principles utilized in triple-resonance NMR spectroscopy. These experiments alone usually are not sufficient to establish complete backbone sequential connectivities, and do not indicate the enormous variety of triple-resonance experiments that have been developed (7). The HNCA and HN(CO)CA provide correlations between 1HN, 15N, and 13C spins. The HNCO and HN(CA)CO experiments provide analogous correlations between the 1HN, 15N, and 13CO spins. These experiments are presented briefly in this section. 7.4.4.1 HNCO The HNCO experiment correlates the amide 1HN and 15N chemical shifts of one amino acid with the 13CO chemical shift of the preceding residue, by using the one-bond 15N–13CO J coupling ( 15 Hz) to establish the sequential correlation (81, 84). The sequential connectivities provided by this experiment are particularly useful when used in conjunction with interresidue connectivities provided by the HN(CA)CO (Section 7.4.4.2) and HCACO (81, 90) experiments. A detailed description of alternative pulse sequences for the HNCO experiment has been given by Muhandiram and Kay (33, 88, 91). The CT-HNCO pulse sequence illustrated in Fig. 7.40 is analogous to the CT-HNCA and CT-HN(CO)CA experiments. The CT-HNCO substitutes an evolution period for the carbonyl spins instead of the HMQC-type magnetization transfer from the 13CO to the 13C spins used in the HN(CO)CA experiment. The important coherences present at times a–e of the pulse sequence are a ¼ 2HN iz Niy ,
½7:124
b ¼ 2Niy C0ði1Þz cosðNi t1 Þ sinðJNCO TÞ sinðJNH TÞ,
½7:125
c ¼ 2Niz C0ði1Þy cosðNi t1 Þ sinðJNCO TÞ sinðJNH TÞ,
½7:126
d ¼ 2Niz C0ði1Þy cosðNi t1 Þ cosðCOi1 t2 Þ sinðJNCO TÞ sinðJNH TÞ, ½7:127 e ¼ H N ix cosðNi t1 Þ cosðCOi1 t2 Þ sinðJNCO TÞ sinðJNH TÞ sinð2JNCO 1 Þ sinð2JNH 3 Þ,
½7:128
638
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS y
1H
t
t f1
15N
t
t1 2 a
13CO
T 2
T 2 f3
t3
e
f2
f1
t
t1 2 b
d1
d2
d3
decouple
f3 t2 2
t2 2
c
d
13Ca
FIGURE 7.40 Pulse sequence for a constant-time HNCO experiment. Values of delays are described in the text. Other experimental details are similar to those given in Fig. 7.31. The phase cycle is 1 ¼ x, x; 2 ¼ 4(x), 4(y), 4(x), 4(y); 3 ¼ 2(x), 2(x); and receiver ¼ x, x, x, x, x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1 and 3, respectively, and the receiver phase, in a TPPI–States manner.
in which 2 ¼ 1/(2JNH), 2 1 ¼ 2( 2þ 3), and 2 3 ¼ 1/(2JNH). Typically, 2 1 1/(3JNCO) (22 ms) instead of 1/(2JNCO) to reduce relaxation losses. The time T must be an odd multiple of 1=ð2JNH Þ [e.g., 5/(2JNH) ¼ 27 ms] to ensure that the 15N magnetization is in-phase with respect to the amide proton spins at the end of the constant-time evolution period. In contrast to the CT-HNCA experiment (Fig. 7.31b), antiphase 15N magnetization present at a rephases with respect to the 1HN spins under the influence of the 15N–1H coupling during the constant-time evolution period. Therefore, evolution of the 2NizC0ði1Þy coherence during t2 does not contain contributions from relaxation of the amide proton spin. The constant-time evolution scheme described here also differs from that described in Section 7.4.2 for the HN(CO)CA experiment, in which the duration of the constant-time period was not limited to be an odd multiple of 1=ð2JNH Þ. Either scheme may be used in each experiment, because if relaxation effects are ignored, the final results are very similar. The sensitivity of this scheme may be further improved by the use of synchronous broadband proton decoupling in a way analogous to that for the decoupled CT-HNCA experiment. 7.4.4.2 HN(CA)CO The HN(CA)CO experiment provides intraresidue correlations between the amide 1H, 15N, and 13CO chemical
639
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
shifts by using the one-bond 15N–13C and 13C–13CO J couplings to transfer coherence (92). In addition, this experiment can also provide sequential connectivities from the 15N spins to the 13CO of the preceding residue via the interresidue two-bond 15N–13C J coupling. When used in conjunction with the HNCO experiment, which gives the sequential correlations only (Section 7.4.4.1), the HN(CA)CO experiment provides a method for sequentially assigning the amide 1H, 15N, and 13CO resonances. A constant-time version of a HN(CA)CO pulse-sequence is illustrated in Fig. 7.41. The important product operator terms at times a–f of the pulse sequence are given by a ¼ 2HN iz Niy ,
½7:129
n o b ¼ 2Niy Ciz 1 ðT Þ þ 2Niy Cði1Þz 2 ðT Þ cosðNi t1 Þ,
½7:130
n o c ¼ 2Niz Ciy 1 ðT Þ 2Niz Cði1Þy 2 ðT Þ cosðNi t1 Þ,
½7:131
n o d ¼ 4Niz Ciz C0iy 1 ðT Þ þ 4Niz Cði1Þz C0ði1Þy 2 ðT Þ cosðNi t1 Þ3 ð2 2 Þ, ½7:132 n e ¼ 4Niz Ciz C0iy cosðCOi t2 Þ1 ðT Þ
o þ 4Niz Cði1Þz C0ði1Þy cosðCOi1 t2 Þ2 ðT Þ cosðNi t1 Þ3 ð2 2 Þ,
½7:133
o n 2 2 2 f ¼ H N ix cosðCOi t2 Þ1 ðT Þ:þ cosðCOi1 t2 Þ2 ðT Þ cosðNi t1 Þ3 ð2 2 Þ, ½7:134 in which ¼ 1=ð2JNH Þ, and the coherence transfer functions, 1(t) and 2(t), are given by [7.100] and 3 ðtÞ ¼ sinðJC CO tÞ cosðJC C tÞ:
½7:135
As with the CT-HNCA experiment, the length of the constant-time evolution period, T, is typically chosen to be between 22 and 28 ms (Section 7.4.1.2). A value of 3.0–3.5 ms is optimal for 2. The CT-HN(CA)CO pulse sequence illustrated in Fig. 7.41 differs slightly from the originally reported sequence (92), in which a refocused INEPT sequence (Section 2.7.7.3) is used to generate in-phase 15N
640
y 1H
t f1
15N
d
t f1
t
t
t3
f2 T 2
t1 2
T 2 f3
13Ca
t1 2 b
T 2 f3
y d2
f4
c
decouple
y d2
d2 f4
T 2
d2
f4 t2 2 d
t2 2 e
FIGURE 7.41 Pulse sequence for a constant-time HN(CA)CO experiment. Values of delays are described in the text. Other experimental details are similar to those given in Fig. 7.31. The phase cycle is 1 ¼ x, x; 2 ¼ 8(x), 8(y), 8(x), 8(y); 3 ¼ 2(x), 2(x); 4 ¼ 4(x), 4(x); and receiver ¼ x, x, x, x, 2(x), x, x, x x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1 and 4, respectively, and the receiver phase, in a TPPI–States manner.
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
a
13CO
d
decouple
641
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
magnetization prior to the constant-time evolution period, and to convert in-phase 15N magnetization back into detectable 1H magnetization immediately prior to acquisition. The experiment discussed here achieves the same magnetization state at times b and e (ignoring relaxation effects), but does so in less total evolution time and with four less 1808 pulses; for these reasons, the sequence presented here is likely to be more sensitive than the originally proposed sequence. The main shortcoming of the HN(CA)CO experiment is the low sensitivity that results from rapid relaxation of the transverse 13C magnetization during the delays 2. For 2H/13C/15N-labeled proteins, TROSY–HN(CA)CO experiments are substantially more sensitive because of the smaller dipolar 2H–13C interaction (88). For 13 C/15N-labeled proteins, the straight-through (HCA)CONH experiment is more sensitive than the HN(CA)CO experiment (93–95).
7.4.5 CORRELATIONS
WITH THE
C/H SPINS
The triple-resonance experiments discussed thus far provide sequential connectivities along the peptide backbone. The HCCH experiments (Section 7.3) provide connectivities of the aliphatic side chains of individual amino acid residues. Complete assignments can be obtained if the backbone assignments and the side chain assignments can be connected using either the 1H or the 13C spins. Congestion in the 1 13 H – C region of the spectrum often renders this approach difficult. One solution to the problem combines HCCH-type magnetization transfer from the side chain to the 13C spin (using either COSY or TOCSY transfers) with a ‘‘straight-through’’ magnetization transfer from the 13C spin to the amide moiety of the following residue (96–105). The side chain assignments are connected thereby to the more highly resolved amide 1HN and 15N spins. Two such experiments, CBCA(CO)NH and CBCANH, are discussed in the following sections. These straight-through experiments have out-and-back analogs, the HN(CO)CACB and HNCACB experiments. The HNCACB is discussed in the following sections as well. These experiments frequency label the 15N magnetization as one of the dimensions of the 3D spectrum. The CBCACO(N)H (106) and CO_H(N)CACB (107) experiments frequency label the 13CO magnetization rather than the 15N magnetization to resolve resonance correlations overlapped in CBCA(CO)NH and HNCACB spectra. Correlations to side chain resonances beyond 1H and 13C are discussed in Sections 9.1.6 and 9.1.7. The topology of the amino acid spin system affects coherence transfer in the CBCA(CO)NH, CBCANH, and HNCACB experiments
642
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
more strongly than do the triple-resonance experiments that only correlate backbone spins. In the following discussion, b is the number of 13C nuclei in the spin system, g is number of resonant aliphatic 13 C nuclei in the spin system, m is the number of 1H spins, and n is the number of 1H spins in the spin system. The value of b is 0 for glycine and 1 for all other amino acids; the value of g is 0 for alanine, aspartic acid, asparagine, cysteine, and serine, 2 for valine and isoleucine, and 1 for all other residues (aromatic 13C spins may be perturbed by the aliphatic 13C pulses); m is 2 for glycine and 1 for all other amino acids; and n is one for valine, isoleucine, and threonine and 2 for all other amino acids. 7.4.5.1 CBCA(CO)NH The CBCA(CO)NH experiment correlates both the 13C and the 13C resonances of an amino acid residue with the amide 1H and 15N resonances of the following residue (97). These correlations are extremely useful if significant 13C–1H chemical shift degeneracy exists. In addition, the 13C and 13C chemical shifts provide information on the type of amino acid preceding each amide (Chapter 10). As with the HCA(CO)N and HN(CO)CA experiments, interresidue correlations are established by transferring coherence via the intervening 13 CO spin. The pulse sequence for the CBCA(CO)NH experiment, which incorporates two constant-time evolution periods, is illustrated in Fig. 7.42. The relevant components of the density operator at the indicated time points in the pulse sequence are given by a ¼ 2Hiz Ciy 2Hiz Ciy ,
½7:136
b ¼ Cix cosðCi t1 Þ cosb ðJC C TAB Þmm ð 1 Þ þ 2Ciy Ciz cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ, i
½7:137 c ¼ Cix cosðCi t1 Þ cosb ðJC C TAB Þmm ð 1 Þ 2Ciz Ciy cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ, i
½7:138 d ¼ 2Ciy C0iz cosðCi t1 Þ cosb ðJC C TAB Þmm ð 1 Þ1A ð 2 Þ þ 2Ciy C0iz cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ1B ð 2 Þ, i
½7:139
1H
t1
t1 f1
13Ca/b
d1
decouple-1
f1 t1 2
TAB 2
t2
t2
t3
h
f2 TAB 2
a
t1 2
d2
d2
b
c
d
13Ca
d3 f3
f4 d4
d4
13CO e
15N
d5
decouple-2
decouple
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
y
f5
f
f5
f6 TN 2
t2 2
TN 2
t2 2
decouple
g
643
FIGURE 7.42 Pulse sequence for the CBCA(CO)NH experiment. Values of delays are described in the text. Rounded bars represent selective 1808 pulses applied to 13CO spins. Other experimental details are similar to those given in Fig. 7.31. The phase cycle is 1 ¼ x, x; 2 ¼ 8(x), 8(y), 8(x), 8(y); 3 ¼ 4(x), 4(x); 4 ¼ x plus the off-resonance phase error calculated using [3.88] (see text); 5 ¼ 2(x), 2(x); 6 ¼ 8(x), 8(x); and receiver ¼ x, x, x, x, 2(x, x, x, x), x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1 and 5, respectively, and the receiver phase, in a TPPI–States manner.
644
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
e ¼ 2Ciz C0iy cosðCi t1 Þ cosb ðJC C TAB Þmm ð 1 Þ1A ð 2 Þ 2Ciz C0iy cosðC t1 Þ sinðJC C TAB Þ i
g
cos ðJC C TAB Þnn ð 1 Þ1B ð 2 Þ,
½7:140
n f ¼ 2C0iy Nðiþ1Þz cosðCi t1 Þ cosb ðJC C TAB Þmm ð 1 Þ1A ð 2 Þ o þ cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ1B ð 2 Þ i
2 ð 3 , 4 Þ,
½7:141
n b g ¼ 2HN ðiþ1Þz Nðiþ1Þy cosðCi t1 Þ cos ðJC C TAB Þmm ð 1 Þ1A ð 2 Þ þ cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ1B ð 2 Þ
o
i
cosðNiþ1 t2 Þ sinðJNCO TN Þ2 ð 3 , 4 Þ,
½7:142
n b h ¼ H N ðiþ1Þx cosðCi t1 Þ cos ðJC C TAB Þmm ð 1 Þ1A ð 2 Þ þ cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ1B ð 2 Þ
o
i
cosðNiþ1 t2 Þ sinðJNCO TN Þ2 ð 3 , 4 Þ,
½7:143
in which 2 1 ¼ 1/(2JCH), TAB 6.6 ms to maximize coherence transfer for both 13C and 13C spins, TN ¼ 22 ms, and 5 ¼ 2 2 ¼ 1/(2JNH). The magnitudes of the coherence transfer functions, n( 1) and m( 1), are optimized for 1 ¼ 2.2 ms ([7.25] and Fig. 7.7). The magnitudes of the coherence transfer functions, 1A ð 2 Þ ¼ cosn ð2JC C 2 Þ sinð2JC CO 2 Þ,
½7:144
1B ð 2 Þ ¼ sinð2JC C 2 Þ sinð2JC CO 2 Þ, are optimized simultaneously by setting 2 3.7 ms. The coherence transfer function 2 ð 3 , 4 Þ is 2 ð 3 , 4 Þ ¼ sinð2JC CO 3 Þ sinð2JNCO 4 Þ,
½7:145
in which 4 1=ð6JNCO Þ to 1=ð4JNCO Þ, as required to minimize relaxation losses, and 3 ¼ 1=ð4JC CO Þ ( 4.5 ms). In the original description of
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
645
this experiment, a value of 4 ¼ 11.4 ms was demonstrated to be suitable for a 31- kDa protein (97). All 13C/ pulses in the CBCA(CO)NH experiment are applied near the center of the 13C and 13C chemical shift range in order to maximize excitation of the 13C and 13C spins, and the power of these pulses is adjusted in order p toffiffiffiffiffiminimize their effect on the 13COpspins (i.e., for 908 ffiffiffi pulses, B1 ¼ = 15, and for 1808 pulses, B1 ¼ = 3; Section 3.4.1). The two 1808(13C) pulses in this experiment are applied in the center of the 13C chemical shift region to maximize their effect on these nuclei, and the power of these pulses ispadjusted such that they do not ffiffiffi perturb the 13CO spins (i.e., B1 ¼ = 3; Section 3.4.1). At time f the position of the 1H carrier (and hence the frequency of the 1H broadband decoupling) is shifted from the water resonance to the center of the amide region. The second shaped selective 1808(13CO) pulse, applied immediately before time b, serves to eliminate phase error induced by the offresonance effects of the first such pulse on the transverse 13C magnetization (Section 3.4.1), and thus ensures pure cosinusoidal modulation in the t1 dimension. The 1808(13C) pulse applied between time points e and f results in a change in the 13CO phase due to the off-resonant effect of this pulse (Section 3.4.1). This phase error, which can be approximated using [3.88], can be compensated for by an adjustment of the phase of the 908(13CO) pulse immediately following time f (6); the phase of this pulse should be set to x plus the off-resonance phase error, in order to fully transfer 13CO magnetization to the 15N spins. Selected F2(15N) slices from a CBCA(CO)NH spectrum of ubiquitin are shown in Fig. 7.43. Two resonances are observed at the F1(13C) frequencies of the 13C and 13C spins of residue i and the F3(1HN) frequency of residue i þ 1 for all amino acid residues except glycine (which has only a single resonance corresponding the 13C spin). With relatively minor modifications, the CBCA(CO)NH experiment can be converted to an experiment that correlates the 1HN and 15N resonances of one residue with the 1H and 1H resonances of the preceding residue; such a pulse scheme has been called the HBHA(CBCACO)NH experiment (96). However, this experiment alone does not provide unambiguous assignment of the 1H and 1H resonances to amino acid type, because such assignments generally cannot be made on the basis of 1H chemical shifts alone. 7.4.5.2 CBCANH The CBCANH experiment correlates the 13C and 13C resonances with the amide 1H and 15N resonances of the same residue and the amide 1H and 15N resonances of the succeeding residue
646
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
a
CfJ031
b
0
CfJV26
C')
CfJV17
CfJV17
CfJN60
0
.;-
reo,
C"G53 «) HN R54
C"G53. HN R54
0
L{)
C"N60 HN 161
C"V1?
C"l6? HN H68
HN E18
lC"V26 HN F2 = 118,6 ppm K2?
9
u: OlC"V1? C"031 HN HN E18 032
F2=119,Q,ppm
9
8
E0.
.e,
0
<0
0 I"-
8
F3 (ppm)
FIGURE 7.43 CBCA(CO)NH spectrum of ubiquitin, Selected (adjacent) F 1C3C)F3C'H N ) slices from a 3D CBCA(CO)NH spectrum of 15Njl3C_labeled ubiquitin at F 2C5N) chemical shifts of (a) 118,6 ppm and (b) 119,0 ppm, The interresidue cross-peaks are labeled,
via the I JcaN and 2JCaN couplings, respectively (98), For proteins up to "'20 kDa, this experiment alone can provide virtually complete sequential assignment of the I H N , 15N, l3C''', and 13C.B resonances, because in addition to the sequential connectivities, the 13CQ' and 13C.B chemical shifts provide information on the amino acid type. The pulse sequence for the CBCANH experiment is illustrated in Fig. 7.44. As with the closely related CBCA(CO)NH experiment already discussed, the CBCANH experiment utilizes two constant-time evolution periods. The evolution in the CBCANH experiment up to time c (Fig. 7.44) is identical to that found for the CBCA(CO)NH experiment (Fig. 7.42). The magnetization present at times a, b, and c in Fig. 7.44 is therefore described by operators [7.136], [7.137], and [7.138], respectively. The relevant components of the density operator at the time points d-g
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
647
in the pulse sequence are given by n d ¼ 2Ciy Niz cosðCai t1 Þ cosb ðJC C TAB Þmm ð 1 Þ1A ð 2 Þ
o þ cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ1B ð 2 Þ i n þ 2Ciy Nðiþ1Þz cosðCai t1 Þ cosb ðJC C TAB Þmm ð 1 Þ2A ð 2 Þ o þ cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ2B ð 2 Þ , i
e ¼
n
2Ciz Niy
½7:146 b
cosðCai t1 Þ cos ðJC C TAB Þmm ð 1 Þ1A ð 2 Þ
o þ cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ1B ð 2 Þ i n 2Ciz Nðiþ1Þy cosðCai t1 Þ cosb ðJC C TAB Þmm ð 1 Þ2A ð 2 Þ o þ cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ2B ð 2 Þ , i
n
½7:147
b a f ¼ 2HN iz Niy cosðCi t1 Þ cos ðJC C TAB Þmm ð 1 Þ1A ð 2 Þ
o þ cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ1B ð 2 Þ i cos Ni t2 1 ðTN Þ n b a þ 2HN ðiþ1Þz Nðiþ1Þy cosðCi t1 Þ cos ðJC C TAB Þmm ð 1 Þ2A ð 2 Þ o þ cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ2B ð 2 Þ i cos Niþ1 t2 2 ðTN Þ, n
½7:148
b a g ¼ H N ix cosðCi t1 Þ cos ðJC C TAB Þmm ð 1 Þ1A ð 2 Þ
o þ cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ1B ð 2 Þ i cos Ni t2 1 ðTN Þ n b a þ HN ðiþ1Þx cosðCi t1 Þ cos ðJC C TAB Þmm ð 1 Þ2A ð 2 Þ o þ cosðC t1 Þ sinðJC C TAB Þ cosg ðJC C TAB Þnn ð 1 Þ2B ð 2 Þ i cos Niþ1 t2 2 ðTN Þ: ½7:149
648
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS y
1
H
t1
t1 f1
13Ca/b
d1
decouple-1
f1
d3
decouple-2
TAB 2
a
t2
t3
g
f2 t1 2
t2
TAB 2
t1 2 b
d2
d2 c
d
13Ca
13CO f3
15N
decouple
f3
f4 TN 2 e
t2 2
TN 2
t2 2
decouple
f
FIGURE 7.44 Pulse sequence for the CBCANH experiment. Values of delays are described in the text. Rounded bars represent selective 1808 pulses applied to 13CO spins. Other experimental details are similar to those given in Fig. 7.31. The phase cycle is 1 ¼ x, x; 2 ¼ 4(x), 4(y), 4(x), 4(y); 3 ¼ 2(x), 2(x); 4 ¼ 8(x), 8(x); receiver ¼ x, x, x, x, x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 1 and 3, respectively, and the receiver phase, in a TPPI–States manner.
The coherence transfer functions 1A ð 2 Þ, 2A ð 2 Þ, 1B ð 2 Þ, and 2B ð 2 Þ are given by 1A ð 2 Þ ¼ sinð21JC N 2 Þ cosð22JC N 2 Þ cosn ð2JC C 2 Þ, 2A ð 2 Þ ¼ cosð21JC N 2 Þ sinð22JC N 2 Þ cosn ð2JC C 2 Þ, 1B ð 2 Þ ¼ sinð21JC N 2 Þ cosð22JC N 2 Þ sinð2JC C 2 Þ,
½7:150
2B ð 2 Þ ¼ cosð21JC N 2 Þ sinð22JC N 2 Þ sinð2JC C 2 Þ: The magnitudes of these transfer functions are optimized simultaneously by setting 2 to 11 ms, as indicated in Fig. 7.45. The coherence transfer functions 1 ðTN Þ and 2 ðTN Þ are given by [7.100]. As indicated by Fig. 7.32, 1 ðTN Þ is maximized by setting TN to between 22 and 28 ms. Following time d, the position of the 1H carrier and hence the frequency of the 1H broadband decoupling is shifted from the water resonance to the center of the amide region. As usual, the 1808(13C/) pulse power is adjusted such that it does not perturb the 13CO spins. Each given amide proton will have correlations to its intraresidue 13 C and 13C nuclei and to the 13C and 13C of the preceding residue.
649
7.4 3D TRIPLE-RESONANCE EXPERIMENTS 0.6 0.4
ΓkX(d2)
0.2 0 –0.2 –0.4 –0.6 –0.8 0
5
10
15
20
25
30
35
d2 (ms)
FIGURE 7.45 Plots of the CBCANH coherence transfer functions, 1A( 2) (22, thick), 2A( 2) (—, thin), 1B( 2) (- - -, thick), and 2B( 2) (- - -, thin) [7.150], for 1 JC N1 , 2 JC N2 , and 1 JC C coupling constants of 10, 7, and 35 Hz, respectively. The dashed vertical line at 11 ms indicates the optimal value of 2 to maximize all four transfer functions simultaneously.
With the exception of glycine residues, for which the exponent b is 0, the 1A ð 2 Þ and 2A ð 2 Þ functions have signs opposite to the sign of 1B ð 2 Þ and 2B ð 2 Þ (Fig. 7.45). This feature is helpful in the final spectrum because the 13C resonances of glycines and all 13C resonances have opposite phase relative to the other 13C resonances. This feature of the CBCANH is particularly useful for discriminating between the 13C and 13 C resonances of serine and threonine residues, which resonate in the same spectral region. A CBCANH spectrum of ubiquitin is shown in Fig. 7.46. Identification of the intra- and interresidue connectivities on the basis of resonance intensities in the CBCANH experiment is subject to the same uncertainties as discussed for the HNCA experiment (Section 7.4.1). As with the CBCA(CO)NH experiment, relatively minor modifications can be made to the CBCANH experiment, to produce an experiment that provides correlations to the 1H and 1H resonances rather than to the 13C and 13C resonances; this experiment has been called the HBHA(CBCA)NH experiment (99). In contrast to the
650
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
b Cb E18
Cb R56
Cb K27
Cb
Cb H68
I61
Cb E18
Cb R54
30
a
Cb D32 Cb L67
Ca L67 Ca H68
Ca R54 HN R54
Ca V17 Ca K27 HN N H E18 K27
Ca I61 HN
HN H68
50
Ca N60
Ca E18
Ca G53
Ca E18 Ca V17 HN E18
Ca R54 HN Ca D32 R54 N H D32
F1 (ppm)
Ca G53
60
Cb N60
40
Cb V17
F2 = 118.6 ppm 9
8
F2 = 119.0 ppm 9
70
I61
8
F3 (ppm)
FIGURE 7.46 CBCANH spectrum of ubiquitin. Selected (adjacent) F1(13C)– F3(1HN) slices from a 3D CBCANH spectrum of 15N/13C-labeled ubiquitin at F2(15N) chemical shifts of (a) 118.6 ppm and (b) 119.0 ppm. Correlations to 13C nuclei and the 13C of glycines are plotted with multiple contours while those to the remaining 13C nuclei, which have opposite phase, are plotted with a single contour. Cross-peak assignments are indicated.
HBHA(CBCACO)NH experiment (96), the 1H and 1H resonances in an HBHA(CBCA)NH spectrum are easily identified, as they have opposite phase (the 1H resonances of glycine residues also have phase opposite to those of all other 1H resonances). 7.4.5.3 HNCACB Shortly after development of the straightthrough CBCANH experiment, the out-and-back HNCAB experiment was developed as an alternative approach for obtaining correlations between the amide moiety and the 13C and 13C spins. The HNCACB has two advantages compared with the earlier approach: (i) the time for which transverse 13C magnetization is present is shorter, resulting in increased sensitivity, and (ii) the experiment is applicable to larger proteins in which carbon sites have been perdeuterated. A pulse sequence for the HNCACB experiment is shown in Fig. 7.47. This pulse sequence shares many aspects of the decoupled HNCA experiment discussed in Section 7.4.1.3.
651
7.4 3D TRIPLE-RESONANCE EXPERIMENTS y
1
H
t
d1
t
d1
decouple f2
f3 d2
13Ca/b b
c
d2 d
t 1 t1 2 2
d2
e
f
t
t3
h
f5
f5
f4
f3
t
d2 g
13 a C
13CO f1
f6
f1 T 2
15N
T 2
T 2
t2 2
T 2
t2 2
decouple
a
FIGURE 7.47 Pulse sequence for the HNCACB experiment. Values of delays are described in the text. Rounded bars represent selective 1808 pulses applied to 13 CO spins. Other experimental details are similar to those given in Fig. 7.31. The phase cycle is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ y; 4 ¼ y; 5 ¼ x; 6 ¼ 4(x), 4(x); receiver ¼ x, x, x, x, x, x, x, x. Quadrature detection during t1 is obtained by TPPI–States protocol applied to pulse phases 2, 3, and the receiver. Quadrature detection during t2 is obtained by TPPI–States protocol applied to pulse phases 6 and the receiver. The phase rotation induced by the nonresonant effects of the 1808 pulse applied to the 13CO spins during t1 can be compensated by adding a correction to pulse phases 4 and 5 (Section 3.4.1).
At point a, the density operator is given by a ¼ 2HN iz Niy
½7:151
for ¼ 1/(4JNH). This operator is converted during the subsequent T period to b ¼ 2Niy Ciz 1 ðT Þ 2Niy Cði1Þz 2 ðT Þ
½7:152
at point b, in which 1 ðT Þ and 2 ðT Þ are given in [7.100]. As in the decoupled HNCA experiment, T ¼ 22–28 ms and 1 ¼ 1/(2JNH). Antiphase coherence with respect to the 15N spin is converted to antiphase coherence with respect to the 13C spins at point c by the simultaneous 908(15N) and 908(13C) pulses: c ¼ 2Niz Ciy 1 ðT Þ þ 2Niz Cði1Þy 2 ðT Þ:
½7:153
652
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
During the following 2 2 element, the JC C scalar coupling is active because the 1808 pulse is nonselective for the 13C and 13C spins (in practice, this pulse frequently is applied as a composite element to obtain the best possible inversion). Evolution under the scalar coupling Hamiltonian yields, at point d, d ¼ 2Niz Ciy 1 ðT Þ cosbi ð2JC C 2 Þ þ bi 4Niz Cix Ciz 1 ðT Þ sinð2JC C 2 Þ 2 ðT Þ cosbi1 ð2JC C 2 Þ 2Niz Cði1Þy þ bi1 4Niz Cði1Þx Cði1Þz 2 ðT Þ sinð2JC C 2 Þ,
½7:154
in which bi is the number of C spins within the ith spin system (bi ¼ 0 for glycine residues and 1 for other residues). The relative magnitudes of the two- and three-spin operators are determined by the value of 2. If 2 ¼ 1=ð8JC C Þ, then the magnitudes of the operators are approximately equal and correlations to both the 13C and the 13C spins are observed in the final spectrum. If 2 ¼ 1=ð4JC C Þ, then the magnitudes of the two-spin operators are approximately zero and correlations to 13 C spins are observed in the final spectrum with maximum magnitude (ignoring the effects of relaxation). However, the longer 2 delay incurs significant relaxation losses from transverse relaxation of the 13C spins and, except for very small proteins, is most often utilized for perdeuterated proteins. The 90y pulse following point d generates the coherences e ¼ 2Niz Ciy 1 ðT Þ cosbi ð2JC C 2 Þ bi 4Niz Ciz Cix 1 ðT Þ sinð2JC C 2 Þ 2 ðT Þ cosbi1 ð2JC C 2 Þ 2Niz Cði1Þy bi1 4Niz Cði1Þz Cði1Þx 2 ðT Þ sinð2JC C 2 Þ,
½7:155 which evolve under the free-precession Hamiltonian during the t1 period. The 90y pulse after the t1 period yields, at point f, f ¼ 2Niz Ciy 1 ðT Þ cosbi ð2JC C 2 Þ cosðCi t1 Þ cosbi ðJC C t1 Þ þ bi 4Niz Cix Ciz 1 ðT Þ sinð2JC C 2 Þ cosðC t1 Þ cosbi ðJC C t1 Þ i
2 ðT Þ cosbi1 ð2JC C 2 Þ cosðCi1 t1 Þ cosgi ðJC C t1 Þ 2Niz Cði1Þy Cði1Þz 2 ðT Þ sinð2JC C 2 Þ cosbi1 ðJC C t1 Þ þ bi1 4Niz Cði1Þx
cosðC t1 Þ cosbi1 ðJC C t1 Þ cosgi1 ðJC C t1 Þ, i1
½7:156
653
7.4 3D TRIPLE-RESONANCE EXPERIMENTS
in which gi is the number of C spins within the ith spin system. The maximum value of t1 is set relatively short (510 ms) to avoid resolving homonuclear 13C scalar coupling multiplets. The preceding operators are refocused during the second 2 2 period to yield, at point g, g ¼ 2Niz Ciy 1 ðT Þ cosbi ðJC C t1 Þfcos2bi ð2JC C 2 Þ cosðCi t1 Þ bi sin2 ð2JC C 2 Þ cosðC t1 Þ cosgi ðJC C t1 Þg i
þ 2Niz Cði1Þy 2 ðT Þ cosbi1 ðJC C t1 Þfcos2bi1 ð2JC C 2 Þ cosðCi1 t1 Þ bi1 sin2 ð2JC C 2 Þ cosðC t1 Þ cosgi1 ðJC C t1 Þg: i1
½7:157 The relative signs of the terms in brackets indicate that the correlation peaks for 13C and 13C spins have opposite signs in the final HNCACB spectrum. This feature facilitates distinguishing such correlations. However, unlike most triple-resonance experiments, if 2 ¼ 1=ð8JC C Þ, then the initial signal intensity is approximately zero when t1 ¼ 0. The remainder of the pulse sequence frequency labels the coherences with the 15N chemical shift during a constant-time evolution period and transfers the resulting coherence to the 1HN spin for detection. The final operator of interest prior to acquisition, at point h, is 2 2bi bi h ¼ H N ix cosðNi t2 Þf1 ðT Þ cos ð2JC C 2 Þ cosðCi t1 Þ cos ðJC C t1 Þ
bi 21 ðT Þ sin2 ð2JC C 2 Þ cosðC t1 Þ cosbi ðJC C t1 Þ cosgi ðJC C t1 Þ i
þ 22 ðT Þ cos2bi1 ð2JC C 2 Þ cosðCi1 t1 Þ cosbi1 ðJC C t1 Þ bi1 22 ðT Þ sin2 ð2JC C 2 Þ cosðC t1 Þ cosbi1 ðJC C t1 Þ i1
gi1
cos
ðJC
C
t1 Þg: ½7:158
An HNCACB spectrum of ubiquitin is shown in Fig. 7.48. Identification of the intra- and interresidue connectivities on the basis of resonance intensities in the HNCACB experiment is subject to the same uncertainties discussed for the HNCA experiment (Section 7.4.1). In combination, the HBHA(CBCA)NH, HBHA(CBCACO)NH, CBCANH, and CBCA(CO)NH experiments can provide complete sequential assignments of the 1HN, 15N, 1H, 13C, 1H, and 13C resonances for proteins up to about 20 kDa. A major limitation of the CBCANH and HBHA(CBCA)NH experiments, however, is that they are relatively insensitive. The sensitivity of these two experiments
654
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
a
b
Cb N25 Cb I13 Cb L73
Cb
I44
Cb I13
Cb I44 Ca G47
Ca G10
Cb L43
30
Cb N25 Cb L73
Cb K11
Ca G47
Ca G10
50
Cb L43
Cb K48
K11
40
Cb K48
Cb Cb R74 V26
Cb
Ca L43 Ca
Ca I44
Ca L73 Ca
Ca K48 Ca
Ca K11
Ca R74
9
60
N25 I13 R74 Ca T14 Ca V26
8
Cb T14
7 F3 (ppm)
9
F2 = 121.6 ppm
8
70
Cb T14
F1 (ppm)
Cb Cb R74 V26
7
FIGURE 7.48 HNCACB spectrum of ubiquitin. Selected F1(13C)–F3(1HN) slices from 600-MHz 3D HNCACB spectra of 15N/13C-labeled ubiquitin at an F2(15N) chemical shift of 121.6 ppm. (a) 2 ¼ 3.5 ms to obtain correlations to both 13 C and 13C spins. (b) 2 ¼ 7.0 ms to obtain correlations primarily to the 13 C spins. Correlations to 13C nuclei are plotted with multiple contours while those to the 13C nuclei, which have opposite phase, are plotted with a single contour. Cross-peak assignments are indicated.
is limited by the transverse relaxation rate of 13C magnetization. For larger proteins (420 kDa), observation of a complete set of interresidue correlations is therefore unlikely. For larger proteins that have been perdeuterated at carbon sites, the HNCACB and HN(CO)CACB experiments (5) are preferable (see Chapter 9).
7.4.6 ADDITIONAL CONSIDERATIONS EXPERIMENTS
FOR
TRIPLE-RESONANCE
Unless otherwise noted above, 13C (13CO) pulses are applied with the transmitter frequency in the middle of the 13C (13CO) region of the spectrum. The field strengths of the 13C 908 and 1808 pulses
655 pffiffiffiffiffi arepadjusted to minimally excite the 13CO spins (i.e., B1 ¼ = 15 or ffiffiffi = 3 for 908 and 1808 pulses, respectively, in which is the frequency difference between the 13C and 13CO spectral regions; Section 3.4.1). Similarly, the 13CO pulses are applied as selective shaped pulses or as weak rectangular pulses, with field strengths adjusted such that the 13C spins remain unperturbed. Nonresonant effects of the 13CO pulses are compensated using the techniques presented in Section 3.4.1 and as discussed in the descriptions of individual triple-resonance experiments. Frequency discrimination in indirectly detected dimensions usually is achieved using the TPPI–States protocol (Section 4.3.4.1). The phase shifts are applied to all the pulses preceding the evolution period that are applied to the spin whose coherence is being frequency labeled. Normally, the initial value of the sampling delay is adjusted to exactly 1/(2SW) as described in Section 3.3.2.3. Most triple-resonance experiments (particularly constant-time versions) yield in-phase absorptive lineshapes. Consequently, the discussion of processing HSQC spectra (Section 7.1.2.4) is applicable. The data are truncated in the t1 and t2 dimensions of triple-resonance experiments and linear prediction or maximum entropy reconstruction usually will be utilized. A secondary problem is that most experiments using a non-constant-time frequency labeling of the 13C spins contain a contribution from passive coupling to the 13C spins. In these cases, the total acquisition time is limited to less than 1/(2JCC) to avoid resolution of the scalar coupling and loss of sensitivity. The effect of the coupling can be eliminated by multiplication of the interferogram by the function cos–1(JCCt1) or cos–1(JCCt2) as relevant prior to linear prediction or maximum entropy reconstruction. The preceding analyses of triple-resonance NMR experiments have focused on 3D versions of the experiments; however, most of the 3D experiments can be converted into 4D experiments by inserting an additional evolution period. Generally, equivalent information can be obtained from a pair of complementary three-dimensional NMR experiments or a single four-dimensional experiment. The greater resolution in the indirect dimensions of 3D experiments, compared with 4D experiments, facilitates more accurate determination of resonance frequencies during the assignment process (Chapter 10). However, 4D triple resonance experiments have the advantage that assignments are obtained from a single spectrum, whereas the two 3D experiments would be acquired at different times (with possible slight variations in conditions). In practice, a mix of 3D and 4D experiments typically would be used in obtaining resonance assignments. 7.4 3D TRIPLE-RESONANCE EXPERIMENTS
656
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
7.5 Measurement of Scalar Coupling Constants Homo- and heteronuclear three-bond J coupling constants are important for establishing local backbone and side chain conformations in proteins; in particular, information on the backbone dihedral angle and the side chain torsion angle 1 can be obtained from measurement of such coupling constants. Measurement of 3 JHN H and 3 JH H from homonuclear spectra either by direct methods (i.e., direct measurement or iterative fitting of in-phase or antiphase splittings; Section 6.2.1.5) or from homonuclear E.COSY spectra (Section 6.3.3) was discussed in the preceding chapter. For larger proteins, accurate measurement of homonuclear J couplings by these methods is difficult because the couplings involved are smaller than the proton resonance linewidths. In recent years, a large number of alternative experiments to measure three-bond J coupling constants from isotopically labeled protein samples have been published, elucidating how to circumvent these problems. The new methods may be classified into three distinct categories: (i) direct measurement of resolved J couplings from heteronuclear-edited spectra (108, 109), (ii) E.COSY-like methods for measurement of unresolved J couplings, in which a well-resolved 1J coupling in one dimension of an nD experiment allows resolution of two components of the cross-peaks that are separated by the 3J coupling of interest (110–132), and (iii) quantitative J correlation, in which the coupling constant is determined from the intensity ratio of two cross-peaks (133–141). A great deal of similarity exists among the different published experiments within these three experimental categories. The most common schemes for measurement of homo- and heteronuclear threebond J coupling constants to date are based on the E.COSY principle; an example of this class of experiment, designed to measure 3 JHN H coupling constants, is discussed in the following section. An alternative method for measurement of 3 JHN H coupling constants using the quantitative J correlation class of experiments is discussed later.
7.5.1 HNCA-J EXPERIMENT The first example of a heteronuclear triple-resonance correlation experiment to measure 3 JHN H coupling constants using the heteronuclear E.COSY principle was an HNCA-J experiment (111, 112). Indeed, this was also the first HNCA-type experiment to be published. Further modification of the original sequence (113–117) has led to the HNCA-J experiment illustrated in Fig. 7.49 (110, 118). The basic
7.5 MEASUREMENT
OF
657
SCALAR COUPLING CONSTANTS
y 1
H
t
d1
t f1
–x t
decouple f2
f1
15N
d2 a
d 2–
d2 f4
e
f3 t2 2
t3
t
d3
d 4+
t2 4
t2 4
decouple
d
f4
13Ca
t1 b
e
decouple c
13CO
FIGURE 7.49 Pulse sequence for the HNCA-J experiment. Values of delays are described in the text. Other experimental details are similar to those given in Fig. 7.31. The phase cycle is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ 8(x), 8(y), 8(x), 8(y); 4 ¼ 4(x), 4(x); and receiver ¼ x, x, x, x, 2(x, x, x, x), x, x, x, x. Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 4 and 2, respectively, and the receiver phase, in a TPPI–States manner.
principle behind this E.COSY-like experiment is identical to that described previously for the homonuclear E.COSY experiment (Section 6.3.3): the 13C and 1HN spins are correlated without disturbing the spin state of the 1H nucleus. As a result, the large one-bond JC H coupling resolved in one dimension allows the 3 JHN H coupling to be measured in another dimension. The (optional) 15N dimension simply improves the spectral resolution by separating the 13C–1HN correlations into a third dimension. A more detailed product operator description of the HNCA-J experiment is given in the following discussion. The HNCA-J experiment illustrated in Fig. 7.49 begins by transferring magnetization originating on amide protons to their directly bonded 15N nuclei via an INEPT sequence. For 2 ¼ 1/(2JNH), the magnetization present at time a is represented by a ¼ 2HN z Ny :
½7:159
A second INEPT sequence transfers the 15N magnetization to its coupled 13C spins. In addition, the 15N magnetization refocuses to become in-phase with respect to its attached proton during the delay 1 ¼ 1/(2JNH). Application of synchronous broadband 1H decoupling (WALTZ-16 or DIPSI-2, for instance) after 1 ensures that the 15N spin
658
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
remains decoupled from its attached proton. As with the previously discussed triple-resonance experiments (Section 7.4.6), the power and duration of the 13C pulses are adjusted so that they do not perturb the 13 CO spins (Section 3.4.1). Focusing on the intraresidue correlations, and ignoring the interresidue correlations, the relevant component of the density operator at time b is given by b ¼ 2Nz Cy 1 ð2 2 Þ,
½7:160
in which the coherence transfer function 1(2 2) is given by [7.100]. For the purpose of measuring the 3 JHN H coupling, 2 2 is adjusted to optimize 1(2 2) for intraresidue coherence transfer only; after allowing for the additional relaxation term, expð2 2 R2N Þ, 2 2 is typically set to 20–30 ms. At this stage, [7.160] is rewritten using a mixture of Cartesian operators for the 15N spin and single-element operators for the 13C and 1 H spins (Section 2.7.2) to give b ¼ Nz ½Cy H ðÞ Cy H ðÞ1 ð2 2 Þ,
½7:161
in which H() and H () correspond to the ji and ji spin states of the 1H nucleus, respectively. Thus, Cy H ðÞ and Cy H ðÞ represent the 13C spins whose attached 1H spins are in the ji and ji states. Using [2.218], Cy H ðÞ and Cy H ðÞ can be expanded to give the equivalent Cartesian representations: Cy H ðÞ ¼ Cy ðHz þ 12EÞ ¼ Cy Hz þ 12Cy
½7:162
Cy H ðÞ ¼ Cy ðHz 12EÞ ¼ Cy Hz 12Cy :
½7:163
and
Evolution of the single-element operators can be followed either by using the product operator rules or, more simply, by direct matrix manipulation using the matrix representations given in Table 2.2, together with the relationships given in [7.162] and [7.163]. Following time b, the 13C chemical shift evolves without proton decoupling during t1. The application of 1808(15N) and 1808(13CO) pulses at the midpoint of the t1 evolution period ensures decoupling of these spins; however, the passive scalar coupling interactions between 13 C and 13C spins evolve during this period. The 1808(13C) and
7.5 MEASUREMENT
OF
SCALAR COUPLING CONSTANTS
659
1808(13CO) pulses applied following t1 and prior to time c provide a method to compensate for the off-resonance effect of the first 1808(13CO) pulse on the transverse 13C magnetization (Section 3.4.1). The 1808(15N) pulse applied in the middle of the e delay decouples the 13 C and 15N spins. Setting the delay e equivalent to the initial value of t1(t1(0)) ensures that no phase correction is necessary in the F1 dimension; need only be long enough to accommodate the 1808(13CO) pulse. The relevant components of the density operator present at time c are described by n c ¼ Nz Cy H ðÞ cos½ðC þ JC H Þt1 o Cy H ðÞ cos½ðC JC H Þt1 cosðJC C t1 Þ1 ð 2 Þ:
½7:164
The frequencies of the two multiplet components now are separated in the F1 dimension by the large one-bond JC H scalar coupling constant. During the remainder of the experiment, the spin states of the 1H nuclei effectively are unchanged, and the two correlations observed in the F1 dimension are treated separately. Following time c, the antiphase 13C magnetization is transferred back to 15N by the simultaneous 908 13C and 15N pulses. During the following constant-time evolution period of length 2 2, in which 2 ¼ 3 þ 4, and 2 4 ¼ 1/(2JNH), the antiphase 15N magnetization refocuses with respect to 13C (during 2 2) and dephases with respect to its attached 1HN (during 2 4). The 15N chemical shift evolution proceeds during t2 only. The magnetization present at time d is thus represented by d ¼ Ny HN z H ðÞ cos½ðC þ JC H Þt1 2 HN z H ðÞ cos½ðC JC H Þt1 cosðN t2 Þ cosðJC C t1 Þ1 ð2 2 Þ: ½7:165 The final set of pulses represents a reverse INEPT sequence that transfers the antiphase magnetization back into observable 1HN magnetization. Note that the inclusion of the final 908(1H) pulse results in zero net rotation of the 1H spins (and all other protons not directly bound to 15N) between times b and e. In addition, this pulse also returns any 1HN magnetization that is not aligned along the x-axis to the z-axis, and thus effectively purges phase errors resulting from the 3 JHN H coupling that is active during the delays (Section 7.1.1.2).
660
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
For 2 ¼ 1/(2JNH), the magnetization present at time e is described by e ¼ 12 HN x H ðÞ cos½ðC þ JC H Þt1 N Hx H ðÞ cos½ðC JC H Þt1 cosðN t2 Þ cosðJC C t1 Þ21 ð2 2 Þ: ½7:166 During acquisition, 15N spins are decoupled from the 1HN spins. Decoupling of the 13C spins (13C and 13CO) also is desirable in order to remove the effects of the small 2 JHN C , 2 JHN CO , and 3 JHN C couplings. The coherences represented by e evolve under the chemical shift Hamiltonian during t3 to yield an observable signal, after forming the trace with the observation operator, proportional to 3 1 N 4 cos½ðC þ JC H Þt1 exp½iðH JHN H Þt3 þ cos½ðC JC H Þt1 exp½iðHN þ 3JHN H Þt3 cosðN t2 Þ cosðJC C t1 Þ21 ð2 2 Þ:
½7:167
From [7.167], two components corresponding to the ji and ji spin states of 1H are observed for each intraresidue F1(13C)–F3(1HN) cross-peak. The F1 displacement of the two components corresponds to 1JCH and the F3 displacement corresponds to 3 JHN H . The E.COSY principle requires that the 1H nuclear spin does not change spin states between the t1 and t3 time periods. In practice, nuclear spin relaxation exchanges the 1H spin between states ji and ji with two consequences: (i) relaxation during the t3 period results in self-decoupling of the spins (Section 5.4.2), and (ii) relaxation during the fixed time period between b and d results in the superposition of the E.COSY and natural multiplet structures in the final spectrum. Both effects mediated by spin relaxation reduce the size of the apparent 3 JHN H scalar coupling constant. Measurement of 3 JHN H by the heteronuclear E.COSY technique is illustrated for a cross peak from a selected F2(15N) slice in the HNCA-J spectrum of ubiquitin in Fig. 7.50. The asymmetry in the peak shape is a consequence of relaxation between points b and d in the pulse sequence, as discussed above.
7.5.2 HNHA EXPERIMENT An alternative method to the HNCA-J experiment for measurement of 3 JHN H coupling constants relies on a quantitative analysis of the
7.5
661
MEASUREMENT OF SCALAR COUPLING CONSTANTS 0
10
a
•
E18 •
o
°R54
(8.0)
oH68 G
0
10 10
(8.6)
(9.0)
E0-
o (J)
S
u::
o K27 •
0
(3.0)
0
/61
G (5.9)
F2 = 118.9 ppm
9.0
I
I
I
8.5
8.0
7.5
10
F3 (ppm)
"? ,....
C
lO
0
N
lO
lO
N
lO
0
'" to
lO
'" lO
8.70
8.65
F3 (ppm)
40
20
0
-20
-40
Hz
7.50 (a) A selected F,(13ca)-F3eHN) slice, at an F,esN) chemical shift of 118.9 ppm, from a HNCA-J spectrum of lsN/13C-labeled ubiquitin. The intraresidue cross-peaks are labeled with the residue name and the measured 3JHNl-I" coupling constants are in parentheses. The weaker, unlabeled cross-peaks are due to interresidue correlations. The E.COSY-Iike cross-peak pattern is highlighted in panel b for the cross-peak of Glu 18. The 3JHNH" coupling constants were determined as illustrated in panel c, which shows rows taken through the maxim of the two cross-peak components of Glul8. The peak displacements were measured following inverse Fourier transformation, zerofilling to 16,384 points, and retransformation. FIGURE
662
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS f1
1H
d1 a
15N
f3
f2 d2–d1
t2 2
d2
b
f4
c
f7
d3+
t1 4
d3–
t1 4
f5
t2 2 f8
d2–d1
d2 d
d 3+
f6
e
t1 4
t3
d1 f
d 3–
t1 4
decouple
FIGURE 7.51 Pulse sequence for the HNHA experiment. The delay 1 is set to 4.5 ms, the delay 2 is set to 13.05 ms, and 2 3 ¼ 2 2 – 1. Other experimental details are similar to those given in Fig. 7.31. The phase cycle is 1 ¼ x; 2 ¼ 4(x), 4(x), 4(y), 4(y); 3 ¼ x, x; 4 ¼ 2(x), 2(x); 5 ¼ 4(x), 4(y), 4(x), 4(y); 6 ¼ y, y; 7 ¼ x; 8 ¼ 16(x), 16(y); and receiver ¼ 4(x), 8(x), 4(x), 4(x), 8(x), 4(x). Quadrature detection in the t1 and t2 dimensions is achieved by incrementing independently the phases 3 and 7, respectively, and the receiver phase, in a TPPI–States manner.
diagonal to cross-peak intensity ratio in an 15N-edited 1HN–1H correlation experiment. This technique is referred to as the HNHA experiment because the intraresidue 1HN, 15N, and 1H resonances are correlated by using the pulse sequence illustrated in Fig. 7.51 (133). A description of this experiment using the product operator formalism is given in the following discussion. The initial part of the pulse sequence transfers 1HN magnetization to heteronuclear multiple-quantum coherence in an HMQC-type manner, giving, at time b, b ¼ 2HN x Ny sinðJNH 1 Þ:
½7:168
In addition, the transverse 1HN magnetization present at time a dephases due to homonuclear 1HN–1H J coupling that is active for a total duration 2 2, between times a and c. Chemical shift evolution of the 15N spins occurs in a constant-time period between points b and c (2 3) for a duration of t1/2. The multiple-quantum coherence does not evolve under the influence of the active 1HN–15N coupling during the constanttime period (Section 2.7.5). The magnetization present at time c can therefore be represented by n o 3 N 3 N 2 Þ 4H H Ny sinð2 J N 2 Þ N cosð2 J c ¼ 2HN y H H H H x y z cosðN t1 =2Þ sinðJNH 1 Þ:
½7:169
7.5 MEASUREMENT
OF
SCALAR COUPLING CONSTANTS
663
The subsequent 908(1H) pulse converts the antiphase 1HN magnetization to antiphase 1H magnetization. Following a short t2 evolution period, the antiphase 1H terms are converted back to antiphase 1HN magnetization to give, at time d, 3 d ¼ 2HN x Ny cosðHN t2 Þ cosð2 JHN H 2 Þ o 3 N 4HN H N cosð t Þ sinð2 J Þ cosðN t1 =2Þ sinðJNH 1 Þ: H 2 H H 2 y z y ½7:170 During the following rephasing period, between time points d and f, the 15N chemical shift evolution is continued for an additional period t1/2, the 1HN–1H J coupling is again active, and the 15N–1HN MQ coherence is converted back to observable 1HN magnetization. The observable magnetization present at time f is represented by 2 2 3 3 f ¼ HN y cosðHN t2 Þ cos ð2 JHN H 2 Þ þ cosðH t2 Þ sin ð2 JHN H 2 Þ cosðN t1 Þ sin2 ðJNH 1 Þ:
½7:171
The 908(1H) purge pulse applied immediately before acquisition ensures that only in-phase 1HN magnetization contributes to the observed signal. This analysis has neglected the effects of heteronuclear JNH two- and three-bond couplings that are active for limited parts of the pulse sequence. In the first part of the sequence (i.e., prior to t2 between times b and c), these couplings are active for a fraction corresponding to 2 3 t1/2 1. In the latter part of the experiment, the JNH couplings are active for a time 2 3 þ t1/2 1 if t1/2 5 1, and for 2 3 t1/2 þ 1 if t1/2 4 1. The effects of these couplings are identical for both terms in [7.171] and can be accounted for by the inclusion of additional cosine terms; for instance, for t1/2 5 1, the 2JNH coupling contributes the following terms to the observed magnetization: cos½2 JNH ð2 3 t1 =2 1 Þ cos½2JNH ð2 3 þ t1 =2 1 Þ:
½7:172
Analogous expressions can be written for t1/2 4 1, and for the 3 JNH couplings. In practice, however, these coupling constants are small, such that 2 3 1=JNH , and therefore have little effect on the observed intensity. Equation [7.171] indicates that two peaks, an autocorrelation or ‘‘diagonal’’ peak at F1(15N)–F2(1HN)–F3(1HN) and a cross-peak at F1(15N)–F2(1H)–F3(1HN), with opposite phase will be observed for each
664
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
amino acid spin system (except glycine). The lineshapes of these peaks in the F1 and F3 dimensions are determined by identical factors. The intrinsic linewidths of the diagonal and cross-peaks in the F2 dimension, however, are determined by the relaxation rates of the transverse 1HN and 1H magnetization present during t2. If the lineshapes of the diagonal and cross-peaks in F2 dimension are assumed to be identical (but see later for a discussion of relaxation rates), the intensity ratio of these peaks then provides a measure of the magnitude of the 3 JHN H coupling constant, Scross sin2 ð23 JHN H 2 Þ ¼ tan2 ð23JHN H 2 Þ: ¼ cos2 ð23 JHN H 2 Þ Sdiagonal
½7:173
In practice, the intensity of the cross-peak resonance is reduced relative to the ‘‘diagonal’’ resonance because the antiphase component of the 4HN y Hz Ny operator leads to faster relaxation during the 2 2 periods compared with the pure multiple-quantum 2HN x Ny operator. Ignoring cross-correlation, to a first approximation, the relaxation rates of the two operators are given by R2 ð2HN x Ny Þ ¼ R2HN þ R2MQ ,
½7:174
R2 ð4HN y Hz Ny Þ ¼ R2HN þ R2MQ þ R1H :
½7:175
The time dependence of the in-phase and antiphase operators during the 2 2 delays, including relaxation effects, is a problem identical to that presented in Section 5.4.2 for relaxation in a scalar-coupled spin system. Using the results presented in Section 5.4.2, h i 8 92 2 1=2 < = tan 1 Scross h i , ¼ ½7:176 :ð1 2 Þ1=2 þ tan ð1 2 Þ1=2 ; Sdiagonal in which the dimensionless parameters, and , are given by ¼ 23JHN H 2 : ¼ R1H = 23JHN H ,
½7:177
Equation [7.176] reduces to [7.173] as expected if approaches zero. If R1H is known from experimental measurements or from calculations, then [7.176] can be solved numerically to determine values of 3 JHN H that are more accurate than those determined from [7.173]. A graph of the differences between the actual value of 3 JHN H and the apparent values calculated using [7.173] is shown in Fig. 7.52. As shown, the effect of
7.6 MEASUREMENT
OF
665
RESIDUAL DIPOLAR COUPLING CONSTANTS
1.0
2p 3JHNHa(app) d2
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1.0
2p 3JHNHa d2
FIGURE 7.52 Effects of relaxation on the HNHA experiment. The apparent coupling constant calculated using [7.173] is plotted (in dimensionless units) versus the actual coupling constant calculated (also in dimension using [7.176] less units). Results are shown for ¼ R1H = 2JHN H equal to (—) 0, (– –) 0.1, (- - -) 0.2, ( – ) 0.4 and ( ) 0.8.
relaxation is always to reduce the apparent scalar coupling constant measured using [7.173]. A selected F1(15N) slice of an HNHA spectrum of ubiquitin is shown in Fig. 7.53. Results obtained for 3 JHN H from COSY, HNCA-J, and HNHA experiments are summarized in Table 7.3. As shown, the values of 3 JHN H obtained from COSY spectra are consistently larger than values obtained from HNCA-J or HNHA spectra. Values obtained from the HNHA experiment using [7.176] are approximately 7.5% larger than values obtained using [7.173].
7.6 Measurement of Residual Dipolar Coupling Constants Residual dipolar couplings (RDCs) between pairs of NMR active nuclei provide structural and dynamical information that is not strictly local in nature. In particular, RDCs between directly bonded nuclei provide restraints on the orientation of the internuclear vector in the molecular alignment frame (Section 10.2.1.4).
666
6.0
F1 (ppm)
4.0
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
I61
8.0
R54
E18
K27
F1 = 118.9 ppm
H68
9.0
8.5
8.0
7.5
F3 (ppm)
FIGURE 7.53 HNHA spectrum of ubiquitin. A selected F2(1H)–F3(1HN) slice, at an F1(15N) chemical shift of 118.9 ppm, from an HNHA spectrum of 15N-labeled ubiquitin acquired with 1 ¼ 4.5 ms and 2 2 ¼ 26.1 ms. The intraresidue peaks are labeled with their residue names. The cross-peaks have phase opposite to the phase of ‘‘diagonal’’ peaks. From the ratios of cross-peak to ‘‘diagonal’’ peak intensity, the following uncorrected 3 JHN H coupling constant values were obtained: His68, 8.9 Hz; Glu18, 8.2 Hz; Lys27, 3.1 Hz; Arg54, 8.5 Hz; Ile61, 6.4 Hz.
Dipolar couplings, although a major source of structural information in solid-state NMR, are not normally observed for proteins in solution because rotational diffusion in an isotropic solution environment averages the traceless dipole coupling tensor to zero. However, in 1982, Bothner-By and co-workers demonstrated that small heme-like molecules, possessing large anisotropic magnetic susceptibilities, adopted preferred orientations with respect to the main static field (142). This so-called molecular alignment demonstrated that dipolar couplings would not always average to zero in solution NMR studies. In 1995, Prestegard and co-workers, studying the protein
7.6 MEASUREMENT
OF
667
RESIDUAL DIPOLAR COUPLING CONSTANTS
TABLE 7.3 3
Residue Glu18 Lys27 Asp32 Arg54 Ile61 His68
JHN H Scalar Coupling Constantsa
COSY
HNCA-J
HNHA (uncorrected)
HNHA (corrected)
9.3 4.8 4.3 9.7 6.9 9.8
8.0 3.0 3.3 8.6 5.9 9.0
8.2 3.1 3.4 8.5 6.7 8.9
8.8 3.4 3.7 9.2 7.3 9.6
a
All values are given in Hertz. Scalar coupling constants were determined from COSY spectra by line fitting (Section 6.2.1.5). HNCA-J results were measured from the spectrum as illustrated in Fig. 7.50. Uncorrected HNHA results were calculated from [7.173]. Corrected HNHA results were calculated from [7.176], assuming R1(H) ¼ 6.4 s–1.
cyanometmyoglobin at high magnetic field strengths (up to 750 MHz), demonstrated that this alignment mechanism resulted in measurable one-bond 1HN–15N RDCs (143). Cyanometmyoglobin, because of the heme moiety, has a very highly anisotropic paramagnetic susceptibility. In fact, most proteins have a measurable, albeit very small, anisotropic magnetic susceptibility tensor. Bax and co-workers have accurately measured one-bond 1HN–15N dipolar couplings in human ubiquitin (144). In this diamagnetic protein, magnetic alignment results from the sum of the anisotropic magnetic susceptibility contributions of the backbone peptide bonds and the aromatic side chains. The 1HN–15N RDCs were determined by fitting resonance intensities from a series of 1JNH amplitude-modulated 2D spectra collected at 1H frequencies of 360, 500, and 600 MHz. The measured residual couplings in ubiquitin are very small, ranging from 0.09 to þ0.10 Hz. A major breakthrough in the measurement of dipolar couplings was realized with the introduction of anisotropic media for macromolecular alignment (145). The first such method used a dilute aqueous liquid crystalline phase of discotic phospholipid micelles consisting of a 1:2.9 ratio of dihexanoyl phosphatidylcholine (DHPC) and dimyristoyl phosphatidylcholine (DMPC). These lipids form bicelles and a liquid crystalline phase over a narrow concentration and temperature range (145). Macromolecules dissolved in this medium experience a small degree of alignment due to steric and electrostatic interactions with the bicelle phase. The degree of alignment is adjusted, by variation in the concentration of the liquid crystalline phase, so that only dipolar
668
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
couplings between closely spaced nuclei give measurable dipolar couplings. If alignment is too strong, then the resulting spectra become impossible to interpret because of the large number of RDCs present, the degradation of the lineshape due to unresolved RDCs and anisotropic chemical shifts, and the increased transverse relaxation rate constants (which are essentially population averages of the rate constants for the free protein and the protein interacting with the bicelle). RDCs for proteins properly aligned in such media are several orders of magnitude larger than are couplings measured based on the alignment caused by the inherent magnetic susceptibility anisotropy of a diamagnetic protein. Numerous media for obtaining tunable molecular alignment have been introduced and a comprehensive listing is provided in the review article by Prestegard and co-workers (146). No single alignment medium is universal and the search for alignment media suitable for a given protein is an empirical process. Many bicellar media based on various cocktails of amphiphilic molecules have been described as providing improved physical properties, including increased chemical stability, broader temperature ranges over which alignment is obtained, and modified surface electrostatics, compared to DMPC/DHPC bicelles. The alignment media developed by Ru¨ckert and Otting (147) based on n-alkyl-poly(ethylene glycol)/n-alkyl alcohol mixtures meet these objectives and are compatible with many proteins. Two effective approaches for obtaining molecular alignment — filamentous phage and polyacrylamide gels — are distinctly different from the bicellar media, and are discussed briefly here. The Pf1 filamentous bacteriophage has been widely adopted as an alignment medium (148, 149). The Pf1 bacteriophage particle has a diameter of 60 A˚ and a length of 20,000 A˚. The coat protein, which packages the phage DNA, forms an -helical structure that is parallel to the long axis of the phage particle. This organization is believed to be the reason for the large magnetic susceptibility anisotropy and the alignment of the particle in the magnetic field. Phage particles have many favorable properties for use as alignment media. The particles stably align over a range of concentrations, pH, temperatures, and salt concentrations. In addition, they are fully aligned at all magnetic fields used for protein NMR studies. Furthermore, the solutions formed using precious protein samples are stable over long periods of time and the phage and protein can be separated and retrieved by ultracentrifugation (148). Molecular alignment also can be achieved using compressed or stretched polyacrylamide gels (150, 151). Anisotropy results from the mechanical forces of stretching or compressing the gel and as a consequence, unlike the other media already discussed,
669
7.6 MEASUREMENT OF RESIDUAL DIPOLAR COUPLING CONSTANTS
alignment is independent of the orientation with respect to the static magnetic field. Procedures for compressing or stretching the gel inside the NMR tllbe have been described (150-152). As described by [2.328], the RDC between two nuclear spins has the same form as the weak scalar coupling Hamiltonian; thus, the RDC simply adds to the conventional scalar coupling splitting observed in an NMR spectrum. Figure 7.54 illustrates the effect of molecular alignment on the apparent coupling and lineshape in an 'H- 15 N HSQC spectrum of ~bit]liitin, recorded with?ut decouplihg durin¥ I,. In the absence of blcelles, the peak separatIOn results [l'om the J NH scalar coupling with a magnitude of approximately 93 Hz (the coupling constant is known to be negative). As the concentration of bicelles is increased, the apparent peak separation, given by I J NH + D NH , changes and val~les both smaller and larger than' J NH are observed, depending on whet\1er the RDC is positive or negative for a particular N-H moiety.
a
b
H68
0 V5
I
I
eo I I
0
I
02
I I 192.6 Hz I
C» 144
9.3
co
~
•
9.2 9.1
I I 1107.1 Hz
110 1.4 Hz I I
193 .7 Hz I
192 .6 Hz I
-=
c
Cll:ll>
I
0
I I 93.5 HZ I
• I
9.0 8.9
1100.9 Hz I .. I I
C3
cG>
•
§ls-
c:P
I cez:;IQl>
I 84.6 Hz I
~
0
I I 1108.0 Hz I
.
I 199.2 Hz I
iii'
.d=>
0I
79.4 Hz I I
0
0 9.3 9.2 9.1 9.0 8.9 1H (ppm)
<:=
I 1105.6 Hz I
I
o
N
I
9.3
9.2 9.1
9.0 8.9
FIGURE 7.54 Residual dipole couplings in ubiquitin. Regions of the 'H- 15 N HSQC spectra of ubiquitin, recorded in the absence of I H decoupling in the 15N dimension, are shown for three degrees of molecular alignment. (a) Spectrum recorded in isotropic solution. The peak. separations between doublet components, marked with dashed lines, correspond to IJNH . (b) Spectrum recorded in 4.5% (w/v) bicelles, consisting of a 30: 10: I molar ratio of DMPC, DHPC, and cetyl-trimethyl ammonium bromide (CTAB). (c) Spectrum recorded in 8% (w/v) bicelles. The peak separations in panels band c correspond to the sum of the I J NH and residual dipolar coupling. Reprinted with permission from A. Bax, Weak alignment offers new NMR opportunities to study protein structure and dynamics, Protein Sci. 12, 1-16 (2003). Protein Science copyrighted © 1992-2004 by The Protein Society.
670
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
The magnitude of the RDCs scales linearly over the range of concentrations of bicelles for the three spectra shown in Fig. 7.54. As the degree of alignment increases, 1H–1H dipolar couplings also increase and eventually become comparable to or larger than the 3JHH scalar couplings. As a result, the apparent linewidth in the 1H dimension increases, leading to reduced sensitivity and increased overlap. For this example, the lineshape is only slightly degraded in the spectrum recorded for the sample containing 4.5% volume fraction of the liquid crystalline phase, while the RDCs, on the order of 10 Hz, are easily measurable from the difference in splitting between the isotropic and aligned samples. The increased degradation in the lineshape in the 1H dimension of the spectra shown in Fig. 7.54, compared with the 15N dimension, is the reason that measuring the RDCs from the splitting in the 15N dimension usually is preferable, even though the 1H acquisition dimension has higher digital resolution. Because the RDC adds to the scalar coupling constant, experimental techniques discussed in Section 7.5 for measuring heteronuclear scalar coupling constants and in Chapter 6 for measuring homonuclear scalar coupling constants are equally applicable for measuring RDCs. In particular, numerous techniques based on the E.COSY (frequency domain) and quantitative J-correlation (time-domain) approaches have been developed. Pulse sequence development for measuring RDCs is directed toward maximizing precision and accuracy, particularly for large proteins and for small coupling constants. Experimental methods for measuring RDCs have been tabulated by Prestegard and co-workers (146). The simple method of recording an HSQC spectrum without decoupling during either the t1 or t2 evolution periods, used in Fig. 7.54, doubles the number of resonance correlations in the spectrum, and resonance overlap represents a severe limitation in practical applications. One approach to reduce spectral crowding separates the two doublet components into different subspectra using the IPAP (in-phase/ antiphase) technique introduced by Bax and co-workers (153). A pulse sequence for the 1H–15N IPAP HSQC experiment is shown in Fig. 7.55. Product operator analysis of the IP and AP experiments shows that the density operators immediately prior to acquisition are given by, respectively, IP ¼ cosðN t1 Þ cosðJapp NH t1 ÞIx , app AP ¼ expðR2S Þ sinðJapp NH Þ sinðN t1 Þ sinðJNH t1 ÞIx ,
½7:178
7.6 MEASUREMENT
OF
f1
f1
f4
I f2
f2
S
t
g1
∆
∆
2
2
–x
–x
t2
t
t
decouple
g4
g4
f3
t
g1
671
RESIDUAL DIPOLAR COUPLING CONSTANTS
t1
g2
g3
Grad
FIGURE 7.55 Pulse scheme of the IPAP 15N–1H HSQC experiment. Thin and thick rectangular bars represent 908 and 1808 pulses, respectively. Pulses are applied with x-phase unless the phase is indicated above the bar. Short, wide rectangles are selective soft pulses applied at the frequency of the water resonance. Field gradient pulses are used for artifact and water suppression. The sequence element shown in brackets is used only in the experiment for generating the antiphase spectrum and is omitted for generating the in-phase spectrum. The low-power 908 pulses surrounding the final 1H 1808 pulse are part of the WATERGATE solvent-suppression scheme. The delay durations are 2 ¼ 1/(2JIS) and ¼ 1/(2JIS). The phase cycle for the in-phase experiment is 1 ¼ y, y; 2 ¼ 2(x), 2(x); and receiver ¼ x, 2(x), x. The phase cycle for the antiphase experiment is: 1 ¼ y, y; 2 ¼ 2(y), 2(y); 3 ¼ 4(x), 4(y), 4(x), 4(y); 4 ¼ 8(x), 8(x); and receiver ¼ x, 2(x), x, x, 2(x), x. Quadrature detection in the t1 dimension is obtained by TPPI–States applied to 2 for the in-phase experiment or to 2 and 3 simultaneously for the antiphase experiment.
1 in which Japp NH ¼ JNH þ DNH . The AP data set is scaled by an empirical 1 and the resulting data sets are added factor ½expðR2S Þ sinðJapp NH Þ and subtracted to yield app add ¼ fcosðN t1 Þ cosðJapp NH t1 Þ þ sinðN t1 Þ sinðJNH t1 ÞgIx
¼ cosð½N Japp NH t1 ÞIx app sub ¼ fcosðN t1 Þ cosðJapp NH t1 Þ sinðN t1 Þ sinðJNH t1 ÞgIx
¼ cosð½N þ Japp NH t1 ÞIx :
½7:179
672
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
These two density operators result in two spectra, one of which contains only the upfield doublet component and the other of which contains only the downfield doublet component. For spectra recorded at high static magnetic field strengths, CSA/ dipole relaxation interference causes the linewidth of the upfield doublet component to broaden relative to the downfield component. This effect, although barely evident in Fig. 7.54, is dramatically illustrated in Fig. 7.10 for spectra recorded at 800 MHz for the protein calbindin D28k and, of course, is the basis for the TROSY technique. The broadening and loss of sensitivity of the upfield doublet component limits the precision with which the RDCs can be measured directly from the observed splitting. For 15N-labeled or 13C/15N-labeled proteins, i.e., proteins that have not been deuterated at carbon sites, a simple approach to address this problem measures the frequency difference between the resonance position in a TROSY spectrum and the resonance position in a decoupled HSQC spectrum as (1JNH þ DNH)/2 (25). The narrower linewidth of the resonance signal in the decoupled HSQC experiment, compared to the upfield component of the 1H–15N doublet, shown in Fig. 7.10, increases the precision of the measured RDCs, even though the apparent splitting is reduced by a factor of two. For 2H/15N-labeled or 2H/13C/15N-labeled proteins, i.e., proteins that have been deuterated at carbon sites, a superior approach incorporates the techniques of accordion spectroscopy and J-scaling into the TROSY experiment. A pulse sequence for one such method, called coupling-enhanced TROSY (CE-TROSY), is shown in Fig. 7.56. Two spectra are acquired, one with ¼ 0, corresponding to the conventional TROSY spectrum, and another with set to a value between 0 and 1. The apparent splitting between resonance signals in the two spectra is given by (1JNH þ DNH). During the period t1, the resonance signal relaxes with the average rate constant R2S , because the 1H 1808 pulse interchanges the I and I spin states, rather than the relaxation rate constant for the desired TROSY component. Thus, overly large values of lead to decreased resolution and sensitivity for large proteins. Eventually, of course, for larger proteins, resonance overlap in 2D spectra becomes too severe and 3D approaches are necessary to accurately measure RDCs in large partially aligned 2H/13C/15N-labeled proteins. The HNCO experiment, discussed in Section 7.4.4.1, is the most sensitive and highly resolved 3D experiment for obtaining backbone resonance correlations. This experiment forms the basis for pulse sequences designed for measuring RDCs for nuclear spins in the polypeptide backbone. Pulse sequences incorporating the IPAP, TROSY/ decoupled HSQC, and J-scaling approaches into the HNCO experiment,
7.6 MEASUREMENT
OF
y
f3
y
y
–x
I
f1 S
–x
f1
t
t
g1
g1
673
RESIDUAL DIPOLAR COUPLING CONSTANTS
t2
f2 k t1 2
g2
k t1 2
t1
t
±g2 g3
t
t
t
g3
g4
g4
Grad
FIGURE 7.56 Pulse sequence for the CE-TROSY experiment incorporating J-scaling. Thin and thick rectangular bars represent 908 and 1808 pulses, respectively. Pulses are applied with x-phase unless the phase is indicated above the bar. Short, wide rectangles are selective soft pulses applied at the frequency of the water resonance. Field gradient pulses are used for artifact and water suppression. For ¼ 0, bracketed pulse sequence elements are omitted, the second gradient g2 is negative, and the resulting pulse sequence is identical to the water flip-back TROSY experiment shown in Fig. 7.11b. Phase cycling is identical to that discussed in the caption to Fig. 7.11. When is nonzero, the bracketed pulse sequence elements are included, and the second gradient g2 is positive. In order to select for the proper TROSY resonance, the phase cycles for pulse 1 are interchanged between the two data sets acquired for quadrature detection and Rance–Kay processing.
as discussed for 2D correlation spectroscopy, have been reported in the literature (146). For example, Yang and co-workers have introduced a suite of TROSY-HNCO triple-resonance pulse sequences that measure the one-bond 1HN–15N, 15N–13C0 , and 13C0 –13C, the two-bond 1 N 13 0 H – C , and the three-bond 1HN–13C RDCs (154), and new variants continue to be developed (155). The precision and accuracy of the IPAP, TROSY/decoupled HSQC, and TROSY J-scaling 2D and 3D methods have been evaluated (156).
References 1. S. W. Fesik, E. R. P. Zuiderweg, Q. Rev. Biophys. 23, 97–131 (1990). 2. G. M. Clore, A. M. Gronenborn, Prog. NMR Spectrosc. 23, 43–92 (1991). 3. A. Bax, S. Grzesiek, Acc. Chem. Res. 26, 131–138 (1993).
674
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
4. L. P. McIntosh, F. W. Dahlquist, Q. Rev. Biophys. 23, 1–38 (1990). 5. T. Yamazaki, W. Lee, C. H. Arrowsmith, D. R. Muhandiram, L. E. Kay, J. Am. Chem. Soc. 116, 11655–11666 (1994). 6. D. M. LeMaster, Q. Rev. Biophys. 23, 113–174 (1990). 7. M. Sattler, J. Schleucher, C. Griesinger, Prog. NMR Spectrosc. 34, 93–158 (1999). 8. R. R. Ernst, G. Bodenhausen, A. Wokaun, ‘‘Principles of Nuclear Magnetic Resonance in One and Two Dimensions,’’ pp. 1–610. Clarendon Press, Oxford, 1987. 9. Z. Serber, C. Richter, V. Do¨tsch, ChemBioChem 2, 247–251 (2001). 10. A. Bax, M. Ikura, L. E. Kay, D. A. Torchia, R. Tschudin, J. Magn. Reson. 86, 304–318 (1990). 11. G. Bodenhausen, D. J. Ruben, Chem. Phys. Lett. 69, 185–189 (1980). 12. L. Mu¨ller, J. Am. Chem. Soc. 101, 4481–4484 (1979). 13. T. J. Norwood, J. Boyd, J. E. Heritage, N. Soffe, I. D. Campbell, J. Magn. Reson. 87, 488–501 (1990). 14. K. Pervushin, R. Reik, G. Wider, K. Wu¨thrich, Proc. Natl. Acad. Sci. U.S.A. 94, 12366–12371 (1997). 15. A. Bax, R. Freeman, J. Magn. Reson. 44, 542–561 (1981). 16. M. Rance, G. Wagner, O. W. Sørensen, K. Wu¨thrich, R. R. Ernst, J. Magn. Reson. 59, 250–261 (1984). 17. A. Bax, A. F. Mehlkopf, J. Smidt, J. Magn. Reson. 35, 373–377 (1979). 18. J. Santoro, G. C. King, J. Magn. Reson. 97, 202–207 (1992). 19. G. W. Vuister, A. Bax, J. Magn. Reson. 98, 428–435 (1992). 20. A. G. Palmer, J. Cavanagh, R. A. Byrd, M. Rance, J. Magn. Reson. 96, 416–424 (1992). 21. A. Hammarstro¨m, G. Otting, J. Magn. Reson., Ser. A 109, 246–249 (1994). 22. J. Cavanagh, J. Keeler, J. Magn. Reson. 77, 356–362 (1988). Kupcˇe, H. Matsuo, G. Wagner, in ‘‘Biological Magnetic Resonance, 16: Modern 23. E. Techniques in Protein NMR’’ (N. R. Krisha, L. Berliner, eds.), pp. 149–194. Kluwer Academic, New York, 1999. 24. D. P. Burum, R. R. Ernst, J. Magn. Reson. 39, 163–168 (1980). 25. M. Ottiger, A. Bax, J. Am. Chem. Soc. 120, 12334–12341 (1998). 26. T. Untidt, T. Schulte-Herbru¨ggen, B. Luy, S. J. Glaser, C. Griesinger, O. W. Sørensen, N. C. Nielsen, Mol. Phys. 95, 787–796 (1998). 27. J. Cavanagh, A. G. Palmer, P. E. Wright, M. Rance, J. Magn. Reson. 91, 429–436 (1991). 28. A. G. Palmer, J. Cavanagh, P. E. Wright, M. Rance, J. Magn. Reson. 93, 151–170 (1991). 29. J. Cavanagh, M. Rance, Annu. Rep. NMR Spectrosc. 27, 1–58 (1993). 30. J. Schleucher, M. Sattler, C. Griesinger, Angew. Chem. Int. Ed. Engl. 32, 1489–1491 (1993). 31. M. Rance, Bull. Magn. Reson. 16, 54–67 (1994). 32. M. Akke, P. A. Carr, A. G. Palmer, J. Magn. Reson., Ser. B 104, 298–302 (1994). 33. D. W. Yang, L. E. Kay, J. Biomol. NMR 13, 3–10 (1999). 34. T. Schulte-Herbru¨ggen, O. W. Sørensen, J. Magn. Reson. 144, 123–128 (2000). 35. D. Nietlispach, J. Biomol. NMR 31, 161–166 (2005). 36. R. H. Griffey, A. G. Redfield, Q. Rev. Biophys. 19, 51–82 (1987). 37. K. Pervushin, R. Riek, G. Wider, K. Wu¨thrich, J. Am. Chem. Soc. 120, 6394–6400 (1998). 38. E. Miclet, D. C. Williams, G. M. Clore, D. L. Bryce, J. Boisbouvier, A. Bax, J. Am. Chem. Soc. 126, 10560–10570 (2004). 39. V. Tugarinov, R. Sprangers, L. E. Kay, J. Am. Chem. Soc. 126, 4921–4925 (2004). 40. S. Grzesiek, A. Bax, J. Am. Chem. Soc. 115, 12593–12594 (1993). 41. Y.-C. Li, G. T. Montelione, J. Magn. Reson., Ser. B 101, 315–319 (1993).
7.6 MEASUREMENT
OF
RESIDUAL DIPOLAR COUPLING CONSTANTS
675
42. J. Stonehouse, G. L. Shaw, J. Keeler, E. D. Laue, J. Magn. Reson., Ser. A 107, 178–184 (1994). 43. B. A. Messerle, G. Wider, G. Otting, C. Weber, K. Wu¨thrich, J. Magn. Reson. 85, 608–613 (1989). 44. G. W. Vuister, R. Boelens, R. Kaptein, R. E. Hurd, B. John, P. C. M. van Zijl, J. Am. Chem. Soc. 113, 9688–9690 (1991). 45. L. E. Kay, P. Keifer, T. Saarinen, J. Am. Chem. Soc. 114, 10663–10665 (1992). 46. G. Kontaxis, J. Stonehouse, E. D. Laue, J. Keeler, J. Magn. Reson., Ser. A 111, 70–76 (1994). 47. C. Griesinger, O. W. Sørensen, R. R. Ernst, J. Magn. Reson. 84, 14–63 (1989). 48. E. R. P. Zuiderweg, S. W. Fesik, Biochemistry 28, 2387–2391 (1989). 49. D. Marion, P. C. Driscoll, L. E. Kay, P. T. Wingfield, A. Bax, A. M. Gronenborn, G. M. Clore, Biochemistry 28, 6150–6156 (1989). 50. D. Marion, L. E. Kay, S. W. Sparks, D. A. Torchia, A. Bax, J. Am. Chem. Soc. 111, 1515–1517 (1989). 51. S. Talluri, G. Wagner, J. Magn. Reson., Ser. B 112, 200–205 (1996). 52. M. Ikura, L. E. Kay, R. Tschudin, A. Bax, J. Magn. Reson. 86, 204–209 (1990). 53. G. M. Clore, A. Bax, A. M. Gronenborn, J. Biomol. NMR 1, 13–22 (1991). 54. J. Cavanagh, M. Rance, J. Magn. Reson. 96, 670–678 (1992). 55. S. W. Fesik, E. R. P. Zuiderweg, J. Magn. Reson. 78, 588–593 (1988). 56. V. V. Krishnamurthy, J. Magn. Reson., Ser. B 106, 170–177 (1995). 57. K. E. Ko¨ve´r, V. J. Hruby, D. Uhrin, J. Magn. Reson. 129, 125–129 (1997). 58. G. M. Clore, A. M. Gronenborn, Annu. Rev. Biophys. Chem. 20, 29–63 (1991). 59. T. Frenkiel, C. Bauer, M. D. Carr, B. Birdsall, J. Feeney, J. Magn. Reson. 90, 420–425 (1990). 60. M. Ikura, A. Bax, G. M. Clore, A. M. Gronenborn, J. Am. Chem. Soc. 112, 9020–9022 (1990). 61. L. E. Kay, G. M. Clore, A. Bax, A. M. Gronenborn, Science 249, 411–414 (1990). 62. D. R. Muhandiram, G. Y. Xu, L. E. Kay, J. Biomol. NMR 3, 463–470 (1993). 63. G. M. Clore, L. E. Kay, A. Bax, A. M. Gronenborn, Biochemistry 30, 12–18 (1991). 64. G. W. Vuister, G. M. Clore, A. M. Gronenborn, R. Powers, D. S. Garrett, R. Tschudin, A. Bax, J. Magn. Reson., Ser. B 101, 210–213 (1993). 65. G. Zhu, A. Bax, J. Magn. Reson. 98, 192–199 (1992). 66. L. E. Kay, M. Ikura, G. Zhu, A. Bax, J. Magn. Reson. 91, 422–428 (1991). 67. L. E. Kay, M. Ikura, A. Bax, J. Am. Chem. Soc. 112, 888–889 (1990). 68. A. Bax, G. M. Clore, P. C. Driscoll, A. M. Gronenborn, M. Ikura, L. E. Kay, J. Magn. Reson. 87, 620–627 (1990). 69. M. Ikura, L. E. Kay, A. Bax, J. Biomol. NMR 1, 299–304 (1991). 70. E. T. Olejniczak, R. X. Xu, S. W. Fesik, J. Biomol. NMR 2, 655–659 (1992). 71. A. Bax, G. M. Clore, A. M. Gronenborn, J. Magn. Reson. 88, 425–431 (1990). 72. L. E. Kay, M. Ikura, A. Bax, J. Magn. Reson. 91, 84–92 (1991). 73. L. E. Kay, G.-Y. Xu, A. U. Singer, D. R. Muhandiram, J. D. Forman-Kay, J. Magn. Reson., Ser. B 101, 333–337 (1993). 74. A. J. Shaka, C. J. Lee, A. Pines, J. Magn. Reson. 77, 274–293 (1988). 75. M. Kadkhodaie, O. Rivas, M. Tan, A. Mohebbi, A. J. Shaka, J. Magn. Reson. 91, 437–443 (1991). 76. A. Mohebbi, A. J. Shaka, Chem. Phys. Lett. 178, 374–378 (1991). 77. S. P. Rucker, A. J. Shaka, Mol. Phys. 68, 509–517 (1989). 78. M. Rance, J. Magn. Reson. 74, 557–564 (1987).
676
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
79. G. M. Clore, A. Bax, P. C. Driscoll, P. T. Wingfield, A. M. Gronenborn, Biochemistry 29, 8172–8184 (1990). 80. M. Ikura, L. E. Kay, A. Bax, Biochemistry 29, 4659–4667 (1990). 81. L. E. Kay, M. Ikura, R. Tschudin, A. Bax, J. Magn. Reson. 89, 496–514 (1990). 82. V. Tugarinov, R. Muhandiram, A. Ayed, L. E. Kay, J. Am. Chem. Soc. 124, 10025–10035 (2002). 83. V. Tugarinov, L. E. Kay, J. Am. Chem. Soc. 125, 13868–13878 (2003). 84. S. Grzesiek, A. Bax, J. Magn. Reson. 96, 432–440 (1992). 85. F. Delaglio, D. A. Torchia, A. Bax, J. Biomol. NMR 1, 439–446 (1991). 86. B. T. Farmer II, L. D. Venters, L. D. Spicer, M. G. Wittekind, L. Mu¨ller, J. Biomol. NMR 2, 195–202 (1992). 87. M. Salzmann, G. Wider, K. Pervushin, K. Wu¨thrich, J. Biomol. NMR 15, 181–184 (1999). 88. J. P. Loria, M. Rance, A. G. Palmer, J. Magn. Reson. 141, 180–184 (1999). 89. A. Bax, M. Ikura, J. Biomol. NMR 1, 99–104 (1991). 90. R. Powers, A. M. Gronenborn, G. M. Clore, A. Bax, J. Magn. Reson. 94, 209–213 (1991). 91. D. R. Muhandiram, L. E. Kay, J. Magn. Reson., Ser. B 103, 203–216 (1994). 92. R. T. Clubb, V. Thanabal, G. Wagner, J. Magn. Reson. 97, 213–217 (1992). 93. R. Folmer, G. Otting, J. Biomol. NMR 16, 229–233 (2000). 94. R. Bazzo, D. O. Cicero, G. Barbato, J. Magn. Reson., Ser. B 110, 65–68 (1996). 95. F. Lo¨hr, H. Ru¨terjans, J. Biomol. NMR 6, 189–197 (1995). 96. S. Grzesiek, A. Bax, J. Biomol. NMR 3, 185–204 (1993). 97. S. Grzesiek, A. Bax, J. Am. Chem. Soc. 114, 6291–6293 (1992). 98. S. Grzesiek, A. Bax, J. Magn. Reson. 99, 201–207 (1992). 99. A. C. Wang, P. J. Lodi, J. Qin, G. W. Vuister, A. M. Gronenborn, G. M. Clore, J. Magn. Reson., Ser. B 105, 196–198 (1994). 100. G. T. Montelione, B. A. Lyons, S. D. Emerson, M. Tashiro, J. Am. Chem. Soc. 114, 10974–10975 (1992). 101. S. Grzesiek, J. Anglister, A. Bax, J. Magn. Reson., Ser. B 101, 114–119 (1993). 102. R. T. Clowes, W. Boucher, C. H. Hardman, P. J. Domaille, E. D. Laue, J. Biomol. NMR 3, 349–354 (1993). 103. B. A. Lyons, G. T. Montelione, J. Magn. Reson., Ser. B 101, 206–209 (1993). 104. T. M. Logan, E. T. Olejniczak, R. X. Xu, S. W. Fesik, J. Biomol. NMR 3, 225–231 (1993). 105. M. Wittekind, L. Mueller, J. Magn. Reson., Ser. B 101, 201–205 (1993). 106. B. A. Lyons, M. Tashiro, L. Cedergren, B. Nilsson, G. T. Montelione, Biochemistry 32, 7839–7845 (1993). 107. E. W. Sayers, D. A. Torchia, J. Magn. Reson. 153, 246–253 (2001). 108. N. Astrof, C. Bracken, J. Cavanagh, A. G. Palmer, J. Biomol. NMR 11, 451–456 (1998). 109. L. E. Kay, A. Bax, J. Magn. Reson. 86, 110–126 (1990). 110. M. Billeter, D. Neri, G. Otting, Y. Q. Qian, K. Wu¨thrich, J. Biomol. NMR 2, 257–274 (1992). 111. R. Weisemann, H. Ru¨terjans, H. Schwalbe, J. Schleucher, W. Bermal, C. Greisinger, J. Biomol. NMR 4, 231–240 (1994). 112. G. Montelione, G. Wagner, J. Am. Chem. Soc. 111, 5474–5475 (1989). 113. G. Montelione, G. Wagner, J. Magn. Reson. 87, 183–188 (1990). 114. P. Schmieder, V. Thanabal, L. P. McIntosh, F. W. Dahlquist, G. Wagner, J. Am. Chem. Soc. 113, 6323–6324 (1991). 115. G. Wagner, P. Schmieder, V. Thanabal, J. Magn. Reson. 93, 436–440 (1991). 116. O. W. Sørensen, J. Magn. Reson. 90, 433–438 (1990).
7.6 MEASUREMENT
OF
RESIDUAL DIPOLAR COUPLING CONSTANTS
677
117. J. C. Madsen, O. W. Sørensen, P. Sørensen, F. M. Poulsen, J. Biomol. NMR 3, 239–244 (1993). 118. S. Seip, J. Balbach, H. Kessler, Angew. Chem. Int. Ed. Engl. 31, 1609–1611 (1992). 119. S. Seip, J. Balbach, H. Kessler, J. Magn. Reson., Ser. B 104, 172–179 (1994). 120. C. Griesinger, U. Eggenberger, J. Magn. Reson. 97, 426–434 (1992). 121. G. T. Montelione, M. E. Winkler, P. Rauenbuehler, G. Wagner, J. Magn. Reson. 82, 198–204 (1989). 122. Y. Karimi-Nejad, J. M. Schmidt, H. Ru¨terjans, H. Schwalbe, C. Griesinger, Biochemistry 33, 5481–5492 (1994). 123. G. Wider, D. Neri, G. Otting, K. Wu¨thrich, J. Magn. Reson. 85, 426–431 (1989). 124. M. Kurz, P. Schmieder, H. Kessler, Angew. Chem. Int. Ed. Engl. 30, 1329–1331 (1991). 125. A. S. Edison, W. M. Westler, J. L. Markley, J. Magn. Reson. 92, 434–438 (1991). 126. E. R. P. Zuiderweg, S. W. Fesik, J. Magn. Reson. 93, 653–658 (1991). 127. R. X. Xu, E. T. Olejniczak, S. W. Fesik, FEBS Lett. 305, 137–143 (1992). 128. U. Eggenberger, Y. Karimi-Nejad, H. Thu¨ring, H. Ru¨terjans, C. Griesinger, J. Biomol. NMR 2, 583–590 (1992). 129. S. D. Emerson, G. T. Montelione, J. Am. Chem. Soc. 114, 354–356 (1992). 130. S. D. Emerson, G. T. Montelione, J. Magn. Reson. 99, 413–420 (1992). 131. M. Sattler, H. Schwalbe, C. Griesinger, J. Am. Chem. Soc. 114, 1126–1127 (1992). 132. M. D. Sørensen, S. M. Kristensen, J. J. Led, O. W. Sørensen, J. Magn. Reson., Ser. A 103, 364–368 (1993). 133. G. W. Vuister, A. Bax, J. Biomol. NMR 2, 401–405 (1992). 134. G. W. Vuister, A. Bax, J. Am. Chem. Soc. 115, 7772–7777 (1993). 135. P. R. Blake, M. F. Summers, M. W. W. Adams, J.-B. Park, Z. H. Zhou, A. Bax, J. Biomol. NMR 2, 527–533 (1992). 136. A. Bax, G. W. Vuister, S. Grzesiek, F. Delaglio, A. C. Wang, R. Tschudin, G. Zhu, Meth. Enzymol. 239, 79–105 (1994). 137. S. Grzesiek, G. W. Vuister, A. Bax, J. Biomol. NMR 3, 487–493 (1993). 138. G. W. Vuister, A. C. Wang, A. Bax, J. Am. Chem. Soc. 115, 5334–5335 (1993). 139. G. W. Vuister, T. Yamazaki, D. A. Torchia, A. Bax, J. Biomol. NMR 3, 297–306 (1993). 140. A. Bax, D. Max, D. Zax, J. Am. Chem. Soc. 114, 6923–6925 (1992). 141. S. Grzesiek, M. Ikura, G. M. Clore, A. M. Gronenborn, A. Bax, J. Magn. Reson. 96, 215–221 (1992). 142. S. J. Archer, M. Ikura, D. A. Torchia, A. Bax, J. Magn. Reson. 95, 636–641 (1991). 143. C. Gayathri, A. A. Bothner-By, P. C. M. Van Zijl, C. Maclean, Chem. Phys. Lett. 87, 192–196 (1982). 144. J. R. Tolman, J. M. Flanagan, M. A. Kennedy, J. H. Prestegard, Proc. Natl. Acad. Sci. U.S.A. 92, 9279–9283 (1995). 145. N. Tjandra, S. Grzesiek, A. Bax, J. Am. Chem. Soc. 118, 6264–6272 (1996). 146. N. Tjandra, A. Bax, Science 278, 1111–1114 (1997). 147. J. H. Prestegard, C. M. Bougault, A. I. Kishore, Chem. Rev. 104, 3519–3540 (2004). 148. M. Ru¨ckert, G. Otting, J. Am. Chem. Soc. 122, 7793–7797 (2000). 149. M. R. Hansen, L. Mueller, A. Pardi, Nat. Struct. Biol. 5, 1065–1074 (1998). 150. G. M. Clore, M. R. Starich, A. M. Gronenborn, J. Am. Chem. Soc. 120, 10571–10572 (1998). 151. Y. Ishii, M. A. Markus, R. Tycko, J. Biomol. NMR 21, 141–151 (2001). 152. H.-J. Sass, G. Musco, S. J. Stahl, P. T. Wingfield, S. Grzesiek, J. Biomol. NMR 18, 303–309 (2000). 153. J. J. Chou, S. Gaemers, B. Howder, J. M. Louis, A. Bax, J. Biomol. NMR 21, 377–382 (2001).
678
CHAPTER 7 HETERONUCLEAR NMR EXPERIMENTS
154. M. Ottiger, F. Delaglio, A. Bax, J. Magn. Reson. 131, 373–378 (1998). 155. D. W. Yang, R. A. Venters, G. A. Mueller, W. Y. Choy, L. E. Kay, J. Biomol. NMR 14, 333–343 (1999). 156. V. Vijayan, M. Zweckstetter, J. Magn. Reson. 174, 245–253 (2005). 157. G. Kontaxis, G. M. Clore, A. Bax, J. Magn. Reson. 143, 184–196 (2000). 158. A. S. Edison, F. Abildgaard, W. M. Westler, E. S. Mooberry, J. L. Markley, Meth. Enzymol. 239, 3–79 (1994).
CHAPTER
8 EXPERIMENTAL NMR RELAXATION METHODS
The basics of the theory of spin relaxation in NMR spectroscopy are presented in Chapter 5 and aspects of relaxation important for the design of particular NMR experiments are discussed in Chapters 6 and 7. This chapter describes spin relaxation experiments for characterizing dynamic properties of molecules over a range of different time scales with atomic resolution (1). Applications of these methods include investigations of backbone and side chain conformational dynamics of multiple functional states of proteins in order to characterize folding, stability, binding, and catalysis (2). The development of new experiments for probing molecular dynamic properties is an active area of investigation, and novel methods continue to be described in the literature on a regular basis. The experiments described in this chapter largely report on the local dynamics of a single spin or a pair of directly bonded spins. Relaxation interference between different stochastic Hamiltonians results from correlated dynamic processes (Sections 5.2.1 and 5.45). Development of comprehensive and sensitive methods for investigating correlated motions in proteins represents a particularly important area of current research (3). The time scale of a dynamic process that can be characterized by spin relaxation methods depends directly on the magnitude of the variation in the spin Hamiltonian modulated by the dynamic process.
679
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Thus, motions on picosecond–nanosecond time scales are accessible to spin relaxation resulting from modulation of dipole–dipole (DD), chemical shift anistropy (CSA), and quadrupolar Hamiltonians, as discussed in Section 5.4. Motions on microsecond–millisecond time scales are accessible to spin relaxation resulting from modulation of isotropic chemical shifts, that is, from chemical exchange phenomena, as discussed in Section 5.6. The magnitude of the variation in isotropic scalar coupling constants in proteins, on the order of 10 Hz, is too small to have proved useful as a direct relaxation probe of molecular dynamics in macromolecules; however, observation of averaged values of scalar coupling constants provides evidence of dynamic processes that are too fast to serve as efficient relaxation mechanisms, as discussed in Section 5.6.2. A number of comprehensive reviews of these methods and applications are available (4–9).
8.1 Pulse Sequences and Experimental Methods Pulse sequences for measuring spin relaxation usually consist of five building blocks: preparation, relaxation, frequency labeling, mixing, and acquisition. In some cases, frequency labeling may occur before the relaxation period or pairs of building blocks may be combined. Insertion of the relaxation period is the only difference from other types of twodimensional NMR spectroscopy, discussed in Section 4.1. Thus, preparation, frequency labeling, and mixing steps can be performed using the range of approaches described in other chapters. For homonuclear 1H experiments, the preparation period normally consists of a 908 pulse and the mixing period consists of a 908 pulse or a TOCSY sequence (Section 6.5). For heteronuclear experiments, the preparation period consists of an INEPT (Sections 2.7.7.2 and 2.7.7.3) or NOE (Section 5.5) transfer step from 1H spins to the nucleus of interest and the mixing period consists of INEPT, PEP (Section 7.1.3.2), or TROSY (Chapter 7) sequences. The density operator present after the preparation period provides the initial condition for the relaxation period, T. Two-dimensional Fourier transformation yields a frequency-domain spectrum in which the relaxation information is encoded in the intensities or lineshapes of the resonance signals. The prototypical situation that arises in the design of experimental methods for measuring relaxation rate constants consists of two eigenoperators, A and B, with eigenfrequencies A and B that are subject to autorelaxation, with rate constants A and B, respectively,
8.1 PULSE SEQUENCES
AND
681
EXPERIMENTAL METHODS
and cross-relaxation, with rate constant : d hAiðtÞ iA A ¼ dt hBiðtÞ
iB B
hAiðtÞ hAi0 , hBiðtÞ hBi0
½8:1
in which hAiðtÞ ¼ TrfAðtÞg, and hAi0 is the equilibrium expectation value; similar expressions hold for B. The formal solution to this equation is, for a relaxation period T,
hAiðTÞ iA A hAi0 ¼ þ exp T hBiðTÞ hBi0
iB B
hAið0Þ hAi0 : hBið0Þ hBi0 ½8:2
Aspects of these equations have been discussed at length in Chapter 5 (in particular, Sections 5.1.2, 5.5.2, and 5.6) and are recapitulated here. If A and B are not secular, typically due to resolved scalar coupling and chemical shift interactions, so that |A B| A, B, and , then the effects of vanish and each basis operator relaxes independently with rate constants A and B. If A and B are secular, then relaxation is biexponential with rate constants that depend on the eigenvalues of the relaxation rate matrix. Multiexponential intensity decay curves are difficult to resolve experimentally and accurate measurements of A, B, and based on [8.2] are not easily achieved. In the initial rate regime,
hAiðTÞ hAið0Þ iA A þT hBiðTÞ hBið0Þ
iB B
hAið0Þ hAi0 : hBið0Þ hBi0 ½8:3
If selective excitation or inversion of the operator A is possible, so that hBið0Þ ¼ hBi0, then A is obtained from the initial decay of hAiðTÞ and is obtained from the initial buildup of hBiðTÞ. Analogous results are obtained if selective perturbation of B is possible. Experimental manipulations during T commonly are utilized to affect the relaxation behavior predicted by [8.1] and [8.2]. Such manipulations fall into two categories: (i) application of rf fields that introduce a new time dependence to spin evolution and alter whether operators are secular and (ii) pulse sequence elements that provide a desired average relaxation behavior. Examples of these approaches are given in the following discussion, assuming that A and B are secular with A ¼ B ¼ 0 in the rotating frame.
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As an example of the first category, if the operator B is saturated (Section 5.1.2), then A relaxes independently according to dhAiðtÞ ¼ A hAiðtÞ hAiSS , ½8:4 dt in which the steady-state expectation value of A is given by hBi0 ss 0 hAi ¼ hAi 1 þ : A hAi0
½8:5
As a second example, if the operators A and B are spin-locked, either by application of a continuous-wave rf field or by a train of pulses, then the operators can become secular in the rotating frame of reference and ROESY-like cross-relaxation pathways become active (Section 5.4.3). As an example of the second category, if two experiments are recorded in which hAi1 ð0Þ ¼ hAi2 ð0Þ ¼ hAið0Þ, and with similar expressions for B, then the average relaxation decay obtained by subtracting the two data sets is described by 1 hAi1 ðTÞ hAiðTÞ hAið0Þ hAi2 ðTÞ A ¼ , ¼ exp T hBiðTÞ hBið0Þ hBi2 ðTÞ B 2 hBi1 ðTÞ ½8:6 and the contribution of the equilibrium values of the basis operators has been eliminated. As a second example, consider the sequence T/2–U–T/2, in which 1 0 U¼ ½8:7 0 1 represents an inversion operation for B and an identity operation for A. If T 1, then, ignoring the equilibrium expectation values, the average evolution during time T is given by T A T A 1 1 U MðTÞ ¼ U exp U exp Mð0Þ 2 B 2 B A T T A U exp Mð0Þ ¼ exp U1 2 2 B B A T 1 A U Uþ Mð0Þ exp 2 B B A 0 exp T Mð0Þ, ½8:8 0 B
8.1 PULSE SEQUENCES
AND
EXPERIMENTAL METHODS
683
in which U–1M(T) ¼ ½hAiðTÞ, hBiðTÞT . In this case, the effect of crossrelaxation is suppressed to first order in time and the resulting equations then yield independent, monoexponential relaxation decays for both operators A and B. More complex pulsing schemes can provide a higher degree of suppression if necessary (10). In the preceding examples, relaxation of the desired operator is monoexponential, either by nature of the secular approximation or by pulse sequence manipulations. Consequently, the intensity of the resonance signal is described by hAiðTÞ ¼ hAið0Þ expðA TÞ:
½8:9
Most commonly, the rate constant is determined by curve fitting this equation to a time series hAiðTÞ recorded for a range of values of T. A minimum of two time points is required, in which case, A ¼ ð1=TÞ ln½hAið0Þ=hAiðTÞ:
½8:10
Strategies for optimal sampling have been investigated thoroughly (11, 12). In accordion spectroscopy (13–16), the relaxation delay is varied in concert with the frequency-labeling period according to T ¼ kt1, in which k is a constant. The relaxation rate constant is encoded into the time decay of the t1 interferogram, or the F1 linewidth. A powerful approach for measuring cross-relaxation rate constants utilizes the sequence T / 2–U–T / 2, in which the propagator U interchanges A and B (17): 0 1 U¼ : ½8:11 1 0 The average evolution over the sequence is ( " #) ( " #) T A T A 1 1 U MðTÞ ¼ U exp U exp Mð0Þ 2 B 2 B ( " # ) ( " #) T 1 A T A U exp Mð0Þ ¼ exp U 2 2 B B ( " # " # !) A A T U1 exp Uþ Mð0Þ 2 B B ( " #) Mð0Þ, ½8:12 exp T
684
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
in which ¼ ½A þ B =2. In this equation, U–1M(T) ¼ ½hBiðTÞ, hAiðTÞT ; therefore, the amplitude hAiðT Þ is encoded into the intensity of the operator B in the final spectrum, and vice versa. If the initial state of the density operator is proportional to A and the magnitude of hAiðTÞ and hBiðTÞ are recorded in separate experiments after the relaxation period, then hBiðTÞ=hAiðTÞ ¼ tanhðTÞ:
½8:13
The propagator U may reflect evolution under the scalar coupling Hamiltonian or a designed pulse sequence element. Equation [8.12] is symmetrical, so identical results are obtained if the initial state of the density operator is proportional to B, except that A and B are swapped in [8.13]. Performing the experiment twice with the different initial conditions can be useful in compensating for imperfections in the experimental methods, a technique that has been called ‘‘symmetrical reconversion’’ (18). The hallmark of relaxation interference is that different components of scalar-coupled multiplets relax at different rates. Accordingly, interference relaxation rate constants can be obtained from analysis of the linewidths or intensities of multiplets in frequency-domain NMR spectra, as an alternative to the time-domain analysis of [8.13] (19–21). If the scalar couplings are resolved, then the multiplet components are nonsecular with respect to each other and relax independently (Sections 5.4.2 and 5.4.5). For simplicity, a scalar-coupled doublet is considered. Evolution is described by [5.146]:
d Iþ S ðtÞ iJIS þ R2 þ xy þ
¼ dt I S ðtÞ 0
0 iJIS þ R2 xy
þ I S ðtÞ þ
: I S ðtÞ ½8:14
At the end of a constant-time evolution period of length T, the intensities of the two doublet components are given by
Iþ S ðTÞ ¼ 12 Iþ ð0Þ exp R2 þ xy T , þ
I S ðTÞ ¼ 12 Iþ ð0Þ exp R2 xy T ,
½8:15
in which hIþ ið0Þ ¼ hIþ S ið0Þ þ hIþ S ið0Þ is the total initial intensity of the doublet — that is, the in-phase magnetization. Accordingly, the relaxation interference rate constant is obtained from the measured
8.2 PICOSECOND–NANOSECOND DYNAMICS
685
doublet intensities as xy
þ I S ðTÞ 1 ln þ : ¼ hI S iðTÞ 2T
½8:16
This approach is particularly powerful for more complex spin systems, such as 13CH2 moieties (20), or multiple-quantum coherences (19, 21). The preceding manipulations will be utilized in various combinations in the pulse sequences described next. In many of these methods, the number of pulses applied during T will increase as T increases. Consequently, systematic errors result if the pulses have unintended effects on the density operator. For example, application of increasing numbers of 1H pulses can lead to progressive saturation of the water resonance. Saturation transfer from water to amide 1HN spins by amide proton solvent exchange will occur during the recycle delay for fastexchanging amide sites. As a result, the initial value of the 1HN magnetization will depend on T, and relaxation rate constants will be overestimated. These errors can be avoided by using gradients to completely dephase the water resonance at the start of the recycle delay, at a cost of reduced sensitivity for fast-exchanging spins (Section 3.5), by using recycle delays 43/R1w, in which R1w is the longitudinal relaxation rate constant for the water resonance, or by using carefully optimized water flip-back methods (Section 3.5). Most of the pulse sequences illustrated in the following discussions will use the first approach; consequently, the experiments shown for 15N backbone amide nuclear spins also are applicable to other X nuclei in isolated IS spin systems. Water flip-back approaches for spin relaxation methods are discussed elsewhere (22). As another example, heating of the sample may occur as T increases because the amount of rf power deposited into the sample increases with the number of applied pulses or with the length of spinlocking and decoupling sequences. Because relaxation rate constants are highly temperature sensitive, severe systematic errors can result. To avoid such errors, the field strength of applied rf fields may need to be reduced and compensatory pulses should be applied during the recycle delay to ensure that the rf power deposited into the sample is independent of T (23).
8.2 Picosecond^Nanosecond Dynamics Techniques for probing picosecond–nanosecond time scale motions rely upon the DD and CSA relaxation mechanisms for 1H, 13C, and 15N
686
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
spins and the quadrupolar relaxation mechanisms for 2H (¼ D) spins. The following sections describe experiments for measuring relaxation rate constants for backbone amide 15N spins, 2H spins in side chain CH2D groups, and backbone carbonyl 13C spins.
8.2.1 EXPERIMENTAL METHODS RELAXATION
FOR
15
N LABORATORY-FRAME
Experimental methods have been developed extensively for R1, R2, and NOE measurements of backbone and side chain 15N nuclear spins in proteins (22, 24–26). Illustrative pulse sequences for measuring backbone amide 15N spin relaxation in proteins are shown in Fig. 8.1. These experiments are based on the HSQC pulse sequence (Section 7.1.1) and are easily modified for larger proteins using the TROSY technique (27). These sequences can be adapted to measurements of side chain amide 15N spin relaxation (24), and for 1H–13C IS spin systems in which 13C–13C scalar coupling and dipolar interactions are negligible, such as in naturally abundant (28), fractionally enriched (29), or alternately labeled (30) samples. The inversion-recovery technique (31) is used to measure the spin– lattice relaxation rate constant for longitudinal magnetization, R1. This experiment uses a refocused INEPT sequence for the preparation period. Thus, the density operator at the beginning of the relaxation period, T, is proportional to Sz. Phase alternation of 1 in Fig. 8.1a alternates the sign of the Sz magnetization and consequently suppresses the contribution to relaxation from the equilibrium or steady-state magnetization. Decoupling of the 1H spins during the relaxation period T is used to suppress 1H–15N dipolar cross-relaxation and 1H–15N DD/15N CSA relaxation interference (32). Decoupling is accomplished by applying a train of 1808 pulses spaced at short, typically 5-ms intervals. The intensity decay is described by hSz iðTÞ ¼ hSz ið0Þ expðR1 TÞ:
½8:17
Either the Carr–Purcell–Meiboom–Gill (CPMG) (33, 34) or the R1 (35) technique is used to measure the spin–spin relaxation rate constant for transverse magnetization, R2. Even though rf fields are applied to the S spins during T in these pulse sequences, the average amplitudes of the fields are sufficiently weak that Jð! !e Þ ¼ Jð!Þ for the dipole– dipole and CSA relaxation mechanisms, and the relaxation rate constants are not affected by the rf fields. This situation is different for slower chemical exchange processes discussed in Section 8.3. The CPMG
8.2 PICOSECOND–NANOSECOND DYNAMICS
687
R2 experiment shown in Fig. 8.1b uses a refocused INEPT sequence for the preparation period. Thus, the density operator at the beginning of the relaxation period, T, is proportional to Sx. Cross-relaxation due to 1 H–15N DD/15N CSA relaxation interference is suppressed by applying 1808 1H pulses. These pulses are applied after an even-numbered spin echo so as not to interfere with the compensating effects of the CPMG pulse train (36, 37). The intensity decay is described by hSx iðTÞ ¼ hSx ið0Þ expðR2 TÞ,
½8:18
in which T ¼ 2n cp, 2n is the number of spin echo periods, and cp is the length of a single spin echo period. An even number of spin echo periods must be used to obtain a degree of compensation for refocusing pulse imperfections. The in-phase Sx operator evolves partially into the antiphase operator 2IzSy under the scalar coupling Hamiltonian during cp. The relaxation rate constant for heteronuclear antiphase magnetization includes a contribution from longitudinal relaxation of the coupled 1H spin and thus is greater than the rate constant for in-phase magnetization (Section 5.4.2). Assuming that the initial density operator at the beginning of a spin echo period is proportional to Sx and JIS R2, then the phenomenological transverse relaxation rate constant measured in a CPMG experiment is given by R2 cp R2 þ "Rext , ½8:19 in which Rext is the longitudinal relaxation rate constant for dipole– dipole interactions of the 1HN spin with remote 1H spins and " ¼ 0.5 [1 sinc[2JIS cp)]. Parenthetically, if the initial density operator is proportional to 2IzSy, then the phenomenological relaxation rate constant is obtained by substituting 1 " for " in [8.19]. To minimize contributions from antiphase magnetization to R2, cp 5 1/(4JIS); typically, values of cp 1 ms are utilized. CPMG measurements of R2 also are affected by off-resonance effects due to rotation of magnetization components (and consequent introduction of a degree of R1 relaxation) during the refocusing pulses (38, 39). Numerical calculations suggest that systematic decreases in R2 in the range of 3–10% are possible. To minimize these errors, the refocusing pulse length should be as short as possible and 510% of the spin echo delay. Off-resonance effects can be corrected approximately by numerical simulations of the Bloch equations [1.28]. Measured values of R1 and R2 together with known values of !1 and are used to simulate the magnetization decay curve. The apparent value of R2
688
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS a
1H
y ∆
∆
∆
f1
f1
15N
f2
y
∆
y ∆
Decouple y t1
T
t2
y y -x y ∆ ∆ ∆ d d
f3 f4 f4 t3
GARP
Grad Ge
b
1H
y
y ∆
∆ f1
f1
15N
f2
y
∆
Gd
∆
∆
tcp
tcp
tcp
2
t ′1 t′2
2
y y -x y ∆ ∆ ∆ d d
f3 f4 f4 t3
GARP
n
T Grad
Ge
c
1H
15N
y ∆
f2
y
y ∆
∆ f1
Gd
f1
∆
c ∆+c
T Spin-lock
t ′1 t ′2
f3 f4 f4 t3
y y -x y ∆ ∆ ∆ d d
GARP
Grad Ge
d
f2
1H
y ∆
Saturate 15N
f1
t1
t2
f3 f4 f4 t3
y ∆ ∆
Gd
y -x y ∆ d d
GARP
Grad Ge 15
Gd
FIGURE 8.1 Pulse sequences for N (a) R1, (b) R2, (c) R1, and (d) NOE spin relaxation measurements in 15N backbone amide spin systems. R1, R2, and R1 intensity decay curves are recorded by varying the relaxation period T in a series of two-dimensional experiments. The NOE is measured by recording one spectrum with saturation of 1H magnetization and one spectrum without saturation. Narrow and wide bars depict 908 and 1808 pulses, respectively. All pulses are x-phase unless otherwise indicated. Decoupling during acquisition is
8.2 PICOSECOND–NANOSECOND DYNAMICS
689
obtained from fitting the decay curve is used to correct the measured value (38). In the R1 experiment shown in Fig. 8.1c, the initial density operator is proportional to S0z , in which S0z is the magnetization component locked along the direction of the effective field in the rotating frame of reference (Section 5.2.3). The initial density operator is created using a modified refocused INEPT sequence element. The 2IzSy magnetization created after the first S spin 908 pulse is converted to Sx cosð2 Þ þ Sy sinðs Þ
½8:20
during the subsequent evolution periods. The 908 S spin pulse prior to T converts this density operator to Sx cosð2 Þ þ Sz sinð2 Þ ¼ S0z
½8:21
for ¼ 1 / (2!1), in which !1 is the amplitude of the spin-locking rf field. This method relies on zthe identity tan ¼ for small and is accurate to within 5% for |/!1| 5 0.4, corresponding to a tilt angle of 4688. Alignment by means of an adiabatic sweep also is possible (see later,
FIGURE 8.1—Continued achieved with the GARP-1 sequence (117). Decoupling during T in the R1 experiment is performed using composite pulse decoupling or a train of 1H 1808 pulses. The R2 CPMG experiment typically uses cp ¼ 1 ms and T ¼ 2n cp, in which n is an integer. Spin locking during the R1 experiment is performed using a continuous-wave rf field with amplitude !1 and frequency centered in the spectral range of interest. Saturation during the NOE experiment is performed using composite pulse decoupling or a train of 1H 1808 pulses. Delays are ¼ 1 / (4JIS), ¼ 1 / (2!1), 1 ¼ þ t1 / 2, 2 ¼ (1 2 / t1max) t1 / 2, 3 ¼ (1 t1 / t1max) , t1max is the maximum value of the t1 labeling period, 01 ¼ 1 þ , 02 ¼ 2 þ , and 4 Gd. The delay is used to orient the initial magnetization along the direction of the effective field in the rotating reference frame (118). A semi-constant-time frequency-labeling period is used to increase resolution in the indirect dimension (42, 43). Conventional frequency labeling is obtained by setting 2 ¼ t1 / 2 and 3 ¼ . The phase cycle is 1 ¼ x, x; 2 ¼ 4(x), 4(x); 3 ¼ x, x, y, y, x, x, y, y; 4 ¼ x; receiver ¼ x, x, x, x. The unlabeled gradients are used to suppress unwanted coherences and pulse imperfections (119). PEP gradient coherence selection is achieved with gradients Gd and Ge. Echo/antiecho signals are recorded in separate experiments by inverting the amplitude of Ge and phase 4 (120). The 1 and receiver phases are inverted for each t1 increment to shift axial peaks to the edge of the spectrum (121).
690
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
Fig. 8.12c). Cross-relaxation due to 1H–15N DD/15N CSA relaxation interference is suppressed by applying 1808 1H pulses at T/4 and 3T/4 (40). The intensity decay is described by 0
Sz ðTÞ ¼ Sz0 ð0Þ expðR1 TÞ: ½8:22 The relaxation rate constants are corrected for the off-resonance tilted field using the expression R1 ¼ R1 cos2 þ R2 sin2 :
½8:23
The steady-state NOE technique described briefly in Section 5.5 is used to measure the {1H}–15N NOE. The NOE experiment shown in Fig. 8.1d records two spectra, one with saturation of the 1H spins and one without saturation during the recycle delay. Assuming that the recycle delay is 1 / R1, in which R1 is the spin–lattice relaxation rate constant for the 15N spin, then the signal intensities in the two experiments are given by hSz iunsat ¼ hSz i0 ,
hSz isat ¼ ð1 þ IS ÞhSz i0 ,
½8:24
and the steady-state NOE is given by NOE ¼ hSz isat =hSz iunsat ¼ ð1 þ IS Þ,
½8:25
in which IS is the NOE enhancement [5.150]. Saturation is achieved with a composite pulse decoupling sequence such as WALTZ-16 or GARP-1. During the control experiment, a continuous-wave rf field of the same amplitude as the decoupling field strength is applied off-resonance to ensure that any heating effects are identical between experiments. The main concern in this experiment is that equilibrium magnetization must be restored in the control experiment. The 1HN and 15 N magnetizations are saturated at the beginning of the recycle delay; consequently, at a minimum, the recycle delay must be greater than 5/R1 to ensure that 499% of the equilibrium magnetization is restored. The NOE between the 1HN and 15N spins couples the recovery of 15N magnetization to that of 1HN. In highly deuterated proteins, the 1HN R1 may be significantly reduced and slow recovery of 1HN magnetization retards recovery of 15N magnetization. Amide proton solvent exchange also complicates this experiment because the very small water 1H, R1 ¼ 0.25 s–1 (T1 ¼ 4 s), is communicated to the 1HN spin by saturation transfer. Efficient water flip-back techniques are helpful in minimizing the degree of saturation transfer; however, even so, recycle delays 43T1w may be necessary to obtain
8.2 PICOSECOND–NANOSECOND DYNAMICS
691
accurate results or the degree of saturation must be quantified empirically (41). Because the NOE experiment begins with 15N magnetization, rather than a polarization transfer step from 1HN spins, the overall sensitivity of this experiment is much lower than the R1 and R2 experiments. The pulse sequences shown in Fig. 8.1 use a semi-constant-time evolution period for frequency-labeling (42, 43) and PEP sequences for the mixing period. The semi-constant-time period provides some of the resolution advantages of a constant-time period, used extensively in Chapter 7, when suitably long delays do not naturally exist in the pulse sequence. As shown in Fig. 8.1, the total length of the evolution period is t1 2 t1 t1 1 þ 2 þ 3 ¼ þ þ 1 þ 1 t1max 2 2 t1max 2 : ½8:26 ¼ 2 þ t1 1 t1max When t1 ¼ 0, the evolution period has length 2; when t1 ¼ t1max, the evolution period has length t1max. Evolution under the chemical shift Hamiltonian during the evolution period is given by t1 2 t1 t1 1 þ 2 3 ¼ þ þ 1 1 ½8:27 ¼ t1 t1max 2 2 t1max and evolution under the scalar coupling Hamiltonian is given by t1 2 t1 t1 1 2 þ 3 ¼ þ þ 1 þ 1 ¼ 2: ½8:28 2 t1max t1max 2 The values of these delays are incremented according to 1 ¼ t1 , 2 t1 2 ¼ 1 , t1max 2 t1 3 ¼ , t1max
½8:29
in which t1 ¼ 1/SW1 is the desired t1 increment. Thus, the period 2, required in the pulse sequence to evolve in-phase to antiphase magnetization, is utilized for t1 frequency labeling as well. The effective
692
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
decay constant observed for the t1 interferogram is R2eff ¼ R2 ð1 2=t1max Þ;
½8:30
consequently, a degree of resolution enhancement is obtained. The semiconstant-time approach can be adapted for many of the pulse sequences presented in the following sections and in Chapter 7. Conventional frequency labeling is obtained in the pulse sequences of Fig. 8.1 by setting 2 ¼ t1 / 2 and 3 ¼ . Examples of 15N relaxation data for ubiquitin are shown in Fig. 8.2. The derived relaxation rate constants for ubiquitin are shown in Fig. 8.3. The median uncertainties in R1, R2, and NOE are 1.5%, 2.8%, and 2.8%, respectively.
8.2.2 EXPERIMENTAL METHODS INTERFERENCE
FOR
15
N RELAXATION
Cross-relaxation due to 1H–15N DD/15N CSA relaxation interference is suppressed in each of the preceding pulse sequences. Measurement of the 1H–15N DD/15N CSA relaxation interference rate constant for transverse magnetization, xy, is useful for investigation of the variability of the CSA (44, 45) and for identifying sites subject to chemical exchange (46). A pulse sequence for measuring this rate constant is shown in Fig. 8.4. The initial density operator is proportional to 2IzSy. This operator evolves into Sx coherence under the scalar coupling Hamiltonian. If T ¼ n / JIS, in which n is an integer, then the autorelaxation rate constants for in-phase and antiphase magnetization are averaged identically according to [8.19]. Consequently, the decay of magnetization is described by [8.12]. The experiment is performed twice. The first time, the intensity of the antiphase operator is recorded; the second time, the intensity of the in-phase operator, created by crossrelaxation, is recorded. The relaxation interference rate constant is calculated by [8.13]. Examples of xy for ubiquitin are shown in Fig. 8.4. The median uncertainty in xy is 1.1%; the larger scatter in the measured values, compared with R2 (Fig. 8.2), arises from site-specific variation in the 15N CSA (44, 45). Other pulse sequences have been described for measuring CSA/ dipole (10, 17, 47, 48), CSA/CSA (49–51), and dipole/dipole (20) relaxation interference rate constants. Use of relaxation interference relaxation rate constants to augment conventional R1, R2, and NOE measurements promises to significantly impact the information content of spin relaxation investigations of proteins.
693
8.2 PICOSECOND–NANOSECOND DYNAMICS 1
0
a
0.5
Intensity (arb. units)
c 0
–1 0
0.2
0.4
0.6
0.8
1 1
1
b
d
0.5
0
0 0
0.1
0.2 T (s)
0.3
8.0 7.8 1H (ppm)
FIGURE 8.2 The 15N relaxation data for ubiquitin recorded at 500 MHz and a temperature of 298 K. (a) R1 intensity decays for residues Ile30 () in stable secondary structure and Arg74 ( ) located near the C-terminus. Spectra were recorded with the pulse sequence of Fig. 8.1a. Solid lines are the best fits to [8.17], yielding R1 ¼ 2.58 0.03 and 1.87 0.04 for Ile30 and Arg74, respectively. (b) R2 intensity decays for residues Ile30 () and Arg74 ( ). Spectra were recorded with the pulse sequence of Fig. 8.1b. Solid lines are the best fits to [8.18], yielding R2 ¼ 5.98 0.07 and 2.97 0.08 for Ile30 and Arg74, respectively. (c) The slice parallel to the F1 dimension is shown for residue Arg76 located at the C-terminus taken from the NOE (saturated) spectrum recorded with the pulse sequence of Fig. 8.1d. (d) The slice parallel to the F1 dimension is shown for residue Arg76 taken from the control (unsaturated) spectrum recorded with the pulse sequence of Fig. 8.1d. The NOE ¼ 1.33 0.01 is calculated from the ratio of the peak intensities according to [8.25].
8.2.3 EXPERIMENTAL METHODS FOR 13CH2D METHYL LABORATORY-FRAME RELAXATION Experimental techniques have been developed to measure deuterium quadrupolar relaxation rate constants (see Chapter 5, Table 5.9) in
694
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
3
R1 (s–1)
2
1
a 0
R2 (s–1)
8
6
4
2
b 0 1
NOE
0
–1
c –2 0
10
20
30
40 Residue
50
60
70
80
FIGURE 8.3 The 15N relaxation rate constants for ubiquitin recorded at 500 MHz and a temperature of 298 K. Shown are (a) R1, (b) R2, and (c) NOE data as a function of residue position. The secondary structure of ubiquitin is depicted above the graphs, with strands shown as filled arrows and helices shown as open rectangles.
695
8.2 PICOSECOND–NANOSECOND DYNAMICS
a 1H
∆
f3
15N
f1
∆
ta
T/2 T/2 f3
f2 tb
f4
y y
y -x
∆ ∆ ∆ d d
∆
f5 f6 f6 y y ∆
GARP
Grad Ge 8
b 6 0.5
hxy (s–1)
<Sy>(t)/<2IzSy>(t)
0.75
Gd
0.25
4 2
0
0.05
0.1
T (s)
0.15
0
c 10
20
30
40
50
60
70
80
Residue
FIGURE 8.4 The 15N CSA/1H–15N dipolar relaxation interference. (a) A pulse sequence for measuring xy is shown. Narrow and wide bars correspond to 908 and 1808 pulses, respectively. Solid bars represent rectangular pulses while open bars correspond to composite (908– x 908) y 908 and composite (908– y 908– x 908) y 1808 pulses. All pulses are applied with x-phase unless specified otherwise. Decoupling during acquisition is achieved with the GARP sequence (117). Delay durations are ¼ 1/(4JIS), T ¼ n / JIS, 4 Gd, and t1 is the labeling period. Two experiments are performed for each value of the relaxation period T. In the first experiment, the composite 1H 908 pulse, designated by the narrow open bar, is included, a ¼ and b ¼ þ t1 / 2. In the second experiment, the composite 908 pulse is absent, a ¼ þ t1 / 2 and b ¼ t1 / 2. The phase cycle is 2 ¼ 4(x), 4(x); 3 ¼ 4(x), 4(x); 4 ¼ 8(x), 8(x); 1 ¼ y, y, y, y; 5 ¼ x, y, x, y; 6 ¼ x; receiver ¼ x, x, x, x, 2(x, x, x, x), x, x, x, x. The unlabeled gradients are used to suppress unwanted coherences and pulse imperfections. PEP gradient coherence selection is achieved with gradients Gd and Ge. Echo/antiecho signals are recorded in separate experiments by inverting the amplitude of Ge and phase 6 (120). The 3 and receiver phases are inverted for each t1 increment to shift axial peaks to the edge of the spectrum (121). (b) Data are shown for residues Ile30 () in stable secondary structure and Arg74 ( ) located near the C-terminus of ubiquitin. Solid lines are the best fits to [8.13], yielding xy ¼ 6.35 0.14 s1 and 2.84 0.02 s1 for Ile30 and Arg74, respectively. (c) Values of xy are shown as a function of residue position for ubiquitin. Data were recorded at 500 MHz and a temperature of 298 K.
696
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
13
CH2D methyl groups (20, 52–54). Pulse sequences for 2H R1 and R1 relaxation measurements are shown in Fig. 8.5 (52). The R1 technique is used to measure the spin–spin relaxation rate constant because short, high-quality 2H 1808 pulses necessary for CPMG methods are difficult to
a
f2
y 1H
∆
∆ f1
f1
13C
∆
t1/2
2∆
TC
TC
f3
f5
y
∆
y
TC+t1/2
TC–t1/2
WALTZ
f4 2H
t
WALTZ
T
t
Tr
WALTZ
Grad
b
f2
y 1H
∆
∆ f1
13C
∆
t1/2
2∆ f1 TC
TC
f3
f5
y
∆
y
TC+t1/2
TC–t1/2
WALTZ
f4 2H
WALTZ
t
T SLy
t
WALTZ
Tr
Grad
FIGURE 8.5 Pulse sequences for (a) R1 and (b) R1 spin relaxation measurements in CH2D spin systems (D ¼ 2H). Intensity decay curves are recorded by varying the relaxation period T in a series of two-dimensional experiments. Narrow and wide bars depict 908 and 1808 pulses, respectively; shaped pulses are shown as rounded bars. The 908 shaped pulse with phase cycle 3 is selective for the 13C spectral region. The 1808 shaped pulses applied in the 2TC constant-time periods are band selective for the aliphatic region of the 13C spectrum. All pulses are x-phase unless otherwise indicated. Decoupling is achieved using WALTZ-16 (122). Delays are ¼ 1/(4JCH), TC ¼ 1/(2JCC), ¼ 1 / (4JCD), Tr ¼ Tmax T, Tmax is the maximum value of T, and t1 is the labeling period. The phase cycle is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ x, x; 4 ¼ 4(x), 4(x); 5 ¼ 8(x), 8(x), receiver ¼ x, x, –x, x, 2(x, x, x, x), x, x, x, x. The unlabeled gradients are used to suppress unwanted coherences and pulse imperfections (119). Coherence selection is achieved with the TPPI–States scheme applied to 5 (121).
8.2 PICOSECOND–NANOSECOND DYNAMICS
697
obtain on solution triple-resonance NMR probes. Pulse sequences for the relaxation rate constant for quadrupolar order, R1Q, represented by the operator 3D2z 2, and the relaxation rate constant for antiphase coherence, represented by the operator DþDz þ DzDþ, are shown in Fig. 8.6 (53). In each of these cases, the quadrupolar relaxation of the 2H spin dominates relaxation, and other relaxation contributions are small (52, 53, 55). As discussed elsewhere, the relaxation rate constant for double-quantum coherence also can be measured experimentally; however, this relaxation rate constant contains contributions from other relaxation effects that require consideration (53). The ways in which 2H and 13C methods for investigating methyl group dynamics are used have been compared (56, 57). Similar approaches can be applied to 15 NHD amide moieties (58). The experiments begin with an INEPT sequence to generate a density operator 2IzSy. This operator evolves during the subsequent period 2TC under the 1H–13C, 2H–13C, and 13C–13C scalar coupling Hamiltonians. The value of 2TC ¼ 1 / JCC so that the effects of the JCC coupling are refocused. The value of JCH JCD (D ¼ 2H); consequently, after the delay 2, the following operators are present: Cx 2Iz Cy 4Iiz Ijz Cx
ðCHD2 isotopomerÞ ðCH2 D isotopomerÞ ðCH3 isotopomer, i 6¼ jÞ
The operator present depends on the number of deuterons incorporated into the methyl group. The CH2D isotopomers are selected by the 908– x 908 2 pulse sandwich. Evolution under the JCD scalar coupling Hamiltonian during the 2TC period yields Iz Sy ! Iz Sy 1 D2z þ Iz Sy D2z cos½2JCD Iz Sx Dz sin½2JCD : ½8:31 The first term on the right-hand side of [8.31] represents the central line of the 13C triplet and does not evolve under the scalar coupling Hamiltonian. The second and third terms represent the outer lines of the triplet that are either in-phase or antiphase with respect to the central line, respectively. Because is much shorter in the pulse sequences of Fig. 8.5 than of Fig. 8.6, increased sensitivity is obtained by using WALTZ-16 2H decoupling to maintain in-phase 13C coherence for 2TC . This suppresses relaxation contributions from 2H R1 processes in the same manner that similar 1H decoupling is used in decoupled HSQC and triple-resonance experiments (Chapter 7). The 13C selective
698
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
a
f2
y
1H
∆
∆
f1
13C
∆
2
f1 TC
2H
t1
2∆ TC
y f5 TC+t1/2 TC–t1/2
y f3
t/2
t/2
∆
WALTZ
Tr
Grad A
b
B
c 1H
1H
13C
13C
f4
2H
2H
T
Grad
f4 f 6
f7 T/2
T/2
Grad A
B
A
B
FIGURE 8.6 Pulse sequences for spin relaxation measurements for (b) quadrupolar order (3D2z 2) and (c) antiphase coherence (DþDz þ DzDþ), in CH2D spin systems (D ¼ 2H) obtained by inserting the bracketed segments into sequences in panel a. Intensity decay curves are recorded by varying the relaxation period T in a series of two-dimensional experiments. Narrow and wide bars depict 908 and 1808 pulses, respectively; shaped pulses are shown as rounded bars. The 908 shaped pulse with phase cycle 3 is selective for the 13C spectral region. The 1808 shaped pulses applied in the 2TC constant-time periods are band selective for the aliphatic region of the 13C spectrum. The rectangular open bar in panel c designates a 458 pulse. All pulses are x-phase unless otherwise indicated. Decoupling during acquisition is achieved using WALTZ16 (122). Delays are ¼ 1 / (4JCH), TC ¼ 1 / (2 JCC), ¼ 1 / (2JCD), Tr ¼ Tmax T, Tmax is the maximum value of T, and t1 is the labeling period. (b) The phase cycle is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ x, x; 4 ¼ 4(08), 4(458), 4(908), 4(1358); 5 ¼ 8(x), 8(–x); receiver ¼ x, –x, –x, x, 2(–x, x, x, x), x, x, x, x. (c) The phase cycle is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ x, x; 4 ¼ 4(x), 4(x); 5 ¼ 16(x), 16(x); 6 ¼ 8(x), 8(x); 7 ¼ 2(x), 2(x); receiver ¼ 2(x, x, x, x), 4(x, x, x, x), 2(x, x, x, x). The unlabeled gradients are used to suppress unwanted coherences and pulse imperfections (119); relative signs of gradients to achieve optimal water suppression are discussed elsewhere (53). Coherence selection is achieved with the TPPI–States scheme applied to 3 (121).
699
8.2 PICOSECOND–NANOSECOND DYNAMICS
pulse, with phase cycle 3, is used to suppress contributions from long-range 13C–13C scalar coupling interactions (52, 53). In the R1 and R2 measurements, a density operator proportional to IzSy(1 2D2z ) IzSxDz is selected by setting ¼ 1 / (4JCD); only the desired operator IzSxDz is retained by the phase cycle of the 2H 9084 pulse. For the other experiments, a density operator proportional to IzSy(1 2D2z ) is selected by setting ¼ 1 / (2JCD). In Fig. 8.6a, the IzSyD2z operator is selected using a double-quantum filter following the relaxation period T. In Fig. 8.6b, the IzSy (DþDz þ DzDþ) operator is generated by applying a 458 pulse to the IzSy(1 2D2z ) operator and contributions from other operators are suppressed by phase cycling 6 and 7. The decay of magnetization during T contains contributions from relaxation of the IzSz operators as well as of the desired 2H operators. For example, in the R1 experiment, the relaxation of the IzSzDz operator is recorded, rather than the relaxation of Dz. These additional contributions are compensated by the Tr ¼ Tmax T period using the constant-relaxation-time approach (59). The effective relaxation decay for the entire pulse sequence is given by IðTÞ ¼ Ið0ÞeRT T eRzz ðTmax TÞ ¼ Ið0ÞeRzz Tmax eTðRT Rzz Þ ,
½8:32
in which RT is the effective relaxation rate constant during the period T and Rzz is the relaxation rate constant for 2IzSz two-spin order during Tr. The relaxation rate constant RT is well approximated by the independent sum of the desired 2H relaxation rate constant and Rzz. Consequently, the effective relaxation constant measured in these pulse sequences does not contain a contribution from Rzz. Finally, the operators D2z , and (1 2D2z ) have the same quadrupolar relaxation rate constants as for quadrupolar order, 3D2z 2, because the identity operator does not contribute to relaxation, as shown by [5.70]. Examples of 2H R1 and R1 relaxation rate constants for ubiquitin are shown in Fig. 8.7. The median uncertainties in R1 and R2 are 6.1% and 5.5%, respectively.
8.2.4 EXPERIMENTAL METHODS RELAXATION
FOR
13
CO LABORATORY-FRAME
Techniques have been reported for measurements of 13C spin relaxation in carbonyl groups (60–63). The dipolar cross-relaxation rate constant between 13CO and 13C spins, C CO , is unaffected by the 13CO CSA or by dipolar interactions with 1H spins and consequently is easier
700
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS 40
a
R1 (s–1)
30
20
10
0 100
b 80
R2 (s–1)
60 40 20
0
10
20
30
40
50
60
70
80
Residue 2
FIGURE 8.7 The H relaxation rate constants for CH2D methyl groups in ubiquitin (56). (a) R1 and (b) R1 rate constants are shown as a function of residue position. Data were recorded at a field of 600 MHz and a temperature of 303 K. Data for Ile are shown as grey (black) bars for the ( ) methyl group; data for Leu are shown as grey (black) for the 1 ( 2) methyl group, and data for Val are shown as grey (black) for 1 ( 2) methyl group.
to interpret than are 13CO R1 or R2 measurements (64). The crossrelaxation rate constant can be measured by a transient NOE experiment (60) or by a steady-state NOE measurement combined with an R1 measurement (63). The latter approach is illustrated in Fig. 8.8. These experiments are based on the HNCO experiment (Section 7.4.4.1) and are similar in design to the 15N R1 and NOE experiments. The principal experimental consideration is that pulses used to saturate the 13C spins must be sufficiently band selective so as to leave the 13CO spins unperturbed. Carbonyl relaxation is important for providing additional detail on backbone motions in proteins, because both the amide 15N and the 13CO spins report on motions of the peptide plane (47, 65, 66).
701
8.2 PICOSECOND–NANOSECOND DYNAMICS
a 1H 15N 13CO
y ∆ ∆ f1
y y y
-y
y 2∆
2∆ ∆ ∆ ∆ ∆ d d
WALTZ
f1 TN TN 2 2 f2 f2 y TC TC T 2 2
f4 f4 y y TN–t1 TN+t1 2 2 f3 y
GARP
TC TC 2 2
13C
Grad Ge
b
-y 2∆ ∆ ∆ ∆ ∆ d d WALTZ f2 f2 f4 f4 y y TN–t1 TN+t1 GARP 2 2 f3 y f1 TC TC 2 2 y
1H 15N 13CO 13C
Gd
y y y
Saturate
Grad Ge
Gd
FIGURE 8.8 Pulse sequences for measuring 13CO (a) R1, and (b) {13C}–13CO NOE. The R1 intensity decay curves are recorded by varying the relaxation period T in a series of two-dimensional experiments. The NOE is measured by recording one spectrum with saturation of 13C magnetization and one spectrum without saturation. Narrow and wide bars depict 908 and 1808 pulses, respectively; water flip-back pulses are shown as rounded bars (123). All pulses are x-phase unless otherwise indicated. Decoupling during acquisition is achieved with the GARP sequence (117); decoupling during T is performed using WALTZ-16 (122). Saturation during the NOE experiment is performed using a train of selective 13C pulses. Delays are ¼ 1/(4JNH), TN ¼ 1 / (4JNCO), TC ¼ 1 / (4JNCO), and 4 Gd. The phase cycle is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ 4(x), 4(x); 4 ¼ x; receiver ¼ x, x, x, x. The unlabeled gradients are used to suppress unwanted coherences and pulse imperfections (119). PEP gradient coherence selection is achieved with gradients Gd and Ge. Echo/ antiecho signals are recorded in separate experiments by inverting the amplitude of Ge and phase 4 (120). The 1 and receiver phases are inverted for each t1 increment to shift axial peaks to the edge of the spectrum (121).
702
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
8.3 Microsecond^Second Dynamics Techniques for probing microsecond–second time scale motions rely upon chemical exchange processes that modulate the isotropic chemical shift of 1H, 13C, and 15N spins. The main techniques, ZZ-exchange (or NOESY) spectroscopy, CPMG relaxation dispersion, and R1 relaxation dispersion, are applicable to millisecond–second, millisecond, and microsecond–millisecond time scales, respectively. Classical lineshape analysis is applicable roughly to the same microsecond–millisecond time scales accessible to CPMG and R1 relaxation dispersion techniques. The basic theory of chemical exchange in NMR spectroscopy is described in Section 5.6. The simple two-site exchange mechanism is assumed throughout this section. The initial step in characterizing a kinetic process by NMR spectroscopy is identifying spins subject to chemical exchange. If site populations are sufficiently large and exchange is slow on the chemical shift time scale, then additional resonances arising from the additional conformational or chemical states are observed in the NMR spectra. These resonances usually are identified during initial efforts to obtain sequence-specific resonance assignments. NOESY, ROESY, and ZZexchange spectroscopy, discussed in the following sections, are used to confirm that the additional resonances arise from chemical exchange processes. However, only a single set of resonances is observed in the NMR spectra if exchange is fast on the chemical shift time scale or if site populations are sufficiently low that the minor components are not observable in the slow exchange limit. In such circumstances, exchange effects are recognized by an increase in the transverse relaxation rate constant relative to the value expected for dipolar and CSA interactions for the observed resonance signals: 0
R2 ¼ R2 þ Rex , 0
½8:33
in which R2 refers to either the relaxation rate constant for an individual site (for slow exchange) or to the population-averaged relaxation rate constant (for fast exchange), and Rex is the excess contribution to R2 due to exchange. The exact relaxation rate constant for the dominant resonance signal is given by [5.183]. For fast exchange [5.186], Rex ¼ p1p2!2 / kex and, assuming p2 0.1 and ! 5000 s–1, then Rex 2.5 s–1 for rates kex 5 106 s–1. This result represents a rough upper bound on kinetic processes in proteins that can be characterized through methods based on chemical exchange broadening.
8.3 MICROSECOND–SECOND DYNAMICS
703
The value of R2 in Eq. [8.33] can be approximated either by R2 , the free-precession damping constant including effects of magnetic field inhomogeneity, or by R2HE, the damping constant observed for a Hahn spin echo experiment. The value of R2 is obtained from the resonance full-width at half-height linewidth, FWHH, using R2 ¼ FWHH, or equivalently, from the decay of the time-domain free induction decay or interferogram (in multidimensional NMR spectra) (67). The size of the inhomogeneity contribution to R2 can be estimated using spins not affected by the exchange process. The value of R2HE is obtained using the sequence T / 2–1808–T / 2 incorporated into either a one-dimensional or a multidimensional NMR experiment (68). The 1808 pulse must be selective to refocus homonuclear scalar coupling interactions, if any such interactions are significant. Usually, spectra are recorded for relaxation delays equal to 0 and T and then R2HE is calculated using [8.10]. The echo time, T, should be long enough that kexT 1; otherwise, the effects of exchange on the echo decay are partially suppressed. In either approach, additional decoupling sequences may need to be applied during the free-precession evolution period or during the spin echo delay to prevent evolution into antiphase coherences subject to additional relaxation mechanisms (36, 37). An example of a pulse sequence for a WALTZ-16 decoupled Hahn echo experiment for 15N spins is shown in Fig. 8.9. This experiment is very similar to the decoupled HSQC experiment (Section 7.1.1.3). The value of T/4 is adjusted so that an integral number of WALTZ-16 supercycles (Section 3.4.3) are performed in order to minimize effects of the decoupling sequence on the quality of the spin echo (37, 69). 0 The value of R2 in [8.33] is obtained by one of the following approaches: (i) analysis of R1, R2, and NOE data for 13C or 15N spins using the model-free formalism (70, 71), (ii) use of the magnetic field dependence of R2 – R1/2 to determine J(0), the spectral density function at zero frequency (72), or (iii) use of 1H–15N DD/15N CSA relaxation interference rate constants (10, 45). Using the last approach, R02 ¼ kxy ,
½8:34
in which k is a constant of proportionality that can be calculated theoretically using expressions in Chapter 5 (Tables 5.5 and 5.8 and [5.144]), or determined empirically for residues in the protein of interest known to be unaffected by chemical exchange processes. Values of Rex determined from [8.33] and [8.34] are shown in Fig. 8.10. Values of R2HE were determined using the experiment shown in Fig. 8.9, and values of
704
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
1H
y ∆
y ∆
f1 X
f2
Waltz-16 ∆
∆
f1 T/4
T/2
a
T/4 t1
t1 2
y ∆ f 3 f4 f4 ∆
y y -x y ∆ ∆ ∆ d d
GARP
Grad Ge
Gd
FIGURE 8.9 Pulse sequence for an in-phase Hahn echo pulse sequence for identifying chemical exchange in small to medium-sized proteins. The pulse sequence is applicable to isolated IS spin systems. Narrow and wide bars correspond to 908 and 1808 pulses, respectively. All pulses are applied with x-phase unless specified otherwise. Decoupling during acquisition is achieved with the GARP sequence (117). Decoupling during the spin echo periods, shown as open boxes, is performed using WALTZ-16 (122). Delay durations are ¼ 1/ (4JIS), ta1 ¼ þ t1 / 2, and 4 Gd. Two experiments are performed: once with T ¼ 0 (and without WALTZ-16 decoupling) and once with T/4 chosen to be an integral number of WALTZ-16 supercycles. The phase cycling is 1 ¼ y, y; 2 ¼ x, x, x, x; 3 ¼ x, x, x, x; 4 ¼ x; receiver ¼ x, x, x, x. The unlabeled gradients are used to suppress unwanted coherences and pulse imperfections (119). PEP gradient coherence selection is achieved with gradients Gd and Ge. Echo/antiecho signals are recorded in separate experiments by inverting the amplitude of Ge and phase 4 (120). The 1 and receiver phases are inverted for each t1 increment to shift axial peaks to the edge of the spectrum (121).
xy were taken from Fig. 8.4. Chemical exchange line broadening of the amide 15N spins of residues 23, 25, 55, and 70 of ubiquitin is evident.
8.3.1 LINESHAPE ANALYSIS Lineshape analysis is a classical approach for analyzing exchange processes in NMR spectroscopy (73, 74). Applications to proteins include chemical exchange arising from folding (75–77), ligand binding (78–80), and oligomerization (81). In each of these applications, the lineshapes can be altered by varying experimental parameters, such as temperature, ligand concentration, denaturant concentration, or protein concentration, which enables a global analysis of lineshapes to be performed as functions of the adjustable parameters. Typically, lineshape analysis is applicable to exchange processes in biological macromolecules with kex 105 s–1.
705
8.3 MICROSECOND–SECOND DYNAMICS 8 23
a 70
hxy (s–1)
6
25
4
2
0
4
8
12
16
R2HE (s–1) 8
b
Rex (s–1)
6 4 2 0
0
10
20
30
40
50
60
70
80
Residue 15
FIGURE 8.10 Values of N Rex for ubiquitin. (a) values of xy taken from Fig. 8.4 are plotted versus R2HE determined using the experiment shown in Fig. 8.9 with T ¼ 57.9 ms. (b) Rex was calculated from [8.33] and [8.34] and an empirical value of k ¼ 1.57. Values of Rex are shown as a function of residue position.
The NMR spectrum for an exchanging system is given by the Fourier transformation of [5.172] or [5.181]. Computer optimization of the parameters in these equations is utilized to fit the experimental lineshapes. In practical applications, particularly for 1H lineshapes, the magnetization components should be regarded as representing individual resonance lines of scalar coupled multiplets. If scalar coupling
706
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
constants are modulated by the exchange process, then a more general theoretical approach based on a density operator formalism is required (82).
8.3.2 ZZ-EXCHANGE SPECTROSCOPY Slow chemical exchange processes can be studied by monitoring the exchange of longitudinal magnetization between sites (Section 5.6.1) if the population of the minor site A2 is large enough and exchange is slow enough to generate observable resonance signals for both sites A1 and A2 (83). In addition, kex must not be much less than R1 for the exchanging sites; otherwise, the signals decay due to relaxation faster than population transfer. Investigations of chemical exchange using 1H NMR spectroscopy are complicated by the coexisting transfer of magnetization through the nuclear Overhauser effect, although the opposite signs of the NOE and rotating-frame Overhauser effect (ROE) for macromolecules can be used to approximately suppress magnetization transfer through dipolar relaxation (84). Isotopic enrichment of macromolecules enables X-nucleus ZZ-exchange to be utilized instead of 1H magnetization transfer. Longitudinal relaxation is much slower for 15 N and 13C spins than for 1H spins in macromolecules, and X-nucleus longitudinal magnetization is unaffected by 1H–1H cross-relaxation. Thus, although either exchange of longitudinal magnetization (85, 86) or two-spin order (86, 87) can be monitored, the relaxation rate constant for two-spin order is less favorable than for X-nucleus longitudinal magnetization. For biological macromolecules, X-nucleus R1 values typically are on the order of 1 s–1; consequently, ZZ-exchange measurements are applicable to chemical exchange processes with 1 s–1 kex 103 s–1. A three-dimensional 1H–X–1H pulse sequence for a ZZ-exchange measurement is shown in Fig. 8.11; as described in the figure caption, either 1H–1H or X–1H two-dimensional pulse sequences can be derived from the three-dimensional sequence and employed if the resonances due to exchanging spins are sufficiently well-resolved (85, 88). The evolution of longitudinal magnetization during the mixing time, T, is described by the modified Bloch–McConnell equations ([5.171] or [5.175]). Cross-peak intensity can be observed at T ¼ 0 if exchange is fast enough to transfer magnetization between sites during the INEPT periods. If 1H magnetization is frequency labeled during t1, then exchange effects must be considered between points A–B and C–D in Fig. 8.11. If only X magnetization is frequency labeled, during t2, then exchange effects must be considered between points C–D in Fig. 8.11.
707
8.3 MICROSECOND–SECOND DYNAMICS 1H
f1
t1
a t1
2
f2
X
f1 y
f3
∆
Decouple f2 a t2
A
B C
y t2 2
∆
y T
∆
∆ ∆
∆ GARP
D
Grad
FIGURE 8.11 Pulse sequence for X nucleus ZZ-exchange NMR spectroscopy. The pulse sequence is applicable to isolated IS spin systems. Narrow and wide bars depict 908 and 1808 pulses, respectively. All pulses are x-phase unless otherwise indicated. Decoupling during the relaxation delay to suppress heteronuclear dipole–dipole cross relaxation, and dipole/CSA cross-correlation can be performed using WALTZ-16 (122) or a train of 1H 1808 pulses at 5-ms intervals. Decoupling during acquisition is achieved with the GARP sequence (117). Delays are ¼ 1 / (4JIS), t1a ¼ þ t1 / 2, t2a ¼ þ t2 / 2, and T is the mixing time. A two-dimensional X–1H correlation spectrum is obtained by setting t1 ¼ 0; a two-dimensional 1H–1H correlation spectrum is obtained by setting t2 ¼ 0. The phase cycle is 1 ¼ x, x; 2 ¼ 4(x), 4(x); 3 ¼ x, x, y, y, x, x, y, y; and receiver ¼ x, x, x, x. The gradients are used to suppress unwanted coherences and pulse imperfections (119). Frequency discrimination is obtained by shifting the phase of the receiver, 1 and 2 according the TPPI– States protocol (121).
The effects of exchange during the INEPT periods can be approximated by including the length of the INEPT sequences, 2, in the mixing time, provided that |R011 R012 | kex and |R021 R022 | kex for the X nucleus and |R021 R022 | kex for the 1H spin, in which R01i and R02i are the spin– lattice and spin–spin relaxation rate constants for spins in the ith site. If these conditions do not hold, then the effects of exchange during the INEPT periods must be analyzed numerically. If a three-dimensional experiment is performed, then exchange during the t2 evolution period must be analyzed numerically unless t2maxkex 1.
8.3.3 R1 ROTATING-FRAME RELAXATION METHODS In an R1 experiment, magnetization is spin-locked in the rotating frame by application of an rf field (89, 90). R1 experiments are sensitive to chemical exchange processes if values of !e near kex are achievable experimentally. Typically, R1 experiments are limited to
708
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
values of kex 105 s–1. Sample heating and limitations on the maximum value of !1, usually 3 104 s–1 (5 kHz), tolerated by the spectrometer probe and amplifiers during sustained spin-locking pulses are the principal experimental constraints on !e. Furthermore, exchange processes with kex 4 106 s–1 result in small exchange line broadening unless ! is very large. The R1 measurements on biological macromolecules can be performed using near-resonance (often called on-resonance for convenience) (40, 91) or off-resonance (59, 92, 93) rf fields. In the former case, the rf transmitter frequency is positioned close to the resonances of interest (or in the middle of the NMR spectrum) and the minimum value of !1 is large enough to ensure that 4 688 over the spectral region of interest. The relaxation dispersion curve is obtained by varying !1. In the latter case, the rf transmitter is positioned far enough off-resonance to ensure that 5 688. The relaxation dispersion curve is obtained by varying !1 or the carrier offset, or both, in order to vary !e. Theoretical expressions for the R1 relaxation rate constant in the presence of chemical exchange have been presented elsewhere (89, 90, 94–96). For two-site chemical exchange in the fast-exchange limit (89, 90), R1 ¼ R1 cos2 þ R2 sin2
0
¼ R1 cos2 þ R2 sin2 þ sin2
p1 p2 !2 kex , k2ex þ !2e
½8:35
0
in which R1 and R2 are the population average relaxation rate constants in the absence of exchange. When site populations are highly skewed, p1 p2, then (95, 96) R1 ¼ R1 cos2 þ R2 sin2
0
¼ R1 cos2 þ R2 sin2 þ sin2
p1 p2 !2 kex k2ex þ !21 þ 22
½8:36
is valid for all time scales. When [8.36] rather than [8.35] is applicable, relaxation dispersion depends on the resonance offset of the minor, usually unobservable, species. Thus, the value of 2 can be determined from the resonance offset dependence of R1. Pulse sequences for near-resonance (40) and off-resonance (59, 92, 93) R1 experiments for X spins are shown in Fig. 8.12. The nearresonance pulse sequence, Fig. 8.12a, is designed for rf field strengths as low as !1/(2) ¼ 1.7JIS. To prevent evolution under an effective scalar coupling Hamiltonian, the 1H 1808 pulses used to decouple 15N CSA/1H–15N dipolar relaxation interference are applied only when the
709
8.3 MICROSECOND–SECOND DYNAMICS
a
y
1H
∆
∆ f1
f1
15N
f2
y ∆
∆
y ∆
y a t1
t1 2
y y -x y ∆ ∆ ∆ d d
f 3 f4 f4 ∆
GARP
Grad Ge
b
c
1H
y 15N
Grad
-y
y
c T/4 c t c
-y T/2
y
Gd
1H
-y
c t c T/4 c
15N
t
A
T
B
t
Grad
FIGURE 8.12 Pulse sequences for measuring 15N R1 using (b) near-resonance and (c) off-resonance experiments obtained by inserting the bracketed segments into the sequence in panel a. Narrow and wide bars depict 908 and 1808 pulses, respectively. All pulses are x-phase unless otherwise indicated. Decoupling during acquisition is achieved with the GARP sequence (117). Delays are ¼ 1/(4JIS), ta1 ¼ þ t1/2, T is the relaxation delay, and 4 Gd. For a constant-relaxation-time experiment, ¼ (Tmax T) / 2; for a conventional experiment, is long enough to encompass any pulses and gradients. The 1808 pulse during in panel c is omitted for a conventional experiment. (b) The spin lock rf frequency is positioned in the center of the spectral region of interest, and ¼ 1 / (2!1) serves to align the magnetization along the direction of the effective field (118). (c) The spin lock rf frequency is set outside the spectral region of interest. The adiabatic sweep A rotates z-magnetization to the direction of the effective field and the adiabatic sweep B rotates the spin locked magnetization back to the z-axis. The phase cycle is 1 ¼ x, x; 2 ¼ 4(x), 4(x); 3 ¼ x, x, y, y, x, x, y, y; 4 ¼ x; receiver ¼ x, x, x, x. The unlabeled gradients are used to suppress unwanted coherences and pulse imperfections (119). PEP gradient coherence selection is achieved with gradients Gd and Ge. Echo/antiecho signals are recorded in separate experiments by inverting the amplitude of Ge and phase 4 (120). The 1 and receiver phases are inverted for each t1 increment to shift axial peaks to the edge of the spectrum (121).
710
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
magnetization is stored along the z-axis. Gradient pulses following the 1808 1H pulses are used to dephase any residual transverse coherences. For rf field strengths satisfying !1 / (2) 4 2.7JIS, the R1 pulse sequence of Fig. 8.1c is satisfactory as well. The off-resonance pulse sequence, Fig. 8.12b, is identical to the R1 pulse sequence of Fig. 8.1c except that the magnetization is aligned along the direction of the off-resonance effective field before the spin-locking period and is returned to the z-axis after the spin-locking period using adiabatic sweeps (92). The dependence of R1 on R1 in [8.23] can be simplified by the constant-relaxation-time approach (59). If ¼ (Tmax T) / 2 in Fig. 8.12, then the effective relaxation rate constant is given by Reff ¼ R1 R1 ¼ R1 cos2 þ R2 sin2 R1 ¼ ðR2 R1 Þ sin2 : ½8:37 Chemical exchange line broadening of the amide 15N spins of residues 23, 25, 55, and 70 of ubiquitin has been identified from elevated values of transverse relaxation rate constants measured using CPMG (Fig. 8.3) and Hahn echo techniques (Fig. 8.10). A relaxation dispersion curve for residue 25 in ubiquitin is shown in Fig. 8.13.
22
R2 (s–1)
20 18 16 14 12 0
2
4
6
8
10
we2 (s–2)
FIGURE 8.13 The 15N R1 relaxation dispersion for residue Asn25 in ubiquitin (124). Values of R2 were obtained from R1 using [8.23]. Data were recorded using both near-resonance and off-resonance pulse sequences shown in Fig. 8.12 at 500 MHz ( ) and 600 MHz () at a temperature of 280 K. The solid line is the best fit of [8.35], yielding kex ¼ (2.3 0.2) 104 s–1.
8.3 MICROSECOND–SECOND DYNAMICS
711
8.3.4 CPMG RELAXATION METHODS In a CPMG experiment, the relaxation of transverse magnetization is observed during a (180821808)n spin echo sequence, in which cp ¼ 2 is the spacing between 1808 pulses and n is an integer (33, 34). CPMG experiments are sensitive to chemical exchange processes if values of 1/ cp near kex are achievable experimentally. CPMG experiments in proteins typically are applicable to chemical exchange processes with values of kex 104 s–1 as a result of experimental constraints on the minimum value of cp 0.1–1 ms. Sample heating at high pulsing rates is a major experimental limitation. In addition, theoretical analyses of relaxation during CPMG experiments assume that the refocusing pulses have negligible duration (97–99); at pulse duty cycles 410%, this assumption may not be valid (93). Theoretical expressions for the effective R2 relaxation rate constant observed in a CPMG experiment in the presence of chemical exchange have been presented elsewhere (89, 97, 98, 100). For the case of two-site exchange, a general expression for the transverse relaxation rate constant for site 1 (p1 4 p2), R2(1/ cp), is given by (89) 1 1 0 1 kex cosh ½Dþ coshðþ Þ D cosð Þ , R2 1=cp ¼ R2 þ 2 cp ½8:38 in which " #1=2 1 þ 2!2 D ¼ 1 þ , 2 ð 2 þ 2 Þ1=2 1=2 i1=2 cp h ¼ þ 2 þ 2 , 2
½8:39
where ¼ k2ex þ !2 and ¼ 2!kex ðp1 p2 Þ. In the fast-exchange limit, [8.38] simplifies to 2 tanh kex cp =2 p1 p2 !2 0 1 R2 1=cp ¼ R2 þ : ½8:40 kex cp kex The maximum value of cp for CPMG experiments applied to X spins is limited by evolution under the one-bond heteronuclear scalar coupling Hamiltonian, which interconverts in-phase and antiphase magnetization during the spin echo period [8.19]. The range of cp values that can be utilized is expanded by CPMG experiments designed to
712
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
a
U
y y 1H
X
∆ ∆
f1
∆
f1
f2
∆
∆
y t
a t t t1
t
A
y y y
B
C
t1 2
b
∆
GARP
∆ ∆
f1
∆
f1
f2
y y y
∆
e
t
t
t1
t 2
a t1
z
c ∆
Saturate
f1
f3
∆
∆ ∆
y f2
y t
t
t
Gd
2n U
1H
rec.
GARP
Ge
G 2n
d
e ∆ ∆ d
f3 f4 f4 y y
y
y t
Gd
2n
U
y y
X
f 3 f4 f4 y y
Ge 2n
X
d
D
G
1H
rec.
∆ ∆ ∆ d
t
t1 2
+z
t1 2
rec.
f4 z
WALTZ
G 2n
2n
FIGURE 8.14 Pulse sequences for relaxation-compensated CPMG experiments in (a) IS, (b) I2S, and (c) I3S spin systems. Narrow and wide bars depict 908 and 1808 pulses, respectively. All pulses are x-phase unless otherwise indicated. Decoupling during acquisition is achieved with the GARP or WALTZ-16 sequences (117, 122). Delays are ¼ cp / 2, ¼ 1 / (4JIS), and 4 Gd; (a) t1a ¼ þ t1 / 2; (b) " ¼ 1/ (8JIS), t1a ¼ þ t1 / 2, and 4 Ge; (c) ¼ 1 / (10JIS).
8.3 MICROSECOND–SECOND DYNAMICS
713
average in-phase and antiphase magnetization. Relaxation-compensated CPMG pulse sequences for IS (101), I2S (102), and I3S (103) are shown in Fig. 8.14. In these experiments, in-phase and antiphase coherences are interchanged during the period U in order to eliminate any cpdependent effects arising from differential relaxation rates. The principles of the relaxation-compensated pulse sequences are illustrated for the simplest case of an IS spin system. The pulse sequence is shown in Fig. 8.14a. Following the initial INEPT period, the density operator at point A is proportional to 2IzSy. Between points A and B, the relaxation of the density operator is described by [8.19], with "AB ¼ 0.5 [1 þ sinc[2JIS cp)] (36, 37). The period U converts antiphase 2IzSy coherence to in-phase Sy coherence and suppresses CSA/dipolar cross-correlation to first order. Gradients G4 eliminate any imperfections in the 1808 pulses, and the 1H 908 pulse at the end of the U period converts any residual antiphase magnetization to multiple-quantum coherence, which is not detected. Between points C and D, the relaxation of the density operator is described by [8.19], with "CD ¼ 0.5 [1 sinc[2JIS cp)]. The total decay of the magnitude of the density operator between points A and D is given by
Ið8ncp Þ ¼ I0 exp 4ncp R2 ð1=cp Þ þ "AB Rext
exp 4ncp R2 ð1=cp Þ þ "CD Rext
¼ I0 exp 8ncp R2 ð1=cp Þ þ Rext =2 , ½8:41 in which the efficiency of U is incorporated into I0. Thus, the effective transverse relaxation constant for relaxation mechanisms other than chemical exchange is independent of cp. After the final spin echo period, the magnetization is then frequency labeled during t1 and returned back FIGURE 8.14—Continued Unlabeled gradients are used to suppress unwanted coherences and pulse imperfections (119). In panels a and b, the phase cycle is 1 ¼ x, x; 2 ¼ 4(x), 4(x); 3 ¼ x, x, y, y, x, x, y, y; 4 ¼ x; receiver ¼ x, x, x, x. Gradient coherence selection are achieved with gradients Gd and Ge. Echo/antiecho signals are recorded in separate experiments by inverting the amplitude of Gd and phase 4 (120). The 1 and receiver phases are inverted for each t1 increment to shift axial peaks to the edge of the spectrum (121). In panel c, the phase cycle is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ 8(x), 8(x); 4 ¼ 4(x), 4(y), 4(x), 4(y); receiver ¼ x, x, x, x, x, x, x, x. Frequency discrimination during t1 is obtained by shifting the phase of 2 and the receiver according the TPPI–States protocol (121).
714
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
to the I spins for detection by the refocused INEPT period. One drawback of the illustrated sequence is that the minimum relaxation delay is 4 cp. Conventional CPMG pulse sequences for measuring R2 are self-compensating for evolution of the scalar coupling if cp ¼ m/JIS, in which m is an integer (104). Hahn spin echo measurements, selfcompensated CPMG experiments, and relaxation-compensated CPMG experiments can be used in combination to cover the widest possible range of cp values (104). For I2S and I3S spin systems, relaxation is multiexponential and constant-relaxation-time experiments are necessary to obtain accurate results (102, 103). In this approach, relaxation rate constants are determined from measurements recorded with a relaxation delay of 0 and T using [8.10]. The relaxation dispersion curve is generated by varying cp such that T is constant. If relaxation is multiexponential due to, for example, multiple dipole–dipole interactions, then X hAiðTÞ ¼ exp Rex 1=cp T aj exp Rj T hAið0Þ, ½8:42 j
in which ai and Ri are the amplitude and rate constant for the ith term in the multiexponential expansion, respectively, and Rex (1/ cp ) is 0the exchange damping constant, given, for example, by R2 1=cp R2 in [8.38]–[8.40]. Thus, using [8.10], an effective relaxation rate constant is defined: Reff 1=cp ¼ ð1=TÞ ln½hAið0Þ=hAiðTÞ " # X ½8:43 ¼ Rex 1=cp ð1=TÞ ln aj exp Rj T : j
The second term does not depend on cp and consequently represents a constant offset to the relaxation dispersion curve and is determined as part of the curve-fitting analysis. A pulse sequence for a relaxation-compensated CPMG experiment for 1H spins is shown in Fig. 8.15 (105). This experiment is applicable to 1 H spins that do not have significant homonuclear scalar coupling interactions; for example, the 3JHNH scalar coupling interaction is eliminated in uniformly 15N/2H-enriched proteins. In this experiment, the 15N frequency-labeling period is placed before the CPMG element. As a result, the 1H spins relax as unlike spins, even in the limit of fast pulsing (106). In addition, any ROESY cross-peaks that do arise from cross-relaxation when pulsing is fast are not degenerate with the main HSQC peaks, unless the 15N shifts are degenerate. This experiment is
715
8.3 MICROSECOND–SECOND DYNAMICS f2
y y 1H
15N
y
∆ ∆ f1
U
y
∆ ∆ f1
t1
t1
2
2
f3
tcp
f5 d f4 f4 d
tcp
GARP
Grad 2n
2n 1
FIGURE 8.15 Pulse sequence for an amide H CPMG experiment in 15N/2Hlabeled proteins. Narrow and wide bars depict 908 and 1808 pulses, respectively. All pulses are x-phase unless otherwise indicated. Decoupling during acquisition is achieved with the GARP or WALTZ-16 sequences (117, 122). Delays are ¼ 1 / (4JIS), and is longer than the included gradients. Unlabeled gradients are used to suppress unwanted coherences and pulse imperfections (119). The phase cycle is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ 8(x), 8(x); 4 ¼ 4(x), 4(y), 4(x), 4(y); receiver ¼ x, x, x, x, x, x, x, x. Frequency discrimination during t1 is obtained by shifting the phase of 1 and the receiver according the TPPI–States protocol (121).
performed using the constant-relaxation-time approach described previously to reduce the effects of multiexponential 1H dipole–dipole relaxation on the measured dispersion curves.
8.3.5 CHEMICAL EXCHANGE SPECTROSCOPY
IN
MULTIPLE-QUANTUM
If two spins are affected by the same chemical exchange kinetic process, then the chemical shift changes for the two spins resulting from transitions between sites will be correlated. This correlation gives rise to exchange effects that can either broaden or narrow resonance lineshapes for multiple-quantum coherences (107). In essence, for multiplequantum coherences, the value of ! in the equations given here and in Section 5.6 should be replaced by the difference in multiple-quantum (MQ) frequencies, !MQ; thus, for double-quantum (DQ) and zeroquantum (ZQ) coherences, respectively, !DQ ¼ !I þ !S and !ZQ ¼ !I !S, in which !I (!S) is the chemical shift difference
716
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
between sites A1 and A2 for the I (S) spin. Hahn spin echo pulse sequences for measuring the difference between the relaxation rate constants for DQ and ZQ coherences have been developed for 1H–15N backbone amide moieties (108) and for 13C–13C moieties (109) in proteins. In some cases, selection of the narrower of the DQ or ZQ coherences (depending on the relative chemical shifts) results in improved resolution and sensitivity of multidimensional NMR spectra (107, 110). A pulse sequence for MQ relaxation in 1H–15N backbone amide moieties is shown in Fig. 8.16. The experiment creates 2IySy coherence at the beginning of the relaxation period T and detects 2IySy coherence in one experiment and 2IxSx coherence in the second experiment. The
cross-relaxation rate constant is obtained from h2Ix Sx i(T) / 2Iy Sy (T) using [8.13]. The relaxation of these operators is described by d h 2Ix Sx iðtÞ h 2Ix Sx iðtÞ R2 ¼ : ½8:44 2Iy Sy ðtÞ R2 dt 2Iy Sy ðtÞ Recall from Section 2.7.5 that
h 2Ix Sx iðtÞ 1 1 DQx ðtÞ ¼ 2Iy Sy ðtÞ 1 1 ZQx ðtÞ gives
RDQ 1 1 1 ¼ 2 1 1 0 R2 " RDQ þ RZQ =2 ¼ RDQ RZQ =2 Thus, 2 ¼ RDQ RZQ ¼ RMQ . For two-site chemical exchange in the
R2
0
0 1 1 RZQ 1 1 # RDQ RZQ =2 : RDQ þ RZQ =2
½8:45
½8:46
fast-exchange limit (108),
0
RMQ ¼ RDQ RZQ ¼ RDQ RZQ þ 4p1 p2 !I !S =kex ,
½8:47
0 R2
in which R1 and are the population average relaxation rate constants in the absence of exchange. When p1 p2, RMQ is given by (2, 111) 0
0
RMQ ¼ RDQ RZQ þ
4p21 p2 !I !S k3ex 2 p41 k4ex þ 2p21 k2ex !2I þ !2S þ !2I !2S ½8:48 0 RZQ
0 RDQ
typically is small; for all time scales. The difference consequently, the sign of RMQ gives the relative sign of the chemical
717
8.3 MICROSECOND–SECOND DYNAMICS
a 1H
15N
y ∆
f2
y ∆
f1
f1 T/2
T/2
f4
f2 f3 t1
y ∆
y y -x y ∆ ∆ ∆ d d
f 5 f6 f 6 e
e
GARP
Grad Ge
b
c
0.50
0.25
0
Gd
30
∆RMQ (s–1)
<2IySy>(t)/<2IxSx>(t)
0.75
0.02
0.04
T (s)
0.06
20
10
0
10
20
30
40
50
60
70
80
Residue
FIGURE 8.16 Multiple-quantum relaxation in ubiquitin. (a) Pulse sequence for measuring 1H–15N RMQ. Narrow and wide bars correspond to 908 and 1808 pulses, respectively. All pulses are applied with x-phase unless specified otherwise. The shaped pulse is a band-selective 1808 pulse centered on the amide 1H spectral region. Delay durations are ¼ 1 / (4JIS), 4 Gd, and, " 4 Ge. Two experiments are performed for each value of the relaxation period T. In the first experiment, 2 ¼ y, and 2IxSx magnetization is detected. In the second experiment, 2 ¼ x, and 2IySy magnetization is detected. Decoupling is achieved with the GARP sequence during acquisition. The phase cycling is 1 ¼ y, y; 3 ¼ x, x, x, x; 4 ¼ x, x, x, x; 5 ¼ x; receiver ¼ x, x, x, x. The unlabeled gradients are used to suppress unwanted coherences and pulse imperfections (119). PEP gradient coherence selection is achieved with gradients Gd and Ge. Echo/antiecho signals are recorded in separate experiments by inverting the amplitude of Ge and phase 6 (120). The 1 and receiver phases are inverted for each t1 increment to shift axial peaks to the edge of the spectrum (121). (b) Relaxation decay curves for residues 23 () and 70 ( ) in ubiquitin, known to be subject to chemical exchange broadening (124). Data were recorded at 500 MHz and a temperature of 280 K. Solid lines are the best fits to [8.17], yielding RMQ ¼ 25.2 1.5 s–1 and 7.2 0.4 s–1 for residues 23 and 70, respectively. (c) Values of RMQ are shown as a function of residue position.
shift differences for I and S spins (108). This information, which cannot be obtained from single-quantum experiments, is helpful for mechanistic interpretations of exchange phenomena. Examples of the measurement of RMQ for ubiquitin are shown in Fig. 8.16. Residue 25 exhibits a large exchange broadening in SQ
718
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
experiments (Fig. 8.10), but a small value of RMQ; consequently, !H must be close to zero for the exchange process affecting this residue. In contrast, residue 23 shows large exchange broadening in both SQ and MQ experiments; consequently, both !H and !N must be significant. For a heteronuclear spin system (S ¼ 15N or 13C and I ¼ 1H), the HMQC experiment records the average of the DQ and ZQ evolution frequencies, while the HSQC experiment records the single-quantum frequency (Section 7.1). When p1 p2, the difference between the frequencies recorded in the HSQC and HMQC experiments is given by (2, 112) ¼ HSQC HMQC # " p1 p2 !S !2I k2ex 3p21 k2ex þ !2I !2S ¼ 2 : p21 k2ex þ !2S p41 k4ex þ 2p21 k2ex !2I þ !2S þ !2I !2S ½8:49 As shown by Skrynnikov and co-workers (112), the second bracketed term is positive for a wide range of applicable parameters. Therefore, the sign of is the same as the sign of !S. Thus, provided that exchange is not in the fast limit, the sign of !S can be determined simply by comparing HSQC and HMQC spectra recorded under identical conditions.
8.3.6 TROSY-BASED APPROACHES Hahn echo (113) and CPMG (106, 114, 115) pulse sequences have been modified to select the narrow TROSY (Chapter 7) doublet component during the relaxation, indirect labeling, and detection periods. These modifications facilitate the quantification of exchange line-broadening contributions in larger macromolecules because the relative conformational exchange contribution to the phenomenological relaxation rate constant is enhanced and the improved resolution and sensitivity of TROSY 1H–15N correlation spectra are obtained. Fig. 8.17 shows a TROSY pulse sequence for measuring relaxation of the two scalar-coupled doublet components to simultaneously determine Rex and xy (113). During the spin echo period of Fig. 8.17a, the signal intensities of the narrow and wide components
719
8.3 MICROSECOND–SECOND DYNAMICS
a
b
T 1
H
f3
y
y ∆ ∆
15
N
f1 f1
f2
y t1
A
y
∆ ∆ ∆
y y
x
x
T/2
e
∆
rec.
f4
f1 y y f2
e z
B
A
B
Grad Ge Ge
Gd
FIGURE 8.17 TROSY-based methods for detecting 15N Rex in large, deuterated proteins. The sequence shown in panel a detects relaxation of the narrow doublet component during the Hahn spin echo period he ¼ 2 when the grey pulse element is (908x 908y 908y 908x ) and detects relaxation of the broad doublet component when this element is (908x 90y8 908y 908). x Relaxation of longitudinal two-spin order is detected if the elements between points A and B in panel a are replaced with pulse elements in panel b; the gray element in panel b is 1 (908x 908y 908y 908). x These pulses are applied at the center of the amide H region to minimize off-resonance effects. Other delays are ¼ 1 / 4JIS, x 4 Ge, " ¼ – / 2, 4 Gd. All pulses have x-phase unless otherwise stated. Phase cycles are 1 ¼ x, x; 2 ¼ x, x, x, x; 3 ¼ y; 4 ¼ x; receiver phase ¼ x, x, x, x. Gradients Ge and Gd are used for coherence selection; other gradients are for artifact suppression. Echo/antiecho quadrature detection is achieved by inverting 3, 4, and the sign of gradient Ge. The 1 and receiver phases are inverted for each t1 increment to shift axial peaks to the edge of the spectrum (121).
of the 15N doublet, Sþ I ðTÞ and Sþ I ðTÞ, respectively, decay according to h i þ
S I ðTÞ ¼ Ið0Þ exp R2 T , h i ½8:50 þ
S I ðTÞ ¼ Ið0Þ exp R2 þ 2xy T , in which the initial intensity I(0) is identical for both doublet components, and 0
R2 ¼ R2 xy þ Rext =2 þ Rex :
½8:51
720
CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
Cross-relaxation between the two doublet components is suppressed by choosing T ¼ n/JIS, where n is a positive The experiment is þinteger.
I ð T Þ is detected and once performed twice: once in which the signal S
in which the signal Sþ I ðTÞ is detected. Thus,
xy ¼ ð1=2TÞ ln Sþ I ðTÞ= Sþ I ðTÞ : ½8:52 z Sz The relaxation rate constant R2 R2I =2 is obtained by recording a 1 third spectrum using the sequence shown in Fig. 8.17b. During the relaxation period of length T/2, the signal decays according to z Sz h2Iz Sz iðT=2Þ ¼ Ið0Þ exp R2I T=2 : ½8:53 1
Therefore,
z Sz =2 ¼ ð1=TÞ ln h2Iz Sz iðT=2Þ= Sþ I ðTÞ : R2 R2I 1
½8:54
The relaxation rate constant for two-spin order is given approximately by 15 z Sz ¼ R1 þ RH R2I 1 . For proteins with a molecular mass 420 kDa, the N 1 R1 is negligibly small. Thus, Rex can be determined from z Sz =2 ðk 1Þxy : Rex ¼ R2 R2I 1
½8:55
A minimum of three two-dimensional NMR spectra are sufficient to determine Rex using this approach. 15 Figure N 8.18 shows a CPMG pulse sequence used for measuring 1 R2 1=cp as a function of cp for the narrow component of the H–15N scalar-coupled doublet (115). The pulse sequence is designed to suppress cross-relaxation between the doublet components during the CPMG relaxation period, which would otherwise contribute to the apparent cp dependence of the phenomenological relaxation rate constants. As shown by [5.146], the cross relaxation rate constant DD R =2 for free-precession evolution. In a CPMG experi0 ¼ RDD 2I 2IS ment, the cross-relaxation rate constant also depends on cp because the average overlap of the SþI and SþI operators depends on the relative phase angles acquired by the operators during the spin echo period (37). Thus, the effective cross-relaxation rate constant ¼ 0 sinc(J cp). In the limit where cp !0, !0, and as cp ! 1, ! 0. Suppression of the effects of cross-relaxation is achieved by the pulse sequence element U, which uses the S3CT sequence element (116) to selectively invert one of the doublet components at a time T/2. Thus, U is given by [8.7] and the decay of magnetization is described by [8.8].
721
8.3 MICROSECOND–SECOND DYNAMICS U
1H
–y
15
N
f3
–y –y ∆
∆ f1 f1
∆
∆
f4 –x ∆
y f2
y
–x –x ∆
∆ y
–x
–x ∆
rec.
f5
t1 tcp
tcp
Grad 2n
2n
FIGURE 8.18 Pulse sequence for the 15N TROSY–CPMG experiment. Narrow and wide bars depict 908 and 1808 pulses, respectively. Short, solid bars are water-selective 908 pulses. All pulses are x-phase unless otherwise indicated. The delays are ¼ cp / 2 and ¼ 1 / (4JIS). The relaxation period is T ¼ 4n cp. The data are acquired using the PEP scheme in which echo and antiecho FIDs are acquired for each t1 point (125). The phase cycle for the first FID is 1 ¼ 4(x), 4(–x); 2 ¼ –y, y, x, –x; 3 ¼ y; 4 ¼ y; 5 ¼ x; receiver ¼ x, x, y, y, –x, x, y, y. The phase cycle for the second FID is 1 ¼ 4(x), 4(x); 2 ¼ y, y, x, x; 3 ¼ y; 4 ¼ y; 5 ¼ x; receiver ¼ x, x, y, y, x, x, y, y. Gradients are used to suppress unwanted coherences and pulse imperfections (119) and perform water flip-back solvent suppression (123).
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
A. G. Palmer, J. Williams, A. McDermott, J. Phys. Chem. 100, 13293–13310 (1996). A. G. Palmer, Chem. Rev. 104, 3623–3640 (2004). P. Pelupessy, S. Ravindranathan, G. Bodenhausen, J. Biomol. NMR 25, 265–280 (2003). V. A. Daragan, K. H. Mayo, Prog. NMR Spectrosc. 31, 63–105 (1997). M. W. F. Fischer, A. Majumdar, E. R. P. Zuiderweg, Prog. NMR Spectrosc. 33, 207–272 (1998). L. E. Kay, Nat. Struct. Biol. 5, 513–517 (1998). A. G. Palmer, Curr. Opin. Struct. Biol. 7, 732–737 (1997). A. G. Palmer, C. D. Kroenke, J. P. Loria, Meth. Enzymol. 339, 204–238 (2001). A. G. Palmer, Annu. Rev. Biophys. Biomolec. Struct. 30, 129–155 (2001). C. D. Kroenke, J. P. Loria, L. K. Lee, M. Rance, A. G. Palmer, J. Am. Chem. Soc. 120, 7905–7915 (1998). J. A. Jones, J. Magn. Reson. 126, 283–286 (1997). G. H. Weiss, J. A. Ferretti, Prog. NMR Spectrosc. 20, 317–335 (1988). A. M. Mandel, A. G. Palmer, J. Magn. Reson., Ser. A 110, 62–72 (1994). L. E. Kay, J. H. Prestegard, J. Magn. Reson. 77, 599–605 (1988). P. A. Carr, D. A. Fearing, A. G. Palmer, J. Magn. Reson. 132, 25–33 (1998).
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CHAPTER 8 EXPERIMENTAL NMR RELAXATION METHODS
16. 17. 18. 19. 20. 21.
G. Bodenhausen, R. R. Ernst, J. Am. Chem. Soc. 104, 1304–1309 (1982). N. Tjandra, A. Szabo, A. Bax, J. Am. Chem. Soc. 118, 6986–6991 (1996). P. Pelupessy, G. M. Espallargas, G. Bodenhausen, J. Magn. Res. 161, 258–264 (2003). B. Reif, M. Hennig, C. Griesinger, Science 276, 1230–1233 (1997). D. W. Yang, A. Mittermaier, Y. K. Mok, L. E. Kay, J. Mol. Biol. 276, 939–954 (1998). B. Reif, A. Diener, M. Hennig, M. Maurer, C. Griesinger, J. Magn. Res. 143, 45–68 (2000). N. A. Farrow, R. Muhandiram, A. U. Singer, S. M. Pascal, C. M. Kay, G. Gish, S. E. Shoelson, T. Pawson, J. D. Forman-Kay, L. E. Kay, Biochemistry 33, 5984–6003 (1994). A. C. Wang, A. Bax, J. Biomol. NMR 3, 715–720 (1993). J. Boyd, J. Magn. Reson., Ser. B 107, 279–285 (1995). K. T. Dayie and G. Wagner, J. Magn. Reson., Ser. A, 111, 121–124 (1994). N. J. Skelton, A. G. Palmer, M. Akke, J. Ko¨rdel, M. Rance, W. J. Chazin, J. Magn. Reson. Ser. B 102, 253–264 (1993). G. Zhu, Y. Xia, L. K. Nicholson, K. H. Sze, J. Magn. Reson. 143, 423–426 (2000). A. G. Palmer, M. Rance, P. E. Wright, J. Am. Chem. Soc. 113, 4371–4380 (1991). A. J. Wand, R. J. Bieber, J. L. Urbauer, R. P. McEvoy, Z. Gan, J. Magn. Reson. Ser. B 108, 173–175 (1995). D. M. LeMaster, D. M. Kushlan, J. Am. Chem. Soc. 118, 9263–9272 (1996). R. L. Vold, J. S. Waugh, M. P. Klein, D. E. Phelps, J. Chem. Phys. 48, 3831–3832 (1968). J. Boyd, U. Hommel, I. D. Campbell, Chem. Phys. Lett. 175, 477–482 (1990). H. Y. Carr, E. M. Purcell, Phys. Rev. 94, 630–638 (1954). S. Meiboom, D. Gill, Rev. Sci. Instrum. 29, 688–691 (1958). J. W. Peng, V. Thanabal, G. Wagner, J. Magn. Reson. 94, 82–100 (1991). L. E. Kay, L. K. Nicholson, F. Delagio, A. Bax, D. A. Torchia, J. Magn. Reson. 97, 359–375 (1992). A. G. Palmer, N. J. Skelton, W. J. Chazin, P. E. Wright, M. Rance, Mol. Phys. 75, 699– 711 (1992). D. M. Korzhnev, E. V. Tischenko, A. S. Arseniev, J. Biomol. NMR 17, 231–237 (2000). A. Ross, M. Czisch, G. C. King, J. Magn. Reson. 124, 355–365 (1997). F. Massi, C. Wang, E. Johnson, M. Rance, A. G. Palmer, J. Am. Chem. Soc. 126, 2247–2256 (2004). D. Idiyatullin, V. A. Daragan, K. H. Mayo, J. Magn. Res. 153, 138–143 (2001). S. Grzesiek, A. Bax, J. Biomol. NMR 3, 185–204 (1993). T. M. Logan, E. T. Olejniczak, R. X. Xu, S. W. Fesik, J. Biomol. NMR 3, 225–231 (1993). D. Fushman, N. Tjandra, D. Cowburn, J. Am. Chem. Soc. 120, 10947–10952 (1998). C. D. Kroenke, M. Rance, A. G. Palmer, J. Am. Chem. Soc. 121, 10119–10125 (1999). C. Wang, A. G. Palmer, Magn. Reson. Chem. 41, 866–876 (2003). M. W. F. Fischer, L. Zeng, Y. Pang, W. Hu, A. Majumdar, E. R. P. Zuiderweg, J. Am. Chem. Soc. 119, 12629–12642 (1997). R. Ghose, K. Huang, J. H. Prestegard, J. Magn. Reson. 135, 487–499 (1998). T. J. Norwood, M. L. Tillett, L.-Y. Lian, Chem. Phys. Lett. 300, 429–434 (1999). M. Pellecchia, Y. Pang, L. Wang, A. V. Kurochkin, A. Kumar, E. R. P. Zuiderweg, J. Am. Chem. Soc. 121, 9165–9170 (1999). M. Tessari, G. W. Vuister, J. Biomol. NMR 16, 171–174 (2000). D. R. Muhandiram, T. Yamazaki, B. D. Sykes, L. E. Kay, J. Am. Chem. Soc. 117, 11536–11544 (1995). O. Millet, D. R. Muhandiram, N. R. Skrynnikov, L. E. Kay, J. Am. Chem. Soc. 124, 6439–6448 (2002).
22.
23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53.
8.3 MICROSECOND–SECOND DYNAMICS 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81.
82. 83. 84. 85. 86. 87. 88. 89. 90. 91.
723
N. R. Skrynnikov, O. Millet, L. E. Kay, J. Am. Chem. Soc. 124, 6449–6460 (2002). D. Yang, L. E. Kay, J. Magn. Reson., Ser. B 110, 213–218 (1996). A. L. Lee, P. F. Flynn, A. J. Wand, J. Am. Chem. Soc. 121, 2891–2902 (1999). R. Ishima, A. P. Petkova, J. M. Louis, D. A. Torchia, J. Am. Chem. Soc. 123, 6164–6171 (2001). K. Pervushin, G. Wider, K. Wu¨thrich, J. Am. Chem. Soc. 119, 3842–3843 (1997). M. Akke, A. G. Palmer, J. Am. Chem. Soc. 118, 911–912 (1996). F. Cordier, B. Brutscher, D. Marion, J. Biomol. NMR 7, 163–168 (1996). K. Dayie, G. Wagner, J. Am. Chem. Soc. 119, 7797–7806 (1997). J. Engelke, H. Ru¨terjans, J. Biomol. NMR 9, 63–78 (1997). L. Zeng, M. W. F. Fischer, E. R. P. Zuiderweg, J. Biomol. NMR 7, 157–162 (1996). P. Allard, T. Ha¨rd, J. Magn. Reson. 126, 48–57 (1997). S. F. Lienin, T. Bremi, B. Brutscher, R. Bru¨schweiler, R. R. Ernst, J. Am. Chem. Soc. 120, 9870–9879 (1998). T. Wang, S. Cai, E. R. P. Zuiderweg, J. Am. Chem. Soc. 125, 8639–8643 (2003). M. Akke, J. Liu, J. Cavanagh, H. P. Erickson, A. G. Palmer, Nat. Struct. Biol. 5, 55–59 (1998). L. C. Wang, Y. X. Pang, T. Holder, J. R. Brender, A. V. Kurochkin, E. R. P. Zuiderweg, Proc. Natl. Acad. Sci. U.S.A. 98, 7684–7689 (2001). A. J. Shaka, J. Keeler, Prog. NMR Spectrosc. 19, 47–129 (1987). A. M. Mandel, M. Akke, A. G. Palmer, J. Mol. Biol. 246, 144–163 (1995). L. E. Kay, D. A. Torchia, A. Bax, Biochemistry 28, 8972–8979 (1989). I. Q. H. Phan, J. Boyd, I. D. Campbell, J. Biomol. NMR 8, 369–378 (1996). B. D. N. Rao, Meth. Enzymol. 176, 279–311 (1989). J. Sandstrom, ‘‘Dynamic NMR Spectroscopy.’’ Academic Press, London, 1982. R. E. Burton, R. S. Busby, T. G. Oas, J. Biomol. NMR 11, 355–359 (1998). M. E. Holtzer, G. L. Bretthorst, D. A. d’Avignon, R. H. Angeletti, L. Mints, A. Holtzer, Biophys. J. 80, 939–951 (2001). B. Kuhlman, D. L. Luisi, P. A. Evans, D. P. Raleigh, J. Mol. Biol. 284, 1661–1670 (1998). D. Kern, G. Kern, G. Scherer, G. Fischer, T. Drakenberg, Biochemistry 34, 13594–13602 (1995). U. Gu¨nther, T. Mittag, B. Schaffhausen, Biochemistry 41, 11658–11669 (2002). T. Mittag, B. Schaffhaussen, U. L. Gu¨nther, J. Am. Chem. Soc. 126, 9017–9023 (2004). D. S. Korchuganov, S. B. Nolde, M. Y. Reibarkh, V. Y. Orekhov, A. A. Schulga, Y. S. Ermolyuk, M. P. Kirpichnikov, A. S. Arseniev, J. Am. Chem. Soc. 123, 2068–2069 (2001). J. I. Kaplan, G. Fraenkel, ‘‘NMR of Chemically Exchanging Systems.’’ Academic Press, New York, 1980. J. J. Led, H. Gesmar, F. Abildgaard, Meth. Enzymol. 176, 311–329 (1989). J. Fejzo, Z. Zolnai, S. Macura, J. L. Markley, J. Magn. Reson. 88, 93–110 (1990). N. Farrow, O. Zhang, J. D. Forman-Kay, L. E. Kay, J. Biomol. NMR 4, 727–734 (1994). G. T. Montelione, G. Wagner, J. Am. Chem. Soc. 111, 3096–3098 (1989). G. Wider, D. Neri, K. Wu¨thrich, J. Biomol. NMR 1, 93–98 (1991). H. Wang, Y. Hea, C. D. Kroenke, S. Kodukulab, J. Storch, A. G. Palmer, R. E. Stark, Biochemistry 41, 5453–5461 (2001). D. G. Davis, M. E. Perlman, R. E. London, J. Magn. Reson. Ser. B 104, 266–275 (1994). C. Deverell, R. E. Morgan, J. H. Strange, Mol. Phys. 18, 553–559 (1970). T. Szyperski, P. Luginbu¨hl, G. Otting, P. Gu¨ntert, K. Wu¨thrich, J. Biomol. NMR 3, 151–164 (1993).
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92. F. A. A. Mulder, R. A. de Graaf, R. Kaptein, R. Boelens, J. Magn. Reson. 131, 351–357 (1998). 93. S. Zinn-Justin, P. Berthault, M. Guenneugues, H. Desvaux, J. Biomol. NMR 10, 363–372 (1997). 94. D. Abergel, A. G. Palmer, Concepts Magn. Reson. 19A, 134–148 (2003). 95. O. Trott, A. G. Palmer, J. Magn. Reson. 154, 157–160 (2002). 96. O. Trott, D. Abergel, A. G. Palmer, Mol. Phys. 101, 753–763 (2003). 97. A. Allerhand, E. Thiele, J. Chem. Phys. 45, 902–916 (1966). 98. J. P. Carver, R. E. Richards, J. Magn. Reson. 6, 89–105 (1972). 99. W. T. Sobol, Magn. Reson. Med. 21, 2–9 (1991). 100. J. Jen, J. Magn. Reson. 30, 111–128 (1978). 101. J. P. Loria, M. Rance, A. G. Palmer, J. Am. Chem. Soc. 121, 2331–2332 (1999). 102. F. A. A. Mulder, N. R. Skrynnikov, B. Hon, F. W. Dahlquist, L. E. Kay, J. Am. Chem. Soc. 123, 967–975 (2001). 103. N. R. Skrynnikov, F. A. A. Mulder, B. Hon, F. W. Dahlquist, L. E. Kay, J. Am. Chem. Soc. 123, 4556–4566 (2001). 104. O. Millet, J. P. Loria, C. D. Kroenke, M. Pons, A. G. Palmer, J. Am. Chem. Soc. 122, 2867–2877 (2000). 105. R. Ishima, D. A. Torchia, J. Biomol. NMR 25, 243–248 (2003). 106. R. Ishima, P. T. Wingfield, S. J. Stahl, J. D. Kaufman, D. A. Torchia, J. Am. Chem. Soc. 120, 10534–10542 (1998). 107. M. Rance, J. Am. Chem. Soc. 110, 1973–1974 (1988). 108. K. Kloiber, R. Konrat, J. Biomol. NMR 18, 33–42 (2000). 109. D. Fruh, J. R. Tolman, G. Bodenhausen, C. Zwahlen, J. Am. Chem. Soc. 123, 4810–4816 (2001). 110. K. Pervushin, J. Biomol. NMR 20, 275–285 (2001). 111. C. Wang, A. G. Palmer, J. Biomol. NMR 24, 263–268 (2002). 112. N. R. Skrynnikov, F. W. Dahlquist, L. E. Kay, J. Am. Chem. Soc. 124, 12352–12360 (2002). 113. C. Wang, M. Rance, A. G. Palmer, J. Am. Chem. Soc. 125, 8968–8969 (2003). 114. J. Boisbouvier, B. Brutscher, J.-P. Simorre, D. Marion, J. Biomol. NMR 14, 241–252 (1999). 115. J. P. Loria, M. Rance, A. G. Palmer, J. Biomol. NMR 15, 151–155 (1999). 116. M. D. Sørensen, A. Meissner, O. W. Sørensen, J. Biomol. NMR 10, 181–186 (1997). 117. A. J. Shaka, P. B. Barker, R. Freeman, J. Magn. Reson. 64, 547–552 (1985). 118. T. Yamazaki, R. Muhandiram, L. E. Kay, J. Am. Chem. Soc. 116, 8266–8278 (1994). 119. A. Bax, S. S. Pochapsky, J. Magn. Reson. 99, 638–643 (1992). 120. L. E. Kay, P. Keifer, T. Saarinen, J. Am. Chem. Soc. 114, 10663–10665 (1992). 121. D. Marion, M. Ikura, R. Tschudin, A. Bax, J. Magn. Reson. 85, 393–399 (1989). 122. A. J. Shaka, J. Keeler, T. Frenkiel, R. Freeman, J. Magn. Reson. 52, 335–338 (1983). 123. S. Grzesiek, A. Bax, J. Am. Chem. Soc. 115, 12593–12594 (1993). 124. F. Massi, M. J. Grey and A. G. Palmer, Protein Sci. 14, 735–742 (2005). 125. M. Rance, J. P. Loria, A. G. Palmer, J. Magn. Reson. 136, 92–101 (1999).
CHAPTER
9 LARGER PROTEINS AND MOLECULAR INTERACTIONS
A hallmark of the historical development of biological NMR spectroscopy is the continued increase in the size of the molecular species amenable to investigation. The first part of this chapter presents methods for resonance assignments and structure determination applicable to proteins with molecular masses of 420–30 kDa. These techniques are modifications of the experiments presented in Chapter 7 for smaller proteins. Increases in the size limits for solution NMR spectroscopy also facilitate investigation of molecular interactions and complex formation by proteins. The second part of this chapter discusses some of the many NMR techniques for detecting and quantifying intermolecular interactions between proteins and ligands. Efficiency in data acquistion remains an important concern. The final part of this chapter briefly discusses emerging methods for recording spectral data more rapidly.
9.1 Larger Proteins The multidimensional triple-resonance NMR techniques described in Chapter 7 allow characterization of proteins up to a molecular mass
725
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AND
MOLECULAR INTERACTIONS
of 20–30 kDa. In addition to the resolving power afforded by multiple frequency dimensions, the central premise behind such experiments is that transfer of magnetization occurs much more efficiently through large heteronuclear scalar couplings and 13C–13C homonuclear couplings than through small 1H–1H homonuclear scalar couplings (Table 7.2). Concomitant short magnetization transfer periods are employed to maximize sensitivity by keeping transverse relaxation losses to a minimum. However, as molecular masses increase beyond 20 kDa, proteins become progressively more difficult to study by these established methods (1–7). The NMR spectra of larger proteins contain more resonances and, because of longer rotational correlation times and correspondingly faster transverse relaxation, these resonances have larger linewidths. Consequently, resonance congestion escalates rapidly with increased molecular size. Increased transverse relaxation rate constants also decrease overall sensitivity, because less magnetization survives through pulse sequence delays to be detected by the receiver. Accordingly, for larger proteins, standard 13C/15N-triple-resonance NMR experiments become ineffective. The following sections discuss approaches for overcoming these limitations [for a review, see Permi (8)].
9.1.1 PROTEIN DEUTERATION Dramatically reducing the length of INEPT magnetization transfer periods, or other pulse sequence elements that rely on evolution under the scalar coupling Hamiltonian, in heteronuclear NMR experiments is not feasible, because delays of length 1/(2JIS) are required to ensure efficient coherence transfer from a given spin to its scalar-coupled partners (Sections 2.7.7.2 and 5.1.1). However, transverse relaxation rate constants can be reduced in larger proteins by eliminating (or avoiding) unfavorable relaxation pathways. The TROSY experiment discussed in Section 7.1.3 provides one illustration of this principle; the crossrelaxation-induced polarization transfer (CRIPT) technique uses relaxation interference rather than scalar coupling evolution to obtain coherence transfer (9, 10). Applications of optimal control theory to develop relaxation-optimized coherence transfer schemes are an active area of research (11). Sensitivity of 13C/15N double- and triple-resonance NMR experiments used for assignment of protein backbone resonances is affected most adversely by increases in the transverse relaxation rate constants for 13C and 1HN spins. The 1H–13C dipolar interaction dominates relaxation of 13C spins for carbon atoms with directly attached hydrogen atoms (12, 13). Relaxation of 1HN spins has a
9.1 LARGER PROTEINS
727
substantial component (40%) arising from dipolar interactions with proximal aliphatic 1H spins (14). Consequently, significant increases in both resolution and sensitivity can be realized by eliminating these specific relaxation pathways, thereby reducing R2 for both 13C and 1HN spins. This can be achieved by replacing most, if not all, carbon-bound hydrogen atoms in a protein with deuterium (2H or D) (15–22). The two labeling strategies in common usage are referred to as (i) perdeuteration and (ii) random fractional deuteration. Complete deuteration, or perdeuteration, replaces 99% of all carbon-bound hydrogen atoms with deuterium atoms. Random fractional deuteration describes the distribution of deuterium throughout all protein molecules in the sample. For example, in a 70% randomly deuterated protein, on average over the entire sample, 70% of the protons have been replaced by deuterium, but not the same 70% in each molecule (23–25). In either case, proteins are produced biosynthetically by modifications of the overexpression protocols used to produce nondeuterated isotopically enriched proteins (15, 26). Perdeuterated proteins are obtained by growth of bacterial hosts on media containing perdeuterated carbon sources, such as perdeuterated [13C] glucose or [99% 2H, 13C, 15N] algal hydrolysate medium, in 99% D2O solution. To first approximation, random fractional deuteration is achieved by simply expressing the protein in minimal media containing the appropriate v/v ratio of D2O/H2O (14, 17, 20, 25). Average enrichment levels of 60–80% typically are obtained by growth on 99% D2O with protonated glucose carbon sources. With this protocol, isotopic enrichment may not be completely random, and enrichment at the more acidic C carbon typically is higher than the average. Frequently, cell growth rates are significantly slower and protein expression levels are lower in D2O solution. In addition, bacterial cells may require adaptation to growth in D2O (13). In distinction to either perdeuteration or random fractional deuteration, a third approach, termed SAIL (stereo-array isotope labeling), sterospecifically replaces 1H atoms with 2H atoms to provide optimal simplification of amino acid spin systems (27). This technique is highly promising, but not yet in general use. In some instances, biosynthetic metabolic pathways can be utilized to produce specified labeling patterns from labeled precursors added to growth media (Section 9.1.9) (28–32). For either perdeuterated or random fractionally deuterated proteins, 1 N H spins are reintroduced through amide proton solvent exchange by dissolving the protein in H2O buffer. Reintroduction of 1HN spins is slow for amide moieties that are highly protected from solvent exchange and a reduction in sensitivity for these residues may
728
CHAPTER 9 LARGER PROTEINS
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MOLECULAR INTERACTIONS
be observed. Unfolding of the protein to obtain complete exchange should be performed only if the subsequent refolding results in complete regeneration of the functional protein structure (33). For the purposes of the following discussion, 1HN spins are assumed to be fully present at all amide sites.
9.1.2 RELAXATION IN PERDEUTERATED AND RANDOM FRACTIONALLY DEUTERATED PROTEINS Deuteration, when combined with 13C/15N isotopic labeling, produces a protein often referred to as triply labeled (2H/13C/15N). Because the deuteron magnetogyric ratio ( D) is 6.5-fold smaller than the 1 H magnetogyric ratio ( H), a deuteron is [IH(IH þ 1)/ID(IDþ1)]( H/ D)2 16-fold less effective (where IH ¼ 1/2 and ID ¼ 1 are the spin quantum numbers of the 1H and 2H nuclei, respectively) than a proton at causing dipolar relaxation of the directly attached heteronucleus and surrounding 1H nuclei. Figure 9.1 displays calculated T2 relaxation times as a function of the rotational correlation time, c, for backbone nuclei in both a fully protonated protein and its perdeuterated counterpart. The calculations include both CSA and dipolar contributions (17, 34). These calculations show that perdeuteration dramatically increases the T2 (¼ 1/R2) relaxation times for 13C (13) and 1HN (14) spins, but does not affect the T2 relaxation of in-phase 15N and 13CO magnetization to significant extents. Figure 9.2 shows the average T2 and T1 relaxation times as a function of deuteration level for a protein with correlation time of 12 ns (at 600 MHz). The calculations include both CSA and dipolar contributions (25). The T2 relaxation times of both 1HN and 1H spins increase with increasing level of random fractional deuteration, with 1HN T2 showing a rapid increase for deuteration levels of 50% and above. At the same time, increases in 1HN T1 are small up to a deuteration level of 80%. These calculations suggest that ‘‘out-and-back’’ experiments, in which magnetization originates and is detected on the 1HN spin, will be effective for proteins that are 450% deuterated. Indeed, a 70% random fractionally deuterated protein was used to develop a suite of triple-resonance experiments for the backbone assignment of triply labeled proteins (13). Perdeuteration provides the largest increase in T2 relaxation times and consequent enhanced sensitivity and resolution in 1HN-detected 13C/15N triple-resonance correlation experiments, particularly for obtaining 1HN, 15N, 13C, and 13CO backbone and 13 C side chain assignments (17, 22, 35–37). Optimization of the recycle
729
9.1 LARGER PROTEINS
a
160
b 60
T2 (ms)
120 40 80 20 40
0
8
10
12
14
0 8 16 tc (ns)
10
12
14
16
FIGURE 9.1 Calculated T2 relaxation times versus the isotropic rotational correlation time for backbone nuclei in both a fully protonated protein (H) and its perdeuterated counterpart (D) (17). (a) () 13C(D), (---) 13C (H), (—) 13CO(D), (- - - -) 13CO(H). (b) () 1HN(D), (---) 1HN(H), (—) 15N(D), (- - - -) 15N(H). The T2(15N) relaxation times were calculated with the following parameters: N ¼ 160 ppm, rN-HN ¼ 1.02 A˚, rN-H ¼ 2.12 A˚, rN-CO ¼ 1.49 A˚, and rN-C ¼ 1.49 A˚. The T2(13CO) relaxation times were calculated with the following parameters: CO ¼ 102 ppm, rCO-C ¼ 1.54 A˚, rCO-N ¼ 1.49 A˚, rCO-H ¼ 2.16 A˚, and rCO-HN ¼ (2.24 A˚, 3.30 A˚). The T2(C) relaxation times were calculated with the following parameters: C ¼ 29.5 ppm for Gly and 21.5 ppm for Ala, rC -D ¼ 1.05 A˚, rC -H ¼ 1.09 A˚, rC -H ¼ 2.16 A˚, rC -CO ¼ 1.54 A˚, rC -N ¼ 1.49 A˚, rC -C ¼ 1.54 A˚ (Ala only), and rC -HN ¼ (2.24 A˚, 3.00 A˚). The T2(1HN) relaxation times represent an average of values from -helices and parallel -sheets. For an -helix, each 1HN spin was assumed to interact with four other amide 1HN spins at the following distances: 2.8, 2.8, 4.2, and 4.2 A˚. For a parallel -sheet, each amide 1HN spin was assumed to interact with three other amide 1HN spins at the following distances: 4.2, 4.2, and 4.0 A˚. In both cases, CSA contributions to the amide proton linewidth for a correlation time c ¼ 11.4 ns were estimated at 3.8 Hz.
delay to reflect longer T1 relaxation times may be necessary, particularly for perdeuterated proteins.
9.1.3 SENSITIVITY
FOR
PERDEUTERATED PROTEINS
For the gradient-enhanced, sensitivity-enhanced 1H–15N HSQC experiment (Section 7.1.4.2), the theoretical increase in sensitivity, Sr(D/H), due to perdeuteration, is calculated, using transfer functions
730
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
70
a 60
T2 (ms)
50
40
30
20
10
b
1.0
0.8
T1 (s)
0.6
0.4
0.2
0.0 0.0
0.20
0.4
0.6
0.8
1.0
fractional deuteration
FIGURE 9.2 Calculated average (a) transverse and (b) longitudinal relaxation times as a function of deuteration level, for (—) 1HN, (- - - -) 1H, (– - –) 15N, and (– – –) 13C nuclei in a protein with a correlation time c ¼ 12 ns at a static field strength of 14.1 T. Complete protonation of amide moieties is assumed. Calculations for 13C assume that the heteronucleus of interest is protonated. An increase is observed in the transverse and longitudinal relaxation times of all nuclei as the level of deuteration increases; however, the increase is much more pronounced for 1HN in both cases.
731
9.1 LARGER PROTEINS
that include both appropriate scalar coupling and transverse relaxation contributions, to be (17) Sr ðD=HÞ ¼ SrH ðD=HÞ SrN ðD=HÞ,
½9:1
in which D D SrH ðD=HÞ ¼ exp½ð2 H þ 22 ÞRH 2H ð2 þ 22 ÞR2H ( ) D D D D exp½2 D ðRD 2MQ þ R2H Þ þ exp½2 ðR2H þ R1H Þ H H H H exp½2 H ðRH 2MQ þ R2H Þ þ exp½2 ðR2H þ R1H Þ
sin2 ð2JNH D Þ=fsin2 ð2JNH H Þ cos2 ð2JHN H H Þg; ½9:2
SrN ðD=HÞ ¼
D exp½21 ðR2N
H R2N Þ
( H ) D R2N ð1 exp½t1max R2N Þ D
X ¼ ðJNH Þ1 tan1 ðJNH =RX 2H Þ, X
H
,
½9:3
R2N ð1 exp½t1max R2N Þ
X R2N ¼ ðRX 2N þ R2NH Þ=2,
½9:4 ½9:5
X X X where X ¼ {D, H}; RX 2H , R2N , R2NH , and R2MQ are the transverse 1 N relaxation rate constants for in-phase H magnetization, in-phase 15N magnetization, antiphase 15N magnetization, and 1H–15N multiplequantum coherence, respectively, in protonated (X ¼ H) or deuterated (X ¼ D) backgrounds; and RX 1H is the longitudinal relaxation rate constant for 1HN magnetization in protonated (X ¼ H) or deuterated (X ¼ D) backgrounds. The delay ¼ X is the optimal INEPT transfer delay adjusted for relaxation, and the delays 1 and 2 encompass the coherence-selection gradients, as was shown in Fig. 7.12. The expression D for Sr(D/H) is approximate because differences in RH 1H and R1H are neglected during the recycle delay. Venters and co-workers measured the relative sensitivity for perdeuterated and fully protonated human carbonic anhydrase II (HCA II), a protein of 29 kDa, c ¼ 11.4 ns at 308C (22). The average gain in sensitivity was a factor of 1.7 for -helix residues and 2.8 for -sheet residues. Using values of t1max ¼ 51.0 ms, 1 ¼ 5.5 ms, 2 ¼ 1.2 ms, D ¼ H ¼ 2.5 ms, JNH ¼ 92 Hz, and JHN H ¼ 4.5 Hz for an -helix and 9.0 Hz for a -sheet in [9.1] yields a theoretical gain in the signal-to-noise ratio in the gradient-enhanced, sensitivity-enhanced 1H–15N HSQC
732
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
experiment of 1.4 for residues in -helices and 3.0 for residues in -sheets. These theoretical results are in good agreement with the experimental measurements.
9.1.4 2H ISOTOPE SHIFTS Replacing a hydrogen atom with a deuteron causes isotope effects on the chemical shifts of nuclei within as many as four covalent bonds of the site of substitution (38–40). As a result, 13C chemical shifts in a fully protonated protein, C(H), will be different than those in a perdeuterated protein, C(D) (17, 22, 35, 40, 41). To convert C(H) to C(D), or vice versa, the 2H isotope effect, C(2H) must be calculated for each carbon atom in a perdeuterated protein. Venters et al. have measured C(2H) and C(2H) for all residue types by comparing the 13C chemical shifts in perdeuterated and protonated HCA II (17, 22). A statistical analysis suggests that 2H isotope effects can be predicted for 13C and 13C chemical shifts in perdeuterated proteins, as shown in Table 9.1. The values in Table 9.1 are consistent with average values of 1C(2H) ¼ 0.25 and 2C(2H) ¼ 0.1 ppm measured by Gardner and co-workers (41). Isotope effects are cumulative, and so, for 13C nuclei at positions surrounded by many deuterons, such as those in the middle of long amino acid side chains, total isotope shifts of over 1 ppm can be expected. For backbone nuclei, isotope shifts on the order of 0.3 ppm
TABLE 9.1 Total Deuterium Isotope Shifts for Residue Asn, Asp, Ser, His, Phe, Trp, Tyr, Cys Lys, Arg, Pro Gln, Glu, Met Ala Ile Leu Thr Val
13
C and
13
C Nucleia
C (ppm)
C (ppm)
0.55 0.69 0.69 0.68 0.77 0.62 0.63 0.84
0.71 1.11 0.97 1.00 1.28 1.26 0.81 1.20
C(2H) (one bond) ¼ 0.29 0.05 ppm; 2C(2H) (two bond) ¼ 0.13 0.02 ppm; C(2H) (three bond) ¼ 0.07 0.02 ppm; 1Cgly(2H) ¼ 0.39 0.04 ppm (measured separately).
a1 3
733
9.1 LARGER PROTEINS
for 15N and 0.5 ppm for 13C for highly deuterated proteins are typical. In addition, the 13C deuterium isotope shifts appear to be weakly dependent on secondary structure (42). These predictable properties allow the dependable transfer of 13C chemical shift assignments from perdeuterated to protonated molecules. Thus, estimated 13 C side chain isotope shifts can be used to assign 13C/15N- and 13C/13Cseparated NOESY spectra recorded on protonated protein samples. In addition, the corrected 13C (H) and 13C (H) chemical shifts can be used to accurately discern -helical and -strand components of secondary structure (43, 44). Perdeuteration ensures a homogeneous 2H isotopic environment for 13C and 15N nuclei and guarantees that losses in both sensitivity and resolution arising from the multibond 2H isotope effect are minimized (3, 22, 38). In contrast, random fractional deuteration of uniformly 13C-labeled proteins gives rise to all possible CHnDm isotopomers for each aliphatic carbon moiety. The 13C chemical shift of a particular aliphatic carbon depends not only on its isotopomeric state (n and m values), but also on the isotopomeric state of aliphatic carbon groups both one and two bonds removed. Because of this multibond 2H isotope effect, random fractional deuteration produces a distribution of 13C chemical shifts for aliphatic carbons, especially methylene and methyl groups, thereby decreasing both sensitivity and resolution in heteronuclear NMR experiments. The 13C broadening induced by 2H isotope effects is more evident for experiments recorded with high resolution in the 13C indirect dimension of multidimensional NMR spectra. This point is somewhat ironic. Deuteration increases 13C T2 so that higher resolution data can be collected, but random fractional deuteration introduces 2H isotopomeric resonance shifts that adversely affect resolution. This problem can be addressed, to some degree, by employing 4D rather than 3D heteronuclear experiments. As a rule, 4D 13C experiments are recorded with a small number of increments in 13C indirect dimensions. Consequently, any 2H isotope effects that are smaller than the limited digital resolution of the 4D spectrum will not be as apparent (17, 22).
9.1.5 EXPERIMENTS FOR 1HN, 15N, 13C, 13C, ASSIGNMENTS IN DEUTERATED PROTEINS
AND
13
CO
Generally, only minor modifications need to be made to the conventional 3D and 4D heteronuclear pulse sequences discussed in Chapter 7 for use on deuterated proteins. The most important change
734
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
is that composite pulse deuterium decoupling must be applied whenever transverse 13C magnetization is present, so that evolution of the scalar coupling interaction between the deuteron and its attached carbon is eliminated and scalar relaxation of the second kind, due to short 2H T1 relaxation times, is minimized (18). Longer 13C T2 relaxation times allow incorporation of pulse sequence elements that would normally result in excessive sensitivity losses for protonated protein samples. In particular, constant-time evolution periods on the order of 1/1JCC (25–30 ms) can be used to remove passive homonuclear scalar coupling interactions when indirectly recording carbon chemical shifts (Sections 7.1.1.4 and 7.1.3.1). The resulting line narrowing increases both resolution and sensitivity and has been discussed in Section 7.1.3.1. Substitution of deuterons increases the T1 values of the remaining 1 H spins (1HN in the case of perdeuteration), with this effect particularly evident for perdeuterated proteins. Because a majority of experiments for generating backbone assignments in deuterated proteins begin with magnetization originating on 1HN spins, increases in 1HN T1 values are detrimental to overall sensitivity. Estimating HN T1 values by 1D inversion recovery methods (Section 3.8.2.5) is recommended so that appropriate relaxation delays can be employed between successive scans (Section 3.6.2.5). Techniques for optimizing recovery of magnetization between transients have been discussed (45–47). A number of experiments are available for obtaining the assignments of 1HN, 15N, 13C, 13C, and 13CO resonances in deuterated proteins (2, 13, 18, 20, 25, 35, 37, 48). Rather than review all available or popular pulse sequences for the study of deuterated proteins, the following discussion focuses on illustrative examples in order to highlight the most salient features of pulse sequence design for use with deuterated proteins. A particularly useful suite of triple-resonance experiments includes the CT-HNCA, CT-HN(CO)CA, HN(CA)CB, and HN(COCA)CB, (13, 49). The magnetization transfer pathways for these experiments are, respectively, 1
JNH
1
JNC =2 JNC
1
JNC =2 JNC
15 13 13 15 HN i ! Ni ! Ci ðt1 Þ= Ci1 ðt1 Þ ! Ni ðt2 Þ JNH
! 1 HN i ðt3 Þ; 1
JNH
½9:6 JC CO
JNCO
JC CO
15 13 HN !13 Ci1 ðt1 Þ !13 COi1 i ! Ni ! COi1 JNCO
JNH
! 15 Ni ðt2 Þ ! 1 HN i ðt3 Þ,
½9:7
735
9.1 LARGER PROTEINS 1
JNH
1J
2 NC = JNC
JCC
15 13 13 13 13 HN i ! Ni ! Ci = Ci1 ! Ci = Ci1 ðt1 Þ 1
JCC
JNC =2 JNC
JNH
! 13 Ci =13 Ci1 !15 Ni ðt2 Þ ! 1 HN i ðt3 Þ, 1
JNH
JNCO
JC CO
½9:8
JCC
15 13 13 13 HN i ! Ni ! COi1 ! Ci1 ! Ci1 ðt1 Þ JCC
JC CO
JNCO
JNH
! 13 Ci1 ! 13 COi1 ! 15 Ni ðt2 Þ ! 1 HN i ðt3 Þ:
½9:9
The magnetization transfer pathways for these pulse sequences are essentially the same as for the equivalent sequences used for protonated protein. Full product operator analyses for these experiments are provided in Sections 7.4.1, 7.4.2, and 7.4.5.3. The experimental pulse sequences presented here use water flip-back solvent suppression techniques and PEP gradient- and sensitivity-enhanced HSQC reverse polarization schemes for obtaining 1HN–15N correlations. As for other triple-resonance experiments (Section 7.4.1.5), the HSQC pulse sequence elements can be replaced by TROSY sequences (Section 7.1.3.3) for increased sensitivity and resolution for larger proteins at high static magnetic field strengths. 9.1.5.1 Constant-Time HNCA for Deuterated Proteins Figure 9.3 shows a constant-time (CT) HNCA pulse scheme for correlating 1HN and 15N nuclei with both inter- and intraresidue 13C nuclei in deuterated proteins. This experiment is very similar to the pulse sequence shown in Fig. 7.31d and only important differences are discussed. The 13C shift evolution (t1) occurs during the constant-time period 2TC 1/1JCC, thereby eliminating the effects of one-bond 13C–13C scalar couplings. During the constant-time period, when transverse 13C operators are present, 2H decoupling is employed. The 1H decoupling during 2TC is not needed for perdeuterated proteins, but may be required, depending on molecular size, for random fractionally deuterated proteins. For large proteins, 1H decoupling can be omitted, even for fractionally deuterated samples, because signals from the 1H-bound 13 C spins will decay rapidly during the 2TC period. Decoupling sequences must be interrupted when gradients are applied. The 908 pulses flanking the 1H decoupling periods ensure that the water magnetization is along the z-axis during application of gradient pulses, but is spinlocked during decoupling. The 908 pulses flanking the deuterium decoupling periods ensure that the deuterium magnetization is spin-locked during decoupling to minimize disturbance of the 2H lock signal.
–y
y
y
y
ta
1H
ta f1
tb
y
–y y
y
–y
decouplex
–y y decx
decouplex
decouplex
f1
f3 TN
15N
–y
t TN – 2 2
TN f2
tb
f4
ta f5
t3 ta
ta
ta
z
z
y
t TN + 2 2
decouple
f2 t1 2
13Ca
13CO
t Tc – 1 2
Tc
decouple
y 2H
–y decouplex
1
2
23
4
y
–y decx
5 5
6
78
89
9
10
Grad
FIGURE 9.3 Pulse sequence for a 3D constant-time HNCA experiment for use on deuterated proteins. Thin and thick bars represent 908 and 1808 pulses, respectively. Pulses are applied along the x-axis unless noted. The 1H pulse shown as a short wide bar is a soft 908 selective-water pulse. Pulses bracketing the 1H and 2H decoupling periods are at the same field strength as the decoupling sequence. Decoupling sequences are turned off during gradient pulses. Delays are 2 a ¼ b 1/(2JNH), 2TN 22–28 ms; 2Tc 1/1JCC; and is long enough to accommodate the enclosed gradient. The phase cycle is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ x; 4 ¼ 4(x), 4(x); 5 ¼ x; and receiver ¼ x, x, x, x. Gradients 7 and 10 are used to select the appropriate 15N coherence transfer pathway. Quadrature detection in the F1 dimension is performed using the TPPI–States method applied to 2 and the receiver. Frequency discrimination in the F2 dimension and sensitivity enhancement is achieved as described in Section 7.1.3.2. For each t2 increment, N- and P-type coherences are obtained by recording two data sets. Gradient 7 and the phase 5 are inverted in the second data set. For each t2 increment, 3 is incremented by 1808 in concert with the receiver.
9.1 LARGER PROTEINS
737
This general pulse scheme can be used for studying both perdeuterated and random fractionally deuterated proteins (taking into account the comments concerning 1H decoupling). A CT-HN(CO)CA 15 experiment, providing solely interresidue connections between1 HN i , Ni , 13 and Ci1 spins, is recorded to complement the CT-HNCA. 9.1.5.2 HN(CA)CB for Deuterated Proteins The chemical shifts of the 13C spins are assigned using HNCACB experiments (Section 7.4.5.3). Figure 9.4 shows a general pulse sequence for recording HNCACB-type data on deuterated proteins and is similar to Fig. 7.47. The product operator analysis in Section 7.4.5.3 shows that the intensities of 13C correlation peaks are proportional to cos2(21JCCTC) and the intensities of 13C correlation peaks are proportional to sin2 (21JCCTC). In the conventional HNCACB experiment (Section 7.4.5.3), the delay TC is normally set to 1/(81JCC), so that both 13C and 13C peaks are obtained with approximately equal intensities; however, each set of peaks has only one-half of the maximum possible intensity. In principle, optimal transfer to the 13C spins can be achieved by setting TC 1/(41JCC). In this case, 13C transverse magnetization is present for 4TC 1/1JCC 28 ms. In fully protonated larger proteins, this approach is impractical because efficient 13C relaxation losses during these delays degrade sensitivity. In deuterated samples, the 13C T2 is significantly longer and a value of 4TC 1/1JCC can be used to transfer all magnetization to the 13C spins, thereby increasing sensitivity for these correlations. For this reason, the experiment is referred to as an HN(CA)CB experiment (13). The pulse sequence shown in Fig. 9.4 is appropriate for random fractionally deuterated proteins. For perdeuterated proteins, 1H decoupling does not need to be applied between the first and last 908 pulses on 13C spins. The HN(CA)CB sequence shown in Fig. 9.4 uses conventional incrementation for the t1 evolution period, rather than a constant-time evolution t1 period. Experiments that measure 13C chemical shifts using constant-time periods on the order of 1/1JCC suffer from sensitivity losses, compared to non-constant-time experiments, unless both 13C and 13C nuclei involved in the coherence transfer process are perdeuterated (13). The fraction of 13C and 13C spin pairs that are perdeuterated decreases rapidly as the level of fractional deuteration is reduced. For example, if a protein is randomly deuterated to a fraction f, without any biased deuteration patterns resulting from biosynthetic pathways, then the fraction of perdeuterated 13CD–13CD2 moieties is f 3 for an amino acid residue with a -methylene group. Consequently, nonconstant-time versions of the HN(CA)CB [and the complementary
1H
y
y
ta ta f1
tb
–y y
y
–y
y
–y decx
decouplex
y
decouplex
–y y decouplex
decx
f1
f5 t TN – 2 2
TN f2
f2
f3
f3
tc
13Ca /b
tc
13CO
tb
ta
f6 t TN + 2 2
f7
ta ta
t3 ta
z
z
y decouple
f4 t1
tc
tc
decouple
y
2H
–y
y
decx
1 2
23
4
–y decouplex
5 5
y
–y decx
6 6
7
89
9 10
10
11
MOLECULAR INTERACTIONS
FIGURE 9.4 Pulse sequence for a 3D HN(CA)CB experiment for use on deuterated proteins. Thin and thick bars represent 908 and 1808 pulses, respectively. Pulses are applied along the x-axis unless noted. The short wide rectangle is an 1H selective 908 water pulse. Pulses bracketing the 1H decoupling periods are at the same field strength as the decoupling sequence. These pulses ensure that the water magnetization is along the z-axis during application of gradient pulses, but is spin-locked during decoupling. Decoupling sequences are turned off during gradient pulses. Delays are discussed in the text, but the small delay is to accommodate gradients. The phase cycle is 1 ¼ x, x; 2 ¼ 2(x), 2(x); 3 ¼ 4(y), 4(y); 4 ¼ 8(y), 8(y); 5 ¼ x; 6 ¼ 4(x), 4(x); 7 ¼ x; receiver ¼ x, x, x, x. Gradients 8 and 11 are used to select the appropriate 15N coherence transfer pathway. Quadrature detection in the F1 dimension is performed using the TPPI–States method applied to 2, 3, and the receiver. Frequency discrimination in the F2 dimension and sensitivity enhancement is achieved as described in Section 7.1.3.2. For each t2 increment, N- and P-type coherences are obtained by recording two data sets. Gradient 8 and the phase 7 are inverted in the second data set. For each t2 increment, 5 is incremented by 1808 in concert with the receiver.
AND
Grad
CHAPTER 9 LARGER PROTEINS
TN
15N
–y
738
–y y
739
9.1 LARGER PROTEINS
HN(COCA)CB experiment] are recommended for random fractionally deuterated proteins (13). Constant-time versions of the HN(CA)CB and HN(COCA)CB experiments can be used for perdeuterated proteins, as described for the CT-HNCA experiment in Section 9.1.5.1. 9.1.5.3 Other Experiments for Resonance Assignments For determining 13CO assignments, straightforward HNCO experiments, correlating 13COi–1, 15Ni, and 1 HN i spins, and HN(CA)CO experiments, correlating 13COi, 15Ni, and 1 HN i spins, are performed as described in Sections 7.4.4.1 and 7.4.4.2. The 3D/4D HN(COCA)NH spectrum allows the direct connection of adjacent amide groups along the protein backbone and has the advantage that the spectra are extremely easy to interpret (18, 50–52). This experiment utilizes the coherence transfer pathway, 1
JNH
JNCO
JC CO
15 HN ! 13 COi1 !13 Ci1 i ðt1 Þ ! Ni ðt2 Þ JNC
JNH
! 15 Ni1 ðt3 Þ ! 1 HN i1 ðt4 Þ,
½9:10 15
Ni, 1 HN to generate sequential correlations between the 1 HN i , i1 , and 15 Ni–1 backbone atoms. The utility of this experiment, for fully protonated proteins, has been limited predominantly by the large 13C relaxation rates arising from dipolar coupling between the 13C and 1 H spins. A significant improvement in sensitivity is realized for perdeuterated proteins because the 13C T2 relaxation times are notably increased. Deuterium decoupling should be applied in the HN(CA)CO and HN(COCA)NH experiments when transverse 13C magnetization is present. A complementary set of experiments that begin with 1H/1H spins has been proposed and therefore these experiments are not useful for application to perdeuterated proteins (25). These 4D HBHACBCANH and HBHACBCA(CO)NH experiments provide correlations between 1H/1H spins and intra- and interresidue amide moieties in random fractionally deuterated systems. The experiments are adversely affected by 13C isotopomeric broadening, although this effect can be alleviated to some extent by reducing the digital resolution in the 13C dimensions of 4D experiments. To ensure the highest sensitivity, experiments on deuterated proteins should be performed at the highest possible static magnetic field strength. However, transverse relaxation rates of 13CO spins increase approximately with the square of the static magnetic field strength due to the CSA relaxation mechanism. As a result, the sensitivity of
740
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
NMR experiments that transfer magnetization to 13CO spins begins to decrease as the static field increases (for either protonated or deuterated proteins). For example, the usual approach uses the pair of HNCA and HN(CO)CA spectra to obtain sequential assignments; however, the HNCA experiment can be recorded at the largest available field (900 MHz), but the HN(CO)CA experiment usually is recorded at a static magnetic field strength of 14.1 T (600 MHz). Several HNCA-based approaches that minimize or completely avoid the time during which transverse 13CO magnetization is present have been proposed. Using HNCA methods to obtain sequential connectivities has the inherent problem that the coupling constants 1 JNC and 2 JNC are similar in size. Consequently, both intra- and interresidue correlations may be observed and can be difficult to differentiate. In one method to circumvent this difficulty, spin state selection is used to distinguish between the intra- and interresidue connectivities (53). A second method suppresses the coherence transfer pathway that uses the 1 JNC scalar coupling interaction (54). A different approach uses a combination of intraresidue and double-quantum (DQ) HNCA experiments (55). The intraresidue HNCA experiment correlates the 1 HN i and 15 Ni spins exclusively with the 13 Ci spin. The DQ HNCA experiment 15 correlates the 1 HN Ni spins with the sum of the frequencies of the i and 13 13 13 Ci and Ci1 . The sequential correlation between 1 HN i and Ci1 spins 13 is established by subtraction of the matching Ci shift from the DQ resonance frequency. The preceding sections have discussed general methods, and caveats, for obtaining backbone 1HN, 15N, 13C, and 13C assignments in larger proteins (420 - 30 kDa). Perdeuteration, rather than random fractional deuteration, of proteins is warranted to obtain maximum sensitivity in out-and-back experiments originating and terminating on the 1HN spins, particularly for experiments that utilize transfer between multiple aliphatic 13C spins.
9.1.6 SIDE CHAIN
13
C ASSIGNMENTS
IN
DEUTERATED PROTEINS
Correlations to 13C spins further down the amino acid side chain than C can be made by inserting additional magnetization transfer steps into the HN(CA)CB experiment. For example, addition of a pair of 13 C–13C pulse-interrupted free-precession (COSY-type) steps before and after t1 ( c–180– c in Fig. 9.4) produces an HN(CACB)CG experiment that correlates the 13C spin to the amide moiety (22, 33). The sequence can be extended in similar fashion to obtain correlations to C and even C" positions in the side chain. These HN(CX)nCY experiments also
741
9.1 LARGER PROTEINS
are useful for editing the spectra of particular amino acids based on the branch point of the side chain. For example, Ile and Leu have similar 13 C chemical shifts, but branch at different locations (Ile at C, Leu at C ). As a result, Leu residues give stronger correlations than do Ile residues in an HN(CACB)CG experiment (17). However, each additional COSY-style pulse element requires a delay of length of 1/(2JCC), resulting in reduced sensitivity for experiments that correlate spins farther along the side chain. For moderately sized 13C/15N labeled proteins, side chain 13C assignments are most commonly obtained from the HCCH–TOCSY (Section 7.4.4.2) experiment or the (H)CC(CO)NH–TOCSY experiment (56). These experiments also can be used with random fractionally deuterated proteins to obtain side chain 13C assignments; however, rapid relaxation of the residual proton-bound 13C spins reduces sensitivity and restricts the usefulness of these experiments. For example, if a protein is randomly deuterated to a fraction f, without any biased deuteration patterns resulting from biosynthetic pathways, then the fraction of 13 C D–13CD2–13C HD moieties is (1 – f ) f 4 for an amino acid residue with - and -methylene groups. Consequently, the probability is high that the 13C–13C TOCSY transfer will pass magnetization through a protonated 13C or 13C site, quickly reducing the sensitivity of the experiment for correlations to the 13C spin and to spins farther along the side chain. Protein samples prepared using higher levels of deuteration or perdeuteration cannot be used to overcome this disadvantage because HCCH–TOCSY and (H)CC(CO)NH–TOCSY experiments begin from magnetization of a carbon-attached 1H spin. The CC(CO)NH–TOCSY experiment is a modification of the (H)CC(CO)NH–TOCSY that begins from 13C, rather than 1H, magnetization (57). The concomitant fourfold reduction in sensitivity, due to the smaller magnetogyric ratio of 13C, is compensated by the increase in 13C T2 at each position in the side chain (57). In theory, sensitivity gains as large as a factor of seven are obtained from the reduced relaxation losses for methylene groups in perdeuterated, compared to fully protonated, proteins (22, 58–60). However, the 13C T1 is longer by approximately a factor of two in perdeuterated proteins (Fig. 9.2); therefore, longer recycle delays are necessary, which reduces the overall signal-to-noise ratio per unit time. The pulse sequence for the CC(CO)NH–TOCSY experiment is shown in Fig. 9.5. This experiment utilizes the coherence transfer pathway, 13
JCC
JC CO
JNCO
JNH
Ci ðt1 Þ ! 13 Ci !13 COi ! 15 Niþ1 ðt2 Þ ! 1 HN iþ1 ðt3 Þ,
½9:11
decouple SL
f1
f2
y
t1 2
13C
t1 2
TOCSY
40ppm
y
y
tc
f3
y
–y
y
ta
ta
z
z
y
t CN decouple
–x
y
tCN
13CO
y
f5
ta
∆C
tc –x
56ppm
–x
ta
tCN
–y decx
decx 1
1
2
2 3 4
45
6 7
78
8
9
Grad
AND
MOLECULAR INTERACTIONS
FIGURE 9.5 Pulse sequence for the 3D gradient-enhanced, sensitivity-enhanced CC(CO)NH–TOCSY experiment for use on deuterated proteins. Thin and thick bars represent 908 and 1808 pulses, respectively. Pulses labeled SL are spin lock purge pulses. Pulses are applied along the x-axis unless noted. Initially, 13C pulses are broadband and centered at 40 ppm. Where noted, the 13C transmitter is shifted to 56 ppm (middle of 13C region) and subsequent pulses are 13C selective. Delays: c 1=ð4JC CO Þ (slightly less to account for relaxation); CN ¼ 1/(4JNCO); C ¼ 1=ð4JC CO Þ; a 1/(4JNH); b 1/(2JNH); and long enough to accommodate gradients. The phase cycle is 1 ¼ 4(y), 4(y); 2 ¼ 8(x), 8(x); 3 ¼ 2(x), 2(x); 4 ¼ x, x; 5 ¼ x; receiver ¼ x, 2(x), x, x, 2(x), 2(x), 2(x), x, x, 2(x), x. Gradients 6 and 9 are used to select the appropriate 15 N coherence transfer pathway. Quadrature detection in the F1 dimension is performed using the TPPI–States method applied to 1 and the receiver. Frequency discrimination in the F2 dimension and sensitivity enhancement are achieved as described in Section 7.1.3.2. For each t2 increment, N- and P-type coherences are obtained by recording two data sets. Gradient 6 and the phase 5 are inverted in the second data set. For each t2 increment, 4 is incremented by 1808 in concert with the receiver.
CHAPTER 9 LARGER PROTEINS
f4 y t t2 t CN – 2 2 2
15N
2H
tb
742
t3
y x y
1H
9.1 LARGER PROTEINS
743
to obtain correlations between the side chain 13C spins of residue i with the amide group of residue i þ 1. The experiment begins by recording side chain 13C chemical shifts during t1. The evolution period is followed by a z-filtered 13C–13C TOCSY isotropic mixing sequence. The isotropic mixing time is chosen as for the HCCH–TOCSY experiment (Section 7.3.3). The magnetization that is transferred to the 13C spin during the TOCSY period subsequently is transferred in a ‘‘straight-through’’ fashion to the 13CO, 15N, and 1HN spins. Figure 9.6 shows strips taken from a CC(CO)NH–TOCSY spectrum of perdeuterated 2H/13C/15N calbindin D28k (30 kDa). Each strip shows the side chain 13C chemical shifts of several lysine residues correlated with the amide 15N and 1HN shifts of the succeeding residue.
9.1.7 SIDE CHAIN 1H ASSIGNMENTS Two approaches are available for obtaining the assignment of side chain 1H resonances in larger proteins, although neither is ideal. In one approach, 13C(D) side chain assignments are obtained from a CC(CO)NH–TOCSY recorded on a perdeuterated protein sample. The 13 C(D) side chain chemical shifts are used to estimate 13C(H) side chain chemical shifts using 2H isotope shift information (Table 9.1). A 4D HCC(CO)NH–TOCSY is recorded on a random fractionally deuterated or fully protonated sample to obtain 1H side chain assignments. In the other approach, 13C(H) carbon and 1H proton side chain shifts are obtained directly from a 4D HCC(CO)NH–TOCSY recorded on a random fractionally deuterated or fully protonated sample. The advantage of the second method is that 13C(H) shifts do not have to be estimated. However, full assignment of 13C(H) carbon side chain shifts might not be achieved because of relaxation losses arising from the short 13C T2 values. As discussed in Section 9.1.4, line narrowing observed in the spectra of random fractionally deuterated proteins is offset by the isotopomer broadening effect. These effects can be minimized by (i) using higher levels of deuteration (e.g., 70%) and/or (ii) using 4D experiments with a limited number of points in the carbon dimensions. Overall, applying both approaches may prove fruitful. The limited data that are extracted easily from the 4D HCC(CO)NH– TOCSY experiment recorded on a fully protonated sample can be used to confirm information obtained from the spectra recorded on random fractionally deuterated samples, particularly in corroborating 2H isotope shifts. In studies of 30-kDa calbindin D28k, 50% of the side chain 1H assignments were made from a 4D HCC(CO)NH–TOCSY experiment on a fully protonated sample.
CHAPTER 9 LARGER PROTEINS T135 N=120.6 ppm
M186 N=121.5 ppm
K134
K185
AND
MOLECULAR INTERACTIONS
L247 N=124.6 ppm 15
744
20
K246
Cg
Cg Cd
Cb
Cb
Ce
Ce
45 70
65
60
Ca
55
50
13C
Ce
40
35
Cb
Cd
30
Cd
25
Cg
7.8
7.7
9.1
9.0
9.0
8.9
1HN
FIGURE 9.6 Strip plots from a 3D CC(CO)NH–TOCSY on perdeuterated 2 H/13C/15N calbindin D28k. The strips show side chain 13C chemical shifts of selected lysine residues throughout the protein correlated with the 1HN and 15 N chemical shifts of the i þ 1 residue. The TOCSY mixing time is 20 ms. The spectrum was recorded on a 600-MHz NMR spectrometer.
745
9.1 LARGER PROTEINS
9.1.8 NOE RESTRAINTS
FROM
DEUTERATED PROTEINS
1
H–1H NOEs are the primary data used to generate restraints for protein structure determination by NMR spectroscopy. As discussed previously, sizeable gains in sensitivity are attainable for many tripleresonance scalar correlation experiments by using proteins perdeuterated at nonexchangeable proton sites. Although the use of perdeuterated proteins is optimal for obtaining backbone and side chain 13C assignments, only 1HN–1HN NOEs can be obtained from such a sample. Other NOEs must be obtained from fully protonated, random fractionally deuterated, or selectively protonated protein samples. 9.1.8.1 4D HN–HN 15N/15N-Separated NOESY Experiment Perdeuteration decreases both the amide 1HN R1 and R2 relaxation rate constants. Taking advantage of these characteristics, up to a sevenfold increase in signal-to-noise ratio can be realized for a perdeuterated protein relative to its protonated counterpart in a 4D 1HN–1HN NOESY experiment (22, 58–60). Additionally, perdeuteration significantly reduces spin diffusion effects, allowing the use of notably longer NOE mixing times, up to several hundred milliseconds (17, 33, 40, 58–60). As a result of the longer mixing times, NOEs over greater distances (up to 8 A˚) potentially can be obtained. However, even in perdeuterated proteins, short distances between amide 1HN spins in -helices can result in spin diffusion; recording a series of NOESY spectra for different mixing times can aid in detecting such effects. A gradient-enhanced, sensitivity-enhanced 4D 15N/15N-separated NOESY experiment for use on a 2H/13C/15N-labeled protein is shown in Fig. 9.7 (59, 60). This experiment is very similar to the 3D variant described in Section 7.2.4.1. The overall flow of magnetization is described by 1
JNH
JNH
JNH
JNH
NOE
15 1 N 1 N HN A ! NA ðt1 Þ ! HA ðt2 Þ ! HB
! 15 NB ðt3 Þ ! 1 HN B ðt4 Þ:
½9:12
The frequency axes in the 4D spectrum correspond to F1 ¼ (15NA), 15 1 N F2 ¼ (1 HN A ), F3 ¼ ( NB), F4 ¼ ( HB ). The resulting 4D spectrum correlates amide group A to a spatially close amide group B, via the NOE. The 4D 15N/15N-separated NOESY experiment also can be used with random fractionally deuterated proteins. Spin diffusion effects are also reduced compared to fully protonated samples, but are not reduced to the same degree as for perdeuterated proteins.
t + t2
t
t
y t
t
f1
f3 t1 2
15N
t1 2
t
t
f4 t3 2
t t+ 2 2
t3 2
e
y
t
t
z
t4 z
y
e
decouple
13C 1
12
3 4
4
5 6
6
7
8 9
9 10
10
11
Grad
AND
MOLECULAR INTERACTIONS
FIGURE 9.7 Pulse sequence of the gradient-enhanced, sensitivity-enhanced 4D 15N/15N-separated NOESY experiment with water flip-back solvent suppression. Deuterium decoupling is not required, because transverse 13C magnetization is never present. Thin and thick bars represent 908 and 1808 pulses, respectively. Low-power selective 1H pulses, applied to the H2O resonance, are shown as short rectangles. Pulses are applied along the x-axis unless noted. The delays are 1/(4JNH); m is the NOESY mixing time; " and are long enough to accommodate the encompassed gradients. The phase cycle is 1 ¼ x, x; 2 ¼ x; 3 ¼ x; 4 ¼ x; receiver ¼ x, x. Gradient 5 is applied at the end of the mixing time to defocus any residual transverse magnetization after radiation damping has returned the H2O signal to the z-axis. Gradients 8 and 11 are used to select the appropriate 15N coherence transfer pathway. Quadrature detection in the F1and F2 dimensions is performed using the TPPI– States method applied to 1 and the receiver, and 2 and the receiver, respectively. Frequency discrimination in the F3 dimension and sensitivity enhancement is achieved as described in Section 7.1.3.2. For each t3 increment, N- and P-type coherences are obtained by recording two data sets. Gradient 8 and the phase 4 are inverted in the second data set. For each t3 increment, 3 is incremented by 1808 in concert with the receiver.
CHAPTER 9 LARGER PROTEINS
f1
tm
t
y
y
746
f2
y
–y y 1H
747
9.1 LARGER PROTEINS
The pulse sequence shown in Fig. 9.7 was used to record a 4D HN–HN 15N/15N-separated NOESY spectrum on perdeuterated 2 H/13C/15N calbindin D28k using a mixing time of 175 ms. Figure 9.8 shows a slice from this 4D 15N/15N-separated NOESY spectrum. The slice is taken at the 15N/1HN shift(s) of Val62, and the autocorrelation peak of Val62 can be seen in the spectrum at F1¼ F3 ¼ 118.2 ppm (15N) and F2 ¼ F4 ¼ 9.12 ppm (1H). All other correlations are NOE crosspeaks from the Val62 1HN spin.
V62 N,HNF1=118.2 ppm, F4=9.12 ppm:
110
9.1.8.2 13C/15N-, 13C/13C-, and 15N/15N-Separated NOESY Experiments on Random Fractionally Deuterated Proteins At first sight, random fractionally deuterated proteins appear to be ideal candidates for establishing general 1H–1H NOE contacts using heteronuclear-separated NOESY experiments such as the 3D and 4D
120
Q64HN,N
V62HN,N
D63HN,N
K59HN,N
(F3 ppm)
T60HN,N
15N
115
Y65HN,N
125
F61HN,N
G66HN,N 11.0
10.5
10.0
9.5
9.0 1HN
8.5
8.0
7.5
7.0
(F2 ppm)
FIGURE 9.8 A 2D slice taken from a 4D 15N/15N-separated NOESY on perdeuterated calbindin D28k. The data were taken using the pulse sequence described in Fig. 9.7 and a mixing time of 175 ms. The spectrum was recorded on an 800-MHz NMR spectrometer. The slice is taken at the 15N/1HN (F1/F4) shift(s) of V62, and the autocorrelation peak of V62 can be seen in the spectrum. Other peaks in the spectrum are NOEs between the amide group of V62 and the indicated amide groups.
748 13
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
C/15N-, 13C/13C-, and 15N/15N-separated NOESY experiments discussed in Section 7.2.4 (25, 58–62), because, in contrast to perdeuterated molecules, side chain protons are available, albeit at reduced concentrations. However, because the sample is a mixture of different isotopomers, significant variations in relaxation rates may exist between a specific 1H spin in one protein molecule and the same 1H spin in a different protein molecule with a different deuterium composition (63). These variations pose difficulties for the interpretation of crosspeak intensities in NOESY spectra of random fractionally deuterated proteins. The relaxation times of 1H spins increase as proteins are more highly deuterated, which leads to increased intensity of NOE crosspeaks. On the other hand, increasing the level of deuteration dilutes the population of carbon-bound 1H spins, thereby reducing the intensity of any NOE peak to or from a carbon-bound 1H spin. Nietlispach, Laue, and co-workers discuss these effects (25) and reach the following conclusions. First, the intensities of HN–HN NOE peaks rise with increasing level of deuteration. Second, the intensities of HN–H NOE peaks do not drastically change up to a deuteration level of 50%; at this level the intensities start to decrease. The dilution effect is compensated at low deuteration levels by the favorable relaxation of the remaining 1H spins, but at higher enrichment levels the dilution effect dominates. Third, NOE peaks between two H spins show a decrease in intensity as the level of deuteration increases, because the reduction in 1H spin population offsets any advantage gained by increases in the relaxation times of the 1H spins. From a practical standpoint, the variation in relaxation rate constants for different molecules in a random fractionally deuterated sample means that NOE cross-peak volumes do not necessarily provide accurate estimates of internuclear distances. Use of average relaxation rate constants calculated by taking population-weighted averages of the rate constants for each isotopomer has been suggested as one approach for treating this problem (25). This approach is most useful if the distribution of relaxation rate constants for different isotopomers is narrow, compared to the magnitudes of the relaxation rate constants. A quantitative analysis of cross-relaxation in a random fractionally deuterated system has been discussed (63). In this approach, a series expansion is derived that, to second order, expresses cross-peak volume at a specific mixing time independently of autorelaxation or external relaxation effects. In principle, this treatment removes the effect of the degree of deuteration and allows more accurate crossrelaxation rate constants to be established.
749
9.1 LARGER PROTEINS
In light of these difficulties, random fractionally deuterated proteins have not been widely used for obtaining quantitative 1H–1H NOE distance restraints for structure calculations. NOESY data acquired on random fractionally deuterated proteins are perhaps better employed to generate global folds, rather than atomic-resolution structures. The nD-separated NOESY pulse sequences for use with random fractionally deuterated proteins are essentially the same as their counterparts for fully protonated samples (Section 7.2), with the addition of deuterium decoupling when transverse carbon magnetization is present (25). A gradient- and sensitivity-enhanced 4D 13C/15N-separated NOESY experiment is shown in Fig. 9.9. The overall flow of magnetization is 1
JCH
JCH
Ha ! 13 Cðt1 Þ ! 1 Ha ðt2 Þ NOE
JNH
JNH
! 1 HN ! 15 Nðt3 Þ ! 1 HN ðt4 Þ,
½9:13
in which 1Ha is an aliphatic, carbon-attached 1H spin. This experiment is particularly useful in the study of selectively protonated, otherwise perdeuterated, proteins, as discussed in the following sections.
9.1.9 SELECTIVE PROTONATION Because the overall success of a structure determination relies on obtaining side chain distance restraints, especially those involving residues in the hydrophobic core of the protein, a ‘‘middle-ground’’ approach for obtaining 1H–1H NOEs in larger proteins is available. This technique involves selectively reintroducing 1H spins into nonexchangeable sites of otherwise perdeuterated proteins. NOESY experiments performed on selectively protonated aliphatic methyl groups (Ile, Leu, Val, Ala) and aromatic side chain rings (Phe, Tyr, Trp) in otherwise perdeuterated proteins provide very useful long-range distance restraints. To date, the majority of applications of selective protonation has involved methyl 1H spins, because these spins, although exhibiting somewhat limited chemical shift dispersion, have long T2 relaxation times, even in 13C-labeled molecules. Kay and co-workers have developed methyl-TROSY techniques, based on the HMQC pulse sequence (Section 7.1.1.1), that provide additional resolution and sensitivity (64). Methyl groups also are abundant in the hydrophobic cores of proteins, allowing additional NOE connectivities to be identified and used as restraints for structure calculations.
750
CHAPTER 9 LARGER PROTEINS f1
1H
ta
f2 t1 2
t2 2
t2 2
tm tb
y
t bSL
tb
tb tb
t4 tb
z
z
t1 2
decouple
f3 t3 2
15N
13CO
decouple
2H
decouple
1 2
MOLECULAR INTERACTIONS
y
ta
13C
AND
t3 2
e
f4
y
6 7
7 8
e
decouple
2
34
4
5
8
9
Grad
FIGURE 9.9 Pulse sequence of the gradient-enhanced, sensitivity-enhanced 4D 13 C/15N-separated NOESY experiment. Thin and thick bars represent 908 and 1808 pulses, respectively. The spin lock pulse applied to the H2O resonance is denoted SL. Pulses are applied along the x-axis unless noted. The delays are a 1/(4JCH); b 1/(4JNH); m is the NOESY mixing time; " and are long enough to accommodate the encompassed gradients. The phase cycle is 1 ¼ 458; 2 ¼ x, x; 3 ¼ x; 4 ¼ x; receiver ¼ x, x. Gradient 3 is applied at the end of the mixing time to defocus any residual transverse magnetization after radiation damping has returned the H2O signal to the z-axis. Gradients 6 and 9 are used to select the appropriate 15N coherence transfer pathway. Quadrature detection in the F1and F2 dimensions is performed using the TPPI– States method applied to 2 and the receiver, and 1 and the receiver, respectively. Frequency discrimination in the F3 dimension and sensitivity enhancement are achieved as described in Section 7.1.3.2. For each t3 increment, N- and P-type coherences are obtained by recording two data sets. Gradient 6 and the phase 4 are inverted in the second data set. For each t3 increment, 3 is incremented by 1808 in concert with the receiver.
Rosen et al. developed a protocol using 1H/13C pyruvate as the sole carbon source for biosynthesis of 2H/13C/15N triply labeled proteins with selective protonation at the methyl groups of Ala, Val, Leu and Ile-2 (29). Goto et al. developed a protocol using [3-2H], 13 C -ketoisovalerate, and 13C -ketobutyrate to supplement [2H/13C] glucose in bacterial growth media to obtain selective protonation
9.1 LARGER PROTEINS
751
of Leu-, Val-, and Ile-1 methyl groups; this labeling pattern is abbreviated as ILV (30, 31). The level of methyl protonation achieved in these protocols is 490%. Elegant experiments for assigning methyl resonances in methyl-protonated, otherwise perdeuterated, proteins have been developed by Kay and co-workers. Both straight-through and out-and-back experiments are used to correlate the methyl 13C and 1 H resonances with previously assigned backbone 15N or 13CO spins (65). These samples allow HN–HN, HN–CH3, and CH3–CH3 NOEs to be measured using 4D 15N, 15N-edited NOESY; 4D 13C, 15N-edited NOESY; and 13C, 13C-edited NOESY experiments, respectively (25, 58–62). The limited number of 1H spins in the protein again means that spin diffusion effects are minimized, allowing the use of longer NOE mixing times in these experiments. Figure 9.10 shows slices from 3D 13C, 13C-edited NOESY; 4D 13C, 15 N-edited NOESY; and 4D 15N, 15N-edited NOESY spectra for an ILV methyl-protonated, otherwise perdeuterated, sample of calbindin D28k. In the 3D 13C, 13C-edited NOESY spectrum (Fig. 9.10a), the autocorrelation peak for the -CH3 group of Val62 can be seen at F1 ¼ F2 ¼ 22.9 ppm (13C) and F3 ¼ 1.38 ppm (1H). All other cross-peaks arise from NOE interactions with the Val62 1H spins. The 4D 13C, 15Nedited NOESY in Fig. 9.10b shows NOEs from the Val62 1HN spin to a variety of - and -methyl groups in the specifically protonated ILV sample. The slice is taken at the 15N and 1HN shifts of Val62. The 4D 15 N, 15N-edited NOESY in Fig. 9.10c shows the autocorrelation peak for the 1HN–15N group of Val62 at F1¼ F3 ¼ 118.2 ppm (15N) and F2 ¼ F4 ¼ 9.12 ppm (1H). All other peaks are NOE peaks from the Val62 1 N H spin. The value of methyl-selective protonation in an otherwise perdeuterated environment has been demonstrated in the determination of global folds from a limited set of HN–HN, HN–CH3, and CH3–CH3 NOE restraints (41, 66). To some degree, however, both secondary structure and topology play roles in determining the quality of the structures that can be obtained from these limited NOE sets. This limitation is more evident for highly helical proteins, because long distances sometimes separate 1HN spins on adjacent helices (41). The relatively sparse NOEs available from perdeuterated and selectively protonated samples can be augmented by RDCs and anisotropic chemical shift changes measured in weakly aligned samples. Techniques for measurements of RDCs are described in Section 7.6. Kay and co-workers have determined the global fold of the 723-residue, 82-kDa enzyme malate synthase G using 746 HN–HN (99 long-range), 428 CH3–CH3 (386 long-range), and 357 HN–CH3 (142 long-range)
21
CHAPTER 9 LARGER PROTEINS
V62Cg: F1=22.9 ppm
AND
MOLECULAR INTERACTIONS
V62 N,HN: F1=118.2 ppm, F4=9.12 ppm V81Hg,Cg V62Hg,Cg
20
752
22
26
23
1.4
1.2
1.0
0.8 1HC
0.6
0.4
0.0
V62 N,HN: F1=118.2 ppm, F4=9.12 ppm
L43Cd
T60HN,N Q64HN,N
26
V62HN,N
120
L44Cd
D63HN,N
K59HN,N
125
F61HN,N
(F3 ppm)
115
Y65HN,N
25
L43Cd
0.2
(F2 ppm) 110
1.6
28
b
13C
24
L36Hd,Cd
L44Hd,Cd
(F2 ppm)
L54Cd
13C
L43Hd,Cd
L43Hd,Cd
V62Cg
24
L54Hd,Cd
V62Hg,Cg
15N
V62Cg
(F1 ppm)
22
V81Hg,Cg
27
a 1.5
1.4 1H C
(F3 ppm)
1.3
c 11.0
G66HN,N 10.5
10.0
9.5 1HN
9.0
8.5
8.0
7.5
7.0
(F2 ppm)
FIGURE 9.10 The 2D slices taken from (a) 3D 13C/13C-separated NOESY, (b) 4D 13C/15N-separated NOESY, and (c) 4D 15N/15N-separated NOESY for a calbindin D28k sample that is specifically protonated at the methyl groups of isoleucine, leucine, and valine and otherwise perdeuterated. The spectra were recorded on an 800-MHz NMR spectrometer. See text for details.
NOEs measured in isotropic solution together with 415 1HN–15N RDCs and 300 13CO anisotropic chemical shift changes measured upon alignment of the protein using Pf1 phage (67). Aromatic residues also are prevalent in the hydrophobic cores of proteins, and including distance restraints derived from aromatic 1 H NOEs can increase the accuracy of structures calculated from low-density NOE data sets (68–70). A biosynthetic method has been described for the selective proton labeling of Phe, Tyr, and Trp residues in perdeuterated proteins, using natural-abundance shikimic
753
9.2 INTERMOLECULAR INTERACTIONS
acid (32). This approach selectively protonates aromatic rings, while the labeling of the backbone - and side chain -positions can be controlled independently via the D2O and glucose added to the growth medium.
9.2 Intermolecular Interactions Proteins interact with a variety of different types of ligands, including other proteins, peptides, DNA, RNA, small molecules, and metal ions. NMR methods together with differential isotope labeling strategies allow spectral information to be selectively acquired for individual, noncovalently linked constituents of multicomponent complexes (5–7, 71–80). Using these approaches, NMR spectroscopy can be used to detect binding of ligands to proteins, to identify the sites of ligand interactions on the surfaces of proteins, and to determine atomic-resolution molecular structures of protein–ligand complexes. The present discussion assumes that isotopically enriched molecules are available and that detailed structural information about the complex is to be obtained. Numerous techniques for detecting whether a given ligand interacts (weakly) with a target protein have been described, many of which do not require isotopic enrichment. These methods often are used in screening for potential lead compounds for pharmaceutical applications, rather than in structural studies [for reviews, see Carlomagno (81), Homans (82), and Pellechia et al. (83)].
9.2.1 EXCHANGE REGIMES Whether a protein binds to its target ligand in a weak, intermediate, or strong fashion dictates not only the quality of the resulting NMR spectra, but also the information that those spectra can unambiguously provide. With this in mind, the chemical exchange regime for the system must be established before more detailed investigations can be undertaken (84). The following discussion follows the theoretical principles presented in Section 5.6. The simplest kinetic scheme for binding of a ligand (L) to a protein (P) is described as a second-order exchange process, kf
P þ L Ð PL; kr
½9:14
754
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
in which kf is the rate of the forward (association) reaction, kr is the rate of the reverse (dissociation) reaction, and the chemical exchange lifetime, ex ¼ 1/kex, is defined by kex ¼ kf ½L þ kr :
½9:15
The three exchange regimes of interest are defined by Slow exchange : Intermediate exchange : Fast exchange :
kex j!j, kex j!j, kex j!j,
in which ! ¼ !P !PL, and !P and !PL are the resonance frequencies of a nuclear spin in the uncomplexed and complexed protein, respectively. The exchange regime for the protein–ligand system under investigation is determined by monitoring the changes in the protein NMR spectrum during a titration with ligand. The simplest approach monitors a well-resolved resonance in, for example, a 1D 1H spectrum or a 2D 1H–15N correlation spectrum, with the proviso that the chosen peak must be detectably affected by the protein–ligand interaction. As the concentration of ligand is increased, changes in the spectrum (and the specific resonance in particular) provide information on the chemical exchange regime. In general, the following changes in the spectra potentially are observed: 1. A new protein resonance appears that increases in intensity but does not alter chemical shift as the ligand concentration increases. At the same time, the intensity of the resonance for the unliganded species decreases in intensity. Changes in linewidth are small. These observations imply a slow exchange process. 2. The protein resonance broadens significantly as ligand concentration is increased until it disappears when the total ligand concentration is approximately one-half that of the total protein concentration. The protein signal reappears at much higher ligand concentrations (perhaps in a different location). These observations imply an intermediate exchange process. 3. The protein resonance broadens notably at lower ligand concentrations, but becomes narrower at higher ligand concentrations. These observations imply a moderately fast exchange process. 4. The protein resonance does not change significantly in intensity or linewidth, but the chemical shift changes continuously as the ligand concentration increases. These observations imply a fast exchange process.
755
9.2 INTERMOLECULAR INTERACTIONS
a
1.0
[PL]/[P]T
0.20
Rex/|∆w|
0.15
b
0.5
0.10 0
0.5
1.5
2.0
2.5
3.0
[L]T/[P]T
0.05
0
1.0
0.5
1.0
1.5 [L]T/[P]T
2.0
2.5
3.0
FIGURE 9.11 Ligand-induced contributions to chemical exchange linebroadening. (a) The chemical exchange contribution to transverse relaxation rate constant, Rex, and (b) the degree of saturation of the protein binding site are shown as a function of total ligand concentration, [L]T, for (—) [P]T ¼ 1000Kd and kr ¼ 0.01!, (- - -) [P]T ¼ 100Kd and kr ¼ 0.2!, (– – –) [P]T ¼ 10Kd and kr ¼ !, and (– - – -) [P]T ¼ 2Kd and kr ¼ 5!.
To provide a quantitative basis for these observations, calculated contributions to transverse relaxation resulting from chemical exchange are shown in Fig. 9.11. Rates of exchange are very dependent on solution conditions, including temperature, pH, and ligand concentration. These parameters must be known accurately and studies should be performed over a wide range of precisely monitored conditions. Studies on proteins subject to intermediate exchange processes are particularly challenging, because line-broadening effects render resonances difficult, if not impossible, to detect. As a practical rule of thumb, the intermediate exchange regime frequently will be encountered when the dissociation constant (Kd) for the protein–ligand interaction is on the order of 10–100 M.
9.2.2 PROTEIN–LIGAND BINDING INTERFACES The exquisite sensitivity of NMR spectral parameters, including chemical shifts and relaxation rate constants, to molecular environments allows facile detection of intermolecular interactions of proteins with ligands. A large number of approaches have been developed
756
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
to efficiently identify the sites on a protein surface that interact with ligands. In some instances, the NMR observables can be used to guide modeling of the protein–ligand complex even in the absence of intermolecular structural restraints based on NOEs (85, 86). A selection of techniques in widespread use is discussed in the following sections. 9.2.2.1 Chemical Shift Mapping Perturbation of chemical shifts as a result of complex formation provides a highly sensitive tool for identification of protein binding sites. This procedure is referred to as chemical shift mapping and has been utilized in a variety of ways, including defining contacts between macromolecules and highthroughput screening of small ligands to determine structure–activity relationships (SARs) (87). Chemical shift mapping is especially valuable for studying weaker protein–ligand interactions in fast exchange (Kd 4 106 M), because crystallization of such complexes is often difficult or impossible. Furthermore, large conformational changes of the protein, which could result in extensive changes to the NMR spectrum, usually do not occur in such weak binding interactions. HSQC, TROSY, and other 1H–15N correlation spectra commonly are used for chemical shift mapping because these spectra are highly sensitive and often well-resolved. The binding surface on an 15N-labeled protein is identified by titrating the unlabeled ligand into a solution of the protein and following the associated spectral changes in the 1H–15N correlation spectrum. Assignments for a complex in the fast-exchange regime are obtained by tracking the 1HN and 15N resonances from their positions in the initial spectrum of the uncomplexed protein as the titration proceeds. When resonance positions no longer change with increasing ligand concentration, the protein is presumed to be saturated with ligand. The stoichiometry of protein–ligand binding is inferred from the titration endpoint. If the chemical exchange between the free and ligand-bound protein is slow, indicating a tightly bound complex, and/or if a large conformational change occurs upon binding, then chemical shift mapping is not as straightforward as in the fast-exchange limit, because reassignment of the protein resonances in the complex by standard approaches may be necessary. Chemical shift changes often are quantified using the weighted chemical shift change per residue, (88), ¼ ½2HN þ ð0:1N Þ2 1=2 ,
½9:16
in which HN and N are the changes in chemical shifts of 1HN and 15N, respectively, during complex formation, measured in units
9.2 INTERMOLECULAR INTERACTIONS
757
of parts per million. The scaling factor of 0.1 is calculated from the ratio of the magnetogyric ratios ( N/ H 0.1). The scaling factor empirically scales the 15N chemical shift changes to be more equivalent to the 1H chemical shift changes. If changes in 13C chemical shifts are being monitored, a scaling factor of C/ H 0.25 is used. Scaling factors also can be estimated from the distribution of nucleus-specific chemical shift ranges in proteins (88). The dissociation constant for 1:1 binding equilibrium, Kd, can be estimated by fitting the observed chemical shift changes to the following equation: n o1=2 2 ¼ max ½LT þ ½PT þ Kd ½LT þ ½PT þKd 4½LT ½PT =ð2½PT Þ, ½9:17 in which is the observed chemical shift change at a given total ligand concentration, [L]T, (relative to the resonance frequency in the absence of ligand), max is the change in chemical shift at saturation and [P]T is the total protein concentration. However, because relatively high sample concentrations are required for NMR spectroscopy, dissociation constants are not easily measured for high-affinity complexes. Theoretical aspects of the determination of dissociation constants by NMR spectroscopy have been discussed (89, 90). Resonances exhibiting large chemical shift changes upon ligand binding are assumed to reside at the binding interface. However, resonances in the spectrum may be shifted because the associated nuclei are proximal to the protein–ligand interface or because conformational changes upon binding have resulted in altered local magnetic environments, even for nuclear spins distant from the interaction site. These other local structural effects that would result in concomitant shift changes cannot be disregarded a priori, so caution is necessary to prevent overinterpretation of chemical shift perturbations. The key to discriminating between these two cases lies in establishing intermolecular NOE connectivities between the protein and the ligand (Section 9.2.4). Nevertheless, mapping of interfaces in this fashion is an excellent initial diagnostic. 9.2.2.2 Cross-Saturation Labeling strategies can be used together with cross-saturation methods to map interfaces in protein–ligand complexes (91). The method is generally applicable to large protein– protein complexes and, with minor modifications, to other protein– macromolecular ligand interactions — e.g., appropriately labeled DNA and RNA ligands (92–94).
758
CHAPTER 9 LARGER PROTEINS
aliphatic frequency
water frequency
MOLECULAR INTERACTIONS
f3
y
y
AND
t2 1H
saturation f1
15N
t
1
f1
f2
t
1
t1
2
–2
t
t
t
t
3
3
4
4
Grad
FIGURE 9.12 TROSY pulse sequence for using cross-saturation to determine protein–ligand interfaces. Thin and thick bars represent 908 and 1808 pulses, respectively. The pulse depicted as an open bar is a 1808 pulse crafted to leave the water magnetization unperturbed. Pulses are applied along the x-axis unless noted; ¼ 1/(4JNH). Phase cycle is as follows: 1 ¼ x, y, x, y; 2 ¼ y; 3 ¼ y; receiver ¼ x, y, x, y for the first FID. The second FID is acquired with 1 ¼ x, y, x, y; 2 ¼ y; 3 ¼ y; receiver ¼ x, y, x, y. Depending on the spectrometer, y and y phases may need to be interchanged. Band-selective saturation is applied in the middle of the aliphatic 1H spectral region during the recycle delay. Frequency discrimination in the F1 dimension is achieved as described in Section 7.1.3.3.
The simplest example is provided by a complex between two proteins. Protein A, regarded as the ligand, is produced unlabeled and fully protonated. Protein B, the protein whose binding interface is to be defined, is uniformly labeled with 2H and 15N. Protein A possesses a high density of 1H spins, while protein B possesses a low density of 1 H spins. The physical basis of the technique is saturation transfer. The technique utilizes a variant of the water flip-back 1H–15N TROSY experiment (Section 7.1.4.2), as shown in Fig. 9.12. The experiment begins by irradiating the aliphatic region of the 1H NMR spectrum (0–3 ppm), while leaving the amide, aromatic, and water 1H spins unperturbed. Such discrimination can be achieved by using a bandselective decoupling scheme such as WURST or CHIRP (95–97). Because protein B does not contain aliphatic 1H spins, only protein A
9.2 INTERMOLECULAR INTERACTIONS
759
initially is affected by the low-power rf field. Efficient spin diffusion through the high-density 1H spin network causes rapid saturation of the aromatic and amide 1H resonances of protein A. Thus, within a short period of time, the 1H spectrum of protein A is fully saturated, while the amide 1HN spectrum of protein B is nearly unperturbed. However, because spin diffusion is effective for 1H spins in close proximity, this condition does not persist. Although protein B is not directly affected by the irradiation, saturation of resonances in protein A is transferred across the intermolecular interface to proximal amide 1HN spins in protein B by the NOE. The low density of 1H spins in protein B minimizes spin diffusion within the spin network of protein B and restricts saturation transfer effects to 1HN spins in the intermolecular interface. The reduction in intensities of 1HN resonances in protein B is measured by comparison to a reference spectrum acquired without irradiation of the aliphatic spectral region. Residues showing the largest difference in resonance intensity due to cross-saturation from protein A are identified as being located at the interface. The method just described, although useful, has drawbacks, principally because cross-saturation is detected by the labile backbone amide 1HN spins. In order to make identification of the intermolecular interface as accurate as possible, the experiment should be performed in 90% D2O to further isolate the amide 1HN spins and reduce spin diffusion in protein B. As a direct consequence, the overall sensitivity of the experiment is reduced approximately 10-fold. In addition, relatively few HN atoms may be located in intermolecular interfaces, particularly for proteins with hydrophobic binding sites. An improved version of this basic experiment takes advantage of the observation that methyl groups frequently are located at intermolecular interfaces (98). Cross-saturation between a fully protonated protein molecule and a methyl-protonated, otherwise perdeuterated, ligand offers significant sensitivity advantages. Perdeuteration ensures the required low 1H spin density, methyl 1H spins are nonexchangeable, and spin diffusion is intrinsically reduced by the favorable R1 relaxation rate constants of the methyl 1H spins. 9.2.2.3 Transverse Relaxation and Amide Proton Solvent Exchange Provided that the time scale for chemical exchange does not approach the very slow or very fast limits, then enhanced line broadening (or increase in R2) frequently is observed for the resonances of nuclei at the binding interface of the protein–ligand complex. This effect is most easily observed for small to medium-sized proteins, because exchange results in only a small fractional increase in line
760
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
broadening (or R2) for larger proteins with relatively broad lines in the absence of ligand. The advantage of this method, compared to mapping chemical shift perturbations, is that binding-induced structural rearrangements usually do not significantly increase linewidths or R2 relaxation rate constants (provided that ligand binding is not associated with a change in oligomerization state of the protein). Mapping of a protein–ligand interface also can be achieved by comparing the rates of amide–solvent hydrogen–deuterium exchange for the free and bound forms of the protein. 1HN and 15N protein assignments must be known for the free and bound protein, and the largest effects usually are obtained for complexes in the slow exchange regime. Amide exchange rates are measured as described in Section 10.2.1.5. Amide 1HN spins in the complex that show slower exchange rates with solvent (enhanced protection) likely define regions of the protein involved in binding, because these spins are protected from solvent to an additional degree in the complex.
9.2.3 RESONANCE ASSIGNMENTS FOR PROTEIN COMPLEXES
AND
STRUCTURAL RESTRAINTS
The exchange regime for a protein–ligand complex governs the atomic resolution structural information that can be obtained for the protein, the ligand, and the protein–ligand interface. Again, the fast and slow chemical exchange regimes are the most amenable to detailed investigation. If chemical exchange is slow (strong binding), then a stoichiometric complex between protein and ligand is suitable (in practice, the ligand may be titrated slightly past the stoichiometric ratio). The NMR resonances reflect the stoichiometric bound conformations of the protein and the ligand. In this regime, the complete structure of the protein– ligand complex can be determined based on intraprotein, intraligand, and interprotein–ligand NOE restraints (Section 9.2.4.2). If exchange is fast, then the resulting NMR spectra characterize the average properties of the nuclear spins over an ensemble of possible states, depending on the protein:ligand ratio. The structure of the protein in the complex can be determined if the NMR observables are dominated by the properties of the protein in the complex, rather than by those of the free protein. To ensure this limit, sufficient excess ligand must be provided to saturate the protein binding site. In some cases, limited solubility of ligands may preclude reaching this limit. If exchange is fast, then protein–ligand NOEs usually will not be obtained.
9.2 INTERMOLECULAR INTERACTIONS
761
In this regime, the transferred NOE can be used to determine structural characteristics of the ligand in its complexed form (99–103). The techniques described in Section 9.2.3 subsequently can be used to model the ligand–protein complex. Resonance assignments and structural restraints for the protein component of a protein–ligand complex are obtained by recording the spectrum of the protein while suppressing resonances arising from the ligand, which otherwise would cause ambiguities and overlap. Selective observation of the NMR spectrum of one component of a complex is made possible by differential incorporation of isotopic labels into the protein and ligand. In the experimental design considered herein, the protein is labeled, typically with 15N and 13C, and the ligand is unlabeled. In some cases, different isotopes are incorporated into the protein and the ligand (i.e., one component of the complex might be labeled with 13C and the other component might be labeled with 15N). In either situation, the protein and ligand molecules can be regarded as independent from an NMR viewpoint. 9.2.3.1 Assignments and Structures of Proteins in Protein–Ligand Complexes If the protein component of a protein–ligand complex is doubly labeled with 15N/13C or triply labeled with 2H/13C/15N, then the standard multidimensional heteronuclear NMR experiments described in Chapters 7 and 9 (see Section 9.1) are employed to obtain backbone and side chain assignments, stereospecific assignments, and torsion angle restraints solely for the protein in its complexed form. Four types of NOE cross-peaks can be expected for a binary complex consisting of labeled (15N/13C) protein (P) and unlabeled (12C/14N) ligand (L): Intramolecular: Pð15 N=13 CÞ ! Pð15 N=13 CÞ, Lð12 N=14 CÞ ! Lð12 N=14 CÞ: Intermolecular: Pð15 N=13 CÞ ! Lð12 N=14 CÞ, Lð12 N=14 CÞ ! Pð15 N=13 CÞ: ½9:18 For structure determination of the protein, those NOEs that provide structural information for the labeled protein alone are of interest. NOEs between 1H spins of the labeled protein are easily established because only those signals that originate or terminate on the unlabeled ligand must be suppressed. However, conventional 3D 15N- or 13 C-edited NOESY experiments (Sections 7.2.1 and 7.2.3) ordinarily will not suffice, because these experiments generally only employ one heteronuclear filter. In the final NOESY spectrum, NOEs appear between 1H spins attached to isotopically labeled nuclei and all other
762
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
proximal 1H spins, whether they are attached to isotopically labeled nuclei or not. Consequently, both intra- and intermolecular NOEs are present in the spectrum. To ensure suppression of cross-peaks generated by intermolecular NOEs, NOESY experiments must select for heteronuclear coherences both before and after the NOESY mixing time. The only NOEs observed occur between 1H nuclei that are directly attached to isotopically labeled heteronuclei (those in the protein). For example, the 3D 1H–15N HMQC–NOESY–HMQC experiment (Section 7.2.4.1) can be used to provide NOEs between amide 1HN spins in the protein alone. The only modification to the sequence required is to introduce 13C 1808 decoupling pulses at t1/2 and t2/2, assuming a doubly labeled protein. Similarly, the 4D 13C/15N HMQC–NOESY–HMQC experiment (Section 7.2.4.2) provides NOEs between 13C-bound 1H spins and 15 N-bound 1H spins exclusively, rejecting any interference from unlabeled ligand 1H spins. The 4D 13C/13C HMQC–NOESY–HMQC (Section 7.2.4.3), with 15N decoupling applied throughout the sequence, can be used to generate NOEs between 13C-bound protons within the protein exclusively. The 13C 3D HSQC–NOESY experiment can be used to generate intraprotein NOE correlations for a protein–ligand complex if the protein is labeled with 13C, while the ligand is perdeuterated, but not labeled with 15N/13C. This is straightforward if the ligand is another protein that can be easily obtained biosynthetically by overexpression. In 100% D2O buffer, intermolecular 1H–1H NOEs cannot be generated and only intramolecular 1H(13C)–1H(13C) NOEs are observed. 9.2.3.2 Isotope-Edited/Filtered NOESY to Define Intermolecular Interfaces Accurately identifying and determining the atomic resolution structure of the interface in a protein–ligand complex requires establishing unambiguous intermolecular restraints between the protein and ligand. The following discussion focuses on isotope-filtered techniques for establishing NOE connectivities between the protein and the ligand; however, RDCs (104–106) and trans-hydrogen scalar coupling interactions (107) also potentially provide intermolecular restraints. Isotope-filtered methods have been reviewed (108, 109). Identifying intermolecular NOEs depends on techniques to selectively observe the NMR spectra of individual components of the protein–ligand complex based upon the differential incorporation of isotopic labels into the components of a molecular complex. In the present discussion, the protein is assumed to be labeled and the ligand is assumed to be unlabeled; in practice, the opposite labeling strategy
763
9.2 INTERMOLECULAR INTERACTIONS
might be used as well (particularly for investigations of protein–nucleic acid complexes). The pulse sequence for separating the NMR spectrum into individual subspectra of isotopically enriched or natural-abundance components is referred to as an isotope filter and is constructed from a heteronuclear spin echo difference experiment. The pulse sequence can be used to either select signals from the isotopically labeled species while suppressing signals from nonlabeled entities, normally referred to as isotope editing, or can be used to reject signals from the isotopically labeled species while retaining signals from nonlabeled entities, normally referred to as isotope filtering (109–121). The most straightforward experiment for these applications is referred to as a half-filter. Half-filters often are combined with full NOESY experiments to generate subspectra that contain different types of NOE cross-peaks. The basic half-filter is (122) 1
H: X:
90x – – 180x – –, 90x 90 ,
½9:19
in which ¼ 1/(21JIS) Because 1JCH and 1JNH are significantly larger than 3JHH, evolution of the 1H–1H scalar coupling interaction during is considered negligible. The pulse sequence is executed twice, once with ¼ x and once with ¼ x. The 1H spins that are not coupled to a heteronucleus (designated by the operator K) do not experience the effects of the pulses on the X nucleus (designated by the operator S). Evolution through the half-filter yields Kz ! Ky
½9:20
for both transients. The 1H spins that are coupled to a heteronucleus (designated by the operator I) evolve under the heteronuclear coupling Hamiltonian and experience the effects of the pulses on the X nucleus. Evolution through the half-filter yields 2 Ix
1
JIS
Iz ! Iy ! Iy cosð1 JIS Þ þ 2Ix Sz sinð1 JIS Þ Ix , 2 Sx , 2 S
! Iy cosð1 JIS Þ sgnð Þ2Ix Sz sinð1 JIS Þ 1J
IS
! Iy ½cos2 ð1 JIS Þ sgnð Þ sin2 ð1 JIS Þ 2Ix Sz ½1 þ sgnð Þ sinð1 JIS Þ cosð1 JIS Þ,
½9:21
764
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
in which sgn( ) is þ1 when ¼ x and 1 when ¼ x. Adding the two transients together selects for resonances of 1H nuclei not coupled to X nuclei, yielding an isotope-filtered spectrum: Kz þ Iz ! Ky þ Iy cos2 ð 1JIS Þ:
½9:22 1
Subtracting the two scans selects for resonances of H nuclei that are coupled to a heteronucleus, yielding an isotope-edited spectrum: Kz þ Iz ! Iy sin2 ð 1JIS Þ:
½9:23
The antiphase operators have been ignored in writing [9.22] and [9.23] because these terms normally are suppressed or rendered unobservable by other pulse sequence elements in the complete experiment. The level of suppression of the S-bound I-spin magnetization achieved in the isotope-filtered subspectrum is proportional to cos2(1JIS). Filtering is perfect for a nominal coupling constant 1 0 JIS ¼ 1=ð2Þ. For other scalar coupling constants, the intensity of the residual Iy magnetization that survives the filter is given by " ¼ cos2 ð1JIS Þ ¼ cos2 ð1JIS =½21JIS0 Þ ¼ sin2 ðJ=½21 JIS0 Þ ð2 =4ÞðJ=1JIS0 Þ2 ,
½9:24
in which J ¼ 1JIS 1JIS0 . The filter depends on two delays of nominal length 1=ð21JIS0 Þ and results in filter breakthrough errors proportional to J2, for small variations in the scalar coupling constant. A filter with this property is referred to as a ‘‘second-order J-filter.’’ As shown by [9.23], variation in the scalar coupling constants reduces the sensitivity of isotope-edited experiments, but does not result in breakthrough peaks. In actual practice, the two transients for the isotope filter in [9.19] might be acquired in a different fashion. For the first transient, the 90 x 90 x 13C pulse combination is replaced by a composite 1808 pulse to minimize pulse imperfections and effects of resonance offset. For the second transient, the 90 x 90 x 13C pulse combination is omitted. Breakthrough peaks in the isotope-filtered subspectrum are particularly troublesome for 1H–13C isotope-filtered experiments. Perfect discrimination between 1H atoms bound to 13C or 12C atoms relies on the delay being perfectly matched to the 1JCH coupling. This coupling constant is far from uniform. For example, aromatic 13C nuclei have 1 JCH in the range 160–180 Hz, 13C nuclei attached to nitrogen or oxygen atoms have 1JCH 140 Hz, and 13C nuclei in a methyl group have 1 JCH 125 Hz. Therefore, the isotope-filter delay cannot be tuned
765
9.2 INTERMOLECULAR INTERACTIONS
simultaneously for all 1JCH values. The resulting unwanted signals in the subspectra increase overlap and cause ambiguities in spectral interpretation. In principle, any 1H NMR experiment can be modified to incorporate the half-filter element. For example, an !1 half-filtered NOESY experiment is obtained by replacing the first 908 pulse in an 1H 2D NOESY pulse sequence with the half-filter element (112). The basic experiment is written as follows: 1
H: X:
90x ––180x ––t1 =2–t1 =2–90x –m –90x –receiver 90x 90
180x
decouple
½9:25
Appropriate construction of the phase cycle to subtract alternate scans ensures that only 1H spins attached to, for instance, 15N enter into the t1 period. In this case, the resulting spectrum would include only NOE cross-peaks having resonance frequencies of 1HN in the indirect dimension, but all proximal 1H spins in the direct dimension. The complementary !2 half-filtered NOESY (123) replaces the final 908 pulse in an 1H NOESY experiment with the half-filter element and provides a full complement of NOEs in the indirect dimension and, with suitable phase cycling, only 1HN spins in the direct dimension. The !1 and !2 half-filter experiments are combined in the doublehalf-filter experiment (117). A pulse sequence for a 2D 1H–1H NOESY experiment with a 13C(F1, F2)-double-half-filter is shown in Fig. 9.13. The sequence consists of two half-filter elements surrounding a NOESY experiment. The two pairs of 90 x 90 1 and 90 x 90 3 pulses on the 13C spin are phase cycled independently in combination with the receiver to generate four data sets that are recorded separately. Linear combinations of the four sets of data provide for an elegant editing scheme that allows each of the four types of NOEs listed in [9.18] to be independently observed. Table 9.2 summarizes the processing scheme and the information content of the four sets of subspectra. Intermolecular NOEs between the protein and the ligand are present in two of the subspectra, reflecting magnetization transfer originating on a protein 1 H spin or originating on a ligand 1H spin. If the protein assignments are known, then the precise location of the protein interface can be determined. A practical issue concerns 1D versus nD isotope-edited NMR experiments. Sometimes, a 1D isotope-edited experiment would, in principle, suffice for a given application (for example, monitoring the resonances of a small labeled ligand bound to a large unlabeled protein). In practice, the large natural-abundance 13C signal from the protein can
766
CHAPTER 9 LARGER PROTEINS φ1 1H
φ2
φ3
ψ1
m
ψ2 t1 2
13C
MOLECULAR INTERACTIONS
φ4 τ
τ
τ
AND
φ5 τ
t2 τ
ψ3 t1 2
decouple
FIGURE 9.13 Basic pulse sequence for a 13C-double-half-filter NOESY experiment. Thin and thick bars represent 908 and 1808 pulses, respectively. Pulses are applied along the x-axis unless noted. Phases 1– 5 and 2 are all alternated independently from x to x; concomitantly, the receiver phase is alternated with each phase shift of a 908 pulse. Four combinations of this sequence are acquired: (1) 1 ¼ 3 ¼ x; (2) 1 ¼ x, 3 ¼ x; (3) 1 ¼ x, 3 ¼ x; and (4) 1 ¼ 3 ¼ x. Subspectra are generated as noted in Table 9.2.
TABLE 9.2 Processing Schemes and Information Content for Double-Half-Filtered NOESY Linear combinationa 1þ2þ3þ4 123þ4 12þ34
1þ234
a
Subspectrum type
Information content
C(F1)–13C(F2), doubly filtered 13 C(F1)–13C(F2), doubly selected 13 C(F1)–13C(F2), F1 selected–F2 filtered
Intramolecular NOEs between 1 H spins in unlabeled ligand Intramolecular NOEs between 1 H spins in 13C-labeled protein Intermolecular NOEs between 1 H spins in unlabeled ligand (F2) and 1H spins in 13C-labeled protein (F1) Intermolecular NOEs between 1 H spins in 13C-labeled protein (F2) and 1H spins in unlabeled ligand (F1)
13
13
C(F1)–13C(F2), F1 filtered–F2 selected
Phase cycles used for data sets 1–4 are given in the caption to Fig. 9.13.
767
9.2 INTERMOLECULAR INTERACTIONS
be a major impediment. In such cases, a 2D (or higher dimensionality) experiment is needed to disperse the natural-abundance signals so that the ligand resonances can be observed (124). The half-filter sequence given in [9.19] is a difference experiment and, as such, relies on extremely high spectrometer stability (just like phase cycling). Isotope filtration that does not rely on a phase-cycled difference experiment is obtained by incorporating spin lock purge pulses or gradient pulses into the pulse sequence elements. A variation of the basic half-filter that uses purge pulses to obtain isotope filtering is represented as follows (125): 1
H:
90x –=2–180x –=2–½spin locky
X:
180x
90x
½9:26
After a total delay of ¼ 1/(21JXH), prior to the spin lock pulse and the 908(S) pulse, evolution of magnetization is described by Kz þ Iz ! Ky þ Iy cosð1JIS Þ 2Ix Sz sinð1JIS Þ:
½9:27
The 908(S) pulse converts the antiphase coherence to multiple-quantum coherence and the spin lock applied to the 1H spins both dephases the multiple-quantum coherence and retains in-phase 1H magnetization. The total time for this filter (neglecting the duration of the spin lock) is one-half that of the basic half-filter; thus, losses due to relaxation also are reduced. In contrast to [9.19], no phase cycling is required for this filter sequence. For the simple half-filter element described in [9.26], the level of suppression of the S-bound I-spin magnetization is dependent on cos(1JIS). The intensity of the residual Iy magnetization that survives the filter is given by 0 0 " ¼ cosð1JIS Þ ¼ cosð1JIS =½21JIS Þ ¼ sinðJ=½21JIS Þ 0 ð=2ÞðJ=1JIS Þ:
½9:28
The filter depends on only the one delay, and results in filter breakthrough errors proportional to J, for small variations in the scalar coupling constant. Thus, this filter is referred to as a ‘‘first-order J-filter.’’ Various schemes are available to increase the degree of suppression achieved by isotope filters, particularly for 1H–13C spin pairs; first-order filters normally are sufficient for application to amide 1H–15N spin pairs.
768
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
A simple example of a second-order filter using spin lock purge pulses is obtained from the addition of two first-order filters (48, 118, 125): 1 13
H: 90x –1 =2–180x –1 =2–½spin-locky –2 =2–180x –2 =2–½spin-locky C:
180x
90
180x
90 90
½9:29
The values of 1 and 2 are set to accommodate different 1JCH couplings, e.g., 1¼ 3.57 ms (1JCH ¼ 140 Hz) and 2 ¼ 4.0 ms (1JCH ¼ 125 Hz). The overall level of suppression during this sequence is given by cos(1JCH 1) cos(1JCH 2). Higher order filters provide efficient suppression of undesired magnetization over a wider range of 1JCH scalar coupling constants; however, the sequences are longer and relaxation losses are greater. A simple second-order filter that utilizes pulsed field gradients is 1
H:
13
90x – – 180x – –
C:
Grad:
90x g1
90x g1
½9:30
Magnetization for 1H spins not coupled to a 13C spin is unaffected by the 13C pulses and forms a gradient echo. Magnetization for 1H spins coupled to a 13C spin evolves into antiphase coherence during the first delay , is converted to multiple-quantum coherence by the first 13 C 908 pulse, and is dephased by the second gradient g1. Residual Iy magnetization evolves into antiphase coherence during the second delay and is converted to multiple-quantum coherence by the second 13C 908 pulse. The overall level of suppression during this sequence is given by cos2(1JCH). Replacing the hard 1808(13C) pulses in half-filter elements by designed adiabatic inversion pulses is an elegant approach to increase the level of suppression (126, 127). The scalar coupling constants 1JCH are roughly linearly correlated with 13C chemical shifts. Thus, by judicious choice of the sweep rate and field strength of the adiabatic inversion pulse (Section 3.4.6), suppression errors caused by the mismatch of the filter timing with respect to 1JCH are minimized. Another method providing improved suppression takes advantage of the similarity between composite pulse rotations and broadband polarization transfer and does not rely on relationships between 13C shifts and 1JCH values (128). The scaling for this third-order filter is proportional to
9.3 METHODS
FOR
RAPID DATA ACQUISITION
769
cos3(1JCH), in which ¼ 1/(21JCH). This filter has been shown to provide good suppression over a range 115 to 165 Hz (when is set for 1 0 JCH ¼ 140 Hz) and, as such, provides excellent purging for all 1H atoms attached to 13C atoms in a protein, apart from those in histidine imidazole groups. Isotope-filter elements can be implemented in any dimension of a multidimensional experiment as deemed appropriate. For example, an F1-edited, F3-filtered 13C 3D HMQC–NOESY experiment has been used to assign intermolecular NOEs between isotopically labeled and unlabeled components of a complex (129).
9.3 Methods for Rapid Data Acquisition As the discussions in Chapters 6–8 make clear, a large number of NMR spectra might be recorded in the course of a single investigation of the structure, dynamics, and interactions of a protein. Recording the necessary multidimensional NMR data sets represents a significant investment of spectrometer time; therefore, efficiency in collecting data is an important issue in biomolecular NMR spectroscopy. In particular, the inherent sensitivity of NMR spectrometers has steadily increased as a result of developments in magnet, probe, and amplifier technology and the total number of transients that must be recorded to achieve a given signal-to-noise ratio decreases with each improvement in spectrometer sensitivity. Many modern gradient-enhanced heteronuclear multidimensional NMR experiments already use minimal phase cycling, typically consisting only of isotope editing and quadrature detection. In this limit, the total length of the NMR experiment is determined by the requirements of the Nyquist sampling theorem and the maximum evolution times in indirect dimensions needed for resolution. Further reduction in total experimental data acquisition times requires new methods for recording NMR spectra. Certain techniques that potentially provide two- to fourfold reductions in experimental data acquisition time already are in common use. Techniques for optimizing recovery of magnetization between transients allow individual transients to be recorded more rapidly (45–47). Aliasing in 13C frequency dimensions enables long evolution times to be achieved in a smaller number of increments, which enables highly resolved spectra to be acquired more rapidly (Sections 7.1.2.3 and 7.1.5). Linear prediction and maximum entropy reconstruction methods can improve resolution for spectra recorded with limited digital resolution (Section 3.3.4). In some cases, more than one type of correlation can be
770
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
recorded in a single NMR experiment. The simultaneous 13C/15N-edited NOESY experiment is an important example in biomolecular NMR (130). Atreya and Szyperski review many of these approaches (131). Emerging methods seek to minimize experimental data acquisition times by dramatically reducing the number of transients that must be recorded to obtain the necessary spectral resolution. These methods avoid the strictures imposed by the conventional sampling strategy in which multidimensional NMR experiments are recorded by systematically and independently incrementing each of the indirect evolution delays. Three of these methods, nonuniform sampling, GFT, and projection–reconstruction, are briefly discussed here in order to provide an introduction to the field. These and other approaches, including filter diagonalization (132–134), single-scan NMR (135, 136), and Hadamard spectroscopy (137–139), have been discussed in reviews by Freeman and Kupcˇe (140, 141).
9.3.1 NONUNIFORM SAMPLING The most intuitive approach relaxes the requirement that indirect frequency dimensions are sampled at regular increments. This requirement is imposed by the conventional fast Fourier transformation algorithm (Section 3.3.1); however, this sampling strategy means that transients with low signal-to-noise ratios are recorded whenever one or more of the indirect evolution periods are long. Nonuniform sampling uses more extensive recording of data for smaller values of the indirect evolution delays (when sensitivity is high) and less extensive recording for larger values of the evolution delays (when sensitivity is low). The sampling scheme is designed to maximize resolution and signalto-noise ratio within a given total experimental data acquisition time (142). Nonuniform sampled data cannot be processed by conventional Fourier transformation algorithms. Maximum entropy reconstruction (142), Fourier transform algorithms for nonequispaced data (143), and multidimensional decomposition (144) have been used to process spectra recorded with nonuniform sampling in indirect evolution periods. Two- to sixfold reductions in experimental data acquisition times have been demonstrated (142, 144). A particularly powerful illustration of this approach, combined with other techniques discussed in Section 9.1, is provided by a 4D 13C, 13C-edited NOESY spectrum recorded using a methyl-protonated, otherwise perdeuterated, sample of malate synthase G (723 residues, 82 kDa) (145). Data acquisition used methyl-TROSY techniques and nonuniform sampling to optimize
9.3 METHODS
FOR
RAPID DATA ACQUISITION
771
sensitivity and resolution. The spectrum was processed using multidimensional decomposition.
9.3.2 GFT-NMR SPECTROSCOPY The G-matrix Fourier transform NMR (GFT-NMR) was introduced by Kim and Szyperski in 2003 (146) and is comprehensively reviewed by Atreya and Szyperski (131) and by Freeman and Kupcˇe (140). GFT employs the so-called reduced dimensionality (RD) principle in which multiple indirect evolution periods are incremented simultaneously, rather than independently (147–155). The initial report of GFTNMR has been followed by additional developments and demonstrations (156–159) as well as applications to protein assignments and structure determination (160, 161). A GFT-NMR experiment co-evolves m indirect evolution periods of an nD experimental pulse sequence using the RD technique. The resulting data set is referred to as an (n, n m þ 1)D GFT-NMR spectrum. For example, t1 and t2 indirect evolution times are defined by t1 ¼ t, t2 ¼ kt, in which k is a constant that allows for different spectral widths in the different frequency dimensions (k ¼ 1 in the simplest implementation), in a (3, 2)D GFT-NMR experiment. A GFTNMR data set consists of 2m RD NMR experiments in which sineand cosine-modulated data for each of the co-evolved indirect dimensions are collected by modulating the phases of the appropriate pulses in the sequence. For example, in a (3, 2)D GFT-HNCA experiment, four data sets are acquired in which the 908 pulse before t1 and the 908 pulse before t2 have phases {x, x}, {x, y}, {y, x}, and {y, y}; these are the same sets of phase shifts that would be required for quadrature detection in the two indirect evolution periods of a conventional data acquisition. The resulting time-domain signals in the indirect dimensions are given by sA ðtÞ ¼ cosð1 tÞ cosð2 ktÞ,
sC ðtÞ ¼ sinð1 tÞ cosð2 ktÞ;
sB ðtÞ ¼ cosð1 tÞ sinð2 ktÞ,
sD ðtÞ ¼ sinð1 tÞ sinð2 ktÞ:
½9:31
The GFT method unravels the chemical shift information from the intertwined indirect evolution times by addition or subtraction of the individual sets of frequency-domain data. In the present example, the two spectra, represented by sA ðtÞ þ sD ðtÞ ¼ cosð½1 k2 tÞ,
sB ðtÞ sC ðtÞ ¼ sinð½1 k2 tÞ, ½9:32
772
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
represent a phase-sensitive 2D data set in which resonance signals appear at frequency positions 1 k2 in the indirect dimension. The two spectra, represented by sA ðtÞ sD ðtÞ ¼ cosð½1 þ k2 tÞ, sB ðtÞ þ sC ðtÞ ¼ sinð½1 þ k2 tÞ, ½9:33 represent a phase-sensitive 2D data set in which resonance signals appear at frequency positions 1 þ k2 in the indirect dimension. In principle, the resonance frequencies for cross-peaks in a conventional 3D experiment are obtained equivalently in (3, 2)D GFT-NMR by adding and subtracting the frequencies of the resonance pairs in these two 2D spectra. In practice, identifying the corresponding resonances in congested spectra is facilitated by acquiring additional lower dimensionality spectra with fewer co-evolved evolution periods. In the example being discussed, a conventional 2D spectrum that records only the frequency 1 in the indirect dimension enables the cross-peaks with frequencies 1 k2 and 1 þ k2 to be identified unambiguously by symmetry with respect to the cross-peak with frequency 1. Thus, in this example, a total of six 2D data sets is acquired to generate the information that would be obtained from a conventional 3D experiment with k sampled points in the third dimension (requiring acquisition of a total of 2k 2D spectra for quadrature detection). For k ¼ 32, the GFT-NMR data set requires only one-tenth the number of 2D acquisitions. The 2D RD data sets for GFT-NMR may need to be acquired with increased numbers of points in the indirect dimensions to achieve sufficient resolution and/or increased numbers of transients to achieve acceptable signalto-noise ratios compared to the individual 2D planes from the conventional 3D spectrum. Nonetheless, even accounting for these issues, considerable savings in data acquisition time are achievable by GFT-NMR. As indicated by the preceding example, the GFT method provides large reductions in data acquisition times for higher dimensionality experiments. Resonance signal overlap, as is often the case in the spectra of larger proteins, can result in difficulties in obtaining clean frequency separation in GFT data sets. Enhanced resolution can be obtained by increasing the number of frequency dimensions; however, each additional dimension in GFT-NMR splits the resonance signals into a doublet and sensitivity drops accordingly. GFT has been incorporated into a highly efficient streamlined protocol for structure determination of proteins 520 kDa (161).
9.3 METHODS
FOR
RAPID DATA ACQUISITION
773
9.3.3 PROJECTION–RECONSTRUCTION Kupcˇe and Freeman have introduced projection–reconstruction (PR), a method widely used in disparate fields, including X-ray tomography, medical imaging, and astronomy to the field of protein NMR spectroscopy (162–168). Kupcˇe and Freeman demonstrated that full 3D and 4D spectra can be reconstructed from a small set of 2D spectra recorded at various ‘‘tilt angles.’’ Using the terminology of reduced dimensionality (GFT), these experiments are generally referred to as (3, 2)D PR-NMR and (4, 2)D PR-NMR data sets. The PR method, like GFT, co-evolves each of the indirect evolution times. The relative sizes of the incrementable evolution delays are chosen to generate a projection of the spectrum onto a plane inclined at a predefined angle (the ‘‘tilt angle’’). PR is based on a Fourier transform theorem that states that a section through the origin of a twodimensional time-domain signal, s(t1, t2), inclined at an angle, , with respect to the t1 axis, transforms as the projection of the corresponding frequency-domain spectrum S(F1, F2) onto an axis through the origin inclined at the same angle. For (3, 2)D PR-NMR experiments, t1 and t2 indirect evolution times are defined by t1 ¼ t cos , t2 ¼ t sin to generate a projection onto a plane rotated around the F3 axis, subtending an angle with respect to the F1–F3 plane [the extension to (4, 2)D PR-NMR experiments is straightforward]. For each tilt angle in a (3, 2)D PR-NMR experiment, two projections are collected (positive and negative angles), both of which provide independent information about cross-peaks in the spectrum. The key is to record enough tilted planes to define the position of cross-peaks accurately, but not so many that the time required experimentally becomes large. Criteria for this choice have been discussed (169, 170). In PR-NMR, unlike GFT-NMR, the data are reconstructed to form a conventional NMR spectrum; therefore, chemical shift information does not need to be unraveled by multiple additions and subtractions. In addition, cross-peaks are not split into pairs by each additional co-evolved dimension; therefore, sensitivity is not substantially degraded by adding additional dimensions. A number of different algorithms have been proposed to reconstruct the NMR spectrum from a set of projections, and research in this area continues. The Lower-Value (LV) reconstruction algorithm, introduced by Kupcˇe and Freeman, is well-suited for data sets containing relatively few highly dispersed signals (165). However, the intensities of the reconstructed peaks are determined by the signal-to-noise ratio of the weakest of the projections, and the signal-to-noise ratio of the
44 HNCACB
HACANH
54
H114
59
H114
S115
S115
c
44
b
69
a
49
G71
54
H(N), N: G71
G71
Ca (ppm)
64
H(N), N: S115
49
HACA(CO)NH
D70
D70
64
59
D70
e 70
60
50
40
Cb (ppm)
30
20
f 7
6
5
Ha (ppm)
4
3
69
d
7
6
5
4
3
Ha (ppm)
FIGURE 9.14 Projection reconstruction NMR spectra for calbindin D28k (170). Shown are 2D slices extracted from (4, 2)D PR (a, d) HNCACB, (b, e) HACANH, and (c, f) HACA(CO)NH spectra recorded using a 0.9 mM 1H/13C/15N sample of calbindin D28k at 508C. The spectra were recorded on an 800-MHz NMR spectrometer. The slices correspond to 1H and 15N chemical shifts of residues (a–c) S115 and (d–f) G71. Data were reconstructed to 4D (64 64 128 738 complex points) using the HBLV method (bin size ¼ 8) from three orthogonal planes and five tilt angles (eight data sets total).
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reconstructed spectrum, do not benefit from the cumulative signal recorded across all of the projections. Thus, the LV algorithm is not optimal when the signal-to-noise ratio for the individual projections is low, as often could be the case for larger proteins. Backprojection (BP), proposed by Kupcˇe and Freeman as a solution to sensitivity problems associated with PR-NMR (167), enjoys the benefit of signal accumulation and retains as much signal as would be available from a conventional spectrum recorded for the same length of time as that used to collect the entire set of projections. This reconstruction algorithm was demonstrated by the reconstruction of a 3D HNCO spectrum from 18 projections (167). The Hybrid Backprojection-LowerValue (HBLV) algorithm, introduced subsequently by Venters and coworkers, accumulates signal intensity without significant artifacts and appears particularly useful for the accurate reconstruction of sensitivitylimited data (170). As an illustration of these new approaches for data acquisition and processing, Fig. 9.14 shows 2D slices extracted from (4, 2)D PR HNCACB (Fig. 9.14a, d), HACANH (Fig. 9.14b, e), and HACA(CO) NH (Fig. 9.14c, f ) spectra recorded for calbindin D28k. The measuring time for these PR-NMR datasets was 33 hr.
References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
L. E. Kay, J. Magn. Reson. 173, 193–207 (2005). V. Tugarinov, P. M. Hwang, L. E. Kay, Annu. Rev. Biochem. 73, 107–146 (2004). R. A. Venters, R. Thompson, J. Cavanagh, J. Mol. Struct. 602, 275–292 (2002). J. Fiaux, E. B. Bertelsen, A. L. Horwich, K. Wu¨thrich, Nature 418, 207–211 (2002). G. M. Clore, A. M. Gronenborn, Curr. Opin. Chem. Biol. 2, 564–570 (1998). G. M. Clore, A. M. Gronenborn, Trends. Biotechnol. 16, 22–34 (1998). G. M. Clore, A. M. Gronenborn, Nat. Struct. Biol. (Suppl.) 4, 849–853 (1997). P. Permi, Annu. Rep. NMR Spectrosc. 53, 245–296 (2004). R. Riek, G. Wider, K. Pervushin, K. Wu¨thrich, Proc. Natl. Acad. Sci. U.S.A. 96, 4918–4923 (1999). C. Dalvit, J. Magn. Reson. 97, 645–650 (1992). D. P. Frueh, T. Ito, J. S. Li, G. Wagner, S. J. Glaser, N. Khaneja, J. Biomol. NMR 32, 23–30 (2005). D. T. Browne, G. L. Kenyon, E. L. Packer, H. Sternlicht, D. M. Wilson, J. Am. Chem. Soc. 95, 1316–1323 (1973). T. Yamazaki, W. Lee, C. H. Arrowsmith, D. R. Muhandiram, L. E. Kay, J. Am. Chem. Soc. 116, 11655–11666 (1994). M. A. Markus, K. T. Dayie, P. Matsudaira, G. Wagner, J. Magn. Reson., Ser. B 105, 192–195 (1994). R. A. Venters, C. C. Huang, B. T. Farmer, R. Trolard, L. D. Spicer, C. A. Fierke, J. Biomol. NMR. 5, 339–344 (1995).
776
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
16. D. M. LeMaster, Prog. NMR Spectrosc. 26, 371–419 (1994). 17. B. T. Farmer, R. A. Venters, in ‘‘Modern Techniques in Protein NMR’’ (N. R. Krishna, L. J. Berliner, eds.), Vol. 16, pp. 75–120, Kluwer/Plenum, New York, 1998. 18. N. Murali and B.D.N. Rao, J. Magn. Reson. Ser. A. 118, 202–213 (1996). 19. D. M. Kushlan, D. M. LeMaster, J. Biomol. NMR 3, 701–708 (1993). 20. K. H. Gardner, L. E. Kay, Annu. Rev. Biophys. Biomol. Struct. 27, 357–406 (1998). 21. L. E. Kay, K. H. Gardner, Curr. Opin. Struct. Biol. 7, 722–731 (1997). 22. R. A. Venters, B. T. Farmer, C. A. Fierke, L. D. Spicer, J. Mol. Biol. 264, 1101–1116 (1996). 23. D. M. LeMaster, Meth. Enzymol. 177, 23–43 (1989). 24. D. M. LeMaster, Annu. Rev. Biophys. Biophys. Chem. 19, 243–266 (1990). 25. D. Nietlispach, R. T. Clowes, R. W. Broadhurst, Y. Ito, J. H. Keeler, M. Kelly, J. Ashurs, H. Oschkinat, P. J. Domaille, E. D. Laue, J. Am. Chem. Soc. 118, 407–415 (1996). 26. J. Marley, M. Lu, C. Bracken, J. Biomol. NMR 20, 71–75 (2001). 27. T. Torizawa, M. Shimizu, M. Taoka, H. Miyano, M. Kainosho, J. Biomol. NMR 30, 311–325 (2004). 28. T. Szyperski, D. Neri, B. Leiting, G. Otting, K. Wu¨thrich, J. Biomol. NMR 2, 323–334 (1992). 29. M. K. Rosen, K. H. Gardner, R. C. Willis, W. E. Parris, T. Pawson, L. E. Kay, J. Mol. Biol. 263, 627–636 (1996). 30. N. K. Goto, K. H. Gardner, G. A. Mueller, R. C. Willis, L. E. Kay, J. Biomol. NMR 13, 369–374 (1999). 31. N. K. Goto, L. E. Kay, Curr. Opin. Struct. Biol. 10, 585–592 (2000). 32. S. Rajesh, D. Nietlispach, H. Nakayama, K. Takio, E. D. Laue, T. Shibata, Y. Ito, J. Biomol. NMR 27, 81–86 (2003). 33. K. L. Constantine, L. Mueller, V. Goldfarb, M. Wittekind, W. J. Metzler, J. Yanchunas, J. G. Robertson, M. F. Malley, M. S. Friedrichs, B. T. Farmer, J. Mol. Biol. 267, 1223–1246 (1997). 34. T. Yamazaki, R. Muhandiram, L. E. Kay, J. Am. Chem. Soc. 116, 8266–8278 (1994). 35. K. H. Gardner, L. E. Kay, in ‘‘Modern Techniques in Protein NMR’’ (N. R. Krishna, L. J. Berliner, eds.), Vol. 16, pp. 27–74, Kluwer/Plenum, New York, 1998. 36. V. Tugarinov, R. Muhandiram, A. Ayed, L. E. Kay, J. Am. Chem. Soc. 124, 10025–10035 (2002). 37. F. A. Mulder, A. Ayed, D. Yang, C. H. Arrowsmith, L. E. Kay, J. Biomol. NMR. 18, 173–176 (2000). 38. P. E. Hansen, Prog. NMR Spectrosc. 20, 207–255 (1988). 39. Z. Majerski, M. Zuanic, B. Metelko, J. Am. Chem. Soc. 107, 1721–1726 (1985). 40. D. S. Garrett, Y. J. Seok, D. I. Liao, A. Peterkofsky, A. M. Gronenborn, G. M. Clore, Biochemistry 36, 2517–2530 (1997). 41. K. H. Gardner, M. K. Rosen, L. E. Kay, Biochemistry 36, 1389–1401 (1997). 42. D. M. LeMaster, J. C. LaIuppa, D. M. Kushlan, J. Biomol. NMR 4, 863–870 (1994). 43. D. S. Wishart, B. D. Sykes, F. M. Richards, J. Mol. Biol. 222, 311–333 (1991). 44. D. S. Wishart, B. D. Sykes, J. Biomol. NMR 4, 171–180 (1994). 45. M. Deschamps, I. D. Campbell, J. Magn. Reson, 178, 206–211 (2006). 46. K. Pervushin, B. Vo¨geli, A. Eletsky, J. Am. Chem. Soc. 124, 12898–12902 (2002). 47. P. Schanda, B. Brutscher, J. Am. Chem. Soc. 127, 8014–8015 (2005). 48. M. Sattler, J. Schleucher, C. Griesinger, Prog. NMR Spectrosc. 34, 93–158 (1999). 49. T. Yamazaki, W. Lee, M. Revington, D. L. Mattiello, F. W. Dahlquist, C. H. Arrowsmith, L. E. Kay, J. Am. Chem. Soc. 116, 6464–6465 (1994). 50. C. Bracken, A. G. Palmer, J. Cavanagh, J. Biomol. NMR 9, 94–100 (1997). 51. H. Matsuo, E. Kupcˇe, H. Li, G. Wagner, J. Magn. Reson. 111, 194–198 (1996).
9.3 METHODS
FOR
RAPID DATA ACQUISITION
777
52. Z. Y. Sun, D. P. Frueh, P. Selenko, J. C. Hoch, G. Wagner, J. Biomol. NMR 33, 43–50 (2005). 53. P. Permi, A. Annila, J. Biomol. NMR 20, 127–133 (2001). 54. A. Meissner, O. W. Sørensen, J. Magn. Reson. 150, 100–104 (2001). 55. D. Nietlispach, Y. Ito, E. D. Laue, J. Am. Chem. Soc. 124, 11199–11207 (2002). 56. B. A. Lyons, M. Tashiro, L. Cedergren, B. Nilsson, G. T. Montelione, Biochemistry 32, 7839–7845 (1993). 57. B. T. Farmer, R. A. Venters, J. Am. Chem. Soc. 117, 4187–4188 (1995). 58. P. Tsang, P. E. Wright, M. Rance, J. Am. Chem. Soc. 112, 8183–8185 (1990). 59. R. A. Venters, W. J. Metzler, L. D. Spicer, L. Mueller, B. T. Farmer, J. Am. Chem. Soc. 117, 9592–9593 (1995). 60. S. Grzesiek, P. Wingfield, S. Stahl, D. Kaufman, A. Bax, J. Am. Chem. Soc. 117, 9594–9595 (1995). 61. C. Zwahlen, K. H. Gardner, S. P. Sarma, D. A. Horita, R. A. Byrd, L. E. Kay, J. Am. Chem. Soc. 120, 7617–7625 (1998). 62. D. R. Muhandiram, G. Y. Xu, L. E. Kay, J. Biomol. NMR 3, 463–470 (1993). 63. Z. Zolnai, N. Juranic, S. Macura, J. Biomol. NMR 3, 463–470 (1993). 64. V. Tugarinov, R. Sprangers, L. E. Kay, J. Am. Chem. Soc. 126, 4921–4925 (2004). 65. V. Tugarinov, L. E. Kay, J. Am. Chem. Soc. 125, 13868–13878 (2003). 66. G. A. Mueller, W. Y. Choy, D. Yang, J. D. Forman-Kay, R. A. Venters, L. E. Kay, J. Mol. Biol. 300, 197–212 (2000). 67. V. Tugarinov, W. Y. Choy, V. Y. Orekhov, L. E. Kay, Proc. Natl. Acad. Sci. U.S.A. 102, 622–627 (2005). 68. B. O. Smith, Y. Ito, A. Raine, S. Teichman, L. ben-Tovim, D. Nietlispach, R. W. Broadhurst, T. Terada, M. Kelly, H. Oschkinat, T. Shibata, S. Yokoyama, E. D. Laue, J. Biomol. NMR 8, 360–368 (1996). 69. G. M. Clore, M. R. Starich, C. A. Bewley, M. Cai, J. Kuszewski, J. Am. Chem. Soc. 121, 6513–6514 (1999). 70. A. Medek, E. T. Olejniczak, R. P. Meadows, S. W. Fesik, J. Biomol. NMR 18, 229–238 (2000). 71. A. J. Wand, J. H. Short, Meth. Enzymol. 239, 700–717 (1994). 72. G. M. Clore, A. M. Gronenborn, Prog. Biophys. Mol. Biol. 62, 153–184 (1994). 73. G. M. Clore, A. M. Gronenborn, Crit. Rev. Biochem. Mol. Biol. 30, 351–385 (1995). 74. T. Ha¨rd, Q. Rev. Biophys. 32, 57–98 (1999). 75. A. M. Petros, S. W. Fesik, Meth. Enzymol. 239, 717–739 (1994). 76. H. Wu, L. D. Finger, J. Feigon, Meth. Enzymol. 394, 525–545 (2005). 77. E. R. Zuiderweg, Biochemistry 41, 1–7 (2002). 78. A. J. Wand, S. W. Englander, Curr. Opin. Biotechnol. 7, 403–408 (1996). 79. D. Nietlispach, H.R. Mott, K.M. Stott, P.R. Nielsen, A. Thiru, E. D. Laue, in ‘‘Protein NMR Techniques’’ (A. K. Downing, ed.), Vol. 278, pp. 255–288, Humana Press Inc., Totowa, NJ, 2005. 80. D. E. Wemmer, P. G. Williams, Meth. Enzymol. 239, 739–767 (1994). 81. T. Carlomagno, Annu. Rev. Biophys. Biomol. Struct. 34, 245–266 (2005). 82. S. W. Homans, Angew. Chem. Int. Ed. Engl. 343, 290–300 (2004). 83. M. Pellecchia, D. S. Sem, K. Wu¨thrich, Nat. Rev. Drug Discov. 1, 211–219 (2002). 84. L. Y. Lian, I. L. Barsukov, M. J. Sutcliffe, K. H. Sze, G. C. Roberts, Meth. Enzymol. 239, 657–700 (1994). 85. C. Dominguez, R. Boelens, A. M. Bonvin, J. Am. Chem. Soc. 125, 1731–1737 (2003). 86. U. Schiebor, M. Vogtherr, B. Elshorst, M. Betz, S. Grimme, B. Pescatore, T. Langer, K. Saxena, H. Schwalbe, ChemBioChem 6, 1891–1898 (2005).
778
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
87. S. B. Shuker, P. J. Hajduk, R. P. Meadows, S. W. Fesik, Science 274, 1531–1534 (1996). 88. B. T. Farmer, K. L. Constantine, V. Goldfarb, M. S. Friedrichs, M. Wittekind, J. Yanchunas, J. G. Robertson, L. Mueller, Nat. Struct. Biol. 3, 995–997 (1996). 89. L. Fielding, Tetrahedron 56, 6151–6170 (2000). 90. C. S. Wilcox, in ‘‘Frontiers in Supramolecular Organic Chemistry and Photochemistry’’ (H.-J. Schneider, H. Du¨rr, eds.), pp. 123–143, VCH Verlag, Weinheim, 1991. 91. H. Takahashi, T. Nakanishi, K. Kami, Y. Arata, I. Shimada, Nat. Struct. Biol. 7, 220–223 (2000). 92. V. Do¨tsch, Meth. Enzymol. 339, 343–357 (2001). 93. M. Allen, L. Varani, G. Varani, Meth. Enzymol. 339, 357–376 (2001). 94. J. Qin, O. Vinogradova, A. M. Gronenborn, Meth. Enzymol. 339, 377–389 (2001). Kupcˇe, R. Freeman, J. Magn. Reson., Ser. A 115, 273–276 (1995). 95. E. 96. R. Fu, G. Bodenhausen, Chem. Phys. Lett. 245, 415–420 (1995). 97. R. Fu, G. Bodenhausen, J. Magn. Reson., Ser. A 117, 324–325 (1995). 98. H. Takahashi, Miyazawa, Y. Ina, Y. Fukunishi, Y. Mizukoshi, H. Nakamura, I. Shimada, J. Biomol. NMR, 34, 167–177 (2006). 99. G. M. Clore, A. M. Gronenborn, J. Magn. Reson. 48, 402–417 (1982). 100. G. M. Clore, A. M. Gronenborn, J. Magn. Reson. 53, 423–442 (1983). 101. N. R. Nirmala, G. M. Lippens, K. Hallenga, J. Magn. Reson. 100, 25–42 (1992). 102. G. M. Lippens, C. Cerf, K. Hallenga, J. Magn. Reson. 99, 268–281 (1992). 103. T. Polenova, T. Iwashita, A. G. Palmer, A. E. McDermott, Biochemistry 36, 14202–14217 (1997). 104. N. U. Jain, T. J. Wyckoff, C. R. Raetz, J. H. Prestegard, J. Mol. Biol. 343, 1379–1389 (2004). 105. G. M. Clore, C. D. Schwieters, J. Am. Chem. Soc. 125, 2902–2912 (2003). 106. M. A. McCoy, D. F. Wyss, J. Am. Chem. Soc. 124, 2104–2105 (2002). 107. F. Lohr, G. N. Yalloway, S. G. Mayhew, H. Ru¨terjans, ChemBioChem. 5, 1523–1534 (2004). 108. A. L. Breeze, Prog. NMR Spectrosc. 36, 323–372 (2000). 109. G. Otting, K. Wu¨thrich, Q. Rev. Biophys. 23, 39–96 (1990). 110. R. H. Griffey, A. G. Redfield, J. Magn. Reson. 65, 344–348 (1985). 111. R. H. Griffey, A. G. Redfield, Q. Rev. Biophys. 19, 51–82 (1987). 112. G. Otting, H. Senn, G. Wagner, K. Wu¨thrich, J. Magn. Reson. 70, 500–505 (1986). 113. H. Senn, G. Otting, K. Wu¨thrich, J. Am. Chem. Soc. 109, 1090–1092 (1987). 114. M. A. Weiss, A. G. Redfield, R. H. Griffey, Proc. Natl. Acad. Sci. U.S.A 83, 1325–1329 (1986). 115. S. W. Fesik, R. T. Gampe, T. W. Rockway, J. Magn. Reson. 74, 366–371 (1987). 116. A. Bax, M. A. Weiss, J. Magn. Reson. 71, 571–575 (1987). 117. G. Wider, C. Weber, R. Traber, H. Widmer, K. Wu¨thrich, J. Am. Chem. Soc. 112, 9015–9016 (1990). 118. M. Ikura, A. Bax, J. Am. Chem. Soc. 114, 2433–2440 (1992). 119. A. M. Petros, M. Kawai, J. R. Luly, S. W. Fesik, FEBS Lett. 308, 309–314 (1992). 120. C. Weber, G. Wider, B. von Freyberg, R. Traber, W. Braun, H. Widmer, K. Wu¨thrich, Biochemistry 30, 6563–6574 (1991). 121. P. J. M. Folkers, R. H. A. Folmer, R. N. H. Konings, C. W. Hilbers, J. Am. Chem. Soc. 115, 3798–3799 (1993). 122. R. Freeman, T. H. Mareci, G. A. Morris, J. Magn. Reson. 42, 341–345 (1981). 123. G. Wider, C. Weber, K. Wu¨thrich, J. Am. Chem. Soc. 113, 4676–4678 (1991). 124. M. Rance, P. E. Wright, B. A. Messerle, L. D. Field, J. Am. Chem. Soc. 109, 1591–1593 (1987).
9.3 METHODS
FOR
RAPID DATA ACQUISITION
779
125. G. Gemmecker, E. T. Olejniczak, S. W. Fesik, J. Magn. Reson. 96, 199–204 (1992). Kupcˇe, R. Freeman, J. Magn. Reson. 127, 36–48 (1997). 126. E. 127. C. Zwahlen, P. Legault, S. F. J. Vincent, J. Greenblatt, R. Konrat, L. E. Kay, J. Am. Chem. Soc. 119, 6711–6721 (1997). 128. A. C. Stuart, K. A. Borzilleri, J. M. Withka, A. G. Palmer, J. Am. Chem. Soc. 121, 5346–5347 (1999). 129. W. Lee, M. J. Revington, C. Arrowsmith, L. E. Kay, FEBS Lett. 350, 87–90 (1994). 130. Y. Xia, A. Yee, C. H. Arrowsmith, X. Gao, J. Biomol. NMR 27, 193–203 (2003). 131. H. S. Atreya, T. Szyperski, Meth. Enzymol. 394, 78–108 (2005). 132. J. Chen, V. A. Mandelshtam, A. J. Shaka, J. Magn. Reson. 146, 363–368 (2000). 133. V. A. Mandelshtam, J. Magn. Reson. 144, 343–356 (2000). 134. H. Hu, A. A. De Angelis, V. A. Mandelshtam, A. J. Shaka, J. Magn. Reson. 144, 357–366 (2000). 135. L. Frydman, A. Lupulescu, T. Scherf, J. Am. Chem. Soc. 125, 9204–9217 (2003). 136. L. Frydman, T. Scherf, A. Lupulescu, Proc. Natl. Acad. Sci. U.S.A. 99, 15858–15862 (2002). Kupcˇe, J. Magn. Reson. 163, 56–63 (2003). 137. R. Freeman, E. Kupcˇe, J. Magn. Reson. 162, 158–165 (2003). 138. R. Freeman, E. Kupcˇe, J. Magn. Reson. 162, 300–310 (2003). 139. R. Freeman, E. Kupcˇe, J. Biomol. NMR 27, 101–113 (2003). 140. R. Freeman, E. Kupcˇe, Concepts Magn. Reson. 23A, 63–75 (2004). 141. R. Freeman, E. 142. D. Rovnyak, J. C. Hoch, A. S. Stern, G. Wagner, J. Biomol. NMR 30, 1–10 (2004). 143. D. Marion, J. Biomol. NMR 32, 141–150 (2005). 144. T. Luan, V. Jaravine, A. Yee, C. H. Arrowsmith, V. Y. Orekhov, J. Biomol. NMR 33, 1–14 (2005). 145. V. Tugarinov, L. E. Kay, I. Ibraghimov, V. Y. Orekhov, J. Am. Chem. Soc. 127, 2767–2775 (2005). 146. S. Kim, T. Szyperski, J. Am. Chem. Soc. 125, 1385–1393 (2003). 147. T. Szyperski, G. Wider, J. H. Bushweller, K. Wu¨thrich, J. Am. Chem. Soc. 115, 9307–9308 (1993). 148. T. Szyperski, G. Wider, J. H. Bushweller, K. Wu¨thrich, J. Biomol. NMR 3, 127–132 (1993). 149. K. Ding, A. M. Gronenborn, J. Magn. Reson. 156, 262–268 (2002). 150. J. P. Simorre, B. Brutscher, M. S. Caffrey, D. Marion, J. Biomol. NMR 4, 325–333 (1994). 151. B. Brutscher, J. P. Simorre, M. S. Caffrey, D. Marion, J. Magn. Reson. 105, 77–82 (1994). 152. T. Szyperski, D. C. Yeh, D. K. Sukumaran, H. N. Moseley, G. T. Montelione, Proc. Natl. Acad. Sci. U.S.A. 99, 8009–8014 (2002). 153. B. Bersch, E. Rossy, J. Coves, B. Brutscher, J. Biomol. NMR 27, 57–67 (2003). 154. W. Kozminski, I. Zhukov, J. Biomol. NMR 26, 157–166 (2003). 155. F. Lohr, H. Ru¨terjans, J. Biomol. NMR 6, 189–197 (1995). 156. H. S. Atreya, T. Szyperski, Proc. Natl. Acad. Sci. U.S.A. 101, 9642–9647 (2004). 157. Y. Shen, H. S. Atreya, G. Liu, T. Szyperski, J. Am. Chem. Soc. 127, 9085–9099 (2005). 158. H. S. Atreya, A. Eletsky, T. Szyperski, J. Am. Chem. Soc. 127, 4554–4555 (2005). 159. Y. Xia, G. Zhu, S. Veeraraghavan, X. Gao, J. Biomol. NMR 29, 467–476 (2004). 160. G. Liu, J. Aramini, H. S. Atreya, A. Eletsky, R. Xiao, T. A. Acton, L. C. Ma, G. T. Montelione, T. Szyperski, J. Biomol. NMR 32, 261–261 (2005). 161. G. Liu, Y. Shen, H. S. Atreya, D. Parish, Y. Shao, D. Sukumaran, R. Xiao, A. Yee, A. Lemak, A. Bhattacharya, T. A. Acton, C. H. Arrowsmith, G. T. Montelione, T. Szyperski, Proc. Natl. Acad. Sci., U.S.A. 102, 10487–10492 (2005). Kupcˇe, J. Biomol. NMR 25, 349–354 (2003). 162. R. Freeman, E.
780 163. 164. 165. 166. 167. 168. 169. 170.
CHAPTER 9 LARGER PROTEINS
AND
MOLECULAR INTERACTIONS
Kupcˇe, J. Biomol. NMR 27, 383–387 (2003). R. Freeman, E. Kupcˇe, J. Am. Chem. Soc. 125, 13958–13959 (2003). R. Freeman, E. Kupcˇe, J. Am. Chem. Soc. 126, 6429–6440 (2004). R. Freeman, E. Kupcˇe, J. Biomol. NMR 28, 391–395 (2004). R. Freeman, E. Kupcˇe, R. Freeman, Concepts Magn. Reson. 22A, 4–11 (2004). E. Kupcˇe, R. Freeman, J. Magn. Reson. 172, 329–332 (2005). E. B. E. Coggins, R. A.Venters, P. Zhou, J. Am. Chem. Soc. 126, 1000–1001 (2004). R. A. Venters, B. E. Coggins, D. Kojetin, J. Cavanagh, P. Zhou, J. Am. Chem. Soc. 127, 8785–8795 (2005).
CHAPTER
10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION, AND OTHER APPLICATIONS
The first nine chapters of this book have focused on the theory of NMR spectroscopy and on the experimental methods used for multidimensional NMR spectroscopy of proteins. Analyses of the information contained in the NMR spectra of proteins are the subjects of this final chapter. Exhaustive coverage of these topics would comprise an entire additional text; instead, this chapter has three aims: (i) to indicate the types of analyses commonly performed using NMR spectra of proteins, (ii) to rationalize usage of the experiments described in Chapters 6–9 for a thorough investigation of proteins by NMR spectroscopy, and (iii) to provide a bibliography of the primary literature to augment the discussions of aims (i) and (ii).
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10.1 Resonance Assignment Strategies In the initial stage of any investigation by NMR spectroscopy, each resonance in the NMR spectrum must be associated with a specific nucleus in the molecule under investigation. Resonance assignments must be sequence specific: each resonance must be assigned to a spin in a particular amino acid residue in the protein sequence. NMR spectroscopy provides three types of information useful for spectral assignments: through-bond interactions (via scalar couplings), throughspace interactions (via dipolar couplings), and chemical environment (via isotropic chemical shifts). The strategies employed for resonance assignments depend upon whether only homonuclear 1H NMR spectra are available (unlabeled proteins) or whether 13C and 15N heteronuclear correlation spectra are available (isotopically labeled proteins).
10.1.1 1H RESONANCE ASSIGNMENTS 1
FOR
UNLABELED PROTEINS
The procedures for obtaining H resonance assignments are based upon the following critical observation: with few exceptions, correlations resulting from 1H–1H scalar couplings normally are only observed between 1H nuclei separated by two or three bonds in proteins. Crosspeaks in 1H homonuclear correlation NMR spectra occur between 1H spins within the same amino acid residue or spin system. Cross-peaks do not occur between 1H spins in different residues, because the interresidue 4 JHNiþ1 Hi coupling constant is negligible. Therefore, scalar correlation experiments, such as COSY (Section 6.2), MQF-COSY (Section 6.3), MQ spectroscopy (Section 6.4), and TOCSY (Section 6.5), are used to identify resonance positions within each amino acid spin system, and the NOESY experiment (Section 6.6) is used to sequentially connect the amino acid spin systems. Two-dimensional NOESY and TOCSY experiments also can be combined to yield homonuclear 3D experiments, as described in Section 6.7. Initially, 1H resonances are categorized on the basis of their chemical shifts. As indicated in Fig. 10.1, for the vast majority of residues in a protein, backbone amide 1HN spins resonate between 10.0 and 7.0 ppm, aromatic 1H spins resonate between 8.0 and 6.5 ppm (with the exception of Trp and His 1H"1 spins), backbone 1H spins resonate between 6.0 and 3.5 ppm, aliphatic side chain methine and methylene 1 H spins resonate between 3.5 and 1.0 ppm, and methyl 1H spins (other than Met 1H" resonate at chemical shifts less than 2.0 ppm. Random coil 1 H chemical shifts have been determined for unstructured peptides (1–3).
783
10.1 RESONANCE ASSIGNMENT STRATEGIES
backboneHN
methyl aromatic aliphatic side-chainHN Ha
10
8
6 1H
4
2
0
chemical shift (ppm)
FIGURE 10.1 Chemical shift ranges observed for the various types of 1H resonances in ubiquitin. The spectrum was recorded in H2O solution using presaturation and a Hahn echo (Section 3.6.4.2) following the acquisition pulse. The residual water resonance was removed from the FID by convolution (Section 3.5.4) prior to Fourier transformation.
The dependence of 1H chemical shifts on protein secondary structure is discussed elsewhere (4–11). Figure 10.2 plots the distribution of 1H chemical shifts (averaged over all observed conformations) for proteins in a database compiled by BioMagResBank (12). The backbone amide 1HN signals are usually the best resolved set of resonances within a protein 1H NMR spectrum; thus, the 1H and side chain resonance positions are most readily determined in scalar correlation experiments by the observation of direct and relayed crosspeaks to the backbone amide spins (13, 14). In theory, most 1H 2D spectra are symmetric about the F2 ¼ F1 diagonal (multiple-quantum spectra being the principal exceptions); hence, a particular crosspeak involving an amide 1HN spin is observable above (F2 4 F1) or below (F2 5 F1) the diagonal. In real spectra, this symmetry often is broken due to experimental artifacts, the most obvious of which is usually a streak of noise at the F2 frequency of the water resonance that
784 a
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION Ala
a
Arg
a
b b
d
Asn
a
b
Asp
a
b a
Cys
g
b
Gln
a
g
Glu
a
g
b b
a
Gly a
His
b a
Ile Leu
a
Lys
a
Met
a
b
a
Thr
b g2
b b
a
Tyr
b a
Val 6
5
g
b g
a
Trp
d
e
d a
Ser
b
d
b
a
Pro
d
g
g
g2 b
e
a
Phe
b
g1
b
4
3 1
2
g 1
0
H (ppm)
FIGURE 10.2 (a) Aliphatic and (b) nonaliphatic side chain 1H chemical shifts. The mean and standard deviations for each 1H nucleus type were obtained from the database of chemical shifts at BioMagResBank (www.bmrb.wisc.edu). The bars extend one standard deviation in either direction from the mean value. The chemical shifts of backbone amide 1HN spins show limited dependence on amino acid type [sequence-dependent random coil chemical shifts are given in Schwarzinger et al. (1) and Braun et al. (143)] and have a mean of 8.3 ppm and a standard deviation of 0.7 ppm.
785
10.1 RESONANCE ASSIGNMENT STRATEGIES
b
e
Arg
h d
Asn
e
Gln
d2
e1
His
z
Lys
d e
Phe z3
h2
e3
e1
Trp
z
d1
z2 Tyr
d 11
10
9
8 1H
7
e
6
(ppm)
FIGURE 10.2—Continued
arises from incomplete solvent suppression. In unfavorable circumstances, ridges emanating from the central stripe extend 1.0 ppm or more from the water resonance and obscure many cross-peaks in this region. The distortions at the F1 frequency of the water resonance are rarely comparable in width or intensity to the F2 distortions. This asymmetry means that cross-peaks involving amide protons are most readily observed in the ‘‘above diagonal’’ region. Concentrating on these cross-peaks also has the advantage of combining the high digital resolution of the F2 dimension with the high chemical shift dispersion of the amide 1HN resonances. For these reasons, the discussion of R.COSY, DR.COSY, and TOCSY experiments performed in H2O in Chapter 6 focused on the side chain to amide proton correlation crosspeaks occurring above the diagonal. The principal process of determining 1H resonance assignments is known as the sequential assignment strategy, and has been developed by Wu¨thrich and co-workers (15). This strategy is summarized in Fig. 10.3. The first stage of analysis makes use of 1H–1H scalar couplings to establish sets of 1HN, 1H, and aliphatic side chain resonances that belong to the same amino acid residue spin system. A protein of N residues has N distinct backbone-based spin systems. The N spin
786
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION Sequential Assignment Strategy Correlation spectra detailed analysis Amino acid type classification
Main Chain Directed Approach Correlation spectra brief analysis HN Ha Hb Coupling units main chain NOE pattern analysis
sequential NOE analysis
Complete Sequential Assignments semi-quantitative NOE /J analysis
Secondary structure and Global fold
Backbone Sequential Assignments
Secondary Structure
detailed analysis of correlation spectra Complete Assignments long-range NOEs Global fold
FIGURE 10.3 Comparison of the steps involved in the sequential assignment method (left) and the main chain directed method (right) of 1H resonance assignment for unlabeled proteins. Both schemes require knowledge of the primary sequence and make use of the same experimental data, although this information is utilized at different stages of the assignment process.
systems are assigned an amino acid type (or one of several possible types) based on the coupling topology and resonance chemical shifts, as described in Fig. 10.4. Aromatic 1H spins of tyrosine, phenylalanine, tryptophan, and histidine residues; side chain amide protons of glutamine and asparagine residues; the side chain guanidinium group of arginine residues; and the methyl group of methionine residues are not scalar coupled to the remainder of the side chain and consequently comprise distinct spin
787
10.1 RESONANCE ASSIGNMENT STRATEGIES amino acid type
Glycine
Hb shifts only
Hb Hg shifts
Hb Hg Hd shifts
Hb Hg Hd shifts and intraresidue NOEs
GLY
GLY
GLY
GLY
Alanine
ALA
ALA
ALA
ALA
Threonine
THR
THR
THR
THR
VAL ILE
VAL
VAL
ILE
ILE
LEU
LEU
LONG
ARG
ARG
Valine Isoleucine Leucine Arginine
LONG
Lysine
LYS
Glutamine
LYS GLN
Glutamate
5-SPIN
5-SPIN
GLU
Methionine
MET
Cysteine
CYS ASP
Aspartate Asparagine Phenylalanine
ASN 3-SPIN
3-SPIN
3-SPIN
PHE
Tyrosine
TYR
Tryptophan
TRP
Histidine
HIS
Serine
SER
SER
SER
SER
FIGURE 10.4 Categorization of spin systems based on knowledge of chemical shifts and spin topology. Each box indicates a unique category containing the amino acid types shown on the left. The four columns of categories indicate how spin systems can be more finely subdivided as chemical shifts further along the side chain are included in the analysis. The shading indicates that assignment of threonine and serine spin systems may be ambiguous due to the partial overlap of the 1H chemical shifts of these residues with 1H of three-spin systems. The first three category columns indicate categorizations possible using correlation spectra only; intraresidue NOEs are needed to perform the most complete categorization shown in the fourth column. Categorization in this fashion assumes that correlations to methyl groups can be differentiated from other interactions on the basis of chemical shift, intensity, or cross-peak fine structure.
788
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
systems. Association of the side chain and backbone resonances of these amino acid residues has to be made on the basis of intraresidue NOE correlations. For example, the most intense NOE correlations to 1H spins of tyrosine or phenylalanine residues are generally from intraresidue 1H spins, because these nuclei are always less than 2.8 A˚ apart (15). As a second example, the side chain amide 1H nuclei trans to the side chain carbonyl oxygen (1HN2 in asparagine and 1HN"2 in glutamine) are always closer to the other side chain protons than are the cis 1H nuclei (1HN1 in asparagine and 1HN"1 in glutamine). Therefore, the 1H resonance with the more intense intraresidue NOEs is assigned to the trans position (16). In the second stage of the assignment process, every spin system is assigned to a particular residue within the polypeptide chain by using through-space dipolar coupling (NOE) interactions to sequentially connect the spin systems identified from scalar correlations. Statistical analysis of hydrogen atom locations inferred from x-ray crystal structures of proteins has shown that a majority of the short distances between 1HN, 1H, and 1H nuclei are between residues adjacent in the primary sequence (17). Thus, identification of intense NOEs from 1 N 1 H , H , and/or 1H of one spin system to 1HN of a second spin system suggests that the two spin systems are adjacent in the primary sequence, with the first spin system nearer to the N-terminus of the protein (i.e., the spin systems correspond to residue i and residue i þ 1, respectively). The sequential NOEs commonly are given the short-hand notation dNN, dN, and dN, respectively (15, 18). Identification of a series of sequential NOE interactions places several spin systems in the order i, i þ 1, i þ 2, . . . , i þ n, as illustrated for ubiquitin in Fig. 10.5. As more spin systems are connected, the sequence of spin systems eventually will match a unique section of the primary amino acid sequence of the protein (which must be known a priori); at this point, the spin systems are assigned sequence specifically. If the spin system types are well characterized (i.e., the majority of side chain resonance positions have been identified), then alignment of four or five spin systems usually is sufficient to achieve sequence-specific assignment. The observation of 1HN–1HN, 1H–1HN, or 1H–1HN NOEs is not limited to sequential interactions, and may also occur between nonsequential residues as a result of secondary or tertiary structure in the protein (17). The resulting ambiguity in the assignment process is reduced by the identification of sequential 1HN–1HN, 1H–1HN, and 1H–1HN NOEs. Additionally, the sequential ordering of spin systems must match the primary sequence; 1HN–1HN, 1H–1HN, or 1H–1HN NOEs between spin systems that are never adjacent in the
789
10.1 RESONANCE ASSIGNMENT STRATEGIES
HO H
O
N
CH3 H
CH
N
N H
H
O
H
O
N
H
H
H3C
CH3
+ NH 3 Lys6
Thr7
Leu8
Ile13
Thr14
Leu15
Arg54
Thr55
Leu56
Ser65
Thr66
Leu67
FIGURE 10.5 The use of backbone–backbone NOEs to obtain sequence-specific 1 H resonance assignments. In the top portion of the figure, the dNN, dN, and dN connecting the three spin systems together are shown by the curved lines. Observation of a threonine prior to a leucine residue occurs four times in ubiquitin, hence such pairs of spin systems cannot be assigned sequence specifically. Once a lysine residue preceding the threonine is identified, the tripeptide segment can only arise from residues 6, 7, and 8. Due to the similarity of 1H and 1H shifts of lysine, isoleucine, and arginine residues, 1H chemical shifts must be identified in the first residue of this tripeptide to distinguish between Lys6, Ile13, and Arg54, as described in Fig. 10.4.
primary sequence must result from longer range contacts. In the limit, the assignments encompass all spin systems, and self-consistency is the best measure of the validity of the results. Tabulations of the relative intensities of the sequential NOEs customarily are presented graphically to demonstrate that complete, self-consistent sequential assignments have been achieved, as shown in Fig. 10.6.
790
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
P
20 S N T
I
E
N V
K
A
K
30 I Q D K
E
G I
P
dNN (i,i+1) daN (i,i+1)
*
dbN (i,i+1)
*
daN (i,i+2)
daN (i,i+3)
daN (i,i+4) 3J N a H H
*
FIGURE 10.6 Summary of sequential NOEs observed for a portion of ubiquitin. The sequence is displayed along the top of the box, and the bars connect residues between which sequential NOEs are observed. The thickness of the bars indicates the intensity of the NOE. Commonly, medium-range NOEs and coupling constant data are included in such tabulations to help identify regions of regular secondary structure. Downward-pointing arrows, vertical lines, and upward-pointing arrows indicate 3 JHN H < 6:0 Hz, 6:0 Hz < 3 JHN H < 8:0 Hz, and 3 JHN H > 8:0 Hz, respectively. Asterisks indicate data that were not observed due to resonance overlap. This particular section of ubiquitin contains a helix from residues Thr22 to Glu34, as indicated by the intense dNN NOEs, medium-range NOEs, and small coupling constants.
An alternative strategy, known as the main chain directed (MCD) approach, has been developed by Englander and Wand (19, 20). In the MCD approach, scalar coupling connectivities are used initially to identify 1HN–1H–1H units only. Assignment of the spin systems by amino acid type is not attempted. Next, the 1HN–1H–1H units are aligned sequentially by systematically searching the NOESY spectrum for patterns of sequential NOEs. Different elements of secondary structure give rise to specific patterns of NOEs (15), and a search is made for these motifs in the following order: helix, antiparallel sheet, parallel sheet, turns, and loops. Once all of the backbone coupling
10.1 RESONANCE ASSIGNMENT STRATEGIES
791
units have been aligned sequentially and categorized by secondary structural element, determination of the amino acid type of several side chains permits the defined elements of secondary structure to be aligned with the primary sequence. The sequential and MCD methods are compared schematically in Fig. 10.3. The advantage of the MCD approach is that the sometimestroublesome identification of amino acid type from scalar coupling data is not attempted initially, so that at least sequential backbone assignments can be made. Eventually, however, complete assignments of the side chains are needed because NOEs to all resonances must be assigned to determine precise three-dimensional structures. Frequently, elements of both the sequential and the MCD assignment approaches are combined. The initial identification of spin systems in the sequential approach is rarely complete due to problems of resonance degeneracy, spectral artifacts, and absent cross-peaks (due to small coupling constants). However, even limited knowledge of amino acid types can assist in the MCD sequential assignment process: once several residues have been connected sequentially, identification of the next residue is trivial if only one of the candidates is the correct type of amino acid. Further, sequential assignment of the backbone resonances facilitates assignment of the side chain resonances because the expected spin system topologies (amino acid types) are known. Computer automated analysis of 1H 2D NMR spectra of proteins to obtain resonance assignments has been discussed for some years. Both fully automated (21–24) and semiautomated (25–28) approaches have been described. Resonance degeneracy and incomplete correlations in the 1H spectra are the main obstacles to automation. Homonuclear three-dimensional spectra potentially can circumvent resonance overlap (29–32). However, as discussed in Section 10.1.2, most recent efforts to automate the resonance assignment process have made use of the increased resolution afforded by heteronuclear NMR spectroscopy of isotopically labeled proteins. [For reviews of automation in NMR spectroscopy, see Baran et al. (33), Altieri and Byrd (34), Gu¨ntert (35), and Gronwald and Kalbitzer (36).] The largest proteins for which nearly complete 1H sequential assignments can be obtained by homonuclear methods alone are usually in the range of 10–12 kDa [for examples, see Chazin et al. (37) and Williamson et al. (38)]. The upper limit depends somewhat on topology because proteins with substantial -sheet content usually display better dispersion of 1H resonances than do predominantly -helical proteins. Homonuclear 1H assignments of ubiquitin have been obtained by both the sequential assignment and the MCD strategies (20, 39).
792
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
10.1.2 HETERONUCLEAR RESONANCE ASSIGNMENTS ISOTOPICALLY LABELED PROTEINS
FOR
The conventional sequential or main chain directed assignment strategies also are used to obtain sequence-specific assignments from 3D heteronuclear NMR spectra. The principal advantage of using 3D heteronuclear-edited NOESY and TOCSY (Section 7.2), rather than homonuclear 2D experiments, for resonance assignments is the significant reduction in cross-peak overlap. The 3D 1H–15N NOESY– HSQC experiment is used to identify sequential through-space dN, dN, and dNN connectivities. Amino acid spin systems are identified by observation of direct and relayed through-bond connectivities between the 1HN spins and the 1H and aliphatic side chain protons using the 3D 1H–15N TOCSY–HSQC experiment (Section 7.2.2). Alternatively, complete side chain assignments also can be obtained from HCCH–COSY and HCCH–TOCSY experiments (Section 7.3). In this case, the side chain spin systems are connected with the backbone 1HN and 15N resonances via 1HN–15N–1H and other correlations observed in an 1H–15N TOCSY–HSQC spectrum, or by using correlations observed in one or two of several triple-resonance experiments (Section 7.4) [e.g., HNCA, HN(CO)CA, HNCACB, CBCA(CO)NH, HBHA(CBCA)NH, or HBHA(CBCACO)NH]. The heteronuclear-edited NOE-based sequential assignment method has been successfully applied to a number of proteins, with molecular masses up to 20 kDa (40–46). The triple-resonance experiments introduced in Section 7.4 offer an alternative to the NOE-based strategy for sequentially assigning 1HN, 15 N, 13CO, 1H, 13C, 1H, and 13C resonances. Using these experiments, sequential correlations are established via the relatively uniform and well-resolved heteronuclear one-bond and two-bond couplings, without any prior knowledge of spin system types. Side chain assignments are completed using the TOCSY–HSQC, HCCH–COSY, and HCCH–TOCSY experiments. Potential errors that arise from misassignment of sequential and long-range connectivities in the NOE-based procedures are avoided because assignments are based solely on predictable through-bond scalar correlations. Triple-resonance approaches were developed by Bax and co-workers to overcome difficulties in obtaining resonance assignments for calmodulin by homonuclear techniques (47). Calmodulin (16.7 kDa) is largely -helical and has very narrow chemical shift distributions for both 1HN and 1H spins. For example, the combined use of the HNCA (Section 7.4.1) experiment, which provides intraresidue (together with some
10.1 RESONANCE ASSIGNMENT STRATEGIES
793
sequential) correlations between 1HN, 15N, and 13C resonances, and the HN(CO)CA experiment (Section 7.4.2), which gives solely interresidue correlations between the 1HN and 15N resonances of one residue and the 13C resonance of the preceding residue, provides an obvious route to sequential assignment. Each 13C resonance is linked to both its intraresidue and sequential 1HN and 15N resonances. Ambiguities caused by chemical shift degeneracy are solved by using additional experiments that provide alternative correlations. For instance, the HNCO (Section 7.4.4.1) and HN(CA)CO (Section 7.4.4.2) experiments correlate the 1HN and 15N resonances with both the intraresidue and the sequential 13CO signals (rather than the 13C spins). To ‘‘align’’ the backbone sequential assignments with the protein amino acid sequence, side chain amino acid spin systems are identified from HCCH–COSY, HCCH–TOCSY, and 1H–15N TOCSY–HSQC experiments. A different set of experiments, CBCA(CO)NH [or HN(CO)CACB] and HNCACB (or CBCANH), together with the closely related HBHA (CBCACO)NH and HBHA(CBCA)NH experiments (Section 7.4.5), offers an alternative sequential assignment strategy for proteins with 13 C and 1H chemical shift degeneracy. Using these experiments, the 1HN and 15N resonances are correlated with the intraresidue and sequential 13C and 13C (or 1H and 1H) resonances. Information regarding amino acid type can be obtained from the 13C and 13C chemical shifts (see later). The HCC(CO)NH–TOCSY (48–52) and CC(CO)NH–TOCSY (53) experiments, discussed in Sections 9.1.6 and 9.1.7, also provide sequential and spin system assignment information. These experiments are similar in principle to the CBCA(CO)NH and HNCACB experiments, except that 13C isotropic mixing is used instead of COSY-type transfer to relay aliphatic side chain magnetization to the 13C nucleus. Sequential and intraresidue correlations are obtained between backbone 1HN and 15N resonances and the side chain (either 13C or 1H) resonances. Finally, banks of experiments have been designed to achieve resonance assignments for particular side chain topologies, such as aromatic amino acids (54, 55), side chain 1 H–15N groups (56), arginine amino acids (57), and proline amino acids (58, 59). Increasingly, resonance assignments of proteins with molecular masses of 420 kDa are obtained using 2H/13C/15N triply labeled proteins to reduce losses due to efficient 1H–13C dipole–dipole relaxation, as discussed in Section 9.1. Triple-resonance experiments for backbone resonance assignments discussed in Section 9.1.5 are based on out-and-back transfers from the 1HN spin. Assignments are obtained
794
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
primarily from three pairs of three-dimensional experiments: HNCA and HN(CO)CA, HNCACB and HN(CO)CACB, and HNCO and HN(CA)CO experiments. These experiments also have four-dimensional variants, such as HNCOCA (60). Assignments of aliphatic and aromatic side chains using combined analysis of 13C/15N doubly labeled and 2H/13C/15N triply labeled proteins, random fractionally deuterated 2 H/13C/15N proteins, or selectively protonated proteins have been described in Sections 9.1.6, 9.1.7, and 9.1.9. Nearly complete backbone and methyl resonance assignments of malate synthase G, with 723 amino acid residues and a molecular mass of 81.4 kDa, have been achieved by Kay and co-workers (61, 62). Information on amino acid type is obtained from 13C chemical shift data. Random coil 13C chemical shifts have been determined for unstructured peptides (1, 2, 63). The dependence of 13C chemical shifts on protein secondary structure is discussed elsewhere (11, 64–66). The distributions of aliphatic and aromatic 13C chemical shifts for different amino acid residues compiled in a database at BioMagResBank (12) are plotted in Fig. 10.7. The characteristic 13C and 13C chemical shifts of alanine, threonine, serine, and glycine residues (Fig. 10.7) allow ready identification of these amino acid types. Clearly, knowledge of other aliphatic 13C chemical shifts can also be used to assign a given spin system to a unique or limited number of possible amino acid types. This information, coupled with the alignment of sequentially connected spin systems with the known amino acid sequence, leads to unambiguous assignment. The relative simplicity and predictability of triple-resonance 3D and 4D spectra used for protein assignment purposes makes these experiments particularly amenable to automated or semiautomated analysis. Current efforts at automated assignment generally begin with automatic peak picking. The reduced resonance overlap in 3D and 4D triple-resonance spectra, relative to 2D homonuclear spectra, increases the reliability of this process; however, the final peak lists usually must be edited (by the spectroscopist) to distinguish ‘‘real’’ resonance peaks from spectral artifacts. The peak lists are searched automatically to find expected intraresidue and interresidue correlations, and the spin systems are identified and sequentially ordered according to these correlations. Information regarding spin system type, which may be obtained from 13C and 13C chemical shifts values, for instance, is incorporated and the ordered spin systems are aligned with the known amino acid sequence. Current state-of-the-art automatic assignment methods have been reviewed (33–36, 67).
795
10.1 RESONANCE ASSIGNMENT STRATEGIES
a
a
Ala
b
a
Arg
b
d a
Asn
b
a
Asp
g
b
Cys
a
box
b red
Gln
a
g
b
a
Glu
g
b
a
Gly a
His
b
a
Ile
g1 g 2
b a
Leu Lys
a
Met
a
e
b
Thr
e
g
a g2
a
Trp
a
Tyr
a
b b
a
Val 80
b
b
d
b
Ser
70
d g
b
a
Pro
d
b g
a
Phe
g
b
d
60
50 13C
40 (ppm)
b
g
30
20
10
FIGURE 10.7 (a) Aliphatic and (b) aromatic 13C chemical shifts. The mean and standard deviations in the mean for each nucleus type were obtained from the database of chemical shifts at BioMagResBank (www.bmrb.wisc.edu). The bars extend one standard deviation in either direction from the mean value. Two ranges are included for 13C of cysteine, due to the significant effect that disulfide bond formation has on these chemical shifts. 13CO chemical shifts have a mean and standard deviation of 176.1 ppm and 2.3 ppm, respectively.
796
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
b e1
His
d2 d
Phe
e
z
z3
d1
Trp
h2
e3
d
Tyr 140
z2 e
130
120 13C
110
(ppm)
FIGURE 10.7—Continued
10.2 Three-Dimensional Solution Structures Sequence-specific assignment of NMR resonances typically requires many different experiments and a substantial effort from the spectroscopist. However, mere achievement of resonance assignments usually does not reveal much of interest about the structure, dynamics, and interactions of the protein under investigation. Assignments are a necessary prerequisite to achieving more biochemical goals. This section discusses procedures for determining the three-dimensional structures of proteins with the aim of highlighting the information that can be obtained from the NMR experiments described in Chapters 6–9. Structure determination (68), characterization of protein dynamics (69), and assessment of protein–ligand interactions (70) by NMR spectroscopy have been reviewed.
10.2.1 NMR-DERIVED STRUCTURAL RESTRAINTS Essentially all parameters that can be measured by NMR spectroscopy are sensitive in some, more-or-less complex, manner to molecular conformation; therefore, quantification of these parameters permits structural analysis by NMR spectroscopy. At present, dipolar crossrelaxation (NOE) rate constants, scalar coupling constants, isotropic chemical shifts, and residual dipole–dipole coupling constants (RDCs) are the most commonly utilized parameters for protein structure determination. Structural restraints derived from NOE interactions and scalar coupling constants have formed the basis of protein structure determination by NMR spectroscopy, beginning with the report of the structure of proteinase inhibitor IIA from bull seminal plasma in 1985 (71).
10.2 THREE-DIMENSIONAL SOLUTION STRUCTURES
797
In recent years, restraints derived from isotropic chemical shifts (66) and RDCs (72) have become extensively utilized. Other NMR parameters, including amide proton–solvent exchange protection factors (73), transhydrogen bond scalar coupling constants (74), and paramagnetic effects (75–77), also provide structural restraints in systems for which these data are available. 10.2.1.1 NOE Distance Restraints The most important NMRobservable parameter used in determining protein structure is the NOE. The dipolar cross-relaxation rate constant is proportional to the inverse sixth power of the distance between two interacting 1H spins (Section 5.5). In the initial rate approximation, NOE cross-peak intensities are proportional to the cross-relaxation rate constants. Thus, if one interproton distance, rref, is known (e.g., from covalent geometry), then another, unknown interproton distance, ri, is determined by the relationship (ignoring internal mobility) ri ¼ rref ðSref =Si Þ1=6 ,
½10:1
in which Sref and Si are the cross-peak intensities. In practice, NOESY mixing times short enough to satisfy the initial rate approximation usually are impractical because the cross-peak intensities then have low signal-to-noise ratios. For longer mixing times, the intensities of NOESY cross-peaks are no longer directly proportional to the crossrelaxation rate constants between the interacting spins both because the time evolution of magnetization transfer even between two isolated spins is exponential and because magnetization is transferred among spins in multiple steps via spin diffusion (Section 6.6.1.2). A majority of the restraints used to calculate the three-dimensional structure of a protein come from many hundreds of NOE cross-peaks, and are represented as bounds on the separation of pairs of 1H atoms. Without recourse to complex calculations (78), precise 1H–1H separations cannot be determined from the NOE cross-peak intensities; instead, NOE cross-peaks typically are grouped on the basis of their intensities into several categories (e.g., strong, medium, and weak). Each category is associated with an upper bound separation between the interacting spins (e.g., 2.7, 3.3, and 5.0 A˚). The cross-peak volume limits and the upper bound distances for the categories are estimated from the NOE intensities observed for protons of known covalent geometry (e.g., geminal methylene protons, or vicinal protons in aromatic rings) or between protons in regions of regular secondary structure (e.g., sequential 1H–1HN, cross-strand 1H–1H and
798
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
cross-strand 1H–1HN NOEs in -sheets, or 1H–1HN medium-range NOEs in helices). Conservative upper bound distances compensate for cross-peaks affected by spin diffusion or partial overlap; otherwise, failure to allow for such artifacts results in structures that have large violations of the input restraints, high energies, and artificially high precision (79). Usually, the lower bound separations for pairs of protons are set to the sum of the van der Waals radii. The precision to which a structure can be determined is directly related to the number of experimental restraints used in the structure calculation. Structures of low resolution can be obtained with as few as five restraints per residue, while the most precise structures obtained from 1H data alone have up to 15 restraints per residue. For the latter cases, the root-mean-square deviation (RMSD) of backbone atoms from the mean structure may be as low as 0.5 A˚ for the well-defined regions of secondary structure. The use of 15N or 13C labeling usually allows many more 1H–1H NOEs to be unambiguously identified, increasing the number of restraints per residue to between 20 and 25, and leading to the highest precision structures (backbone RMSD in the range of 0.3 to 0.5 A˚). Stereospecific assignments of prochiral groups [especially CH2 groups, and (CH3)2CH groups of valine and leucine residues] are critical to obtaining structures of high precision (80). Coverage, defined as the fraction of theoretically observable NOEs that are actually utilized in the structure calculation, has been emphasized as an important determinant of structure quality (81). Structure determination requires that the accuracy of the experimental NOE intensities must be high. NOESY spectra must have very flat base lines (even slight rolls or tilts in the baseline can have dramatic effects on cross-peak volumes) and be otherwise free of spectral artifacts; such topics are discussed in Chapters 6 and 7. 10.2.1.2 Dihedral Angle Restraints from Scalar Coupling Constants As was first described by Karplus (82), the magnitude of a 3J scalar coupling constant is a function of the dihedral angle formed by the three covalent bonds: 3
J ¼ A cos2 þ B cos þ C:
½10:2
The constants A, B, and C depend upon the particular nuclei involved, and is the dihedral angle. Historically, dihedral angle restraints for and 1 dihedral angles have been derived only from 3 JHN H and 3 JH H coupling constants, respectively (83–85). Recently,
10.2 THREE-DIMENSIONAL SOLUTION STRUCTURES
799
numerous experiments have been developed that allow measurement of 13C–13C, 13C–15N, 1H–15N, and 1H–13C three-bond coupling constants in isotopically enriched proteins [Section 7.5; reviewed in Vuister et al. (86)]. Accurate parameterization of the Karplus equation is necessary in order to relate measured values of 3J coupling constants to dihedral angle ranges. Numerous attempts have been made to refine the original Karplus approach to account for the chemical nature of the atoms involved in the coupling (87). However, such theoretical approaches lead to complex relationships that are not generally applicable. Instead, semiempirical methods of obtaining values for the constants A, B, and C in [10.2] have been more successful. The most relevant values for the constants have been derived by correlation of observed 3J values with the corresponding dihedral angles measured in protein structures determined by x-ray crystallography or NMR methods (84, 88–92). Recently, the availability of large data sets of coupling constants have allowed application of sophisticated mathematical and statistical methods to the optimization of the parameters of the Karplus equation (93–95). Calibration of the Karplus equation and determination of dihedral angle restraints are complicated by relaxation-induced self-decoupling that reduces the apparent coupling constant in larger proteins (Section 5.4.2) (96) and by intramolecular motions that average the values of 3J over distributions of dihedral angles (97). Motional averaging means that useful restraints sometimes cannot be derived from accurately measured coupling constants (98). For example, extensive backbone motion results in a 3 JHN H 7:0 Hz in short, unstructured linear peptides. Thus, 3 JHN H coupling constants observed in the range 6.0–8.0 Hz generally are not utilized as protein structural restraints, because they may reflect motional averaging of multiple conformations. The closer 3J is to one of the extrema expected from the Karplus relationship, the less the likelihood of significant motional averaging and the more accurate the resulting restraint. Examples of the Karplus parameters that have been reported in the literature for 3 JHN H are given in Table 10.1 and plotted in Fig. 10.8a. Experimental data for ubiquitin are compared to the theoretical curves in Fig. 10.8b. Given the shape of the Karplus curve, shown in Fig. 10.8, as many as four different conformations can yield the same value of 3J, although some of the possible orientations may be sterically unfavorable. The degeneracy inherent in the Karplus relationship is alleviated if more than one scalar coupling constant sensitive to a given dihedral angle is measured. As shown in Table 10.2 and Fig. 10.8c, d,
800
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
TABLE 10.1 Parameterization of the Karplus Equation for 3 JHN H a B
C
Ref.
1.4 1.4 1.3 1.76 1.42 1.05 1.31
1.9 2.4 1.5 1.60 1.55 0.65 1.56
(84) (88) (89) (90) (137) (93) (94)
A 6.4 6.0 6.7 6.51 7.09 7.90 7.13 a
The value of ¼ 608.
10
a
b
(Hz)
4
3J
6
HNHa
8
2 0 10
(Hz) IS
4
3J
6
c
3J N a H H
8
d 3J
COHa
3J HNCO 3J
HNCb
3J
COCb
2 3J
0 –180
–120
–60
0
60
120
180 –180
–120
–60
0
60
COCO
120
180
f (°)
FIGURE 10.8 Karplus curves describing the variation of 3 JCOCO , 3 JCOH , 3 JCOC ,3 JHN CO ,3 JHN H , and 3 JHN C with backbone dihedral angle . Curves were calculated using [10.2]. (a) Curves for 3 JHN H are calculated using parameter values given in Table 10.1 and are taken from various studies: (- - - -) (84), (– – –) (88), (– - –) (89), (— — —) (90), (– –) (137), (– - - - –) (93), and (—) (94). (b) Curves for 3 JHN H are calculated using parameter values given in Table 10.1 and are taken from three studies: (– - –) (137), (– – –) (93), and (—)(94). Solid points are experimental values taken from PDB file 1d3z (129). (c) and (d) Parameter values are given in Tables 10.1 and 10.2 and are taken from one study (94).
801
10.2 THREE-DIMENSIONAL SOLUTION STRUCTURES
TABLE 10.2 Parameterization of the Karplus Equation for a Coupling constant 3
JCOCO
3
JCOH
3
3
JHN CO
3
3
JCOC
JHN H
JHN C
A
B
C
1.36 1.51 0.86 1.30 0.12 3.72 3.76 1.05 3.84 0.14 1.74 2.72 0.80 2.52 0.33 4.29 4.41 0.81 4.19 0.30 7.09 7.90 1.02 7.13 0.34 3.06 2.90 0.80 3.26 0.23
0.93 1.09 1.11 0.93 0.06 2.18 1.63 0.56 2.19 0.10 0.57 0.31 0.52 0.49 0.33 1.01 1.36 1.03 0.99 0.18 1.42 1.05 0.54 1.31 0.13 0.74 0.56 0.52 0.87 0.24
0.60 0.52 0.39 0.64 0.03 1.28 0.89 0.60 1.20 0.11 0.25 0.39 0.37 0.51 0.12 0.00 0.24 0.37 0.03 0.05 1.55 0.65 0.58 1.56 0.34 0.13 0.18 0.37 0.10 0.08
08
þ1208
1208
1808
608
þ608
Ref. (137) (93) (94) (137) (93) (94) (137) (93) (94) (137) (93) (94) (137) (93) (94) (137) (93) (94)
a
The value of ¼ þ .
up to six 3J scalar coupling constants depend upon the backbone dihedral angle. In contrast to the abundance of scalar coupling constants that depend on , only the relatively small 3 JH N (referring to the 1 Hi–1 15Ni spin pair) depends on the backbone dihedral angle . The Karplus parameters for this coupling constant are given in Table 10.3 and plotted in Fig. 10.9. Experiments for measuring relaxation interference have been proposed as an alternative approach for determining (99). The side chain dihedral 1 historically has been characterized using 3 JH H scalar coupling constants; as for , the availability of isotopically enriched proteins has allowed measurement of many additional scalar coupling constants that depend on 1. Consensus values of the parameters of the Karplus equation for scalar coupling constants sensitive to 1 are given in Table 10.4 and the Karplus curves for 3 JH H2 and 3 JH H3 (the two 1H spins for an amino acid with a C methylene group)
802
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
TABLE 10.3 Parameterization of the Karplus Equation for 3 JH N a A 0.88 1.0 a
The value of ¼
B
C
Ref.
0.61 0.65
0.27 0.15
(138) (139)
1208.
0
–1.0
3
JHaN (Hz)
–0.5
–1.5
–2.0 –180
–120
–60
0
60
120
180
ψ (°)
FIGURE 10.9 Karplus curve describing the variation of 3 JH N with backbone dihedral angle . Curves were calculated using [10.2]. Parameter values are given in Table 10.3 and are taken from two studies: (—) (138) and (- - -) (139).
are plotted in Fig. 10.10. Dynamic averaging within and between rotameric states is more severe for side chain dihedral angles and the Karplus parameters for side chain dihedral angles are more dependent on the nature of the amino acid side chain than for the backbone dihedral angles (95). As a result of the preceding considerations, dihedral angle restraints enforced during structural calculations must include all of the valid solutions to the Karplus curve and usually are defined conservatively (usually not restrained to ranges less than 608 or 1208) to allow for limited motional averaging of the observed coupling constant.
803
10.2 THREE-DIMENSIONAL SOLUTION STRUCTURES
TABLE 10.4 Parameterization of the Karplus Equation for 1 a Coupling constant 3
JH H
3
JH C
3
JNH
3
JCOH
3
JNC JCOC
3
Ref.
A
B
C
9.5 9.5 10.2 7.23 7.1 5.34 4.4 2.30 7.2 4.02 1.29 2.31
1.6 1.0 1.8 1.37 1.0 0.96 1.2 0.75 2.0 1.58 0.49 0.87
1.8 1.4 1.9 2.22 0.7 0.79 0.1 1.07 0.6 1.32 0.37 0.55
(92) (91) (83) (95) (140) (95) (141) (95) (142) (95) (95) (95)
1208/08
1208 1208/–1208 08/1208 08 1208
a
The value of ¼ þ . When two values of are given, the first corresponds to H2 and the second to 1H3.
1
15
3J HaHb
(Hz)
10
5
0 –180
–120
–60
0
60
120
180
c1 (°)
FIGURE 10.10 Karplus curves describing the variation of 3 JH H2 (thin lines) and 3 JH H3 (thick lines) with side chain dihedral angle 1. Curves were calculated using [10.2]. Parameter values are given in Table 10.4 and are taken from various studies: (- - -) (92), (– – –) (91), (– - –) (83), and (—) (95).
804
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
10.2.1.3 Dihedral Angle Restraints from Isotropic Chemical Shifts Isotropic chemical shifts are exquisitely sensitive to local molecular conformation, but this extreme sensitivity also complicates the interpretation of chemical shifts in atomic detail. Fortunately, the dependence of chemical shifts of backbone nuclei, particularly 1H, 13CO, 13C, and 13C, on secondary structure is well-established (11, 100). Thus, the secondary chemical shift, defined as the observed value of the shift minus the value expected for the same residue in a random coil peptide [a compilation of random coil shifts is given by Schwarzinger and co-workers (1)], exhibits characteristic patterns for regular elements of secondary structure. This correlation forms the basis for the chemical shift index (CSI) method for identifying elements of secondary structure in proteins (10, 64). The TALOS program compares observed chemical shifts to a database of proteins with 1H, 13CO, 13C, 13C, and 15N resonance assignments and high-resolution structures to obtain dihedral angle restraints for incorporation into structure calculations (66). 10.2.1.4 Restraints from Residual Dipolar Coupling Constants Recently, a new class of structural restraint has been introduced that is not strictly local in nature and which represents a major advance in NMR structural studies. These restraints are based upon the measurement of residual dipolar couplings (RDCs) between pairs of NMR active nuclei in partially aligned molecules. Theoretical origins of RDCs are discussed in Section 2.8 and experimental approaches for measuring RDCs are discussed in Section 7.6. For directly bonded pairs of nuclei, for which the bond length is known, RDCs depend only on the orientation of the bond vector in the molecular alignment frame, as shown by [2.326], and described by the polar angles {, }. Approaches for incorporating restraints from RDCs into structure calculations continue to be developed and the optimal approach is not established at present. Two problems arise in the interpretation of RDCs: (i) the axial and rhombic components of the alignment tensor, Aa and Ar, and the Euler angles that define the orientation of the alignment frame in the laboratory frame must be determined and (ii) a single RDC can arise from more than one set of {, } angles. Thus, a minimum of five measured RDC values are required to define the alignment tensor (alternatively, [2.318]–[2.321] show that five measurements are needed to define the five independent elements of the Saupe order matrix). In addition, a given RDC is consistent with eight combinations of the polar angles, corresponding to all combinations of {, } and {, , , þ }. Thus, a single measured RDC does not provide a unique restraint.
10.2 THREE-DIMENSIONAL SOLUTION STRUCTURES
805
How these problems are solved has given rise to different methods for utilizing RDCs in structure calculations. The most straightforward method for incorporating RDC data into structure calculation protocols is by direct refinement of the orientation of individual bond vectors against the measured values of the RDCs (101). In this case, the orientation of each individual bond vector is changed to satisfy dipolar couplings as structures are calculated. Direct refinement has been shown to improve the accuracy and precision of structures when used in conjunction with nearly complete sets of NOE, coupling constant, and chemical shift data (101). This approach can be used with limited sets of RDCs; for example, many applications use only the backbone N–H RDCs. However, direct refinement requires that relatively high-quality initial structures have been determined from NOE, scalar coupling, and other restraints, because many local minima are encountered when refining RDCs (102, 103). Bax and Grishaev discuss the difficulties that can result from refining against too limited a set of RDCs, such as the 1H–15N RDCs alone (104). The second common approach treats the peptide plane as a fixed structural unit (103). For each plane the goal is to use at least five RDCs to define the Euler angles (, , ) that transform from an initial frame to the peptide alignment frame. Each peptide alignment frame must coincide with the global alignment coordinate system, effectively rotating each peptide into its correct orientation in the overall structure. The five dipolar couplings measured with the HNCO-based experiments described in Section 7.6 are sufficient to solve for the transformation of each peptide plane into the alignment frame. A difficulty with this approach is that some method is needed to distinguish between the eight possible orientations of the peptide plane consistent with the RDCs. Trial structures calculated from NOE restraints can be used to serve this purpose. 10.2.1.5 Hydrogen Bond Restraints from Amide Proton–Solvent Exchange Slow rates of amide exchange (relative to the intrinsic rates for unstructured peptides; Chapter 3, Fig. 3.36) are associated with shielding of amide 1HN atoms from solvent, and most commonly result from hydrogen bonding interactions (73). Amide exchange rates are usually measured in one of two ways, depending on the rate of exchange. When the rate is comparable to or faster than the spin–lattice relaxation rate (kex 4 0.1 s–1), the rate constant is most easily determined from a saturation transfer experiment (analogous to the transient NOE experiment discussed in Section 5.1.2) (105, 106). For slower rates
806
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
(kex 5 0.01 s–1), exchange usually is measured by rapidly transferring the protein from H2O into D2O solution, and repeatedly acquiring homonuclear TOCSY (Section 6.5.5) or 1H–15N HSQC (Section 7.1.1.2) spectra to observe the decrease in amide proton resonance intensities with time. Observation of a slow amide proton exchange rate implies that the 1HN atom may be involved in a hydrogen bond, but does not identify the atoms acting as hydrogen bond acceptors (and cannot exclude the possibility that the reduced exchange rate results from steric effects rather than hydrogen bonding). Hydrogen bond restraints have a large impact on the nature and precision of the resulting structures and are usually only enforced in well-defined regions of regular secondary structure, in which only one possible hydrogen bond acceptor is consistent with the NOE data. 10.2.1.6 Hydrogen Bond Restraints from Trans-Hydrogen Bond Scalar Coupling Constants Recently, trans-hydrogen bond scalar coupling interactions have been observed in nucleic acids (107, 108) and proteins (74) by NMR spectroscopy. In proteins, the N–H O¼C hydrogen bond between backbone amide and carbonyl moieties is characterized by 3h JNC0 , 2h JHC0 , and 3h JHC trans-hydrogen bonds with absolute magnitudes of 0–1 Hz. These can be detected by modifications of conventional triple-resonance experiments. For example, evolution due to the 3h JNC0 scalar coupling interaction can be detected in an HNCO experiment in which the 15 Ni –13 C0 i1 dephasing time is set to 2=JNC0 ¼ 133 ms (for a 15-Hz coupling constant) (74). The cross-peaks obtained from the usual one-bond scalar coupling between the 15 Ni –13 C0 i1 nuclei are suppressed and the cross-peaks from the transhydrogen bond coupling interactions are observed. Variants of these experiments have been performed on proteins up to 30 kDa in mass. The observation of trans-hydrogen cross-peaks unambiguously identifies the acceptor group for the hydrogen bond, not possible with amide exchange measurements, and the size of the coupling constant is a strong function of the geometry of the hydrogen bond (109).
10.2.2 STRUCTURE DETERMINATION Details of the local backbone geometry can be obtained by an extension of the sequential assignment process; the relative intensities of dNN, dN, and dN NOE cross-peaks and the measurement of the backbone 3 JHN H are required. The observation of intense dNN NOEs and small 3 JHN H coupling constants (56.0 Hz) are indicative of helical or turn sections of polypeptide (e.g., see Fig. 10.6); observation of
807
10.2 THREE-DIMENSIONAL SOLUTION STRUCTURES
intense dN, weak dNN, and dN NOEs and large 3 JHN H coupling constants (48.0 Hz) are indicative of extended -strands of polypeptide (15). The combination of sequential NOE and 3 JHN H coupling constant data with medium-range and a few long-range NOEs is capable of providing details of the regions of regular secondary structure within the protein. The elements of secondary structures can be connected together to give a crude view of the global fold by the identification of a few key long-range NOEs, as shown for ubiquitin in Fig. 10.11. Thus, without recourse to extensive calculations, important structural results (albeit of low absolute resolution) can be obtained in a straightforward manner. A variety of methods have been developed to calculate atomicresolution protein structures using restraints derived from experimental NMR data [for reviews, see Gu¨ntert (35, 68) and Grishaev and
34 11
17
1
7
71
N
64 23
C 40
45
50
59
48 56
FIGURE 10.11 Definition of the global fold using long-range NOEs. Previously identified elements of regular secondary structure (see Fig. 10.6), can be arranged in a low-resolution global fold by the observation of a few key longrange NOEs involving protons at the termini of the strands and helices. This figure depicts the strands of -sheet and sections of helix present in ubiquitin as black arrows and white cylinders, respectively.
808
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
Llina´s (67)]. Importantly, NMR data do not uniquely define the threedimensional structure of a protein or other biological macromolecule, because the restraints are included as ranges of allowed values, the data contain experimental uncertainties, and only a sparse subset of all possible restraints are observable. To increase the efficiency and accuracy of structure calculations, the experimentally derived restraints normally are supplemented by restraints specifically imposed to enforce proper covalent structure of the protein, including bond lengths, bond angles, and other elements of standard covalent geometry (chirality and the planarity of aromatic rings and peptide units). Protocols for structure determination aim to find coordinates for the protein atoms that will satisfy the input restraints in an unbiased fashion while exploring all of the regions of conformational space compatible with these restraints. Because of these considerations, structure calculations are repeated many times to determine an ensemble of (low energy) structures consistent with the input NMR data. Thus, a ‘‘good’’ ensemble of structures minimizes violations of the input restraints and maximizes the RMSD between members of the ensemble (79, 110). The two most common approaches to generation of structures are distance geometry (DG) and restrained molecular dynamics (rMD). Historically, DG was the first approach utilized for structure determination; at an intermediate stage of development, DG frequently was used to generate initial structures for subsequent refinement by rMD methods. In modern approaches for structure determination, rMD has become the predominant technique. However, other approaches to structure determination continue to be pursued and future developments can be expected (111). Popular implementations of DG use either the metric matrix algorithm (112, 113) or the variable target function approach (114, 115). Distance geometry determines ensembles of three-dimensional structures consistent with an incomplete set of distance restraints. The restraints are incomplete because not all distances can be characterized (the NOE is limited to distances less than approximately 5 A˚) and because the distance restraints are not known precisely. The metric matrix algorithms in particular tend to be computationally expensive as the size of the protein increases. Restrained molecular dynamics algorithms use either Cartesian or torsion-angle coordinate systems (68). Torsion-angle rMD has become the preferred method due to advances in computational algorithms. In either approach, molecular dynamics force fields are supplemented by pseudo-energy terms based on the NMR-derived restraints (116, 117).
10.2 THREE-DIMENSIONAL SOLUTION STRUCTURES
809
These potentials drive the structure toward a conformation that will reduce the violation of the restraints during a forced heat-up and cool-down annealing cycle. The most computationally efficient implementations of the rMD method use a simplified force field in which bond length, bond angle, and repulsive van der Waals terms are retained (electrostatic and attractive van der Waals terms are ignored), and are referred to as dynamical simulated annealing (SA) (118). Due to advances in computational power, structures determined using simplified force fields now frequently are refined using complete force fields and including explicit or implicit solvent models (119, 120). As has already been discussed (Sections 10.1.1 and 10.1.2), considerable efforts have been made to partially or fully automate the process of resonance assignments. Extensive efforts also are being made to automate the process of structure determination. Most automated structure calculation programs take as input a (sufficiently complete) list of resonance assignments and one or more lists of crosspeak positions and volumes from nD NOESY spectra. The programs then automatically assign the NOESY cross-peaks and calculate the three-dimensional structure of the protein. Current state-of-the-art methods for automated structure determination have been reviewed (33–36, 67). A comparison of ‘‘conventional’’ structure determination protocols with an optimized pipeline consisting of fast data acquisition (Section 9.3), automated resonance assignments, and automated structure calculation has been reported by Szyperski and co-workers (121). Assessment of the resultant structures is an important aspect of any structure determination by NMR spectroscopy. The RMSD of the final ensemble of structures is a poor measure for judging the outcome of a structure calculation. Instead, quality of the calculated threedimensional structures is assessed by two general measures: (i) how well the structures agree with the experimental restraint data and (ii) the geometrical reasonableness of the structures — for example, compared with expected stereochemistry derived from high-resolution x-ray crystal structures [for a review, see Spronk et al. (122)]. Consistency with experimental results commonly is judged by examining restraint pseudo-energies from the structure refinement, root-mean-square violations of restraints, and maximum consistent violations within the ensemble of structures. Stereochemical quality commonly is judged by quantifying the distributions of backbone and side chain dihedral angles, the number of van der Waals steric clashes, and packing of core residues. Programs such as PROCHECK (123), PROCHECK-NMR
810
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
(124), and WHATIF (125) are widely utilized for structure assessment. A number of new approaches for structure assessment have been developed using concepts from information theory (126, 127). Structures also can be validated independently against sets of experimental data that were not used in the structure calculation. Both RDCs (128) and anisotropic chemical shifts (129) have been suggested for this purpose by defining a figure of merit, or Q factor. A powerful approach for structure assessment, and potentially refinement, is known as complete relaxation matrix analysis (CORMA) (78), or back-calculation. Once three-dimensional structures have been calculated, interproton distances can be calculated from the atomic coordinates, the dipolar relaxation rate matrix can be calculated from the distances, and the theoretically expected NOE intensities can be calculated from the rate matrix using the results of Chapter 5 to generate theoretically expected NOESY spectra. After suitable scaling, the calculated NOE intensities are compared to the experimentally observed NOE intensities to judge how well the structures reproduce the experimental data. Usually, the agreement is judged by a figure of merit, or R factor (130). The difference between observed and back-calculated NOE intensities also can be included as a pseudo-energy term during rMD calculations (131, 132). Thus, the structures are driven during the calculation to minimize the R factor and maximize the agreement between the structures and the experimental NOE intensities. This process requires a complete relaxation matrix analysis to be performed at every step of the annealing cycle and therefore is computationally demanding. In addition, the dynamical model must be capable of reproducing the effects of molecular motion known to be present in proteins, such as fast rotation of methyl groups and 1808 flips of aromatic side chains (Section 5.3) (133) and systematic deficiencies in the experimental data (134). However, back-calculation provides one of the few ways to independently assess the accuracy of protein structures derived from NMR data. As an example, the structure of ubiquitin determined by NMR spectroscopy by Bax and co-workers is shown in Fig. 10.12. This structure was determined from 2727 NOEs, 98 dihedral angle restraints derived from homo- and heteronuclear J couplings, and 372 RDC restraints (129). The high degree of agreement between experimental RDCs and values calculated from the final structure is shown in Fig. 10.13. These RDCs were used for the structure refinement and are not independent figures of merit for the stucture determination. The structure quality was assessed using the Q factor calculated from the measured and calculated anisotropic carbonyl
10.2 THREE-DIMENSIONAL SOLUTION STRUCTURES
811
chemical shifts (129), 2 P c i1=2 h P 2 , Q¼
½10:3
c are the measured and predicted differences in in which and 13 the CO chemical shift measured in isotropic and anisotropic solution, and the summation extends over the backbone residues. The value of the Q factor is 16%; for comparison, the Q factor is 25% if the structure is calculated without inclusion of the RDCs and is 23% for the 1.8-A˚ crystal structure of ubiquitin (135). Ubiquitin also has been the subject of many investigations of protein conformational dynamics by NMR spectroscopy, including studies of spin relaxation, RDCs, and amide proton–solvent exchange. As only one example of this voluminous literature, the backbone amide order parameters determined from 15N spin relaxation rate constants are shown in Fig. 10.14 (136). As shown, and consistent with the ensemble of structures shown in Fig. 10.12, the backbone is highly constrained,
FIGURE 10.12 The NMR-derived structure of ubiquitin. (a) Backbone chain trace for the first member of the ensemble of 10 structures in PDB file 1d3z. (b) Superposition of the 10 members of the structure ensemble showing backbone and side chain heavy atoms. Side chain atoms are not shown for the disordered C-terminal residues 71–77 of ubiquitin for clarity.
812
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
10
N ~ -0
2
a
2 c
b 0
0
Q)
0
-1
~U)
-2
0-10 -2 -20 -20
-10
0
10
-3 -3
-2
-1
0
-4 2 -4
1
0
-2
D~66 (Hz)
D~~PI (Hz)
2
SlIpl
Dcaco (Hz)
FIGURE 10.13 (a) I H _ 15 N, (b) 13CO_15N, and (c) 13CQ'_13CO residual dipolar coupling constants for ubiquitin at 304 K using 5% w/v bicelles (3: 1 DMPC:DHPC). Experimental values (129) are compared to values calculated for the first member of the ensemble of structures shown in Fig. 10.12.
..•", .•....,-._ ~
0.8
·"tIl·
~·Y. til
• • • ~
.
.....
•••••
0.6
•
N
U)
0.4
0.2
o
10
20
30
40
50
60
70
80
Residue FIGURE 10.14 Backbone 15N generalized order parameters for ubiquitin. Experimental values (136) are shown as a function of sequence position. Secondary structure elements are indicated above the graph.
with reduced order parameters observed primarily in loops and at the C-terminus. The backbone amide order parameters - or more simply, the heteronuclear {' H}- 15 N steady-state NOE (Section 8.2.1) - aid in distinguishing between disordered regions in the ensemble of calculated structures that reflect conformational fluctuations in solution
813
10.3 CONCLUSION
and disordered regions that arise from insufficient restraints in the structure calculation.
10.3 Conclusion This text has two objectives: (i) presentation of the theory of NMR spectroscopy applicable to studies of biological macromolecules and (ii) explication of the types of NMR experiments utilized for assignments of resonances, determinations of three-dimensional structures, and investigations of dynamics and interactions of proteins. Knowledge of the three-dimensional structure of a protein, whether determined by x-ray crystallography or by NMR spectroscopy, is the beginning of structural biology and biophysics of protein function, not the endpoint. Proteins are not rigid isolated molecules, and understanding protein function depends upon a detailed knowledge of structure, dynamics, and interactions with other biomolecules. As indicated briefly in Chapters 8 and 9, NMR spectroscopy is a powerful approach for investigations of the intramolecular dynamical properties (fluctuations about the time-average structures) of proteins and of structural and kinetic aspects of intermolecular interactions. The experiments discussed in this text have a certain universality (the topology of amino acid spin systems and secondary structure elements serve as unifying features) and are widely used in studies of proteins by NMR spectroscopy. In the interest of brevity, this text does not discuss the complete wealth of NMR experiments that have been developed and continue to be developed for investigations of proteins and other biological macromolecules. Nonetheless, the theoretical and experimental principles presented herein are equally valid and important for the understanding and optimization of all experiments for highresolution NMR spectroscopy of biomolecules in solution. Since the first reports of the observation of nuclear magnetic resonance absorption in condensed phase in 1946, nuclear magnetic resonance has developed into a major technique in biology, chemistry, and medicine. Yet, surveying the developments of the past decade suggests that the ‘‘Golden Age’’ of biological NMR spectroscopy shows no end. Continued advances in sample preparation, magnet and rf technology, NMR spectroscopic techniques, and computational methods promise to make the next 60 years as exciting as the last.
814
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
References 1. S. Schwarzinger, G. J. A. Kroon, T. R. Foss, P. E. Wright, H. J. Dyson, J. Biomol. NMR 18, 43–48 (2000). 2. S. Schwarzinger, G. J. Kroon, T. R. Foss, J. Chung, P. E. Wright, H. J. Dyson, J. Am. Chem. Soc. 123, 2970–2978 (2001). 3. A. Bundi, K. Wu¨thrich, Biopolymers 18, 285–297 (1979). 4. D. S. Wishart, B. D. Sykes, Meth. Enzymol. 239, 363–392 (1994). 5. D. S. Wishart, B. D. Sykes, F. M. Richards, J. Mol. Biol. 222, 311–333 (1991). 6. K.-H. Gross, H. R. Kalbitzer, J. Magn. Reson. 76, 87–99 (1988). 7. D. C. Dalgarno, B. A. Levine, R. J. P. Williams, Biosci. Rep. 3, 443–452 (1983). 8. A. Pastore, V. Saudek, J. Magn. Reson. 90, 165–176 (1990). 9. M. P. Williamson, Biopolymers 29, 1423–1431 (1990). 10. D. S. Wishart, B. D. Sykes, F. M. Richards, Biochemistry 31, 1647–1651 (1992). 11. D. S. Wishart, D. A. Case, Meth. Enzymol. 338, 3–34 (2001). 12. B. R. Seavey, E. A. Farr, W. M. Westler, J. L. Markley, J. Biomol. NMR 1, 217–236 (1991). 13. W. J. Chazin, P. E. Wright, Biopolymers 26, 973–977 (1987). 14. W. J. Chazin, P. E. Wright, J. Mol. Biol. 202, 623–636 (1988). 15. K. Wu¨thrich, ‘‘NMR of Proteins and Nucleic Acids,’’ pp. 1–292. Wiley, New York, 1986. 16. A. D. Kline, W. Braun, K. Wu¨thrich, J. Mol. Biol. 204, 675–724 (1988). 17. M. Billeter, W. Braun, K. Wu¨thrich, J. Mol. Biol. 155, 321–346 (1982). 18. K. Wu¨thrich, Biopolymers 22, 131–138 (1982). 19. S. W. Englander, A. J. Wand, Biochemistry 26, 5953–5958 (1987). 20. D. L. Di Stefano, A. J. Wand, Biochemistry 26, 7272–7281 (1987). 21. G. J. Kleywegt, R. M. N. J. Lamerichs, R. Boelens, R. Kaptein, J. Magn. Reson. 85, 186–197 (1989). 22. F. M. van de Ven, J. Magn. Reson. 86, 633–644 (1990). 23. J. Xu, S. K. Strauss, B. C. Sanctuary, L. Trimble, J. Magn. Reson., Ser. B 103, 53–58 (1994). 24. Z. Z. Chen, T. Jiang, G. H. Lin, J. J. Wen, D. Xu, Y. Xu, in ‘‘Lecture Notes in Computer Science’’ (R. Guigo´, D. Gusfield, eds.), Vol. 2452, pp. 83–96. Springer-Verlag GmbH, Heidelberg, 2002. 25. P. J. Kraulis, J. Magn. Reson. 84, 627–633 (1989). 26. M. Billeter, V. J. Basus, I. D. Kuntz, J. Magn. Reson. 76, 400–415 (1988). 27. C. Eccles, P. Gu¨ntert, M. Billeter, K. Wu¨thrich, J. Biomol. NMR 1, 111–130 (1991). 28. G. J. Kleywegt, R. Boelens, R. Kaptein, J. Magn. Reson. 88, 601–608 (1990). 29. C. Cieslar, T. A. Holak, H. Oschkinat, J. Magn. Reson. 89, 184–190 (1990). 30. G. W. Vuister, R. Boelens, A. Padilla, R. Kaptein, J. Biomol. NMR 1, 421–438 (1991). 31. R. Bernstein, A. Ross, C. Cieslar, T. A. Holak, J. Magn. Reson., Ser. B 101, 185–188 (1993). 32. H. Oschkinat, T. A. Holak, C. Ciesler, Biopolymers 31, 699–712 (1991). 33. M. C. Baran, Y. J. Huang, H. N. Moseley, G. T. Montelione, Chem. Rev. 104, 3541–3556 (2004). 34. A. S. Altieri, R. A. Byrd, Curr. Opin. Struct. Biol. 14, 547–553 (2004). 35. P. Gu¨ntert, Prog. NMR Spectrosc. 43, 105–125 (2003). 36. W. Gronwald, H. R. Kalbitzer, Prog. NMR Spectrosc. 44, 33–96 (2004). 37. W. J. Chazin, M. Rance, P. E. Wright, J. Mol. Biol. 202, 603–622 (1988). 38. R. A. Williamson, G. Mortorell, M. D. Carr, G. Murphy, A. J. P. Docherty, R. B. Freedman, J. Feeney, Biochemistry 33, 11745–11759 (1994).
10.3 CONCLUSION
815
39. P. L. Weber, S. C. Brown, L. Mueller, Biochemistry 26, 7282–7290 (1987). 40. P. C. Driscoll, G. M. Clore, D. Marion, P. T. Wingfield, A.M. Gronenborn, Biochemistry 29, 3542–3556 (1990). 41. G. M. Clore, A. Bax, P. C. Driscoll, P. T. Wingfield, A. M. Gronenborn, Biochemistry 29, 8172–8184 (1990). 42. M. D. Carr, B. Birdsall, T. A. Frenkiel, C. J. Bauer, J. Jimenez-Barbero, V. I. Polshakov, J. E. McCormick, G. C. K. Roberts, J. Feeney, Biochemistry 30, 6330–6341 (1991). 43. R. T. Clubb, V. Thanabal, C. Osborne, G. Wagner, Biochemistry 30, 7718–7730 (1991). 44. W. J. Fairbrother, J. Cavanagh, H. J. Dyson, A. G. Palmer, S. L. Sutrina, J. Reizer, M. H. Saier, P. E. Wright, Biochemistry 30, 6896–6907 (1991). 45. W. J. Fairbrother, A. G. Palmer, M. Rance, J. Reizer, M. H. Saier, P. E. Wright, Biochemistry 31, 4413–4425 (1992). 46. B. J. Stockman, N. R. Nirmala, G. Wagner, T. J. Delcamp, M. T. DeYarman, J. H. Freisheim, Biochemistry 31, 218–229 (1992). 47. M. Ikura, L. E. Kay, A. Bax, Biochemistry 29, 4659–4667 (1990). 48. G. T. Montelione, B. A. Lyons, S. D. Emerson, M. Tashiro, J. Am. Chem. Soc. 114, 10974–10975 (1992). 49. S. Grzesiek, J. Anglister, A. Bax, J. Magn. Reson., Ser. B 101, 114–119 (1993). 50. R. T. Clowes, W. Boucher, C. H. Hardman, P. J. Domaille, E. D. Laue, J. Biomol. NMR 3, 349–354 (1993). 51. B. A. Lyons, M. Tashiro, L. Cedergren, B. Nilsson, G. T. Montelione, Biochemistry 32, 7839–7845 (1993). 52. T. M. Logan, E. T. Olejniczak, R. X. Xu, S. W. Fesik, J. Biomol. NMR 3, 225–231 (1993). 53. B. T. Farmer, R. A. Venters, J. Am. Chem. Soc. 117, 4187–4188 (1995). 54. T. Torizawa, A. M. Ono, T. Terauchi, M. Kainosho, J. Am. Chem. Soc. 127, 12620–12626 (2005). 55. J. J. Prompers, A. Groenewegen, C. W. Hilbers, H. A. M. Pepermans, J. Magn. Reson. 130, 68–75 (1998). 56. B. T. Farmer, R. A. Venters, J. Biomol. NMR 7, 59–71 (1996). 57. T. Yamazaki, S. M. Pascal, A. U. Singer, J. D. Forman-Kay, L. E. Kay, J. Am. Chem. Soc. 117, 3556–3564 (1995). 58. F. Lohr, S. Pfeiffer, Y. J. Lin, J. Hartleib, O. Klimmek, H. Ru¨terjans, J. Biomol. NMR 18, 337–346 (2000). 59. V. Kanelis, L. Donaldson, D. R. Muhandiram, D. Rotin, J. D. Forman-Kay, L. E. Kay, J. Biomol. NMR 16, 253–259 (2000). 60. D. Yang, L. E. Kay, J. Am. Chem. Soc. 121, 2571–2575 (1999). 61. V. Tugarinov, R. Muhandiram, A. Ayed, L. E. Kay, J. Am. Chem. Soc. 124, 10025–10035 (2002). 62. V. Tugarinov, L. E. Kay, J. Am. Chem. Soc. 125, 13868–13878 (2003). 63. R. Richarz, K. Wu¨thrich, Biopolymers 17, 2133–2141 (1978). 64. D. S. Wishart, B. D. Sykes, J. Biomol. NMR 4, 171–180 (1994). 65. S. Spera, A. Bax, J. Am. Chem. Soc. 113, 5490–5492 (1991). 66. G. Cornilescu, F. Delaglio, A. Bax, J. Biomol. NMR 13, 289–302 (1999). 67. A. Grishaev, M. Llina´s, Meth. Enzymol. 394, 261–295 (2005). 68. P. Gu¨ntert, Prog. NMR Spectrosc. 31, 145–237 (1998). 69. A. G. Palmer, Chem. Rev. 104, 3623–3640 (2004). 70. T. Carlomagno, Annu. Rev. Biophys. Biomol. Struct. 34, 245–266 (2005). 71. M. P. Williamson, T. F. Havel, K. Wu¨thrich, J. Mol. Biol. 182, 295–315 (1985). 72. N. Tjandra, A. Bax, Science 278, 1111–1114 (1997). 73. G. Wagner, Q. Rev. Biophys. 16, 1–57 (1983).
816
CHAPTER 10 SEQUENTIAL ASSIGNMENT, STRUCTURE DETERMINATION
74. Y.-X. Wang, J. Jacob, F. Cordier, P. Wingfield, S. J. Stahl, S. Lee-Huang, D. A. Torchia, S. Grzesiek, A. Bax, J. Biomol. NMR 14, 181–184 (1999). 75. V. Gaponenko, S. P. Sarma, A. S. Altieri, D. A. Horita, J. Li, R. A. Byrd, J. Biomol. NMR 28, 205–212 (2004). 76. I. Bertini, C. Luchinat, M. Piccioli, Meth. Enzymol. 339, 314–340 (2001). 77. J. Iwahara, C. D. Schwieters, G. M. Clore, J. Am. Chem. Soc. 126, 5879–5896 (2004). 78. J. W. Keepers, T. L. James, J. Magn. Reson. 57, 404–426 (1984). 79. T. Havel, in ‘‘Proteins: Structure, Dynamics, Design’’ (V. Renugopalakrishnan, P. R. Carey, I. C. P. Smith, S.-G. Huans, A. L. Storer, eds.), pp. 110–115. ESCOM Science Publishers, Leiden, Holland, 1991. 80. P. Gu¨ntert, W. Braun, M. Billeter, K. Wu¨thrich, J. Am. Chem. Soc. 111, 3997–4004 (1989). 81. J. F. Doreleijers, M. L. Raves, T. Rullmann, R. Kaptein, J. Biomol. NMR 14, 123–132 (1999). 82. M. Karplus, J. Phys. Chem. 30, 11–15 (1959). 83. V. F. Bystrov, Prog. Nucl. Magn. Reson. 10, 41–82 (1976). 84. A. Pardi, M. Billeter, K. Wu¨thrich, J. Mol. Biol. 180, 741–751 (1984). 85. G. Wagner, W. Braun, T. F. Havel, T. Schaumann, N. Go, K. Wu¨thrich, J. Mol. Biol. 196, 611–639 (1987). 86. G. W. Vuister, M. Tessari, Y. Karimi-Nejad, B. Whitehead, in ‘‘Biological Magnetic Resonance, 16: Modern Techniques in Protein NMR’’ (N. R. Krisha, L. Berliner, eds.), pp. 195–257. Kluwer Academic, New York, 1999. 87. M. J. Minch, Concepts Magn. Reson. 6, 41–56 (1994). 88. L. J. Smith, M. J. Sutcliffe, C. Redfield, C. M. Dobson, Biochemistry 30, 986–996 (1991). 89. S. Ludvigsen, K. V. Andersen, F. M. Poulsen, J. Mol. Biol. 217, 731–736 (1991). 90. G. W. Vuister, A. Bax, J. Am. Chem. Soc. 115, 7772–7777 (1993). 91. C. M. Deber, D. A. Torchia, E. R. Blout, J. Am. Chem. Soc. 93, 4893–4897 (1971). 92. A. DeMarco, M. Llina´s, K. Wu¨thrich, Biopolymers 17, 617–636 (1978). 93. J. M. Schmidt, M. Blu¨mel, F. Lo¨hr, H. Ru¨terjans, J. Biomol. NMR 14, 1–12 (1999). 94. M. Habeck, W. Rieping, M. Nilges, J. Magn. Reson. 177, 160–165 (2005). 95. C. Pere´z, F. Lo¨hr, H. Ru¨terjans, J. Schmidt, J. Am. Chem. Soc. 123, 7081–7093 (2001). 96. G. S. Harbison, J. Am. Chem. Soc. 115, 3026–3027 (1993). 97. R. Bru¨schweiler, D. Case, J. Am. Chem. Soc. 116, 11199–11200 (1994). 98. S. G. Hyberts, W. Ma¨rki, G. Wagner, Eur. J. Biochem. 164, 625–635 (1987). 99. B. Reif, A. Diener, M. Hennig, M. Maurer, C. Griesinger, J. Magn. Reson. 143, 45–68 (2000). 100. D. Sitkoff, D. A. Case, Prog. NMR Spectrosc. 32, 165–190 (1998). 101. G. M. Clore, A. M. Gronenborn, N. Tjandra, J. Magn. Reson. 131, 159–162 (1998). 102. M. W. F. Fischer, J. A. Losonczi, J. L. Weaver, J. H. Prestegard, Biochemistry 38, 9013–9022 (1999). 103. G. A. Mueller, W. Y. Choy, D. Yang, J. D. Forman-Kay, R. A. Venters, L. E. Kay, J. Mol. Biol. 300, 197–212 (2000). 104. A. Bax, A. Grishaev, Curr. Opin. Struct. Biol. 15, 563–570 (2005). 105. S. Forse´n, R. A. Hoffman, Acta Chem. Scand. 17, 1787 (1963). 106. S. Forse´n, R. A. Hoffman, J. Chem. Phys. 39, 2892 (1963). 107. A. J. Dingley, S. Grzesiek, J. Am. Chem. Soc. 120, 8293–8297 (1998). 108. K. Pervushin, A. Ono, C. Ferna´ndez, T. Szyperski, M. Kainosho, K. Wu¨thrich, Proc. Natl. Acad. Sci. U.S.A. 95, 14147–14151 (1998). 109. G. Cornilescu, B. E. Ramirez, M. K. Frank, G. M. Clore, A. M. Gronenborn, A. Bax, J. Am. Chem. Soc. 121, 6275–6279 (1999). 110. S. Hyberts, M. S. Goldberg, T. F. Havel, G. Wagner, Protein Sci. 1, 736–751 (1992).
10.3 CONCLUSION
817
111. W. Rieping, M. Habeck, M. Nilges, Science 309, 303–306 (2005). 112. G. M. Crippen, T. F. Havel, ‘‘Distance Geometry and Molecular Conformation,’’ pp. 1–58. Research Studies Press, Taunton, England, 1988. 113. T. F. Havel, Prog. Biophys. Mol. Biol. 56, 43–78 (1991). 114. W. Braun, N. Go, J. Mol. Biol. 186, 611–626 (1985). 115. P. Gu¨ntert, W. Braun, K. Wu¨thrich, J. Mol. Biol. 217, 517–530 (1991). 116. G. M. Clore, A. T. Bru¨nger, M. Karplus, A. M. Gronenborn, J. Mol. Biol. 191, 523–551 (1986). 117. A. T. Bru¨nger, G. M. Clore, A. M. Gronenborn, M. Karplus, Proc. Natl. Acad. Sci. U.S.A. 83, 3801–3805 (1986). 118. M. Nilges, G. M. Clore, A. M. Gronenborn, FEBS Lett. 239, 129–136 (1988). 119. B. Xia, V. Tsui, D. A. Case, H. J. Dyson, P. E. Wright, J. Biomol. NMR 22, 317–331 (2002). 120. J. P. Linge, M. A. Williams, C. A. Spronk, A. M. Bonvin, M. Nilges, Proteins 50, 496–506 (2003). 121. G. Liu, Y. Shen, H. S. Atreya, D. Parish, Y. Shao, D. K. Sukumaran, R. Xiao, A. Yee, A. Lemak, A. Bhattacharya, T. A. Acton, C. H. Arrowsmith, G. T. Montelione, T. Szyperski, Proc. Natl. Acad. Sci. U.S.A. 102, 10487–10492 (2005). 122. C. A. E. M. Spronk, S. B. Nabuurs, E. Krieger, G. Vriend, G. W. Vuister, Prog. NMR Spectrosc. 45, 315–337 (2004). 123. R. A. Laskowski, M. W. MacArthur, D. S. Moss, J. M. Thornton, J. Appl. Cryst. 26, 283–291 (1993). 124. R. A. Laskowski, J. A. C. Rullmann, M. W. MacArthur, R. Kaptein, J. M. Thornton, J. Biomol. NMR 8, 477–486 (1996). 125. G. Vriend, J. Mol. Graph. 8, 52–56 (1990). 126. S. B. Nabuurs, E. Krieger, C. A. Spronk, A. J. Nederveen, G. Vriend, G. W. Vuister, J. Biomol. NMR 33, 123–134 (2005). 127. Y. J. Huang, R. Powers, G. T. Montelione, J. Am. Chem. Soc. 127, 1665–1674 (2005). 128. A. Bax, Protein Sci. 12, 1–16 (2003). 129. G. Cornilescu, J. L. Marquardt, M. Ottiger, A. Bax, J. Am. Chem. Soc. 120, 6836–6837 (1998). 130. C. Gonzalez, J. A. C. Rullmann, A. M. J. J. Bonvin, R. Boelens, R. Kaptein, J. Magn. Reson. 91, 659–664 (1991). 131. P. Yip, D. A. Case, J. Magn. Res. 83, 643–648 (1989). 132. A. M. J. J. Bonvin, R. Boelens, R. Kaptein, J. Biomol. NMR 1, 305–309 (1991). 133. H. Liu, P. D. Thomas, T. J. James, J. Magn. Reson. 98, 163–175 (1992). 134. M. J. Dellwo, D. M. Schneider, A. J. Wand, J. Magn. Reson., Ser. B 103, 1–9 (1994). 135. S. Vijay-Kumar, C. E. Bugg, W. J. J. Cook, J. Mol. Biol. 194, 531–544 (1987). 136. N. Tjandra, S. E. Feller, R. W. Pastor, A. Bax, J. Am. Chem. Soc. 117, 12562–12566 (1995). 137. J.-S. Hu, A. Bax, J. Am. Chem. Soc. 119, 6360–6368 (1997). 138. A. C. Wang, A. Bax, J. Am. Chem. Soc. 117, 1810–1813 (1995). 139. F. Lo¨hr, J. M. Schmidt, S. Maurer, H. Ru¨terjans, J. Magn. Reson. 153, 75–82 (2001). 140. J. Schmidt, J. Magn. Reson. 124, 310–322 (1997). 141. A. DeMarco, M. Llina´s, K. Wu¨thrich, Biopolymers 17, 2727–2742 (1978). 142. A. J. Fischman, D. H. Live, H. R. Wyssbrod, W. C. Agosta, D. Cowburn, J. Am. Chem. Soc. 102, 2533–2539 (1980). 143. D. Braun, G. Wider, K. Wu¨thrich, J. Am. Chem. Soc. 116, 8466–8469 (1994).
TABLE OF SYMBOLS
hAi [A, B] A Ay A A Akk Ar Aqk
Aqkp
expectation value of an operator A [2.19], [2.47] commutator of operators A and B [2.26] matrix or operator A adjoint of a matrix or operator A Commutation superoperator of A axial component of the alignment tensor [2.323] principal value of the alignment tensor [2.323] rhombicity of the alignment tensor [2.323] irreducible spherical tensor of rank k [2.303] tensor spin operator [5.53] basis tensor operators of Aqk [5.54]
A(!) amn(t)
B, B(t) B0 B0 B1 Beff(t) Br Br
819
absorptive Lorentizian lineshape [3.29] mixing coefficient for transfer of coherence from spin or site m to spin or site n [4.18], [5.19], [5.175], [5.180] magnetic field vector static magnetic field strength reduced static magnetic field strength [1.19] rf magnetic field amplitude effective magnetic field in rotating frame [1.13] effective magnetic field in rotating frame [1.18] magnitude of the effective magnetic field in rotating frame [1.20]
820 Brf(t) Bj C C(n)
rf magnetic field [1.15] jth basis operator capacitance [3.2] irreducible tensor of rank n [2.298] C200 ðÞ orientational correlation function for rotation of an isotropic sphere [5.97] CðÞ stochastic correlation function [5.93] c0(t) function of physical constants and spatial variables for lattice-spin system coupling [5.93] DIS (residual) dipolar coupling constant [2.326], [2.327] D2mn ðLP , Wigner rotation LP ,LP Þ matrices, Table 2.4 D(!) dispersive Lorentzian lineshape [3.30] dN sequential 1H-1HN NOE connectivity dN sequential 1H-1HN NOE connectivity dNN sequential 1HN-1HN NOE connectivity d00 c20 [5.94] 2 dmn ðÞ reduced rotation matrices, Table 2.4 E identity matrix or operator Fþ observation operator [2.128] Fz non-selective z-rotation [4.33] Fq2 components of irreducible spherical tensor of second rank [2.305] Fqk ðtÞ random function of spatial lattice variables [5.53]
TABLE
OF
SYMBOLS
Fourier transformation of s(t) [3.14] frequency bandwidth of fc time-domain signal fn Nyquist frequency [3.6] Gx,y,z pulsed field gradient strengths H Hamiltonian operator H0 time-independent Hamiltonian in absence of applied RF fields [2.154], [5.47] H1(t) stochastic Hamiltonian [5.47] He effective Hamiltonian in rotating frame [2.66], [2.97] HJ scalar coupling Hamiltonian [2.154] Hrf(t) rf magnetic field Hamiltonian [2.93] Hz Zeeman Hamiltonian [2.93], [2.154] h ¼ h=2 Planck’s constant divided by 2p I spin angular momentum quantum number I angular momentum vector [1.1] Ix, Iy, Iz Cartesian angular momentum operators Iz ¼ mh z-component of the angular momentum vector [1.2] I0, I þ, I shift basis operators I0{} zero-order modified Bessel function [3.46] I , I, single element basis I þ, I operators I þ, I shift (raising and lowering) angular momentum operators F{s(t)}
TABLE
I þ(rs), I (rs) i i J J(t) n
Jij, Jij
J(!) j j(!)
jq(!) jqmn ð!Þ
k kij
kB kex k1
k–1
L
OF
821
SYMBOLS
single transition shift p ffiffiffiffiffiffiffi operators 1 unit vector along Cartesian x-axis scalar coupling tensor bulk angular momentum vector [1.10] n-bond scalar coupling constant between spins Ii and Ij orientational spectral density function [5.95] unit vector along Cartesian y-axis spectral density function for isotropic phase in the high temperature limit [5.90] spectral density function [5.63] cross spectral density function for the mth and nth relaxation interactions [5.77] unit vector along Cartesian z-axis microscopic rate constant for exchange between the ith and jth chemical species [5.165] Boltzmann constant k1 þ k–1 forward microscopic rate constant for two-site chemical exchange [5.156] reverse microscopic rate constant for two-site chemical exchange [5.156] inductance [3.1]
^ LðtÞ
Liouvilian superoperator [5.48] equilibrium M0 magnetization [1.7] Mr molecular mass Mx, My, Cartesian component of Mz the magnetization vector M(t) bulk magnetic moment or magnetization vector [1.11] Mr(t) bulk magnetization vector in the rotating frame [1.17] Mþ(t) complex magnetization [1.40] m magnetic quantum number [1.2] N number of spins per unit volume NA Avogadro’s number nw unit vector directed along a rotation axis w P projection operator [2.40] Pn(cos) Legendre polynomials Pnm(cos) associated Legendre polynomials P() rf pulse with phase and on-resonance rotation angle [3.96], [3.106] P(t) probability density [2.3] P, P, spin state populations P, P P() probability density for lattice-spin system coupling [2.45] p(, , ) orientational probability distribution [2.314] R resistance [3.1] pulse element for composite pulse decoupling
822 R Rex
RI()
R1
R1 R2 R2 R2MQ
RC Rw()
Rinhom RIS (I, S)
RL rij rH rw SW
TABLE
relaxation rate matrix [5.14] contribution to transverse relaxation from chemical exchange relaxation rate constant for spin I in a tilted rotating frame [5.138] spin-lattice or longitudinal relaxation rate constant [1.24] rotating-frame autorelaxation rate constant spin-spin or transverse relaxation rate constant [1.26] inhomogeneously broadened transverse relaxation rate constant relaxation rate constant for heteronuclear multiple quantum coherence [7.9] cross rate constant [5.21] rotation matrix for rotation around the w axis by an angle [1.34], [2.102] inhomogeneous line broadening constant cross-relaxation rate constant for spins I and S in a tilted rotating frame [5.140] leakage rate constant [5.21] distance between spins i and i hydrodynamic radius [1.44] thickness of hydration layer [1.45] spectral width [3.9]
Sc(t1,t2)
OF
SYMBOLS
cosine-modulated signal [4.44] Sm order parameter [2.319], [2.321] SN(t1,t2) N-type signal [4.47] SP(t1,t2) P-type signal [4.46] Ss(t1,t2) sine-modulated signal [4.45] S2 square of generalized order parameter [5.103] S/N signal-to-noise ratio S(!), S( ) frequency domain signal se(t) signal envelope function s(t) time-domain signal T inversion operator [2.110] Tr{A} trace of A T1 ¼ 1/R1 spin-lattice or longitudinal time constant T2 ¼ 1/R2 spin-spin or transverse relaxation time constant t sampling interval [3.6] t0 initial sampling delay [3.27] tmax maximum acquisition time for 1D NMR t1, t2, . . . direct acquisition time or indirect evolution time t1,max, maximum value of t1, t2,max, . . . t2, . . . U unitary operator, matrix or propagator u vector u(!) imaginary part of complex frequency domain spectrum [1.43] V voltage [3.180] volume V specific volume [1.45] v vector v(!) real part of complex frequency domain spectrum [1.42]
TABLE
OF
823
SYMBOLS
W0, WI, transition frequencies WS, W2 [5.10], [5.118] W(, , ) potential of mean force [2.314] Yq2 [(t)] modified spherical harmonic function (Table 5.1) Z complex impedence [3.2] pulse rotation angle [1.23] ji spin state with m ¼ 1/2 {, , } Euler angles Ernst angle [3.189] ae spin state with m ¼ –1/2 magnetogyric ratio [1.3] !^ relaxation superoperator [5.68] rs rate constant for crossrelaxation between basis operators Br and Bs [5.73] FWHH full-width-at-half-height linewidth in Hertz ! resonance frequency difference between chemical species in twosite exchange process !FWHH full-width-at-half-height linewidth
chemical shift (ppm) [1.51]
ij Kronecker delta function [2.16] asymmetry of the shielding tensor [1.50] IS NOE enhancement [5.150] w viscosity of the solvent [1.44] xy transverse interference relaxation rate constant [5.145]
z
0 1 {!} l x , y , z 0 I
r
(t),
eq
NOE
IS , IS
ROE
IS
longitudinal interference relaxation rate constant [5.144] tilt angle [1.21] or strong coupling parameter [2.157] zero-order phase correction first-order phase correction frequency-dependent phase correction nuclear magnetic moment vector [1.3] Cartesian components of the magnetic moment [1.3], [2.28]-[2.30] permeability of free space frequency in units of Hertz spin-lattice relaxation rate constant for spin I [5.10] isotropic nuclear shielding [1.48] Cartesian nuclear shielding tensor [2.299] density operator [2.46] nuclear shielding anisotropy [1.49] equilibrium spin density operator or matrix [2.123] cross-relaxation rate constant between spins I and S [5.10], [5.150] rotating frame crossrelaxation rate constant for spins I and S [5.141]
824
ij
k ¼ zz
? ¼ ð xx þ
yy Þ=2 c
e m p RD {, } (t) (t) (),
*
TABLE
components of nuclear shielding tensor [1.46], [2.299] parallel component of nuclear shielding tensor perpendicular component of nuclear shielding tensor rotational correlation time of a molecule [1.44] effective correlation time for internal motions mixing time pulse length radiation damping time constant [3.162] backbone dihedral angles time-dependent phase of wavefunction [2.6] wavefunction or state function [2.1] basis function, eigenfunction, or stationary state function complex conjugate of offset or chemical shift
:
:(t) ¼ {(t), (t)} s w1 u ! !e !0 !1 !r
!NR !pivot !rf
OF
SYMBOLS
diagonal matrix of chemical shift frequencies diji polar angles
magnetic susceptibility tensor [2.329] side-chain dihedral angle angular velocity frequency in angular units, s–1 effective frequency in rotating frame Larmor resonance frequency [1.14] rf magnetic field strength effective angular frequency in rotating frame [1.22] non-resonance frequency shift [3.86] pivot frequency for phase correction rf magnetic field frequency relative resonance frequency for an X spin [3.194]
LIST OF F IGURES
Figure 1.1. Figure 1.2. Figure Figure Figure Figure Figure Figure Figure Figure
1.3. 1.4. 1.5. 1.6. 1.7. 2.1. 2.2. 2.3.
Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure
2.4. 2.5. 2.6. 2.7 2.8. 3.1. 3.2. 3.3. 3.4. 3.5.
Angular momentum Orientations of magnetic fields in the rotating reference frame Effect of an applied rf field on magnetization Effects of on-resonance pulses One-pulse NMR experiment Resonance linewidths Energy levels for AX spin system Multiple quantum transitions for an IS spin system Spin states and spectrum for a three-spin IRS system Geometrical representations of rotations in an operator space Single-transition basis operators for an IS spin system Operator rotations Transformations of product operators Energy-level diagram for S3CT pulse sequence Coordinate transformations by Euler angles Block diagram of an NMR spectrometer Diagram of a superconducting magnet Probe assembly Nyquist theorem Experimental scheme for quadrature detection
825
826
LIST
Figure Figure Figure Figure Figure
3.6. 3.7. 3.8. 3.9. 3.10.
Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure
3.11. 3.12. 3.13. 3.14. 3.15. 3.16. 3.17. 3.18. 3.19. 3.20. 3.21. 3.22. 3.23. 3.24. 3.25. 3.26. 3.27.
Figure 3.28. Figure 3.29. Figure 3.30. Figure 3.31. Figure 3.32. Figure 3.33 Figure 3.34 Figure 3.35. Figure 3.36. Figure 3.37. Figure 3.38.
OF
FIGURES
Illustration of quadrature detection Adsorptive and dispersive Lorentzian lineshapes Window functions for apodization Digital resolution enhancement Digital resolution enhancement of ubiquitin 1H NMR spectrum Phase dependence of lineshapes Phase corrections Baseline distortions from phase corrections Baseline distortions from a corrupt FID Linear prediction using the HSVD algorithm Linear prediction of an HSQC spectrum Maximum entropy reconstruction of an HSQC spectrum Off-resonance effects for 908 and 1808 pulses Resonance lineshapes for off-resonance 908 pulses Excitation profiles for 1808 pulses Selective pulse shapes Selective 908 pulses Selective 1808 pulses Phase-modulated pulse Vector representations of phase-modulated pulses Magnetization trajectories during an adiabatic pulse Pulse shape and excitation profile for a WURST-20 adiabatic sweep Scalar coupling scaling factor for WALTZ decoupling sequences Scalar coupling scaling factor for GARP-1 and SUSAN-1 decoupling sequences Scalar coupling scaling factor for MPF-7 and MPF-9 decoupling sequences Scalar coupling scaling factors for WURST-40 decoupling sequence Cycling sidebands for WALTZ-16, GARP-1, and SUSAN-1 decoupling sequences Cycling sidebands for MPF-7 and MPF-9 decoupling sequences Cycling sidebands for WURST decoupling sequences Dephasing of transverse magnetization by pulsed field gradients Intrinsic backbone HN exchange rates Binomial excitation profiles Water suppression using spin lock purge pulses
LIST
OF
FIGURES
Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure
3.39. 3.40. 3.41. 3.42. 3.43. 3.44. 3.45. 3.46. 3.47. 3.48.
Figure Figure Figure Figure Figure Figure
3.49. 3.50. 3.51. 3.52. 3.53. 3.54.
Figure Figure Figure Figure Figure
4.1. 4.2. 4.3. 4.4. 4.5.
Figure Figure Figure Figure Figure Figure
4.6. 4.7. 4.8. 4.9. 4.10. 4.11.
Figure 4.12. Figure 4.13. Figure Figure Figure Figure
4.14. 4.15. 4.16. 4.17.
Figure 4.18.
827
Water suppression using field gradient pulses Excitation profiles for water-selective pulses Experimental examples of water suppression techniques Post-acquisition solvent suppression by low-pass filtration Tuning circuits Shimming protocol Gradient image of transmitter/receiver coil volume Pulse sequences for z-axis gradient shimming Typical z-axis shim maps Pulse sequences for indirect calibration of heteronuclear pulse lengths Ernst angle 1 H inversion recovery spectra of ubiquitin Pulse sequence for 1D jump-return Hahn echo experiment Estimating 1H R2 relaxation rate constants Hahn-echo 1H NMR spectra Pulse-acquire and Hahn-echo 1H NMR spectra of ubiquitin 1 H NMR spectra of a hexapeptide and ubiquitin General scheme for two-dimensional NMR spectroscopy Dependence of the NMR signal on the t1 evolution period Schematic two-dimensional NMR spectrum for two spins Coherence transfer pathway for a double quantum experiment Pulse phase Receiver phase Pulse and receiver phase Selection of double quantum coherence Vectorial picture of phase cycling Two-pulse segment used to generate zero quantum coherence EXORCYCLE Selection of a coherence transfer pathway by pulsed field gradients Frequency discrimination in the F1 dimension Comparison of absorptive and phase twisted lineshapes Folding and aliasing in the F1 dimension Schematic generation of a three-dimensional NMR experiment Development of a three-dimensional data set from a two-dimensional data set
828
LIST
OF
FIGURES
Figure 4.19. Schematic representation of a four-dimensional NMR experiment Figure 4.20. Visualization of a four-dimensional experiment Figure 5.1. Transitions and associated rate constants for a two-spin system Figure 5.2. Time dependence of matrix coefficients for longitudinal relaxation Figure 5.3. Magnetization decays for inversion recovery experiments Figure 5.4. Redfield kite Figure 5.5. Relative orientations of laboratory and tilted reference frames Figure 5.6. Spectral density functions for an isotropic rotor Figure 5.7. Transitions and associated eigenfrequencies for a two-spin system Figure 5.8. Relaxation rate constants for 1H-15N dipolar spin system Figure 5.9. Population transfer due to chemical exchange Figure 5.10. Chemical exchange for a two-site system Figure 6.1. One-dimensional 1H spectrum of ubiquitin Figure 6.2. Pulse sequence and coherence level diagram for the COSY experiment Figure 6.3. Variation in the peak height of an anti-phase absorptive doublet Figure 6.4. Sections of the 1HN-1H region of COSY spectra for ubiquitin Figure 6.5. Effect of t1 window functions on COSY spectra Figure 6.6. Five fingerprint regions of the COSY spectrum of ubiquitin Figure 6.7. Leucine, valine and isoleucine methyl fingerprint region of the COSY spectrum of ubiquitin Figure 6.8. Alanine and threonine methyl fingerprint region of the COSY spectrum of ubiquitin Figure 6.9. 1HN-1H fingerprint region of the COSY spectrum of ubiquitin Figure 6.10. Linewidth dependence of a Lorentzian anti-phase absorptive doublet lineshape Figure 6.11. Linewidth dependence of peak height of a Lorentzian anti-phase absorptive doublet Figure 6.12. Linewidth dependence of peak separation for anti-phase absorptive Lorentzian doublet Figure 6.13. Measurement of the peak separation for absorptive and dispersive anti-phase Lorentzian doublet
LIST
OF
FIGURES
829
Figure 6.14. Cross-peaks for Gly10 in the COSY-35 spectrum of ubiquitin Figure 6.15. Two sections of the pre-TOCSY COSY spectrum of ubiquitin Figure 6.16. Pulse sequence and coherence level diagram for the R.COSY experiment Figure 6.17. R.COSY spectrum of ubiquitin Figure 6.18. Pulse sequence and coherence level diagram for the DR.COSY experiment Figure 6.19. DR.COSY spectrum of ubiquitin Figure 6.20. Pulse sequence and coherence level diagram for 2QF-COSY experiment Figure 6.21. Pulse sequence and coherence level diagram for gradientenhanced 2QF-COSY experiment Figure 6.22. 2QF-COSY spectrum of ubiquitin acquired with a short recycle delay Figure 6.23. Cross-peaks obscured by ‘‘double diagonal’’ artifacts Figure 6.24. Comparison of the S/N ratio in COSY and 2QF-COSY spectra Figure 6.25. Comparison of cross-peaks close to the diagonal in COSY and 2QF-COSY Figure 6.26. Gradient-enhanced 2QF-COSY spectrum of ubiquitin Figure 6.27. Methyl fingerprint region in 2QF and 3QF COSY spectra of ubiquitin Figure 6.28. 1H-1H cross-peaks in 2QF-COSY and 3QF-COSY spectra of ubiquitin 0 00 Figure 6.29. 1H-1H and 1H-1H cross-peaks of Glu24 in the 3QF-COSY spectrum of ubiquitin Figure 6.30. Schematic view of partially overlapped cross-peaks in 2QF- and 3QF-COSY spectra Figure 6.31. Schematic view of cross-peaks for a three-spin system in 2QF-COSY, 3QF-COSY, and E.COSY spectra Figure 6.32. Section of the E.COSY spectrum of ubiquitin Figure 6.33. Comparison of the COSY-35 and E.COSY spectra of ubiquitin Figure 6.34. Pulse sequence and coherence level diagram for 2Q and 3Q experiments Figure 6.35. Example of the appearance of 2Q cross-peaks arising from two-spin systems Figure 6.36. Lineshapes of single and double anti-phase Lorentzian lines
830
LIST
OF
FIGURES
Figure 6.37. Contributions of initial I1 magnetization to peaks observed in the 2Q spectrum of a three-spin system Figure 6.38. Schematic representation of the peaks expected in the 2Q spectrum of a 1HN-1H-1H2 spin system Figure 6.39. 1HN-1H region of the 2Q spectrum of ubiquitin Figure 6.40. 2Q spectrum of ubiquitin showing remote glycine peaks 0 00 at F1 ¼ 1H þ 1H Figure 6.41. Remote and direct peaks for CH-CH2 fragments in the 2Q spectrum of ubiquitin Figure 6.42. Cross-peaks between 1H of Glu64 and its side-chain resonances in 2Q, TOCSY, and 2QF-COSY spectra of ubiquitin Figure 6.43. Spectral simplification achieved for three coupled spins in a 3Q spectrum of ubiquitin Figure 6.44. Section of the 3Q spectrum of ubiquitin showing aromatic side-chain resonances Figure 6.45. Pulse sequence and coherence level diagram for the TOCSY experiment Figure 6.46. Pulse sequence and coherence level diagram for a TOCSY experiment using excitation sculpting Figure 6.47. Efficiency of coherence transfer between two scalar coupled spins under the influence of different HartmannHahn mixing schemes Figure 6.48. TOCSY cross-peak intensity as a function of isotropic mixing time Figure 6.49. TOCSY spectra of ubiquitin acquired with different mixing times Figure 6.50. Rapid acquisition of TOCSY spectra for ubiquitin Figure 6.51. Pulse sequence and coherence level diagram for the NOESY experiment Figure 6.52. Pulse sequence and coherence level diagram for the NOESY experiment using excitation sculpting Figure 6.53. NOESY spectra of ubiquitin acquired with different mixing times Figure 6.54. Pulse sequence and coherence level diagram for the JR-NOESY experiment Figure 6.55. NOESY spectra of ubiquitin with presaturation and JR water suppression Figure 6.56. NOESY spectra of ubiquitin with excitation-sculpting water suppression Figure 6.57. Pulse sequence and coherence level diagram for the relayed NOESY experiment
LIST
OF
FIGURES
831
Figure 6.58. Comparison of NOESY and relayed NOESY spectra of ubiquitin Figure 6.59. Pulse sequences and coherence level diagram for ROESY experiments Figure 6.60. ROESY spectra of ubiquitin Figure 6.61. Pulse sequence and coherence level diagram for a ROESY experiment using excitation sculpting Figure 6.62. Pulse sequence and coherence level diagram for a homonuclear 3D NOESY-TOCSY experiment Figure 7.1. Pulse sequences for HMQC, HSQC and constant-time HSQC experiments Figure 7.2. 1H-15N HMQC, HSQC and constant-time HSQC spectra of 15N-labeled ubiquitin Figure 7.3. Interferograms and lineshapes for 1H-15N HMQC, HSQC and constant-time HSQC spectra Figure 7.4. Schematic illustration of halving the F1 spectral width in a 1H-13C heteronuclear correlation experiment Figure 7.5. 1H-13C HSQC spectra of uniformly 15N/13C-labeled ubiquitin Figure 7.6. Pulse sequences for decoupled HSQC, sensitivityenhanced PEP HSQC, and TROSY experiments Figure 7.7. Refocused INEPT coherence transfer functions for InS spin systems Figure 7.8. Energy-level representation of reverse polarization transfers for decoupled HSQC, TROSY and PEP-HSQC experiments Figure 7.9. 1H-15N decoupled HSQC, PEP HSQC, and TROSY spectra of 15N-labeled ubiquitin Figure 7.10. 1H-15N multiplet structure for calbindin D28k Figure 7.11. Pulse sequences for fast HSQC and TROSY experiments incorporating water-flipback solvent suppression Figure 7.12. Pulse sequences for PFG-PEP-HSQC and PFG-TROSY experiments Figure 7.13. Pulse sequences for the 1H-13C constant-time HSQC experiment Figure 7.14. 1H-13C HSQC and constant-time HSQC spectra of 15 N/13C-labeled ubiquitin Figure 7.15. Expansions of the 1H-13C HSQC and constant-time HSQC spectra of 15N/13C-labeled ubiquitin Figure 7.16. Pulse sequence for the 3D 1H-15N NOESY-HSQC experiment
832
LIST
OF
FIGURES
Figure 7.17. Schematic illustration showing the relationship between a 3D heteronuclear-edited spectrum and 2D homonuclear and heteronuclear correlation spectra Figure 7.18. Selected F1(1H)-F3(1HN) regions from a 3D 1H-15N NOESY-HSQC spectrum of 15N-labeled ubiquitin Figure 7.19. Selected F1(1H)-F3(1H) regions from a 3D 1H-13C NOESY-HSQC spectrum of 13C-labeled ubiquitin Figure 7.20. Pulse sequence for the 3D 1H-15N TOCSY-HSQC experiment Figure 7.21 1H-15N TOCSY-HSQC spectrum of 15N-labeled ubiquitin Figure 7.22 Pulse sequence for the HSQC-NOESY experiment Figure 7.23. Pulse sequence for the 3D 1H-15N HMQC-NOESYHMQC experiment Figure 7.24. Pulse sequence for the 4D 13C/15N HMQC-NOESYHMQC experiment Figure 7.25. Pulse sequence for the 4D 13C/13C HMQC-NOESYHMQC experiment Figure 7.26. Pulse sequences for 3D HCCH-COSY experiments Figure 7.27. Selected F1(1H)-F3(1H) slice from a constant-time HCCHCOSY spectrum of 13C-labeled ubiquitin Figure 7.28. Pulse sequence for the 3D HCCH-TOCSY experiment Figure 7.29. Isotropic mixing for 13C spins in isoleucine residues Figure 7.30. Selected regions from F2(13C) slices of the 3D HCCHTOCSY spectrum of 13C-labeled ubiquitin Figure 7.31. Pulse sequences for the HNCA experiment Figure 7.32. Plots of the HNCA coherence transfer functions, k(T) Figure 7.33. Plots of the HNCA coherence transfer functions, k( )cos(pJNH ) Figure 7.34. Selected F2(13C)-F3(1HN) slice from an HNCA spectrum of ubiquitin Figure 7.35. Pulse sequence for the constant-time HN(CO)CA experiment Figure 7.36. Selected F2(13C)-F3(1HN) slice from a HN(CO)CA spectrum of ubiquitin Figure 7.37. Pulse sequence for the constant-time H(CA)NH experiment Figure 7.38. Plots of the H(CA)NH coherence transfer function, (2 2) Figure 7.39. Selected F1(1H)-F3(1HN) slice from an H(CA)NH spectrum of ubiquitin Figure 7.40. Pulse sequence for the constant-time HNCO experiment
LIST
OF
FIGURES
833
Figure 7.41. Pulse sequence for the constant-time HN(CA)CO experiment Figure 7.42. Pulse sequence for the CBCA(CO)NH experiment Figure 7.43. CBCA(CO)NH spectrum of ubiquitin Figure 7.44. Pulse sequence for the CBCANH experiment Figure 7.45. Plots of the CBCANH coherence transfer functions Figure 7.46. CBCANH spectrum of ubiquitin Figure 7.47. Pulse sequence for the HNCACB experiment Figure 7.48. HNCACB spectrum of ubiquitin Figure 7.49. Pulse sequence for the HNCA-J experiment Figure 7.50. Selected F1(13C)-F3(1HN) slice from a HNCA-J spectrum of ubiquitin Figure 7.51. Pulse sequence for the HNHA experiment Figure 7.52. Effects of relaxation on the HNHA experiment Figure 7.53. HNHA spectrum of ubiquitin Figure 7.54. Residual dipole couplings in ubiquitin Figure 7.55. Pulse sequence for the IPAP 15N-1H HSQC experiment Figure 7.56. Pulse sequence for the CE-TROSY experiment incorporating J-scaling Figure 8.1. Pulse sequences for 15N R1, R2, R1 and NOE relaxation measurements 15 Figure 8.2 N relaxation data for ubiquitin 15 Figure 8.3 N relaxation rate constants for ubiquitin Figure 8.4. 15N CSA/1H-15N dipolar relaxation interference Figure 8.5. Pulse sequences for 2H R1 and R1 relaxation measurements in CH2D spin systems Figure 8.6. Pulse sequences for 2H quadrupolar order and antiphase coherence relaxation measurements in CH2D spin systems Figure 8.7. 2H relaxation rate constants for CH2D methyl groups in ubiquitin Figure 8.8. Pulse sequences for 13CO R1 and {13C}-13CO NOE relaxation measurements Figure 8.9 Pulse sequence for the in-phase Hahn-echo experiment Figure 8.10. Values of 15N Rex for ubiquitin Figure 8.11. Pulse sequence for the X nucleus ZZ-exchange experiment Figure 8.12. Pulse sequences for 15N R1 relaxation measurements Figure 8.13. 15N R1 relaxation dispersion for residue Asn 25 in ubiquitin Figure 8.14. Pulse sequences for relaxation-compensated CPMG experiments Figure 8.15. Pulse sequence for the 1HN CPMG experiment in 15N-2H labeled proteins
834
LIST
OF
FIGURES
Figure 8.16. Multiple quantum relaxation in ubiquitin Figure 8.17. TROSY-based methods for detecting 15N Rex in large, deuterated proteins Figure 8.18. Pulse sequence for the 15N TROSY-CPMG experiment Figure 9.1. T2 relaxation times for fully protonated and perdeuterated proteins Figure 9.2. Transverse and longitudinal relaxation times as a function of deuteration level in proteins Figure 9.3. Pulse sequence for the 3D constant-time HNCA experiment Figure 9.4. Pulse sequence for the 3D HN(CA)CB experiment Figure 9.5. Pulse sequence for the 3D gradient-enhanced, sensitivityenhanced CC(CO)NH-TOCSY experiment Figure 9.6. 3D CC(CO)NH-TOCSY spectrum of perdeuterated 2 H/13C/15N-labeled calbindin D28K Figure 9.7. Pulse sequence for the gradient-enhanced, sensitivityenhanced 4D 15N/15N-separated NOESY experiment Figure 9.8. 4D 15N/15N-separated NOESY spectrum of ILV-protonated, otherwise perdeuterated, calbindin D28K Figure 9.9. Pulse sequence for the gradient-enhanced, sensitivityenhanced 4D 13C/15N separated NOESY experiment Figure 9.10. 3D 13C/13C-separated NOESY, 4D 13C/15N-separated NOESY, and 4D 15N/15N-separated NOESY spectra of ILV-protonated, otherwise perdeuterated, calbindin D28K Figure 9.11. Ligand-induced contributions to chemical exchange linebroadening Figure 9.12. TROSY pulse sequence for using cross-saturation to identify protein-ligand interfaces Figure 9.13. Basic pulse sequence for the 13C double-half-filtered NOESY experiment Figure 9.14. Projection reconstruction NMR spectra of 13C/15Nlabeled calbindin D28K Figure 10.1. 1H chemical shift ranges for ubiquitin Figure 10.2. Aliphatic and non-aliphatic side chain 1H chemical shifts Figure 10.3. Assignment methods for unlabeled proteins. Figure 10.4. Categorization of spin-systems using chemical shifts and spin topology Figure 10.5. Backbone-backbone NOEs for sequence-specific 1H resonance assignments Figure 10.6. Summary of sequential NOEs observed for a portion of ubiquitin Figure 10.7. Aliphatic and aromatic 13C chemical shifts
LIST
OF
FIGURES
835
Figure 10.8. Karplus curves describing the variation of 3 JCOCO , 3 JCOH , 3 JCOC , 3 JHN CO , 3 JHN H , and 3 JHN C with backbone dihedral angle Figure 10.9. Karplus curve describing the variation of 3 JH N with backbone dihedral angle Figure 10.10. Karplus curves describing the variation of 3 JH H2 and 3 JH H3 Figure 10.11. Definition of the global fold using long-range NOEs Figure 10.12. NMR-derived structure of ubiquitin Figure 10.13. Residual dipolar coupling constants for ubiquitin Figure 10.14. Backbone 15N generalized order parameters for ubiquitin
LIST OF TABLES
Table 1.1. Table 2.1. Table 2.2. Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table Table
2.3. 2.4. 3.1. 3.2. 4.1 4.2 4.3 4.4 4.5 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8.
Properties of selected nuclei. Rotation properties of angular momentum operators Product operators in the Cartesian basis for a two-spin-1/2 system Product operators in the shift basis for a two-spin system 2 Reduced rotation matrices dmn ðÞ Shim coil spherical harmonic functions Indirect chemical shift references Selection of double-quantum coherence Rejection of zero-quantum coherence Distinguishing p ¼ þ1 from p ¼ –3 Rejecting p ¼ –3 Quadrature detection methods Modified second order spherical harmonics Spatial functions for relaxation mechanisms Tensor operators for the dipolar interaction Commutator relationships Relaxation rate constants for IS dipolar interaction Tensor operators for the CSA interaction Tensor operators for the spin-1 quadrupolar interaction CSA relaxation rate constants
837
838
LIST
Table 5.9. Table 6.1. Table 6.2. Table 7.1. Table 7.2. Table 7.3 Table 9.1. Table 9.2. Table Table Table Table
10.1. 10.2. 10.3. 10.4.
OF
TABLES
Relaxation rate constants for the spin-1 quadrupolar interaction Phase cycles for the E.COSY experiment Lineshapes in a 2Q spectrum Processing 4D 13C/15N and 13C/13C-edited NOESY data sets Triple resonance experiments used for sequential resonance assignment 3 JHN H scalar coupling constants Deuterium isotope shifts for 13C and 13C nuclei Processing schemes and information content for doublehalf-filtered NOESY Parameterization of the Karplus equation for 3 JHN H Parameterization of the Karplus equation for Parameterization of the Karplus equation for 3 JH N Parameterization of the Karplus equation for w1
SUGGESTED R EADING
The literature concerning NMR spectroscopy contains a number of texts that amplify and complement the present text. A selection of these sources is listed below.
Biomolecular NMR Spectroscopy Bertini, I. and Luchinat, C., ‘‘NMR of paramagnetic molecules in biological systems,’’ pp. 1–319, Benjamin/Cummings, Menlo Park, CA, 1986. Clore, G. M. and Gronenborn, A. M. (Eds.), ‘‘NMR of proteins,’’ pp. 1–307, CRC Press, Boca Raton, 1993. Croasmun, W. R. and Carlson, R. M. K. (Eds.), ‘‘Two-dimensional NMR spectroscopy. Applications for chemists and biochemists,’’ 2nd edition, pp. 1–958, VCH Publishers, New York, 1994. Downing, A.K. (Ed.), ‘‘Protein NMR Techniques,’’ 2nd edition, pp. 1–487, Humana Press, Totowa, 2004. (Methods in Molecular Biology, Vol. 278). Evans, J. N. S., ‘‘Biomolecular NMR Spectroscopy’’, pp. 1–464, Oxford University Press, Oxford, 1995. James, T. L. and Oppenheimer, N. J. (Eds.), Methods in Enzymology, 239, 1–813 (1994). James, T. L., Do¨tsch, V., and Schmitz, U. (Eds.), Methods in Enzymology, 338, 1–480 (2001). James, T. L., Do¨tsch, V., and Schmitz, U. (Eds.), Methods in Enzymology, 339, 1–454 (2001). James, T. L., Do¨tsch, V., and Schmitz, U. (Eds.), Methods in Enzymology, 394, 1–672 (2005). Roberts, G. C. K. (Ed.), ‘‘NMR of Macromolecules,’’ pp. 1–399, IRL Press, Oxford, 1993. Rule, G. S. and Hitchens, T. K., ‘‘Fundamentals of Protein NMR Spectroscopy,’’ pp. 1–530, Springer, Dordrecht, 2006. Wu¨thrich, K., ‘‘NMR of Proteins and Nucleic Acids,’’ pp. 1–292, Wiley, New York, 1986.
839
840
SUGGESTED READING
NMR Spectroscopy Abragam, A., ‘‘Principles of Nuclear Magnetism,’’ pp. 1–599, Clarendon Press, Oxford, 1961. Canet, D., ‘‘Nuclear magnetic resonance. Concepts and methods,’’ pp. 1–260, John Wiley & Sons, Chichester, 1996. Corio, P. L., ‘‘Structure of high-resolution NMR spectra,’’ pp. 1–548, Academic Press, New York, 1967. Ernst, R. R., Bodenhausen, G., and Wokaun, A., ‘‘Principles of nuclear magnetic resonance in one and two dimensions,’’ pp. 1–610, Clarendon Press, Oxford, 1987. Farrar, T. C. and Harriman, J. E., ‘‘Density matrix theory and its applications in NMR spectroscopy,’’ pp. 1–211, Farragut Press, Madison, WI, 1992. Freeman, R., ‘‘Spin Choreography. Basic steps in high resolution NMR,’’ pp. 1–391, Oxford University Press, Oxford, 1998. Goldman, M., ‘‘Quantum description of high-resolution NMR in liquids,’’ pp. 1–268, Clarendon Press, New York, 1988. Grant, D. M. and Harris, R. K., ‘‘Encyclopedia of Nuclear Magnetic Resonance’’, pp. 1–6490, Wiley and Sons, Chichester, 2002. Harris, R. K., ‘‘Nuclear Magnetic Resonance Spectroscopy,’’ pp. 1–260, Longman Scientific and Technical, Harlow, 1986. Hoch, J. C. and Stern, A. S., ‘‘NMR data processing,’’ pp. 1–196, Wiley-Liss, New York, 1996. Hore, P. J., ‘‘Nuclear magnetic resonance,’’ pp. 1–90, Oxford University Press, Oxford, 1995. Hore, P. J., Jones, J. A. and Wimperis, S., ‘‘NMR: The toolkit,’’ pp. 1–85, Oxford University Press, Oxford, 2000. Keeler, J., ‘‘Understanding NMR Spectroscopy’’, pp. 1–459, Wiley and Sons, Chichester, 2005. Kimmich, R., ‘‘NMR Tomography, Diffusometry and Relaxometry,’’ pp. 1–526, SpringerVerlag, Berlin, 1997. In particular, Part IV: Analytical NMR Toolbox, pp. 395–496. Kowalewski, J. and Ma¨ler, L., ‘‘Nuclear Spin Relaxation in Liquids: Theory, Experiments, and Applications,’’ pp. 1–426, Taylor & Francis, Boca Raton, 2006. Levitt, M., ‘‘Spin Dynamics’’, pp. 1–686, Wiley and Sons, Chichester, 2001. Mateescu, G. D. and Valeriu, A., ‘‘2D NMR. Density matrix and product operator treatment,’’ pp. 1–195, Prentice-Hall, Englewood Cliffs, 1993. Munowitz, M., ‘‘Coherence and NMR,’’ pp. 1–289, Wiley and Sons, New York, 1988. Mehring, M., ‘‘Principles of high resolution NMR in solids,’’ 2nd edition, pp. 1–342, Springer-Verlag, Berlin, 1983. Neuhaus, D. and Williamson, M. P., ‘‘The nuclear Overhauser effect in structural and conformational analysis,’’ 2nd edition, pp. 1–619, Wiley-VCH, New York, 2000. Slichter, C. P., ‘‘Principles of Magnetic Resonance,’’ pp. 1–655, Springer-Verlag, Berlin, 1990. Van de Ven, F. J., ‘‘ Multidimensional NMR in liquids. Basic Principles and experimental methods’’, pp. 1–399, Wiley-VCH, Weinheim, 1995.
Quantum Mechanics Blum, K., ‘‘Density matrix theory and applications,’’ pp. 1–217, Plenum Press, New York, 1981. Levine, I. N., ‘‘Quantum Chemistry,’’ pp. 1–566, Allyn and Bacon, Boston, 1983. McQuarrie, D. A., ‘‘Quantum Chemistry,’’ pp. 1–517, University Science Books, Mill Valley, CA, 1983. Merzbacher, E., ‘‘Quantum Mechanics,’’ pp. 1–621, Wiley & Sons, New York, 1970.
I NDEX
1–9 1 binomial sequence, 224–226, 1-3-3227f 1808 () pulse, 16, 17f. See also Adiabatic pulse Bloch-Siegert shift for, 171 catenation of, 607, 641 composite, 174–179, 178f, 203–205 decoupling by, 94–96, 203–204 excitation null for, 170 excitation profile for, 167f, 178f EXORCYCLE for, 266, 309, 313 Gaussian, 182f, 184f imperfect, 176, 178f, 309, 547 inversion by, 16, 17f, 167f, 169–170 off-resonance effects for, 167f, 169, 178f nonresonant phase shift for, 171 pulsed field gradients and, 313–314 R-SNOB, 182f, 184f, 186 RE-BURP, 182f, 184f, 186 refocusing by, 93–94, 265–266, 703 selective, 181, 182f, 184f, 186 sinc, 182f, 184f spin-echo and, 93–94
1D (One-dimensional) 1H NMR, 263–267, 406–409 excitation sculpting, 229f, 230, 233f inversion-recovery, 258–259, 260f jump-return, 224–225, 227f, 233f, 261f one-pulse acquire, 263, 265, 267f peptide, of, 272f spin-echo, Hahn, 259, 228f, 229f, 261f, 265–267, 267f, 407f, 783f ubiquitin, of, 150f, 233f, 235f, 260f, 261f, 267f, 272f, 407f, 783f ubiquitin, denatured, of, 407f water-flipback, 229f, 231–232 WATERGATE, 229f, 230 2D (Two-dimensional) NMR, 273–280. See also individual NMR experiments 3D NMR, compared with, 327–229, 525–526 2D spectrum, schematic, 277 2Q. See Double-quantum coherence 2Q (Double-quantum) NMR, 465–481 cosy, comparison with, 475–476 cross-peak fine structure in, 466–471 direct peaks in, 469 experimental protocol for, 473–474
841
842 2Q (Double-quantum) NMR (Continued ) 1 a 1 b H - H fingerprint region in, 478, 479f lineshapes for, 467–471, 470f, 471f, 472t processing of, 474 product operator analysis for, 466–473 pulse sequence for, 465f remote peaks in, 469, 476–477, 478f three-spin system, of, 467–468 two-spin system, of, 466–467, 467f ubiquitin, of, 467f, 477f, 478f, 479f, 480f 2QF-COSY (Double-quantum filtered COSY), 437–449 aliasing and folding in, 444 artifact suppression in, 444–445, 445f cross-peak fine structure in, 441–444, 456f experimental protocol for, 444–445 fingerprint regions in, 418–419, 419f, 446 lineshapes in, 440, 444 processing of, 445–446 product operator analysis for, 440–440 pulse sequence for, 437f, 439f S/N ratio and, 447–448, 448f three-spin system, of, 442–444 two-spin system, of, 440–442 ubiquitin, of, 445f, 446f, 449f, 450f, 453f, 454f, 462f, 480f 3D (Three-dimensional) NMR, 525, 581. See also individual NMR experiments 2D NMR, compared with, 327–329, 525–526 4D NMR, compared with, 330, 655 deuteration and, 733–734 3D spectrum, schematic, 328f, 329f, 586f 3Q. See Three-quantum coherence 3Q (Three-quantum) NMR, 481–486 experimental protocol for, 483 product operator analysis for, 481–483 processing of, 483
INDEX pulse sequence for, 465f ubiquitin, of, 484f 3QF-COSY, 449–455 cross-peak fine structure in,456f experimental protocol for, 451–452 fingerprint region in 1 a 1 b H - H fingerprint region, 453–455, 454f, 455f methyl, 453f forbidden cross-peaks, 453 product operator analysis for, 449–451 processing of, 452 pulse sequence for, 437f ubiquitin, of, 453f, 454f, 455f, 462f 4D NMR. See also individual NMR experiments 3D NMR, compared with, 330, 655 deuteration and, 733–734 4D spectrum, schematic, 330f 908 (/2) pulse, 16, 17f, 19f excitation null for, 169–170 off-resonance effects for, 166–169, 167f, 168f selective, 183f A AB spin system, 26, 62–64. See also Two-spin system Absorptive antiphase lineshape, 413f, 421, 423f, 424–426, 424f, 425f, 470f COSY and, 411–412 representation in spectra, 406 Absorptive lineshape, 18, 139–140, 140f, 317, 320f Acquisition period, 17, 19f, 56, 124, 271, 275f, 279, 280. See also Indirect evolution period initial delay for, 266 maximum, 144–145, 158 Active nuclei, 2, 3t Active scalar coupling, 91, 282–283, 412 ADC (Analog-to-digital converter), 115f, 123, 127–129, 321 Adiabatic condition, 192 Adiabatic following, 189–190 Adiabatic full-passage (AFP), 199 Adiabatic half-passage (AHP), 199 Adiabatic pulse, 189–201, 709f
843
INDEX CHIRP, 197 dephasing ZQ coherence by, 314 half-filter, in, 768–769 hyperbolic secant, 198 magnetization trajectories during, 195f sech/tanh, 198 WURST, 198–199, 200f Adiabatic isotropic mixing, 491–493, 492f Adiabatic rapid passage, 189–190 Adiabatic relaxation, 336–337, 349–350 Adiabatic spin decoupling, 209–212, 213f, 216f, 758 Adjoint operator, 32 AFP (Adiabatic full passage), 199 Aggregation, 19, 236, 408 AHP (Adiabatic half-passage), 199 Aliasing and folding, 125–126, 323–326, 325f, 550f, 551f 2QF-COSY, in, 444 13 C spectrum, of, 551f, 769 heteronuclear correlation and, 549–552 multidimensional NMR, in, 323–326 15 N spectrum, of, 551f Allowed transition, 25, 25f, 77f Amide proton exchange backbone correlations missing due to, 418 exchange cross-peaks from, 512–513 hydrogen bond restraints from, 805–806 intrinsic rates of, 223–224, 224f line broadening by, 406 measuring, 499–500 perdeuteration and, 727–728 protein-ligand interface and, 760 saturation transfer by, 223, 409, 573, 685 scalar coupling and, 401 Amino acid residue chemical shifts in proteins, 783f-785f, 795f-796f random coil, 23, 407f, 408, 782, 794, 804 scalar coupling constants for, 800–803 Amino acid spin system, 787 1 H chemical shifts, 784f-785f 1 H-1H isotropic mixing in, 487–493, 492f, 495f
HCCH, spin topology, for, 602 C chemical shifts, 795f-796f 13 C-13C isotropic mixing in, 611–612, 611f triple-resonance, spin topology, for, 615t-617t, 641–642 Amplifier, rf, 120, 253 Amplitude modulated signal, 316 Amplitude, rf, 9–10, 120, 171–173 Analog signal, 123, 127–129, 156 Analog-to-digital converter (ADC), 115f, 123, 127–129, 321 Angle Euler, 104, 107, 108f, 804–805 polar, 110, 365, 368, 804–805 Angular momentum, 2–4 See also Magnetic quantum number bulk, 5 intrinsic, 29–30 nuclear spin, 2–3, 3t orientation of, 3–4 quantum mechanical, 29 quantum number, 2 vectors, 4, 4f Angular momentum operator, 46, 80 commutation relationships for, 46 eigenfunctions of, 37, 47 N-spin, 60 Pauli spin matrix for, 46 rotation properties of, 51t, 81f two-spin, Cartesian basis for, 70 Anisotropic chemical shift, 112, 668, 751–752, 810–811. See also Chemical shift anisotropy Anisotropic diffusion. See Diffusion Anisotropic magnetic susceptibility, 110, 666–667 Annealing, simulated, 809 Anticonnected transition, 460 Antiphase lineshape, 89–90, 413, 413f, 421, 423f, 424–426, 424f, 425f, 470f COSY and, 411–412 representation in spectra, 406 Antiphase lineshape, double, 443–444, 453, 463, 469–470, 470f, 472t Apodization, 138, 143–151. See also Apodization function; individual NMR experiments 13 C-13C scalar coupling, removal of, 655 13
844 Apodization (Continued ) constant time NMR, in, 161, 543–546, 545f COSY and, 415–416 examples of, 146f, 149f, 150f, 417f heteronuclear correlation, in, 552 matched filtering, for, 144, 145, 147–148 multidimensional NMR, in, 327 peak integrals influenced by, 148, 151 resolution enhancement, for, 147–148 signal-to-noise ratio, and, 158 truncation artifacts with, 145 Apodization function cos–1(JCCt), 655 cosine bell, 145, 146f Dolph-Chebyshev, 144 exponential, 147, 149f, 150f Hamming, 145, 146f Hanning, 145 Kaiser, 145, 146f, 149f, 150f Lorentzian-to-Gaussian, 147–148, 149f, 150f maximum resolution, 148, 149f sine bell, phase-shifted, 147, 414 Apparent scalar coupling. See Scalar coupling, apparent Aromatic ring, 749 flip of, 369, 480, 810 local magnetic fields for, 408 selective protonation of, 749, 752–753 scalar coupling constants for, 764 Aromatic spin system, 480, 485–486 aliasing and folding for, 444, 483 chemical shifts of, 782, 783f, 785f, 796f decoupling of, 607–608 fingerprint region for, 418, 419f NOE calibration using, 797 Artifact axial peak, 309–310, 323–324 baseline, 130, 154–158, 156f, 157f, 226, 264f, 265, 298, 307–308, 497 breakthrough peak, J-filter, 764 clipped FID, 156–157 dispersive tails, 411–412, 417f double-diagonal, 444–446, 445f phase cycle, 310 pseudo-diagonal, 597
INDEX pulse breakthrough, 161 repetition rate, 258, 310–311, 444–445 ridges, 416, 493, 785 steady-state, 258, 310–311 t1 noise, 262, 310, 556 truncation, 143–148, 146f, 157, 265 water stripe, 416, 477 Artifact suppression, 307–310, 313–314 in heteronuclear correlation, 546–549 phase cycling, by, 307–310, 313, 437f, 546–549 pulsed field gradient, by, 313–314, 577f, 584, 598 spin lock purge pulses, by, 228–230 Artifact suppression, examples of 2QF-COSY, 445f, 446f 4D HMQC-NOESY-HMQC, 598–599 HSQC, 547–548 Assignments, resonance. See Resonance assignments Attenuation, rf power, of, 253 Audiofrequency filter. See Filter, audiofrequency Audiofrequency signal, 132 Autocorrelation function, 138, 348 Autocorrelation peak. See Diagonal peak Automated assignment, 791, 794, 809 Autorelaxation, 339, 340, 357, 359, 362, 375, 381–383, 692. See also Cross-relaxation; Relaxation AX spin system, 25f, 26, 62–64. See also Two-spin system Axial peak, 309–310, 323–324 B B0 field gradient. See Pulsed field gradient B1 field. See rf field Back transfer, 507, 527–528 line, 528 peak, 527 plane, 528 Backbone conformation, 806–807 fingerprint region, 418, 419f resonance assignments for, 785–794, 786t, 789f, 790f
845
INDEX Back-calculation, 810 Backprojection (BP), 775 Baker-Campbell-Hausdorff (BCH) relationship, 43, 45–46 Bandpass filter, 131, 135–136 Bandwidth filter, 126 noise, 126 receiver, 119, 123 signal, 124–125 Baseline correction, 131, 136, 497 Baseline distortion, 130, 154–158, 156f, 157f, 226, 264f, 265, 298, 307–308, 497 phase correction and, 156f, 322, 324–326 Basic pancreatic trypsin inhibitor (BPTI), 241–242 Basis expansion, 32–33, 47, 78–79 Basis ket, 38 Liouville von Neumann equation and, 41 matrix representation and, 47 Basis normalization, 30, 79 Basis operator space, 78–79 Basis operator, 80–83 Cartesian, 70, 71t, 80 shift, 70, 72, 73t single element, 80, 82 single transition, 81, 83f, 100 spherical, 70–74, 73t Basis transformation, 47–50, 64 Bayesian analysis, 159 BCH (Baker-Campbell-Hausdorff relationship), 43, 45–46 Biexponential relaxation, 343–345, 681 Bilinear operator, 91, 287–288 Binomial pulse sequence, 224–227, 227f BioMagResBank, 783, 784f, 794, 795f BIP (Broadband inversion pulse) 179, 197 Bloch equations, 7–16 chemical exchange and, 392–393 derivation of, 7–8, 12–13 free-precession, for, 14, 17 integration of, 347 laboratory reference frame, in, 7–8, 13 Larmor frequency in, 8 limitations of, 23–24 McConnell equations and, 393
pulses in, 9, 11 relaxation and, 337–338 rotating frame, in, 14 rotating frame transformation and, 10 rotation angle and, 15 rotation axis and, 15 Bloch vector models. See Bloch equations Bloch, Wangsness, and Redfield (BWR), 350. See also Relaxation theory Bloch-McConnell equations, 393, 706 Bloch-Siegert shift, 9, 170 Body diagonal, 527 Boltzmann distribution, 5, 12, 333, 335, 350 Boltzmann equation, 5, 55, 107 Boltzmann probability, 5, 55, 107 Bound water, 512–513 BP (Backprojection), 775 BPTI (Basic pancreatic trypsin inhibitor), 241–242 Bra, 37–38, 47, 81 Broadband decoupling. See Decoupling, phase modulated Broadband inversion pulse (BIP), 179, 197 Broadband isotropic mixing. See Isotropic mixing Brownian motion, 365, 366. See also Diffusion Brownian rotational diffusion, 21 Buildup series, 510 Bulk angular momentum, 5, 7 Bulk magnetic moment, 5, 7, 16 Bulk magnetization, 9, 11 1808 pulse and, 16 equilibrium, 12 rf pulse and, 11 vector, 9, 12f detection of, 9 Burg linear prediction, 161 BWR (Bloch, Wangsness, and Redfield), 350. See also Relaxation theory C 13
Ca and 13CO, treated as different nuclear species, 549, 654–655 decoupling, 170–172, 549 nonresonant phase shift, 170–171, 628
846 13
Ca and 13CO (Continued ) rf field strength, adjusting, 169–170, 549 13 a 13 b C - C decoupling, 549 13 C chemical shift. See Chemical shift, 13 C 13 C isotopic labeling, 533, 727 13 C-13C correlation (HCCH), 601–606, 740–743. See also CC(CO)NH-TOCSY HCCH-COSY; HCCH-TOCSY; HCC(CO)NH-TOCSY 13 b 13 a C - C correlation (CBCA), 641–642, 734–735. See also CBCANH; CBCA(CO)NH; HNCACB; HCC(CO)NH-TOCSY; CC(CO)NH-TOCSY Calbindin D28k CC(CO)NH-TOCSY spectrum, 744f description, vi ILV NOESY spectrum, 752f 15 N/15N separated NOESY spectrum, 747f PR HACA(CO)NH spectrum, 774f TROSY, multiplet structure in, 572 Calibration NOESY, of, 506–508,510, 797 pulse width, 252–257 temperature, 236–237 CAMELSPIN (Cross-relaxation appropriate for mini-molecules emulated by locked spins). See ROESY Capacitance, 119 Carrier frequency, 9 Carr-Purcell-Meiboom-Gill (CPMG), 686–699, 688f, 702, 711–715, 712f, 720–721, 721f Cartesian basis, 70, 71t, 80 Cartesian operator, 37, 47, 70, 81f, 82, 90–91 basis, 70, 71t, 80 single-element operators and, 82 Cartesian tensor, 102, 103, 109 Catenation of 1808 pulses, 607, 641 Causality principle, 142 CBCACO(N)H, 641 CBCANH, 617t, 641, 645–650 product operator analysis for, 646–648
INDEX pulse sequence for, 648f ubiquitin, of, 650f CBCA(CO)NH, 616t, 641, 642–645 pulse sequence for, 643f product operator analysis for, 642, 643–645 resonance assignment and, 793 ubiquitin, of, 646 CC(CO)NH-TOCSY, 741 calbindin D28k, of, 744 pulse sequence for, 742f CE-TROSY. See Coupling-enhanced TROSY Characteristic frequency. See Eigenfrequency Chemical exchange, 391–402. 702–721, See also Amide proton exchange Bloch equations for, 392–393 chemical shift time scale in, 392, 399–400, 753–754 coalescence, in, 399, 754 CPMG and, 711 cross-relaxation, similarity to, 397 fast, 392, 399, 401–402,754–755 intermediate, 399, 401, 754–755 intermolecular, 394, 401, 753–754, 755f intramolecular, 393, 401–402 isolated spins, for, 392–400 ligand concentration and, 755f lineshape analysis, for, 704–706 McConnell equations for, 393 MQ NMR, in, 715–718 NOESY and, 397, 511 population transfer and, 395–397, 397f, 706–707 relaxation rate constant for, 398–399, 708, 711, 755f ROESY and, 397, 522 R1 and, 708 scalar coupled spin systems, 392, 401–402 scalar relaxation and, 388 self-decoupling by, 381 slow, 392, 399, 401–402, 754–755 TOCSY, 499 two-site system, 392, 400f, 755f longitudinal magnetization, 394–397
INDEX rate matrix, 393, 395, 397 transverse magnetization, 395, 397–399, 708, 711 755f ZZ-exchange, 706–707 Chemical exchange lifetime, 392, 399–400, 753–754 Chemical kinetics, 392 Chemical reactions, 391–394. See also Chemical exchange Chemical shift, 21–22 anisotropic, 112, 668, 751–752, 810–811 average population weighted, 392, 399 13 C resonances, of assignments, use in, 794 proteins, in, 795f-796f random coil, in, 794 secondary structure dependence, 23, 804 conformation-dependent secondary, 23, 782–783, 804 degeneracy of, 476, 593 2D NMR, alleviating by, 278 3D heteronuclear-edited NMR, alleviating by, 327–328, 329f, 586f 3D homonuclear NMR, alleviating by, 525, 528 4D heteronuclear-edited NMR, alleviating by, 329–330, 330f, 601, 733 triple-resonance NMR, alleviating by, 792 dispersion, 179, 236, 272, 405, 407f, 408–409, 498, 525, 749, 785, 791 ethylene glycol, of, 236 1 H resonances, of proteins, in, 783f-785f random coil, in, 23, 408, 782 secondary structure dependence, 23, 804 Hamiltonian and, 51, 102, 107 isotropic, 22 dihedral angle restraints from, 804 perturbation mapping of, 756–757 measurement of, 23 methanol, of, 236 nuclear shielding and, 22, 102, 107 perdeuteration and, 732
847 referencing of, 23, 262–263 indirect, 263t tensor, 22, 102, 107 time scale, 392, 399–400, 753–754 water, of, 237 Chemical shift anisotropy (CSA), 22, 383–384, 571. See also Anisotropic chemical shift principal axis for, 349–350, 365, 387 relaxation, 349–350, 383–384, 502–506, 680 relaxation interference, 386–387 relaxation rate constants for, 385t spatial functions for, 366t spherical harmonic functions for, 350t tensor operators and, 384t Chemical shift asymmetry, 22 Chemical shift index (CSI), 804 Chemical shift mapping, 756–757 Chemical shift time scale, 392, 399–400, 753–754. See also Chemical exchange CHIRP, 197–198, 210–211 Circular polarization, 9 CITY, 493 Clipped FID, 156–157 Closure theorem, 38 13 CO and 13Ca, treated as different nuclear species, 549, 654–655 decoupling, 170–172, 549 nonresonant phase shift, 170–171, 628 rf field strength, adjusting, 169–170, 549 Coalescence, resonance, of, 399, 754. See also Chemical exchange Cogwheel phase cycling, 307 CO_H(N)CACB, 641 Coherence, 70–77. See also Doublequantum coherence; Singlequantum coherence; Zero-quantum coherence multiple quantum (MQ), 90–92, 473 phase cycling and, 292–295, 298 phase shift of frequency discrimination, for, 320 rf pulse, by, 298 pulsed field gradient (PFG), 311 three-quantum (3Q), 442, 449–551, 481–482 three-spin single-quantum, 450–451
848 Coherence, dephasing of pulsed field gradient, by, 217–221, 311 relaxation, by, 335–336, 346–350 rf inhomogeneity, by, 227–230, 364, 488 static magnetic field inhomogeneity, by, 18 Coherence level. See Coherence order Coherence level diagram, 293–295 Coherence order, 293–295 change in, 294, 298 heteronuclear, 314, 546–547 maximum, 293, 307 relaxation and, 358, 372 signed, 311, 316 Coherence order selective (COS), 560. See also PEP Coherence selection, 292–293. See also individual NMR experiments phase cycle, by, 298–307 pulsed field gradients, by, 311–313, 312f, 438, 574, 576–578 Coherence transfer, 92–93. See also Magnetization transfer; individual NMR experiments continuous (CW) rf field, by, 286–287 COSY-type, 281–284, 290–291 DCOSY, 291–292 heteronuclear, 290–291 INEPT, 96–98 refocused, 98–99, 556f in-phase coherence transfer. See also Magnetization transfer COSY-type, 98–99, 283–284, 605–606, 632–633 DCOSY, 291–292 TOCSY-type, 284–289, 486–496, 608–612 isotropic mixing, by, 287–288, 487–493, 492f, 495f, 611f NOE, by, 289–290 pulse-interrupted free precession, by, 280–284 RDC, by, 291–292 rf pulse, by, 92, 294 through-bond, 280 COSY-type, 281–284, 290–291 scalar coupling, by, 782 TOCSY-type, 284–289, 290–291
INDEX through-space, 280 NOE, by, 289–290 RDC, by, 291–292 spin-state selective (S3CT), 99, 100, 101f TOCSY-type, 284–289, 290–291 triple-resonance, 615t-617t Coherence transfer amplitude, 281–282. See also individual NMR experiments Coherence transfer pathway, 293–295, 295f. See also individual NMR experiments EXORCYCLE and, 309f phase cycle, and, 298–307 pulsed field gradients and, 311–313, 312f Coherent averaging theory, 205 Coil gradient, 217, 219–220, 244 Helmholtz, 117 rf, 116f, 117–119, 118f shim, 115–117, 115f, 116f, 220, 239–241, 240t, 250–251, 251t Z0, 117 Commutation superoperator, 45, 351, 353 Commutator, 35, 40, 46, 372t Commuting operators Baker-Campell-Hausdorff (BCH) relationship for, 43 exponential matrix of, 43 simultaneous eigenfunctions, of, 34–35 Complete relaxation matrix analysis (CORMA), 810 Complex Fourier transformation, 317, 319, 321. See also Fourier transformation Composite pulse, 174–179. See also Decoupling 1808 (), 174–179, 178f, 203–205 BIP, 179 optimal control theory and, 179 Composite pulse decoupling, 203–209. See also Decoupling Computer-aided assignments, 791, 792, 796 Conformation, protein. See Structure determination
INDEX Conformational exchange. See Chemical exchange Conformation-dependent secondary chemical shift, 23, 804 Connected transition, 76, 456, 460 Connectivity, 784–791, 789f, 790f, 792–794 Constant-time (CT) evolution. See also individual NMR experiments digital resolution in, 544, 549–550 evolution period for, 543–544, 549–550 homonuclear decoupling, 543 13 C-13C decoupling in, 578–579 lineshape in, 543–546, 545f 1 H-13C HSQC product operator analysis for, 578–581 pulse sequence for, 579f ubiquitin, of, 581f, 582f HSQC pulse sequence for, 537 product operator analysis for, 543–544 relaxation in, 543–544 ubiquitin, of, 544f, 545f linear prediction and, 161, 162f, 163f relaxation in, 543–544 semi-constant time, 553, 691–692 Constraint. See Restraint Continuous Fourier transformation, 136–137, 151. See also Fourier transformation Continuous wave (CW) rf field, 9, 201–204 Continuous signal, 124 Contraction of bra and ket, 37 Convolution. See Apodization Convolution difference low pass filter, 232–234, 235f Convolution theorem, 137–138, 143–145 Coordinate system, convention for, 10, 58, 121–122 CORMA (Complete relaxation matrix analysis), 810 Correlated Spectroscopy. See COSY Correlation function. See Stochastic function Correlation integral, 138 Correlation theorem, 138
849 Correlation time effective, 368 internal, 368 Hamiltonian, for, 352 rotational, 18–19, 20f, 21, 336, 352, 363, 366, 729f COS (Coherence order selective), 560. See also PEP Cos–1(JCCt) apodization, 655 Cosine bell apodization, 145, 146f Cosine-modulated signal, 132, 315, 316, 320, 561, 562, 564, 771 COSY (Correlation Spectroscopy), 409–429. See also 2QF-COSY; 3QF-COSY; COSY-b; DCOSY; DR.COSY; E.COSY; HCCHCOSY; pQF-COSY; P.COSY; P.E.COSY; Pre-TOCSY COSY; R. COSY apodization for, 147 cross-peak fine structure in, 412 experimental protocol for, 412–414 fingerprint regions in aromatic, 418, 419f 1 a 1 b H - H fingerprint region, 418, 419f 1 N 1 a H - H fingerprint region, 418–419, 419f, 422f methyl, 418, 419f, 420f, 421f lineshape in, 411–412, 420–422, 423f processing of, 415–418 product operator analysis for, 281–284, 410–412 pulse sequence for, 410f scalar coupling constants, measuring from, 420–426, 423f, 424f, 425f self-cancellation in, 284, 414, 420–424, 423f, 424f ubiquitin of, 445f, 446f, 449f, 450f, 453f, 454f, 462f, 480f variants of, 426–429 COSY-35, 426–428, 463, 464f COSY-b, 426–428, 463 COSY-type coherence transfer, 281–284, 290–291 Counter-rotating rf field, 9, 50, 170 Coupling, dipolar. See Dipolar coupling; Residual dipolar coupling Coupling, residual dipolar. See Residual dipolar coupling
850 Coupling, scalar. See Scalar coupling Coupling-enhanced TROSY (CETROSY), 672–673, 673f CPMG. See Carr-Purcell-Meiboom-Gill CRIPT (Cross-relaxation-induced polarization transfer), 726 Cross-correlation. See Relaxation interference Cross-peak, 277–278, 277f. See also individual NMR experiments 3D NMR and, 529f, 586f amino acid spin system and, 782 fine structure COSY, 412, 414 E.COSY, 456f, 459f pQ NMR, 469–473 pQF-COSY, 452, 453–455, 456f forbidden, 453, 483 homonuclear 3D, in, 527–528 Cross-polarization, 280, 286, 290. See also Isotropic mixing; TOCSY Cross-rate constant, 342 Cross-relaxation, 340, 346, 357, 362–363, 372–375, 680–685. See also NOE; NOESY; ROE; ROESY dipolar, 20f, 359–360, 365–366, 370–383, 377t. 346, 390, 797 forbidden cross-peaks and, 360, 453, 483 laboratory frame, 357, 359, 362, 374–375, 390 NOE and ROE, compensating for, 391, 491, 493 rotating frame, 381–383, 390 secular approximation and, 358, 360–363, 372, 376, 380, 683 through-space correlation by, 289–290 Cross-relaxation appropriate for minimolecules emulated by locked spins (CAMELSPIN). See ROESY Cross-relaxation-induced polarization transfer (CRIPT), 726 Cross-saturation, 757–759, 758f Cross-spectral density function, 359, 369–370 Cryogenic probe, 119–120, 222, 252 CSA. See Chemical shift anisotropy CSA relaxation. See Chemical shift anisotropy
INDEX CSA/dipolar relaxation interference, 359–360, 373, 385–387, 570–572, 672, 686, 692, 695f, 703–704, 713 CSI (Chemical shift index) 804 Curvature, baseline, 154–158, 156f, 157f CW (continuous wave) rf field, 9, 201–204 Cyanometmyoglobin, 667 Cyclically ordered phase sequence (CYCLOPS) , 266, 307–308 Cycling sidebands, 203, 211–217, 214f, 215f CYCLOPS (Cyclically ordered phase sequence), 266, 307–308 D D2O, 21, 222–223, 224f, 259, 260f, 406, 420, 444, 461, 474, 500, 573, 588, 727, 753, 759, 806 Damped oscillator, 139. See also FID; Interferogram Data processing, 129, 136–165, 263–267 HSVD and, 161 LPSVD and, 161 NMR spectrometer, in, 136 DCOSY (Dipolar correlation spectroscopy), 292 Decay constant, 139, 399, 400, 692 Decibel (dB), 253 Decimation, 129 Deconvolution of N/P signals, 576 Deconvolution of orthogonal signals, 562, 564, 567, 576 Decoupled HSQC. See HSQC, decoupled Decoupled inversion recovery, 345, 346f Decoupling, 96, 98, 181, 201–217. See also Isotropic mixing; individual decoupling sequences 1808 pulses, by, 96, 201, 203, 204 HSQC, in, 541, 547–548 selective, 170–172, 549, 579 adiabatic, 209–212 antiphase magnetization, effect on, 98–99, 556 asynchronous, 203 composite pulse, 203–209 constant-time evolution, in, 543, 578–581
INDEX cycling sidebands, 212–216 1 H, 552–553, 555–557, 570f, 626–627 HSQC, in, 555, 697 omitting, effects of, 596, 618 recommendations for, 216–217 relaxation and, 364 scalar coupling and, 96, 98 scaling factor, 204 selective, 271 self-, 381, 660, 799 semiselective, 549 supercycle, 205 synchronous, 555, 627 theory of, 201–204 Decoupling in the presence of scalar interaction. See DIPSI Degenerate NOE, 515, 516f, 593, 598 Degeneracy, resonance. See Chemical shift Degenerate transition, 357, 358f, 361 Denaturation, 407f, 408, 409 Density matrix, 37, 41. See also Density operator BCH and, 43, 45 closure theorem and, 38 density operator and, 41 diagonal elements, 55, 74, 333 Dirac notation and, 37–38 off-diagonal elements, 74 rotating frame transformation and, 43 similarity transformation and, 44 trace of, 39 Density matrix formalism, 29, 78, 401 Density operator, 41, 68, 361 BCH and, 43, 45 diagonal elements of, 333 equilibrium, 55–56, 333 high temperature approximation for, 55 semi-classical relaxation, in, 352 two-spin system, for, 68 expansion of, 77–80, 294 interaction frame, in, 351 Liouville von Neumann equation and, 42 off-diagonal elements of, 333, 336 relaxation of, 351–359 rotating frame, in, 43 time-evolution of, 41–43, 280
851 transformation of, 79 Dephasing, coherence, of. See Coherence, dephasing of DEPT (Distortionless Enhancement by Polarization Transfer), 290 Derivative theorem, 137 Detected nucleus, 535 Detection, indirect, 535 Detection operator, 55–56, 89, 308 Detection, phase sensitive. See Frequency discrimination; Quadrature detection Detection, proton, 535 Detection, quadrature. See Frequency discrimination; Quadrature detection Detection, single-channel, 132 Detector, phase sensitive, 120–123, 132–136 Deuteration, 533, 726–728, 730f, 733, 748, 759 CT-HNCA and, 735–737 experiments in, 733–744 HNCACB and, 737–739 HNCO and, 739 HN(COCA)NH and, 739 NOE restraints from, 745–749 relaxation and, 730f side chains and, 740–743 static magnetic field and, 739 Deuteration, complete. See Perdeuteration Deuteration, fractional, 533, 727–733, 730f, 737, 740, 745 Deuterium, 117, 222, 727, 748 decoupling, 697, 734, 735, 749 isotope shifts and, 732f, 733 lock circuit, 117 relaxation rate constants and, 693–694 shimming and, 241, 252 Deuteron, 2, 222, 500, 697, 728, 732, 734. See also Deuterium DG (Distance geometry), 808 DHPC. See Dihexanoyl phosphatidylcholine Diagonal anti-, 520 double, 445f, 446f pseudo-, 471f, 475f, 597
852 Diagonal peak, 277, 290, 452–453, 475, 593, 597 Diagonalization, 63, 90, 341 exponential matrix, of, 341 Hamiltonian, of, 50, 63 Diamagnetic molecule, 102, 111, 370 Diamagnetic protein, 667, 668 Differential equation coupled, 357 evolution in time, 41 first order, 339, 341 harmonic, 84 integration of, 352 Diffusion. See also Spin diffusion Brownian rotational, 21, 335, 365–367 rotational anisotropic, 19, 367, 370 Brownian motion, in, 21, 335, 365–368 correlation time, for, 19–21, 352, 366, 370 isotropic, 19–21, 365–367 relaxation, by, 352, 366 Diffusion, spin. See Spin diffusion Digital filter, in NMR spectrometer, 126–131 Digital phase shifting, 438 Digital Quadrature Detection (DQD), 136 Digital resolution. See Resolution, digital Digital resolution enhancement, 149f, 150f Digital signal processing, 124, 129–130, 135,142, 223 Digitizer, 123, 128, 133f Digitizer phase, 298, 308 Dihedral angle, 455, 656 restraint, 798–804 Karplus, curve, dependence on, 388, 798–803 time-dependent, 388, 799–800 Dihedral angle restraint from isotropic chemical shift, 804 from scalar coupling constant, 798–803 Dihexanoyl phosphatidylcholine (DHPC), RDC and, 667 2,2-dimethyl-2-silapentane-5-sulfonic acid (DSS), 262–263, 263t
INDEX Dimyristoyl phosphatidylcholine (DMPC), RDC and, 667 Dioxane, 262 Dipolar correlation spectroscopy (DCOSY), 292 Dipolar coupling, 24 coherence transfer under, 291 Hamiltonian and, 80, 110–111 heteronuclear, 290–291 homonuclear, 289 like spins, for, 381–383 magnetization transfer by, 289, 346, 502–510 measurement of, 665 principal orientation of, 365 spatial variables for, 366t spherical harmonic functions for, 350t tensor operators for, 372t, 382 Dipolar coupling constants, restraints from, 804–805 Dipolar field, 371 Dipolar-CSA relaxation interference, 359–360, 373, 385–387, 570–572, 672, 686, 692, 695f, 703–704, 713 Dipolar relaxation, 20f, 359–360, 365–366, 370–383, 377t, 386, 502, 571–572, 680, 685 See also Cross-relaxation; NOE; NOESY; ROE; ROESY Dipole-dipole (DD) relaxation. See Dipolar relaxation DIPSI (Decoupling in the presence of scalar interaction), 364, 490–493 HCCH-TOCSY and, 610 HSQC and, 555 pulse sequences and, 490 DIPSI-2, 364, 490–493 DIPSI-2 relaxation-compensated (DIPSI-2rc) sequence, 486, 493, 496, 515, 590 DIPSI-2rc. See DIPSI-2 relaxationcompensated sequence DIPSI-3, 611 Dirac notation, 37 basis operators and, 80 closure theorem and, 38 density matrix and, 37–38 Liouville von Neumann equation and, 41
853
INDEX Schro¨dinger equation and, 41 Direct peak, 469 2Q and, 471, 475f, 479f 3Q and, 484f, 486, 487f Direct product, 59–61 Discrete Fourier transformation, 140–143, 154, 158. See also Fourier transformation Discrete signal, 124 Dispersion, chemical shift, 179, 236, 272, 405, 407f, 408–409, 498, 525, 749, 785, 791 Dispersive lineshape, 18, 139–140, 140f, 317 Dispersive antiphase lineshape, 425f, 469, 470f, 471, 474, 476–477 Dispersive tails, 235f, 319, 412, 417f, 431, 434f, 442, 444, 471, 475, 490, 505 Displacement vector, 459 Distance constraint, 507–508, 510, 522. See also Distance restraint Distance geometry (DG), 808 Distance, lower bound, 798 Distance, upper bound, 510, 522, 797, 798 Distance restraint, 499, 502, 510, 527, 745, 749, 751–752, 756, 760, 762, 796–798, 805, 807–808. See also Distance constraint Distortionless Enhancement by Polarization Transfer (DEPT), 290 DMPC. See Dimyristoyl phosphatidylcholine Dolph-Chebycheff window, 144–145 Double difference isotope filter, 547 Double quantum NMR. See 2Q NMR Double-half-filter. See Isotope filter Double-quantum (DQ, 2Q) coherence, 74, 75, 82, 83f, 101f, 293, 295f, 301, 312, 441, 443, 480, 697. See also 2Q NMR; Coherence bilinear product operators and, 90–91 chemical exchange and, 715 chemical shifts and, 91 HMQC and, 540, 718 NOE and, 505 selection of, 301f, 302t
Double-quantum filtered COSY. See 2QF-COSY Double-quantum (DQ) splitting, 468–469 Double-quantum (DQ) transition, 76, 82–83, 346 Double-relayed COSY. See DR.COSY Doubly absorptive lineshape, 319 Doubly antiphase absorptive lineshape, 415, 470, 483 Doubly antiphase dispersive lineshape, 469–471, 477 Doubly dispersive lineshape, 319 Doubly rotating tilted frame, 363–364, 381–383 DQ. See Double-quantum (DQ) coherence DQD (Digital Quadrature Detection), 136 DR.COSY (Double-relayed COSY), 433–436, 785 absorptive antiphase lineshape in, 434 pulse sequence for, 435f ubiquitin, of, 436f DSS (2,2-dimethyl-2-silapentane-5sulfonic acid), 262, 263 Dynamic frequency shift, 356–357 Dynamic range, 123, 128, 135, 156, 203, 221, 223, 265 Dynamical simulated annealing (SA), 809 Dynamics, internal, 334, 365–370, 409, 499 aromatic ring flips, 369, 810 chemical shifts, effect on. See Chemical exchange methyl rotation, 369, 406, 810 NOE, effect on, 369 scalar coupling, effect on, 799, 802 E E.COSY (Exclusive correlation spectroscopy), 455–463 displacement vector, 459 experimental protocol for, 460–461 heteronuclear, 656–657, 660, 670 lineshape asymmetry, 660 phase cycles for, 461t processing of, 461–462
854 E.COSY (Continued ) product operator analysis for, 457–458 self-decoupling in, 660 ubiquitin, of, 462f, 464f, 661f variants of, 463 weight factors for, 457 Eigenbra, 46–47 Eigenfrequency, 353, 360 Eigenfunction, 31 Hamiltonian, of, 50 Hamiltonian commutation superoperator, of, 353 Heisenberg uncertainty relationship and, 35 Hermetian operator, of, 32 simultaneous, 34–35 stationary state, 34 strong coupling Hamiltonian, for, 62–63 Eigenket, 38, 46–47 Eigenoperator, 353–354, 361 Eigenstate, 63, 74, 81–82. See also Stationary state coherence order and, 293–294 Eigenvalue, 31–34, 341, 362, 395, 398 Eigenvalue equation, 31–34 Electric field gradient tensor, 365, 366t, 384 Electric quadrupole moment. See Quadrupole moment Energy, 4, 31 Energy level, 4–7, 24–26, 62–64, 82–83, 335, 338 Enhancement, NOE, 344–345, 389–391 Ensemble of 3D structures, 808–813, 811 Envelope function, 144 Equilibrium chemical, 393 dissociation constant, 757 orientational, 106–107, 368 thermal density operator, 55–56, 75, 294 lattice, of, 335–336 magnetization, 5, 17f phase cycling and, 294, 306, 310–311 relaxation to, 257–258, 333, 336, 350, 352
INDEX Ernst angle, 257–258, 258f Ethylene glycol, temperature calibration using, 236 Euler angles, 104, 107, 108f, 804–805 Evolution. See Liouville von Neumann equation; Product operator formalism Evolution period, indirect. See Indirect evolution period Exchangeable proton. See Amide proton exchange Excitation null, 169–170 Excitation sculpting, 230, 233f in 3D NOESY-TOCSY spectra, 526f in NOESY, 504f in TOCSY, 489f Exclusive correlation spectroscopy. See E.COSY Exclusive COSY. See E.COSY EXORCYCLE, 266, 309, 313 Expectation value, 33–35, 40–41, 48, 79 of magnetic moment, 35–37 Exponential apodization, 147, 149f, 150f Exponential of matrix, 341 Exponential operator, 42–43, 45–46 Extreme narrowing, 367, 384, 391 F F1, 276 F2, 274 Faraday’s law, 9 Fast exchange, 392, 399, 401–402,754–755. See also Chemical exchange Fast Fourier Transformation (FFT), 142. See also Fourier transformation FFT (Fast Fourier Transform), 142. See also Fourier transformation FID (Free induction decay), 17, 19f, 134f clipping of, 156–157 distortion by filter response, 127, 157 interferogram and, 274 quantization noise in, 127–129 shimming, used for, 241 Field gradients. See Pulsed field gradients
INDEX Filamentous phage, RDC and, 668, 752 Filter. See also Isotope filter analog, 126–127 audiofrequency, 115f, 123, 132–133,133f, 157 baseline distortions, 156–158, 493–494 phase delay, 127 transient response, 127, 131, 157, 266 bandpass, 135–136 digital, 129–131, 232–234, 235f first-order J, 767 half, 763–769, 766f, 766t low-pass, 133, 235f second-order J, 764, 768 Filter diagonalization, 770 Filter function. See Apodization function Filter, isotope. See Isotope filter Filtering. See Apodization Fine structure. See Cross-peak, fine structure Fingerprint region, 418–421 aromatic, 418, 419f 1 a 1 b H - H fingerprint region, 418, 419f, 453–455, 454f, 455f, 478, 479f 1 N 1 a H - H fingerprint region, 418–419, 419f, 422f methyl, 418, 419f, 420f, 421f, 453f Finite impulse response (FIR), 129 FIR (Finite impulse response), 129 First-order phase correction, 153f Flip angle, pulse, 11, 52 Flip-back, 231–232, 574–578, 575f saturation transfer and, 690–691 Flip-flip transition, 75, 76f, 83f, 339, 346 Flip-flop spectroscopy. See FLOPSY Flip-flop transition, 75, 76f, 83f, 339, 346 FLOPSY (Flip-flop spectroscopy), 490–491, 492f HCCH-TOCSY and, 610–611 FM (Frequency modulation frame), 191–192, 196 Folding. See Aliasing and folding Forbidden cross-peak, 360, 453, 483 Force field, molecular dynamics, 808–809
855 Four-dimensional NMR. See 4D NMR Fourier analysis, 175, 180, 335 Fourier transform algorithms complex, 317, 319, 321 continuous, 136–137, 151 discrete, 141–142, 151, 154, 158 fast (FFT), 142, 770 inverse, 137, 600t, 661f linearity of, 137 periodicity of, 125 real, 141, 317, 318f, 319, 321–322 scaling first point in, 156 Fourier transform theorems, 137–138 Fourier transformation, 131 alternatives to, 159–165 apodization in, 143 baseline roll and, 157 causality in, 142–143 damped oscillator, of, 139, 142, 316–317 Dolph-Chebycheff window and, 144 FID, of, 17–18, 19f frequency discrimination and, 318f Lorentzian lineshape and, 140f, 317 nonuniform sampling and, 770–771 sinc-wiggles with, 157 truncation artifacts and, 143–148, 146f, 157, 265 zero-filling in, 142 Fractional deuteration, 533, 727–733, 730f, 737, 740, 745 Free induction decay. See FID Free precession, 14, 17 coherence order, conservation of, 294 evolution in the product operator formalism, 84–85 Hamiltonian, 61–62, 64, 538, 602 pulse-interrupted, 280–284 Frequency audio, 132, 135 carrier, 9 intermediate, 135 Larmor, 8 Nyquist, 124–126, 126f transmitter, 9 Frequency, characteristic. See Eigenfrequency
856
INDEX
Frequency discrimination, 132–135, 133f, 134f, 315–326, 318f. See also Quadrature detection hypercomplex (States), 315, 320–321, 323t N/P selection for, 316–317, 322, 576 PEP (preservation of equivalent pathways), 560–566, 574–578 phase cycling, by, 320–322 pQ NMR, in, 473 pulsed field gradient, by, 322–323, 574–578 Redfield’s method for, 321 time-proportional-phase-incrementation (TPPI), 315, 321–322, 323t TPPI-States, 322, 323t, 324–326, 325t, 655 Frequency labeling, 279–280 Frequency modulation (FM) frame, 191–192, 196 Frequency, resonance, 8, 11, 125 132, 237, 324 Frequency shift dynamic, 356–357 nonresonant, 9, 170–172 off-resonance, 9, 170–172 Frequency-shifted pulse, 181–182, 184–189 Frequency-shifting theorem, 137–138 Full-width-at-half-height (FWHH), 18, 139–140, 147–148, 408, 703 FWHH. See Full-width-at-half-height G GARP (Globally Optimized Alternating-phase Rectangular Pulse), 206, 208f, 214–215, 214f, 217, 690 Gaussian lineshape, 148, 149f, 150f Gaussian selective pulse, 182f, 183f, 184f Geminal proton NOE calibration, for, 797 spin-diffusion in, 522 GFT-NMR (G-matrix Fourier Transform NMR spectroscopy) 771–772 Global fold, 749, 751, 786f, 807, 807f Globally Optimized Alternating-phase Rectangular Pulse. See GARP
G-matrix Fourier transform (GFT) NMR spectroscopy, 771–772 Gradient echo, 218, 230, 311–312, 768 Gradient enhancement, heteronuclear correlation and, 574–578 Gradient, pulsed field. See Pulsed field gradient Gradient shimming, 242, 244–252 3D profile edge shimming 252 pulse sequence for, 248f shim map for, 251f Z1 profile shimming, 252 Gradient-enhanced (PFG) HNCA, 620f, 627–628 Gradient-enhanced (PFG) HSQC, 574, 576–578, 577f Gradient-enhanced (PFG) TROSY, 574, 576–578, 577f Gradient-enhanced (PFG) TROSYHNCA, 620f, 628 Gram-Schmidt process, 32 Group theory, 62, 361, 373 H 1
H 3D NMR, 525–529 experimental protocol for, 526–527 processing of, 527 pulse sequence for, 526 variants of, 528–529 1 H chemical shifts. See Chemical shift, 1 H 1 H decoupling, 552–553, 555–557, 570f, 626–627 1 H detection, 535 1 H linewidth, 406, 408, 407f 1 H NMR spectrum. See 1D 1H NMR 1 H relaxation rate constants, 20f, 729f, 730f 1 H resonance assignments, 782–791 1 a 1 b H – H fingerprint region, 418, 419f, 453–455, 454f, 455f, 478, 479f 1 N 1 a H - H fingerprint region, 418–419, 419f, 422f 2 H. See Deuterium HACANH, 774f, 775 HACA(CO)NH, 774f, 775 Hahn echo, 264f, 265–266, 267f, 704f 2D NMR, in, 488f, 503f baseline distortions, 264f, 265–266
INDEX chemical exchange and, 718, 719f jump-return and, 261f phase, 265–266, 264f pulse sequence for, 228f, 229f, 265–266, 704f, 719f spin-spin relaxation, for, 259, 261f solvent suppression, 228f, 229f, 232, 263–267 ubiquitin, of, 267f, 407f, 783f Half-filter. See Isotope filter Half-Gaussian selective pulse, 182f, 183f Hamiltonian, 30, 34, 102–104, 106, 351 averaging of, 102–112 chemical shift, 102, 107, 349 commutation superoperator, 351, 353 diagonal, 49–50 diagonalization, of, 49–50, 62–63 dipolar coupling principal orientation of, 365 spatial variables for, 366t spherical harmonic functions for, 350t tensor operators for, 372t, 382 residual, 80, 110–111, 291 effective, 44–45, 50–51 eigenbase and, 43, 55. See also Eigenfunction eigenfunction of, 34, 49–50 eigenvalue of, 34, 49–50 free precession, 61–62, 84, 602 interaction frame, in, 351–352, 363 isotropic chemical shift, 51, 84, 107 in isotropic phase, 107 Liouville von Neumann equation and, 42 magnetic field, 50 pulse, 85–86 quadrupolar, 350t, 366t, 385t, 680 relaxation and, 351 residual dipole coupling, 80, 110–111, 291 rf, 50, 85–86 rotating frame, in, 43–45, 363 scalar coupling and, 61–67, 107, 287 strong, 26, 61–64, 90, 284 weak, 64, 84 similarity transformation of, 44–45 stationary states of, 34, 49–50 stochastic, 335, 351, 353 strong coupling, 61–62
857 time dependent, 351 time independent, 31, 34, 42, 351 transformation under rotation of, 104, 108 truncation of, 104, 106, 108 weak coupling, 64, 84 Zeeman, 50, 57, 363 Hamiltonian superoperator, 351, 353 Hamming apodization, 145, 146f, 148 Hankel singular value decomposition (HSVD), 161, 162f, 163f, 498 Hanning apodization, 145 Harmonic signal, 132 Hartmann-Hahn condition, 286–289. See also Isotropic mixing; TOCSY isotropic mixing sequences and, 288 HBHACBCA(CO)NH, 739 HBHACBCANH, 739 HBHA(CBCACO)NH, 645 HBHA(CBCA)NH, 649 HBLV (Hybrid Backprojection Lower Valve), 775 HCA II (Human carbonic anhydrase II), 731–732 HCACO, 637 H(CA)NH, 615t, 632–636 product operator analysis for, 633–636 pulse sequence for, 633f ubiquitin, of, 636f (H)CC(CO)NH-TOCSY, 741, 743 HCCH-COSY, 601–608 constant-time pulse sequence for, 603f product operator analysis for, 607–608 ubiquitin, of, 609f product operator analysis for, 604–607 pulse sequence for, 603f HCCH-TOCSY, 601–603, 608–612 Isotropic mixing in, 611f product operator analysis for, 608–610 pulse sequence for, 610f ubiquitin, of, 612f HCC(CO)NH-TOCSY, 743 Heat bath. See Lattice Heisenberg uncertainty principle, 35, 336
858 Helmholtz coil, 117 Hermitian operator, 31–32, 39, 41 Heteronuclear coherence order, 314, 546–547 Heteronuclear coherence transfer, 290–291 Heteronuclear correlation NMR, 533–581. See also individual NMR experiments aliasing and folding, in, 546, 549–552, 550f artifact suppression in, 546–549 13 C-13C scalar coupling in, 548–549 isotope filter, in, 546–547 gradient enhancement and, 573–578 initial sampling delay in, 325–326 phase cycling and, 546–549 processing of, 552 quadrature detection and, 546 sensitivity of, 534–535 solvent suppression in, 573–574 Heteronuclear E.COSY, 656–660, 661f, 670 Heteronuclear Multiple-Quantum Coherence. See HMQC Heteronuclear NMR experiments, 533 Heteronuclear scalar coupling constants, 617t Heteronuclear single-quantum coherence. See HSQC Heteronuclear-edited NMR, 581—601, 745–753. See also individual NMR experiments Heteronuclear-edited NOESY. See individual NMR experiments Heteronuclear-edited TOCSY. See individual NMR experiments High resolution 3D structure, 806–813 High temperature limit, 5, 56, 356, 365 Hilbert space, 32, 55 Hilbert transform, 143, 357 HMQC (Heteronuclear MultipleQuantum Coherence), 535–540. See also HSQC, TROSY chemical exchange and, 718 EXORCYCLE and, 547 isotope filter in, 546–547 jump-return in, 226 lineshape in, 536, 538, 546, 549 phase cycle for, 546–547
INDEX processing for, 552 product operator analysis for, 536, 538–539 pulse sequence for, 537f relaxation in, 539–540, 545–546, 545f ubiquitin, of, 544f, 545f HMQC-NOESY-HMQC, 593–601, 762. See also HMQC-NOESY-HSQC; HSQC-NOESY-HSQC 13 C/13C, 597–599, 598f, 600t 13 C/15N, 595–597, 596f 15 N/15N, 594–595, 594f processing, 599–600, 600t pulse sequence for, 594f, 596f, 598f HMQC-NOESY-HSQC, 597 HNCA, 614, 615t, 618–628, 734–737 constant-time, 625–626 decoupled, 626–627 deuteration and, 735–737 DQ, 740 gradient-enhanced, 627–628 HN(CO)CA and, 629, 740 intraresidue and interresidue connectivities, distinguishing, 740 13 C-13C scalar coupling in, 622–623 pulse sequence for, 619f–620f, 736f product operator analysis for, 618, 621–627 relaxation in, 623, 625–626 sensitivity of, 625–626 TROSY, 628 ubiquitin, of, 624f HNCACB, 641–642, 650–654. See also CBCANH, HN(CA)CB pulse sequence for, 651f, 738f product operator analysis for, 651–653 resonance assignment and, 793–794 ubiquitin, of, 654f, 774f HN(CA)CB, 737–739. See also CBCANH, HNCACB deuteration and, 737–739 pulse sequence for, 738f HN(CACB)CG, 740 HN(CA)CO, 616t, 639–641 pulse sequence for, 640f sensitivity of, 641
INDEX HNCA-J, 656–660 3 JHNHa measured by, 660, 665, 667t pulse sequence for, 657f product operator analysis for, 657–660 relaxation in, 660 ubiquitin, of, 661f, 667t HNCO, 637–638 product analysis for, 637–638 pulse sequence for, 638f HN(CO)CA, 629–632 intraresidue and interresidue connectivities, distinguishing, 740 product operator analysis for, 629–631 pulse sequence for, 630f of ubiquitin, 632f HN(CO)CACB, 641. See also HN(COCA)CB resonance assignment and, 793–794 HN(COCA)CB, 734–735, 738–739 HN(COCA)NH, 739 HN(CX)nCY, 740–741 HNHA, 660, 662–665 3 JHNHa measured by, 664–665, 667t product operator analysis for, 662–664 pulse sequence for, 662f relaxation in, 664, 665f ubiquitin of, 666f, 667t HOHAHA. See Homonuclear Hartmann-Hahn Homogeneity, magnetic field, 18, 115–116, 223, 238–239, 241, 245–247 Homonuclear 3D NMR. See 1H 3D NMR Homonuclear chemical shifts. See Chemical shifts, 1H Homonuclear Hartmann-Hahn (HOHAHA), 280–281, 286, 288–289, 486. See also isotropic mixing; TOCSY Homonuclear scalar coupling constants, 401, 504, 535, 538, 539, 541, 543, 583, 670, 703, 714, 726, 734, 798–802, 800f, 800t, 801t, 802f, 802t, 803f, 803f, 806–807
859 Homospoil pulse, 220, 574 HSQC (Heteronuclear single-quantum correlation), 535. See also HMQC; TROSY 13 C scalar coupling and multiplet structure, 548–549 chemical exchange and, 718 chemical shift mapping and, 756 constant-time, 543–544 pulse sequence for, 537 product operator analysis for, 543–544 relaxation in, 543–544 ubiquitin, of, 544f, 545f decoupled, 553–560 energy level diagram for, 559f pulse sequence for, 554f product operator analysis for, 553, 555–560 relaxation in, 557–558 ubiquitin, of, 570f fast, 573–574, 575f folding and aliasing in, 449–452, 550f, 551f 1 H-13C product operator analysis for, 578–581 pulse sequence for, 579f ubiquitin, of, 581f, 582f interferogram for, 545f IPAP, 670–672, 671f phase cycling and artifact suppression, 546–548 processing, 552 product operator analysis for, 540–542 pulse sequence for, 537f, 554f RDC measured from, 669–670, 669f relaxation in, 338, 542, 543–544, 557–558, 562–563 sensitivity-enhanced (PEP), 560–566 energy level diagram for, 559f pulse sequence for, 554f product operator analysis for, 560–566 relaxation in, 562–563 ubiquitin, of, 570f ubiquitin of, 162f, 163f, 165f, 544f, 545f, 551f, 581f
860
INDEX
HSQC-NOESY, 591–593, 762 pulse sequence for, 592f HSQC-NOESY-HSQC. 745–753. See also HMQC-NOESY-HMQC; HMQC-NOESY-HSQC 15 N/15N-separated, 745–747 pulse sequence for, 746f calbindin D28k, of, 747f, 752f 13 C/15N-separated, 747–749 pulse sequence for, 750f calbindin D28k, of, 752f 13 C/13C-separated, 747–749 calbindin D28k, of, 752f HSQC-TOCSY, 591–593 HSVD (Hankel singular value decomposition) linear prediction, 161, 162f, 163f, 498 Human carbonic anhydrase II (HCA II), 731–732 Hybrid Backprojection Lower Valve (HBLV), 775 Hydration layer, 21 Hydrodynamic radius, 21 Hydrogen bond restraints from amide proton-solvent exchange, 805–806 from trans-hydrogen bond scalar coupling constants, 806 Hydrogen exchange. See Amide proton exchange Hydroxyl proton, 512–513 Hypercomplex (States) frequencydiscrimination, 315, 320, 322–325, 511, 550, 564, 576 I Identity operator, 38, 55–56 IF (Intermediate frequency), 135 Impedance, 118–119, 237 Impurity, detecting, 409 Incrementable delay, 273, 279, 326–327. See also Indirect evolution period Indirect detection, 535 Indirect evolution period, 274, 275f, 278–280. See also Acquisition period constant time, 543–541, 578–581 initial delay, for, 172, 325–326, 546 semi-constant time, 553, 691–692
Indirect evolution time, 273, 279, 326–327. See also Indirect evolution period Indirect magnetization transfer. See Spin diffusion Inductance, 119 INEPT (Insensitive Nuclei Enhanced by Polarization Transfer), 96–99. See also PEP; TROSY homonuclear scalar coupling in, 541–542 HSQC and, 540–542, 543, 553 optimal transfer, 338 refocused, 98–99, 553, 554f, 555–560, 556f, 633 relaxation in, 338 solvent suppression in, 573–574 Inhomogeneous broadening, 18, 534–544, 545f Initial rate approximation, 343–345, 507, 681, 797 Initial rate regime, 343–345, 507, 681, 797 In-phase coherence transfer. See also Magnetization transfer COSY-type, 98–99, 283–284, 605–606, 632–633 DCOSY, 291–292 TOCSY-type, 284–289, 486–496, 608–612 In-phase lineshape, 88–90 InS spin system, 553, 555–557, 556f, 598 Insensitive Nuclei Enhanced by Polarization Transfer. See INEPT Integrated intensity, 148, 151, 505–506 Interaction frame, 351–352 Interference, between relaxation interactions. See Relaxation interference Interference, destructive, 284, 453, 508, 572 Interference effects. See Relaxation interference Interferogram, 124, 144–145, 274–276, 275f. See also FID heteronuclear correlation and, 545f Intermediate exchange, 399, 401, 754–755. See also Chemical exchange
861
INDEX Intermediate frequency (IF), 135 Intermolecular chemical exchange, 394, 401, 753–754, 755f Intermolecular interactions, 753–760 Internal mobility. See Dynamics, internal Interresidue NOE, 788–790, 789f-790f Intramolecular dynamics. See Dynamics, internal Intramolecular chemical exchange, 393, 401–402 Intrinsic angular momentum, 29–30 Intrinsic amide-solvent exchange rate, 224f Inverse detection, 535 Inverse Fourier transformation, 137. See also Fourier transformation Inversion operator, 52–53, 66 Inversion, population, 16, 57 Inversion recovery, 258–259, 342–346, 686, 688f IPAP HSQC, 670–672 pulse sequence for, 671f IRS spin system. See Three-spin system IS spin system. See Two-spin system Isochromats, 218, 247 Isolated spin, 6, 8, 15, 58, 93, 181, 337, 392–400 Isolated two-spin approximation, 507 Isotope editing, 763 Isotope filter, 762–769 breakthrough peaks, 764–765 double difference, 547 double half-filtered NOESY, 765, 766f, 766t, 767 first-order J-filter, 767 half-filter, 763–764 adiabatic, 768 composite, 768–769 gradient, 767, 768 spin-lock, 768 heteronuclear correlation, in, 546–547, 584 second-order J-filter, 764, 768 signal selection by, 763 signal suppression by, 763 third-order J-filter, 768–769 !1 half-filtered NOESY, 765 !2 half-filtered NOESY, 765 Isotope shifts, 732–733, 732t
Isotopic labeling, 533, 727 Isotopomers, 697, 733, 743, 748 Isotropic chemical shift, 22. See also Anisotropic chemical shift dihedral angle restraints from, 804 perturbation mapping of, 756–757 Isotropic mixing, 287–289. See also decoupling; TOCSY coherence transfer efficiency and, 492f, 495f, 611f Hartmann-Hahn condition and, 288 RDCs and, 291–292 relaxation in, 364 suppression of ROE peaks, 491, 493 Isotropic mixing sequences, 287–288, 291, 490–493. See also individual mixing sequences Isotropic rotor spectral density function, 366 IUPAC-IUBMB-IUPAB Inter-Union Task Group on the Standardization of Data Bases of Protein and Nucleic Acid Structures Determined by NMR Spectroscopy, 262 J J coupling. See scalar coupling J filter. See Isotope filter JR.NOESY. See Jump-return NOESY Jump-return, 224–226, 233f, 235f excitation profiles of, 227f Hahn echo experiment, in, 261f Jump-return NOESY (JR.NOESY), 511–514 pulse sequence for, 512f ubiquitin, of, 513 K Kaiser apodization function, 144–145, 146f, 148, 149f, 150f Karplus equation, 798 curves, 800f, 802f, 803f parameterization of, 799, 800t, 801t, 802t, 803t Ket, 37–38, 47 Kinetics. See Chemical kinetics
862
INDEX
Kramers-Kro¨nig relations, 143, 357 Kronecker delta, 32 L Labels, isotopic, 526, 727, 761–763 Labile proton. See Amide proton exchange Laboratory reference frame, 4, 7–8, 102, 104, 106, 107, 108f, 364f, 368, 387 Larmor frequency, 8. See also resonance frequency Lattice, 40, 55, 335 Lattice jump model, 368 Leakage rate constant, 342 Levinson-Durbin linear prediction, 161 Lifetime, 2, 336, 368, 392, 754 Ligand. See Protein-ligand binding Like spins, 360–363, 382 Linear prediction, 160,-161, 234 4D NMR, in, 599–600, 600t HSQC, of, 162f, 163f mirror image, 161, 162f, 163f Linear prediction singular value decomposition (LPSVD), 161 Line-broadening. See Apodization; Linewidth Lineshape. See also Apodization; Linewidth absorptive antiphase lineshape, 413f, 421, 423f, 424–426, 424f, 425f, 470f absorptive lineshape, 18, 139–140, 140f, 317, 320f analysis, 684–685, 704–706 chemical exchange, 400f, 507–506 constant time evolution in, 543–546, 545f dispersive lineshape, 18, 139–140, 140f, 317 dispersive antiphase lineshape, 425f, 470f, 471, 474, 476–477 doubly absorptive, 319, 469 doubly antiphase absorptive, 415, 470, 483 doubly antiphase dispersive, 469–471, 477
doubly dispersive, 319 full-width-at-half-height (FWHH), 18, 139–140, 147–148, 408, 703 Gaussian, 148, 149f, 150f homogeneous, 18, 544 Lorentzian, 18, 139, 140, 140f, 155, 317 inhomogeneous, 18, 544, 545f in-phase, 88–90 off-resonance, 168f phase-twisted, 319, 320f shimming and, 242 Linewidth chemical exchange, 400f, 755f Gaussian, 148 1 H NMR, 406, 408, 407f HMQC, in, 539–540 HSQC, in, 542, 557 Lorentzian, 18–21, 20f, 139–140, 140f, 729f, 730f measurement of, 259, 261f TROSY, in, 387, 571–572 ubiquitin, for, 21, 259, 261f, 408 Liouville operator space, 78–80 Liouville von Neumann equation, 41–43, 79, 351, 356–357 Lipari-Szabo model free formalism, 368–369 Lock, field-frequency, 115f, 116–117 adjustment of, 241 shimming and, 241 Longitudinal relaxation. See spin-lattice relaxation Longitudinal two-spin order, 314, 519, 574, 699, 706, 720 Lorentzian lineshape, 18, 139, 140, 140f, 155, 317 Lorentzian spectral density function, 367, 367f Lorentzian-Gaussian transformation, 496–497 Lowering operator, 52, 72, 81. See also Shift operator Lower-value (LV), 773, 775 Low-pass filter, 232–234, 233f LPSVD (Linear prediction singular value decomposition), 161
863
INDEX M Magnet, superconducting, 114–117, 116f, 238 Magnetic equivalence, 353, 361, 412, 483 Magnetic field. See also rf field amplitude of, 8 anisotropic, 383, 408 effective, 8, 10, 16 homogeneity of, 115–117, 223, 233, 238, 241–244 imaging of, 244, 252 inhomogeneity of, 18, 117, 172, 239, 242, 245, 247–251, 293, 311, 408, 703 local, 22, 199, 247, 335–336, 367, 383, 387 measure of, 8 resolution, dependence on, 116 secondary, 22 shimming, 117, 223, 238–252 spherical harmonic expansion of, 239–240 static, 4–5, 8, 10, 18, 116, 169, 238 stochastic, 336, 345, 367 strength of, 4, 8, 10, 116 vector, 4, 102 Magnetic moment, 3–5 Bloch model and, 7 equation of motion, 7 expectation value of, 35–37 free precession, 14 Larmor frequency and, 8, 37 precession, 8–9 Magnetic quantum number, 3, 24–25, 59. See also Coherence order; Selection rule coherence and, 74–75, 293 product basis, in, 59–62 spin-1/2 nuclei, for, 24–25 spin quantum number and, 2 Magnetic susceptibility, anisotropic, 110, 666–667 Magnetization. See also Bulk magnetization; Transverse magnetization adiabatic pulse, during, 194, 195f average, 399
complex, 56, 70, 347 equilibrium, bulk, 5, 8–9, 11 longitudinal, 5, 12–13, 257, 293–294 observable, 55, 68–69 sense of precession, 58 steady-state, 389, 686 Magnetization transfer, 280–292. See also in-phase coherence transfer; NOESY; ROESY; TOCSY Magnetogyric ratio, 3, 3t, 8, 97, 119, 122, 371, 405, 534, 728 Magnitude mode, 246f Main chain directed (MCD) assignment, 786f, 790–792 Master equation, relaxation and, 351–363. See also Relaxation theory Matching. See Tuning and matching Matrix density operator representation with, 39, 41, 62 direct product, 58–61 exponential, 43, 52, 69, 341, 504, 518 rotation operator representation with, 52–54, 65–67 spin operator representation with, 46–50 trace, 39–40 unitary, 44, 341 Matrix representation, 39, 41, 46, 65 basic kets and, 46 exponential operator, of, 43, 69, 341 pulse operator, of, 66 rotation operator, of, 52–54, 65–67 two-spin system, in, 59 Maximum entropy reconstruction, 159–165, 655, 769–770 4D NMR and, 599–600 homonuclear 3D, 527 HSQC and, 165f McConnell equations, 393, 706. See also Chemical exchange MCD (Main chain directed) assignment, 786f, 790–792 Medium range NOE, 790f, 798, 807 Methanol, temperature calibration and, 236 Methyl group assigning, 751
864 Methyl group (Continued ) coherence transfer for, 556, 606 dynamics, 697 fingerprint region for, 418, 419f, 420f, 421f, 453f forbidden cross-peaks and, 453 interference in, 360 isotopomers, 733 linewidths for, 406 magnetic equivalence and, 361 relaxation and, 259, 361, 369, 453, 483, 696, 700f rotation and, 369, 406, 810 selectively protonated, 749–752, 759 TROSY, 749, 770 ubiquitin, in, 408 Methylene group 2Q spectrum and, 478 3Q spectrum and, 485 3QF-COSY and, 452–453 coherence transfer for, 556f, 606 DR.COSY and, 436 E.COSY and, 456–457 interference in, 360 isotopomers of, 733 Karplus curves for, 801–802, 803f NOESY and, 522, 797 R.COSY and, 433 ROESY and, 522 sensitivity and, 535, 741 spin diffusion in, 522 stereospecific assignment of, 502, 522 TOCSY and, 498, 502 Methyl-TROSY, 749, 770 Metric matrix distance geometry, 808 Microsecond-millisecond time scale, 680, 702 Microsecond-second dynamics, 702–721 Millisecond time scale, 702 Millisecond-second time scale, 702 Mixed state, 40 Mixing coefficients, 287, 488, 490, 610 Mixing period, 273–280, 324, 328f, 330f, 525, 680. See also Mixing time 2Q and, 471 NOESY and, 499, 506, 515, 593 R.COSY and, 432 ROESY and, 520
INDEX TOCSY and, 486, 490, 491, 494–496, 499 Mixing sequences, 490–493. See also Isotropic mixing sequences chemical exchange and, 481, 493, 522 CITY and, 493 DIPSI and, 490–493 FLOPSY and, 490–492 Hartmann-Hahn, 280, 286–288, 492f MLEV and, 490–492 NOE and, 491, 493 ROE and, 491, 493 WALTZ and, 490–492 WURST and, 491–493 Mixing time, 273, 284, 289, 344. See also Mixing period; individual NMR experiments MLEV, 206, 208–209, 491, 492f mixing sequences and, 490–492 MLEV-17, 490–491, 492f Molecular dynamics, internal. See Dynamics, internal Molecular dynamics, restrained (rMD), 808–810 Molecular reference frame, 367–368 MPF, 209, 210f, 215, 215f MQ. See pQ NMR MQF (Multiple-quantum filtered) 460 MQF-COSY. See pQF-COSY Multidimensional NMR. See also individual NMR experiments 2D (Two-dimensional), 273–280 3D NMR, compared with, 327–328, 328f, 329f 3D (Three-dimensional), 327–328, 525–526, 581 2D NMR, compared with, 327–328, 328f, 329f, 525 4D NMR, compared with, 329, 613–614 4D (Four-dimensional), 329–330, 330f 3D NMR, compared with, 329, 613–614 sensitivity in, 326–327 Multiexponential relaxation, 341, 453, 483, 681, 714–715 Multiple conformations, 799 Multiple pulse. See Composite pulse; Decoupling
865
INDEX Multiple quantum coherence (MQ), 91–92, 473 Multiple quantum echo, 221 Multiple quantum filtered COSY. See pQF-COSY Multiple quantum filtration. See p-quantum filtration Multiple-quantum (MQ) NMR, 90, 147, 279, 463–486, 465f, 536 chemical exchange in, 715–718 HMQC and, 536, 540 HSQC and, 541, 561 operators and, 90–92, 255, 288 scalar coupling and, 91–92 Multiple-quantum relaxation, in ubiquitin, 717f Multiple-quantum transition, 76f Multiple-spin echo, 221 Multiplet, 23–27, 76–77, 98, 201, 271, 284, 380, 401–402, 455, 535, 548, 571, 572f, 684. See also Cross-peaks, fine structure; Scalar coupling Lorentzian, 425 Multiplicity, 555 Multiplicity filter, 562 Mutual resolved coupling, 452 N 15
N relaxation rate constants, ubiquitin, of, 694f, 695f, 705f, 710f, 717f Natural abundance, 3t, 405, 568, 763, 765, 767 Net magnetization transfer. See In-phase coherence transfer; Magnetization transfer NMR active nuclei, 2, 3t NMR chemical shift time scale, 392, 399–400, 753–754 NMR spectrometer, 114. See also Probe; Shim system components of, 115f computer in, 123 data acquisition by, 124 digital filters in, 126–131 lock, field frequency, in, 115f, 116–117 oversampling in, 127–131 quadrature detection in, 132–136, 133f, 134f
receiver in, 115f, 123 rf transmitter in, 115f, 120–122 NOE (Nuclear Overhauser effect), 680. See also Cross-relaxation; NOESY; ROE accuracy of, 798 calibration of, 510, 597–598 classifications intraresidue, 787f, 788 long range, 786f, 806–807 medium range, 790f, 797–798, 806–807 sequential, 786f, 788–790, 789f, 790f, 806–807 coverage, 798 deuteration and, 745, 748–749 difference, 389–390 distance dependence of, 797–798 enhancement, 345, 389–390 global fold and, 807f heteronuclear, 290, 694f mixing sequences and, 491, 493 protein-ligand binding and, 761–762 rate constant, 390–391, 796–797 ROESY, contribution to, 520 steady-state, 389–390, 690, 700, 812 structural restraints and, 510, 797–798 transferred, 761 transient, 289, 344–345, 390–391, 700, 805 ubiquitin for, 694f, 789f, 790f NOESY (Nuclear Overhauser Effect Spectroscopy), 289–290, 502–517. See also Cross-relaxation; Isotope filter; Jump-return NOESY; ROESY; ZZ-exchange; individual NMR experiments baseline in, 506, 509 buildup series for, 510 chemical exchange and, 397, 511, 522 deuteration and, 745, 748–749 double half-filtered, 765, 766f, 766t, 767 excitation sculpting and, 513–514, 514f experimental protocol for, 506–510 Hahn echo and, 502–503, 503f initial rate regime for, 507
866 NOESY (Nuclear Overhauser Effect Spectroscopy) (Continued ) mixing period in, 502, 506–508, 510, 515 phase cycling in, 509, 511–512 presaturation in, 506, 511, 513f processing of, 510 product operator analysis for, 503–506 protein-ligand binding and, 761–762 pulse sequence for, 503f, 504f, 512f, 515f recycle delay in, 506, 514 relayed, 515–517 pulse sequence for, 515f ubiquitin, of, 516f resonance assignment and, 786, 788–792, 789f, 790f spin diffusion in, 506–508 ubiquitin, of, 509f, 513f, 514f, 516f variants, of, 511–517 !1 half-filtered, 765 !2 half-filtered, 765 zero-quantum peaks in, 490, 505, 508–509 z-filtration in, 490, 508–509 ZQ peaks in, 490, 505, 508–509 NOESY-HSQC, 582–589. See also HSQC-NOESY; HMQC-NOESYHMQC; HMQC-NOESY-HSQC; HSQC-NOESY-HSQC pulse sequence for, 584f product operator analysis for, 582–584 schematic of, 586f ubiquitin for, 587f, 589f NOESY-NOESY, 528–529 NOESY-TOCSY, 525–529, 526f Noise, 119, 123, 126–129, 131–132, 158, 163, 253. See also Signal-to-noise ratio Noise, t1, 262, 453 Nonadiabatic relaxation processes, 336–337, 349, 350 Nonlinear nonresonant phase shift. See Phase shift, nonresonant Nonresonant frequency shift, 170–172 Nonresonant phase shift. See Phase shift, nonresonant Nonsecular, 355, 361–362, 380, 399, 684
INDEX Nonselective inversion recovery, 258–259, 260f, 344, 346f Normalization, 30, 49, 79 N/P selection, 316–317, 322, 576. See also Frequency discrimination N-type signal, 316–317, 319, 322 Nuclear magnetic moment, 3–6 Nuclear magnetism, 2, 5–6 angular momentum and, 3 magnetogyric ratio and, 3 nuclear spin states and, 5–6 Nuclear Overhauser effect. See NOE Nuclear Overhauser effect (NOE) classifications intraresidue, 787f, 788 long range, 786f, 806–807 medium range, 790f, 797–798, 806–807 sequential, 786f, 788–790, 789f, 790f, 806–807 Nuclear Overhauser Effect Spectroscopy. See NOESY Nuclear shielding, 22, 51, 102, 107 Nuclear spin angular momentum. See Angular momentum Nuclei active, 2, 3t properties of, 3t quadrupolar, 2 Nutation axis, 120–122 Nyquist frequency, 124–126, 126f O Observable operator, 32, 55–56, 89–90, 300, 308 transition, 6, 25 Observable quantity, 31–34, 41 Observation operator, 55–56, 300, 308 Off-resonance 1808 () pulse, 167f, 169, 178f 908 (/2) pulse, 166–169, 167f, 168f phase shift, 168–171 pulse, 165–169 Offset baseline, 155–157, 157f, 307–308, 326 resonance, 10, 58, 121 !1 half-filter. See Isotope filter One-dimensional NMR. See 1D 1H NMR
867
INDEX One-pulse acquire experiment, 16–18, 19f, 56–58, 263, 265 recycle delay, for, 257–258, 258f signal-to-noise of, 158–159 two-spin system, for, 68–70 ubiquitin, of, 150f, 233f, 235f, 267f, 272f One-spin operators, 46, 70 On-resonance pulses, 16, 17f, 51, 85–86 rf field, 11, 12f Operator. See also Angular momentum operator; Basis operator; Cartesian operator; Density operator; Hamiltonian; Product operator; Tensor Operator; Trace adjoint, 32 bilinear, 91, 287–288 compatible, 34 commuting, 34 direct product, 60–61 equilibrium density, 55–56, 75, 294 expansion of, 78–79 exponential, 42–43, 45–46 Hermitian, 31–32, 39, 41 identity, 38, 55–56 inversion, 52–53, 66 matrix representation, 39, 41, 43, 46–47 MQ, 90–92 observable, 32, 55–56, 89–90, 300, 30 observation, 55–56, 300, 308 one-spin, 46, 70 orthogonal, 79 Pauli spin, 46–50 projection, 39, 74 rotation, 50–54, 65–67, 84, 85f shift, 70, 72, 73t, 75, 80–81, 90, 293, 316, 356 similarity transformation of, 44–45, 48, 341 single element, 80–83, 99, 353, 450, 658 single transition shift, 81, 83f, 100, 294, 460 space, 78–80 spherical tensor, 103–104, 353 two-spin, 70, 71t, 73t unitary, 44
Optimal control theory, 179, 726 Order parameter Saupe, 109 Lipari-Szabo, 368–370 ubiquitin, for, 812f Orientational correlation function. See Stochastic correlation function Orientational spectral density function. See Spectral density function Orthogonal operator, 79 Orthogonality, 32, 79. See also Normalization Orthonormal set, 32–33, 38 Orthonormality, 32 Oscillator, damped, 139. See also FID; Interferogram Oscilloscope, 237, 238f, 253 Out-and-back coherence transfer, 613 Out-and-stay coherence transfer, 632 Oversampling, 126–129, 266 P Paramagnetic relaxation, 365, 370 Paramagnetic effects, 797 Parseval’s theorem, 138 Parts per million (ppm), 23, 262 PAS. See Principal axis system Pascal triangle, 412 Passive scalar coupling. See Scalar coupling Pauli spin matrices, 37, 46 P.COSY (Purged Correlation Spectroscopy), 427–428 Peaks. See also Cross-peaks; Direct peaks; Remote peaks back-transfer, 527 breakthrough peaks, J-filter, 764 diagonal, 277, 290, 452–453, 475, 593, 597 zero-quantum, 490, 505–506, 508–509 suppression of, 314, 508–509 Peakshape. See Lineshape Peak-to-peak voltage, 253 P.E.COSY (primitive E.COSY), 427 PEP (Preservation of equivalent pathways), 500–502, 560–572. See also individual NMR experiments 13 C heteronuclear correlation, in, 598
868 PEP (Preservation of equivalent pathways) (Continued ) 15 N heteronuclear correlation, in, 554f, 560–566, 570f gradient-enhanced, 574, 575–578, 577f HSQC-TOCSY, in, 593 triple-resonance, in, 620f, 627–628 TROSY, in, 566–570, 570f, 620f, 628 relaxation, 562–563, 568–570 sensitivity enhancement, 562–563 spin multiplicity in, 560–561 TOCSY, in, 500–502, 593 Peptide chemical shifts, random coil, 23, 408, 782, 794, 804 1 H 1D spectrum of, 272f scalar coupling in, 799 test sample, as, 255–256 Perdeuteration, 727 13 a C decoupling and, 734 isotope shifts and, 732–733, 732t relaxation and, 728–729, 729f, 734 sensitivity and, 729–732 PFG. See Pulsed field gradient Phase angle, 10, 37, 120, 166, 167f, 720 Phase correction aliasing and folding in, 125–126 baseline distortion, from, 156–158, 156f, 157f, 324 first-order, 131, 152, 154f, 155 nonresonant phase shift for, 170–172, 621–622, 645 pivot, 152, 155 zero-order, 152, 153f, 154f, 155 Phase cycle, 292–307. See also Coherence transfer pathway; Frequency discrimination; Isotope filter; Quadrature detection; individual NMR experiments artifact suppression in, 307–310 axial peak suppression, 309–310, 323–324 cogwheel, 307 CYCLOPS (Cyclically ordered phase sequence), 266, 307–308 EXORCYCLE, 266, 309, 313 first pulse, for, 306–307 grouping pulses in, 306–307
INDEX heteronuclear correlation, for, 546–548 last pulse, for, 307 limitations of, 310–311 p-quantum, 465, 465f p-quantum filtration, 437f, 438, 461t repetition rate artifacts, 258, 310–311, 444–445 saving time in, 305–307, 310–311 selectivity of, 305 Phase error, 123, 127, 154–156, 170, 266, 539, 622, 645, 659 Phase map, 246–249, 250 Phase modulated signal, 316–317 Phase, NMR spectrum of, 139, 151–158 baseline distortions, 156–158, 156f, 157f frequency-dependent, 151–152, 154f frequency-independent, 151–152, 153f, 154f initial, 139 sampling delay and, 154–156, 156f Phase, pulse, 10–11, 168–171 Phase sensitive. See Frequency discrimination; Quadrature detection Phase shift, resonance nonresonant, 9, 170–172 cosine modulation, compensatory, 172 delays, compensatory, 597, 599 phase shift compensatory, 171–172, 621–622, 645 pulses, compensatory, 172, 579–580 off-resonant pulse, 166–169, 167f, 168f receiver phase by, 295–300, 302–306 rf phase by, 298 Phase, spatially dependent, 218, 220, 311, 576 Phase-incremented pulse (PIP). See Phase-modulated pulse Phase-modulated pulse, 181–189, 185f vector representation of, 190f Phase-twisted lineshape, 319, 320f Phasing, 151–156, 153f, 154f, 156f. See also Phase correction in multidimensional NMR, 323–326
INDEX Picosecond-nanosecond dynamics, 685–701 15 N backbone amide, 686–692, 688f, 693f-695f 13 CH2D methyl, 693, 696–699, 696f, 698f, 700f 13 CO backbone, 699–700, 701f PIP (Phase-incremented pulse). See Phase-modulated pulse Pivot, 152, 155 Planck’s law, 6 Polar angle, 110, 365, 368, 804–805 Polarization transfer. See Coherence transfer Poles, signal, 160 Polyacrylamide gels, RDC and, 668–669 Population. See also Spin-lattice relaxation Boltzmann distribution, 5, 55–56, 335 density matrix, 55–56 effects of rf pulse on, 75 equilibrium, 5 inversion, 16, 57 Solomon equation for, 339–340 transfer, 98–99, 397f, 706 Population transfer, chemical exchange and, 397f, 706 Post-acquisition signal processing, 232–234 Post-acquisition solvent suppression, 232–234 Power, 253–254 Power spectral density function. See Spectral density function pQ NMR (Multiple quantum NMR), 463–465, 465f. See also 2Q NMR; 3Q NMR pQF COSY (Multiple quantum filtered COSY), 437–440, 437f. See also 2QF-COSY; 3QF-COSY p-quantum filtration, 437–440. See also 2QF-COSY; 3QF-COSY PR. See Projection-reconstruction Preacquisition delay, 265–266, 494 Preamplifier, 115f, 119, 123, 133f, 253 Precession. See Free precession Precision, three-dimensional structures of, 798, 805–806, 809–810 Prediction order, 160
869 Preparation period, 271, 273f, 279–280, 328f, 330f, 680 Presaturation, 223–224, 233f NOESY and, 513f Preservation of equivalent pathways. See PEP Pre-TOCSY, 428–429, 428f, 499 Pre-TOCSY COSY, 428–429, 428f, 499 Primitive Exclusive Correlation Spectroscopy (P.E.COSY), 427 Principal axis system (PAS), 22, 103–104, 108f, 110 Principal component of tensor, 22, 103, 110, 110 Principal value of tensor, 22, 103, 110, 110 PR-NMR (Projection reconstructionNMR), 773–775, 774f Probability density, 30, 40, 107, 109, 111, 350 Probability, transition, 334–335 Probe, 114–120, 115f, 118f. See also rf coil actively shielded gradient, 217, 219–220 cryogenic, 119 impedance, 118–119, 237 quality factor, 118, 159 resonance frequency, 237 tuning and matching, 119, 237–238, 238f PROCHECK, 809–810 PROCHECK-NMR, 809–810 Prochiral groups, assignment of, 502, 798 Product basis, two-spin, 24, 58–61, 62–64 Product operator. See also Basis operator; Cartesian operator; Density operator; Hamiltonian; Operator; Tensor Operator; Trace Cartesian basis, in, 58–61, 70, 71t, 80 shift, 70, 72, 73t, 75, 80–81, 90, 293, 316, 356 single element, 80–83, 99, 353, 450, 658 single transition shift, 81, 83f, 100, 294, 460 transformation of, 84–88, 85f, 87f
870 Product operator formalism, 77–102 evolution of, 84–88, 85f, 87f cascades during, 86,88 chemical shift, 84, 856 commuting operators of, 88 double quantum (DQ) coherence of, 90–93 free precession, 84–86, 87f, 90–92 rotation, 84, 85f scalar coupling, 84–85, 87f, 92 zero quantum (ZQ) coherence of, 90–93 pulses, 85–86, 87f relaxation, 337–338, 357, 359, 362 transformation, 84–88, 85f, 87f Profile edge shimming, 252 Progressively connected transition, 77, 460 Projection operator, 39, 74 Projection-reconstruction (PR), 773–775, 774f Propagator, 45–46, 54, 79–80, 84, 337–338 Protein. See also Ubiquitin; calbindin D28k chemical shifts for, 784f, 785f, 795f, 796f correlation time for, 18, 21 denatured, 407f, 408–409 hydration layer, 21 hydrodynamic radius, 21 linewidth, 20f, 729f, 730f scalar coupling constants for, 800f, 800t, 801t, 802f, 802t, 803f, 803t specific volume, 21 Protein structure. See Structure determination Protein-ligand binding, 753–769. See also Isotope filter amide proton exchange, 760 chemical exchange regime, 753–755 chemical shift mapping, 756–757 cross saturation, 757–759 NOESY and, 761–769 relaxation for detecting, 759–760 Proton decoupling, 552–553, 555–557, 570f, 626–627 Proton detection, 535 Proton exchange. See Amide proton exchange
INDEX Protonation, 749–753 amide 1HN spins, 727–728 aromatic rings, 752–753 ILV, 751 methyl-selective, 749–752 ubiquitin, ILV NOESY, of, 752f Pseudo-diagonal, 471, 475f, 597 Pseudo-energy, from restraints, 808–810 Pseudo-first-order rate constant, 394 P-type signal, 316–317, 319, 322 Pulse, 14–15. See also 1808 pulse; 908 pulse; Adiabatic pulse; Composite pulse; Phase-modulated pulse; Pulsed field gradient; Selective pulse; Shaped pulse broadband inversion (BIP), 179 Cartesian product operators and, 84 coherence order, effect of, 294 evolution during, 166–169 product operator formalism, in, 84–85 excitation null for, 169–170 flip angle, 11, 52 Hamiltonian and, 51 homospoil, 220, 574 length, 11, 171, 253–255 off-resonance, 166–169, 167f, 168f, 169, 178f on-resonance, 16, 17f, 51, 85–86 phase, 10–11, 168–171 purge, 538, 541, 659, 663 read, 224, 226, 511 rf field strength of, 8 rotation angle, 11, 52 rotation operator, 50–54, 65–67, 84, 85f spin lock purge, 173, 227–230, 228f, 232, 573, 585, 767, 768 Pulse programmer, 115f, 120 Pulse sequence, 43 gradient shimming, 248f INEPT, 96–99 pulse width calibration, for, 254–257, 256f out-and-back, 613 out-and-stay, 632 spin-state-selective (S3CT), 99–101 spin echo, 93–96 straight-through, 632
INDEX Pulse sequence, specific experiment heteronuclear CE-TROSY, 673f Cross-saturation, 758f Double half-filtered NOESY, 766f Fast HSQC, 575f 1 H-13C CT-HSQC, 579f HCCH-COSY, 603f HCCH-TOCSY, 610f HMQC, 537f HMQC-NOESY-HMQC, 594f, 596f, 598f HNHA, 662f HSQC, 537f, 554f, 575f, 577f, 579f HSQC-NOESY, 592f HSQC-NOESY-HSQC, 746f, 750f HSQC-TOCSY, 591–593 IPAP HSQC, 671f PEP-HSQC, 554f PFG-PEP-HSQC, 577f PFG-TROSY, 577f NOESY-HSQC, 584f TOCSY-HSQC, 590f TROSY, 554f, 575f, 577f, 673f homonuclear 2Q, 465f 2QF-COSY, 437f, 439f 3Q, 465f 3QF-COSY, 437f COSY, 410f COSY-b, 426–427 DR.COSY, 435f E.COSY, 457, 461t JR.NOESY, 512f NOESY, 503f, 504f, 512f, 515f NOESY-TOCSY, 526f P.COSY, 427–428 pQ NMR, 464–465 pQF-COSY, 437–439 pre-TOCSY COSY, 428–429, 499 R.COSY, 430f relayed NOESY, 515f ROESY, 521f, 524f SCUBA, 429 TOCSY, 488f, 489f 1D 1H 1, 224–227 1-3-3excitation sculpting, 228f Hahn echo, 228f, 229f, 265–266, 704f
871 jump-return, 224–225, 261f one-pulse acquire, 263–264 WATERGATE, 229–230 water flip-back, 229f Relaxation measurement 1D inversion-recovery, 258 13 CH2D, for, 696f, 698f 13 CO, for, 701f CPMG, 712f, 715f Hahn spin echo, 261f, 704f, 719f multiple quantum, 717f 15 N, for, 688f, 695f, 704f, 709f, 717f R1, 709f relaxation interference, 695f TROSY-CPMG, 721f TROSY Hahn spin echo, 719f ZZ-exchange, 707f Triple-resonance CBCA(CO)NH, 643f CBCANH, 648f CC(CO)NH-TOCSY, 742f HBHA(CBCACO)NH, 645 HBHA(CBCA)NH, 649 HCACO, 637 (HCA)CONH, 641 H(CA)NH, 633f HNCA, 619f-620f, 736f HNCACB, 651f, 738f HN(CACB)CG, 740 HN(CA)CO, 640f HNCA-J, 657f HNCO, 638f HN(CO)CA, 630f Pulse width calibration, 252–257 pulse sequences for, 256f Pulse-acquire. See One-pulse acquire experiment Pulsed field gradient probe, actively shielded, 217, 220, 246f, 251f Pulsed field gradient (PFG), 217–221, 311–315 artifact suppression, 313–313, 577f coherence selection, 311–314, 574–578 dephasing by, 219f frequency discrimination by, 322–323, 574–578 limitations of, 314–315 magic-angle, 221, 444
872
INDEX
Pulsed field gradient (Continued ) radiation damping and, 219, 231–232, 440, 514, 587 recovery delay for, 315 shape factor for, 311 solvent suppression using, 229f, 230–232, 573–574 strength of, 218, 221 three-axis, 220, 444 z-axis, 217–218 Pulse-interrupted free precession, 280–284 Pure state, 40, 74 Purged Correlation Spectroscopy (P.COSY), 427–428 Purity, sample, 409, 418 Q Quadrature detection, 117, 132–136, 133f, 134f, 321. See also Frequency discrimination; States; Timeproportional-phase-incrementation; TPPI-States Quadrature glitch, 307–308 Quadrature images, 135–136, 264–265, 307–308 Quadrature pair, 132, 564 Quadrupolar Hamiltonian, 680 spatial functions for, 366t spherical harmonic functions for, 350t tensor operators and, 385t Quadrupolar nuclei, 2, 384 Quadrupolar relaxation, 370–371, 383–384, 385t techniques for, 693, 696–699, 696f, 698f ubiquitin, of, 700f Quadrupole moment, 2, 384. See also Electric field gradient tensor axial symmetry of, 366 principal axis of, 365 spatial function for, 366 spherical harmonic functions for, 350t tensor operators and, 385t Quality factor, 118, 159, 211 Quantitative J-correlation, 656, 660, 662, 670
Quantum mechanical relaxation theory, 351 Quantum mechanics density matrix, 37–40 multispin systems, of, 58–70 postulates of, 29–37 Quantum number, angular momentum, 2. See also Magnetic quantum numbers Quantum statistical mechanics, 40–41 Quarternion formalism, 15, 177 R R1. See Spin-lattice relaxation R1, 382. See also Rotating frame relaxation R2. See Spin-spin relaxation R2 , 18, 408, 544 Radiation damping, 123, 222, 232, 241, 248–249, 440, 511–512, 514, 574, 586–587 Radiofrequency field. See rf field Raising operator. See Shift operator Rance-Kay processing, 563, 564, 567–568 Random coil chemical shifts, 23, 408, 782, 794, 804 Random field. See Stochastic field Random fractional deuteration, 727 isotopomers in, 733 relaxation and, 728–729, 730f, 741 Random function. See Stochastic function R.COSY, 429–436 experimental protocol for, 432 processing of, 432 product operator analysis for, 430–431 pulse sequence for, 730f ubiquitin, of 434f RCT. See R. COSY RD (Reduced dimensionality), 771–772 RDC. See Residual dipolar coupling RE-BURP, 182f, 184f, 186 Read pulse, 224, 511, 528 Real Fourier transformation, 141, 317, 318f, 319, 321–322. See also Fourier transformation
INDEX Receiver, 115f, 123 gating delay, 157–158, 266 phase shift, 297–298 Recovery delay, pulsed field gradient, 315 Rectangle function, 144–145 Recycle delay, 98, 257–258, 258f double diagonal artifacts and, 444–445, 445f Ernst angle and, 257, 258f NOESY, for, 506 phase cycling and, 306 signal-to-noise and, 158, 534 Redfield equation, 350–359. See also Relaxation theory Redfield kite, 358f Redfield’s method for frequency discrimination, 321 Reduced dimensionality (RD), 771–772 Reduced staic field, 10 Reference, chemical shift, 23, 262–263 Reference frame alignment, 108, 804–805 frequency modulation (FM), 191–192, 196 interaction, 351–352 laboratory, 4, 7–8, 102, 104, 106, 107, 108f, 364f, 368, 387 molecular, 107, 367–368 rotating, 7, 10–11, 11f, 43–46, 364f tilted, 10–11, 11f, 363–364, 364f, 381–383 Reference frequency, 132, 135, 322 Refinement, 3D structures of, 805, 809–810 Refocused INEPT, 98–99, 553, 556f. See also INEPT Refocusing, by 1808 pulses, 93–96 Regressively connected transition, 77, 460 Relaxation, 333. See also individual NMR experiments adiabatic, 336, 337, 349–350 autorelaxation, 340, 357, 359, 362, 375, 382–383, 692 average, 364, 380–381, 383, 396, 540, 542, 572, 681–685, 713, 718, 720, 748 biexponential, 343–345, 681 coherence, dephasing by, 336
873 constants in, 337 cross-correlation. See Relaxation interference cross-rate constant, 342 cross-relaxation. See Cross-relaxation chemical shift anisotropy (CSA), 22, 349–350, 383–384, 385t, 386–387, 502–506, 571, 680 decay constant, 139, 399, 400, 692 deuteration and, 728–729, 729f, 730f dipolar, 20f, 359–360, 365–366, 370–383, 377t, 386, 502, 571–572, 680, 685. See also Cross-relaxation; NOE; NOESY; ROE; ROESY dynamic frequency shift, 356–357 extreme narrowing, 367, 384, 391 forbidden, 360, 453, 483 heteronuclear multiple quantum coherence for, 540 heteronuclear single quantum coherence for, 378–381, 542 high temperature limit, 356, 365 inhomogeneous broadening, 18, 534–544, 545f initial rate regime, 343–345, 507, 681, 797 interference. See Relaxation interference 1 H relaxation rate constants, 20f, 729f, 730f laboratory reference frame, 357–358, 363, 364f, 368, 383, 686–701 leakage rate constant, 342 like spins, 360–363, 382 Lipari-Szabo formalism, 368–369 longitudinal. See Spin-lattice relaxation magnetic equivalence in, 353, 360, 391–392 master equation, 351–359 monoexponential, 344–345, 683 multiexponential, 341, 453, 483, 714 15 N relaxation rate constants, ubiquitin, of, 694f, 695f, 705f, 710f, 717f nonsecular contribution to, 355, 361–362, 380, 399, 684 paramagnetic, 365, 370
874 Relaxation (Continued ) perdeuteration and, 728–729, 729f population, 335–336, 339–340. See also Spin-lattice relaxation product operator, for, 337–338, 357–358, 359, 362 quadrupolar, 365, 370–371, 383–384, 385t random fractional deuteration and, 728–729, 730f, 741 rate matrix, 341, 357–358, 358f, 372, 378, 395, 397, 506, 680–684 Redfield kite, 358f rotating frame, 363–364, 364f, 381–383, 390–391 scalar, 370, 381, 387–388, 401, 406, 734 of first kind, 387–388 of second kind, 387–388 secular approximation in, 358, 360–363, 372, 376, 380, 683 self-decoupling by, 381, 401, 660, 799 Solomon equations, 338–346, 375, 503 relaxation interference and, 386 spatial functions for, 366t spin diffusion, 367, 511, 515, 528, 745, 759, 797, 798 NOESY and, 506–508, 522 ROESY and, 522 spin diffusion regime and, 367 spin-lattice. See Spin-lattice relaxation spin-spin. See Spin-spin relaxation superoperator, 356, 361. 363 transverse. See Spin-spin relaxation ubiquitin order parameters for, 812 relaxation decay for, 693f, 710f relaxation dispersion in, 710f relaxation rate constants for, 694f, 695f, 700f, 705f, 717f unlike spins, 360–363, 714 Relaxation interactions and mechanism. See also Chemical exchange chemical shift anisotropy (CSA), 22, 349–350, 383–384, 385t, 386–387, 502–506, 571, 680 principal axis for, 349–350, 365, 387 spatial functions for, 366t spherical harmonic functions for, 350t tensor operators and, 384t
INDEX dipolar relaxation, 20f, 359–360, 365–366, 370–383, 377t, 386, 502, 571, 680. See also Crossrelaxation; NOE; NOESY; ROE; ROESY picosecond-nanosecond dynamics and, 685 principal orientation of, 365 spatial variables for, 366t spherical harmonic functions for, 350t tensor operators for, 372t, 382 TROSY and, 572 paramagnetic, 365, 370 quadrupolar, 370–371, 383–384, 385t spatial functions for, 366t spherical harmonic functions for, 350t tensor operators and, 385t scalar, 370, 381, 387–388, 401, 406, 734 of first kind, 387–388 of second kind, 387–388 Relaxation interference, 359–360, 385–387 CRIPT and, 726 CSA-CSA, 692 CSA-dipolar, 359–360, 373, 385–387, 570–572, 672, 686, 692, 695f, 703–704, 713 dihedral angle and, 801 dipole-dipole, 692 forbidden cross-peaks from, 360, 453, 483 experimental methods for, 683–685, 692–693 multiplet structure and, 684–685 spectral density function for, 369–370 TROSY and, 570–572, 718–721 Relaxation measurements 13 CH2D, for, 696f, 698f 13 CO, for, 699–700, 701f CPMG, 712f, 715f Hahn spin echo, 261f, 704f, 719f inversion recovery, 258–259, 342–346, 686, 688f Multiple quantum, 717f 15 N, for, 688f, 695f, 704f, 709f, 717f NOE difference, 388–390
INDEX NOESY. See NOESY R1, 709f relaxation interference, 695f ROESY. See ROESY steady-state NOE, 388–390, 690–691, 688f, 699–700, 701f transferred NOE, 760–761 transient NOE, 344–345, 346f TROSY-CPMG, 721f TROSY Hahn spin echo, 719f ZZ-exchange, 707f Relaxation mechanisms. See Relaxation interactions and mechanisms Relaxation theory Bloch equations, 12–14, 337–338, 346–350 Bloch-McConnell equations, 393–400 quantum mechanical, 350–351 semi-classical (BMR), 350–364, 358f Solomon equations, 338–346, 343f, 375, 503–504, 506–507 Relaxation-compensated CPMG, 712f, 713–715, 715f Relayed coherence transfer (RCT). See R.COSY Relayed COSY. See R.COSY Relayed NOESY, 515–517 pulse sequence for, 515 ubiquitin, of, 516 Remote peak 2Q and, 469–472, 471f, 472t, 475, 475f, 479f 3Q and, 484–485 Reorientation. See Diffusion Repetition rate artifacts, 258, 310–311, 444–445 Residual dipolar coupling (RDC), 110, 665–673, 810–811 alignment media for, 667–669 coherence transfer using, 291 measuring, 670–673 restraint, from 751, 804–805 ubiquitin, for, 669f, 812f Resistance, 119–120, 253–254 Resolution digital, 141, 158 constant-time evolution, in 544, 549–550
875 semi-constant time evolution, in, 691–692 enhancement, 147–148, 149f, 150f magnetic field, dependence on, 23, 116 Resonance assignment 1 H, 782–791 isotopically labeled proteins, for, 792–794 Resonance degeneracy. See Chemical shift, degeneracy Resonance dispersion, 179, 236, 405, 407f, 408, 785, 791 Resonance frequency, 8, 11, 125 132, 237, 324 Resonance lineshape. See Lineshape; Linewidth Resonance linewidth. See Lineshape; Linewidth Resonance offset, 10, 58, 121 Restrained molecular dynamics (rMD), 808–809 Restraint covalent, 808 dihedral angle, 798–804, 800f, 802f, 803f, 804, 806–807 distance, 510, 522, 751–752, 797–798, 806–807 hydrogen-bond, 805–806, 807 NOE, 507–508, 510, 522, 751–752, 797–798, 806–807 Paramagnetic effects, 797 protein complexes, for, 756, -769 RDC, 804–805 ROE, 522 residue, per, 798 scalar coupling, 798–804, 800f, 802f, 803f, 806–807 violations, 809 rf coil, 116f, 117–119, 118f rf field amplitude of, 8, 9–11, 120 B1, 9, 11f Bloch-Siegert shift of, 9, 170 counter-rotating, 9, 50, 170 Hamiltonian for, 50 inhomogeneity of, 172–174, 178f, 228–230, 364, 517, 519 nonresonant, 9, 170
876 rf field (Continued ) off-resonance, 10–11, 11f, 12f, 166–167 on-resonance, 10, 11f, 12f phase of, 9–10, 120 strength of, 8, 9–11, 120 rf power meter, 237, 238f rf transmitter in, 115f, 120–122 Ridges, 416, 493, 785 Rigid rotor, 366–367 Ring flip, aromatic, 369, 480, 810 Ringing, 265 rMD (restrained molecular dynamics), 808–809 RMSD (Root-mean-square deviation), 798, 808–809 ROE (Rotating frame Overhauser effect), 363–364, 383, 390–391, 491, 493, 502, 517, 520–522. See also Cross-relaxation; NOE ROESY (Rotating-Frame Overhauser Effect Spectroscopy), 517–524. See also Cross-relaxation; NOESY; ROE chemical exchange in, 522, 524 experimental protocol for, 520–521 mixing time for, 520–521 vs. NOESY, 517, 522 product operator analysis for, 517–520 processing of, 521–522 pulse sequence for, 521f, 524f relaxation in, 363–364, 383, 390–391 resonance offset in, 518–520 rf inhomogeneity in, 364, 517 spin diffusion in, 522 ubiquitin, of, 523f variants of, 524 Roll, baseline, 157, 266 Root-mean-square deviation (RMSD), 798, 808–809 Root-mean-square voltage, 253 Rotating frame, 7, 10–11, 11f, 43–46, 364f Rotating frame Overhauser effect. See ROE Rotating frame transformation, 43–46, 50–51, 363–364
INDEX Rotating-Frame Overhauser Effect Spectroscopy. See ROESY Rotating-frame relaxation, 363–364, 364f, 381–383, 390–391 Rotation, convention for, 10, 58, 121–122 Rotation operator, 50–54, 65–67, 84, 85f Rotational correlation time, 20f, 21. See also Diffusion Rotational diffusion. See Diffusion R-SNOB, 182f, 184f, 186 S S3CT. See Spin-state selective coherence transfer SA (Simulated annealing), 809 SAIL (Stereo-array isotope labeling), 727 Sample preparation, 234–236 Samples, test, 255 Sampling, 124–125 decimation in, 129 nonuniform, 770–771 Sampling delay, 139, 154, 156, 161, 172, 324 acquisition, adjusting, 154, 324, 546, 655 indirect evolution, 324, 770 adjusting in, 324–326, 546, 658 compensating with added delay, 597 Sampling interval, 124–125, 140, 158, 265, 320–321 Sampling rate, 124–129, 132, 265 Sampling theorem, 124–126, 126f SAR (Structure-activity relationship), 736 Saturation, 253, 259, 271, 477f, 585, 685, 688f, 689–691, 701f, 755f, 757–759 Saturation transfer, 223–224, 226, 228, 232, 409, 511, 514, 573, 685, 690, 758–759, 805 Saupe order matrix, 109, 804 Scalar coupling, 23–27, 55 2Q and, 474–475 active, 91, 282–284, 412, 456, 458, 473, 536
INDEX apparent, 110, 660, 664–665, 669, 799 average, 402, 799 Bloch equations in, 23–27 chemical exchange and, 401–402 coherence transfer and, 280–283, 288, 338, 410, 492f, 520, 534, 601 CPMG and, 711, 714 dihedral angle and, 656, 799–802, 800f, 802f, 803f dihedral angel restraints from, 455, 656, 798–803, 810 energy of spin states, 24 evolution in the product operator formalism, 84–85 Hamiltonian and, 61–65, 80, 378. See also Hamiltonian, scalar coupling intramolecular dipolar relaxation and, 378–381 Karplus curve, 388, 424, 798–803, 800f, 802f, 803f multiple quantum coherence, evolution of, 92, 536 multiplet, 24, 63, 76, 98, 201, 241–242, 271, 380, 401–402, 548, 571, 578, 684 passive, 91–92, 282, 655 2Q in, 468 2QF-COSY in, 443 3Q in, 482 COSY in, 412, 420 COSY-b in, 426–427 decoupled HSQC in, 555 E.COSY in, 456–463 HCCH-COSY in, 605 PEP in, 561 relaxation and, 370, 387 resolved, 452 restraints derived from, 799–800 sign of, 459 spin echo and, 95 strong, 26–27, 62, 90, 107, 284–286, 292 time-dependent, 387–388 TOCSY and, 486 topology of, 490, 786 trans-hydrogen bond, 762, 806 in two-spin system, 25f, 62 weak, 26, 62, 64
877 Scalar coupling constants, 24, 62, 84, 110, 401–402, 667t COSY and, 420–426 HCCH-TOCSY and, 602 heteronuclear, 617t homonuclear, 401, 504, 535, 538, 539, 541, 543, 583, 670, 703, 714, 726, 734, 798–802, 800f, 800t, 801t, 802f, 802t, 803f, 803f, 806–807 linewidth and, 426 Lorentzian lineshape and, 425 measurement of, 420–426, 656–665 side chains and, 656 TOCSY and, 502 trans-hydrogen bond, 797, 806 Scalar coupling constants, measurements of absorptive and dispersive lineshapes from, 425 COSY, 420–426 COSY-b, 426–427 E.COSY, 455–463 heteronuclear E.COSY, 656 heteronuclear-edited NMR, 590, 656 HNCA-J, 656–660 HNHA, 660–665 nonlinear least squares, 426, 656 quantitative J-correlation, 656 TOCSY, 502 TOCSY-HSQC, 590 Scalar product, 33, 37 Scalar relaxation, 370, 381, 387–388, 401, 406, 734 of first kind, 387–388 of second kind, 387–388 Scaling interferogram, first point of, 155, 156f Schro¨dinger equation, 30, 41. See also Liouville von Neumann equation Hamiltonian and, 30 time-independent, 34 wavefunction and, 30 SCUBA (Stimulated Cross-peaks Under Bleached Alphas), 426, 429 Secondary chemical shift, 23, 804 Secondary magnetic field, 22 Secondary structure, 786, 790 1 N 1 a H - H scalar coupling for, 798–799
878 Secondary structure (Continued ) chemical shifts for, 405, 409, 783, 794, 804 determination of, 806–808 deuterium isotope shift, 733 NOE characteristic for, 797, 806 Secular, 355, 358, 360, 682 Secular approximation, 358, 360–363, 372, 376, 380, 683 SEDUCE-1, 170, 182f, 183f, 217 Selection rule, 6, 25, 358, 449, 452–453, 473, 475, 482–483 p-quantum, 482–483 p-quantum filtration, 457 Selective decoupling, 170, 217, 271, 549, 758 Selective inversion recovery, 343, 346f Selective protonation. See Protonation Selective pulse, 172, 179–181, 217, 227, 228f, 229f, 230–232, 259, 525, 549, 607, 697 Selectivity, phase cycle, 303–304 Self-cancellation, 284, 414, 424, 425, 435, 476, 481, 510 heteronuclear correlation and, 552 Self-decoupling, relaxation by, 381, 401, 660, 799 Semi-selective decoupling, 170, 549 Sensitivity, 6, 98–99, 116–117, 119, 158. See also Noise; Signal-to-noise ratio; individual NMR experiments 3D compared to 4D, 600–601 constant-time, 544 heteronuclear NMR, 534, 552 recycle delay and, 257–258 relaxation and, 534 Sensitivity-enhancement. See PEP Separation of interactions, 278 Separation of variables, 31 Sequence alignment, 788, 790, 794 Sequence specific assignment, 510, 788, 792, 796 Sequential assignment, 515–516, 516f, 590, 614, 646, 653, 740, 785, 789, 791–793, 806 Sequential NOE, 513f, 515, 533, 786f, 788–790, 790f, 807 Shape factor, 220, 311
INDEX Shaped pulse, 170, 171, 183f, 184f, 185f, 220, 228f, 230, 579, 645, 655, 696f, 698f, 717f Shielding tensor, 22, 102, 103 Shift basis operator, 70, 73t, 75 Shift operator, 70, 72, 73t, 75, 80–81, 90, 293, 316, 356 Shifted laminar pulse (SLP). See Phase-modulated pulse Shim system, 117, 238–252 coils and, 239–240, 240t, 250 FID and, 241 gradient, 242–252 maps, 251f profile edge, 252 protocol for, 243f pulse sequences for, 248f values, 251 Shimming, 238–252 Side chain conformation, 522, 656 Side chain torsion angle, 656, 801–803, 803f, 803t Side chains, 19, 436, 667, 791 1 H and, 743 2Q and, 478 3Q and, 485, 486 aliphatic, 641, 784f, 785 CC(CO)NH-TOCSY and, 743 deuteration and, 732, 740–743 E.COSY and, 456 nonaliphathic, 418, 784f, 785, 810 R.COSY and, 433 scalar coupling constants and, 656 TOCSY and, 498, 499 Sidebands. See Cycling sidebands Signal amplitude modulated, 316, 319, 322, 500, 559–560 analog, 123, 127–129, 156 audio frequency, 132 continuous, 124 cosine-modulated, 132, 315, 316, 320, 561, 562, 564, 771 harmonic, 132, 144 high-frequency, 126f, 132, 134f N-type, 316, 319, 322 phase modulated, 316, 565 P-type, 316, 319, 322 sine-modulated, 132, 315, 316, 320–322, 326, 413, 561
INDEX sinusoidal, 124, 132, 160, 254, 337, 411 Signal averaging, 174, 255, 257, 258, 265, 274, 293, 310 Signal-to-noise (S/N) ratio, 119, 123, 143, 144, 148, 149f, 150f, 158–159, 327–328. See also Noise; Sensitivity; individual NMR experiments 2QF-COSY and, 447, 448f COSY and, 414, 415, 448f deuteration and, 729–732, 741, 745 1 H NMR, in, 406–409 pulsed field gradient, 313, 315, 322–323, 576 heteronuclear NMR, 535 linewidth and, 18 matched filter and, 144 multi-dimensional spectrum, in, 326 probe, of, 119 NOESY and, 506, 797 PEP, 501, 560–566 power ratio, 128 preamplifier, of, 123 rapid data acquisition and, 769, 770, 773, 775 TOCSY and, 501 ubiquitin 1D spectrum, in, 406 Similarity theorem, 138 Similarity transformation, 44, 48, 58 Simulated annealing (SA), dynamical, 809 Simultaneous eigenfunctions, 34–35 Sinc function, 145, 146f, 182f Sinc pulse, 171, 182f Sinc-wiggles, 157 Sine bell, 415f, 416, 417f, 426 2QF-COSY and, 446 phase-shifted, 147 Sine-modulated signal, 132, 315, 316, 320–322, 326, 413, 561 Single element basis operator, 80–83, 99, 353, 450, 658 Single quantum relaxation. See Spin-lattice relaxation; Spin-spin relaxation Single transition shift basis operator, 81, 83f, 100, 294, 460 Single-quantum (SQ) coherence, 74, 82, 83f, 101f, 295
879 HSQC and, 535, 543–544, 572 operators and, 82, 88–90 Single-quantum transitions, 76, 77f, 82, 83, 88, 202 Size limitations, for NMR, 405, 533, 613 Skewed unimolecular exchange, 399 SL. See Spin lock Slices of 3D spectrum, 328–329 Slow exchange, 401, 511, 522, 702, 754, 760. See also Chemical exchange Slow tumbling, 367, 391, 569 SLP (Shifted laminar pulse). See Phase-modulated pulse S/N. See Signal-to-noise ratio Soft-hard-soft pulse sequence, 230 Solomon equations, 338–346, 375, 503. See also Cross-relaxation; Dipolar relaxation relaxation interference and, 386 Solomon transition rate, 375 Solvent exchange. See Amide proton exchange; Chemical exchange Solvent suppression, 117 1, 224–227 1-3-3binomial sequences, 224–227, 230 convolution difference low pass filter, 232–235 dynamic range and, 221, 223 flip-back, 231–232, 574–578, 575f, 685, 690–691 Hahn echo, 232, 263–267 heteronuclear NMR experiments and, 573–578 isotope exchange, 222–223 jump-return, 224–226, 227f, 233f, 235f, 261f NOESY and, 503, 511–514 postacquisition, 232–234, 235f presaturation, 181, 223–224, 233f, 409, 418, 428, 448, 473, 475, 499, 506, 511, 513f, 573 pulsed field gradient using, 227–232, 573, 574 saturation transfer during, 223, 226, 228, 232, 409, 573, 690 selective pulses in, 179–180, 227–232 spin lock purge pulse, 173, 227–230, 228f, 232, 573, 585, 767, 768 TOCSY-NOESY and, 528
880 Solvent suppression (Continued ) water flip-back, 231–232, 574–578, 575f, 685, 690–691 WATERGATE, 229f, 230, 232, 574 Solvent, viscosity of, 21, 366, 408 Spatial variables, relaxation for, 353, 365, 368 Spatially dependent phase, 218, 220, 311, 576 Specific volume, 21 Spectral assignments, 782 Spectral density function, 355, 358, 363, 365–370, 367f autocorrelation for, 355 cross-correlation for, 359, 369–370 high-temperature limit in, 365 internal motion for, 369 lattice jump model for, 368 Lipari-Szabo, 368–369 rigid isotropic sphere, for, 366–367 Spectral editing, 546, 580 Spectral simplification, 437, 449, 452, 484 Spectral width, 124–125 2Q experiment and, 473 3Q NMR, 483 13 C, 550–551 Spectrometer. See NMR spectrometer Spectrum, frequency domain, 18, 133–136, 773 Spherical harmonic functions, 106, 239, 240, 349, 365 modified second-order, 350t Spherical tensor operator, 103, 353 Spherical top, 366 Spin angular momentum. See Angular momentum Spin decoupling. See Decoupling Spin diffusion, 367, 511, 515, 528, 745, 759, 797, 798 NOESY and, 506–508, 522 ROESY and, 522 regime, 367 Spin echo, 92–97, 265, 703, 711, 714 chemical shifts and, 94 difference, 763 gradients and, 313, 315, 438 in gradient shimming, 248 Hamiltonian, 95 homonuclear, 96 multiple, 221
INDEX Spin flip, 338 Spin lock (SL), 197, 227–232, 228f, 254, 382, 518. See also Solvent suppression purge pulse, 173, 227–230, 228f, 232, 573, 585, 767, 768 ROESY in, 520–524 rotating-frame relaxation, in, 707–710 trim pulse, 609 Spin operator, 46 Spin quantum number, 2 Spin system, 24, 28, 58. See also Threespin system; Two-spin system AB, 26 AX, 25f, 26 categorization of, 787f density matrix formalism and, 29 InS, 553, 555–556 Spin system assignments, 222, 405, 534, 613, 615t, 725, 739, 761, 782, 792 Spin tickling, 271 Spin-1/2 nuclei, 6, 27, 38, 55, 78, 370 Spin-lattice relaxation, 5, 12, 337, 338, 340, 378, 534, 805 1 H, 535 axial peaks and, 309 Bloch formulation, 12–14, 337 CSA, 20f, 23, 349, 365, 366t, 370–371, 383–386, 680, 685 dipolar, 346, 365, 366t, 370–378 heteronuclear, 535 inversion recovery, measuring using, 258–260, 343–346, 686, 734 quadrupolar, 365, 366t, 370, 383–385, 680, 686, 693, 696, 700f scalar coupled spins, 378 slow, 491 Solomon equations, 338–346, 375, 386, 389 Spin-lattice relaxation rate constant, 5, 12, 337 Spin-spin coupling. See Scalar coupling Spin-spin relaxation, 13, 337. See also Chemical exchange average, 364, 378–381 Bloch formulation, 12–14, 337 CSA, 20f, 23, 349–350, 365, 366t, 370–371, 383, 384t, 385t, 542, 569, 680, 685, 728, 729f, 739
881
INDEX deuteration and, 726–728, 729f, 730f dipolar, 19, 20f, 365, 366t, 370–371, 377–381, 739 heteronuclear, 542, 545, 557 linewidth, measuring from, 18–19, 406–408 quadrupolar, 2, 370, 383–385, 680, 686, 696 random-phase model for, 346–350 scalar, 387–388, 401, 406, 734 scalar coupled spins, 378–381 Spin-spin relaxation rate constant, 13, 337 Spin-state selective coherence transfer (S3CT), 99, 100, 101f Spontaneous emission, 334–335 SQ. See Single-quantum coherence Stability, 116, 236, 660, 679 State function, 30–31, 34, 37. See also Eigenfunction; Wavefunction States (hypercomplex) frequency discrimination, 315, 320, 322–325, 511, 550, 564, 576. See also TPPI, TPPI-States Static magnetic field, 4, 7, 11, 116, 238, 246 Stationary state, 24, 34, 36, 55, 63, 75, 77, 98, 289, 336–337 Statistical mechanics, 40 Steady-state NOE difference experiment, 389–390, 690, 700 Steady-state NOE enhancement, 389 Stereo-array isotope labeling (SAIL), 727 Stereospecific assignments, 502, 522, 761, 798 Stimulated Cross-peaks Under Bleached Alphas (SCUBA), 426, 429 Stimulated emission, 334–335 Stochastic Brownian motion, 335, 365 Stochastic correlation function, 365. See also Spectral density function factoring, 367 internal, 367 overall, 367 rigid isotropic sphere for, 366 Stochastic field, 335–336, 350–351, 365–367 Stochastic Hamiltonian, 345, 351–353, 358, 359, 370, 385, 679 Stochastic process, 336, 348, 355
Stoichiometry, 756 Stokes Law, 21 Straight-through coherence transfer, 632, 643, 743, 751 Strong coupling, 26, 62–64, 66, 90, 217, 285, 488, 611f Strong coupling parameter, 62 Structural restraint. See Restraint Structure determination, 806–811 backbone conformation, 656 back-calculation, 810 CORMA, 810 distance geometry, 808 global fold, 749, 751, 786f, 807, 807f high resolution, 806–813 restrained molecular dynamics, 808–809 secondary structure, 529, 733, 751, 783, 786f, 790, 794, 804, 806–807, 813 side chain conformation, 656 simulated annealing, 809 three-dimensional structure, 808–811, 813 Structure-activity relationship (SAR), 756 STUD, 211 Supercycle, 205, 206, 209, 211, 491, 493–494, 555, 557 Superoperator, 45, 351, 353, 356, 361, 363 Superposition state, 75, 90 SUSAN, 206, 208f, 209, 214, 215 Symmetrical reconversion, 684 Symmetry, 349, 361, 365, 373, 384, 387, 455, 456f, 588, 772, 783 Synchronous decoupling, 555–556 Synthesizer, frequency, 115f, 120, 189 T t1, 274 T1. See Spin-lattice relaxation t1 noise, 262, 453 T2. See Spin-spin relaxation t2, 274 T5M4, 208–209 TALOS, 804 Temperature calibration, 236–237
882 Temperature (Continued ) correlation time, effect on, 21 fluctuations, 310 Tensor, 102–103, 104 chemical shift, 21–22, 102, 107 scalar coupling, 107 Tensor operator Cartesian, 102 CSA interaction and, 384t dipolar interaction and, 372t quadrupolar interaction and, 385t spherical,103–104, 353 Tesla, 8 Tetramethylsilane (TMS), 262 Thermodynamic equilibrium, 55 Three-dimensional (3D) shim map, 252 Three-dimensional NMR. See 3D NMR Three-quantum (3Q) coherence, 442, 449–551, 481–482 Three-spin single-quantum coherence, 450–451 Three-spin system, 75–77, 77f, 91–92, 787f 2Q and, 467–473, 471f 2QF-COSY, and, 442–444 3Q and, 481–483 3QF-COSY, and, 449–451, 457–462 DCOSY, and 292 ROESY, in, 522 spin-diffusion in, 506–507 Through-bond coherence transfer, 280 COSY-type, 281–284, 290–291 scalar coupling, by, 782 TOCSY-type, 284–289, 290–291 Through-space coherence transfer, 280 NOE, by, 289–290 RDC, by, 291–292 Tilt angle, 16 Tilted rotating frame, 363–364, 364f, 381–383 Time-dependent Hamiltonian, 30, 351 Time-independent Hamiltonian, 31, 51, 351 Time-proportional phaseincrementation (TPPI), 315, 321–322, 323t. See also States frequency discrimination; TPPI-States folding and, 324, 325t, 550, 550f initial delay for, 326
INDEX Time-resolved fluorescence spectroscopy, 21 Time-shifting theorem, 138, 152 TMS (Tetramethylsilane), 262 TOCSY (Total Correlation Spectroscopy ), 284–289, 486–502. See also CC(CO)NH-TOCSY; Coherence transfer; HCCHTOCSY; (H)CC(CO)NHTOCSY; HSQC-TOCSY; Isotropic mixing; NOESY-TOCSY; TOCSY-HSQC chemical exchange, 493 coherence transfer functions, 491, 492f, 495f DCOSY, compared with, 292 disadvantages of, 498–499 dispersive lineshape in, 288, 490 experimental protocol for, 493–496 Hamiltonian for, 284, 287 heteronuclear edited NMR and, 582 isotropic mixing sequences and, 287–288, 487–493, 492f, 495f, 611f Lorentzian-Gaussian transformation for, 496–497 mixing period and, 494–496, 495f, 497f pre-TOCSY, 428–429, 428f, 515–517 processing of, 496–498 product operator analysis for, 284–289, 486–489 pulse sequence for, 488f, 489f rapid acquisition, 500, 501f relaxation in, 363–364, 496 relayed, 428–429, 428f, 515–517 ROE peaks in, 491, 493 sample purity, and, 409 sensitivity-enhanced, 500–502 stereospecific assignments, for, 502 ubiquitin, of, 480f, 497f, 501f variants of, 428, 429, 499–502 zero-quantum peaks, 490 z-filtration, 288, 490 TOCSY-HSQC, 589–591 pulse sequence for, 590f resonance assignment and, 792 ubiquitin of, 591 Torque, 7, 220
INDEX Torsion-angle dynamics, 808 Torsion-angle, side-chain, 656, 801–803, 803f, 803t Total Correlation Spectroscopy. See TOCSY TPPI (Time-proportional phaseincrementation), 315, 321–322, 323t. See also States frequency discrimination; TPPI-States TPPI-States, 322, 323t, 324–326, 325t, 655. See also States frequency discrimination; TPPI Trace, 39 basis operator, 79 commutator, 40 cyclic permutation, 40 expectation value as, 39 Transferred NOE, 761 Transformation. See also Fourier transformation basis sets, between, 48 coordinate, 108 interaction frame, 352, 363 product operator, of, 79, 84, 85f, 87f rotating frame, 10, 43–46, 50–51, 363–364 similarity, 44–45, 48, 341 spherical tensors, irreducible, of, 104 tilted, 363–364, 364f, 517–519 unitary, 44–45, 341 Transient NOE experiment, 289, 344, 346f, 390–391, 700, 805 Transient ROE experiment, 390, 391 Transition, conformational. See Chemical exchange Transition rate constant, 345, 375 Transition allowed, 6, 25 anticonnected, 460 connected, 76, 456, 460 double quantum, 75, 76f, 83f flip-flip, 75, 76f, 83f, 339, 346 flip-flop, 75, 76f, 83f, 339, 346 frequency, 6, 8 magnetic dipole, 6 multiple quantum, 76f non-connected, 460 probability, spontaneous, 334–335 progressively connected, 77, 460 regressively connected, 77, 460
883 single quantum, 25f, 75–77, 77f, 82 stochastic field, induced by, 335–336 three-spin system, in, 77f two-spin system, in, 25f, 83f unconnected, 426–427, 463 Zeeman, 6 zero-quantum, 75, 76f, 83f Transmitter power attenuation, 253–254 frequency, 9 rf, 115f, 120–122 triple-resonance experiments, in, 654 tuning and matching, 237–238 Transverse relaxation. See Spin-spin relaxation Transverse Relaxation Optimized Spectroscopy. See TROSY; Relaxation interference Trim pulse. See Spin lock Triple-quantum. See 3Q (Threequantum) Triple-resonance NMR, 613–655. See also individual NMR experiments 3D and 4D, compared, 654–655 experimental parameters, 654–655 experiments, table of, 615t-617t nomenclature, 613 out-and-back, 613 out-and-stay, 632 straight-through, 632 TROSY (Transverse Relaxation Optimized Spectroscopy), 387, 535, 552–572, 574–578. See also Relaxation interference calbindin D28k, of, 572 chemical shift mapping, for, 756 coupling-enhanced TROSY (CE-TROSY), 672–673, 673f CPMG and, 720, 721f cross saturation, for, 758f energy level representation for, 559f Hahn echo and, 718, 719f methyl groups, for, 749, 770 phase cycling and, 122, 555, 567 product operator analysis for, 566–568 pulse sequence for, 554f, 575f, 673f, 758f
884
INDEX
TROSY (Continued ) pulsed field gradients in, 574–578, 577f RDC, for measuring, 672 relaxation and, 386–387, 568–572, 719–720 ubiquitin, of, 570f water flip-back in, 574, 575f TROSY-CPMG, 720, 721f TROSY-HNCA, gradient-enhanced, 620f, 628 Truncation artifacts, 143–148, 146f, 157, 265 Tuning and matching, 237–238, 238f Two-dimensional NMR. See 2D NMR Two-quantum. See 2Q (Doublequantum) Two-spin order, 314, 519, 574, 699, 706, 720 Two-site exchange, 392, 400f, 755f longitudinal magnetization, 394–397 rate matrix, 393, 395, 397 transverse magnetization, 395, 397–399, 708, 711 755f Two-spin system, 25f energy, 24–26, 63, 371–372 Hamiltonian for, 62, 64 isolated, 345, 507 rotation matrices, 66–67 2Q and, 466, 467f Cartesian basis, 70 direct product spaces in, 59 one-pulse experiment for, 68–70 product operators for, 71t, 73t dipolar relaxation rate constants for, 377t scalar coupling in, 25f, 62, 64 transitions for, 25f, 339f, 371f single element basis, 81–82, 83f wavefunctions, 24, 59, 62 Tycko 5-step phase cycle, 208–209, 211 U Ubiquitin chemical shifts in, 783f correlation time of, 21 denatured, 407f, 408 description, vi, ix linewidth for, 408 order parameters for, 812
RDCs for, 669f, 812f relaxation decay for, 693f, 710f relaxation dispersion in, 710f relaxation rate constants for, 694f, 695f, 700f, 705f, 717f sample preparation, ix scalar coupling constants for, 667t, 800f structure, NMR-derived, 807f, 811f Ubiquitin, NMR spectra of heteronuclear HCCH-COSY, 609f HCCH-TOCSY, 612f HMQC, 544f HNHA, 666f HSQC, 544f, 551f, 570f, 581f, 582f, 669f HSQC-NOESY-HSQC, 747f, 752f NOESY-HSQC, 587f, 589f TOCSY-HSQC, 591f TROSY, 570f homonuclear 2Q, 467f, 477f-480f 2QF-COSY, 445f, 446f, 449f, 450f, 453f, 454f, 462f, 480f 3Q, 484f, 487f 3QF-COSY, 453f-455f, 462f COSY, 415f, 417f, 419f-422f, 427f, 449f COSY-35, 427f, 464f DR.COSY, 436f E.COSY of, 462f, 464f JR.NOESY, 513f NOESY, 509f, 513f, 514f, 516f pre-TOCSY COSY, 428f R.COSY, 434f relayed NOESY, 516f ROESY, 523f TOCSY, 480f, 497f, 501f 1D excitation sculpting, 233f Hahn echo, 267f, 407f, 783f inversion recovery, 260f jump-return, 233f, 261f 1D, denatured, 407f one-pulse acquire, 150f, 233f, 235f, 267f, 272f spin-echo, 261f, 267f, 407f, 783f triple-resonance CBCA(CO)NH, 646f
885
INDEX CBCANH, 650f, 774f CC(CO)NH-TOCSY, 744f HACA(CO)NH, 774f HACANH, 774f H(CA)NH, 636f HNCA, 624f HNCACB, 654f HNCA-J, 661f HN(CO)CA, 632f Unconnected transition, 426–427, 463 Unit vector, 5, 9–10, 54, 365 Unitary matrix, 341 Unitary operator, 44 Unitary transformation, 44, 45 Unlike spins, 360–363, 714 V van der Waals, 798, 809 Variable target function distance geometry, 808 Vector space, 32, 33, 38, 60, 79 Vicinal proton, 796–798 Viscosity, 21, 366, 408 Voltage, 124, 237, 253–254 Voltage standing-wave ratio (VSWR), 237 VSWR (Voltage standing-wave ratio), 237 W WALTZ, 205–206, 207f, 209, 214f, 217 synchronous, 555, 627, 657 Hahn echo and, 703 HSQC and, 555 Isotropic mixing sequence, as, 490–491, 492f Saturation by, 690 WALTZ-4, 205, 207f WALTZ-8, 205, 207f WALTZ-16, 205–206, 207f, 214f, 490–491 Water, bound, 512–513 Water, chemical shift, 237 Water flip-back, 231–232, 574–578, 575f saturation transfer and, 690–691 Water radiation damping. See Radiation damping
Water stripe, 416, 477 Water suppression. See Solvent suppression WATERGATE, 229f, 230, 232, 574, 671f Wavefunction, 24, 30–33, 35, 37–38, 40, 47–48, 59, 62–64, 74 Weak scalar coupling, 26, 62, 64, 80, 90, 284, 538 weight factor, E.COSY, 457 WHATIF, 809–810 Wigner rotation matrices, 104–107, 105t Wigner-Eckart theorem, 3 Window function. See Apodization function Windowing. See Apodization WURST, 198–200, 200f isotropic mixing, 491–493, 492f spin decoupling, 211–212, 213f, 216f, 758 Z Zeeman Hamiltonian, 50, 55–57, 355 Zeeman levels, 4, 6 Zeeman transition, 8, 12 Zero-filling, 142–143, 600 Zero-order phase correction, 152, 154f, 322, 326, 416, 563–564 Zero-padding, 142–143, 600 Zero-quantum (ZQ) coherence, 74, 75, 76f, 82, 83f, 90–92, 314 chemical exchange and, 715 relaxation of, 540 scalar coupling and, 92 two-pulse segment for, 306f Zero-quantum peak, 490, 505–506, 508–509 suppression of, 314, 508–509 Zero-quantum splitting, 92 Zero-quantum (ZQ) transition. See Flip-flop transition Z-filtration, 288, 490, 496 Z-gradient, 218–220. See also Pulsed field gradient ZQ. See Zero-quantum coherence Z-rotation, 300 ZZ-exchange, 702, 706–707 pulse sequence for, 707f
TABLE OF CONSTANTS
Avogadro's number
NA
6.022 × 1023
Boltzmann constant
kB
1.381 × 10-23 J K-1
Magnetogyric ratio of proton
γH
2.6752 × 108 T-1 s-1
Permeability of free space
μ0
4π × 10-7 T m A-1 4π × 10-7 T2 m3 J-1
Planck's constant
h
6.626 × 10-34 J s
ħ = h∕2π
1.055 × 10-34 J s
