[3] Two-dimensional exchange spectroscopy of proteins

106

TECHNIQUES

[3]

[3] T w o - D i m e n s i o n a l E x c h a n g e S p e c t r o s c o p y o f P r o t e i n s

By

MACURA, W I L L I A M M. and JOHN L. MARKLEV

SLOBODAN

WESTLER,

Introduction Two-dimensional (2D) exchange spectroscopy J-3 is a very convenient tool for studying dynamic processes in liquids. It is designed specifically for the elucidation of incoherent magnetization transfer processes: chemical exchange and cross-relaxation. However, the 2D exchange experiment also detects coherent magnetization transfer caused by scalar coupling. The main problem in 2D exchange spectroscopy of proteins is to identify and separate various processes of magnetization transfer when all are present in the system studied. Increasing the dimensionality of the experiment to three or more dimensions can be used to separate different processes; however, all of the fundamental concepts are contained within the framework of 2D spectroscopy. In this chapter, we concentrate on 2D implementations of the separation procedures. The techniques described can be generalized to higher dimensions. Basic Experiment In general, dynamic processes are examined in terms of the migration of suitable labels from an initial to a final state. The use of labels that do not interfere with the basic exchange processes allows dynamic processes to be observed under equilibrium conditions. Nuclear magnetization is a very convenient label for the study of exchange processes, not only because it does not perturb observed processes, but also because dozens of fragments can be labeled by their respective resonance frequencies. Pulsed Fourier nuclear magnetic resonance (NMR) spectroscopy allows all resonance frequencies to be recorded simultaneously. Two-dimensional Fourier NMR spectroscopy traces resonance frequencies in two domains, and is an excellent method for studying exchange processes since one domain can be attributed to the initial state and the other to the final state. Cross talk between different frequencies in the time domain indicates that I B. H. Meier and R. R. Ernst, J. Am. Chem. Soc. 101, 6441 (1979). 2 j. Jeener, B. H. Meier, P. Bachmann, and R. R. Ernst, J. Chem. Phys. 71, 4546 (1979). 3 R. R. Ernst, G. Bodenhausen, and A. Wokaun, "Principles of Nuclear Magnetic Resonance in One and Two Dimensions." Oxford Univ. Press, New York, 1987.

METHODS IN ENZYMOLOGY, VOL. 239

Copyright © 1994 by Academic Press, Inc. All rights of reproduction in any form reserved.

[3]

2D EXCHANGESPECTROSCOPY 2D FT

s(t I

t2)

s(~ I, ~2)~<~

3

i

I

i

107 2D FT

(,0 2

a

b

c

Fro. I. Basic pulse sequences for 2D exchange spectroscopy: (a) in the laboratory frame, (b) 2D exchange spectrum, and (c) in the rotating frame.

exchange has taken place between the spin sites with the respective frequencies. The basic experiment of 2D exchange spectroscopy 2'4-6 is shown in Fig. 1a. The first pair of 90° pulses, separated by the time interval tl, creates frequency-labeled longitudinal magnetization; following the second 90° pulse, longitudinal magnetization then migrates between various sites during the mixing time rm. The result of the migration is recorded during the detection time, t2, which begins after the third 90° pulse. By repeating the experiment with systematic incrementation of the evolution period fi, a series of frequency-modulated, nonequilibrium states are created. After magnetization exchange during the mixing time, rm, one detects the signal, s(fi, t2), which on 2D Fourier transformation (FT) of the two-dimensional matrix yields the 2D exchange spectrum S(%, %), as shown in Fig. lb. The diagonal peaks at positions wt = % = coi and % = co2 = coJ indicate the fraction of magnetization that did not migrate during the mixing time. Cross-peaks at symmetrical positions about the diagonal, (co~ = w i, % = coJ) and (co~ = coi, % = o~i), originate from the fraction of magnetization that migrated between sites i and j during the mixing time. Figure lc shows the equivalent pulse scheme used for studying exchange processes in the rotating frame. 7-m As in the above experiment, the 4 S. Macura and R. R. Ernst, Mol. Phys. 41, 95 (1980). 5 A. Kumar, R. R. Ernst, and K. Wiithrich, Biochem. Biophys. Res. Commun. 95, 1 (1980). 6 y. Huang, S. Macura, and R. R. Ernst, J. Am. Chem. Soc. 103, 5327 (1981). v j. Hennig and H. H. Limbach, J. Magn. Reson. 49, 322 (1982). A. A. Bothner-By, R. L. Stephens, J. Lee, C. D. Warren, and R. W. Jeanloz, J. Am. Chem. Soc. 106, 811 (1984). 9 A. Bax and D. G. Davis, J. Magn. Reson. 63, 207 (1985). m L. R. Brown and B. T. Farmer II, this series, Vol. 176, p. 199.

108

TECHNIQUES

[3]

first 90 ° pulse followed by the incrementable delay, tl, creates frequencylabeled transverse magnetization which is projected onto the rotatingframe axis (x or y) by a spin-locking pulse of duration ~'m. While magnetization is spin-locked, exchange takes place. As in the previous case, data acquisition within time domain t2 is repeated with systematic incrementation of the evolution period, tl, generating a time domain signal s(h, t2) which on 2D FT yields the exchange spectrum S(to~, oJ2). For the analysis of cross-peak evolution during the mixing time, it is convenient to represent the 2D exchange spectrum as a matrix of peak volumes a which depends on the mixing time 7m. Forzm = 0, no magnetization exchange takes place, and, consequently, a(0) represents a diagonal matrix with elements proportional to equilibrium populations of the individual spin sites. The volume matrix of the exchange spectrum recorded with a fixed arbitrary mixing time a(~'m) depends on the equilibrium populations a(0), on the mixing time z m, and on the dynamic matrix L2-4: a(~'m) = exp(Lzm)a(0)

(1)

Elements of the dynamic matrix Lij express the rate of magnetization exchange between sites i andj. Exchange spectra in the rotating or laboratory frame have the same general appearance, but their detailed characteristics depend on the nature of L, that is, on the mechanism of magnetization transfer. Incoherent Magnetization Transfer Incoherent magnetization transfer is driven by processes of random molecular motion. Among N sites, the transfer can be described by the system of N linear, coupled differential equations dm(Tm) d,rm = Lm('rm)

(2)

ln(7"m) : exp(LTm)m(0)

(3)

with formal solution

The vector m has elements nixim i, where n is the number of equivalent spins, x is the mole fraction, and m is the deviation from thermal equilibrium of magnetization at site i. L is the dynamic matrix which has elements Lij. Magnetization of spin ½nuclei, with polarization states a,/3, can migrate between the sites i and j by two different mechanisms: by chemical exchange or by cross-relaxation. In the chemical exchange mechanism, the

[3]

2D EXCHANGESPECTROSCOPY

109

magnetization migrates together with the spin, which changes its site (i ~ - j ) but not its polarization: i(a) ~-- j(oO j(/3) ~.~ i(/3) i(a)j(/3) .-~ i(/3)j(a) In cross-relaxation, the spin changes its polarization (a ~ /3) but not its site: i(a) .~ i(/3) j (t3) .~ j (a) i(a)j(/3) .~ i(/3)j(a) Because the net effect of each process is indistinguishable, the individual contributions of each cannot be determined from a single exchange experiment. The dynamic matrix L contains all the relevant data about the exchange rates in a given system. Elements L o of the dynamic matrix contain terms corresponding to the two possible mechanisms: K 0 , the chemical exchange rate, and Rij, the cross-relaxation rate. Lij = K~j - Rij

(4)

In matrix form,

L=K-R

(5)

where K is the kinetic matrix and R is the relaxation matrix. Specialized 2D exchange experiments permit the individual rates to be determined by the use of Eq. (1). In the following sections we describe methods for identifying and separating processes contributing to magnetization transfer in 2D exchange experiments. Cross-Relaxation Cross-relaxation originates from dipole-dipole interactions between pairs of nuclear spins. Random molecular motions of the magnetic moments of spins at one site (resonance frequency) induce magnetic transitions in the spins at another site. The net effect is the exchange of magnetization between the sites. The magnitude of the effect depends on the proximity of the interacting spins and the type and frequency of the motion that modulates their interaction. It also depends on whether the magnetization is longitudinal or transverse, that is, whether magnetization exchange

110

TECHNIQUES

[3]

takes place in the laboratory or in the rotating frame, u The nuclear Overhauser effect (NOE) 12'13 is an experimental manifestation of cross-relaxation. It represents a change in the intensity of one resonance line when the other line is saturated. The effect described originally by Overhauser t4 arose from electron-nuclear cross-relaxation. The n u c l e a r Overhauser effect was described first by Solomon. 11'~5 In two-dimensional exchange experiments (Fig. 1), the nuclear Overhauser effect is manifested by the presence of cross-peaks at the intersection of the frequencies of the interacting spins. Laboratory-frame NOE spectroscopy (NOESY) 2'4'5 is the 2D NMR experiment used to study the NOE in the laboratory frame (Fig. la), and ROESY 8-j° is the 2D NMR experiment used to study the NOE in the rotating frame (Fig. lc). The cross-relaxation rate 0-ii between two groups of equivalent spins at sites i a n d j is given by 4'll .,r

R ~,r ni

II,l?

Rj i nj

ij

o-ij . . . . .

(W2)

n,r

"'

- (W~) "'r

(6)

where W2 and W0 represent two-spin transition probabilities (per unit time) for flip-flip (double quantum) and flip-flop (zero quantum) transitions, respectively. The superscript n denotes the laboratory frame (NOESY) and r the rotating frame (ROESY) system of reference, and ni,j is the number of equivalent spins in the respective sites. Table I summarizes the dependence of the transition probabilities in the two frames on molecular parameters. The dependence on the interspin distances rij is given by 2 1 {/~0~

2 2h2rs6

q = - ~ ~47rJ TiT)

ij

where, for protons, q (sec -2) = 5.69 × 104 rij -6 (nm-6). The dependence on molecular mobility is expressed by the spectral density function J(oJ). For random isotropic motion, --

J(w°)

TC

2 ~ 2 1 + Wo'C

where ~c is the correlation time and w0 is the transition frequency. i1 I. Solomon, Phys. Reo. 99, 559 (1955). J2 j. H. Noggle and R. E. Schirmer, "The Nuclear Overhauser Effect: Chemical Applications." Academic Press, New York, 1971. ~3 D. Neuhaus and M. Williamson, "The Nuclear Overhauser Effect in Structural and Conformational Analysis." VCH Publ., New York, 1989. 14 A. W. Overhauser, Phys. Rev. 91, 476 (1953). t5 I. Solomon and N. Bloembergen, J. Chem. Phys. 25, 261 (1956).

[3]

2D EXCHANGESPECTROSCOPY

11 1

TABLE I RELAXATION PARAMETERS FOR TWO GROUPS OF EQUIVALENT SPINS UNDERGOING ISOTROPIC MOTION IN LABORATORY FRAME AND ROTATING FRAME Parameter

Laboratory frame

Rotating frame

qJ(O) (3/2)qJ(o~) 6qJ(2w) 6qJ(2w) - qJ(O)

(l/4)qJ(O) + (3/4)qJ(2w) (3/4)qJ(¢o) + (3/4)qJ(2w) (9/4)qJ(O) + 3qJ(co) + (3/4)qJ(2oA 2q J(0) + 3qJ(o~)

H

W~ Wt( W~~ o-ii

¢Oo~"c ~> 1

~r0 I/T'I"

- qr c 0 + R~×

+ 2qr c (l/2)njqr c + R i~ + R~×

" 1/Ti = 2nj(W'i + W!) + 2(ni- 1)(Wi[ + Wf) + Rix = 2nj(Wi~+ Wf) + R u + R~x, where R 'i is intragroup relaxation and R~x is relaxation by external sources. For proteins or other macromolecules in a high magnetic field, co0rc >> 1. In this "spin diffusion limit," the cross-relaxation rate in the rotating frame is twice as fast as in the laboratory frame, and the rates are opposite in sign; in other words, 1 o-~ 2 (7) Also, it is important to note that when oJ0rc >> 1, laboratory-frame crossrelaxation takes place independently from the overall relaxation. In the rotating frame, however, cross-relaxation is an integral part of the overall relaxation process. In addition, rotating-frame intragroup relaxation R i~ may become the dominant relaxation mechanism and, in larger proteins, may quench the rotating-frame cross-relaxation.

Chemical Exchange Chemical exchange, in the NMR sense, reflects all processes of interand intramolecular rearrangements in which observed spins change their magnetic environment. 16,17However, for 2D exchange spectroscopy, only slow processes in which the observed spins change their resonance frequencies are suitable. In NMR spectroscopy, slow refers to an exchange rate ki; between sites i and j that is smaller than the difference in the 16 H. S. G u t o w s k y and C. H. Holm, J. Chem. Phys. 25, 1228 (1956). 17 H. M. McConnell, J. Chem. Phys. 28, 430 (1958).

112

TECHNIQUES

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resonance frequencies of the two exchange sites; that is, exchange does not influence the shapes and positions of the individual resonance lines: kij ~ [toi - ~oj[/2~-. The lower limit of the exchange processes is imposed by T~ relaxation of the spins. To be observed at all, the rate of the exchange process must be greater than or comparable to the T1 relaxation rate, I/T~: kij >- Ri, Rj. Unlike cross-relaxation, chemical exchange does not depend on the frame of reference in which it occurs. Chemical exchange processes in proteins include all internal rearrangements, such as slow side-chain rotations and reversible folding, and reversible intermolecular interactions, such as exchange of labile protons and bimolecular associations. These processes, when exhibiting much higher rates than can be studied by exchange spectroscopy (kij >- [(.oi oJjl/27r), can be investigated by NMR relaxation time measurements 18'19 or line shape analysis.Z0, zl -

-

Coherent Magnetization Transfer Coherent magnetization transfer takes place through the network of scalar coupling. It is a prerequisite for most modern NMR experiments. 3 Coherence transfer can be observed in either the laboratory or rotating frames and can contribute to magnetization transfer in 2D exchange experiments carried out in either frame. This type of transfer is usually studied directly by other techniques. 22-24In 2D exchange experiments it is usually considered as a nuisance to be avoided. Several methods have been proposed for the elimination of coherent magnetization transfer in exchange experiments. 25-2s These methods distinguish coherent and incoherent transfer on the basis of the differences in their time evolution. Coherent magnetization transfer takes place at well-defined frequencies, for exam18 L. W. Reeves, in "Advances in Physical Organic Chemistry" (V. Gold, ed.), Vol. 3, p. 187. Academic Press, London, 1965. 19j. Jen, Adv. Mol. Relax. Processes 6, 171 (1974). 20 K. V. Vasavada, J. I. Kaplan, and B. D. N. Ran, J. Magn. Reson. 41, 467 (1980). 21 B. D. N. Rao, this series, Vol. 176, p. 279. 22 M. Rance, W. J. Chazin, C. Dalvit, and P. E. Wright, this series, Vol. 176, p. 114. 23 G. Wagner, this series, Vol. 176, p. 93. 24 A. Bax, this series, Vol. 176, p. 151. 25 S. Macura, Y. Huang, D. Suter, and R. R. Ernst, J. Magn. Reson. 43, 259 (1981). 26 S. Macura, K. Wtithrich, and R, R. Ernst, J. Magn. Reson. 47, 351 (1982). 27 S. Macura, K. Wtithrich, and R. R. Ernst, J. Magn. Reson. 46, 269 (1982). 2s M. Rance, G. Bodenhausen, G. Wagner, K. Wiithrich, and R. R. Ernst, J. Magn. Reson. 62, 497 (1985).

[3]

2D EXCHANGESPECTROSCOPY

113

pie, sin(toi'rm); by contrast, incoherent transfer continues steadily according to Eq. (1). In laboratory-frame NOE experiments, coherent transfer can be eliminated by random variation of the mixing time, within a range that represents a small fraction of the total mixing time. Under these conditions, the sin(toil-m) terms average to zero, while incoherent exchange averages to a nonzero value. Another way of separating the contributions from the two processes is by manipulation of the pulse sequence. Systematic incrementation of the mixing time in concert with the t~ interval causes the coherent cross-peaks to be offset from the original position of incoherent transfer peaks. 27 The same approach can be implemented as a 3D experiment, in which the third domain t 3 is collected during the mixing period. To maintain a constant mixing period, the third dimension can be collected as a constant-time experiment by systematically incrementing the position of a 180° pulse through the mixing period. 26,29On 3D Fourier transformation, the % = 0 plane will contain pure incoherent transfer peaks, whereas the coherent transfer peaks will be shifted along the % axis by their respective zero-quantum coherence frequencies. Elimination of coherence transfer from rotating-frame experiments is more difficult, since the transferred magnetization evolves as sin2 ¢..0i"/m. Modulation of the mixing time is ineffective, since it does not average the coherent signal to zero. However, other differences between the two transfer mechanisms can be used to separate these two effects in rotating frame-experiments. Most notable is the dependence of coherent transfer on the strength and resonance offset of the radio frequency (rf) field BI. Coherent transfer is highly sensitive to the effective field strength; if the field strength is small compared to the frequency difference between the coupled spins, coherent transfer is virtually absent. 9'3°'3~ If, owing to resonance offset effects, the rf field strengths at the frequencies of the coupled nuclei are different, coherence transfer is strongly suppressed. This sensitivity to resonance offset from the B 1 field has been used as a means to suppress coherence transfer artifacts in magnetization exchange experiments. 32 Hwang and Shaka 33 have proposed a multipulse sequence to suppress coherent transfer in the rotating-frame exchange experiment. 29 A. Bax and R. Freeman, J. Magn. Reson. 44, 542 (1981). 30 H. Kessler, C. Griesinger, R. Kerssebaum, K. Wagner, and R. R. Ernst, J. Am. Chem. Soc. 109, 607 (1987). 31 A. Bax, J. Magn. Reson. 77, 134 (1988). 32 j. Cavanagh and J. Keeler, J. Magn. Reson. 80, 186 (1988). 33 T.-L. Hwang and A. J. Shaka, J. Am. Chem. Soc. 114, 3157 (1992).

114

TECHNIQUES

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Interplay of Chemical Exchange and Cross-Relaxation in Two-Dimensional Exchange Spectra of Proteins The introduction of the homonuclear double-resonance NMR experiment 34 facilitated the systematic study of nuclear exchange rates. 35,36 It was noticed that, in many systems, cross-relaxation and chemical exchange take place simultaneously; the overall effect was named the generalized nuclear Overhauser effect.37 Similarly, the term "transferred N O E " has been coined for the cases when the NOE in a small molecule is enhanced by its transient binding to a macromolecule. 38-42 With 2D and n-dimensional (nD) methods, the single term "spin exchange" encompasses all combinations between chemical exchange and cross-relaxation. To grasp the problem, one expands Eq. (1) into a Taylor series: a(,rm)a(O ) - 1

=

exp[LT.rn ] = 1 + L'r m + ½(LT"m)2 + ~(La-m) 3 + . . .

or for a single matrix element

aij(Tm)

1~

aii(O) = Lij"rm + 2 k

(8)

ij: 2

1

3

LikLkj'rm + 6 ~k ~l LikLklLIj~'m 4 - . . .

(9)

To reconstruct the dynamic matrix L, or its elements Lij, by either full matrix analysis, 43-48 by Eq. (1), or by initial buildup rates, 4'49-53 as given 34 W. A. Anderson, Phys, Rev. 102, 151 (1956). 35 S. Fors6n and R. A. Hoffman, J. Chem. Phys. 39, 2892 (1963). 36 R. A. Hoffman and S. Fors6n, J. Chem. Phys. 45, 2049 (1966). 37 j. M. Lawlor and J. P. Warren, J. Magn. Reson. 7, 319 (1972). 38 p. Balaram, A. A. Bothner-By, and J. Dadok, J. Am. Chem. Soc. 94, 4015 (1972). 39 p. Balaram, A. A. Bothner-By, and E. Breslow, J. Am. Chem. Soc. 94, 4017 (1972). 40 A. A. Bothner-By and R. Gassend, Ann. N.Y. Acad. Sci. 222, 668 (1972). 41 G. M. Clore and A. M. Gronenborn, J. Magn. Reson. 53, 423 (1983). 42 G. M. Clore and A. M. Gronenborn, J. Magn. Reson. 48, 402 (1982). 43 j. Bremer, G. L. Mendz, and W. J. Moore, J. Am. Chem. Soc. 106, 4691 (1984). 44 R. Boelens, T. M. G. Koning, and R. Kaptein, J. Mol. Struct. 173, 299 (1988). 45 B. A. Borgias and T. L. James, this series, Vol. 176, p. 169. 46 R. Boelens, T. M. G. Koning, G. A. Van der Marel, J. H. Van Boom, and R. Kaptein, J. Magn. Reson. 82, 290 (1989). 47 B. A. Borgias and T. L. James, J. Magn. Reson. 87, 475 (1990). 48 B. R. Leeflang and M. J. Kroon-Batenburg, J. Biomol. NMR 2, 495 (1992). 49 A. Kumar, G. Wagner, R. R. Ernst, and K. Wiithrich, J. Am. Chem, Soc. 103, 3654 (1981). 50 S. G. Hyberts and G. Wagner, J. Magn. Reson. 81, 418 (1989). 5E A. Majumdar and R. V. Hosur, Biochem. Biophys. Res. Commun. 159, 886 (1989). 52 j. Fejzo, Zs. Zolnai, S. Macura, and J. L. Markley, J. Magn. Resort. 82, 518 (1989). 53 j. Fejzo, Zs. Zolnai, S. Macura, and J. L. Markley, J. Magn. Reson. 88, 93 (1990).

[3]

2D EXCHANGESPECTROSCOPY

115

by Eq. (9), one needs to separate the contributions of chemical exchange and cross-relaxation from Lij according to Eqs. (4) and (5). In cases where full matrix analysis is not practical, the quadratic and higher order terms in Eq. (9), LikL~j, Lik.Lk~Ltj. . . . . which represent two, three, etc., step transfer, respectively, are a source of error in the analysis of the linear term. The nonlinear terms can overwhelm the desired linear term by the magnitude of individual Lkl terms, by a large number of exchange pathways k, l, or by long mixing times (large ~'Zm). Traditionally, the analysis of L in terms of cross-relaxation spectroscopy (as R) or chemical exchange spectroscopy (as K) has been treated separately, by either full matrix analysis or initial buildup rate analysis. Interference between the two processes of magnetization transfer has been avoided, either by careful selection of the system under study (rigid molecules R~j >> Kij, extreme narrowing and/or fast exchange Kij >> R U) or by varying the exchange and cross-relaxation rates by adjusting the sample conditions. Under these conditions, L is easily separated into R and K according to Eqs. (4) and (5). More recently, the separation of the two processes by spectroscopic means has been facilitated by the development of 2D and nD techniques. 54

Multistep Magnetization Transfer Magnetization exchange takes place in single steps, for example, i

j,j ~ k . . . . . For sufficiently long mixing times, the transfer of magnetization over a common partner j, i ~ j ~ k, creates the illusion of i ~ k transfer, even when this process does not exist. In a system undergoing pure chemical exchange, the problem is simplified, since the transfer pathway are restricted by the chemical structure of the system. In macromolecular systems, however, cross-relaxation leads to the immense problem of unrestricted magnetization transfer among all of the protons in the system. If the macromolecule also contains chemical exchange processes, then the problem of unrestricted transfer is even worse since chemical exchange rates can be an order of magnitude larger than the largest crossrelaxation rate.

Spin Diffusion Spin diffusion represents multistep magnetization transfer arising from (fast) cross-relaxation. 55,56 As indicated in Table I, in macromolecules, 54 D. G. Davis and A. Bax, J. Magn. Reson. 64, 533 (1985). 55 A. Kalk and H. J. C. Berendsen, J. Magn. Reson. 24, 343 (1976). 56 S. L. Gordon and K. Wiithrich, J. Am. Chem. Soc. 100, 7094 (1978).

116

TECHNIQUES

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cross-relaxation is not associated with overall relaxation and can take place unrestricted over all available paths. For longer mixing times, magnetization migrates between distant spins by multiple transfer over common cross-relaxation neighbors. For rigid molecules, Kij = 0, and Eq. (9) becomes aij(rm) aii(O) -

1- E R 2 1 Rijrm + 2 k Rik kjrm + - 6 ~ ~ RikRkIRIjr3m + "'"

(10)

where the summation terms indicate multistep transfer. Terms quadratic in rm stem from two-step transfer, cubic terms from three-step transfer, etc. Indirect transfer can lead to an observable (nonzero) cross-peak volume aij even when Rij = 0:

aij(rm) 1 2 = + ~ ~, RikRkjrm + . . . aii(O) k

(11)

Multiple-step transfer will also affect cross-peak intensities when Rij ¢ 0, if the mixing time is sufficiently long so that Rij < ½Ek RikRkjT"m, since the quadratic term is then much larger than the linear term. Several spectroscopic and computational methods have been proposed to deal with spin diffusion. In the initial buildup rate approximation, 4'49 the experiment is performed with a very short mixing time. As long as Rijr m "~ 1, quadratic and higher order terms are negligible in comparison with the linear term. The disadvantages of short mixing times are that NOE contacts are observed only between close spins and that the overall intensity of the cross-peaks is small. By using the quadratic approximation of Eq. (10) 4,50-53 longer mixing times can be used, but a series of 2D experiments with different mixing times must be recorded. Full matrix analysis 43,45,48is generally regarded as the method of choice, but it has a serious disadvantage in that all elements of the spectral volumes must be known; if they are not known, then iterative procedures must be used to solve for the missing r a t e s . 44'46'47 Landy and R a o 57 have shown that full matrix analysis based on a single mixing time experiment cannot reconstruct cross-relaxation rates when there is a large spread in the rate constants. In the course of magnetization transfer, the cross-peaks build up as the diagonal peaks decay. At long mixing times the cross-peaks and diagonal peaks obtain equal intensities before decaying to zero through T~ relaxation. The problem is that during a mixing time that is sufficiently 57 S. B. Landy and B. D. N. Rao, J. Magn. Reson. 83, 29 (1989).

[3]

2D EXCHANGESPECTROSCOPY

117

long to observe moderate or slow magnetization transfer processes, fast magnetization exchange processes equilibrate the cross-peak and diagonal peak intensities. Under these conditions, it is impossible to determine the rate of the fast magnetization exchange since the same data can be obtained from many combinations of rates and external relaxation times. Only the lower limit of the magnetization exchange rate can be estimated. To demonstrate this problem, we have chosen a three-spin system in four different geometries, as shown in Table II. All the cross-relaxation systems yield the same NOESY spectra within the 50% experimental error, typical of observed volumes of small crosspeaks. The cross-relaxation network for the data cannot be unraveled even by a full matrix analysis. This ambiguity arises as a consequence of fast cross-relaxation among close spins during a longer mixing time. In the case shown in Table II, the mixing time of 0.4 sec, which is required to observe finite cross-peaks among the more distant spins, produces almost equal intensities for cross-peak a~2 and diagonal a11; similarly, cross-peaks a13 and a23 show equal intensities. As a consequence, the NOESY spectrum of the linear system is indistinguishable from the spectrum of the triangular system. For the third and fourth systems in Table II, where RI2"rm = 3 × 0.4 = 1.2 -> 1, the problem is less severe but depends critically on the experimental errors. If the product R r m is small and the relative error of cross-peak volumes is large, a full matrix analysis will be unable to distinguish among various structures on the basis of NOESY data recorded at a single mixing time. To discern the magnetization exchange network unambiguously, additional experiments are required.

Chemical Exchange-Mediated Spin Diffusion Proteins exhibit a variety of internal motions which may affect exchange spectra in different ways. We focus here only on motions that are slow on the NMR time scale and hence are suitable for study by 2D exchange spectroscopy. The interplay of slow chemical exchange and cross-relaxation in macromolecules has been anticipated58-6° and experi58 S. B. Landy and B. D. N. Rao, J. Magn. Reson. 81, 371 (1989). 59 j. H. Prestegard and Y. Kim, in "Computational Aspects of the Study of Biological Macromolecules by Nuclear Magnetic Resonance Spectroscopy" (J. C. Hoch, F, M. Poulsen, and C. Redfield, eds.), p. 269. Plenum, New York, 1991. 6o p. F. Yip and D. A. Case, in "Computational Aspects of the Study of Biological Macromolecules by Nuclear Magnetic Resonance Spectroscopy" (J. C. Hoch, F. M. Poulsen, and C. Redfield, eds.), p. 317. Plenum, New York, 1991.

118

[3]

TECHNIQUES

•-~. --: c~

/

N

I

I

0 Z

Z

<

I

e4

I

T I

II

u~ r~

I.... < 'r-

.<

I-

E

O

t-

r~ O ...=

",3"

e~

I r~ 0 Z

I

oO t ' ~ eq

I

~s c~

r~ eq 0

I.-

olo

Z

I

r.

I

¢q

I

[3]

2D EXCHANGE SPECTROSCOPY

119

tr'~

0@@"

i

©@@-

X

o@@ 4

s II

-%

II

E

@@®. @0@e@@.

"5 u~

i

r.¢?

ea

i

@
~'5

r-

s6

1¢5

~0~

TECHNIQUES

120

:2

82

:2 sec-

c: 2.s

8:

[3] o

82 k

-i

c

o

-1

8:

FIG. 2. Aromatic ring and the magnetization exchange network. Spins on the same side, (el,/51) and (e2, 82), exhibit cross-relaxation with the rate o- owing to close proximity; those on the opposite sides, (el, e2) and (81, ~2), exhibit chemical exchange by ring rotation with exchange rate k.

mentally demonstrated. 59,6~-64A particularly useful aspect of it is the transferred NOE, 41'42 which was documented relatively recently. 65'66 One of the simplest examples of chemical exchange in a protein is the flipping of aromatic rings by 180° about the C~-C ~ bond. For symmetrical rings (tyrosine, phenylalanine), such ring flips do not lead to a structural change. However, depending on the resonance frequency and the chemical shift difference between the aromatic nuclei in the two orientations, separate resonances can be observed for nuclei in nominally symmetrical positions, even at flip rates as fast as 100 s e c - t In comparison, typical cross-relaxation rates with ~'c = 6 nsec and rmin = 1.75 ,~ do not exceed 10 sec-i. Thus, slow ring-flip rates can be an order of magnitude faster than the fastest cross-relaxation rate and can influence unrelated crossrelaxation pathways on opposite edges of the ring over a distance of 4 ,~ or more. Such an effect has been predicted by computer simulation67,68 and has been demonstrated experimentally.63,64 For illustration, we use the analytical solution obtained recently for a spin pair undergoing simultaneous chemical exchange and cross-relaxation. 61'62'65 Four protons of a tyrosine ring el, s2, 61, ~2 exhibit pairwise cross-relaxation and chemical exchange as indicated in Fig. 2. The inter6I B. Choe, G. W. Cook, and N. R. Krishna, J. Magn. Reson. 94, 387 (1991). 62 W. Lee and N. R. Krishna, J. Magn. Reson. 98, 36 (1992). 63 j. Fejzo, A. M. Krezel, W. M. Westler, S. Macura, and J. L. Markley, Biochemistry 30, 3807 (1991). 64 j. Fejzo, W. M. Westler, S. Macura, and J. L. Markley, J. Magn. Reson. 92, 195 (1991). 65 G. M. Lippens, C. Cerf, and K. Hallenga, J. Magn. Reson. 99, 268 (1992). 66 N. R. Nirmala, G. M. Lippens, and K. Hallenga, J. Magn. Reson. 100, 25 (1992). 67 S. Macura, J. Fejzo, W. M. Westler, and J. L. Markley, Book of Abstracts, 14th 1CMRBS (Intl. Conf. Magn. Reson. Biol. Systems), Warwick, England (1990) (Abstract). 68 R. Kaptein, T. M. G. Koning, and R. Boelens, in "Computational Aspects of the Study of Biological Macromolecules by Nuclear Magnetic Resonance Spectroscopy" (J. C. Hoch, F. M. Poulsen, and C. Redfield, eds.), p. 349. Plenum, New York, 1991.

[3]

2D EXCHANGESPECTROSCOPY

121

proton distance 8281 (or e182) is approximately 5 ,~, and the respective direct cross-relaxation paths can be safely neglected. The time evolution of the amplitude of the cross-peaks between the different protons of the aromatic ring can be expressed by the system of equations derived for a model in which two spins undergo a conformational exchange between two equally populated sites61: D 18182 = 18¢2 = 7 [ 1 - exp(-2krm)][ 1 + exp(2o'rm)] D I8,8t = 18282= ~-[1 + exp(-2krm)][1 - exp(2o-rm)] D 18182 = 18281= ~-[1 - exp(-2krm)][1 - exp(2orrm)]

(12)

D I8~8i = 18181 = -~-[1 + exp(-2krm)][1 + exp(2o-rm)] where parameter D also represents the total magnetization of site 81, that is, the sum of the elements along the 81 row or column:

aij(rm) ajj(O)

Iijk = Kij,

o-=R

v

n i = nj = 1 and D = exp(-rm/Tl) D = 18182 ~- 18181 + 18182 + I8181 Equation (12) shows that each peak amplitude depends on both chemical exchange and cross-relaxation. However, by a few simple rearrangements of Eq. (2), the effects of the two processes can be separated. (a) Normalization of cross-peaks by diagonal peaks: 8182

= tanh(krm)

I8181

I8181

(13) = tanh( - O'rm)

8181

(b) Normalization of indirect cross-peaks by direct cross-peaks:

8182

8182

= tanh( - trrm)

(14)

8182 -

8182

tanh(krm)

122

TECHNIQUES

[3]

(c) Normalization of the combined cross-peaks by the magnetization of the column or row vectors: l~le2 + lala2

1

D

2

[1 - e x p ( - 2k~'m)]

(15) lelal + Ieta2

D

=

[1 - exp(2O-rm)]

One should note the equivalence between Eqs. (13) and (14). The evolutions of peak volumes and some of their combinations are shown in Fig. 3. In Fig. 3a, cross-peak intensities are shown as a function of the mixing time according to Eq. (12). Figure 3b shows the same

b

0.25

!

0,2 0.15 fe i6 1

0.i 0.05

I

0

I

I

ij 0.8 0.6 0.4 0.2 o o

o.i

0.2

0.3

0.4

o

o.i

0.2

0.3

0.4

"[ m ( S e c )

FIG. 3. Simulation of magnetization exchange in an aromatic ring. Assumptions: Protein correlation time ~'c = 6 nsec at oJ0/27r = 500 M H z (o" = - 1.4 sec-~), ring flip rate k = 14 sec-l). (a) Buildup curves from the cross-peaks according to Eq. (12). (b) Same buildup curves with superimposed Gaussian noise corresponding to S/N = 140 of volume ajj(0). Noise superposition is useful for the visualization of error propagation. (c) Diagonal peak volume and normalized cross-peak intensities. Normalized intensities have the same shape as in (d), except that they are more stable with respect to errors in volume integrals. (d) Reconstruction of elementary exchange rates, k and o-, from the ratio of indirect cross-peak volume versus direct cross-peak volume [Eq. (14)]. The different noise levels reflect different sensitivity to errors in the cross-peak volume integration.

[3]

2D EXCHANGESPECTROSCOPY

123

functions with Gaussian noise superimposed which mimics a signal-tonoise ratio (S/N) of about 140 for the diagonal peak volume at z m = 0. Independent noise of the same peak-to-peak amplitude has been added to every curve in Fig. 3. The noise is uncorrelated and thus faithfully portrays the error propagation in the different combinations of peak volumes. In the presence of superimposed noise as shown in Fig. 3b, the amplitudes of the direct cross-peak el61 and the indirect cross-peak el61 are equal (within the noise level) at Zm = 200 msec owing to the fast chemical exchange process. These data are analogous to the cases shown in Table II. At Z~n --> 200 msec, even full matrix analysis cannot recover the exchange rates of the individual processes. Figure 3c,d shows the cross-peak volumes normalized according to Eqs. (13) and (14), respectively. As expected from Eqs. (13) and (14), the buildup curves are identical in both cases. However, the noise levels (i.e., the effective S/N) are significantly different. As the data in both cases are generated from the same rate constants with the same (uncorrelated) noise level, the difference must come from the difference in the sensitivity to error propagation for the two normalization modes. Although this may not be obvious from Eqs. (13) and (14), it is easy to understand. In the case shown in Fig. 3d, the buildup curves are obtained from cross-peaks exclusively [Eq. (14)]. For short mixing times, cross-peaks have poor S/N, and thus the constant noise propagates into the ratio. The same buildup curves generated by normalization with the diagonal peak [Fig. 3c, Eq. (13)], show much better S/N because the diagonal peak is well defined at the short mixing time and the single cross-peak is the major source of the noise. In principle, the elements of a complex exchange network can be discerned from the peak volumes of 2D exchange spectra recorded at any finite mixing time. In practice, however, because of experimental errors, the exchange network can be studied experimentally only within a narrow range of mixing times. The position and width of the optimal mixing time range depend on several unrelated and/or unknown parameters (S/N, chemical exchange rate, geometry, etc.) and hence must be determined by experiment. Therefore, even if a full matrix analysis is used with observed values for all of the matrix elements, 2D exchange experiments must be collected at several different mixing times in order to remove ambiguities caused by noise. General Principles of Exchange Network Editing An alternative to the analysis of the full magnetization exchange network is to modify the exchange experiments in order to eliminate selected

124

TECHNIQUES

[3]

cross-relaxation or exchange pathways. We define exchange network editing, as any useful modification, by experimental means, of the dynamic rate matrix L. This may involve either the removal of one or more elements of L or the removal of a particular type of contribution from L (R, Rij, or Kij). Many, seemingly different methods for exchange network editing have been proposed. 54'64'69-75 All the methods can be represented by a set of general principles, which are easily rationalized from the scalar forms of Eq. (2) N l~li = Liimi + E j=l

Lijmj

(16)

and Eq. (4) N l~ni = ( g i i

-

Rii)mi + ~, (Kii - Rij)mi j=l

(17)

The auto term, Lii, includes all processes by which magnetization migrates away from the site i, (direct relaxation, cross-relaxation, and chemical exchange). The summation terms represent the individual paths for chemical exchange, Kij, and cross-relaxation, R;j. Exchange network editing refers to the removal of one or more of the summation terms. To modify any term, one must experimentally intervene during the exchange process. In practice, the magnetization, mj, is forced, by experimental modifications, to undergo additional motion during the mixing period Zm. During an editing experiment, one can define an effective magnetization, m~re, and effective exchange rates, K~ff and R,~ft. Chemical exchange rates Kij do not depend on the NMR properties of the system, and therefore K~.ff = Kii. From this fact and the assumption that the detected magnetization is not perturbed by editing, we obtain from Eq. (17): N

th i = (Kii - Reff)mi + ~ (Kij j=l

eft'} ~. eft - Rij ,mj

(18)

69 E. T. Olejniczak, R. T. J. Gampe, and S. W. Fesik, J. Magn. Reson. 67, 28 (1986). 70 W. Massefski, Jr., and A. G. Redfield, J. Magn. Reson. 78, 150 (1992). 71 j. Fejzo, W. M. Westler, S. Macura, andJ. L. Marldey, J. Am. Chem. Soc. 112, 2574 (1990). 72 j. Fejzo, A. M. Krezel, W. M. Westler, S. Macura, and J. L. Markley, J. Magn. Reson. 92, 651 (1991). 73 j. Fejzo, W. M. Westler, S. Macura, and J. L. Markley, J. Magn. Reson. 92, 20 (1991). 74 S. Macura, J. Fejzo, C. G. Hoogstraten, W. M. Westler, and J. L. Markley, Isr. J. Chem. 32, 245 (1992). 75 j. Fejzo, W. M. Westler, J. L. Markley, and S. Macura, J. Am. Chem. Soc. 114, 1523 (1992).

[3]

2D EXCHANGESPECTROSCOPY

125

Because Kq terms are invariant to spin manipulations, the only terms in the summation that one can influence are R ~rf and m~el. Individual summation terms become zero by setting either R ~.ffor m~ff equal to zero. Because magnetization exchange takes place steadily during the mixing time, it is important that the average of the effective values of R ~ or m~tT equal zero, that is, the necessary editing condition is (R~fr) = 0 and/or (m~f~ = 0.

Setting ( m y f) = 0 The effective magnetization (m~ff) can be averaged to zero in three different ways. The first way is by continuous (either selective or band selective) irradiation of a set of selected spins during the entire experiment. 69 The irradiated spins are saturated, and therefore m~rf = 0 at all points during the mixing time. The saturation destroys all of the magnetization for the affected spins and, therefore, no magnetization can be recovered. In the second method, the selective irradiation is applied only during the mixing time. 7° Since "/'m ~-~ T1, irradiated magnetization is not truly saturated, but rather dephased. In principle, the dephased magnetization is recoverable. In the third method, a sequence of phase-alternated 180° pulses is applied during the mixing time. The 180° pulses can be either selective 64 for one mj or semiselective TMfor a whole class of mj values. The magnetization mj is continuously rephased by the 180° pulses and, consequently, is preserved during the experiment. The preserved magnetization rnj can be observed in a subsequent detection period. However, from the viewpoint of noninverted magnetization mi during the mixing time, the time average of the magnetization mj is zero, (m~~) = 0.

Setting ( R ~ f) = 0 The effective cross-relaxation rate for a pair of spins (ij) undergoing motion imposed by sequence of pulses is a weighted average of the longitudinal rate o-~ and the transverse r a t e O'~j76'77: O',~ff = sin Oi sin 0jo-~ + cos Oi cos 0jo-~j

(19)

where the weighting coefficients depend on the angle 0 between the spin polarizations and the external magnetic field. The detailed description of the cross-relaxation depends on the properties of the mixing-pulse 76 D. G. Davis, J. Am. Chem. Soc. 109, 3471 (1987). 77 C. Griesinger and R. R. Ernst, J. Magn. Reson. 75, 261 (1987).

126

[3]

TECHNIQUES

propagator78; however, some limiting cases are instructive. If both magnetizations are parallel to B 0 (Oi = Oj -- 0°), then 0-~ff = o-~.; if they are orthogonal (Oi = 0/ = 90°), then 0"~ff o-~. The effective cross-relaxation rate is zero if one spin is in the laboratory frame (0i = 0 °) and the other is in the rotating frame (Oj = 90°), irrespective of the o-"/0- r ratio [Eq. (19)]. Other angles for which 0"~ff = 0 critically depend on the ratio of cross-relaxation rates in the two frames: (tan Oi tan 0j)~oN=0 -

o-~j

0-b

(20)

For large molecules undergoing isotropic, rigid motion, 0"~/o-~. = - 1/2 [Eq. (17)], and the effective cross-relaxation vanishes if the magnetizations are spin-locked along the angle complementary to the "magic angle," 0~= 0 = 90 ° - 54.740. 71 This angle is obtained from (tan 0 ) J = o

= (__ 0-/j/0-U I n , r ,I/2

(21)

With respect to exchange network editing, Eqs. (19) and (20) are of limited importance, since they can operate only on one or a few spins. For example, a single spin can be detached from a cross-relaxation network by a selective spin-lock experiment. The spin which is locked in the rotating frame does not cross-relax with other spins that remain in the laboratory frame, irrespective of the ratio of cross-relaxation rates. Equation (21), although limited to cases where o,"/o-r < 0, is more useful, since it applies to spins that are locked in parallel. This allows for the manipulation of many spins simultaneously. If the magnetization vectors follow parallel trajectories [Oi(t ) = Oj(t)], while hopping between the laboratory and rotating frames, then the effective cross-relaxation rate constant can be expressed as a time average of the cross-relaxation rates in the respective frames7)'73: 0-elf = _7"r _ 0 - r . + -7n -0"9. ~'m ~J Tm ~J

(22)

where Tr and T n are the residence times of the spins in the respective frames, and ~'m is the total mixing time, ~'m = ~.r -k- Tn. Neglecting the time during pulses, effective cross-relaxation vanishes when T n

O- r

T r

O-n

78 C. Griesinger and R. R. Ernst, Chem. Phys. Lett. 152, 239 (1988).

(23)

[3]

2D EXCHANGESPECTROSCOPY

127

This condition has been used for the complete elimination of cross-relaxation in total correlated spectroscopy (TOCSY) 79'8° and exchange spectra of macromolecules. 71'73 Combined

Effects

We have considered above the simplest cases for the elimination of undesired exchange pathways by setting the time averages of either R~3ff elf to zero. The time averages can be independently manipulated, or m ~i leading to some interesting experiments, by combining the two averages. From Eq. (1) and Eq. (22), a general expression for effective magnetization exchange is obtained when exchange takes place in a combination of the two frames: N E /eft.

N eft

--ij trlij

j=t

=

~ (K(i _

R e f f ] ~ eft -'ij

~trt(]

j=I N

= ~ [ K i j ( f r m } + f " m } ' ) - (frR[im~. +f"R~m~.)]

(24)

j- l

where f , f " are the respective fractions of the total mixing time that the magnetization spends in the rotating and laboratory frames (f~." = r~'n/ ~'m)" If one particular magnetization component, mj., is inverted every time the magnetization goes from one frame to the other, then only one component of the summation is selected. The inversion occurs such that mj = mjn = - m ~ ; from Eq. (24), one obtains Left... ij " ' jeft = K i j m j ( f " - fr) _ m j ( f " R ~

- f rR i rj )

(25)

F o r f n = f~ = ½ (and R~ = -2R~.) Leff 3 /j = _R~f f = _ ~R~j.

(26)

The upshot of this network editing scheme is that the chemical exchange contribution for a given pathway is removed selectively. This result is rather academic because chemical exchange and cross-relaxation do not, in general, share the same pathway. A more realistic situation is one in which chemical exchange and cross-relaxation take place simultaneously but by different paths, interfering with one another only in second or higher order transfers. The individual elements of L in this case are solely due to either chemical exchange, Lij = K~j, or cross-relaxation, L~i = R~j. 79 C. Griesinger, G. Otting, K. Wfithrich, and R. R. Ernst, J . A m . C h e m . S o c . 110, 7870 (1988).

so D. W. Bearden, S. Macura, and L. R. Brown, J. Magn. Reson. 80, 534 (1988).

128

TECHNIQUES

[3]

In a system without chemical exchange (Ko = 0), all elements arise from cross-relaxation. Equation (25) along with the conditions that are used to eliminate cross-relaxation in rigid macromolecules (r n = 2r r or fn = 2/3 a n d f r = 1/3; R,~. = -2R~.) yields Left ij

= - R i jeft = - ~4R ~ jn

(27)

Consequently, even when all cross-relaxation paths are suppressed by the averaging of rotating- and laboratory-frame cross-relaxation, crossrelaxation between a chosen site (j) and all of its neighbors can be observed by selectively inverting thej spin at a time between the laboratoryand rotating-frame cross-relaxation periods. Magnetization exchange between the j site and all other sites is preserved since cross-relaxation in the two frames has the same sign. Such an experiment has been demonstrated .75 Experimental Methods for Elimination and Separation of Transfer Processes Some of the experimental requirements for a given exchange network editing scheme can be difficult or even impossible to achieve. For example, to time average rn~ff to zero in a cross-relaxation network, mj must be inverted at least several times during a given mixing time, typically 50-300 msec. Simultaneously, to achieve frequency selectivity, one has to apply very long pulses, typically 5-50 msec. The duration of the selective pulse often can be of the length of the desired mixing time. Practical implementation of any method requires careful optimization of several experimental parameters. Figure 4 shows the basic pulse schemes by which exchange network editing can be achieved. Only the events during the mixing period are shown. Figure 4a shows a mixing period for the basic 2D exchange experiment (also see Fig. la); in this experiment, longitudinal magnetization exchanges freely by all available mechanisms and paths. In the rotatingframe exchange experiment (Fig. 4b), a continuous (CW) B~ field is applied during the mixing period to spin-lock the magnetization. As in the longitudinal experiment, exchange takes place by all available pathways and mechanisms in the rotating frame. A simple editing sequence, shown in Fig. 4c, consists of a series of selective 180° pulses applied during the mixing time. This sequence inverts a selected region of spins, within the bandwidth of the 180° pulse, at regular intervals r. The effective magnetization of the inverted resonances, as seen from the laboratory frame, averages to zero. The inversion pulses

[3]

2D EXCHANGESPECTROSCOPY

129

a

c

e

b

d

f

spin

A

lock

li°

lio

I FIG. 4. Pulse schemes for various editing methods. Only the events during mixing time are shown. (a) Laboratory-frame 2D exchange experiment (NOESY). (b) Rotating-frame exchange experiment (ROESY). (c) XD.NOESY with single frequency inversion; BD.NOESY if 180° pulses are band-selective. (d) XD.NOESY with multiple frequency inversion. (e) Pure chemical exchange. (f) S.NOESY; BCD.NOESY if 180° pulses are band-selective.

can be selected for a single frequency or semiselective for a given frequency region, depending on the experimental requirements. If the phase and amplitude of the pulses are constant, then with tp = 7"the sequence becomes equivalent to CW irradiation during the mixing time with associated dephasing of the irradiated spins. If, however, the phases of the inversion pulses compensate for pulse imperfections, then the irradiated spin is not dephased and can be recovered. If the 180° pulses in the sequence are frequency switched, as indicated by the different fill patterns in Fig. 4d, then the magnetization m eUof two or more different frequencies can be simultaneously averaged to zero and thus removed from the crossrelaxation network. Figure 4e shows an experiment in which magnetization exchange takes place alternately in the laboratory and rotating frames. The first 90 ° pulse creates longitudinal magnetization, which is frequency labeled during the tl evolution period. This magnetization exchanges in the laboratory frame during the period rn/2. The second 90°x pulse rotates the longitudinal magnetization onto the y axis of the rotating frame where it is then spinlocked by CW irradiation. During the spin-lock time 7"r magnetization exchanges in the rotating frame. Magnetization is then rotated back along the z axis by a 90°_x pulse, and in the subsequent rn/2 period magnetization exchange again takes place in the laboratory frame. This cycle is repeated k times during the mixing period. During this sequence the effective magnetization exchange is L eft = f n L n + f r L r

(28)

130

TECHNIQUES

[3]

and f o r f n / f r = r " l r ~ = 2, in accordance with Eqs. (7) and (23), crossrelaxation is canceled leaving only the chemical exchange portion of L [Eq. (29)]. L elf= K

(29)

Cross-relaxation that takes place in one frame is opposed and canceled by cross-relaxation in the other frame. Chemical exchange survives since it is independent of the reference frame in which it takes place. Finally, Fig. 4f shows a hybrid of the pulse sequences shown in Fig. 4c,e. Surrounding the [90°-spin lock-90 °] element, 180° pulses are inserted to invert a selected band of magnetization. Cross-relaxation between the inverted and noninverted resonances is additive in the two frames and takes place with the effective rate given by Eq. (27). If r n / r r = 2, then cross-relaxation among all of the noninverted spins is suppressed. The trajectories for the spins within the inversion bandwidth are parallel, so cross-relaxation in the two frames is of opposite sign. Cross-relaxation is canceled within the inverted band by the same mechanism as for the noninverted spins when r " / r r = 2. Elimination of Cross-Relaxation from Chemical Exchange Spectra

In laboratory-frame 2D exchange spectra of macromolecules, crossrelaxation and chemical exchange are indistinguishable, since both give rise to positive cross-peaks. 4 In rotating-frame exchange spectra of macromolecules, cross-relaxation and chemical exchange are distinguishable, since cross-peaks of the former are negative with respect to the diagonal whereas those of the latter are positive. 8~Two-step magnetization transfer in the rotating frame also gives rise to positive cross-peaks. 82'83To identify chemical exchange in proteins unambiguously, it is necessary to eliminate cross-relaxation completely, that is, to edit the exchange network according to Eq. (23). In principle, this can be achieved by the experimental scheme depicted in Fig, 4e. Some practical modifications of the same sequence are shown in Fig. 5. Figure 5a is the same sequence as in Fig. 4e except that the phase of the [90°-spin lock-90 °] element is rotated between orthogonal axes. If the period r r is sufficiently short compared to the reciprocal of the chemical shift difference for the resonance involved in cross-relaxation [i.e., rr(A~O)ma×~ 1, where A~o = co0 - oJij], then dephasing of the transverse magnetization can be neglected, and the spin-lock field can be omitted 81 D. G. Davis and A. Bax, J. Magn. Reson. 64, 533 (1985). 82 D. Neuhaus and J. Keeler, J. Magn. Reson. 68, 568 (1986). 83 B. T. Farmer II, S. Macura, and L. R. Brown, J. Magn. Reson. 72, 347 (1987).

[3]

2D EXCHANGESPECTROSCOPY

a

131

b 9(3x

98 -x

c

98 y

98 _y

d 98 x 9 0 - x

98 x

90 -x

90 g

90 -U

e 98 x-qS-x98-x90 x

360 x

360_ x

FIG. 5. Mixing pulse sequences for the elimination of cross-relaxation. (Reprinted from Fejzo eta/. 73)

(Fig. 5b). Cross-relaxation during the pulse must be taken into account whenever the pulse duration is not negligible when compared to ~.n.r. Neglecting resonance-offset effects, magnetization is rotated by the pulse from the z axis into the transverse plane with constant angular velocity. Thus the magnetization spends, on average, half of the pulse time in the laboratory and the other half in the rotating frame. For the sequence in Fig. 5b, the average cross-relaxation times spent in the two frames are ~-" + /90 and Tr "q- /90 (two 90 ° pulses per cycle). Then from Eq. (22) it follows that OrUff --

"rn + /90 'TI + t90 0 -r. + 0-~ ~. n + ~ + t9° ~j T n + T r + /90

(30)

From Eq. (7), one obtains the following condition for the elimination of cross-relaxation by the sequence from Fig. 5b: r " = 2~ r + t90,

(31)

This sequence is very similar to the W A H U H A sequence,84 which is used to keep magnetization at the angle, 0 = cos -~ (1/3vz). In our sequence Fig. 5b the corresponding angle is/3 = 90 ° - 0 = tan -1 (1/2~/z). 84 j. S. Waugh, L. M. Huber, and U. Haeberlen, Phys. Rev. Lett. 20, 180 (1968).

132

TECHNIQUES

[3]

Figure 6 contains spectra of turkey ovomucoid third domain (OMTKY3), a small protein with a molecular weight of 6062, recorded using this sequence along with spectra from the experiments shown in Fig. 4a (laboratory-frame exchange) and Fig. 4b (rotating-frame exchange). At 5°, the tyrosine-31 aromatic ring of OMTKY3 flips slowly on the NMR time scale, thus contributing chemical exchange peaks to the exchange spectra. Chemical exchange cross-peaks between the tyrosine-31 e] and e2 protons (at 7.80 and 6.65 ppm) and 81 and 82 protons (at 7.27 and 6.95 ppm) are observed in both NOESY (Fig. 6a) and ROESY (Fig. 6b) spectra.

a

b

©

:;:::;~ o

0

:;::/

eZ -

0

0 7.50

7.00

7.50

d

C 0

@

7.00

A

B {2. 0-

0 _tt~

t< t3

l i l l l l l [ [ * l l J l l l

7.50

I

l l l l l l l l l ' l

7.00 PPM

''1'

. . . . . . . .

8.00

i .....

''''1

Z50

. . . . . . . . .

ZOO

I''

6.50

PPM

FIG. 6. Comparison of the aromatic region from three 2D cross-relaxation spectra of turkey ovomucoid third domain (OMTKY3): (a) NOESY spectrum obtained with l"rn = 120 msec, (b) ROESY spectrum obtained with a 13 kHz rf field and 60 msec spin lock, (c) pure chemical exchange spectrum collected by using the pulse sequence from Fig. 5b: ~.n = 65 t~sec; z r = 20/xsec; t90 = 19/zsec; 488 cycles within rm = 120 msec. Sample: 15 mM protein in 2H20, pH* 8.1, temperature 5°. (d) Cross sections parallel to oJ2 through the diagonal resonance el (7.80 ppm) from spectra (a)-(c). (Reprinted with permission from Fejzo e t al. 7! Copyright 1990 American Chemical Society.)

0o to

[3]

2D EXCHANGESPECTROSCOPY

133

In the NOESY spectrum (Fig. 6a), all peaks are positive, and no distinction can be made among chemical exchange, direct cross-relaxation, and spin diffusion cross-peaks. In the ROESY spectrum (Fig. 6b), negative crosspeaks clearly indicate direct cross-relaxation peaks, whereas positive cross-peaks are due to either chemical exchange or spin diffusion. However, in the pure chemical exchange spectrum (Fig. 6c) recorded by the pulse sequence shown in Fig. 5b, all cross-peaks due to cross-relaxation are eliminated, and only cross-peaks due to chemical exchange remain. Because both transverse and longitudinal cross-relaxation take place during the 90 ° pulses, the period that the magnetization lies in the transverse plane 7 r can be eliminated, and the mixing pulse sequence can be simplified further to the sequence of Fig. 5c. The delay time q.n during which one-half of the longitudinal cross-relaxation occurs is followed by two sequential 90 ° pulses with opposite phases. Cross-relaxation during the pulses encompasses one-half of the longitudinal and all of the transverse cross-relaxation periods. The condition for the suppression of crossrelaxation [Eq. (31)] is then r" = tg0 and r r = 0. Since symmetrical pulse sequences have more desirable properties, for the compensation of resonance offset and BI field inhomogeneity effects, the sequence of Fig. 5c can be modified to the sequence in Fig. 5d, where the delay r n is set equal to twice the length of a single 90 ° pulse, since half of the longitudinal cross-relaxation is obtained during the pulses. The phases of successive 90 ° pulses are not relevant for incoherent magnetization transfer, and the pulses in the sequence of Fig. 5d can be rearranged to form 180° pulses. By extending this sequence, a phaseinverted 360 ° pulse sequence is obtained. The sequence of Fig. 5e is more desirable than the sequence constructed from 180° pulses, since the phaseinverted 360 ° sequence possesses small rotation angles over a moderate range of imperfections arising from resonance offsets and Bz inhomogeneity. In the pulse sequence of Fig. 5e, the delay time 7 n, which is set equal to the length of a single 360 ° pulse, is followed by an approximately cyclic pulse consisting of two sequential 360 ° pulses with opposite phases. Sequences such as MLEV-4 or WALTZ-4, which have very good cyclic properties, also can be used to suppress cross-relaxation. For these sequences, the delay time 7 n is set equal to one half of the sum of the pulse lengths. Figure 7 shows the aromatic region of the OMTKY3 protein in exchange spectra recorded by the use of the sequences shown in Fig. 5b,d,e. For the mixing pulse sequence of Fig. 5b, with r r = 20/,sec and tg0 = 19 /*sec, Eq. (31) requires that r n = 2.95rk Experimentally, however, we observed that cross-relaxation is canceled at r n = 3.25r r. Computer simulations of two-spin cross-relaxation during this mixing sequence sug-

134

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FIG. 7. Comparison of the aromatic region from four 2D exchange spectra of turkey ovomucoid third domain (OMTKY3). (a) N O E S Y spectrum obtained with ~'m = 120 msec. Only the positive levels are shown. (b-d) Pure chemical exchange spectra collected with the pulse sequence designated (the total mixing time was 120 msec): (b) pulse sequence of Fig. 5b with ~'n = 65 /xsec, t90 = 19/zsec, and r r = 20/xsec, with 488 cycles; (c) pulse sequence of Fig. 5d with tgo = 23.1 /xsec and ~-" = 46.2/xsec, with 866 cycles; (d) pulse sequence of Fig. 5e with tg0 = 93/zsec and ~-" = 93/xsec, with 430 cycles in order to obtain ~'rn = 120 msec. Positive levels are indicated by thin lines and negative levels by thick lines. Sample conditions: 15 m M protein in 2H20, pH* 8.1, temperature 5°. (Reprinted from Fejzo eta/. 73)

gest that effects due to resonance offset and B r field inhomogeneity contribute to an underestimation of the required ~-". The cyclic properties of the sequences of Fig. 5d,e alleviate the requirement for careful calibration of the delay• Spectra collected with the mixing pulse sequences shown in Fig. 5b,d,e are shown in Fig. 7b-d. In all of the spectra, cross-peaks due to cross-relaxation are absent. In the spectrum recorded with the mixing sequence of Fig. 5e (Fig. 7d), strong TOCSY coherence transfer peaks

[3]

2D EXCHANGE SPECTROSCOPY

135

are observed between the e and 8 cross-peaks of phenylalanine-53 (F53). In the spectrum obtained with the mixing sequence of Fig. 5b (Fig. 7b) these peaks are present but have smaller intensity. Finally, in the spectrum recorded with the mixing pulse sequence of Fig. 5d (Fig. 7c) the F53 e/6 peaks are not observed. The data presented in Fig. 8b-d show the extent of suppression of TOCSY peaks in the crowded aliphatic region of the spectrum of OMTKY3 for the mixing pulse sequences shown in Fig. 5b,d,e. The mixing pulse sequence of Fig. 5e does not eliminate all cross-peaks arising from coherence transfer; those close to the diagonal remain (Fig. 8d). The sequences of Fig. 5b,d proved to be the most efficient in the suppression of TOCSY peaks. The pair of negative cross-peaks that remain in spectrum Fig. 8b and the pair of weak positive cross-peaks present in spectrum Fig. 8c a

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FIG. 8. Elimination of coherence transfer from 2D exchange spectra. Comparison of the aliphatic region from four 2D spectra of OMTKY3. Pulse durations, delays, and sample conditions were as in the experiments in Fig. 7. (a) TOCSY spectrum obtained with a 10.6 kHz rf field and r m 40 msec. Only positive levels are shown. (b-d) Pure chemical exchange spectra collected with the pulse sequences designated: (b) pulse sequence of Fig. 5b, (c) pulse sequence of Fig. 5d, (d) pulse sequence of Fig. 5a. Positive levels are indicated by thick lines and negative levels by thin lines. (Reprinted from Fejzo eta/. 73) =

136

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can be ascribed to cross-relaxation that was not canceled owing to the complications introduced by resonance offset effects. None of the pulse sequences appeared to eliminate both coherence transfer and cross-relaxation from chemical exchange spectra, the sequence of Fig. 5d performed the best.

Elimination of Spin Diffusion from Cross-Relaxation Spectra Spin diffusion contributions are serious obstacles to the determination of highly accurate NMR solution structures. 85 In ordinary NOESY (or ROESY) experiments, spin diffusion takes place simultaneously with direct magnetization transfer, but only the latter can be readily interpreted in terms of molecular geometry. Several approaches to the spin diffusion problem have been taken, including the analysis of buildup curves and complete relaxation matrix analysis, which were discussed above. Here we describe two alternative methods. Direct NOESY. Direct NOESY (D.NOESY) is a data processing method for the removal of spin diffusion from cross-relaxation spectra. 7z The method is based on the constant ratio between NOESY and ROESY cross-relaxation rates [Eq. (7)]. The cross-peak intensities in NOESY and ROESY spectra depend on the respective cross-relaxation and autorelaxation rates. Because the autorelaxation rates in the two frames are unrelated (see 1/T~ in Table I), the ratio of cross-peak intensities is not determined by Eq. (7). However, the dependence on autorelaxation rates can be eliminated by normalizing the cross-peaks to the corresponding diagonal in the same experiment. 5z'86 Then 72 aij(Tm) 1 2 Aij(Tm) = aii(7"m) ~-Rij7 m -t- -~ Z RikRkj't'm /- k~ij

(32)

Equation (32) is very similar to Eq. (10). However, here the cross-peak volume at a given mixing time, aij (rm), is divided by the respective diagonal volume at the same mixing time (and not at ~'m = 0), thus eliminating the autorelaxation terms, R;~, R~ (k ~ i, j), from the summation. The approximation in Eq. (32) indicates that this analysis is strictly correct only up to the second order. Substituting Eq. (7), the linear combination [Eq. (33)] of normalized cross-peak volumes Aij(rm) from NOESY and 85 p. D. Thomas, V. J. Basus, and T. L. James, Proc. Natl. Acad. Sci. U.S.A. 88, 1237 (1991). 86 S. Macura, B. T. Farmer II, and L. R. Brown, J. Magn. Reson. 70, 493 (1986).

[3]

2D EXCHANGESPECTROSCOPY

137

ROESY experiments recorded at the same mixing time "/'mdepends only on the direct cross-relaxation rate Rij72: 4A ~.('/'m)

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As a practical matter, the normalized spectra are obtained by dividing the intensity of each column by the intensity of its diagonal element. This procedure assumes that the diagonal peaks do not overlap seriously. The linear combination of normalized NOESY and ROESY spectra, according to Eq. (33), recorded with the same mixing time, yields the hybrid D.NOESY spectrum. Cross-peak volumes in the D.NOESY spectrum depend, to second order, solely on the direct magnetization transfer rates. This method is demonstrated for OMTKY3 in Fig. 9. Cross-peaks in the normal NOESY spectrum of OMTKY3 (Fig. 9a) contain contributions from both direct and multiple-step cross-relaxation. In NOESY, these effects have the same sign and are additive. Since both contributions occur simultaneously, they cannot be separated in a single experiment. The ROESY spectrum of OMTKY3 (Fig. 9b), also contains cross-peaks that are the superposition of direct and multiple-step cross-relaxation. In this case, the two contributions have opposite signs and are subtracted. If the magnitudes of the components are equal, then the cross-peaks are canceled; for example, cross-peak il of Fig. 9a is missing in Fig. 9b. In the ROESY spectrum, cross-peaks with dominant direct magnetization transfer appear inverted (negative sign) with respect to the diagonal peaks, and those with dominant spin diffusion have the same sign (positive) as the diagonal peaks. Therefore, cross-peaks present in the NOESY spectrum that are absent or positive in the ROESY spectrum are clearly influenced by spin diffusion; on the other hand, the influence of spin diffusion on negative cross-peaks cannot be assessed from a single ROESY experiment. A direct cross-relaxation NOESY spectrum (D.NOESY, Fig. 9c) is obtained by combining normalized NOESY and ROESY spectra according to Eq. (33). In comparing the NOESY and D.NOESY spectra, three effects are apparent: (1) cross-peaks whose intensities are equivalent in the two spectra (e.g, cross-peaks d,, d2, and d3) arise from direct magnetization transfer; (2) cross-peaks that have lost one or more contours in the D.NOESY spectrum (e.g., cross-peaks il-i4) indicate the simultaneous presence of direct and indirect magnetization transfer; and (3) cross-peaks present in the NOESY spectrum but not in the D.NOESY spectrum (e.g., cross-peak sO arise from indirect magnetization transfer. The most common way of interpreting 2D NOE data is to classify crosspeaks into categories, for example, "small," "medium," and "large," and

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FIG. 9. Comparison of the amide region of tH NMR exchange spectra (at 500 MHz) of OMTKY3: (a) NOESY spectrum, (b) ROESY spectrum (thick lines indicate positive and thin lines negative levels), (c) D.NOESY spectrum. Sample conditions were 12 mM protein (OMTKY3) in 2H20 containing 0.2 M KC1, pH* 4.0, temperature 300 K. The NOESY mixing time and the ROESY spin-lock period were each set to 90 msec. The NOESY and ROESY acquisitions were interleaved. The original spectra were base-plane corrected. The series of spectra was scaled to a common intensity by reference to peak d~, which is unaffected by spin diffusion as determined from cross-relaxation buildup curves. The same exponential contour levels (1, 2, 4 . . . . . 2") were used with all spectra. Representatives of three classes of cross-peaks are noted in the spectra: dpd3 denote cross-peaks due to direct magnetization transfer, i~-i4 denote cross-peaks with partial contribution from spin diffusion, and Sl and s2 denote pure spin diffusion peaks. (Reprinted from Fejzo et al. n)

then to associate each category with a different distance constraint. In comparing the NOESY and D.NOESY spectra of OMTKY3, one sees that spin diffusion effects cause NOESY cross-peaks classified on this basis to be placed into incorrect categories. The resulting improper distance constraints lead to errors in the calculated structures. Selective NOES Y. In D.NOESY, spin diffusion takes place during the experiment and is removed only later during data processing. An appealing

[3]

2D EXCHANGESPECTROSCOPY

139

alternative approach is to suppress spin diffusion in real time during the experiment, that is, to edit the cross-relaxation network in a suitable way. Several editing sequences have been proposed to constrain crossrelaxation to direct interactions only. TM We describe here the selective NOESY (S.NOESY) sequence, 75which permits evaluation of cross-relaxation between a selected spin (or group of selected, isolated spins) and all of its neighbors. Magnetization transfer between all nonselected spins is prevented, and thus spin diffusion is eliminated. S.NOESY uses the sequence from Fig. 4f, with r n = 2r r. Under these conditions, crossrelaxation between the inverted spin and all of its neighbors takes place with the exchange rate given by Eq. (27), whereas cross-relaxation between all other spins is removed completely. In the example shown in Fig. 10, the sequence is applied to OMTKY3 with selective inversion of amide peaks in the range 9. I-9.8 ppm. When a long mixing time is used, most of the NOESY cross-peaks contain contributions from spin diffusion (Fig. 10a). In contrast, all of the crosspeaks in the S.NOESY spectrum obtained with the same mixing time arise solely from direct magnetization transfer (Fig. 10b). On comparison of Figs. 9c and 10b, it is evident that both D.NOESY and S.NOESY unravel the same pattern of direct cross-relaxation between Tyr 31 H N (~o2 = 9.7 ppm) and its neighbors. Unfortunately, D.NOESY and S.NOESY have disadvantages that depend on the details of the spectrum. The D.NOESY pulse sequence removes the spin diffusion effect over the entire spectrum but fails in regions where diagonal peaks overlap. S.NOESY fails only in regions where diagonal peaks from spatially close neighbors overlap, but the sensitivity is low owing to the long series of rf pulses affecting the observed resonances. Sensitivity loss is inherent to all editing methods which rely on exchange in the rotating frame. This is due mainly to the high autorelaxation rate in the rotating frame.

Elimination of Chemical Exchange Effects from Cross-Relaxation Spectra Selective elimination of chemical exchange effects from NOESY (or ROESY) spectra is difficult since it does not depend on the frame of reference in which it is monitored. The obvious method is to eliminate cross-relaxation in a magnetization exchange spectrum and thus obtain a "pure" chemical exchange spectrum. The chemical exchange contribution can then be "subtracted" from a normal exchange spectrum. Direct elimination of a selected chemical exchange path can be achieved by the pulse sequence shown in Fig. 4f. If ~-" = ~r, then for the inverted resonance

140

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FIG. 10. Cross-relaxation spectra (600 MHz) of OMTKY3 in 2H20 at pH* 4.1 and 25 °. (a) Normal phase-sensitive NOESY spectrum. (b) S.NOESY spectrum obtained with the pulse sequence of Fig. 4f with selective inversion of peaks in the range 9.1-9.8 ppm: 9.20 (Cys 3s HN), 9.37 (Ser 26 HN), 9.39 (Gly 25 HN), and 9.7 ppm (Tyr 31 HN). Both spectra were base-plane corrected and scaled to a common intensity by reference to peak dl, which is unaffected by spin diffusion as determined from cross-relaxation buildup curves. As in the D.NOESY spectrum, three classes of cross-peaks are noted: peaks dl and d2 arise from direct magnetization transfer, peaks i r i 4 contain partial contributions from spin diffusion; and peak st arises from pure spin diffusion. (Reprinted with permission from Fejzo et al. 75 Copyright 1992 American Chemical Society.)

[3]

2D EXCHANGESPECTROSCOPY

141

the effects of chemical exchange in one frame are canceled by the chemical exchange in the other. At the same time, cross-relaxation between the inverted spin and its neighbors takes place by the rate given in Eq. (26).

Elimination of Selected Pathways from Two-Dimensional Exchange Spectra Another method for removal of chemical exchange effects is to eliminate selected chemical exchange pathways. Experiments that remove entire exchange pathways are much more versatile than the ones that remove only cross-relaxation or chemical exchange. Because this class of sequences averages the effective magnetization to zero, the elimination of exchange is indiscriminate for all types of incoherent exchange processes. The basic pulse sequences are shown in Fig. 4c,d. Depending on the selectivity and frequency of the 180° pulses in the mixing period, one can choose to eliminate one or more individual pathways or one or more groups of pathways. Exchange Decoupled NOES Y. The exchange decoupled NOESY (XD.NOESY) (Fig. 4c) was originally proposed for the elimination of fast chemical exchange effects (exchange decoupled) from NOESY spectra, 64 but the method is more general in that all incoherent processes are eliminated. The basic advantage of this experiment over similar experiments 69,7° is the possibility of selectively suppressing more than one exchange pathway by irradiating multiple resonances by frequency switching the rf pulses during the mixing period (Fig. 4d). Figure 11 shows to2 cross-sections along the T y r 31 e 2 resonance of OMTKY3 from three different exchange experiments. As discussed above (Fig. 2), the flipping of the tyrosine ring brings distant spins into contact (el, 62 and e2,81) by chemical exchange-mediated spin diffusion. As shown in Figs. 6a, 7a, and lla, in the normal NOESY experiment, cross-peaks are present from the resonances of the T y r 31 e 2 proton to those of the other three Tyr 31 ring protons: to 81 by direct chemical exchange, to 82 by direct cross-relaxation, and to 81 by two-step magnetization transfer, either cross-relaxation followed by chemical exchange (e 2 --~ 81 ---) 81) or chemical exchange followed by cross-relaxation (e 2 ~ 82 --~ 81). If the e 1 resonance is inverted selectively during the mixing time, the indirect pathway (e 2 --~ e 1 --~ 81) is blocked by decoupling the direct (el ~ 81) pathway. This decreases the intensity of the (e2, 8 0 cross-peak in the XD.NOESY spectrum (Fig. l lb). The cross-peak intensity is not eliminated, since the alternative indirect pathway (e 2 ~ 82 --~ 81) remains active. If, however, resonances el and 62, which mediate indirect transfer, are selectively inverted, both pathways for magnetization exchange between

142

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FIG. 1 1. The to2 cross sections at the frequency of the O M T K Y 3 T y r 31 H ~2 resonance. (a) N o r m a l N O E S Y s e q u e n c e with 7 m = 60 msec. (b) Exchange-decoupled N O E S Y with inversion o f the T y r 3t H "l line during the mixing time (rm = 63 msec) according to the pulse s c h e m e of Fig. 4c. (c) E x c h a n g e decoupled N O E S Y in which both the T y r 31 H ~l and the T y r 31 H e2 lines were inverted according to the pulse s e q u e n c e of Fig. 4d. A train of 12 selective rr pulses (rx = 5.1 m s e c ; r = 0.4 msec) was used during the mixing time (Zm = 70.8 msec). (Reprinted from Fejzo et al. 64)

sites e2 and 81 are suppressed and the indirect magnetization transfer cross-peak (~2, 81) is eliminated completely (Fig. 1lc). The basic disadvantage of the XD.NOESY experiment is that only one (or few) individual pathways are suppressed in a single experiment; therefore, a large number of experiments are needed to analyze the systems completely. Selectively inverting only the desired spins within the limited time frame of the mixing period is difficult technically. The selectivity for inversion is roughly proportional to the inverse of the pulse length. The inversion pulses need to be a small fraction of the mixing time in order to average the magnetization to zero. With a typical mixing time of 100 msec, the length of the inversion pulses is limited to less than 10

[3]

2D EXCHANGESPECTROSCOPY

143

msec, which yields about a 50 Hz bandwidth. More selective inversion pulses require longer times and will not satisfy the averaging requirements. Block Decoupled NOESY. The block decoupled NOESY (BD.NOESY) experiment 74 uses the same pulse scheme as XD.NOESY except that the 180° pulses are band-selective. Band-selective pulses are shorter than selective pulses, and more of them can be accommodated during the mixing time. The experiment is relatively insensitive to the pulse parameters and is very useful for decoupling one group of spins from the other. This approach has been applied successfully for the removal of aromatic resonance-mediated spin diffusion in NOESY spectra TM and for the observation of direct cross-relaxation in OMTKY3. 87 Block Complementary Decoupled NOES Y. When a band-selective 180° pulse is used in the sequence of Fig. 4f, a group of resonances is inverted on going from one frame to the other. Spin magnetizations within the band remain parallel irrespective of the inversions. The same situation holds for the noninverted spins. Among the inverted spins and among the noninverted spins, effective magnetization exchange takes place according to Eq. (24). However, the interaction of the inverted and noninverted spins is reversed on changing the frame of reference. Magnetization exchange between the noninverted and inverted spins is governed by Eq. (25). For z n = 27 r, the total effect is that cross-relaxation among the inverted spins, as well as among the noninverted spins, is suppressed. However, crossrelaxation between the inverted and noninverted spins takes place according to Eq. (27). The whole exchange network is decomposed into two blocks, with cross-relaxation proceeding between the blocks but suppressed within each block. The situation is complementary to the one observed in BD.NOESY, hence the name, block complementary decoupied NOESY (BCD.NOESY). TM The experiment has been demonstrated on OMTKY3 in D20 by the complementary decoupling of a block of aromatic protons from the block of aliphatic protons.88 Besides the general problem of low sensitivity inherent to all rotating-frame experiments with macromolecules, this experiment is also influenced by proton-proton scalar coupling, which has been ignored totally in this analysis. Other Related Methods Many other methods related to the basic 2D exchange experiment have been described in the literature. The scope and variety of these methods 87 C. G. Hoogstraten, W. M. Westler, S. Macura, and J. L. Markley, J. Magn. Reson. B 102, 232 (1993). 88 S. Macura, J. Fejzo, C. G. Hoogstraten, W. M. Westler, and J. L. Markley, 15th Int. Conf. Magn. Reson. Biol. Systems, Jerusalem, Israel, August 16-21 55 (1992).

144

TECHNIQUES

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show the great versatility of the simple magnetization exchange experiment. Bodenhausen has proposed an elegant method for the complete elimination of spin diffusion from 2D exchange spectra. 89'9°In that method the condition (m.~f3 U ~ = 0 is achieved for all spins except for a selected spin pair by synchronous nutation. The experiment works for incoherent magnetization transfer and does not rely on scalar coupling. A further class of exchange experiments has been developed which involves scalar coupling. For the separation of exchange and cross-relaxation in coupled spin systems, zz-exchange spectroscopy 91 has been used. Double-quantum filtered nuclear Overhauser effect spectroscopy (DQFNOESY) 92 is used for the detection of cross-correlation between proton chemical shift anisotropy and dipole-dipole interactions. Triple-quantum filtered (TQF)-NOESY 93'94 and 3QF-T-ROESY (tilted) 95 are used for detecting cross-correlation between pairs of dipole-dipole interactions. Acknowledgments This research was supported by National Institutes of Health Grants RR02301 and GM35976. This study made u s e o f the National Magnetic Resonance Facility at Madison, which is supported in part by NIH Grant RR02301 from the Biomedical Research Technology Program, Division of Research Resources. Equipment in the facility was purchased with funds from the University of Wisconsin, the National Science Foundation Biological Instrumentation Program (Grant DMB-8415048), the NIH Biomedical Research Technology Program (RR02301), the NIH Shared Instrumentation Program (Grant RR02781), and the U.S. Department of Agriculture. We thank Valerie L. Langworthy for editing the manuscript.

89 G. Bodenhausen, "4th Chianti Workshop on Magnetic Resonance: Nuclear and Electron Relaxation, San Miniato (Pisa), Italy, June 2-8, 1991" (unpublished). 9o I. Burghardt, R. Konrat, B. Boulat, S. J. F. Vincent, and G. Bodenhausen, J. Chem. Phys. 98, 1721 (1993). 91 G. Wagner, G. Bodenhausen, N. Mueller, M. Rance, O. W. Soerensen, R. R. Ernst, and K. Wiithrich, J. Am. Chem. Soc. 107, 6440 (1985). 92 C. Dalvit and G. Bodenhausen, Chem. Phys. Lett. 161, 554 (1989). 93 T. E. Bull, J. Magn. Reson. 80, 470 (1988). 94 C. Dalvit and G. Bodenhausen, in "Advances in Magnetic Resonance" (W. S. Warren, ed.), Vol. 14, p. 1. Academic Press, San Diego, 1992. 95 R. Brueschweiler, C. Griesinger, and R. R. Ernst, J. Am. Chem. Soc. 111, 8034 (1989).