Fourier transform NMR pulse methods for the measurement of slow-exchange rates

JOURNAL

OF MAGNETIC

RESONANCE

29,397-417(1978)

Fourier Transform NMR PulseMethods for the Measurement Slow-ExchangeRates I.

ID.

Department

CAMPBELL,

ofBiochemistry

C. M. DOBSON, and Inorganic

R. G. RATCLIFFE,

Chemistry Laboratory, England

AND R. J. P. WILLIAMS

South

Parks

Road,

Oxford

OXI

SQR,

Received June 20.1977 The seven-membered ring of Valium exists in two equivalent conformations. Exchange between the two forms exchanges two coupled nonequivalent hydrogen atoms in the ring and this system has been used to examine NMR Fourier transform pulse methods for measuring rates of slow exchange. Experiments are reported both for the exchanging system (at 320 K) and for the effectively nonexchanging system at a lower temperature (270 K). The selective inversion of one exchanging resonance allows the determination of all the rate and relaxation constants by an iterative fitting of the data to the general solution of the modified Bloch equations. However, the use of a combination of nonselective inversion, selective inversion, and presaturation sequences is shown to provide definitive and consistent values for the rate and relaxation constants without the need for difficult curve fitting to several variables. By this means the rate of exchange in a 100 mA4 Valium solution in deuterochloroform at 320 K is found to be 5.4 k 0.2 set-i. An appendix discusses the application of transverse relaxation measurements and concludes that for the slow exchange in Valium this method of determining the rate of exchange is inferior to the measurement of longitudinal relaxation. INTRODUCTION

Slow-exchange processes can be studied by a variety of NMR methods including analysis of lineshapes (I), double-irradiation procedures (Z-7), and Carr-Purcell sequences (8). The double-irradiation methods depend on the fact that if an exchange process relates two nuclei, then this process provides a mechanism by which the magnetization of one nucleus can affect the magnetization of the other nucleus. This work was pioneered by For&n and Hoffman (2-4, using cw techniques. The method was put on a firm theoretical footing by their demonstration (4) that the modified Bloch equations derived by Solomon (9) for the longitudinal relaxation of a pair of interacting nonequivalent spins are entirely analogous to the McConnell equations (IO) describing chemical exchange. Double-irradiation methods have also been used with FT techniques. Dahlquist et czl. (5) studied the conformational flipping of [2.212,5-pyrolloparacyclophane by selectively inverting one of a pair of exchanging resonances and then following the changes in magnetization of both resonances. The same method has been applied recently to the study of peptide bond isomerization in the dipeptide glycylsarcosine (6). Rate processes in biological molecules have been studied by cross-saturation experiments, in which one 397

0022-2364/78/0293-0397%02.00/0 Copyright 0 1978 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain

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of a pair of exchanging resonances is selectively saturated and the effect on the other resonance is monitored (II, 12). Recently the cross-saturation experiment has been adapted to study the relaxation and pseudo-first-order exchange of NH protons in solution (7). This method is similar to a method used earlier for the study of relaxation in 13C NMR (13). In this paper the application of FT double-irradiation techniques to the study of slow, first-order exchange processes is discussed in detail. An appendix considers briefly the

I

FIG. 1. The 270-MHz ‘H NMR spectrum of 100 mM Valium/CDCl, at 320 K. The conformational equilibrium is represented by the two boat forms of the cycloheptatriene ring system (after (14)). Another equilibrium may be drawn between the two pseudoboat forms of the cycloheptadiene ring system.

application of Carr-Purcell sequences for the same purpose. To illustrate the methods, results are presented for a system in which exchange occurs between two dipolarcoupled sites. Valium, a common pharmaceutical, provides a good example of such a system. The molecule (Fig. 1) has a nonplanar seven-membered benzodiazepine ring which is thought to exist in a boat conformation. There are two equivalent boat conformations and interconversion of the ring exchanges the geminal ring protons. The two ring protons give two doublets in the NMR spectrum, labeled I and S in Fig. 1, and the exchange has been studied previously by analysis of the temperature-dependent lineshapes (14). The rate of exchange has a convenient temperature dependence: at 270 K the rate is negligible, while at 320 K slow exchange occurs. This makes it possible to discuss the application of the methods both in the presence and in the absence of exchange.

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399

THEORY

n discussing the measurement of exchange and relaxation processes by FT irradiation methods, three experiments are considered. In experiment I, both I and S are inverted by a nonselective 180° pulse and the recovery of the magnetization is followed with a 90” observation pulse. This is the ordinary 180°-r-90° sequence for measuring spin-lattice relaxation times. In experiment II, either I or S is inverted by a selective 180” pulse and the return to equilibrium is monitored with a nonselective 90” pulse. In experiment III, either I or S is irradiated with a saturating pulse for a measured time before the observation pulse. This is the presaturation sequence employed in the study of NH exchange in an aqueous solution of tryptophan (7). The interpretation of these experiments requires an understanding of the effects of cross-relaxation and exchange on spin-lattice relaxation. These topics have been extensively discussed elsewhere (e.g., 4, 9, 10, 25) but it is convenient to summarize the relevant equations here. Consider a first-order exchange process, e.g., the conformational change in Valium, defined by

The equilibrium constant, K, is k&l and at equilibrium kJ, = k&3,, where I, and S, are the equilibrium values of the z magnetization of I and 5’. If I and S are homonuclear spins with a dipolar interaction between them, the modified Bloch equations for the exchanging system are (9, JO) dIJdt = +,(I,

- I,) - a(S, - S,) - kJ, + k,S,,

dS,ldt = -ps(S, - S,) - a(I, - I,) - k,S, + k,l,;

81 I31

where 1, and S, are the instantaneous values of the z magnetization of I and S. The quantities a, pr, and ps are relaxation rate constants which are related to the transition probabilities between the energy levels of the dipolar IS system; p1 and ps may be rewritten as p. + pxr and p. + p,,, respectively, where p. describes the dipolar interaction between the spins and other sources of relaxation are described by pxr and PXS~ The general solution of Eqs. [2] and 131is the sum of two exponentials: I z = Aea+t + Be’z-” + I 52, s, =

A@, +- k1 + J.+)e’+’ (k,-

a>

B@, + k1 + l-)e’mt +

(k, - 4

14; + s,.

iSl

The constants A and B are determined by the values of 1, and S, at t = 0. The quantities A+ and il- are defined by A& = +(-4& + k1 + ps + k,) + ([(p, + k,) - (ps +

k,)l’ - 4(k, - a) (a - k,)1\“‘2). IS1

The results of experiments I, II, and III can all be described in terms of Eqs. [4] and IS]. However, the general solution is cumbersome to handle and each of the three

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ETAL.

experiments will now be considered separately to see when useful conclusions can made without recourse to the general solution. Experiment I. In this experiment I and S suffer equal perturbations, so that immediately after the nonselective 180° pulse S, = KI,. IJsing [2f and 131 the ~~~t~~~ recoveries may be written

Lqldtl, = -/?,(I, - I,) - 49, - S,) = -(PI + a-) (I, - I,), EdSJdtl,= -p&T, - S,) - o(l, -I,) = -(ps + OK-“) (S, - S,).

iii i81

By plotting log,(l, - Ioo) against t, where t is the delay between the 180 and 90° pulses, (p, + OK) can be determined from the initial gradient with Eq. 171. Similarly (p, -t OK-~) can be determined from the log@‘, - S,) plot and Eq. 181. If I, = S,, i.e., K = 1, and pr = ps = p, then I, = S, throughout the recovery and an exponential path is followed: dIJdt = -(p + a) (I, -I,), dS,ldt = -(p + o) (S, - S,).

!91 II01

The quantity (p + o) can be measured from the gradients of the linear semilo~a~thmi~ plots. If I, # S, and pI # ps, then S, # KI, throughout the recovery unless (p, + UK) = (p, + OK-‘), and a nonexponential path is followed. The initial rate at which the difference between I, and K-’ S, is established is given by d(I, - K-l S,) dt

Ii

= -(pr - ps i- o[K - K-l]) (I, - 1,).

ith an exact 180° pulse it can be seen from this equation that S, = KI, is a good assumption if 2 l(p, - ps + a[K - K-‘l)lt Q 1. If this condition holds for the complete recovery, Eq. 171 and [81 can be used to describe the return to equilibrium. Experiment II. This is the experiment that was done by Solomon to illustrate the cross-relaxation effect caused by intramolecular dipolar coupling in hydrofluoric aci (3). A selective 180° pulse is applied to either I or S and the recovery of the system i followed. In general the inverted resonance follows a nonexponential recovery while th unperturbed resonance suffers a transient change in intensity which depends on the relaxation and exchange processes. The experimental data can be analyzed by fitting them to Eqs. [41, 1.51,and E61.This method was used to determine pr, ps, k,, and Kin the study of glycylsarcosine (6). However, in some circumstances this iterative ~r~~e~~re can be avoided. If pr = ps = p and K = 1 then the method of Dahlquist et a.l. (5) can be used. With these assumptions two simple exponentials can be obtained from 121and E31:

41,+s,>= -@ dt

41, - s,> dt

+ o)[(I, + s,> - (I, +S,)l,

‘:121

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RATES

401

where (p + a) and (p + 2k - a) can be obtained from the semilogarithmic plots of (I, -+ S,) and (1, - S,) against time. Dahlquist et al. (5) studied the exchange between two equivalent conformations in [2.212,5-pyrolloparacyclophane. In this system K was necessarily 1, CJhad to be 0, and pr = ps was a good assumption. As a result it was possible to determine the two unknowns, p and k (=k,, k,), without using the method of curve fitting. Note that for a slow-exchange process, K can be measured from the relative intensities of I and S: K = S,I-2. If K = 1, then experiment I can provide evidence for or against the assumption p1 = ps. If K # 1, the Dahlquist method of analysis breaks down because (1, - S,) no longer follows an exponential path. Useful information can be obtained irrespective of the values of pl, ps, and K by measuring the initial changes after the selective 180° pulse. Suppose that at t = 0, is inverted by an imperfect 180° pulse while 1, is unperturbed; i.e., at t = 0, S, = -f, S, and 1, = I,, wheref, measures the perfection of the 180° pulse. The initial changes in S, and 1, can be derived from [21 and [31: MS,/dtl,

= (PS + kJ (1 +.&)Soo,

[dl,/dtIi = (a - k,) (1 +f,)S,.

1141 [HI

Thus by measuring the initial changes in S, and 1,: (ps + k,) and (0 - k,) can be determined using Eq. El41 and [151. Note that if k, > a, 1, passes through a minimum during the recovery of S,; while if k, < a, 1, passes through a maximum. The latter includes the transient Overhauser effect observed by Solomon in the nonexchanging (k = 0) ZW system (9). (ps + k,) can also be found by measuring the initial gradient of log,@, - S,) against t. If k, > 0, then S, + KI, toward the end of the nonexponential recovery. Under these conditions the limiting gradient at long t in the same semilogarithmic plot is (p, + a). Equations [ 141 and [ 151 assume 1, = I, and the range oft over which this is valid is determined by [dl,/dtl,. If the selective pulse causes complete inversion of S,, it can be seen from [ 151 that 1, = Z, is only a good approximation if 2l(a - k,)lt < 1; i.e., (Tand k, determine the range oft in which the initial gradient can be measured. In practice this useful range is reduced by the method used for generating a selective pulse. In a real experiment a finite time is required to produce a selective pulse and this allows the change in I, to begin during the inversion of S,. It can be shown that if 1, changes significantly during the selective pulse, then an analysis of the initial rates using [ 141and E151 will underestimate (ps + k,) and (a - k,). Experiment III. This is the presaturation sequence (7, 13) in which either I or S is irradiated for a measured time before the observation pulse. Suppose that at t = 0, S, is reduced from S, to f,S,, where 0 < f’ < 1. If S, is maintained at this value, 1, will change from 1, to a new equilibrium value, I&. From Eqs. [21 and [31 it can be shown that the approach to I& is exponential: dIz/dt = -co, + k,) (I, - I&),

where 1& is defined as

1161

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402

If S is saturated at t = 0, f, = 0, ](I, conveniently rewritten as

ETAL.

- 1&)l is maximized and Eq. [171 may

where qr, the Qverhauser enhancement parameter, is defined by qr = o/p,. If S is not fully saturated at t = 0, the effect on 1, is reduced according to Eq. 1171. Whether or not S is fully saturated, (p, + kl) can be measured from the rate of change of 1, using Eq. [ 161, although the sensitivity of the method decreases if the saturation is incomplete. This is a useful conclusion since it may not be possible to saturate S completely without partially irradiating I when the chemical shifts of I and S
Summary For an IS system in which first-order exchange occurs between sites of different population the relaxation is described by five quantities: pI, ps, 0, k,, and k,. The quantities k1 and k, are related by k1 = Kk, and for a system in slow exchange K can be calculated from I, and S,. Thus the problem of describing the relaxation is reduced to the determination of pr, pa u, and k1 with K as a known quantity. By fitting the experimental data to Eqs. [41, 151, and [61, experiment II provides all four quantities for a first-order exchange process. TABLE

1

SUMMARY OF THE QUANTITIES WHICH CAN BE OBTAINED EXPERIMENTS I, II, AND III WITHOUT THE GENERAL SOLUTION Experiment Experiment

I: II:

Experiment

III:

FROM

p, + UK; ps + UK-’ p, + k,; u - kr (irradiation of I) ps + k,; u - k, (irradiation of S) p + u (irradiation of I or S with pr = ps = p) p + 2k - u (irradiation of I or S; pr = ps = p and K = 1) p, + kI (incomplete saturation of S) ps + k, (incomplete saturation of I) pr + k,; (or + UK)/@, + kJ (saturation of S) ps + k,; (ps + UK-‘)/@, + k,) (saturation of 1)

Table 1 summarizes the information that can be obtained from experiments I, II, and III without using the general solution. For spin 1, pr, a, and k1 are defined by the following measurements: (p, + OK), (pI + k,), (a - k,), and (p, + uK)/(p, + k,). IfK # 1, all three constants can be determined from these experiments, using either I and II or II and III. If K = I, the three constants cannot be determined without some additional information and there are several approaches to this problem: (a) If K = 1 and exchange occurs between two sites with no dipolar interaction, then Q = 0 and p,, ps, and k1 can be obtained from I, II, and III. Exchanging systems of this type have been studied in which CJcould only be zero (5,6) and in which rswas shown to be negligible by isotopic substitution (7, 16). Even if o # 0, under conditions where k % CJit is possible to determine k within small, known error limits by assuming CJ= 0.

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403

(b) If the temperature of a system can be reduced to a value at which the rate of exchange becomes negligible, pr, ps, and 0 can then be measured from the results of I, II, and III. If the nuclear Overhauser effect is measured at such a temperature (III with measurements of 1, and 1A only) and if the temperature dependence can be shown to be small, then the value of o/p1 at the low temperature may be used to determine pr, ps, o, and k1 at a higher temperature where exchange occurs. This method has been applied satisfactorily to Valium (see below). (c) By analogy with experiments on the steady-state NOE, in which Overhauser enhancements are reduced or eliminated by the addition of relaxation probes, it might be expected that such probes could help to separate the NOE (a) and exchange (k,, k,) in the relaxation of an exchanging two-spin system. This does not prove to be the case, as the following argument shows. Equation [ 181 can be rewritten

AIm

(PI+ UK) - (pI + OK) + (k,-

OK)’

With the addition of a relaxation probe, the external contribution to the relaxation is increased and as (pI + oK) increases, Ib, + I,. If I&/I, is measured as a function of (p, + OK), (k, - OK) can be found. However, since (k, - OK) is one of the quantities that can be obtained from I, II, and III this experiment provides no additional information. (d) Finally the transverse relaxation of an exchanging system is also affected by the rate of exchange. The complication of cross-relaxation is avoided here but since the accurate measurement of transverse relaxation rates is difficult, the method is not convenient for obtaining the slow-exchange rate constant kr A brief description of the results obtained for Valium using a Carr-Purcell pulse sequence is given in the Appendix. The various experiments will now be illustrated with results obtained for the two exchangeable hydrogen atoms in the Valium seven-membered ring. In this system o # 0 and K is necessarily 1. The quantities pI, ps, u, and k will be determined at two temperatures without recourse to the general solution and these values will be confirmed by calculating the observed changes in experiment II. MATERIALS

AND

METHODS

Valium (7-chloro- l$dihydro- 1-methyl-5-phenyL2H- 1,4-benzodiazepin-2-one) was obtained from Roche and it was used without further purification. The same 100 mM solution of Valium in deuterochloroform (99.5% D; Hopkin LG.Williams) was used for all experiments. ‘H Fourier transform spectra were recorded using a Bruker 27Ospectrometer with an Oxford Instrument Company superconducting magnet, a Nicolet 1080 computer, and a 293-pulse controller. The same computer was also used for calculating theoretical fits. Experiment I was carried out in the usual way, allowing sufficient delay between the accumulations to enable the IS system to recover completely. In experiment II a pulse length of 30 msec was used for the selective pulse. The bandwidth is approximately proportional to the reciprocal of the pulse length and so the longer the pulse, the more selective it becomes. However, if the pulse on S is made too long, the perturbation of I will be well advanced by the time it is monitored and this should be avoided.

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Experiment III may be divided into two parts and these differ in their objectives. In experiment IIIa, S, was completely saturated with a long pulse (denotedf,) and the new equilibrium value of 1, (IL) was measured. The power and length of the f, pulse were chosen to allow complete saturation of S, and equilibration of I,. The ratio Co, t c)/ (pl + k) was calculated from this experiment. In experiment IIIb, S, was irradiated with an fz pulse of varying length and the establishment of the new equilibrium value for 1, was followed to determine (p, -t k). Since this f, irradiation did not necessarily cause complete saturation of S,, this experiment could not always be used to measure (p, + o)/(p, + k). This presaturation sequence required the following experimental cycle: (i) the f, pulse was applied at the frequency of the S resonance for a time t; (ii) a nonselective observation pulse followed after 1 msec and the FID was collected; (iii) the IS system was allowed to recover before the cycle was repeated. The description of this experiment in the previous section does not require the complete saturation of S, but it does assume that S, is reduced to a new value at t = 0. It has been pointed out (7) that this is incompatible with the mechanism of saturation (17). In practice it seems that if S, is reduced to its new, maintained value in a time which is short relative to the length of the f, pulse then the method is applicable. The experimental results for the tryptophan-water exchange (7) and the data for Valium reported here confirm this conclusion. In experiments II, IIIa, and IIIb care was taken to ensure that the “selective” pulses were properly selective. The selectivity of the irradiation of S, for example, was checked by irradiating under the same conditions at a position which was as far downfield of I as is upfield of I. The selective pulse was only acceptable if this irradiation did not affect . In experiment IIIb the power used for the fi pulse was the greatest that could be used with the shortest pulse employed without introducing nonselective effects. In all these experiments great care was taken to ensure that the data were obtained with a constant response of the spectrometer. To check the stability of the instrument every third spectrum in a given experiment was run under appropriate standard conditions; e.g., in experiment I the “90°-only” spectrum was chosen as the spectrum. If the standard spectrum showed serious instability the experiment stopped and started again after adjusting the spectrometer. If the standard spectrum showed slight fluctuations, then it was often possible to compensate in the analysis for the variable response by using ratios of peak heights as a measure of peak intensity. However, in the results presented in this paper this procedure was only used for the results of one experiment (IIIb at 270 K; Figs. 7, 8). Usually the peak heights, h, and h,, were used as a measure of I, and S,. It was assumed that any perturbation ofl: or S, affected both components of the doublet equally and so the intensity of either doublet was taken as the mean of the peak heights of the two components. Although 1, = S, for Valium, hp # hp and in the analysis of experiment II by Eq. I121 and 1132 it was necessary to scale h, to h, to avoid a small error. Experiments I, II, and III were performed at two temperatures: at 270 and 320 K. In addition Q was measured over the range 230-270 K using experiment IIIa. RESULTS

270 K; No Exchange Experiment I. Figure 2 shows the semilogarithmic plots obtained for I and K. The data fit two well-defined straight lines with slighly different gradients. This

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FIG. 2. Semilogarithmic plots of the recovery of Z, (0) and S, (A) following nonselective inversion (experiment I) at 270 K. The plot for S, has been displaced for clarity.

indicates that pr # pS, but that 2 I(p, - &It << 1 over the range 0 < t < 2 sec. The gradients of these lines give pI + u = 1.55 + 0.02 set-’ and ps + CJ= 1.64 + 0.02 see-“. Experiment II. Figure 3 shows the transient Overhauser effect~on I, and the recovery of S, after a selective 180° pulse had been applied to S. The initial gradients give (?= 0.39 $I 0.05 set-’ from the curve for I, and ps = 1.24 +_0.05 set-’ from the curve for S,

FIG. 3. The changes in Z, and S, following the selective inversion of S (experiment II) at 270 K. The curves are calculated from the general solution to the modified Bloch equations with pr = 1.17 se&; ps = 1.26 set-r; u = 0.38 set-i; Z, = Sm = 1.0; I,, = 1.0; S, = -0.51. The quantities I, and S, are the values of Z, and S, at t = 0.

CAMPBELL

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ETAL.

(Eqs. [ 141 and [15J). Note that the curves drawn in Fig. 3 are the theoretical curves calculated from Eqs. E4l. [51, and [61 (see below) and not the free-drawn curves through the experimental data from which the initial gradients were measured. Figure 4 illustrates the expected nonexponential recovery of S,. From the initial gradient, pS = 1.26 + 0.03 see-’ (Eq. [ 141).

FIG.

expected

4. Semilogarithmic nonexponential

plot of the recovery behavior.

of S, following

FIG. 5. Semilogarithmic plots of the data in Fig. 3. showing and differences (upper line; Eq. [ 131).

selective

the analysis

inversion

by sums

at 270 K showing

(lower

the

line; Eq. [ 121)

Experiment I shows that there is a small difference between pI and pS. However, although the assumption pr = pS is not strictly valid, Fig. 5 shows that the analysis by sums and differences (Eqs. 1121 and [ 131) is still satisfactory. The plot of Iog,{l(l, + S,) - (I, + S,)l/Ia} against time is a good straight line and from the gradient, p + cr = 1.59 _+0.08 see-r. The plot of log,{(I, - S,)/l,} against time is also linear and the gradient gives p - cr = 0.79 + 0.04 set-'.

FT PULSE METHODS

I

01 FIG. 6. The effect at 270 K.

on I, (upper

FOR SLOW-EXCHANGE

I f2 power

line) and S, (lower

I I-O

0 (arbitrary

401

. units)

line) of increasing

I PO

RATES

the power

I 3.0

t (5)

of a 5-set irradiation

of S

I * 4.0

FIG. 7. The effect on I, of increasing the length of a saturating pulse on S (experiment IJIb) at 270 K. In (a) the data are plotted without any correction for the variable response of the spectrometer. The discontinuous plot emphasizes the instability which affected these measurements. In (b) the variable response is corrected by ratioing the magnetization of I to that of E. The curve was calculated from Eq. [ 161 with p, = 1.17 set”; k = 0 set-‘; I0 = 1 .O; Zk = 1.32.

408

ETAL.

CAMPBELL

Experiment III. The most reliable way (18) of doing experiment IIIa, the measurement of (p, + a)/& from the equilibrium value of 1, when S, = 0, is shown in Fig. 6. The magnetization of I and S is plotted against the power of a 5 secf, pulse applied to S. As the power is increased, S, falls and 1, rises to a short plateau. The value of 1, at the plateau is 1& and from the data in Fig. 6, ‘or= 0.32. At higher powers 1, falls from the plateau with the onset of nonselectivity in the S irradiation. The mean of f-I, + I, -A Qas o.zo- O\“< $\ %

O.lO0.05-

b,

\.

. 0\

o.ozo

FIG. 8. (a) and (b) show 7a and b.

the semilogarithmic

I I.0

tls)

plots for the change

1 2.0

c

in Z, correspotiding

to the data in Figs.

several measurements of q, is 0.325 + 0.01. Saturation of 1 to measure (pE +- o>/p, gives qs = 0.30 + 0.01. Figures 7 and gshow the result of experiment IIIb, in which S ias saturated and the effect on I was followed. The scatter in Fig. 7a arose from instability in the response of the spectrometer and it illustrates the importance of doing these.experiments over a period when the response is constant. The scatter is also observed in the intensities of resonances that should be unaffected by the irradiation of S. For example, the intensity of the lowest downfield component of the aromatic region (labeled-E in Fig. 1) parallels the scatter in I while remaining constant. Figure 7b shows that the scatter in Fig. 7a can be reduced by ratioing the magnetization of I to that of E. In this experiment S was completely saturated and r, = 0.32 in both Fig. 7a and Fig. 7b. The gradient of the

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409

semilogarithmic plot in Fig. 8a gives pr = 1.16 F 0.1 set-’ (Eq. [16]) while in Fig. 8b, pf= 1.19 rt 0.08 set-‘. Summary Table 2 summarizes the results obtained at 270 K. The nonexchanging IS system is described by three quantities and these are easily derived from the results of experiments I and IIIa. Thus at 270 K, pI = 1.17 f 0.02 set-‘, ps = 1.26 + 0.02 set -I, and u = 0.38 & 0.01 set-‘. These values are in agreement with the results of experiments II and IIIb and so the different experiments give a consistent description of the IS system. The values were substituted in Eqs. [41, [5l, and 161 to obtain the curves drawn in Figs. 3 and 7b. TABLE 2 SUMMARY OF THE RESULTS OBTAINED FOR VALIUM AT 270 K FROM EXPERIMENTS I, II, AND III Experiment I: Experiment II:

p,+ ps + p + p-

u= 1.55 + 0.02 set-' u = 1.64 & 0.02 set-’ cr = 1.59 + 0.08 set-* U=O.79 t 0.04 set-'

pL0.39-fo.05:::~: = 1.24 + 0.05 Experiment IIIa: Experiment IIIb:

ps = 1.26 q,= 0.325 qs = 0.30 pr = 1.19

+ 0.03 set-’ + 0.01 + 0.01 + 0.08 set-’

320 K; Slow Exchange Experiment I. Figure 9 shows the semilogarithmic plots for I and S at 320 K. Each straight line is defined by 25 points over the range t < 2 set and the gradients give p1 -+c = 0.79 + 0.01 set-’ and ps + cr= 0.77 -t 0.01 set-‘.

FIG. 9. Semilogarithmic plots of the recovery of Z, (0) and S, (a) (experiment I) at 320 K. The plot for S, has been displaced for clarity.

following

nonselective

inversion

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FIG. 10. The changes in Z, and S, following the selective inversion of S (experiment II) at 320 K. The curves are calculated from the general solution to the modified Bloch equations with p, = ps = 0.595 see-‘; u = 0.185 set-‘; Z, = S, = 1.0; I,, = 0.89; S, = -0.5; k = 5.4 set-‘.

Experiment II. The result of applying a selective 180° pulse to S at 320 K is shown in Figs. 10 and 11; I, passes through a well-defined minimum during the nonexp~ne~t~~ recovery of S, indicating that k > o.

FIG. 11. The data of Fig. 10 plotted the theoretical curves.

on a larger

scale to show

the initial

gradients

and the excellent

fit to

FT

PULSE

METHODS

FOR

SLOW-EXCHANGE

I I.0

I 0.5

O.lo

411

RATES

I I.§* t (sl

FIG. 12. Semilogarithmic plot of the recovery of S, following selective inversion expected nonexponential behaviour. The curve is calculated for the same constants two limiting gradients are shown.

at 320 K showingthe used in Fig, 10 and the

E/leasuring the initial gradient to a curve is a crude method of analysis and it is made more difficult here by the rapid change in gradient. From the curve for I,, k - (3= 4.95 k 0.5 set-l (Eq. [ 151) and from the curve for S,, ps + k = 5.55 + 0.5 set-’ (Eq. i 141). Again these gradients were measured from free-drawn curves and not from the calculated curves shown in Fig. 10. Since I, was reduced by 11% during the selective pulse, it is probable that these values are underestimates of the real values. Figure 12 confirms the expected nonexponential recovery of the inverted resonance. The initial gradient is poorly defined and gives ps + k = 5.5 + 0.5 set-’ (Eq. 1141). For t > 0.5 set, 1, = S, (Fig. 10) and the recovery of S, follows another exponential, giving ps + ci = 0.78 + 0.02 set-’ (Eq. I 101). On the evidence of experiment I, pr = pS is a good assumption at 320 K and the analysis by sums and differences is shown in Figs. 13 and 14. log,j[(l, + S,) - (1, + S,)]/d,} against time is a-good straight line and the gradient gives p + o = 0.78 rir:0.02

FIG.

13. Analysis

ofthe

experiment

shown

in Fig.

10 using Eq. [121.

412

CAMPBELL

FIG. 14.

Analysis

of the experiment

ETAL.

shown

in Fig. 10 using Eq. [ 131.

set-’ (Eq. [ 121). log,{ (1, - SJI,} against time is also linear, giving p + 2k - o = 1 I. 1 + 0.3 set-’ (Eq. [131). Note the limited range oft available for the difference plot-the greater p + 2k - cr,the shorter the time taken to reach 1, = S,. Experiment III. The result of experiment IIIa is shown in Fig. 15, in which 1, is plotted against the power of a 5-set f2 pulse applied to S. Under these conditions it was just possible to saturate S without the nonselectivity of the irradiation affecting I. This experiment gives (pl + a)/@, + k) = 0.132; the result of several measurements is a ratio of 0.135 + 0.01. It is not necessary to saturate S completely in experiment IIIb and the graph shown in Fig. 16 was obtained from an experiment in which the ratio of 1; to 1, was 0.15. The gradient of the linear semilogarithmic plot gives p1 + k = 6.0 5 0.2 see-” (Eq. E161).

oFIG. 1.5.

The effect

f, power (arbitrary

on Z, of increasing

the power

units)

of a 5-set irradiation

of S at 320 K.

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413

FIG. 16. The effect on 1, of increasing the length of a saturating pulse on S (experiment IIIb) at 320 K. The cnrve was calculated from Eq. [ 161 with pr = 0.595 set-‘; k = 5.4 set-I; I, = 1.0; Z& = 0.15.

Summary

Table 3 summarizes the results obtained at 320 K. The exchanging IS system is described by four quantities (K = 1) and it was argued earlier that unless the general solution is used to fit experiment II, another piece of information is required to determine the exchange and relaxation constants. However a series of measurements over the range 230-270 K gave vr = 0.3 1 + 0.02; i.e., the NOE between I and S is approximately independent of temperature. With this result it was found tbat the following values satisfy all the data in Table 3: k = 5.4 + 0.2 set-“, p1 = pS = 0.01 see-l, and cr = 0.185 + 0.01 see-‘. These values were substituted in Eqs. [41, [51, and [61 to obtain the curves drawn in Figs. 10, 11, and 12. TABLE 3 SUMMARY OF THE RESULTS OBTAINED FORVALIUMAT 320 KFROMEXPERIMENTS i,II, AND III Experiment I: Experiment II:

p, + u = pS + u = p + u=

0.79 0.77 0.78

* 0.01 set-’ + 0.01 set-’ * 0.02 set-’

5.55

i 0.5 set-’

p+2k-a=ll.l pS + k =

k-u= pS + k= pS + u =

F 0.3 see-’ 4.95 * 0.5 set-’ 5.5

f 0.5 set-’

0.78 + 0.02 set-’ Experiment IIIa: (JJ~+ u)/@, + k) = 0.135 + 0.01 Experiment IIIb: pI + k= 6.0 + 0.2 set-’

DISCUSSION

Slow first-order exchange processes can be studied by experiment II, the selective 180° pulse method, and the exchange and relaxation constants for the IS system can be obtained by fitting the experimental data to Eqs. 141, 151, and [61. The purpose of this

414

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ETAL.

paper has been to show what information can be derived from other experiments an from less complete analyses of experiment II. The conclusion is that at the very least these procedures can provide limits to assist in the curve fitting of experiment II and that at best these procedures allow the complete description of an exchanging system, without the full analysis of experiment II. In the absence of exchange the relaxation of an IS system is described by pr, ps, and CJ.These three quantities can be measured by various combinations of experiments I, II, and III including: I and IIIa; I and IIIb; IIIa and IIIb; I and II; II, with irradiation of1 or S only, assuming pr = ps; II, with separate experiments for the irradiation of1 and S. All four experiments were performed on Valium at 270 K and a completely consistent set of results was obtained. This agreement confirms the theoretical description and practicality of the experiments. This is an important conclusion for IIIb since it provides further experimental support for the presaturation experiment. The advantages and disadvantages of the four experiments may be summarized as follows: Experiment I is useful if the difference between pr and ps is not so great as to cause nonexponential longitudinal relaxation. The experiment uses nonselective pulses an be performed with great accuracy. Experiment II uses a selective 180” pulse and such a pulse is easier to apply, the greater the chemical shift difference between I and S. Also, as the chemical shift difference increases, the length of the selective pulse can be reduced (with a compensating increase in the power of the irradiation) and this increases the sensitivity of the experiment by reducing the change in the magnetization of the nonirradiated resonance during the pulse. If p1 = ps, the experiment is especially convenient because the analysis by the method of sums and differences is applicable and this is more reliable than the initial rate method. If pr # ps, the experiment becomes less attractive and the data are best analyzed by fitting the general solution using the initial gradients as constraints. Experiment IIIa is an easy experiment to do if I and S are well separated in the spectrum, but as the chemical shift difference is reduced it becomes harder to saturate either resonance selectively. Experiment IIIb is more difficult than IIIa in that the problem of nonselectivity in the f, irradiation becomes more a&rte with short pulse lengths. However, this is compensated by the fact that it is not necessary to saturate completely the irradiated resonance. Experiment IIIa and IIIb together can be used to determine pI, ps, and crand this combination of experiments is useful when pr -f ps. In an exchanging IS system, where the exchange occurs by a first-order process, the relaxation is described by pI, ps, u, k,, and K. If exchange occurs between sites which are not dipolar coupled, then B = 0 and p,, ps, and k1 can be measured from experiments, I, II, and III in the same way that pr, ps and u are measured in the ~onex~~a~~in~ coupled system. The quantity K is measured from the ratio SW/Z, for a system in slow exchange. Similarly, if 0 # 0 and K # 1, pr, ps, 0, and k1 can be determined in the same way. However, if (J f 0 and K = 1, pI, ps, u, and k1 cannot be determined from experiments I, II, and III without either a complete analysis of II by curve fitting OYan extra piece of independent information. In the former case the information provided by I and III will assist the curve fitting of II. For Valium at 320 K an extra piece of

FT

PULSE

METHODS

FOR

SLOW-EXCHANGE

RATES

415

information was available: the NOE was found to be temperature independent over the range 230-270 K and this was assumed to hold to 320 K. y this means the rate of exchange was found to be 5.4 + 0.2 see-l at 320 K. The resul obtained by this method were fully supported by the excellent fit predicted in experiment II using the general solution. Thus it can be seen that while experiment II is useful, experiments IIIa and IIIb are by no means valueless. In addition to the comments above experiment III may be usefully applied when the assumption k % (5is justifiable by the accuracy required of k. Finally, experiment III can be applied in situations where II is impossible; in particular the study of pseudo-first-order reactions in which one component of the IS system is in great excess and must be saturated to obtain the FT spectrum. APPENDIX

In the slow-exchange limit, the observed transverse relaxation rate obtained by continuous-wave or single-pulse techniques is given by T;‘=q-l+k, Ml1 where T2 is the observed transverse relaxation time, G is the transverse relaxation time in the absence of exchange, and k is the exchange rate of the observed resonance. Thus k can be evaluated if T2 and c can be measured. Time T!$ includes contributions from magnetic field inhomogeneities and diffusion of the spins. These contributions are not constant and for measuring small exchange rates, k < 10 see-‘, this simple method is unreliable and more sophisticated methods are required. The most successful method is the Carr-Purcell-Meiboom-Gill pulse sequence: 90:-r--( 1go,“-2r),-,-180,0--r-FID (19). This sequence eliminates many of the instrumental contributions to the transverse relaxation rate but there are still considerable practical difficulties in measuring transverse relaxation (20, 21) and the purpose of this appendix is to show that such measurements do not provide a convenient method for obtaining the slow-exchange rate in Valium. For an exchanging system the theory of the above sequence is more complicated than the theory of the continuous-wave experiment because the observed rate of transverse relaxation (R,) depends on the pulse spacing (22). The theory of Gutowsky et al. (8) predicts this dependence for a number of different situations, including exchange between two equally populated coupled sites. For slow exchange the theoretical expression is of the form: decay = modulation term x exponential term. The modulation term is caused by the J coupling between the sites and it becomes more important at longer pulse spacings. If the pulse spacing is reduced to very small values then the J modulation disappears and in addition the exchange contribution to R, should become zero. In other words with r = 0, R, = R; = the transverse relaxation rate in the absence of exchange. Note that because of the instrumental contributions in the more sim experiments R!j # cr. Note also that r = 0 is experimentaily inaccessible and so cannot be determined directly. As the pulse spacing is increased the exchange proc contributes to R, and in the limit of long pulse spacing: R, = R!j + k. However, for a system exchanging between coupled sites, such as Valium, these long values of r a inaccessible because of the J modulation. Because direct measurement of Rt and k) is not possible, it is necessary to consider fitting data from the sequence to the theoretical description.

416

CAMPBELL

ETAL.

At 270 K, R, was measured for I and S in Valium using 22 = 2 msec. smooth exponential decays were obtained with no J modulation, giving R,, = 1.64 & 0.05 set-r and R, = 1.65 & 0.1 set-‘. For Valium it is expected that TI = T, and indeed these measured rates are close to the longitudinal relaxation rates. Note that in some of the experiments reported here, a non-zero-equilibrium magnetization was obtained at long values of 2nz. This was allowed for in calculating R, (20). ate R, was also measured at 320 K and again, while there were sometimes difficulties with phase correction and non-zero-equilibrium values, smooth exponential decays were observed for 22 < 3 msec. The absence of J modulation at these two temperatures does not agree with the prediction based on the theory of Gutowsky et al.

FIG. A 1. Transverse relaxation rates as a function of spectrometer frequency and pulse spacing in the Carr-Purcell-Meiboom-Gill sequence. The arrows indicate the range of measured values without any estimate of the errors. The curves are calculated from the theory of Gutowsky et al. (8).

(8). With a coupling constant of 10.7 Hz and a chemical shift difference of 281 Hz at 270 MHz, the predicted modulation frequency is 0.70 Hz for 2t = 1 msec and 2.47 for 22 = 2 msec. The absence of J modulation corresponds to a value of 2z which is an order of magnitude smaller than the values used and such values are unobtainable on the instrument employed for this work. The discrepancy between theory and experiment is probably caused by pulse imperfections in the sequence. This disagreement has been put to good use in NMR studies of proteins where it has been possible to measure T, values of coupled resonances and to do multiplet selection experiments (22). Figure Al shows the measured values of R, for I and S at 320 K for both 270 and 90 MHz. The data for Z and S have been combined since with the likely errors their rates are indistinguishable. Since no J modulation was observed for 22 < 3 msec, the exponential term in the equations of Gutowsky et al. (8) was used to plot R, as a function of r. The curves are plotted for k = 5.4 set-” and R! = 0.78 set-’ using the

FT

PULSE

METHODS

FOR

SLOW-EXCHANGE

RATES

417

results obtained in the main part of this paper. Note that it is assumed that T, = T,, i.e., Ri = p + 0, on the basis of the measurements at 270 K. This is a useful assumption since it anchors the theoretical curve at r = 0. If the assumption is invalid then the fit of the theoretical curve has to be optimized using two variables, R!j and k, instead of k only. In the present case the validity of the assumption is unimportant. It is clear from the spread of the experimental measurements in Fig. Al and from the poor fit with the calculated curves that the measurement of transverse relaxation rates is not a suitable method for determining the slow-exchange rate constantfor Valium. In the limit of short pulse spacing the sequence used above is similar to the experiment that measures spin-lattice relaxation in the rotating frame (23). The measurement of T1, has been used for exchange studies (e.g., (24, 25)) but this method has not been evaluated in the present work since it is not expected to offer any advantages for slow-exchange measurements. ACKNOWLEDGMENTS This is a contribution from the Oxford support. We also thank J. R. P. Madden, E. W. Gill for the sample of Valium.

Enzyme Group and we thank the Science Research Council for G. R. Moore, and R. Porteous for help with this work, and Dr.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Il. 12. 13. 14. 15. 16. 17. 18. 19. 20. 71. 22. 23. 24. 25.

H. S. GUTOWSKY AND A. SAIKA,J. Chem. Phys. 21,1688 (1953). S. FOR&N AND R. A. HOFFMAN, J. Chem. Phys. 39,2892 (1963). S. FOR&N AND R. A. HOFFMAN, J. Chem. Phys. 40,1189 (1964). R. A. HOFFMAN AND S. FOF&N, J. Chem. Phys. 45,2049 (1966). F. W. DAHLQUIST, K. J. LONGMUIR, AND R. B. DLJVERNET, J. Magn. Resonance 17,406 (1975). J. R. ALGER AND J. H. PRESTEGARD, submitted for publication. I. D. CAMPBELL, C. M. DOBSON, AND R. G. RATCLIFFE, J. Magn. Resonance 27,455 (1977). H. S. GUTOWSKY, R. L. VOLD, AND E. J. WELLS,J. Chem. Phys. 43,4107 (1965). I. SOLOMON, Phys. Rev. 99,559 (1955). H. M. MCCONNELL, J. Chem. Phys. 28,430 (1958). A. G. REDFIELD AND R. J. GUPTA, Cold Spring Harbor Symp. Quant. Biol. 36,405 (197 1). I. D. CAMPBELL, C. M. DOBSON, G. R. MOORE, S. J. PERKINS, AND R. J. P. WILLIAMS, FEBS Lett. 70,96 (1976). R. FREEMAN, H. D. W. HILL, AND R. KAPTEIN, J. Magn. Resonance 7,327 (1972). P. LINSCHEID AND J-M. LEHN, Bull. Sot. Chim. France, 992 (1967). I. D. CAMPBELL AND R. FREEMAN, J. Magn. Resonance 11,143 (1973). S. WAELDER, L. LEE, AND A. G. REDFIELD, J. Amer. Chem. Sot. 97,2927 (1975). D. I. HOULT, J. Magn. Resonance 21,337 (1976). R. FREEMAN, H. D. W. HILL, B. L. TOMLINSON, AND L. D. HALL, J. Chem. Phys. 61,4466 (1974). S. MEIBO~M AND D. GILL, Rev. Sci. Instrum. 29,688 (1958). R. L. VOLD, R. R. VOLD, AND H. E. SIMON, J. Magn. Resonance 11,283 (1973). R. FREEMAN AND H. D. W. HILL, in “Dynamic Nuclear Magnetic Resonance Spectroscopy” (L, Jackman and F. A. Cotton, Eds.), p. 3 1, Academic Press, New York, 1973. I. D. CAMPBELL, C. M. DOBSON, R. J. P. WILLIAMS, AND P. E. WRIGHT, FEBS Lett. 57,96 (1975). R. FREEMAN AND H. D. W. HILL, J. Chem. Phys. 55,198s (1971). B. D. SYKES, J. Amer. Chem. Sot. 91,949 (1969). C. DEVERELL, R. E. MORGAN, AND J. H. STRANGE,MOI. Phys. 18,553 (1970).