NMR hybrid relaxation methods of studying chemical, physical, and spin exchange. III. Hybrid relaxation dispersion

JOURNAL

OF MAGNETIC

39,27.5-290

RESONANCE

(1980)

NMR Hybrid Relaxation Methods of Studying Chemical, Physical, and Spin Exchange. III. Hybrid Relaxation Dispersion MAURICE Department

of Medicine,

Albert

Einstein

EISENSTADT College

of Medicine,

Bronx,

New

York

10461

Received July 19, 1978; revised May 1, 1979 The NMR hybrid relaxation rate for exchanging systems exhibits a dispersion with pulse spacing (HRD), similar to the dependence of spin-echo r2 values on the 180” pulse spacing. We present a qualitative explanation of HRD, and compare it with its T2 analog. Exact expressions are complex, and we show the results of computer calculations to illustrate the dependence of HRD on various experimental parameters. Supplementing standard Ti and 7’a experiments with an HRD measurement allows one more unknown to be derived for two-phase systems, yielding an unambiguous exchange rate. Procedures for curve fitting are described, and illustrated with data on spin exchange for the hemoglobin-water proton system in red cells.

INTRODUCTION

In Ref. (I), hereafter referred to as I, we showed that for systems with exchange, hybrid relaxation rates can differ in the long and short ~~limits. Data taken as a function of 7r can yield an extra parameter, invaluable in separating the many unknowns typical of multicomponent exchanging systems. We call this rr dependence “hybrid relaxation dispersion” (HRD). It is analogous to the spin-echo measurement of T2 as a function of fcp, the 180” pulse spacing (24). Both have the advantage that only an instrumental parameter is varied, the intrinsic relaxation and exchange rates remaining constant. This is in contrast to other variables employed, such as magnetic field (Ho) or temperature (T); since many relaxation parameters vary with Ho, and almost all vary with T, rather than reduce the number of unknowns, this variation may only introduce their HO and T dependences. We will compare HRD with its T2 analog later. To understand HRD qualitatively, consider the two-phase system used previously (I), with intrinsic rates Rib = RZb = RI, = 0, and R 2a very fast. Figure 1 shows the A and B magnetizations as a function of time, for the long and short TVlimits. The spin population in the B phase is taken to be about twice that in the A phase, and we use the spin-exchange case; i.e., k, and k, are replaced by k, and k,, the probabilities per second of spin transfer from A to B, and vice versa. Spin transfer, a noncorporeal transfer of magnetization between A and B spins at the phase interface, is zero during the r2 domain, 275

0022-2364/8O/Ot?O275-16$02.00/O Copyright @ 19KO by Academic Press. Inc. All rights of reproduction in any form reserved. Printed in Great Bntatn

276

MAURICE

EISENSTADT

FIG. 1. Schematic of hybrid relaxation in the long and short 7, limits. Shaded regions depict the behavior of the magnitude of MA, open regions MB.

For the long rr limit (upper part of Fig. l), the initial A phase equilibrium magnetization Mi decays rapidly to zero because of its short T2, but the B phase value Mi is unchanged, since RZb = 0. During rl, there are no intrinsic relaxation mechanisms, but the A and B spins are coupled via the spin exchange rates k, and k,. For long r,, mixing during r1 is complete, and the total remaining magnetization, Mi, is shared by the two phases in proportion to their populations. After IZ repetitions of this cycle, the total remaining magnetization is M(wT,) = (NB/N)“-lM;,

where the spin populations of the phases are defined by NA + NB = N. Defining an exponential decay constant by M(m,) = exp(-@-nr,), where use of the slow root (minus subscript) implies that II is large, we obtain 45- = -T;~ ln(NB/N).

IIll

Thus the slow decay rate approaches zero for long TV,with a rF1 dependence and a magnitude depending on relative populations. The rate-limiting quantity for magnetization decay is rr itself, since the A phase acts as a perfect magnetization sink during r2, and the remaining B phase magnetization is completely equilibrated between A and B during TV. For short r* (lower part of Fig. l), assuming Rza still is fast compared to 7z1 but T~-Kk;‘, k;‘, the transfer of spin from B to A (kJ becomes rate limiting, and the slow decay constant is @- = (1 - p)k,, i.e., the spin-transfer rate, reduced by the fraction of time spent in the 72 mode. The transition between the two cases occurs when TV = k;‘, and provides a measure of the spin-exchange rate. For this simple example, the short TV limit itself gave kt, but Rib and kt can be comparable in real cases, and a measured hybrid relaxation at a single short TV would only give their combination, at best. But the dispersion of & with T* depends on k, in a different way, permitting both Rib and k, to be determined. This will be illustrated in the last section.

HYBRID

RELAXATION

DISPERSION

277

For the case of atomic exchange, k,, the transfer rate from B to A operates during both r2 and 71. In the short rr limit, relaxation is similar to the spin-exchange case, with @- = k,. For rr long, however, it does not go to zero, but approaches a constant pk,. Qualitatively, this is because the magnetization sink operates only during the fraction of time prr; the reequilibration of A and B spins during 71 enhances overall relaxation very little For finite values of the intrinsic relaxation rates, the two limiting forms of the fast and slow hybrid relaxation rates, for the chemical exchange case, are short 7,: 2~AB*=(R12a+kx+RlZb+ky)f(R12a+kx-R12b-ky)

~[1+4k,k,/(R12,+k,-R12b-ky)~l~‘~, whereRIW,

= P&b

+@w.

= 2p+2*

+ 2&J,*

PI

long 7,: 2&2*

=P(R2,+k,+RZbfky)*P(Rza+kx-Rzb-ky)

x [1+4k,k,/(R,,+

kx-Rz,,-

ky)2]1’2

The c$ABlimit corresponds to a simple two-phase exchange formulation, analogous to the formulas for c$~+and &, Eqs. [8] and [12] of I, but with hybrids R12a and R1zb as the intrinsic A and B phase relaxation. The 4r2 limit is a hybridization of separate TX and T2 exchange regimes. A qualitative explanation was given in I. For spin exchange, the same formulas can be used, substituting k, for k, and k, for k,. However, since k, = k, = 0 during the 72 period, values of qk, and qk, must be used in Eq. [2], and k, = k, = 0 in the portion of Eq. [3] multiplied by p. These expressions predict when a significant HRD will occur. For example, in the limit of p or q approaching unity (pure T2 or TI experiments), Eqs. [2] and [3] become identical; there is dispersion only when TI and T2 are mixed. More important is the limit when the population in one phase is much less than that in the other. Then if detailed balancing applies (i.e., chemical equilibrium or spin conservation), the relations

N&x = N&y, NAk, = NBk,

[41

show that if either k,(k,) or k,(k,) is very small, the square-root terms in Eqs. [2] and [3] approach unity, and c$JAB= 412. This implies that there is no dispersion unless back diffusion is significant, since only then can 7r be rate limiting. The optimization of HRD then consists of enhancing the difference between the square-root terms of Eqs. [2] and [3]. In the simple illustration of Fig. 1, the root is unity for dAn, since pRza >>k, or k,. But for 4 12, it can be far from unity in the term

MAURICE

278

EISENSTADT

multiplied by 4, depending on relative values of k, and k, (i.e., relative populations of A and B, Eq. [4]). In all of the work described here, the fast decay rates (bAB+and c&Z+ are dominated so @+ has negligible dispersion. The usefulness of measuring both the fast bypR2,, and slow decays was stressed in (5), hereafter referred to as II. Since we are concerned only with HRD here, we will ignore the fast component in all that follows, and assume the data have already been separated. To optimize HRD, the parameters most readily varied are p (the fraction of T, spent in the T2 mode) and the relative spin populations, u = NJNu. We have not found any simple relations for choosing an optimum p and u. Populations can be varied through stoichiometry, and also through isotopic composition.‘The latter is a particularly good control in aqueous systems; since either k, or k, can be driven to zero (Eq. [4]) through the use of minimal H20, the HRD should vanish. An experimental example is given later. For quantitative interpretation one must be cautious, since the relaxation parameters themselves usually depend on isotopic composition. However, the latter dependence is different from the stoichiometric

I

I1,1111,

-B -

1

1 11,111,

1

I,,,110

SPIN EXCH k“ L

msec

Tla= I00

T,,:lOOO

-

FIG. 2. HRD of slow relaxation rate, for various k;’ values (A, chemical exchange) and k;’ values (B, spin exchange). Regions to the right of the dashed lines are impractical 7r values, for the given k, or k,. The short T, limits are q5AB, long 7r limits q&.

HYBRID

RELAXATION

DISPERSION

279

dependence, which depends on equilibrium constants or relative affinities for each phase, so both variations are valuable. In general the HRD also depends on the values of intrinsic relaxation, Rla, Rza, Rib, and Rzb. Usually these are unknown and not easily or independently manipulable by the experimenter. In the “infinite sink” case (Rza>> RI,, Rzb, RI,,), the measured values of @- are independent of R 2a,but if k, and k, are of the same order as RI, or Rib, pure T1 and T2 measurements, or even measurements of 4*u- and &- are not sufficient to determine all of the unknowns. HRD provides an extra independent relation needed for unambiguous determination of the exchange rates. This is illustrated in the final section. We have not found simple limiting calculations of HRD to be of use in data analysis, and in fact any first-order treatment leads to a vanishing dispersion. Therefore in what follows, we show the results of computer calculations of @dispersions as various relevant parameters are varied. The only generality is that T2 must be much shorter than TX in one of the phases to get appreciable HR.D; a factor of 10 seems a lower limit of usefulness. For ‘H relaxation, this limits the method to

I

IO

100 k;’

IOOL -B _

,

, , ,,),,, cs

,

1000

, msec ,

, ,,,,,,

, , , /,, SPIN EYCH TIo i 100

T,b ~1000

-

FIG. 3. Same calculation as for Fig. 2, but plotted versus k;’ (chemical exchange) and k;’ (spin exchange). Curves for only 7r values 10 and 100 shown. Dispersion with 7r occurs in region where &*a and & diverge.

280

MAURICE

I[

EISENSTADT

IO

100

1000

k;‘, msec

FIG. 4. The effect of various parameters on @- vs k;’ curves. (A, B) Variation of HRD with p; (C, D) variation with u. For (A-D), 7’1, = 100 msec, Tza = 1 msec, T Ib= Tz~= 1OOOmsec. (E, F) Effect of smaller T,,/ T,,.

exchange between a highly mobile phase, such as solvent water, and a large slowly tumbling molecule such as protein, or else a solid-like phase. For other nuclei such as i3C, large TJ T2 ratios are found even for small chemical species. GENERAL

DISPERSION

BEHAVIOR

In Fig. 2 we show calculated curves of @J-vs r,, using typical values of intrinsic relaxation obtained from our study of spin exchange between protein and water protons. Protein relaxation times, in milliseconds, are denoted by u, and water by b. For comparison, we show both the atomic exchange case, Fig. 2A, and spin exchange, Fig. 2B, using the same numbers. The values of p and u are fixed at 0.1 and 0.55, respectively, and curves for various k, and k, are shown. For phase A, T2 cc T,, and there are sizable dispersions, depending on the value of k, or k,. The qualitative features of these dispersions agree with the simple model discussed above. At short TV,the limiting values 4 AB- are almost the same for chemical andspin

HYBRID

RELAXATION

DISPERSION

281

k;‘, msec FIG.

4-Continued.

exchange. For long 7,, the limiting @- (- q&-) are the same only for slow exchange rates; for fast chemical exchange there is almost no dispersion. To determine the expected dispersion and to see the influence of various parameters, it is more useful to plot @- vs k;‘(k;‘), the lifetime in phase B, for various 7,. Values from Fig. 2 are replotted this way in Fig. 3, and we also show q5AB-, &-, C#J-, and &-. The region where q5i2- and q5*n- diverge shows the range of k, values determinable by HRD. The 4i2- and q5*n- curves are constrained to the region between ~$1~and &, the slow decay rates for pure Tl and T2 experiments, demonstrating the need for T2 <
+qP1-.

There is a practical constraint on the amount of dispersion measurable. Since one value of magnetization is measured every rr set, a meaningful relaxation rate can be obtained only if T,Z @I’. With our 3-decade decay measurements, we have found r, = 2@2i to be an upper limit. The dashed lines in Fig. 2 show this limitation; values of O- to their right are considered unmeasurable. The limiting value at long T,, c$~~-,

282

MAURICE

EISENSTADT

k;’ , msec FIG.

4-Continued.

is of course obtainable from pure T1 and T2 measurements, and is invaluable for data analysis, as discussed below. Figure 3B shows a plot of @- vs k;l equivalent to Fig. 2B. For slow exchange, Cpfor both spin and chemical exchange approach the same limit, and curves for all 7, values converge. For fast spin exchange, the curves do not converge, since there is no exchange averaging during the r2 portion of 7,. Although the values of CD-in the short k;’ limit are still a function of TV,they yield no information about kt, and 4p- - d12just displays the simple 7L1 dependence discussed for Fig. 1. In Fig. 4 we show a number of graphs of k;’ dependences, to illustrate the effect of varying the parameters p, u, and R2JRla. For brevity, we only consider chemical exchange. The dispersion limits C#J AB- and (b12- are shown, and to illustrate some practical cases, we give curves for T, = 10 and 100 also. Figures 4A and B show that small values of p are optimum. Not only does this increase c$~~--/c#I~~-,but the absolute magnitude of @- is smaller, allowing a larger range of 7, to be covered experimentally. For large U, there is a great deal of relaxation sink, and the large magnitude of @- again compromises the range of useful TV.The optimum condition is to have only enough A spins to allow 7r to be rate

HYBRID

RELAXATION

283

DISPERSION

limiting. The choices of p and u are interrelated. Figures 4E and F show that RZa/Rl, must be large to observe a large dispersion. A third type of plot is valuable for data analysis. Since 412- is obtainable by combining pure T1 and T2 experiments, and all & curves approach it at long 7,, it can be considered a baseline and subtracted from the measured G-(7,). It is also clarifying to normalize these values to unity, since the maximum magnitude of dispersion, +AB- - CJ~~~-, varies greatly with k, or k,. Hence we define a normalized hybrid relaxation

[51

x =(~--~12-)/(~AB--~12-).

This is shown in Fig. 5 for the same values as those for Figs. 2 and 3. For chemical exchange, the divergence of the curves at short k,’ does not imply great sensitivity, since the absolute magnitude of the dispersion goes to zero in this limit (Fig. 3A). We define T: as the value of rr at x = 0.5. As in the analogous T2 dispersion discussed below, the hope is that TT will provide an independent measure of k, (k,), and be little dependent on relaxation parameters. The latter is true, but the dependence of r,* on the exchange rate is obscure; it is also highly dependent on p and U. These relations are shown in Figs. 6 and 7. In the middle region of Fig. 6, r,* varies roughly as the inverse square root of the exchange rate. The divergence at the extremes probably reflects shape changes of the normalized curves (Fig. 5), since characterizing them by a single parameter 7: depends on all having the same functional form. The relation 7: oc u*‘* is closely I

A

I1

1ll

CHEM EXCH -

X

I.0

X

0.5

0

IO

too

1000

rr , msec FIG. 5. Normalized dispersions, defined in the text, for the same case as Fig. 2. The 7r value at the half-dispersion point for each curve defines 7:.

MAURICE

284

/

I

EISENSTADT

I

111111

I

lll11ll

I

IO

I

I

I

IllIll

1000

100 k; or ki’ , msec

FIG. 6. Dependence of half-dispersion time 7: on exchange rate, for various u values. Spin-exchange cases shown by dashed lines, where they differ from chemical exchange. All curves for p =O.l, T,, = 1 msec, Tr. = 100 msec, Trb = r,, = 1000 msec.

followed, and is useful when varying stoichiometry or isotopic composition. Variation of ~,h with p (Fig. 7) is steep until r2 is long enough for the relaxation sink to “saturate”; thus it is steeper for the set with k;’ = 10 than for kyl = 100. For longer p, T? increases again, since now the 71 period becomes too short to mix the A and B

& -1 i 2

3

.25 4

0 2

100

//

i-l

I

.2

I

I

.4

I

/’

I

.6

I

Id

.8

P FIG. 7. Variation of half-dispersion time 7,* with p, for various u values. Solid curves, chemical exchange; dashed curves, spin exchange. Upper five curves for k;’ = 100 msec, lower five k;’ = 10 msec. Tza = 1 msec, rt. = 100 msec, Tib = T2s = 1000 msec for all curves. Spin and chemical exchange curves are the same for k;’ = 100 msec.

HYBRID

RELAXATION

DISPERSION

285

spins efficiently. The minimum in the T,* vs p curve seems closely related to k, and k, (i.e., u), but thus far we have not used p variation much in our experimental work. For a given k,l, 7: coincides with an inflection point in the curve of @-(7:) vs kyl (Fig. 3A), a fact we have found no use for as yet. The main value of normalized dispersions has been to give a first estimate of the exchange rate for curve fitting; thereafter, fitting @-(7,) - q512is more useful, as discussed in the final section. COMPARISON

WITH

SPIN-ECHO

STUDIES

OF CHEMICAL

EXCHANGE

The dispersion of @- with rr is analogous to the dispersion of the spin-echo decay rate with echo spacing I,. The latter method has been highly developed by many workers (6-8). We discuss similarities and differences with the aid of Fig. 8. The spin-echo study of chemical exchange is a pure Tz measurement. For exchange between two sites, the most general form of T2 decay is the sum of two exponentials, but attention has almost always been focused on the slow decay. Even if the fast decay has appreciable amplitude, the slow component has a much larger dispersion (for the same reasons as Qp-, discussed above), so we only discuss &. Figure 8A shows a typical set of conditions for T2 exchange studies. The intrinsic transverse relaxation rates at the two sites are RZa and Rzb, Rz, being the faster, and the chemical shift between sites is do. The latter can enhance T2 if it is modulated by the random exchange process, and reaches an optimum for k, - do (when Aw >>

ky-’ FIG. 8. Comparison of graphs of the slow decay rate &- vs exchange time for a T2-CPMG experiment, with the variation of the slow decay rate @- in a hybrid relaxation experiment. The various quantities are defined in the text.

286

MAURICE

EISENSTADT

Rza, R&, accounting for the peak in Fig. 8A. The 180” pulses are used to refocus the dephasing effect of magnetic field inhomogeneity, but in the same way they cancel the dephasing effect of Aw. However, if the lifetime in state A or state B is of the order of tcPor shorter, the phase of the nuclear signal is randomized, and the 180 pulses lose their cancellation property. For tc,+ 0, the peak in Fig. 8A vanishes completely, and 4*- goes monotonically from R 2b, the slower intrinsic rate, to PARza+ PBRZb, the fast exchange average. If t,, can be varied from less than to more than ky’, a dispersion in & can be measured, and is optimum if k,’ - Aw. The t,, dependence provides an independent measurement for unraveling the unknown parameters of the system, in general five: R 2a, Rib, do, k,, and PB (= 1 -Pa). The value of t,, halfway through the dispersion, t$, can yield Aw directly if kil is on the slow exchange side of Fig. 8A; on the fast exchange side, it yields kG1 approximately, but is a good starting value for curve fitting. In simple cases, R2,, may be obtainable by measuring T2 for a sample of pure B phase, but in general this is not valid, since the presence of A molecules may strongly perturb B phase relaxation via viscosity or weak binding sites. Various schemes for curve fitting are given by Carver and Richards (9). The analogous hybrid dispersion case is shown in Fig. 8B. The curve of 4*n, the r* + 0 limit, is exactly like the t,, + 0 curve, but varies between the hybrid relaxation of the slow phase, RIZbr and the fast exchange hybrid average PARlza+PBRlzb. However, the dispersive property results from a decrease in @- for longer values of 7,. The half-dispersion value 7: bears no simple relation to k,, as discussed above (Fig. 6). For hybrid relaxation, there are in general six unknowns: RI,, Rza, Rib, R2,,, k,, and Pb (= 1 -PA). But the instrumental parameter p can also be used to vary the relative amounts of Tl and T2 mode relaxation, enabling Rib and Rzb to be obtained separately. Variations of temperature or Larmor frequency are very useful in their own right, but for curve fitting, they may only introduce a new set of unknowns. Frequently they do aid in distinguishing slow from fast exchange. Experimentally, both methods share certain limitations. Both t,-+and 7, have an upper limit, since a measurement must encompass at least a few values of magnetization decay. The minimum practical value of t,, is about 20 to 50 ksec, since rf pulsewidths are one to several microseconds long, the receiver must recover for a few microseconds, and at least 10 psec of precession signal should be observed. For hybrids, the minimum T, would be three times the equivalent tcp, since there are at least three rf pulses in the basic unit. The underlying causes of the two dispersions are quite different, however. In the spin-echo experiment, the strength of the relaxation sink (the average Aw) is varied with t,. In HRD, the strength of the sink, pkza is left untouched (except perhaps through p variation), but variation of rr enables @- to pass from the value where k, is rate limiting to the case where rr itself limits relaxation. CURVE-FITTING

PROCEDURES; APPLICATION CROSS-RELAXATION

TO PROTEIN-WATER

For two-phase exchange, we wish to determine RI,, Rza, RI,,, Rz~, k, (k,), and u = PA/PB, only by varying the instrumental parameters p and 7,, and u, via

HYBRID

RELAXATION

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287

stoichiometry or isotopic substitution. The exchange rate k, may vary with concentration in the case of exchange between two chemical states, or with relative volumes in the physical exchange case, since k;’ is the mean lifetime in state B. For chemical and physical exchange we assume detailed balancing applies; thus k, and k, are related through Eq. [4], i.e., u = k,/k,. For spin exchange, the relation between k, and k, is less clear, because spin exchange depends on second-order kinetics (i.e., joint transition probabilities between pairs of nuclei), and some transitions are not spin conserving. We will consider these questions elsewhere; here we assume that spin is conserved, hence Eq. [4] also applies, and u = k,/k,. For the rest, we will assume u is determinable. In the chemical case, it is the stability constant for the two chemical species, and could in principle be obtained by varying stoichiometry, although it may be quite entangled with the relaxation rates. Comparing stoichiometric and isotopic variations side by side could reveal chemical isotope effects. At any rate, knowledge of an extra parameter can make it possible to obtain u, just as the chemical shift T2 dispersion made it possible to determine an unambiguous coordination number for the solvation of Co’+ by CH30D (10). In the case of protein-water solutions described below, u is simply obtained from the weight concentration and known amino acid composition of the protein. The measured quantities are 4r-, &, and o-(7,) for one or more p values. We assume the data have been decomposed into two separate exponentials, and the fast decay constants #I~+, I$*+, and Cp, are of no further interest. As discussed in II, they can provide several additional parameters for curve fitting, but here we extract all information from the slow components alone. Three slow decay rates are insufficient to determine the five unknown rates. But as discussed above, HRD is significant only if T2 in one phase, Rza, is much faster than RI,, Rib, Rzb, and k,. If p is chosen such that p2R& >>4k,k,, Eq. [2] reduces to ~nB--pRzb+qRlb+k,=p(R2b-Rlb)+Rlb+ky,

C61

CL in the short TV limit. Measurement of +*n- at two values of p determines R2b-Rlb. From the measured & and 42-, 412- is determined; in the fast R2a approximation, Eq. [3] becomes

[71 The curve fitting consists of choosing a pair of values of Rib and k, which give the measured 4*n-; RI, is the only remaining unknown in Eq. [7], and is chosen to give the measured +i2-. The complete dispersion of a-(~,) is then calculated with the formulas of Appendix A in I, plotted as &-c!~~~ vs 7,, and compared with the data. Although the short TVasymptote only determines Rib + k,, the dispersion fits the data only for a unique combination. More refined curve fitting could be used when 4k,k,/p*R& is not negligible, but some approximate value of RZa must be known. In the spin-exchange experiments discussed below, a pure T2 experiment gives separate R2, and R2b, since there is no exchange in the T2 mode. For chemical exchange, measurements of hybrid relaxation in the short 7r limit for several small values of p should yield estimates of GAB+ as well as 4aB-, from which Rlza and hence R2= can be derived.

288

MAURICE

EISENSTADT

For spin exchange, the equivalents of Eqs. [6] and [7] are

[81 [91 and the curve-fitting procedure is as described above. For this case, Rzb is measured, so only one p value is needed. We apply the above to spin exchange between hemoglobin protons and water protons in shrunken, packed red cells at 25°C. Fast atomic exchange at ionizable protein residues would be an insignificant spin-transfer mechanism at our pH of 6.0 (11). The protein concentration was 48 g/ 100 ml, which gave the fraction of protein protons as Pa = 0.32, or u = 0.45. Protein T2 can be approximately measured by direct observation of the free-induction decay. It was probably a distribution of values; an average R i,’ = 1 msec is accurate enough, since it was much faster than all other rates. A CPMG echo train also measures Rzb, which was 33 msec. A pure Tl measurement gave 4;: = 340 msec; hence &2- is determined. Figure 9 shows raw data for HRD. A range of 7, from 4.8 to 342 msec was covered, and for the shorter rr, 5% changes of @- could easily be detected. For the longest T,, however, only three points on the slow decay can be obtained. To test for instrumental artifacts, the measurements were repeated for the single-phase system, Ni(ClG,J2 in H20; T12 showed less than a 5% change over the same rr range.

IO'

616178

I

200

I

400

I

I

600

800

ooo

I , msec.

FIG. 9. Raw HRD data. Magnetization recovery after an initial inverting pulse, for various 7r values. Sample is hemoglobin in packed, shrunken red cells, 48 g/100 ml solution, pure HzO. An initial fast axnponent is observed for most rr values.

HYBRID

RELAXATION

----0025

I I

DISPERSION

289

A

I1111111

I

I

I IIIIII

IO

100

1000

Tr I msec FIG. 10. Lower curve, reduced data from Fig. 9, also using measured &- and &,-, and another set for p = 0.025. Theoretical dispersion curves are for various k,, with other parameters as described in the text. Upper curve, isotopic dilution of the same sample. Only the best-fitting theoretical curves shown. Dispersion vanishes with high dilution.

The lower part of Fig. 10 shows these p = 0.1 data plotted as @(r,) - +i2, and theoretical curves calculated for various combinations of RI,, Rib, and k,. Combining the measured 4AB, &-, and RZb with one choice of k, constrains the values of RI, and RI,,, and yields one unique dispersion curve, as shown, Since all p = 0.1 curves converge to the same short rr limit, the k, value which best fits the data can be read off the graph. The shape of the dispersion data does not follow theory, but a k, between 5 and 7 is the best choice, with 4 as a lower limit. This flattening of the relaxation dispersion is commonly observed in protein-water NMR studies (II, 12), and probably is due to a distribution of correlation times (13); this would give a distribution of k, also. As a check, a few points of a dispersion were measured for p = 0.025. The short T’, asymptotes were not matched with great care (dashed lines) but the best k, value is consistent with the p = 0.1 result. The quantity @(T,) - & is independent of p for large rr. We have not yet developed a systematic scheme for curve fitting, which also would estimate error limits. An excellent control experiment is to substitute D20 for most of the solvent, and measure spin exchange between the protein protons and the residual HDO. As discussed in the Introduction, the dispersion should vanish for minimal HDO, due to negligible back diffusion. The upper portion of Fig. 10 shows data for 30% Hz0 and 5% HzO, for samples otherwise identical to the 100% Hz0 case. Only the best theoretical fits are shown. In the calculations, it was assumed that 20% of the protein protons had exchanged for deuterons during the 24 hr of isotope exchange (14). The behavior with increasing isotopic dilution is as expected; HRD is still appreciable for

290

MAURICE

EISENSTADT

30% H20, but 7: is longer for 30% Hz0 than for 100% Hz0 (Eq. [l] and Fig. 6). For 5% H20, the magnitude of ~$*n - & becomes negligible. The data do not reach the limiting rL1 slope predicted by theory, mainly, we suspect, due to a distribution of k, values. For cases with larger spin transfer, i.e.: Rz, >>k,, k, >>RI,, Rib, the simple form of Eq. [l] would hold, and the extrapolated values of @- - & at T, = 1 would give a direct determination of In(&/N). This could be valuable in studies of spin transfer between a solvent and an adsorbed phase, where the binding constant of the latter is not known. For atomic exchange, this limit does not apply. ACKNOWLEDGMENTS ThisworkwassponsoredbyNIHGrantsGM18298

andHL17571.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

8. 9. 10. Il. 12. 13. 14.

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