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Paramagnetic relaxation in dipolar-coupled homonuclear spin systems
JOURNAL
OF MAGNETIC
RESONANCE
49,
251-210 (1982)
Paramagnetic Relaxation in Dipolar-Coupled Homonuclear Spin Systems JOSEPH Department
of Physics,
GRANOT
Purdue University IUPUI, Indianapolis,
School Indiana
of Science
at Indianapolis,
46223
Received February 16, 1982; revised April 9, 1982 Measurements of the paramagnetic contribution to the longitudinal relaxation of nuclei interacting with a paramagnetic probe can be complicated by effects of chemical exchange and cross relaxation. A theoretical investigation is presented of the relaxation behavior of the longitudinal nuclear magnetization, following either selective or nonselective inversion of the equilibrium magnetization, in a dipolar-coupled homonuclear two-spin system that undergoes exchange between free and bound (to a paramagnetic probe) states. Approximate solutions are derived for a number of limiting cases characterized in terms of the relative rates of exchange, cross relaxation, and spin-lattice relaxation. It is shown that unless the cross-relaxation rate (CRR) is slow compared to the other rates that affect the longitudinal relaxation, reliable and accurate paramagnetic relaxation rates that can yield useful structural information can be obtained only from measurement of the initial rate of recovery of the longitudinal magnetizations. Some advantages of selective over nonselective spin perturbations are pointed out. INTRODUCTION
Effects of cross relaxation, that arise from mutual flips of dipolar-coupled nuclear spins through zero-quantum transitions, have been the subject of considerable interest in recent years (I-6). When the cross-relaxation rate (CRR) is fast relative to the spin-lattice relaxation rate, magnetization can be transferred between the coupled spins before a significant amount cf magnetic energy is transferred to the lattice. Consequently the time dependence of the longitudinal magnetization of each spin is no longer independent of the relaxation of the other spins in the system, resulting in significant deviations of the individual recovery rates from the intrinsic spin-lattice relaxation rates. Furthermore, the time dependence of the longitudinal magnetizations may become nonexponential, and proper longitudinal relaxation rates may actually be undefinable. The CRR increases when the molecular motions become slower; hence cross-relaxation effects may be dominant in macromolecules (7-13). Also, cross relaxation can be particularly effective in homonuclear spin systems since the Zeeman energy that is exchanged in zero-quantum transitions is small for homonuclei. Paramagnetic probes, e.g., metal ions and spin labels, have been used in numerous NMR studies of structural and dynamic properties of chemical and biochemical systems. Determination of the contribution to the nuclear relaxation rate from the interaction with the unpaired spin of the paramagnetic probe can be complicated 251
0022-2364/82/l
10257-14SO2.00/0
Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
258
JOSEPH
GRANOT
by effects of cross relaxation, when the nuclear spins in the system are coupled, and/or by chemical exchange, when the association with the paramagnetic probe is labile. While the theoretical aspects of the chemical exchange effects on the longitudinal nuclear relaxation in the presence of paramagnetic spins are well known (14-18) and are commonly taken into account in analyses of relaxation data, the effects of cross relaxation have been usually neglected. Possible effects of cross relaxation on distances measured from paramagnetic probes have been recently discussed by Andree (5). It should, however, be noted that some of his remarks, e.g., that in the presence of cross relaxation measured distances may be shorter than the true values due to enhanced nuclear relaxation, and that underestimated correlation times lead to overestimated distances, are by no means general and are not always valid. In the present work the effects of cross relaxation on the longitudinal relaxation in a dipolar-coupled homonuclear spin system that also interacts with a paramagnetic probe are examined in some detail. The theory is formulated for two spins, however, the conclusions can in general be applied to multispin systems as well. Conditions under which the longitudinal magnetization has a single-exponential time dependence are examined for a number of limiting cases characterized in terms of the relative rates of chemical exchange, cross relaxation, and spin-lattice relaxation (diamagnetic and paramagnetic). Procedures that can yield accurate paramagnetic relaxation rates in the presence of cross relaxation are discussed. THEORY
Consider a system, at equilibrium, that contains paramagnetic probes and ligand molecules (L) that undergo exchange between a free state (F), and a bound state (B) in which a ligand molecule is associated with a paramagnetic probe, according to the scheme
111 where kF and k, are the first-order exchange rates from the free and the bound states, respectively.’ Let I and J be two spins on L that interact via dipole-dipole interaction with one another, and are sufficiently close to the paramagnetic probe, in the bound state, to have their relaxation rates significantly affected by the paramagnetic interaction.2 The spins I and J may in general be heteronuclear, however, it has been shown (I, 2) that in such cases a single-exponential time dependence of the longitudinal magnetization of each spin can be achieved simply by continuous ’ Note that while ke is an intrinsic rate constant, kF may be concentration dependent, depending on the type of reaction that takes place when L exchanges between the free and the bound states. Thus for example, in a typical case when L associates with a paramagnetic probe P through the bimolecular reaction L + P G LP, with a second-order forward rate constant k,. and a first-order reverse rate constant kow we have ks = koff and kF = k,,[P], where [P] is the concentration of the paramagnetic spins. * Nomenclature: I and J denote nuclear spins and S denotes electronic spin. The subscripts “F” and “B” refer to the free and bound states, respectively. These subscripts may be omitted when the discussion applies to both the free and the bound states. “SL” and “NS” denote selective and nonselective spin perturbations, respectively. CRR denotes cross-relaxation rate.
DIPOLAR-COUPLED
SPIN
259
SYSTEMS
saturation of the other spin. Hence this work will be concerned with homonuclear spins. It is assumed that the spins I and J have nonoverlapping resonances and that their scalar spin-spin interaction is weak, i.e., that their ratio of chemical shift to spin-coupling constant is large. In addition to simplifying the rate equations for the nuclear magnetizations, the assumption of weak scalar coupling also permits selective perturbation of either one of the spins I and J. It is further assumed that any rf perturbation affects equally all the multiplet components of each spin. Following an rf perturbation of the spin system, the recovery of the longitudinal magnetizations toward thermal equilibrium, after the perturbing field has been turned off, is governed by the coupled rate equations ___
=
dt
-uh
+
+
dt
d&F ___
dt
+
UIJ~@h4
-
Mm)
+ ~IJWJLI - Mm)
dMm = -(RIJF
dM,, dt
RlIP
RlJP
+
UIJ)(MJB
-
-
bI~II3
+
MfIF
7
[21
MOB)
+
UIJ(&
-
MOB)
-
&IMJB
+
kFMJF
,
[31
,
[4]
=
-tRIIF
+
ulJ)(“IF
-
MOP)
+
UIJ(MJ,
-
MOP)
-
kFMIF
+
bdfIL3
=
-tRIJF
+
UIJ)(MJF
-
MOF)
+
UIJ(MIF
-
MOF)
-
k&JF
+
‘b&fJB,
[5]
where it is assumed that the equilibrium magnetizations of the two spins are equal, being MOF and M 0s in the free and the bound states, respectively (MOF may not in general be equal to M,,,). Quantities MI and M, are the expectation values of the component of the nuclear magnetization of I and J along the external magnetic field, R,,r and RIJF are the spin-lattice relaxation rates of I and J in the free state (excluding cross-relaxation effects), R rip and RIJp are the contributions to the longitudinal relaxation of I and J due to their interaction with the paramagnetic probe in the bound state (paramagnetic contributions due to second-sphere effects are neglected), and uIJ (=uJr) is the cross-relaxation rate (CRR) between the spins I and J. The characteristic relaxation rates in the system are given by (19) R ,,p = $ ,‘;h2z(z + 1)r;P[3J,(wr) R 1,P U ,J
=
&
y:gv2m
=
6
#i2z(z
+
+
l)r;P[J,(@,
l~~aJ
+ 12J2(‘0r
+
UJ)]
+
o( 01 - us) + 3Jl(~r) + 6J2(~t -
WJ)
-
6J2(01
+
@J)],
[61
K;p,
+ 41 + REP
3
171
[81
with analogous relations for the spin J. In Eqs. [6] to [8], I and S denote nuclear and electronic SpinS, respectively, yI (=-y,) iS the UUClear gyromagnetic ratio, and rIs are the distances between the nuclei I and J and between I and the paramagnetic probe, respectively, wI and ws are nuclear and electronic Larmor frequencies, and J,,(w), n = 0, 1, 2, are spectral density functions. When the motion r,J
(x.dJ)
260
JOSEPH GRANOT
of the spin system is isotropic, J”(w) = ~c/( 1 + n2w2~&), where 7c is the correlation time that governs the longitudinal relaxation. For example, in the bound state -’ = 7;’ -t 7;’ + kg, where TR is a rotational correlation time and 7s is the re;ixation time of the unpaired spin of the paramagnetic probe. The term Kip in Eq. [6] represents any additional mechanism that contributes to the spin-lattice relaxation in the free state. These mechanisms are assumed not to give rise to crossrelaxation effects. The term RsI”p in Eq. [ 71 represents paramagnetic contribution to the relaxation rate due to scalar electron-nuclear hyperfine interaction that is present when direct ligand-probe coordination (i.e., with a covalent character) takes place. Determination of the dipolar term of RIIp is usually a major objective in experiments that monitor the effects of a paramagnetic probe on nuclear relaxation, since the inverse sixth-power dependence of the dipolar term on internuclear distances makes such measurements a powerful tool in structural studies. Bearing in mind that kB/kF = MoF/M oB, Eqs. [2] to [5] can be rewritten in the matrix form ZM=A(M-M,) dt
II91
where r
CIJ
kF
0
-PJB
0
kF
0
-PIF
-PIB u1.l
A=
k, L
0
-
kB
UIJ
@IJ
[lOI
’ 1
-PJF
r&i
MIBMJB
M=
[Ill
MIF MJF-
and where PIB
=
RIIF
+
Rnp
+
PIF
=
&IF
+
UIJ +
flIJ
+
kB
El21
,
[I31
kF,
with analogous relations for the spin J. The off-diagonal elements of A represent the coupling between the longitudinal relaxations of the spins I and J through cross relaxation and chemical exchange. The general solution for the time dependence of the longitudinal magnetization of the spin I or J in the free or the bound state (i.e., for any component of M) is given by (M,, - M(t))/M,
= i
C,e-‘;‘,
[I41
i=l
where the values of Ci are constants that depend on the initial conditions (at t = 0) of the spin system, and -Xi are the eigenvalues of the matrix A which can be obtained by solving the quartic equation [(PIB
+
x)(P.lB
+
A)
-
-
[(PlB
‘&l[(PlF +
x)(P,F
+ +
X)(PJF A)
+
+ (PJB
A)
+
&,I X)(PJF
+
h)lk&B
+
k#,
=
0.
[ 151
DIPOLAR-COUPLED
261
SPIN SYSTEMS
Approximate solutions for a number of limiting cases are discussed below with the objective being the determination of the conditions under which the time dependence of the longitudinal magnetization is single exponential, thereby allowing a single relaxation rate to be defined and to be measured. DISCUSSION
Slow CRR
When the CRR is SLOW,i.e., a:J 4 P~FPJF(since PIB > PIF, PJB> PJF,it also follows that & 6 P~BPJB),Eq. [ 151 reduces to the simplified form [(PJB + XMPIF + A) - ~F~~I[(PJB + XNPJF + A) - h&l
= 0.
Cl61
The relaxation behavior of each nuclear spin can thus be treated separately, with the problem becoming a simple two-site exchange with paramagnetic contribution to the relaxation rate at one site. This problem has been previously dealt with (17, 18) and will not be further discussed here. It should only be noted that conditions for slow CRR require negligibly small contribution of to the off-diagonal elements of the matrix A (Eq. [lo]), but not to the diagonal elements. Hence cross relaxation may still affect the measured relaxation rates, albeit not necessarily the measured net paramagnetic relaxation. (TIJ
Slow Exchange
Under conditions of slow exchange, i.e., when kFkB 4 smaller of P~BP~F and PJBPJF, Eq. [ 151 reduces to [(PIB + X)(PJB + A> - GJI[(PIF + X)(PJF + A) - ~JI = 0.
t171
Hence the nuclear relaxations in the free and the bound states are decoupled, and the relaxation behavior in each state can in principle be independently investigated. To allow measurements of relaxation rates in the bound state the chemical shift separation between the free and the bound signals should be sufficiently large so that distinct free and bound signals can be observed under the above condition of slow exchange. If, however, the free and the bound signals overlap, the contribution to the signal from free species should be made negligibly small, i.e., by introducing enough paramagnetic probes to bind the major part (or all) of the ligand molecules.3 The recovery pattern of the longitudinal magnetization depends not only on the characteristic rates in the system but also on the initial conditions which in turn depend on the type of rf perturbation that has been applied to the system. Two types of perturbations may be particularly useful in relaxation studies of homonuclear systems: (1) nonselective (NS), when all the spins in the system are simultaneously perturbed, e.g., fo!lowing a 180” pulse, at t = 0 MiB = MJB = --Moe and MiF = MJF = -MOF, and (2) selective (SL) (20) when only one spin (e.g., I) is perturbed, e.g., following a 180” pulse, MiB = -MJB = -MOB, and MIF = -MJF 3 In the limit when labels, and substitution
ks - 0, the discus inert ligand-probm:
don presented complexes.
here is also applicable
to nonexchangeable
spin
262
FIG. [22]).
JOSEPH
GRANOT
1. The variation of KNS (Eq. [ 191) and KsL (Eq. (a) KNS, for (Y > 0, (b) KsL, (c) KNS, for (Y < 0.
[20])
as a function
of the parameter
(Y (Eq.
= -MOF. (Under conditions of slow exchange, selective perturbation of the resonance corresponding only to the free or the bound state may also be possible.) Following a nonselective (NS) or selective (SL) inversion of the equilibrium magnetization, the time dependence of the longitudinal magnetization of the spin I in the bound state, obtained by solving Eq. [ 171, can be written as (MOB - M,,)/M,,
[If31
= (1 - KB)e-xlB’ + (1 + KB)e-‘ZB’,
where KES = (aB - l)(l
+ (r;)-1’2,
[I91
KgL = -( 1 + c&-“2, hB,2B
with the + sign corresponding
=
[201
PB -+ (l/‘J)Apdl
to &, a B = APB
=
PB =
+
2 l/2
aB)
[211
,
and ~I/APB
9
PIB -
PJB >
(~/~)(PIB
+
I.221 [231
0,
1241
PJB).
Analogous expressions and definitions for the free state can be obtained by substituting the subscript F for B in Eqs. [ 181 to [24]. The variation of KNS and KsL with (Y is depicted in Fig. 1. Note that since Ape (Eq. [23]) has been arbitrarily defined (without loss of generality) as positive, (Yg acquires the sign of and can be either positive or negative (cf. Eq. [8]). In the forthcoming discussion the relaxation behavior in the bound state will be examined for three limiting cases with positive CRR, and for the case of negative CRR. (i) Slow CRR. When 0 < (Yg< 1, both KFS and KgL approach - 1 (Fig. l), and Eq. [ 181 reduces to the approximate single-exponential form oIJ,
(ibfoB - bf~B)/i!!f~B N 2e-““. Let (1 /T,), and (1 /T,), denote the measured relaxation bound signals, respectively, then
t251 rates of the free and
DIPOLAR-COUPLED
SPIN
263
SYSTEMS
III
0
100
200
1 400
300
v, (MHz) FIG. 2. Frequency dependence of 2q,/RllP for = J = l/2, S = 1, and g = 2, and an isotropic contribution to RllP (Eq. [7]) has been neglected.
several rIs/rIJ ratios. rotational correlation
Calculations time of
were made 10m8 sec. The
R ltp = (~/T,)B - (Rm + uu + kd
with I scalar
I-261
and if the CRR is slow in the free state as well (i.e., (YF6 l), then
R IIP
=
(1/%l
-
(l/T,)F
-
(‘b
-
[271
kF).
However, since the effect of the paramagnetic probe on the nuclear relaxation is usually much larger than that of a nearby nuclear spin, i.e., RIIp % RIlF, the condition of slow CRR in the free state may not necessarily be fulfilled. In such cases when (Yr S 1, the value of RllF + needed in Eq. [26], should be measured from the relaxation behavior in the free state as described below. The exchange contribution to the relaxation, if not negligible, can in principle be determined from temperature effects on the relaxation. The favorable condition of slow CRR in the bound state requires that ApB (=Rllp - RIJP)be large compared to au; hence rIs should differ from (Eq. 7). Also the paramagnetic probe should be sufficiently close to I to allow effective dipolar interaction between the nuclear spin and the unpaired electron spin. From Eqs. [7] and [8] it is also apparent that when the NMR frequency (wi) is increased, tends to increase whereas RIIp decreases. Hence conditions of slow CRR may be more readily achieved at lower NMR frequencies. Consider, for example, a case when R ,ip 9 RlJp, i.e., LYBN and when the scalar contribution to the paramagnetic relaxation is negligible. Figure 2 shows the dependence of 2uIJ/RIIP on the NMR frequency for several ratios, for a system that undergoes isotropic tumbling with a rotational correlation time of lo-’ set that governs both the cross relaxation and the paramagnetic relaxation rates. From Fig. 2 it is evident that while the distance ratio (rIs/rIJ) is the major determinant of (Yg, due to the inverse sixth-power dependence of the relaxation rates on the internuclear separation, the frequency effect is also quite significant. In the present example (Yg varies by almost two orders of magnitude in the frequency range 50 to 400 MHz. (TIJ
rJs
@[J
&J~J/R~~,
rIs/rIJ
264
JOSEPH GRANOT
I 50
0
I 100
I 150
t (ms)
FIG. 3. Semilogarithmic plots of the recovery of the longitudinal magnetization of the spin I in a twospin system following a nonselective or selective inversion, under conditions of slow chemical exchange rate and intermediate CRR: (a) bound state, NS, (b) bound state, SL, (c) free state, NS, (d) free state, SL. Calculations were made using the values Rllp = 30 set-‘, RIJp = 10 set-‘, RllF = 2 set-‘, Rllr = 1 set-‘, ks = kF = 0, Q,, = 20 set-‘. Dashed lines represent the initial rate of recovery following a selective perturbation of I.
Since nuclei-probe distances are beyond our control, the frequency dependence of (Ya allows some flexibility in the setting of the experimental conditions. Unfortunately sensitivity and resolution considerations usually require high NMR frequencies, contrary to the requirement for slow CRR. Figure 1 reveals an important advantage of selective over nonselective spin perturbations, resulting from the fact that conditions for slow CRR are more readily obtainable in selective experiments. Thus, KsL - -1 (within 15%) for (Y ? 0.60, whereas KNS - -1 for (Y Z 0.15, and since (Y increases when the frequency is increased, this four-fold difference allows relaxation measurements to be carried out at significantly higher NMR frequencies, through selective perturbations. For example, from Fig. 2 a four-fold increase in (Ys corresponds to about a two-fold increase in the NMR Frequency. Hence if slow CRR conditions prevail in nonselective experiments up to 200 MHz, the experimental frequency range can be extended through selective experiments to 400 MHz. (ii) Fast CRR. When (Ys 9 1 (and therefore also ayF$= 1 ), KFS I: KfS 1: 1, and IK”F”I N IK”B”j 4 1, and also from Eq. [ 211 we get &B
=
RIB
+
&B
=
RIB
1
2%
[281
,
1291
where RIB
=
(l/2)&~
+
R,IP
+
R,JF
+
RIJP)
+
kB
[301
with analogous relations for the free state. From Eq. [ 181 the time dependence of the longitudinal magnetization in the free and bound states, following nonselective perturbation, will be approximately single exponential with a rate constant equal
DIPOLAR-COUPLED
265
SPIN SYSTEMS
to R,, or RIB, respectively. Since XIB(XIF) is always larger than &(X,,), following a selective perturbation the first term in Eq. [ 181 (which has approximately the same amplitude as the second term) will decay faster than the second term, and after a time period t - ark’ the longitudinal magnetization will approach a single-exponential time dependence with the rate constant being again R,, (or RI,). The relaxation behavior of the longitudinal magnetizations under conditions of fast CRR result from the known fact that cross relaxation equalizes and maintains equal the magnetizations of all the coupled spins in the system. As a result the longitudinal magnetizations of these spins recover with virtually the same relaxation rate (& or R,,) which is a weighted average of all the contributions to the spinlattice relaxation rates of the coupled spins. In other words, fast cross relaxation is a mechanism that effectively equalizes, and keeps equal, the spin temperatures of coupled spins in the system, following any perturbation that increases the spin temperature either locally or throughout the whole system. Once the spin temperature of the coupled spins equilibrates, they all cool down toward the external temperature with the same rate that is maintained uniform throughout the spin system by the fast cross relaxation. Under conditions of fast CRR it is impossible to determine the individual paramagnetic contribution to the relaxation of either I or J, i.e., RIIP or RIJP, respectively, and only their average (l/2)(& + RIJP) can be obtained. This average cannot in general provide useful structural information, since it can only yield the average distance f = [( 1/2)(r;: + r;l)]-“. A single-exponential time dependence of the longitudinal magnetization should be therefore treated with some caution in systems with cross relaxation, and it is imperative to find if this behavior results from slow or fast CRR. A warning sign for fast CRR may be the observation that all or many resonances in the system relax with very similar longitudinal relaxation rates. (iii) Intermediate CRR. When (Ye - 1 the recovery of the longitudinal magnetization, following either a NS or SL perturbation, is in general nonexponential (e.g., Fig. 3). While in principle the relaxation data of a two-spin system can be fitted as a sum of two exponentials (e.g., Eq. [ 18]), such analysis will involve at least three adjustable parameters (&, XIB, X2& and more if the experimental conditions (e.g., pulse lengths) were not ideal, with the resulting “best-fit” parameters being of dubious merit. Clearly when more than two spins are coupled, a multiexponential fit is completely out of question. Useful relaxation rates can, however, be obtained from measurements of initial rates of recovery of the longitudinal magnetization (I). Over an initial time period t < X;; Eq. [ 181 can be approximated by
- [VW3+ b*) - KB&B- bdlt.
OfrIB - MIB)IM3B = 2
The initial recovery is therefore single exponential, turbations, the initial rates are given by
=
&IF
+
R,IP
+
and following
kB
,
[311
NS or SL per-
[321
266
JOSEPH
=
and similarly, if (YF-
GRANOT
&lF
+
RIIP
+
=
&IF
+
b,
=
&IF
+
‘JIJ
UIJ
+
kB
(331
1,
0 1
T1
[341
SL, initial +- kF
[351
IF
with analogous relations for the spin J. These expressions are valid also for a multispin system, in which case should be replaced by a sum of cross-relaxation terms corresponding to all the spins that are dipolarly coupled to I. From the initial rates the paramagnetic contribution to the relaxation can be directly obtained: bIJ
R,,,=(~)::i”-(~~~‘+(kB-kF).
[361
Note the similarity between Eq. [36] and Eq. [27] obtained in the case of slow CRR. Indeed, Eq. [27] can be considered as a special case of Eq. [ 361 with the initial period extending over the whole experimental time range. While initial rates appear to provide rather handy means for determination of paramagnetic relaxation rates when cross-relaxation effects are significant, they have a serious drawback, namely, they may be very difficult to measure accurately. Difficulties can arise, for example, when due to statistical errors the experimental spread in the measured recovery data is large, when the CRR is relatively rapid (hence the initial period is short), or when the initial rate is not very different from the rate at later time. Since the initial rates in SL experiments contain a crossrelaxation term, in general (for cx > 0), ( 1/T1)SL, i”itia’ > ( 1 /T,)NS* i”itia’. Hence in selective experiments initial rates are more readily discernible (in semilogarithmic plots, e.g., Fig. 3), and probably can be more accurately measured. SL experiments also provide virtually the only means to determine the CRR itself (through ( 1 / T,)fiS, initial ). Hence in the case of intermediate CRR, selective rather than nonselective measurements may well be the method of choice. (iv) Negative CRR. In the limit of fast molecular motions, when the probability of double-quantum transitions exceeds that of single-quantum transitions, i.e., 65, > Jo (Eq. [ 8]), becomes negative. Negative CRR reduces the relaxation rates, e.g., through Eqs. [ 121, [ 131, but does not significantly affect the pattern of the recovery of the magnetization, e.g., through the parameter (Y (Eqs. [ 181 to [22]). The values of KsL are independent of the sign of (Y, and to a good approximation so are the values of KsL when ]a] 6 1. When ]CZ]- 1, the time dependence of the magnetization is not single exponential, and the foregoing discussion for the case of positive and intermediate CRR is essentially valid for negative cxas well. When Ia] $ 1 the magnetization has approximately single-exponential time dependence, albeit since for negative (Y, K NS - -1, the recovery rate constant is R,, compared to KNS - +l and a recovery rate constant of RIB (which is independent of for positive ff. gIJ
21alJI
cIJ)
DIPOLAR-COUPLED
SPIN
267
SYSTEMS
Fast Exchange When the chemical exchange between the free and the bound states is fast compared to their chemical shift separation, the free and the bound signals coalesce and a single resonance is observed that may relax with a rate equal to the weighted average of the relaxation rates in the free and the bound states. Fast chemical exchange conditions relative to the longitudinal relaxation require in addition that ka 9 pIB, PJB.A meaningful relaxation rate for the coalescent signal can be obtained when the recovery of the signal, following a perturbation, has a single-exponential time dependence. It has been shown above, in a specific case, that single-exponential approximation is valid during an initial period of recovery. In fact, from Eqs. [2] to [ 51 it follows that the recovery of the longitudinal magnetizations can always be approximated by a single-exponential function, for a time interval following the perturbation that is shorter than the inverse CRR. During this initial period the populations of the energy levels in the spin system are not significantly affected by cross relaxation and the longitudinal magnetizations do not deviate appreciably from their respective equilibrium values. Thus, for example, following a selective inversion of the signal corresponding to the spin I, during the initial period the approximation can be made that MJB N M os, and from Eqs. [2] and [4] the rate equation for the coalescent signal of the spin I is found to be d(MIB
+
MIF) =
dt
-(&IF
+
RIIP
+
+J)(MIB
-
MOB)
-
(&IF
+
‘?J)(MIF
-
MOF).
Since even under conditions of intermediate or fast CRR MiB N -M,, MIF N -MOF during the initial period, using the approximation &fIB
+
MIB
-
MOB
i%fIF
-
hf,,
MOB -
hf,JF
=
kf,-~B
+
and
f9
[381
MOF)r
[391
=
&fOF
[37]
Eq. [37] can be rewritten in the form
d@fIB+ MIF) dt
%’ -[f&F
+
(l
+
-
R,IP
f)(RlIF
+
+
CIJ)
‘?J)l(MIB
+
MIF
-
MOB
-
which yields single-exponential time dependence for Mra + Mir. From Eq. 1391 the initial relaxation rate under conditions of fast exchange is found to be SL, initial
1
=
( 7; 1 IC and the paramagnetic
relaxation R IIP
=
;
Similarly
+
UIJ
+
f&P
[401
rate is therefore given by SL’i”it’a’ - (RIIF + g,,,] . Ic
+ [C
&IF
1
it can be shown that in nonselective experiments
[411
268
JOSEPH GRANOT
t (ms) FIG. 4. Semilogarithmic plots of the recovery of the longitudinal magnetization of the spin I, in a twospin system in the hound state, following a selective inversion, for several values of cross-relaxation rate. (a) B,, = 0, (b) (rr, = 10 set-‘, (c) orI = 40 set-‘, (d) err = 120 set-‘. Other values used in the calculations are as given in Fig. 3. Bold lines represent the regions of initial, single-exponential recovery.
NS, initial
R IIP
=
f
[( 1 ;
Ic
-
R,IF
1 .
[421
The second term in Eqs. [41] and [42], i.e., RlIF + (TIJ or RIIF, can be determined from measurement of the initial rate of recovery in the absence of the paramagnetic spins. Note that since in SL experiments R ,ir + RIIp + (TIJ and RIIp + dIJ are the initial rates in the bound and the free states, respectively (e.g., Eqs. [33] and [35] with kr = kB = 0), Eq. [ 391 indeed yields an initial rate for the coalescent signal that is the weighted average of the initial rates in the free and the bound states. Intermediate
Exchange
When the exchange rate is intermediate relative to the longitudinal relaxation rates, i.e., kB - pm, pJa, but rapid enough to coalesce the free and the bound signals, and the CRR is also intermediate, i.e., (Yg - 1, an approximate expression for the paramagnetic relaxation can be obtained for the case of dilute paramagnetic spins, i.e., f < 1, using again initial recovery rates. By analogy to the expression obtained in the absence of cross relaxation (1%17), the initial relaxation rate of the coalescent signal that corresponds to the spin I, following a selective perturbation is given by ($);riniti”
= ($);ynitia’
From which the paramagnetic
+ [R,IF + R,Ip f, u,J1-’ + k,’ .
relaxation rate is found to be
[431
DIPOLAR-COUPLED
SPIN SYSTEMS
269
This expression is also applicable in nonselective experiments. CONCLUSIONS
Determination of the paramagnetic contribution to the longitudinal nuclear relaxation requires that a unique longitudinal relaxation rate be definable, and also that this relaxation rate is not coupled to the relaxation of other spins in the system. It has been shown here that these requirements are rendered only during an initial period that immediately follows the perturbation of the spin system. In general, the slower the CRR, the longer the initial period in which the magnetization has a single-exponential time dependence (e.g., Fig. 4), and the better the conditions for measurement of the initial rate. When the CRR is slow (i.e., (Y< 1) the initial period may extend beyond the experimental time range. Conditions for a slow crossrelaxation rate can in principle be improved by lowering the NMR frequency (e.g., Fig. 2), by increasing the temperature (to increase the rate of molecular motions), by using a paramagnetic probe that is an effective relaxant, or by chemical modifications (diluting the homonuclei through substitutions with appropriate heteronuclei can reduce or eliminate the cross relaxation). However, chemical modifications are usually very difficult or even impossible to carry out, lowering the NMR frequency is accompanied by loss of sensitivity and resolution, elevated temperatures may bring chemical or thermodynamical destabilization, and an “effective” paramagnetic probe may also cause undesirable excessive line broadening. On the other hand, a relatively simple, though somewhat time-consuming, procedure that involves the application of selective, rather than nonselective, spin perturbations can either improve the conditions for slow CRR at a given NMR frequency or permit the measurements to be carried out at higher NMR frequencies. Selective experiments have further advantage, namely, that the initial rate in such experiments is more readily discernible from the rate at a later time. Therefore when the relaxation behavior is nonexponential, selective experiments are recommended. Regardless of the type of perturbation, accurate determination of initial relaxation rates requires that the recovery of the longitudinal magnetization toward equilibrium be studied in detail, especially during the time period immediately following the perturbation. In addition, the value of the equilibrium magnetization should be determined with an utmost accuracy, and complete recovery of the magnetization should be allowed between successive perturbations. In systems containing paramagnetic spins the latter requirement usually will not introduce significant waiting periods between perturbations since the nuclear spin-lattice relaxation may well be rather short. Finally, the procedures that yield accurate paramagnetic relaxation rates in the presence of cross relaxation also allow accurate determination of the correlation time that governs this relaxation, e.g., through measurements of the relaxation rates at a number of NMR frequencies. This correlation time is not only needed in the calculations of nuclei-probe distances, but can also provide useful dynamic information when the molecular motions are faster than the chemical exchange rate and the unpaired electron spin relaxation rate.
JOSEPH GRANOT
270
REFERENCES 1. 2. 3. 4. 5. 6.
R. FREEMAN, S. WI~EKOEK, AND R. R. ERNST, J. Chem. Phys. 52, 1529 I. D. CAMPBELL AND R. FREEMAN, J. Magn. Reson. 11, 143 (1973). W. E. HULL AND B. D. SYKES, J. Chem. Phys. 63, 867 (1975). A. KALK AND H. J. C. BERENDSEN, J. Magn. Reson. 24, 343 (1976). P. J. ANDREE, J. Magn. Reson. 29, 419 ( 1978). J. H. NOCCLE AND R. E. SCHIRMER, “The Nuclear Overhauser Effect,”
(1970).
Academic Press, New
York, 1971. G. REDFIELD AND R. K. GUPTA, Cold Spring Harbor Symp. Quant. Biol. 36, 405 (1971). D. SYKES, W. E. HULL, AND G. H. SNYDER, Biophys. J. 21, 137 (1978). T. EDZES AND E. T. SAMULSKI, J. Magn. Reson. 31, 207 (1978). S. BRIODO AND D. R. KEARNS, J. Magn. Reson. 41, 496 (1980). H. KOENIG, R. G. BRYANT, K. HALLENGA, AND G. S. JACOB, Biochemistry 17, 4348 (1978). INAGAKI, J. BOYD, I. D. CAMPBELL, N. J. CLAYDEN, E. W. HULL, N. TAMITA, AND R. J. P. WILLIAMS, Eur. J. Biochem. 121,609 (1982). GRANOT, N. ASSA-MUNT, AND D. R. KEARNS, manuscript in preparation. M. MCCONNEL, J. Chem. Phys. 28,430 (1958). Luz AND S. MEIBOOM, J. Chem. Phys. 40, 2686 (1964). E. O’REILLY AND C. P. POOLE, JR., J. Phys. Chem. 67, 1762 (1963). S. LEIGH, JR., J. Magn. Reson. 4, 308 (197 1). C. MCLAUGHLIN AND J. S. LEIGH, JR., J. Magn. Reson. 9, 296 (1973). ABRAGAM, “The Principles of Nuclear Magnetism,” Chap. VIII, Oxford Univ. Press (Clarendon),
7. 8. 9. 10. II. 12.
A. B. H. M. S. F.
13. 14. 15. 16. 17. 18. 19.
J. H. Z. D. J. A. A.
20.
G. A. MORRIS
Oxford, 1973. AND R. FREEMAN,
J. Magn.
Reson.
29,433
(1978).
OF MAGNETIC
RESONANCE
49,
251-210 (1982)
Paramagnetic Relaxation in Dipolar-Coupled Homonuclear Spin Systems JOSEPH Department
of Physics,
GRANOT
Purdue University IUPUI, Indianapolis,
School Indiana
of Science
at Indianapolis,
46223
Received February 16, 1982; revised April 9, 1982 Measurements of the paramagnetic contribution to the longitudinal relaxation of nuclei interacting with a paramagnetic probe can be complicated by effects of chemical exchange and cross relaxation. A theoretical investigation is presented of the relaxation behavior of the longitudinal nuclear magnetization, following either selective or nonselective inversion of the equilibrium magnetization, in a dipolar-coupled homonuclear two-spin system that undergoes exchange between free and bound (to a paramagnetic probe) states. Approximate solutions are derived for a number of limiting cases characterized in terms of the relative rates of exchange, cross relaxation, and spin-lattice relaxation. It is shown that unless the cross-relaxation rate (CRR) is slow compared to the other rates that affect the longitudinal relaxation, reliable and accurate paramagnetic relaxation rates that can yield useful structural information can be obtained only from measurement of the initial rate of recovery of the longitudinal magnetizations. Some advantages of selective over nonselective spin perturbations are pointed out. INTRODUCTION
Effects of cross relaxation, that arise from mutual flips of dipolar-coupled nuclear spins through zero-quantum transitions, have been the subject of considerable interest in recent years (I-6). When the cross-relaxation rate (CRR) is fast relative to the spin-lattice relaxation rate, magnetization can be transferred between the coupled spins before a significant amount cf magnetic energy is transferred to the lattice. Consequently the time dependence of the longitudinal magnetization of each spin is no longer independent of the relaxation of the other spins in the system, resulting in significant deviations of the individual recovery rates from the intrinsic spin-lattice relaxation rates. Furthermore, the time dependence of the longitudinal magnetizations may become nonexponential, and proper longitudinal relaxation rates may actually be undefinable. The CRR increases when the molecular motions become slower; hence cross-relaxation effects may be dominant in macromolecules (7-13). Also, cross relaxation can be particularly effective in homonuclear spin systems since the Zeeman energy that is exchanged in zero-quantum transitions is small for homonuclei. Paramagnetic probes, e.g., metal ions and spin labels, have been used in numerous NMR studies of structural and dynamic properties of chemical and biochemical systems. Determination of the contribution to the nuclear relaxation rate from the interaction with the unpaired spin of the paramagnetic probe can be complicated 251
0022-2364/82/l
10257-14SO2.00/0
Copyright 0 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.
258
JOSEPH
GRANOT
by effects of cross relaxation, when the nuclear spins in the system are coupled, and/or by chemical exchange, when the association with the paramagnetic probe is labile. While the theoretical aspects of the chemical exchange effects on the longitudinal nuclear relaxation in the presence of paramagnetic spins are well known (14-18) and are commonly taken into account in analyses of relaxation data, the effects of cross relaxation have been usually neglected. Possible effects of cross relaxation on distances measured from paramagnetic probes have been recently discussed by Andree (5). It should, however, be noted that some of his remarks, e.g., that in the presence of cross relaxation measured distances may be shorter than the true values due to enhanced nuclear relaxation, and that underestimated correlation times lead to overestimated distances, are by no means general and are not always valid. In the present work the effects of cross relaxation on the longitudinal relaxation in a dipolar-coupled homonuclear spin system that also interacts with a paramagnetic probe are examined in some detail. The theory is formulated for two spins, however, the conclusions can in general be applied to multispin systems as well. Conditions under which the longitudinal magnetization has a single-exponential time dependence are examined for a number of limiting cases characterized in terms of the relative rates of chemical exchange, cross relaxation, and spin-lattice relaxation (diamagnetic and paramagnetic). Procedures that can yield accurate paramagnetic relaxation rates in the presence of cross relaxation are discussed. THEORY
Consider a system, at equilibrium, that contains paramagnetic probes and ligand molecules (L) that undergo exchange between a free state (F), and a bound state (B) in which a ligand molecule is associated with a paramagnetic probe, according to the scheme
111 where kF and k, are the first-order exchange rates from the free and the bound states, respectively.’ Let I and J be two spins on L that interact via dipole-dipole interaction with one another, and are sufficiently close to the paramagnetic probe, in the bound state, to have their relaxation rates significantly affected by the paramagnetic interaction.2 The spins I and J may in general be heteronuclear, however, it has been shown (I, 2) that in such cases a single-exponential time dependence of the longitudinal magnetization of each spin can be achieved simply by continuous ’ Note that while ke is an intrinsic rate constant, kF may be concentration dependent, depending on the type of reaction that takes place when L exchanges between the free and the bound states. Thus for example, in a typical case when L associates with a paramagnetic probe P through the bimolecular reaction L + P G LP, with a second-order forward rate constant k,. and a first-order reverse rate constant kow we have ks = koff and kF = k,,[P], where [P] is the concentration of the paramagnetic spins. * Nomenclature: I and J denote nuclear spins and S denotes electronic spin. The subscripts “F” and “B” refer to the free and bound states, respectively. These subscripts may be omitted when the discussion applies to both the free and the bound states. “SL” and “NS” denote selective and nonselective spin perturbations, respectively. CRR denotes cross-relaxation rate.
DIPOLAR-COUPLED
SPIN
259
SYSTEMS
saturation of the other spin. Hence this work will be concerned with homonuclear spins. It is assumed that the spins I and J have nonoverlapping resonances and that their scalar spin-spin interaction is weak, i.e., that their ratio of chemical shift to spin-coupling constant is large. In addition to simplifying the rate equations for the nuclear magnetizations, the assumption of weak scalar coupling also permits selective perturbation of either one of the spins I and J. It is further assumed that any rf perturbation affects equally all the multiplet components of each spin. Following an rf perturbation of the spin system, the recovery of the longitudinal magnetizations toward thermal equilibrium, after the perturbing field has been turned off, is governed by the coupled rate equations ___
=
dt
-uh
+
+
dt
d&F ___
dt
+
UIJ~@h4
-
Mm)
+ ~IJWJLI - Mm)
dMm = -(RIJF
dM,, dt
RlIP
RlJP
+
UIJ)(MJB
-
-
bI~II3
+
MfIF
7
[21
MOB)
+
UIJ(&
-
MOB)
-
&IMJB
+
kFMJF
,
[31
,
[4]
=
-tRIIF
+
ulJ)(“IF
-
MOP)
+
UIJ(MJ,
-
MOP)
-
kFMIF
+
bdfIL3
=
-tRIJF
+
UIJ)(MJF
-
MOF)
+
UIJ(MIF
-
MOF)
-
k&JF
+
‘b&fJB,
[5]
where it is assumed that the equilibrium magnetizations of the two spins are equal, being MOF and M 0s in the free and the bound states, respectively (MOF may not in general be equal to M,,,). Quantities MI and M, are the expectation values of the component of the nuclear magnetization of I and J along the external magnetic field, R,,r and RIJF are the spin-lattice relaxation rates of I and J in the free state (excluding cross-relaxation effects), R rip and RIJp are the contributions to the longitudinal relaxation of I and J due to their interaction with the paramagnetic probe in the bound state (paramagnetic contributions due to second-sphere effects are neglected), and uIJ (=uJr) is the cross-relaxation rate (CRR) between the spins I and J. The characteristic relaxation rates in the system are given by (19) R ,,p = $ ,‘;h2z(z + 1)r;P[3J,(wr) R 1,P U ,J
=
&
y:gv2m
=
6
#i2z(z
+
+
l)r;P[J,(@,
l~~aJ
+ 12J2(‘0r
+
UJ)]
+
o( 01 - us) + 3Jl(~r) + 6J2(~t -
WJ)
-
6J2(01
+
@J)],
[61
K;p,
+ 41 + REP
3
171
[81
with analogous relations for the spin J. In Eqs. [6] to [8], I and S denote nuclear and electronic SpinS, respectively, yI (=-y,) iS the UUClear gyromagnetic ratio, and rIs are the distances between the nuclei I and J and between I and the paramagnetic probe, respectively, wI and ws are nuclear and electronic Larmor frequencies, and J,,(w), n = 0, 1, 2, are spectral density functions. When the motion r,J
(x.dJ)
260
JOSEPH GRANOT
of the spin system is isotropic, J”(w) = ~c/( 1 + n2w2~&), where 7c is the correlation time that governs the longitudinal relaxation. For example, in the bound state -’ = 7;’ -t 7;’ + kg, where TR is a rotational correlation time and 7s is the re;ixation time of the unpaired spin of the paramagnetic probe. The term Kip in Eq. [6] represents any additional mechanism that contributes to the spin-lattice relaxation in the free state. These mechanisms are assumed not to give rise to crossrelaxation effects. The term RsI”p in Eq. [ 71 represents paramagnetic contribution to the relaxation rate due to scalar electron-nuclear hyperfine interaction that is present when direct ligand-probe coordination (i.e., with a covalent character) takes place. Determination of the dipolar term of RIIp is usually a major objective in experiments that monitor the effects of a paramagnetic probe on nuclear relaxation, since the inverse sixth-power dependence of the dipolar term on internuclear distances makes such measurements a powerful tool in structural studies. Bearing in mind that kB/kF = MoF/M oB, Eqs. [2] to [5] can be rewritten in the matrix form ZM=A(M-M,) dt
II91
where r
CIJ
kF
0
-PJB
0
kF
0
-PIF
-PIB u1.l
A=
k, L
0
-
kB
UIJ
@IJ
[lOI
’ 1
-PJF
r&i
MIBMJB
M=
[Ill
MIF MJF-
and where PIB
=
RIIF
+
Rnp
+
PIF
=
&IF
+
UIJ +
flIJ
+
kB
El21
,
[I31
kF,
with analogous relations for the spin J. The off-diagonal elements of A represent the coupling between the longitudinal relaxations of the spins I and J through cross relaxation and chemical exchange. The general solution for the time dependence of the longitudinal magnetization of the spin I or J in the free or the bound state (i.e., for any component of M) is given by (M,, - M(t))/M,
= i
C,e-‘;‘,
[I41
i=l
where the values of Ci are constants that depend on the initial conditions (at t = 0) of the spin system, and -Xi are the eigenvalues of the matrix A which can be obtained by solving the quartic equation [(PIB
+
x)(P.lB
+
A)
-
-
[(PlB
‘&l[(PlF +
x)(P,F
+ +
X)(PJF A)
+
+ (PJB
A)
+
&,I X)(PJF
+
h)lk&B
+
k#,
=
0.
[ 151
DIPOLAR-COUPLED
261
SPIN SYSTEMS
Approximate solutions for a number of limiting cases are discussed below with the objective being the determination of the conditions under which the time dependence of the longitudinal magnetization is single exponential, thereby allowing a single relaxation rate to be defined and to be measured. DISCUSSION
Slow CRR
When the CRR is SLOW,i.e., a:J 4 P~FPJF(since PIB > PIF, PJB> PJF,it also follows that & 6 P~BPJB),Eq. [ 151 reduces to the simplified form [(PJB + XMPIF + A) - ~F~~I[(PJB + XNPJF + A) - h&l
= 0.
Cl61
The relaxation behavior of each nuclear spin can thus be treated separately, with the problem becoming a simple two-site exchange with paramagnetic contribution to the relaxation rate at one site. This problem has been previously dealt with (17, 18) and will not be further discussed here. It should only be noted that conditions for slow CRR require negligibly small contribution of to the off-diagonal elements of the matrix A (Eq. [lo]), but not to the diagonal elements. Hence cross relaxation may still affect the measured relaxation rates, albeit not necessarily the measured net paramagnetic relaxation. (TIJ
Slow Exchange
Under conditions of slow exchange, i.e., when kFkB 4 smaller of P~BP~F and PJBPJF, Eq. [ 151 reduces to [(PIB + X)(PJB + A> - GJI[(PIF + X)(PJF + A) - ~JI = 0.
t171
Hence the nuclear relaxations in the free and the bound states are decoupled, and the relaxation behavior in each state can in principle be independently investigated. To allow measurements of relaxation rates in the bound state the chemical shift separation between the free and the bound signals should be sufficiently large so that distinct free and bound signals can be observed under the above condition of slow exchange. If, however, the free and the bound signals overlap, the contribution to the signal from free species should be made negligibly small, i.e., by introducing enough paramagnetic probes to bind the major part (or all) of the ligand molecules.3 The recovery pattern of the longitudinal magnetization depends not only on the characteristic rates in the system but also on the initial conditions which in turn depend on the type of rf perturbation that has been applied to the system. Two types of perturbations may be particularly useful in relaxation studies of homonuclear systems: (1) nonselective (NS), when all the spins in the system are simultaneously perturbed, e.g., fo!lowing a 180” pulse, at t = 0 MiB = MJB = --Moe and MiF = MJF = -MOF, and (2) selective (SL) (20) when only one spin (e.g., I) is perturbed, e.g., following a 180” pulse, MiB = -MJB = -MOB, and MIF = -MJF 3 In the limit when labels, and substitution
ks - 0, the discus inert ligand-probm:
don presented complexes.
here is also applicable
to nonexchangeable
spin
262
FIG. [22]).
JOSEPH
GRANOT
1. The variation of KNS (Eq. [ 191) and KsL (Eq. (a) KNS, for (Y > 0, (b) KsL, (c) KNS, for (Y < 0.
[20])
as a function
of the parameter
(Y (Eq.
= -MOF. (Under conditions of slow exchange, selective perturbation of the resonance corresponding only to the free or the bound state may also be possible.) Following a nonselective (NS) or selective (SL) inversion of the equilibrium magnetization, the time dependence of the longitudinal magnetization of the spin I in the bound state, obtained by solving Eq. [ 171, can be written as (MOB - M,,)/M,,
[If31
= (1 - KB)e-xlB’ + (1 + KB)e-‘ZB’,
where KES = (aB - l)(l
+ (r;)-1’2,
[I91
KgL = -( 1 + c&-“2, hB,2B
with the + sign corresponding
=
[201
PB -+ (l/‘J)Apdl
to &, a B = APB
=
PB =
+
2 l/2
aB)
[211
,
and ~I/APB
9
PIB -
PJB >
(~/~)(PIB
+
I.221 [231
0,
1241
PJB).
Analogous expressions and definitions for the free state can be obtained by substituting the subscript F for B in Eqs. [ 181 to [24]. The variation of KNS and KsL with (Y is depicted in Fig. 1. Note that since Ape (Eq. [23]) has been arbitrarily defined (without loss of generality) as positive, (Yg acquires the sign of and can be either positive or negative (cf. Eq. [8]). In the forthcoming discussion the relaxation behavior in the bound state will be examined for three limiting cases with positive CRR, and for the case of negative CRR. (i) Slow CRR. When 0 < (Yg< 1, both KFS and KgL approach - 1 (Fig. l), and Eq. [ 181 reduces to the approximate single-exponential form oIJ,
(ibfoB - bf~B)/i!!f~B N 2e-““. Let (1 /T,), and (1 /T,), denote the measured relaxation bound signals, respectively, then
t251 rates of the free and
DIPOLAR-COUPLED
SPIN
263
SYSTEMS
III
0
100
200
1 400
300
v, (MHz) FIG. 2. Frequency dependence of 2q,/RllP for = J = l/2, S = 1, and g = 2, and an isotropic contribution to RllP (Eq. [7]) has been neglected.
several rIs/rIJ ratios. rotational correlation
Calculations time of
were made 10m8 sec. The
R ltp = (~/T,)B - (Rm + uu + kd
with I scalar
I-261
and if the CRR is slow in the free state as well (i.e., (YF6 l), then
R IIP
=
(1/%l
-
(l/T,)F
-
(‘b
-
[271
kF).
However, since the effect of the paramagnetic probe on the nuclear relaxation is usually much larger than that of a nearby nuclear spin, i.e., RIIp % RIlF, the condition of slow CRR in the free state may not necessarily be fulfilled. In such cases when (Yr S 1, the value of RllF + needed in Eq. [26], should be measured from the relaxation behavior in the free state as described below. The exchange contribution to the relaxation, if not negligible, can in principle be determined from temperature effects on the relaxation. The favorable condition of slow CRR in the bound state requires that ApB (=Rllp - RIJP)be large compared to au; hence rIs should differ from (Eq. 7). Also the paramagnetic probe should be sufficiently close to I to allow effective dipolar interaction between the nuclear spin and the unpaired electron spin. From Eqs. [7] and [8] it is also apparent that when the NMR frequency (wi) is increased, tends to increase whereas RIIp decreases. Hence conditions of slow CRR may be more readily achieved at lower NMR frequencies. Consider, for example, a case when R ,ip 9 RlJp, i.e., LYBN and when the scalar contribution to the paramagnetic relaxation is negligible. Figure 2 shows the dependence of 2uIJ/RIIP on the NMR frequency for several ratios, for a system that undergoes isotropic tumbling with a rotational correlation time of lo-’ set that governs both the cross relaxation and the paramagnetic relaxation rates. From Fig. 2 it is evident that while the distance ratio (rIs/rIJ) is the major determinant of (Yg, due to the inverse sixth-power dependence of the relaxation rates on the internuclear separation, the frequency effect is also quite significant. In the present example (Yg varies by almost two orders of magnitude in the frequency range 50 to 400 MHz. (TIJ
rJs
@[J
&J~J/R~~,
rIs/rIJ
264
JOSEPH GRANOT
I 50
0
I 100
I 150
t (ms)
FIG. 3. Semilogarithmic plots of the recovery of the longitudinal magnetization of the spin I in a twospin system following a nonselective or selective inversion, under conditions of slow chemical exchange rate and intermediate CRR: (a) bound state, NS, (b) bound state, SL, (c) free state, NS, (d) free state, SL. Calculations were made using the values Rllp = 30 set-‘, RIJp = 10 set-‘, RllF = 2 set-‘, Rllr = 1 set-‘, ks = kF = 0, Q,, = 20 set-‘. Dashed lines represent the initial rate of recovery following a selective perturbation of I.
Since nuclei-probe distances are beyond our control, the frequency dependence of (Ya allows some flexibility in the setting of the experimental conditions. Unfortunately sensitivity and resolution considerations usually require high NMR frequencies, contrary to the requirement for slow CRR. Figure 1 reveals an important advantage of selective over nonselective spin perturbations, resulting from the fact that conditions for slow CRR are more readily obtainable in selective experiments. Thus, KsL - -1 (within 15%) for (Y ? 0.60, whereas KNS - -1 for (Y Z 0.15, and since (Y increases when the frequency is increased, this four-fold difference allows relaxation measurements to be carried out at significantly higher NMR frequencies, through selective perturbations. For example, from Fig. 2 a four-fold increase in (Ys corresponds to about a two-fold increase in the NMR Frequency. Hence if slow CRR conditions prevail in nonselective experiments up to 200 MHz, the experimental frequency range can be extended through selective experiments to 400 MHz. (ii) Fast CRR. When (Ys 9 1 (and therefore also ayF$= 1 ), KFS I: KfS 1: 1, and IK”F”I N IK”B”j 4 1, and also from Eq. [ 211 we get &B
=
RIB
+
&B
=
RIB
1
2%
[281
,
1291
where RIB
=
(l/2)&~
+
R,IP
+
R,JF
+
RIJP)
+
kB
[301
with analogous relations for the free state. From Eq. [ 181 the time dependence of the longitudinal magnetization in the free and bound states, following nonselective perturbation, will be approximately single exponential with a rate constant equal
DIPOLAR-COUPLED
265
SPIN SYSTEMS
to R,, or RIB, respectively. Since XIB(XIF) is always larger than &(X,,), following a selective perturbation the first term in Eq. [ 181 (which has approximately the same amplitude as the second term) will decay faster than the second term, and after a time period t - ark’ the longitudinal magnetization will approach a single-exponential time dependence with the rate constant being again R,, (or RI,). The relaxation behavior of the longitudinal magnetizations under conditions of fast CRR result from the known fact that cross relaxation equalizes and maintains equal the magnetizations of all the coupled spins in the system. As a result the longitudinal magnetizations of these spins recover with virtually the same relaxation rate (& or R,,) which is a weighted average of all the contributions to the spinlattice relaxation rates of the coupled spins. In other words, fast cross relaxation is a mechanism that effectively equalizes, and keeps equal, the spin temperatures of coupled spins in the system, following any perturbation that increases the spin temperature either locally or throughout the whole system. Once the spin temperature of the coupled spins equilibrates, they all cool down toward the external temperature with the same rate that is maintained uniform throughout the spin system by the fast cross relaxation. Under conditions of fast CRR it is impossible to determine the individual paramagnetic contribution to the relaxation of either I or J, i.e., RIIP or RIJP, respectively, and only their average (l/2)(& + RIJP) can be obtained. This average cannot in general provide useful structural information, since it can only yield the average distance f = [( 1/2)(r;: + r;l)]-“. A single-exponential time dependence of the longitudinal magnetization should be therefore treated with some caution in systems with cross relaxation, and it is imperative to find if this behavior results from slow or fast CRR. A warning sign for fast CRR may be the observation that all or many resonances in the system relax with very similar longitudinal relaxation rates. (iii) Intermediate CRR. When (Ye - 1 the recovery of the longitudinal magnetization, following either a NS or SL perturbation, is in general nonexponential (e.g., Fig. 3). While in principle the relaxation data of a two-spin system can be fitted as a sum of two exponentials (e.g., Eq. [ 18]), such analysis will involve at least three adjustable parameters (&, XIB, X2& and more if the experimental conditions (e.g., pulse lengths) were not ideal, with the resulting “best-fit” parameters being of dubious merit. Clearly when more than two spins are coupled, a multiexponential fit is completely out of question. Useful relaxation rates can, however, be obtained from measurements of initial rates of recovery of the longitudinal magnetization (I). Over an initial time period t < X;; Eq. [ 181 can be approximated by
- [VW3+ b*) - KB&B- bdlt.
OfrIB - MIB)IM3B = 2
The initial recovery is therefore single exponential, turbations, the initial rates are given by
=
&IF
+
R,IP
+
and following
kB
,
[311
NS or SL per-
[321
266
JOSEPH
=
and similarly, if (YF-
GRANOT
&lF
+
RIIP
+
=
&IF
+
b,
=
&IF
+
‘JIJ
UIJ
+
kB
(331
1,
0 1
T1
[341
SL, initial +- kF
[351
IF
with analogous relations for the spin J. These expressions are valid also for a multispin system, in which case should be replaced by a sum of cross-relaxation terms corresponding to all the spins that are dipolarly coupled to I. From the initial rates the paramagnetic contribution to the relaxation can be directly obtained: bIJ
R,,,=(~)::i”-(~~~‘+(kB-kF).
[361
Note the similarity between Eq. [36] and Eq. [27] obtained in the case of slow CRR. Indeed, Eq. [27] can be considered as a special case of Eq. [ 361 with the initial period extending over the whole experimental time range. While initial rates appear to provide rather handy means for determination of paramagnetic relaxation rates when cross-relaxation effects are significant, they have a serious drawback, namely, they may be very difficult to measure accurately. Difficulties can arise, for example, when due to statistical errors the experimental spread in the measured recovery data is large, when the CRR is relatively rapid (hence the initial period is short), or when the initial rate is not very different from the rate at later time. Since the initial rates in SL experiments contain a crossrelaxation term, in general (for cx > 0), ( 1/T1)SL, i”itia’ > ( 1 /T,)NS* i”itia’. Hence in selective experiments initial rates are more readily discernible (in semilogarithmic plots, e.g., Fig. 3), and probably can be more accurately measured. SL experiments also provide virtually the only means to determine the CRR itself (through ( 1 / T,)fiS, initial ). Hence in the case of intermediate CRR, selective rather than nonselective measurements may well be the method of choice. (iv) Negative CRR. In the limit of fast molecular motions, when the probability of double-quantum transitions exceeds that of single-quantum transitions, i.e., 65, > Jo (Eq. [ 8]), becomes negative. Negative CRR reduces the relaxation rates, e.g., through Eqs. [ 121, [ 131, but does not significantly affect the pattern of the recovery of the magnetization, e.g., through the parameter (Y (Eqs. [ 181 to [22]). The values of KsL are independent of the sign of (Y, and to a good approximation so are the values of KsL when ]a] 6 1. When ]CZ]- 1, the time dependence of the magnetization is not single exponential, and the foregoing discussion for the case of positive and intermediate CRR is essentially valid for negative cxas well. When Ia] $ 1 the magnetization has approximately single-exponential time dependence, albeit since for negative (Y, K NS - -1, the recovery rate constant is R,, compared to KNS - +l and a recovery rate constant of RIB (which is independent of for positive ff. gIJ
21alJI
cIJ)
DIPOLAR-COUPLED
SPIN
267
SYSTEMS
Fast Exchange When the chemical exchange between the free and the bound states is fast compared to their chemical shift separation, the free and the bound signals coalesce and a single resonance is observed that may relax with a rate equal to the weighted average of the relaxation rates in the free and the bound states. Fast chemical exchange conditions relative to the longitudinal relaxation require in addition that ka 9 pIB, PJB.A meaningful relaxation rate for the coalescent signal can be obtained when the recovery of the signal, following a perturbation, has a single-exponential time dependence. It has been shown above, in a specific case, that single-exponential approximation is valid during an initial period of recovery. In fact, from Eqs. [2] to [ 51 it follows that the recovery of the longitudinal magnetizations can always be approximated by a single-exponential function, for a time interval following the perturbation that is shorter than the inverse CRR. During this initial period the populations of the energy levels in the spin system are not significantly affected by cross relaxation and the longitudinal magnetizations do not deviate appreciably from their respective equilibrium values. Thus, for example, following a selective inversion of the signal corresponding to the spin I, during the initial period the approximation can be made that MJB N M os, and from Eqs. [2] and [4] the rate equation for the coalescent signal of the spin I is found to be d(MIB
+
MIF) =
dt
-(&IF
+
RIIP
+
+J)(MIB
-
MOB)
-
(&IF
+
‘?J)(MIF
-
MOF).
Since even under conditions of intermediate or fast CRR MiB N -M,, MIF N -MOF during the initial period, using the approximation &fIB
+
MIB
-
MOB
i%fIF
-
hf,,
MOB -
hf,JF
=
kf,-~B
+
and
f9
[381
MOF)r
[391
=
&fOF
[37]
Eq. [37] can be rewritten in the form
d@fIB+ MIF) dt
%’ -[f&F
+
(l
+
-
R,IP
f)(RlIF
+
+
CIJ)
‘?J)l(MIB
+
MIF
-
MOB
-
which yields single-exponential time dependence for Mra + Mir. From Eq. 1391 the initial relaxation rate under conditions of fast exchange is found to be SL, initial
1
=
( 7; 1 IC and the paramagnetic
relaxation R IIP
=
;
Similarly
+
UIJ
+
f&P
[401
rate is therefore given by SL’i”it’a’ - (RIIF + g,,,] . Ic
+ [C
&IF
1
it can be shown that in nonselective experiments
[411
268
JOSEPH GRANOT
t (ms) FIG. 4. Semilogarithmic plots of the recovery of the longitudinal magnetization of the spin I, in a twospin system in the hound state, following a selective inversion, for several values of cross-relaxation rate. (a) B,, = 0, (b) (rr, = 10 set-‘, (c) orI = 40 set-‘, (d) err = 120 set-‘. Other values used in the calculations are as given in Fig. 3. Bold lines represent the regions of initial, single-exponential recovery.
NS, initial
R IIP
=
f
[( 1 ;
Ic
-
R,IF
1 .
[421
The second term in Eqs. [41] and [42], i.e., RlIF + (TIJ or RIIF, can be determined from measurement of the initial rate of recovery in the absence of the paramagnetic spins. Note that since in SL experiments R ,ir + RIIp + (TIJ and RIIp + dIJ are the initial rates in the bound and the free states, respectively (e.g., Eqs. [33] and [35] with kr = kB = 0), Eq. [ 391 indeed yields an initial rate for the coalescent signal that is the weighted average of the initial rates in the free and the bound states. Intermediate
Exchange
When the exchange rate is intermediate relative to the longitudinal relaxation rates, i.e., kB - pm, pJa, but rapid enough to coalesce the free and the bound signals, and the CRR is also intermediate, i.e., (Yg - 1, an approximate expression for the paramagnetic relaxation can be obtained for the case of dilute paramagnetic spins, i.e., f < 1, using again initial recovery rates. By analogy to the expression obtained in the absence of cross relaxation (1%17), the initial relaxation rate of the coalescent signal that corresponds to the spin I, following a selective perturbation is given by ($);riniti”
= ($);ynitia’
From which the paramagnetic
+ [R,IF + R,Ip f, u,J1-’ + k,’ .
relaxation rate is found to be
[431
DIPOLAR-COUPLED
SPIN SYSTEMS
269
This expression is also applicable in nonselective experiments. CONCLUSIONS
Determination of the paramagnetic contribution to the longitudinal nuclear relaxation requires that a unique longitudinal relaxation rate be definable, and also that this relaxation rate is not coupled to the relaxation of other spins in the system. It has been shown here that these requirements are rendered only during an initial period that immediately follows the perturbation of the spin system. In general, the slower the CRR, the longer the initial period in which the magnetization has a single-exponential time dependence (e.g., Fig. 4), and the better the conditions for measurement of the initial rate. When the CRR is slow (i.e., (Y< 1) the initial period may extend beyond the experimental time range. Conditions for a slow crossrelaxation rate can in principle be improved by lowering the NMR frequency (e.g., Fig. 2), by increasing the temperature (to increase the rate of molecular motions), by using a paramagnetic probe that is an effective relaxant, or by chemical modifications (diluting the homonuclei through substitutions with appropriate heteronuclei can reduce or eliminate the cross relaxation). However, chemical modifications are usually very difficult or even impossible to carry out, lowering the NMR frequency is accompanied by loss of sensitivity and resolution, elevated temperatures may bring chemical or thermodynamical destabilization, and an “effective” paramagnetic probe may also cause undesirable excessive line broadening. On the other hand, a relatively simple, though somewhat time-consuming, procedure that involves the application of selective, rather than nonselective, spin perturbations can either improve the conditions for slow CRR at a given NMR frequency or permit the measurements to be carried out at higher NMR frequencies. Selective experiments have further advantage, namely, that the initial rate in such experiments is more readily discernible from the rate at a later time. Therefore when the relaxation behavior is nonexponential, selective experiments are recommended. Regardless of the type of perturbation, accurate determination of initial relaxation rates requires that the recovery of the longitudinal magnetization toward equilibrium be studied in detail, especially during the time period immediately following the perturbation. In addition, the value of the equilibrium magnetization should be determined with an utmost accuracy, and complete recovery of the magnetization should be allowed between successive perturbations. In systems containing paramagnetic spins the latter requirement usually will not introduce significant waiting periods between perturbations since the nuclear spin-lattice relaxation may well be rather short. Finally, the procedures that yield accurate paramagnetic relaxation rates in the presence of cross relaxation also allow accurate determination of the correlation time that governs this relaxation, e.g., through measurements of the relaxation rates at a number of NMR frequencies. This correlation time is not only needed in the calculations of nuclei-probe distances, but can also provide useful dynamic information when the molecular motions are faster than the chemical exchange rate and the unpaired electron spin relaxation rate.
JOSEPH GRANOT
270
REFERENCES 1. 2. 3. 4. 5. 6.
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