Fast kinetics studied by NMR

007’)

6565’79/1?01-0?57\05.00/0

FAST .KINETICS STUDIED BY NMR PIERRELASZLO

institut de Chimie, Universite de Liege. Sart-Tihnan par 4000, Liege I. Belgium (Receioed 12 March 1979) CONTENTS

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

introduction The Available Formalisms When is a Complex a Complex? Binding and Release of Small Substrates by Bio-molecules Determination of Rates from Pulse Spacings Discourse Conclusion Acknowledgements References

1.INTRODUCTtON

Chemical exchange of a nucleus between different environments influences markedly Tz, thus affecting the lineshape and the linewidth whenever the rates for sampling the Larmor frequencies for the different sites are similar in magnitude to the differences between the Larmor frequencies. This well-known principle led to a myriad of applications of NMR to various rate processes, starting in the early nineteen sixties,“*2) which we will label the conventional dynamic NMR method (or DNMR-I). For a given nuclear species, the difference in Larmor frequency between various molecular sites amounts to at most a few thousand hertz: this conventional NMR method can measure rates in the range of approximately loo-lo4 s- ‘. As stated above, such a narrow range has not stifled the proliferation of kinetic measurements using either complete lineshape analysis, or treatments of a more approximate nature. At roughly the same time, chemical relaxation techniques were successfully developed and promoted by M. Eigen and his school.‘3) These allowed for the successful extension of classical chemical kinetics from the millisecond down into the nanosecond, and even picosecond, time scale. Hence, the widely-held belief is that the “slow” NMR method (DNMR-I) is limited to rather small rate constants,‘4’ below 103-104s- ‘, just like classical chemical kinetics, and that chemical relaxation tech-

257 258 261 263 266 268 268 268 269

niques are mandatory for exploration of the fast kinetics region. in between lo3 and 10” s-i. True, even though NMR was considered to be restricted to rather slow rate processes, it had some potential advantages over chemical relaxation methods in the loo-lo4 s- ’ region : for example, it needs smaller samples (a fraction of a cubic centimetre being sufficient, albeit at relatively high concentrations), and it is applicable to isodynamic processes, such as the bullvalene rearrangement,“) in which the chemical composition of the system does not change with time. Whereas the temperature-jump technique requires high ionic strengths for a 6-8” temperature jump using discharge of a condenser, the NMR method is of course applicable to non-electrolytes. Also, activation parameters and thermodynamic data are more readily accessible by NMR than through chemical relaxation methods. For instance, Strehlow(*a) has used both methods to study the hydration of pyruvic acid, which is acid-catalyzed and obeys the following scheme : CH&OCOO-

+ H30+ + 2H20 F? CHJCOOH + 3HrO +H3.C(OH)rCOOH,2H20 kz3 = k& +

with

k~~(H~O+)

ks2 = ks2 + k&(HxO+)

As seen from Table 1, while the rate constants are more reliable if obtained from chemical relaxation,

TABLE1. Kinetic data for the hydration of pyruvic acid’*@ k’(s-‘1

hydration (li,,) dehydration (ks2)

0.55 + 0.05 0.5 * 0.4 0.22 5 0.02

,

+ ,

k” (M-r s-r)

6.3 & 0.7 5.0 * 1.5 2.5 k 0.3 2.0 k 0.8

E, (kJmol-‘)

AH’(kJmol-‘)

AS’ (J mol-’ K-r)

38 +2 42 * 2

39

-100

72 * 2

70

+4

257

method(ref) them. relaxn. DNMR-I them. relaxn. DNMR-I

(6) (7) (6) (7)

PIERRELASZLO

258

the activation energy and the AH * and AS’ terms are better evaluated by NMR techniques. Such a short summary does not pay full justice to the very respectable body of work performed using NMR line broadening (DNMR-I), among which one finds truly beautiful contributions to chemistry “-2.5.9-1o). However, while there have been legions of rate measurements from line broadenings, very few studies have taken advantage simultaneously of another important time-dependent feature of NMR (DNMR-II): namely, NMR relaxation times which reflect the rotational correlation functions for the molecules. When there is coupling between chemical exchange and the re-orientational motion which affects the correlation function, one gains access to the chemical rate constants. Clearly, the effect will be at a maximum when the chemical rate constants match the inverse of the correlation times: this occurs in the nanosecond to picosecond time range, precisely that believed erroneously by many to be uniquely accessible through the chemical relaxation techniques. This review is an attempt to redress this imbalance; it will summarize the existing theories relating the longitudinal RI and transverse R2 relaxation rates to chemical rate constants (referred to as DNMR-II); it will present a few selected examples of applications, taken mainly from studies of the binding of small substrates to large biomolecuies; and finally it will attempt to provide an explanation for the widespread neglect of the DNMR-II method, compared with DNMR-I and the chemical relaxation techniques. This review refrains from being comprehensive in scope in favour of a more illustrative coverage. Particular attention will be made to areas which have not previously enjoyed the spotlight. The review will be complemented by reference to a number of existing tests: reference to these will be made in the appropriate sections. 2. THE

rates from the standpoint of chemical rate theory. first pointed out by Eyring “r) before it was developed by many others.‘i2-‘5’ Thus. we associate rotational rate constants k, = (r/)-l and kb = (rb)-’ with the above states. Their fractional populations are p/ and pb, such that p,-- ki = pi,*k_, according to the equilibrium condition. The Bloch-McConnell equations in the presence of chemical exchange were first solved by Woessner,“@ who gave explicit expressions for the decay of the magnetization in pulse experiments. Other workers also used the dependence of the transverse relaxation time Tz of the exchanging nuclei upon the radiofrequency pulse separation in a Carr-Purcell sequence to measure the rates of chemical reactions.~‘7-22’ More generally, the rate of molecular rotation serves as a frequency standard for the exchange process/23’ and a number of articles treated the coupling between molecular rotation and chemical exchange during the sixties!24-32) The treatment of Anderson’“3~27~2g’is particularly clear and elegant, and will now be outlined. Whenever re-orientation can be described as a onedimensional Markov process, according to the theorem by Doob ‘33’it must have a correlation function of the form (f(O)‘) - exp (- T/ ( T, (), where rc is a correlation time for the re-orientation and (f(0)2) is the meansquared initial orientation. In order to describe the above two-states system, Anderson”” used a set of joint (composite) probabilities that are Markovian. Let us define the probability PA(t) that the molecule containing the nucleus of interest is botk in the free state and has not re-orientated; it will have the probability P,(t) of both being in the free state and having jumped to a new orientation; likewise, P=(t) is the probability for the molecule to be both in the bound state and not to have re-orientated ; and P,(t) is the probability for the molecule to be both in the bound state and having re-orientated. These probabilities follow the Kolmogorov(34) equations which become:

AVAILABLE FORMALISMS

We want to describe the magnetic resonance for a nucleus undergoing fast chemical exchange between two states, which are not distinguished by any substantial chemical shift difference, but by considerably different relaxation rates. For instance, this nucleus may belong to an ionic species equilibrating between the free state in the solution and the bound state on a slowly re-orientating macromolecule: free’? bound _

dP.4 dt = -(k/

+ k+)P, + k-Pc

(2)

dPs dt=

- k,P,+

k-PI,

(3)

+ k+Pa

(4)

k,P,

dPc dt = -(kb + k-)Pc

dPo = kbPc - k-PI, + k+P,. dt

(5)

(1)

The re-orientational motions in each state correspond to correlation times, r1 and ?(, respectively; these show large temperature dependence and require activation energies. The jump from a stable configuration (potential energy minimum) in the liquid to another stable configuration demands a concomitant rearrangement of the surrounding solvent molecules. This was the basis for an approach of relaxation

The boundary specify

conditions

PA(O)= k-l@+ + k-),

of the problem at t=O

and

P,(O) = k,/(k+

+ k_).

This system of differential equations is solved:

PA(~)= A+ exp(-E,+t)+

A_ exp(-E,_r)

(6)

p&) = B+ exp (-j.,t)

+ B_ exp(-i._r)

(7)

PC(~) = C+ exp(-E.+t)

+ C- exp (-;._t)

(8)

Fast kinetics studied by NMR + D- exp (-i._r)

P,(f) = D+ exp (-i+t)

(9)

259

with ML(~) - Ml..0 = (M,,(O) [0.2 exp (- a1 r)

where 1.* = f(A f B) A=k,+k+

+ 0.8 exp ( - a2t)]

(10)

+k,+k_

B = [(kb - k, + k- - kT)’ + 4c2k+k-]1’2.

(11)

al = PER, + Pb’Rl,r,,,;

(22)

(12)

a2 = PJ-R, + Pb’R1.,,,,w

(23)

Following Sillescu!35’ we introduce the correlation coefficient c, between chemical exchange and rotational jumps. It varies between zero and unity: in the Marshall formalism,‘30’ any transition between the free and bound states is accompanied by a reorientational jump: conversely, in the Anderson-Fryer formalism,‘29) the transitions between states are not accompanied by re-orientation. Consequently, the resulting correlation function can be written:‘26*35)

and

q+ Ii.+ I + 4c!$/i.: +

G(t) = g$[q+

-exp (-/.+t)

+ q- -exp (-;.-r)]

(21)

q+ Ii+

(13)

+

+

1 + w:/z:

q-Ii._ 1 + 4&i.?

q-Ii.1 + c&i.? 1

(25)

with

with 4+=1-q_

q- =

(14)

&(I - tan $)2

1 + (k-/k,)

tan 4 = $(I;, +

.

+ k+ - 2,).

(16) can the For rate

4q+ Ii.+ 1 + (2wo/i.+)2

4q_li_ q-j;._ + 1 + (o~oli-)2 + 1 + (2wo/j._)’ where B is a proportionality coefficient. Under extreme narrowing conditions this reduces to : RI = R2 = B[q+!i+

+ q-/i.-].

(i.+,i.-

1

(17)

% wo)

+ 0.4exp(‘-b2f)] (26)

(15)

tan2 4

From this bi-exponential correlation function, one compute the relaxation rates R1 and R2, using theory of Bloembergen, Purcell and Pound.‘36’ dipolar relaxation, the longitudinal relaxation is :(29)

-+

M.,.(t) = M,-(0)[0.6exp(-b,t)

b, = P,R,

+ P&was,

(27)

b2 = P/R,

f P&,s,ow

cm

“Fast” and “slow” refer to the two distinct relaxation processes undergone by a spin 3 nucleus. Outside extreme narrowing conditions the s-f (or -f -3) relaxation processes occur at a faster rate than the : + -f relaxation. Accordingly, the resonance line is the superposition of two Lorentzian absorptions: a broad line with 60% of the total intensity, corresponding to the -: + -4 and f -+ 3 transitions, and a sharper line with 40% of the total intensity for the -4 + 4 transition. The terms ML(I) and ML,o are magnetizations at times t and zero, respectively, and ML(O) is the maximum magnetization for t = 0. Under extreme narrowing conditions (j,+,i_ 9 wo), these equations become : R 1.61OW-

R t,s1ow -

Rl,m

=

R2,iart

(18)

For quadrupolar relaxation, using the equations provided by Bull’“” for a spin 3,‘2 nucleus undergoing chemical exchange. the relaxation rates are?”

[q+/i.+

+ q-/i.-]

(29)

It is also possible to express the relaxation rates for the system in terms of the relaxation times characteristic of the free and of the bound state, respectively.‘39) There are two important limiting cases: in the fast exchange limit, when chemical exchange is fast with respect to molecular rotation (k + , k - % k,, kb) 71 =

PbT1.b

+

(30)

P/T,,/.

In the slow exchange limit (k+,krelaxation rates (and not the relaxation weighted averages’“‘) according to : RI = 7-C’ =

PbR1.b

+

P/RI./.

G k,,kb), the times) are the

(31)

PIERRELASZLO

260

an infinitely fast exchange process. By this explicit Hamiltonian, in the inclusion of exchange Wennerstrom” ” reproduced the classical results in the slow exchange region, equation (31) for Ti, and equation (33) for T2) :

90-

R2

=

T2-’

=

PbR2.6

+

P$,,,

+

P/P&&‘2

(33)

where Pb P/ Lx = k, = k_ 60-

and

(34)

ACCI = A@ - AU/.

When there is complete randomization of the relaxing interaction on exchange, the effective correlation time is given by :

c; %J-

40-

(Tef~)-’ = l/Tci f l/Tli

where Tii is the relevant correlation time for the exchange process: the mean lifetime in one specific site i. This is determined by all migrations from the ith site, either to alternate sites intra- or intermolecularly or back into the ith site, since the nucleus can conceivably exchange back and forth at a given site before it diffuses away. Wennerstrom’“‘) also showed that, whenever fast chemical exchange occurs with complete retention of the direction of the interaction, the relaxation time is determined by a single correlation time :

30-

20-

IO-

0

02 Fractm

06

04

(35)

06

complexed

FIG. 1. Hypothetical curves showing the dependence of T1 on the fraction of complexed D molecules. The curves

-I

illustrate (a) slow, (b) intermediate, and (c) fast rates of exchange relative to the rotational rate of the DA complex.

This contrasting behaviour is illustrated in Fig. 1 where the curves are calculated for dipolar relaxation in the extreme narrowing limit ;(“) (a) is a slow exchange, (b) a moderate exchange and (c) a fast exchange curve. In the slow exchange case, Ti changes considerably with a small additional increment in the mole fraction of bound molecules, pb. Curve (b) corresponds to the favourable case of commensurate (within a factor 10) rates of chemical exchange and molecular re-orientation, when the former can be evaluated as a function of the latter. All of the above results are obtained by writing down the correlation functions that include chemical exchange, and then deriving the relaxation rates in the standard manner from knowledge of the correlation functions. A more direct approach has been successfully used by Wennerstriim. .141)chemical exchange is introduced into the spin Hamiltonian itself. If H,(t) is the time-dependent part of the Hamiltonian, which corresponds to the interaction responsible for relaxation, it can be written as a sum : HI(f) =

Cf,(tFfi

(32)

‘

where Hi is the Hamiltonian in site i and the functions J(t) have values of unity if the nucleus considered is in site i at time I, and zero otherwise. This assumes

where ri is the molecular motion correlation time characteristic ofsite i, and pi is the fractional population at that site; and the relaxing interaction is a square of a weighted average over all sites : 2

J(W)

=

Yeff(W)

(i > C

Pici

(37)

where yCff(o) is a reduced spectral density with the effective correlation time Teffof equation (36), and the coefficients Ci measure the strength of the relaxing interaction at each site. This last result is important: for instance, the ii2v/az2 term in quadrupolar relaxation should be taken as the ensemble average squared and not, as is often written, the ensemble average of the square. Wennerstr6mC4” has treated explicitly the case of very fast intramolecular exchange between sites A and B as it affects quadrupolar relaxation: if OAB is the angle between the A and B principal axis systems, when 0”s = 0 or 180” chemical exchange does not affect the relaxation time; when Oas = 90”, the maximum effect of exchange on relaxation is reached, amounting to a lengthening of the relaxation time by a factor of four, in the fast exchange limit. Goulon and RivailC42)have provided a critical discussion of the calculation of autocorrelation functions in the presence of chemical exchange, emphasizing the scalar nature of the thermodynamic force driving

Fast kinetics studied by NMR

261

TABLE2. Relaxation rates (Hz) for 1 mole 7; solutions in (a) carbon tetrachloride, (b) chloroform or(c) benzene-d, solution (35”C), in the presence or absence of 1 mole% 12’2y’ Rl Donor acetone benzene mesitylene: Ar CHlr p-methoxybenzene: Ar OCH3 hexamethylbenzene p-dioxane r-butylamine dimethyl sulphoxide

free

with I2

4.6 1.5

0.050’ 0.012b

0.045” 0.012h

no complex no complex

6.0 6.0

0.029” 0.152’

0.034” 0.156”

(weak complex)

8.1 8.1 15.7 14.9 -8000

0.083” 0.154” 0.133’ 0.053” 0.088’ 0.069”

0.077” 0.147” 0.133” 0.05@ 0.588’ 0.133’

the chemical relaxation process, as opposed to the vectorial nature of the other forces inducing diffusional motion. For instance, considering NMR relaxation of quadrupolar nuclei, a chemical reaction may either alter the modulus of the electrostatic field gradient q = (d%/dz2), or tilt the symmetry axis of the field gradient with respect to the external magnetic field. In the former case, the tensorial chemical process degenerates into a scalar phenomenon, the chemical and orientational processes are decoupled from one another. In the latter case, one is led to define spectroscopically active or inactive normal modes of chemical relaxation.‘42) To sum up, there are two classes of formalisms applicable to measurement of NMR relaxation rates in the presence of chemical exchange: either the chemical process alters the magnitude of the relaxing interaction, as in the treatments by McConnell(24) and by Marshall;‘30’ or the relaxing interaction is invariant with respect to the chemical process, as for instance when one considers intra-molecular dipole-dipole interaction between e.g. two protons, which corresponds to the theoretical treatments by Hertz”” and by Anderson.‘27~2g) 3. WHEiV IS A COMPLEX A COMPLEX? Many authors interpret the change of an experimental observable upon modification of the com-

position for a binary (A,B) mixture as evidence for AI? complex formation. For instance, NMR chemical shifts in general. or, more restrictively, aromatic solvent induced shifts (ASS), have been used to determine apparent equilibrium constants K., for complex formation. We have shown in another context that such conclusions can be highly misleading if they are applied to weak complexes with equilibrium constants of the order of unity or below.‘43’ The definition of a “complex” hinges upon the time scale for the measurement. Obviously, fleeting encounters should be excluded. otherwise every molecule in a liquid would constantly undergo complex forma-

Inference

no complex no complex no complex complex complex

tion and dissociation, and the concept of “complex” would be empty of all meaning. Thus, it is necessary to define an AB complex as an entity whose lifetime is at least equal to the rotational correlation time of the more slowly re-orientating partner, say B. Since NMR relaxation times depend markedly upon the ratio of these two times, or equivalently upon the k _ /kb ratio in the notation introduced in the previous section, they constitute one of the very few authentic tests for complex formation. This was pointed out by Anderson:(29*44) a bona fide AB complex undergoes re-orientation as a single, discrete unit. Consider again curve (a) in Fig. 1: in the slow exchange limit (k+, k_ <
laxation rate is given by equation (31), implying a significant increase in the relaxation rate Ri when complex formation occurs. However, if fast exchange prevails (k,, k_ 9 k,, kb) i.e. according to the above definition no AB complex forms, then the relaxation rate R, increases but little with an increasing degree of molecular association. For the t-butylamine-iodine complex, the kf/kb ratio turns out to be of the order of 10. Compare and contrast the data in Table 2:‘29) whereas the intramolecular relaxation rate for protons in the donor molecules acetone, benzene, p-dioxane is virtually unaffected by introduction of an equimolar amount of the iodine I2 acceptor molecule (=fast exchange), it responds considerably to the same perturbation in the cases of t-butylamine, dimethylsulphoxide and (possibly) mesitylene. Hence, one may speak of, for example, a t-butylamine-iodine complex, but one may not speak of an acetone-iodine complex in these carbon tetrachloride or chloroform solutions.(29) From another set of data (Table 3), Anderson’44’ concluded that molecular association in the acetonechloroform and in the benzene-chloroform system does not involve complexes which rotate as a unit, whereas the p-dioxane-chloroform, t-butylaminechloroform, and dimethyl sulphoxide-chloroform systems involve true complex formation.

262

PIERRE LASZLO

TABLE 3. Relaxation rates (Hz) for 1”; solutions. at 35Y?’ (italics denote frue complex formation)

Compound

CCL,

Solvents CDCI, CsDs

CSI

benzene-& dioxane acetone dimethyl sulphoxide r-butylamine

0.012 0.053 0.050 0.071 0.071

0.013 0.048 0.042 0.083 0.09 1

0.011 0.045 0.053 -

0.012 0.083 0.043 0.185 0.132

Very recently, Homer and Coupland’46) have reexamined the benzene-chloroform system in cyclohexane solution. The proton longitudinal relaxation rates in these mixtures cannot be accounted for by complex formation, and must be explained by the preferential solvation of chloroform by benzene rather than by cyclohexane molecuIes,‘45) a model which rationalizes satisfactorily the induced shifts provided the salvation shell contains about 12 molecules of benzenej4@ The question of distinguishing between transient molecular associations and more durable complexes, has also received a considerable degree of careful attention by Hertz and his coworkers. His methods are best illustrated with a specific example, namely, the system. dimethyl formamide (DMF)-benzene.‘47’ The proton magnetic relaxation rate RI is the sum of intra- and intermolecular terms :

and (R,)z/erD/c’, relaxation rates. reduced to unit concentration and diffusivity, is not follo\ved either: they are obtained from measurement of the DMFDMF proton i~ltermolecular relaxation rate. and C6H6 -+ DMF proton intermolecular relaxation rate. Therefore, although one can interpret the changes in chemical shifts in terms of complex formation.‘4”-“’ these relaxation measurements rule it out conclusively!47’ By similar methods and arguments. the chloroformacetone system, which is not describable as a complex, (44Jis a weak association when envisaged in terms of the centre of mass distribution function. but a distinct association when defined in terms of the Hatom pair distribution function.‘52’ Brevard and LehtP 3’ performed an elegant study of donor-acceptor n-rc complexes. They labelled \ CHD group, so that detailed molecules with a / examination of the proton lineshape allows the indirect determination of the quadrupolar relaxation time for the deuteron.‘54-56’ The fluorene-trinitrobenzene system is typical (see Fig. 2). Analysis of the data gives the following values: k, = 9.4 x 10’Os-1,

kb = 1.5 x 10’Os-1,

li_ = I.5 x

10’“M-ls-l, and k_ = 1.5 x 10’“s-‘. Hence, this weak (K, = l.OM-‘) molecular association is not a complex proper, but merely a “collision complex” in the sense of Mulliken and Orgel.“” Given the rapid growth of NMR studies of binding by small substrates to large biomolecules. a bow jifitlr RI = (Rthntra+ (Rlhnter complex forms only if the residence time rb is greater The intramolecular part can be obtained separately than the re-orientational correlation time in the from the deuteron relaxation rate of the deuterated bound state r&b. However, in many actual cases analogue of the molecule under consideration. 5R.bB rb 9 T~,J, where sR,f is the re-orientational By analysing the relevant equations, Hertz et ~1.‘~‘) correlation time of the free agent. These species having show that a necessary condition (albeit not a sufficient shorter-lived associations should not be referred to as condition) for complex formation to occur is a change complexes. of the quantities:

with composition of the binary mixture. The relaxation rate (Rl)$,$, is simpiy the difference between the experimentally-accessible (RI)/,$,relaxation rate, when neither of the DMF or benzene partners is perdeuterated, and the (RI)/;&, when one of the partners has been fully deuterated. The symbol D refers to the diffusion coefficients, and c; and c; are the proton concentrations in the first salvation sphere: obviously, the association can be studied either from the DMF (1) or from the benzene (2) side. The first criterion, as stated above, for existence of a complex, oiz. a significant increase of the intramolecular relaxation rate upon molecular association, is not obeyed by the dimethyl formamide-benzene system. The second crjterion, a marked change with composition of the binary mixture of both the (Rt)F;,:,D/c;

300-

2001

I 05 0

I IO

FIG. 2. Experimental (points) and calculated (curve) dependence ofdeuteron relaxation time Tp on complexed fraction z in the Huorene-trinitrobenzene system.

Fast kinetics 4. BINDING

studied

AND RELEASE OF SMALL SUBSTRATES BY BIO-MOLECULES

Lysozyme binds monosaccharide inhibitors, methyl a- and /I-N-acetyl-D-glucosamidines, hereafter referred to as Me M-and /I-NAcGlc, at subsite C of the active site.‘58*59)Sykes ‘60) has derived the on and 08 rate constants from NMR line broadening studies, taking advantage of the difference in chemical shift between the free and bound states: k+ = 1.4 x 105M-1s-’ (r-anomer) and 1.6 x lo5 M-‘s-’ (J?-anomer); k- = 5.5 x lo3 s- ’ (ol-anomer)and 4.5 x lo3 s- ’ (B-anomer). More recently, Szilagyi and coworkers(61) have provided a thorough study of the microdynamics of these enzyme-inhibitor complexes. They labelled the substrate molecules with deuterium, and synthesized Me a-NAc-dt-Glc, Me a-NAc-d3-Glc,

Me /?-NAc-dt-Glc, and

Me ,&NAc-ds-Glc.

Lineshapes of the spin-coupled protons yield the deuterium longitudinal relaxation ratesc5@ Measurement of quadrupolar relaxation serves as a probe of the purely intramolecular re-orientations of the inhibitor molecules. The longitudinal relaxation rates of the 13C nuclei in these inhibitor molecules provide confirmatory evidence for the k, values of 1.43 x 10” s- ’ for both anomers. Then, application of the Anderson-Fryer formalism,‘29’ which turns out to correspond to the extreme narrowing region for the quadrupolar relaxation, focuses on the single remaining unknown: the value for kb. The authors derive from their data (see Fig. 3) kb = 3.2 x 109s-’ for the lysozyme -2 monosaccharide, and kb = 5.6 x 109s-’ for the lysozyme -p monosaccharide complex, at 30°C. The latter value ties in well with kb = 4.5 x lo9 s- ’ at 40°C determined by proton T2 measure-

ccl

c i IO’

FIG. 3. T,(*H) values’6” as a function of the fraction of bound inhibitor for Me z-(triangles) and /SNAcGlc’s (circles) measured by the bandshape method (open) on the CHZD-CO group or by the FT inversion-recovery technique (full) on the CDj-CO group. Theoretical lines r-anomer. -------- B-anomer) were calculated using (--equation (I 8).

by NMR

263

ment for the analogous complex with lysozyme containing Gd(III) ion attached to the active site.“‘) Comparison of the effective correlation times r5,= kh 1 with the re-orientational correlation time for the protein TV 5 1Oe8 s(63-69) suggests internal rotation of the methyl group within the complex. Detailed examination of the data shows that, whereas in the a-anomer there is fast internal rotation of the acetamido methyl around an axis nearly parallel to the major ellipsoidal axis of lysozyme, an additional degree of rotational freedom characterizes the acetamido methyl group in the complex derived from the /&anomer.(61) Other binding studies concern transport proteins, such as human or bovine serum albumin (BSA). We have found strong binding of maleic acid, both as the mono- and the di-anion to three high-affinity sites.“‘) Assuming full immobilization of the substrate molecules on the protein, and equating TV with TV, leads to k_ < 1.4 x 105s-’ and, from knowledge of the equilibrium constant, k+ - 10” M- ’ s- ‘. Likewise, aspirin binding and release at a single high-affinity site occurs with rates k+ + lo9 M - I s-l and k_ < 2.5 x 106s-‘. These rate constants are obtained by application of the Anderson-Fryer equations.‘29’ Since the fast exchange condition obtains, with rb < Tzb for the complex (known from the limiting linewidths) we can bracket the rate constants for release of the substrate by BSA as follows: 1.4 x loss-’ > k- > 5Os-’ 2.5 x lo6 s- ’ > k- > 142 se1

for maleic acid for aspirin.

Acetrizoate, another strong BSA-binder,” ‘) provides an interesting set of observations: the methyl protons of the free substrate show non-exponential relaxation behaviour in D20 solution ;(72)the non-exponentiality decreases upon addition of BSA, as acetrizoate binds to the protein.‘71) The former observation is far from being unprecedented, even for relatively small molecules : nonexponential relaxation also characterizes, for instance, the methyl protons of tetragastrin,‘73’ and of the methionyl group of Met-enkephalin :‘74)it comes about because of the intervention of cross-correlation terms in the relaxation equations.‘75) When the various H, H inter-nuclear vectors reorient coherently, because they belong to a single structural unit, such as a methyl CH3 group, the cross-correlation terms, representing the correlation between the orientation of an H,H vector at a given time with the orientation of another H,H vector at a later time, can lead to nonexponential relaxation.‘75’ Pumpemik and Azman”@ have calculated the longitudinal relaxation rate RI of the methyl protons for acetrizoate undergoing chemical exchange between the aqueous free state and the bound state on the protein, explicitly taking into account the contribution of chemical exchange. They find no influence of the correlation parameter c (in equation (12) above) on the relaxation curves they calculate. They are able to reproduce the experimental observation of Miranda

PIERRE LASZLO

264

and Hilbers:‘7L*‘Z’ the non-exponentiality of methyl relaxation may indeed be masked, due to chemical exchange, if the substrate re-orientates much more isotropically at the bound site than in the free state. Serum albumin is a “bury-ah” protein and voraciously binds many molecules in the neutral or ionized states. Reuben and Luz”” made a detailed study of the binding and release of the ‘39La(III) cations. Lanthanum-139 is a spin 7/2 nucleus, for which they give the full expressions for the relaxation rates. Binding of La(II1) to BSA manifests itself by a linear increase of the longitudinal relaxation rate RI upon BSA concentration; at constant [BSA], RI is linear with [La&], with a non-zero intercept; furthermore, there is a strong frequency dependence of RI. These findings point to complexation by the protein of a fraction of the La(II1) ions, with fast exchange between the free and bound forms, leading to a _ significant enhancement of the relaxation rate. Such observations are indeed typical of the results often found in studies of ion binding by proteins, where the binders incorporate quadrupolar nuclei. From a plot of Tt vs o& Reuben and Luz(“) determined a correlation time t, for 139La in the bound state, equal to 37 ns, i.e. identical to the re-orientational correlation time ?‘Rfor the protein. Hence, the residence time of La(II1) on BSA has a lower limit of 37ns, and k- < 2.7 x IO’s_‘. By independent evidence, La(II1) would bind to the carboxylate groups, of which there are about 100 on the protein surface. Using the binding to acetate as a model, the authors”‘) determined a quadrupolar coupling constant, appropriate for the interaction of La(II1) with a carboxyl group, of 7.5 MHz. They were thus able to determine the number of occupied sites: La3+ ions bind to 124 carboxyl groups on BSA, which is akin to saying that there is non-specific binding of La3+ with all the free ionized carboxyl groups. The mean equilibrium constant for dissociation at one of these sites K. = 0.46 M. Lastly, the fast exchange condition sets an upper limit for the residence time of La(II1) on the protein : 5E< Tlb = 2.1 x 10m6s. Therefore, the rate constant for release of the lanthanum ions is conveniently bracketed as 4.8 x 10’
< 2.7 x 10’s_‘.

A number of studies have focused on binding by the univalent sodium cation, both because of its biological importance and because of its relatively high NMR receptivity. (‘ai Shchori, Jagur-Grodzinski, Luz, and Shporer(39) have monitored complexation of Na’ by dibenzo-1%crown-6 in DMF solution, at ambient temperatures. They plotted the transverse relaxation rates as a function of reciprocal temperature (Fig. 4). The system studied is a slow exchange case, where R2 = ~$2,.

+ phR2h.

I

2

Temperature. ‘C -9.7 -35 0

210

I

’

/ /

I

I

3.0

I

34

/.

I ‘/ / I/J28 ,’ /

/

/

’ I:T,av,’

/

I

/

/

’

-55.5

I

/

/

I

I

-!

I

I

3.0 l@/r,

I

4.2

I

4.6

K-’

FIG. 4. Semilog plots of 1/Tz for 23Na vs reciprocal absolute temperature in a DMF solution’39’ containing 0.57 M NaSCN and 0.2 M DBC, and of (l/T&e~ for 0.57 M solution of NaSCN: (----) extrapolated, (-_) calculated.

approximately O’C, and again decreases monotonically at higher temperatures. Below -40°C the exchange rate is too small to affect the NMR spectrum; there are two resonances (infinitely slow exchange), of which only the narrow one, due to free sodium in the DMF solution, is observed. In this temperature range, its temperature dependence parallels that for a 0.57M solution of NaSCN. The broad line is not seen. At the other extreme, in the high temperature range corresponding to observation of a single line, the measured relaxation rate is given by eqn (38). if one assumes a negligible difference in the chemical shifts for free and bound sodium, thus making the third term in eqn (33) vanish. It is then straightforward to read off from Fig. 4 the relaxation rate Rzb characteristic of the bound state, and shown as the upper dashed curve. The pseudo first-order rate constant for Na+ release at 25”C, extrapolated to zero ionic strength, is k_ _ 105s-‘.

Deganit79’ uses the same approach in her determination of the kinetics of complexation of sodium ions by the ionophore antibiotic monensin. Just as in the dibenzo-18-crown-6 system, two mechanisticallydistinct processes have to be considered : Na’ + monensin*Naf + monensin.Nas

$ monensin.Na Na+ + monensin.Na*

(38)

It can be seen from Fig. 4 that, with increasing temperature, R2 first decreases, goes through a minimum at approximately -30°C a maximum at

so that the observed ~7’ can be written: I/rr = kl [monensin.Na] + k- [monensin.Na]/[Na*]

(39)

265

Fast kinetics studied by NMR with concentrations of monensin in the 0.125-0.3M range, and a ratio of complex to total sodium of 3/7, Degani’79) found that 1/To = 80 f 10 Hz is invariant in the above concentration range. Hence, the dominant sodium exchange route is the first-order dissociative mechanism, characterized by the rate constant k-. In methanol solution, at 25°C k- = 63s-’ corresponding to

6I

AHi = 43.1 kJmol_’ and AS* = -66.1 Jmol-‘K-l, while k+ = 6.3 x lO’M_‘s-’ is associated to AH’ = -3.3kJmol-i and to AS+ = -16.3Jmol-‘K-i (the latter value points to a very small conformational change of the antibiotic molecule prior to Na* complexation). These kinetic parameters, as obtained by NMR,‘79’ are comparable with those obtained for the interaction of univalent cations with ionophore antibiotics, mainly using chemical relaxation techniques. Grandjean and Laszlo’s0,81i also use sodium-23 nuclear magnetic resonance, to study the binding and release of the sodium cation by pike parvalbumin. The native protein contains two calcium cations per molecule (MW cu. 12,000 daltons). EGTA achieves reversible removal of one calcium ion, and the protein maintains its conformational integrity: the helical content, as measured by circular dichroism, remains constant. This offers the singular opportunity of exploiting the high sensitivity of sodium-23 NMR for the exploration of the deserted calcium-binding site. Addition of Na+ to aqueous solutions of the single calcium protein produces considerable broadening of the 23Na resonance. The relaxation rate enhancement is linear with respect to protein concentration. Neither the native protein, with two calcium ions, nor the denatured protein, with zero calcium ion, give similar effects. That sodium binds to the calcium-binding site is ascertained by competition experiments: Ca2+ displaces Na+, whose linewidth collapses to that characteristic of the free hexaquo ion. The sodium cation undergoes fast exchange between the free and bound states: the linewidth decreases when the temperature increases.(80’ Comparison of the relaxation rates R2 measured at 23.81 and at 62.86MHz yields a re-orientational correlation time for bound sodium T, = 4 + 1 ns@“) identical with the re-orientational correlation time TR = 3.5 i: 0.2 ns for the protein host, as obtained from ’ 3C 7, measurements’**’ and from depolarized light scattering experiments. (83)From the same comparison of the relaxation rates at two different frequencies, we obtain the quadrupolar coupling constant for bound sodium: 1.3 MHz. which is a relatively high value. Both criteria. the equality of sc and TR. and the magnitude of the quadrupolar coupling constant, point to sitebinding at the calcium-binding site. Analysis of the results’81’ (Fig. 5) yields values of the rate constants appropriate for the equilibrium: Na’ + protein s complex. with k_ = 2 x 10’ s- ’ and

‘I-

1

I

I

10-3

10-4

MO. PAB

41, M

FIG. 5. A plot of T2 (23Na) against concentration of complexed sodium, for Na* undergoing fast exchange between the aqueous solution and a binding site on the protein@” (PAB-1 stands for parvalbumin with a single calcium bound, after removal from the native protein of the second calcium by EGTA; the Na+ cation samples the vacant calciumbinding site).

k+ = 0.6 x lo9 M- ‘s-l. These, when compared with rate constants that appear to be appropriate for calcium-binding, serve to explain why calcium-binding is better than sodium-binding by five orders of magnitude in terms of the equilibrium constants: whereas Caz+ is almost fully dehydrated, retaining only one water molecule in the complex, with the attendant gain in translational entropy for the liberated water molecules, Na+ releases fewer water molecules, only two or three, as it binds to the macromolecule.‘sl) Turning now to the anions, the halide ions (especially the biologically-ubiquitous chloride ion) have served as probes into the microdynamics of biomolecules. Since this particular topic is treated at length in an outstanding book,‘84i one short illustrative example will suffice here. Ward(ss*86i has measured considerable enhancement in the transverse relaxation rate for the chloride ion, due to its interaction with carbonic anhydrase, both with the native zinc enzyme and with the cobalt-substituted metalloenzyme. These relaxation enhancements show strong pH-dependence, with a drop in the Cl- linewidth above pH 7. It has been established’8s’ that only a single chloride ion at most binds to the enzyme. The residence time rb s [enzyme]/R1[Cl-],corresponding to k- 2 3 x 10’s_’ at low pH, and (from the known association constant) to k+ 2 lo8 M-’ s-i. Whereas the site for binding of the Cl- ion is believed by Ward to be the metal, in competition with a water molecule,@” Koenig and Brown(s8i prove (by comparison with the proton relaxation rate of HZ0 molecules) that the competing entity is a residue on the protein, rather than a solvent molecule. They have also determined by i3C NMR, from the line broadening for the C02-HCO; system in solutions of the native human carbonic anhydrase, the kinetic parameters that describe the enzymatic interconversion of CO1 and HCO;.‘*9’

266

PIERRE LMZLO

Behr and Lehn(qo’ use deuterium labelling of methyl cinnamate and t-butylphenate anions to follow’s’-56’ their inclusion within the cavity of r-cyclodextrin hosts: they are only able to place an upper limit of 10s s - ’ on the rate of dissociation k _ for the inclusion complexes. Even though these complexes do not dissociate diffusionally, they are rather loose: the dynamic coupling coefficient (defined by Behr and Lehn”” as the ratio of substrate correlation time to host correlation time) varies between 0.16 and 0.26 in the cases studied. So far, in this section, we have presented examples in which the line broadenings arise, or are assumed to arise, solely from binding of a small substrate to a slowly re-orientating macromolecule (relaxation enhancement). In reality, this is an over-simplification, and one should take into account the contribution of exchange broadening contained in the third term of equation (33). This very question, of relaxation enhancement versus exchange broadening, has been the object of a useful dispute centred on the enzyme, a-chymotrypsin. The Gerig group’g1,q2’ have attributed the line broadenings of D and L-tryptophan, and of transcinnamate substrates to the former effect. Sykes,‘q3’ who has studied the binding of another substrate, N-trifluoroacetyl-D-tryptophan, has assumed that exchange broadening is the dominant factor. He has measured the spin-lattice relaxation time in the rotating frame, Tip, as a function of B’, extrapolating to

k_ from a plot of R2-RI

vs the exchange broadening term pjpbAwz. At pH 5.0. k- = 14.9 x lo6 Me’s_‘, k- = 3.9 x 103s-’ ; at pH 7.0, k, = 5.9 x 106M-Is-‘, and k_ = 3.5 x 103s-‘: the 011values k, are greater than those determined by Sykes’93’ by a factor of lo’-103, because Sykes neglected self-association of the enzyme and because he assumed that the enzyme was saturated by the inhibitor!“’

5. DETERMlNATlON OF RATES FROM PC’LSE SPACINGS

The story of NMR determination of the rates for binding and release of inhibitors to r-chymotrypsin does not end there. Spurred by the competition with the groups of Sykes’v3’ and Richards,‘94.95’ Gerig’lo2’ selected yet another experimental method. For a system undergoing exchange between two sites, the apparent value of the transverse relaxation rate Rz depends upon the rate g at which the refocusing 180” pulses are applied, in a Carr-Purcell pulse experiment on a liquid sample. This is a spin-echo experiment, in the version of Carr and Purcell,‘q6’ as modified by Meiboom and Gill. “” Luz and Meiboom”” showed that it could be applied to the determination of rate constants and of the difference in chemical shifts between the two sites in a two-site exchange problem. The observed Rz is given by:‘qs’

Bl=O:

1

1

Tl,Kv

7.1

---=

EO

YAw2/k-

(40)

KD + I

where E” and I0 are the total concentrations of the enzyme and of the substrate inhibitor, and Ka = (k-/k+) the dissociation equilibrium constant for the EI complex. Sykes (93’obtained the parameters K o and Aw by a computer analysis of the dependence of the substrate chemical shift upon initial inhibitor concentration (I’), keeping the initial enzyme concentration (E’) constant. The rates he obtained are kc = 104M-‘s-l and k_ =4.9 x 102s-’ for xchymotrypsin, and kc = 1.6 x 104M-‘s-’ and k- = 16.6 x 10’s_’ for DFP-chymotrypsin. The important point is that the exchange contribution to the linewidth is not necessarily negligible, even in the fast-exchange limit. This conclusion is borne out by the careful reinvestigation by Smallcombe, Ault and Richardsfg4’ ofthe binding between N-trifluoroacetyl-D-tryptophan and ct-chymotrypsin. They find that the relaxation rates RI and R2 do not extrapolate back to the same value RI I = R2, for the pure inhibitor. The difference between the observed relaxation rates is written as R2 - R, = &pbAw2Jk-

(41)

(from combining equations (31), (33) and (34)). It is therefore possible to determinetg5’ the rate constant

where the following equality is assumed: R,, = RZb = R;

(43)

and g is the pulse rate. The difference in the bracket of equation (42) is equal to zero in the fast pulsing limit (g 4 co), and the observed relaxation rate in that limit is independent of the chemical shift difference Ao. Conversely, in the slow pulsing limit, this difference goes to unity: comparison of the observed R2 in the two regimes of slow pulsing and fast pulsing will yield Aw and the rate constants k, and k- .‘98.qv’ A number of authors”OO+‘O1’have provided treatments which remove the restriction enforced by equation (43). As the pulse rate g increases, the decay rate R2 decreases to a limiting value given by :Ilo” lim R2 = :(Rzr + Rzb) -t :(k+ + k_)

4-m - ([:(Rrr

- Rz~,)- ;(k+ + k-)I2 + k+k-;

“’

(44)

Using this technique and explicitly including dimerization of the protein in their treatment of the data, Gerig and Stock”“’ determined rate constants k_ = 2 x lo3 s- ’ (6S”Q and lo4 s- ’ (26°C) for dissociation of the complex between r-chymotrypsin and the Ntrifluoroacetyl-D-tryptophan inhibitor, at pH 6.6. These can be compared with the value of k- =

Fast kinetics

FIG. 6. Dependence

studied

by NMR

267

of T2 on 180” pulse spacing at 23 and 58°C. Solid Knispel and Pintar.“‘“’

3.5 x IO3s- ’ obtained for the same system, at 34°C and pH 7.0 by Smallcombe, Ault and Richards!94) Using the equations obtained by Gutowsky, Vold and Wells’103’ for the decay of the echo train in coupled AB systems, Kaksal and Caglayan”04’ measured the rate of chair-to-chair interconversion in cyclohexane (2.5 x 106s-‘) and in dioxane (1.8 x lo6 s-l), at 20°C. They also measured the proton exchange rate in water: k, = 1.9 x lo6 s- ‘, a value somewhat at variance with the earlier determination by Meiboom’loS’ of 2.2 x lo5 s- ’ and with the more recent result of Meiboom(106) and others.““) The

lines represent best-fit curves.

proton transfer rate in water, k,, has indeed not only been obtained from the dependence of R2 on the pulse spacing sequence (Meiboom dispersion),‘103*‘04’ but also from measurement of the dispersion of the longitudinal relaxation time in the rotating frame, T1,,.~‘06~107) Knispel and Pintar”“‘) have used both methods. From Meiboom dispersion (Fig. 6), they obtained values of k, = 1.8 x lo3 (4”C), 1.4 x lo3 (23”C), 3.6 x lo3 (58”C), and 3.6 x 104s- ’ (94°C). The dispersion of TIP they observed at 58”C, by varying B1 from 5 x 10-6-10-3 tesla (Fig. 7), is consistent with the above value for k, = 3.6 x lo3 s- *.

H,

gauss

FIG. 7. Dependence of Tl ,,on B1 at 58’C. The solid line represents T
the curve calculated using values of Knispel and Pintar.“‘s’

PIERRE LASZLO

268

6. DISCOL’RSE

Do fast kinetics constitute a hidden dimension of NMR? This appears to be the case from the paucity of applications published so far. A number of authors have explicitly drawn attention to this rather remarkable situation. Sykes writes :‘931“the NMR linewidths also contain information about the rate of these exchange reactions . . No attempt has been made to measure therates of these reactions by NMR methods.” Likewise, Gerig and Stock’loZ) remark that “a theoretical discussion of the Carr-Purcell experiment as applied to measurement of exchange rates in biological systems has appeared,” **) but we are unaware of experimental studies which have employed this technique in this context.” A third quotation, from the paper by Kiiksal and Caglayan(io4) makes a similar point: ,“It is the purpose of this paper to give experimental examples of the coupled AB case and determine the first-order exchange rates in cyclohexane, dioxane, and water. This will probably be the first application of the theoretical work of Gutowsky, Void, and Wells.“‘io3) What is the explanation for the relatively small use that has been made of relaxation measurements for the determination of very fast rate constants? Rather than a single explanation, we believe that a number of factors may be responsible. During the sixties, NMR had the universal reputation of being a slow spectroscopy: which is true when compared with optical and vibrational spectroscopies; often, the spectrum is a weighted average of the different co-existing species whose interconversion is significantly faster than the frequency differences between the individual lines characteristic of each component in the mixture. Many of the textbooks listed this as a characteristic feature, and maybe also as a disadvantage, of NMR methods. Furthermore, Professor Manfred Eigen obtained the Nobel Prize for chemistry in 1967 for developing relaxation techniques for measuring rates of fast chemical reactions, in the microsecond-to-nanosecond time scale. Hence, in the public eye, while NMR was discarded as a method too slow for measurement of such very high rates, a whole set of elegant new methods was seen as appropriate for studying fast kinetics. Somewhat better-informed scientists, if questioned, would point to ESR spectroscopy for measurement of relatively high rates; but they would state that NMR methods were limited to measuring rates below a limit of 103- IO4 Hz. Another important factor was the slow development of methods of relaxation time measurements until pulse methods became the norm for highresolution work, with the advent of pulse FourierTransform commercial NMR spectrometers, in the seventies. Perhaps also of significance was the early general availability of computer programs, through organizations such as the QCPE, for DNMR-I but not for DNMR-II. One should recall that throughout this period, of the blooming of DNMR-1 and relative neglect of

DNMR-II, most NMR applications pertained to organic and, to a smaller extent, to organometallic chemistry. It was the time when conformational analysis reached its acme. Thus, it is perhaps not too surprising that the available DNMR-I was grasped as the natural tool for measuring rates of internal rotations, ring inversions, and other rearrangements. This provided a dynamic counterpoint to knowledge of the structural features of static conformations. All such factors being considered, it is then completely normal that DNMR-I rode the wave of a major fashion, and was in great demand for the last IO-15 years. It is somewhat less normal that DNMR-II underwent so little development during the same period. And in the long run? Historians, including historians of science, are accustomed to distinguish between macro- and micro-history. We are dealing here with the micro-historical level, and clearly at this level, science is not necessarily a cumulative process. Perhaps, only the artificial a posteriori look one can have upon the history of a particular discipline makes it appear as a story of unhindered and uninterrupted progress. Just as major new discoveries completely reshape a field, including the previous contributions to it, likewise careful consideration of the sidelines, of little-explored avenues can also give an altogether different representation of a discipline. One last comment is in order: the division between DNMR-I and DNMR-II is less clear-cut than I have made it appear, for simplicity’s sake. There are areas of overlap. Lest the reader should imagine that these two complementary sets of methods be distinguished by the corresponding time scales of rates than can be studied, with DNMR-I constrained to comparatively low rates, I should point out that in favourable cases, Grunwald~109~“o) used this method to measure rate constants up to approximately lO”Hz, for breaking the R3 N... HOH hydrogen bond (at pH _ 1, the neutral species has an extremely small concentration as compared with the corresponding cation). This whole topic of the NMR study of fast proton transfers has already been reviewed.” “i

7. CONCLUSION

This has been a rather brief survey: not only for brevity’s sake, but also because for the above historical reasons the subject matter has been limited. I hope that this article will serve as an introduction to a very useful and to an extremely promising aspect of nuclear magnetic resonance. Thus, it may contribute to filling a void, in bringing about a much greater usage of these DNMR-II methods.

Acknowledgements--The suggestions and comments by Drs. J. E. Anderson, C. Brevard and H. Sillescu have been very

helpful. I also wish to thank my coworkers, whose names appear in the references, for their hard work and good spirit.

269

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