Volume 55, number 1 CHEMICAL PHYSICS LETTERS 1 April 1975

A GENERAL FORMALISM FOR THE ANALYSIS OF NMR RELAXATION MEASUREMENTS

ON SYSTEMS WITH MULTIPLE DEGREES OF FREEDOM

Roy KING and Oleg JARDETZKY Stailford Magtletic Resonance Laboratory. Statlford Unlersitv, Stqnford. California 91305. USA

Received 15 November 1977

A heretofore neglected application of Markov theory to define a general formalism for the analysis of NMR relaxation data on systems with multiple internal motions is proposed. It is noted that the theory predicts a modified lorentzian line shape for systems not at equilibrium.

NMR relaxation measurements on proteins, nucleic acids and other macromolecules are reported with increasing frequency [l-3 ] _ A characteristic of such molecules is that they form well defined structures, in which various degrees of internal motional free- dom may be allowed, in addition to the overall rota- tional and translational diffusion of the structure. The possibilities of analyzing NMR relaxation data in terms of internal motions is, however, limited by the lack of a suitable general theoretical framework_ Relaxation theory as originally developed by Bloembergen et al_ [4j assumes that the nuclei whose relaxation is observed are attached to a rigid body. Subsequent modifications of the theory made notably by Woessner [5,6] and others [7--121 take into account anisotropy of diffusion, one additional degree of internal rotational freedom or a sequential series of rotations along a hydrocarbon chain. The general case of more than two motions of arbitrary complexity has not been treated. Nor is any theoret- ical formulation in use which would allow a system- atic analysis of relaxation data in cases where the amplitude parameters of the motions cannot be defined a priori.

In this note we wish to point out that the theory of Markov processes can serve as the basis of a general formalism for tire anaIysis of NMR relaxation pheno- mena in systems with multiple internal motions. Markov theory has previously been used to account

for the effects of chemical exchange on NMR relaxa- tion [13,14]. Kubo [ 151 has discussed its application to relaxation in a rigid body. The proposal made here may be regarded as a generalization of his treatment.

It is well known that the\calculation of NMR re- laxation parameters is in essence a calculation of spectral deraity functions JF~(u) defined as

J+4= 7 eiwT IFj(c> Fj(t f 711 dr (1) _m

for a dynamic variable Fp with an autocorrelation function F,<t) Fj(t + T)_ For our purposes it is im- material whether Fj is taken to be the time-dependent part of an appropriate interaction hamiltonian [16], or used to denote the total hamiltonian. Our aim is to define the rules for calcu!ating JF~(w) in a system undergoing an arbitrary number of motions of an arbitrary nature.

The soIt? additional assumption necessary to achieve this is that every motion contributing to relaxation is a Markov process. The assumption in its general fomr implies no more than that a probability can be assigned tc each step in the process and that the system has no memory. The basic relationships which can be derived from it cover a very wide range of specific models, and provide a framework for system- atic comparisons between them. The theoretical basis for the calculation, briefly restated here for

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Volume 55, number 1 CHEMICAL PHYSICS LETTERS I April 1978

clarity and internal consistency is as follows: Let X be the state space of a given motion, and

either discrete or a continuous, compact manifold_ (A state space of one or the other kind, and sometimes of both, can be assigned to any physically realizable motion.) Then the probability of transit from state x to y in time t, P&x, y)dy can be used to define pro- jection operators Pt for any complex valued function of the state space, F: X

(2)

It can be shown quite generally that the projection operators Pf obey the Kohnogorov equation [17]

i),=C.?.?r> forz>O, (3)

when Q is the transition operator. A similar relation- ship involving Q*, the complex conjugate of S2, can be shown to hold for f < 0. if X is a discrete space, the elements of the operator SZ can be stated simply as:

~___ linl WJ) - %j I1 t-0 t

and the important relation holds

C-&= csl,. i

(5)

In the continuous case, if the very general condition

lim 1. P,(x. y) dv = 0 , t-o f

I-u-YI>e forall E>t,x

is true, C? is a second order differential operator [ 151

(6)

The solution of the Kolmogorov equation is

P,=enr, fort>O, (7)

with a corresponding term (-CL*) for t < 0 [ 17]_ To express the spectral density J(W), eq. (l), in

terms of the eigenvalues and eigenfunctions of the transition operator SILL; we use the property that if X is discrete or a compact manifold, then the eigen- values of St are discrete [ 17]_

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We denote by X,, Xz the eigenvahres and I&, 0: the eigenfunctions of !L? such that

Qb,, = x*4,

and similarly for Q*, with @,r ,4; = i_

(8s)

The physical significance of X and 4~ is quite appar- ent from these definitions. If Fj is a dynamic variable, & are the amplitude parameters and A,1 the rate param- eters of the motion, X,, = l/~,~ where ~~ is the corre- lation time for the irth individual motion_ To calcuIate J(W) we rewrite F<t)Fi*(f + 7) = !Fp PTFj) and sub- stituting into eq. 1) obtain (’

Jq = 7 eiwr <Fp P, Fj) = <Fj i 1 eiWrP, dT>Fi) -CO

= _-2Re c [or, - i(o + &)I(& Fj)@j. Qni > (9)

n a; + (Cd f p,y

where A,, = 4, f i/In. We can now distinguish two cases: (1) For systems which may be considered to be at or near equilibrium the principle of detailed balancing holds and fi is hermitean (self-adjoint), C! = R*. This will apply to most real cases of interest.

Using the hermitean property Q = sL* , Qn = &:, eq. (9) simplifies to

J+.J) = -2 c I(Fi. @n)12A,l n x2t02 =

in keeping with other well known results [1] _ (2) For nonequilibrium systems 52 f S2* we note that a frequency dependent term appears in the numerator. The line shape is predicted to be a modified rather *ban proper lorentzian, and shifted in frequency to

(0 f 0,). Generalization to multiple motions is now readily

possible. For two motions, the state-space X can be separated into two subspaces Xl and X2 with X, @ X, or generally X = l& Xk_ If the motions are indepen- dent

p,(x, Y> kcl p:txk 3 yk) *

Volume 55, number 1 CHEMICAL PHYSICS LETIERS I April 1978

The assumption of independent motions may at first appear to be unduly restrictive. However, if the motions are weakIy coupled it is possible to take this into account using standard perturbation theory, and obtaining the solution in terms of Bohr expansions for eigenfunctions and eigenvectors [ 181 _

If, on the other hand, the motions are strongly coupled, Q for the combined motion will not be diagonalizable in the separate components sLI and Q2. Strongiy coupIed motions will therefore appear in this formalism as a single motion, which may nevertheless reasonably approximate physical reality.

The result, eq. (1 l), may be applied in one of two ways. If the character of each motion is precisely known, the eigenvector @,, can be derived from the geometric constraints of the motion, leaving An as the sole unknown. In this form eq. (11) is a straight- forward generalization of existing relaxation theory, which invariably assumes a specific motional model. More important, however, is that the formalism defied by eq. (8) and implicit in eq. (11) allows both the eigenvectors Qn and eigenvalues h, to be treated as unknowns. Given a sufficient number of experi- mental measurements eq. <I 1) can thus be solved for both the amplitudes and the frequencies of each motion. This is Iikely to be a practical necessity in many cases. For example, for globular forms of linear polymers with three or four types of motion - e.g., overall and domain rotation and segmental as weU as side-chain flexibility - it will generally be quite im- possible to rigorously define a priori the range of each motion for each observable group. Rather than fore- going all study of such systems by relaxation methods, or analyzing the data with the aid of inappropriate models, one may gain partial insight into the under- lying processes with the aid of the semiphenomenoi- ogical amplitude factors $J,~ and rate factors X,, _ Of particular interest is that these can be used for com- parative purposes, i.e., to distinguish different degrees of motion in different parts of the system under ob- servation. In generai, if n motions are involved

2”+rr- f

measurements will be necessary to obtain a self-con- sistent solution of eq. (11). Considerikg that a single set of 4n and X, should account for the measured values of all relaxation parameters (T1, T2, TIP, NOE) at all frequencies and for all nuclei the experimental problem is of manageable size, especially in systems in which cross-correlation and spin diffusion effects may be neglected or accounted for. The number of unknowns and hence the number of required NMR relaxation measurements can be further reduced, if at least one of the motions is common to ah parts of the system and its parameters can be determined by other methods_ Thus, for example, in the study of biological macromolecules the overall rotation of the molecule and its symmetry can be determined by light scattering and other techniques. Eq. (i 1) then allows the analysis of NMR relaxation data in terms of internal motions only. Computer caiculations permitting comparisons between specific motional models and the detailed analysis of experimental data using the proposed formalism will be given in subsequent reports.

Acknowtedgement

This work was supported under National Science Foundation Grant $PCM-7502814.

References

[I I E. Oldfield, R-S. Norton and A. Allerhsnd, J. Biol. Chem. 250 (1975) 6368.

[21 S.J. Opella, D.J. Nelson and 0. Jarderzky, ACS Sym- posium Series -“34, Magnetic Resonance in Colioid and Interface Science, eds. H.A. Resiny and C-G. Wade (1976) p_ 397.

[3] R. Deslauriers, E. Ralston and R-L. Somorjai, J. Mol. Biol. I I3 (1977) 697.

[4] N. Bloember3en, E.M. Purceil and R.V. Pound, Phys. Rev. 69 (1946) 37.

[S] D.E. Woessner. J. Chem. Phys. 37 (1962) 647. [6 ] D.E. Woessner, B.S. Snowenden Jr. and G-H. Meyer. J.

Chem. Phys. 50 (1969) 719. W-T. Huntress, Advan. %lagn. Reson. 4 (1970) 1. D. WaIIach, J. Chem. Phys. 47 (1967) 5258. K. van Putte, J. Magn. Resort. 2 (1970) 23. Y.R. Levine, P. Partington and G.C.K. Roberts, Mol. Phys. 25 (1973) 497. Y.K. Levine, N.J.M. Birdsall, A.G. Lee, J.C. Metcalfe. P- Part&ton and G-C-K. Roberts, J. Chem. Phys. 60 (1974) 2890.

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Volume 55, number 1 CHEMICAL PHYSICS LETTERS 1 April 1978

1121 F. Noack, im NMR: basic principles and progress, Vol. 3, eds. P. Diehl, E. Fluck and R. Kosfeld (Springer, Berlin, 1971) p_ 83.

[l3] P-W- Anderson, J- Phys. Sot. Japan 9 (1954) 316. [14] C.S- Johnson, Advan. Magn. Reson. 1 (1965) 33. [ 151 R. Kubo, in: Fluctuation, relaxation and resonance in

magnetic systems, ed. J. ter Haar (Oliver and Boyd, Edinburgh, 1962) p. 23.

1161 A. Abragam, The principles of nuclear magnetisni (Oxford Univ. Press, London, 1961j.

[17] A.T. Bharucha-Reid, Elements of the theory of Markov processes and their applications (McGraw-Hi& New York, 1960).

[18] A. Messiah, Quantum mechanics (North-Holland, Amsterdam, 1961).

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