Theory of NMR relaxation in macromolecules: Restricted diffusion and jump models for multiple internal rotations in amino acid side chains

Abstract

The theory required to extract detailed motional information from NMR relaxation times of nuclei in an amino acid side chain containing multiple internal rotation axes attached to a large macromolecule of at least cylindrical symmetry is developed. Emphasis is placed on the analysis of 13C-NMR of protonated carbons where dipolar relaxation is predominant. Extension to other relaxation processes is straightforward. The spectral density from which the relaxation times can be calculated is obtained for various models for the motion of the side chain. The existing theory which assumes that internal rotations are both independent and free is generalized to incorporate excluded volume effects in a heuristic way by restricting the amplitude of the internal rotations. It is found that small amplitude motions are ineffective in causing relaxation. Thus jump models involving a relatively small number of configurations are appropriate to describe the motion of the side chain. The advantage of jump models is twofold: (1) excluded volume effects can be handled simply by including only sterically allowed configurations, and (2) the unrealistic assumption of diffusional models that rotations about different bonds are independent can be removed by considering concerted motions in which the positions of only a small number of atoms are altered. An explicit expression for the spectral density, which is computationally practical when the number of configurations is less than 200, is obtained for a class of jump models in which the dynamics is described by a master equation. A detailed application of the general formalism is made to the case of a lysine side chain whose carbon–carbon backbone is constrained to lie on a tetrahedral lattice. Finally, numerical calculations are presented to illustrate some of the qualitative features of the different models, and the strategy that can be used to obtain motional information from experiment within the framework of these models is discussed.