The eigenmode perspective of NMR spin relaxation in proteins

We developed in recent years the two-body (protein and probe) coupled-rotator slowly relaxing local structure (SRLS) approach for elucidating protein dynamics from NMR spin relaxation. So far we used as descriptors the set of physical parameters that enter the SRLS model. They include the global (protein-related) diffusion tensor, D1, the local (probe-related) diffusion tensor, D2, and the local coupling∕ordering potential, u. As common in analyzes based on mesoscopic dynamic models, these parameters have been determined with data-fitting techniques. In this study, we describe structural dynamics in terms of the eigenmodes comprising the SRLS time correlation functions (TCFs) generated by using the best-fit parameters as input to the Smoluchowski equation. An eigenmode is a weighted exponential with decay constant given by an eigenvalue of the Smoluchowski operator, and weighting factor determined by the corresponding eigenvector. Obviously, both quantities depend on the SRLS parameters as determined by the SRLS model. Unlike the set of best-fit parameters, the eigenmodes represent patterns of motion of the probe-protein system. The following new information is obtained for the typical probe, the (15)N-(1)H bond. Two eigenmodes, associated with the protein and the probe, dominate when the time scale separation is large (i.e., D2 >> D1), the tensorial properties are simple, and the local potential is either very strong or very weak. When the potential exceeds these limits while the remaining conditions are preserved, new eigenmodes arise. The multi-exponentiality of the TCFs is associated in this case with the restricted nature of the local motion. When the time scale separation is no longer large, the rotational degrees of freedom of the protein and the probe become statistically dependent (coupled dynamically). The multi-exponentiality of the TCFs is associated in this case with the restricted nature of both the local and the global motion. The effects of local diffusion axiality, potential strength, and extent of mode-coupling on the eigenmode setup are investigated. We detect largely global motional or largely local motional eigenmodes. In addition, we detect mixed eigenmodes associated with correlated∕prograde or anti-correlated∕retrograde rotations of the global (D1) and local (D2) motional modes. The eigenmode paradigm is applied to N-H bond dynamics in the β-sheet residue K19, and the α-helix residue A34, of the third immunoglobulin-binding domain o