NMR spin relaxation in proteins: The patterns of motion that dissipate power to the bath
Authors: Shapiro Yury E. , Meirovitch Eva 
Published: April 2014

We developed in recent years the two-body coupled-rotator slowly relaxing local structure (SRLS) approach for the analysis of NMR relaxation in proteins. The two bodies/rotators are the protein (diffusion tensor D1) and the spin-bearing probe, e.g., the 15N−1H bond (diffusion tensor, D2), coupled by a local potential (u). A Smoluchowski equation is solved to yield the generic time correlation functions (TCFs), which are sums of weighted exponentials (eigenmodes). By Fourier transformation one obtains the generic spectral density functions (SDFs) which underlie the experimental relaxation parameters. The typical paradigm is to characterize structural dynamics in terms of the best-fit values of D1, D2, and u. Additional approaches we pursued employ the SRLS TCFs, SDFs, or eigenmodes as descriptors. In this study we develop yet another perspective. We consider the SDF as function of the angular velocity associated with the fluctuating fields underlying NMR relaxation. A parameter called j-fraction, which represents the relative contribution of eigenmode, i, to a given value of the SDF function at a specific frequency, ω, is defined. j-fraction profiles of the dominant eigenmodes are derived for 0 ≤ ω ≤ 1012 rad/s. They reveal which patterns of motion actuate power dissipation at given ω-values, what are their rates, and what is their relative contribution. Simulations are carried out to determine the effect of timescale separation, D1/D2, axial potential strength, and local diffusion axiality. For D1/D2 ≤ 0.01 and strong local potential of 15 kBT, power is dissipated by global diffusion, renormalized (by the strong potential) local diffusion, and probe diffusion on the surface of a cone (to be called cone diffusion). For D1/D2 = 0.1, power is dissipated by mixed eigenmodes largely of a global-diffusion-type or cone-diffusion-type, and a nearly bare renormalized-local-diffusion eigenmode. For D1/D2 &gt; 0.1, most eigenmodes are of a mixed type. The analysis is affected substantially by reducing the potential strength from 15 to 5 kBT, and/or allowing for axial D2 with D2,∥/D2,⊥ = 10. The scheme developed is applied to 15N−1H relaxation from the β-sheet residue K19 and the α-helix residue A34 of the third immunoglobulin-binding domain of streptococcal protein G. Previous studies revealed rhombic local potentials with different rhombicity around ${\rm C}_{i - 1}^\alpha {\rm - C}_i^\alpha $Ci−1α−Ciα, and different timescale separation (0.047 for K19 and 0.102 for A34). Here, we find that K19 and A34 dissipate power to the bath through global diffusion, mixed cone-diffusion-related and mixed renormalized-local-diffusion-related motions. At small ω-values, A34 is more effective than K19 in dissipating power. In general, it executes faster cone-diffusion-type, and slower renormalized-local-diffusion-type and local-probe-fluctuation-type motions. K19 experiences faster N−H fluctuations than A34. Eigenmode clustering, experienced by K19 to a larger extent, is observed in the fast-probe-fluctuation regime. New information on the effect of the structural context on N−H bond dynamics has been obtained. The patterns of motion that dissipate NMR-relaxation-related power illuminate protein dynamics from a new perspective. They constitute yet another qualifier of N−H bond dynamics. This study sets the stage for developing ways for enhancing the contribution of desired pathways for power dissipation at selected angular velocities.