Standard Tensorial Analysis of Local Ordering in Proteins from Residual Dipolar Couplings

Abstract

Residual dipolar couplings (RDCs) in proteins arise from independent external medium-related and internal protein-related ordering of the spin-bearing probe. Griesinger et al. developed a method for treating RDCs in proteins. The global ordering is given in the standard manner by a rank 2 tensor specified in a known molecular frame, MF. The local ordering is described by the spherical harmonic ensemble averages, < Y2m (theta, phi)>, m = 0, +/-1, +/-2, also given in MF. From these quantities, a method we call mf-RDC derives the squared generalized order parameter (S-rdc(2)), the amplitude (direction) of the anisotropic disorder, eta ((Phi) over bar), and an approximation, (N-H)(eff), to the average probe orientation, i.e., to the local director. (N-H)(eff) is determined through a where < Y-20 > is maximized. (Phi) over bar is associated with a subsequent frame transformation where < Y-22 + Y2-2 > is maximized. The mf-RDC method was applied previously to N-H and C-C-methyl sites in ubiquitin. In this study, we convert the respective < Y-2m(theta, phi)>'s into a Saupe tensor, which is diagonalized. This is the standard procedure. It yields the eigenvalues, S-xx, S-yy, and S-zz, and the Principal Axis System (PAS) of the rank 2 local ordering tensor, S-1. S-rdc(2), eta and (Phi) over bar can be recast as S-xx, S-yy, and S-zz. The mf-RDC frame transformations are not the same as the conventional Wigner rotation. The standard tensorial analysis provides new information. The contribution of local ordering rhombicity to S-rdc(2) is evaluated. For the alpha-helix of ubiquitin, the main local ordering axis is assigned as C-i-1(alpha) - C-i-j(alpha) for the methyl sites, it is associated with the C-C-methyl axis, as in mf-RDC. Ordering strength correlates with methyl type. The strength (rhombicity) of Si associated with picosecond-nanosecond local motions is reduced moderately (substantially) by nanosecond millisecond local motions. A scheme for analyzing experimental RDCs based on the standard tensorial perspective, which allows for arbitrary orientation of the local director in the protein and of the PAS of SI in the probe, is formulated. frame transformation