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(θ, φ), m = 0, ±1, ±2, also given in MF. From these quantities, a method we call mf-RDC derives the squared generalized order parameter (Srdc2), the amplitude (direction) of the anisotropic disorder, η (Φ̅′), and an approximation, (N–H)eff, to the average probe orientation, i.e., to the local director. (N–H)eff is determined through a frame transformation where Y20 is maximized. Φ̅′ is associated with a subsequent frame transformation where Y22 + Y2–2 is maximized. The mf-RDC method was applied previously to N–H and C–Cmethyl sites in ubiquitin. In this study, we convert the respective Y2m(θ, φ)'s into a Saupe tensor, which is diagonalized. This is the standard procedure. It yields the eigenvalues, Sxx, Syy, and Szz, and the Principal Axis System (PAS) of the rank 2 local ordering tensor, Sl. Srdc2, η, and Φ̅′ can be recast as Sxx, Syy, and Szz. 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 Srdc2 is evaluated. For the α-helix of ubiquitin, the main local ordering axis is assigned as Ci–1α – Ciα; for the methyl sites, it is associated with the C–Cmethyl axis, as in mf-RDC. Ordering strength correlates with methyl type. The strength (rhombicity) of Sl 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 Sl in the probe, is formulated.
