Standard tensorial analysis of local ordering in proteins from residual dipolar couplings

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, 〈Y_2m(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 (S_rdc²), the amplitude (direction) of the anisotropic disorder, η (Φ̄′), 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. Φ̄′ is associated with a subsequent frame transformation where 〈Y₂₂ + Y _2-2〉 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(θ, φ)〉'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_l. S_rdc², η, and Φ̄′ 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_rdc² is evaluated. For the α-helix of ubiquitin, the main local ordering axis is assigned as C _i-1^α - C_i^α; 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 S_l 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 S_l in the probe, is formulated.