Standard Tensorial Analysis of Local Ordering in Proteins from Residual Dipolar Couplings B (original) (raw)

2012, The Journal of Physical Chemistry

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 rdc 2), 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 ⟨Y 20 ⟩ is maximized. Φ̅ ′ is associated with a subsequent frame transformation where ⟨Y 22 + 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 rdc 2 , η, and Φ̅ ′ 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 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.