Conformational Entropy from Restricted Bond-Vector Motion in Proteins: The Symmetry of the Local Restrictions and Relation to NMR Relaxation

Locally mobile bond-vectors contribute to the conformational entropy of the protein, given by S k ≡ S / k = -∫( P eq ln P eq )dΩ - ln∫dΩ. The quantity P eq = exp(- u )/ Z is the orientational probability density, where Z is the partition function and u is the spatially restricting potential exerted by the immediate internal protein surroundings at the site of the motion of the bond-vector. It is appropriate to expand the potential, u , which restricts local rotational reorientation, in the basis set of the real combinations of the Wigner rotation matrix elements, D 0 K L . For small molecules dissolved in anisotropic media, one typically keeps the lowest even L , L = 2, nonpolar potential in axial or rhombic form. For bond-vectors anchored at the protein, the lowest odd L , L = 1, polar potential is to be used in axial or rhombic form. Here, we investigate the effect of the symmetry and polarity of these potentials on S k . For L = 1 ( L = 2), S k is the same (differs) for parallel and perpendicular ordering. The plots of S k as a function of the coefficients of the rhombic L = 1 ( L = 2) potential exhibit high-symmetry (specific low-symmetry) patterns with parameter-range-dependent sensitivity. Similar statements apply to analogous plots of the potential minima. S k is also examined as a function of the order parameters defined in terms of u . Graphs displaying these correlations, and applications illustrating their usage, are provided. The features delineated above are generally useful for devising orienting potentials that best suit given physical circumstances. They are particularly useful for bond-vectors acting as NMR relaxation probes in proteins, when their restricted local motion is analyzed with stochastic models featuring Wigner-function-made potentials. The relaxation probes could also be molecules adsorbed at surfaces, inserted into membranes, or interlocked within metal-organic frameworks.