Abstract
Locally mobile bond-vectors contribute to the conformational entropy of the protein, given by Sk ≡ S/k = −∫(Peq ln Peq)dΩ – ln∫dΩ. The quantity Peq = 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, D0KL. 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 onSk. For L = 1 (L = 2), Sk is the same (differs) for parallel and perpendicular ordering. The plots of Sk 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. Sk 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.
Highlights
We developed the two-body coupled-rotator slowly relaxing local structure (SRLS) approach[16−18] for the analysis of NMR relaxation in proteins.[19−22] In SRLS, the local potential is expanded in the basis set of the real linear combinations of the Wigner rotation matrix elements, D0LK.[16,19]
SRLS limit where the protein motion is frozen is the microscopic-order-macroscopic-disorder (MOMD) approach,[27] which we developed for proteins in the solid state.[28−32] SRLS and MOMD were originally developed for electron spin resonance (ESR) applications in complex fluids and proteins.[16−18,27] In all of these theoretical approaches, the local potential is expressed in terms of real Wigner functions
The local potentials, u, at the site of mobile bond-vectors in proteins have been expressed in terms of the real linear combinations of the Wigner rotation matrix elements, D0LK, with L = 1 or 2
Summary
The traditional method for the analysis of NMR relaxation in proteins is model-free (MF).[13] In the MF formalism, the local spatial restrictions are expressed in terms of the squared generalized order parameter, S2, rather than a potential function. Comparison between the bestfit Wigner function and the POMF indicated that the set of terms with L = 1−4 suffices for obtaining good agreement Using such optimized potentials, new insights into the dimerization of the Rho GTPase binding domain of plexinB1 (in brief, plexin-B1 RBD) were gained.[26] In future work, we plan to incorporate these potentials unchanged into SRLS datafitting schemes. We have at hand explicit axial and rhombic, polar and nonpolar, fairly accurate Wigner-function-made local potentials This constitutes a rich source for conformational entropy derivation.
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