Conformational Entropy from NMR Relaxation in Proteins: The SRLS Perspective

SRLS Analysis of 15N-1H NMR Relaxation from the Protein S100A1: Dynamic Structure, Calcium Binding, and Related Changes in Conformational Entropy

We report on amide (N-H) NMR relaxation from the protein S100A1 analyzed with the slowly relaxing local structure (SRLS) approach. S100A1 comprises two calcium-binding "EF-hands" (helix-loop-helix motifs) connected by a linker. The dynamic structure of this protein, in both calcium-free and calcium-bound form, is described as the restricted local N-H motion coupled to isotropic protein tumbling. The restrictions are given by a rhombic potential, u (∼10 kT), the local motion by a diffusion tensor with rate constant D₂ (∼10⁹ s^-1), and principal axis tilted from the N-H bond at angle β (10-20°). This parameter combination provides a physically insightful picture of the dynamic structure of S100A1 from the N-H bond perspective. Calcium binding primarily affects the C-terminal EF-hand, among others slowing down the motion of helices III and IV approximately 10-fold. Overall, it brings about significant changes in the shape of the local potential, u, and the orientation of the local diffusion axis, β. Conformational entropy derived from u makes an unfavorable entropic contribution to the free energy of calcium binding estimated at 8.6 ± 0.5 kJ/mol. The N-terminal EF-hand undergoes moderate changes. These findings provide new insights into the calcium-binding process. The same data were analyzed previously with the extended model-free (EMF) method, which is a simple limit of SRLS. In that interpretation, the protein tumbles anisotropically. Locally, calcium binding increases ordering in the loops of S100A1 and conformational exchange (R_ex) in the helices of its N-terminal EF-hand. These are very unusual features. We show that they most likely stem from problematic data-fitting, oversimplifications inherent in EMF, and experimental imperfections. R_ex is shown to be mainly a fit parameter. By reanalyzing the experimental data with SRLS, which is largely free of these deficiencies, we obtain-as delineated above-physically-relevant structural, kinetic, geometric, and binding information.

Conformational Entropy from Mobile Bond Vectors in Proteins: A Viewpoint that Unifies NMR Relaxation Theory and Molecular Dynamics Simulation Approaches

A new method for determining conformational entropy in proteins is reported. Proteins prevail as conformational ensembles, p ∝ exp(-u). By selecting a bond vector (e.g., N-H) as a conformation representative, molecular dynamics simulations can provide (relative to a reference structure) p as approximate Boltzmann probability density and u as N-H potential of mean force (POMF). The latter is as accurate as implied by the force field but statistical in character; this limits the insights it can provide and its utilization. Conformational entropy is given exclusively by u. Deriving it from POMFs renders it accurate but statistical in character. Previously, we devised explicit (i.e., analytical but not exact) potentials made of Wigner functions, D_0K^L, with L ≤ 4, which closely resemble the corresponding POMFs in form; hence, they also approach the latter in accuracy. Such potentials can be beneficially characterized/compared in terms of composition, symmetry, and associated order parameters. In this study, we develop a method for deriving conformational entropy from them, which also features the benefits specified above. The method developed is applied to the dimerization of the Rho GTPase-binding domain of plexin-B1. Insights into local ordering, entropy compensation, and features of allostery are gained. In previous work, we developed the slowly relaxing local structure (SRLS) approach for the analysis of NMR relaxation from restricted bond vector motion in proteins. SRLS comprises explicit (restricting) potentials of the kind developed here. It also comprises diffusion tensors describing the local motion and related features of local geometry. The complete model fits experimental data. In future work, the explicit potentials developed here will be inserted unchanged in SRLS-based data fitting, thereby improving the picture of structural dynamics. Given that SRLS is unique in featuring potentials that can closely approach the corresponding POMFs in accuracy, the present study is an important step toward generally improving protein dynamics by NMR relaxation.

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_0K^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.

15N-H-Related Conformational Entropy Changes Entailed By Plexin-B1 RBD Dimerization: Combined Molecular Dynamics/NMR Relaxation Approach

We report on a new method for determining function-related conformational entropy changes in proteins. Plexin-B1 RBD dimerization serves as example, and internally mobile N-H bonds serve as probes. S_k (entropy in units of k_BT) is given by -∫(P_eqlnP_eq)dΩ, where P_eq = exp(-u) is the probability density for probe orientation, and u the local potential. Previous slowly relaxing local structure (SRLS) analyses of ¹⁵N-H relaxation in proteins determined linear combinations of D₀₀²(Ω) and (D₀₂²(Ω) + D_0-2²(Ω)) (D_0K^L(Ω) represents a Wigner rotation matrix element in uniaxial local medium) as "best-fit" form of u. SRLS also determined the "best-fit" orientation of the related ordering tensor. On the basis of this information the coefficients (in the linear combination) of the terms specified above are determined with molecular dynamics (MD) simulations. With the explicit expression for u thus in hand, S_k is calculated. We find that in general S_k decreases, i.e., the local order increases, upon plexin-B1 RBD dimerization. The largest decrease in S_k occurs in the helices α₁ and α₂, followed by the α₂/β₆ turn. Only the relatively small peripheral β₂ strand, β₂/α₁ turn, and L3 loop become more disordered. That α-helices dominate ΔS_k = S_k(dimer) - S_k(monomer), a few peripheral outliers partly counterbalance the overall decrease in S_k, and the probability density function, P_eq, has rhombic symmetry given that the underlying potential function, u, has rhombic symmetry, are interesting features. We also derive S² (the proxy of u in the simple "model-free (MF)" limit of SRLS) with MD. Its conversion into a potential requires assumptions and adopting a simple axial form of u. Ensuing ΔS_k(MF) profiles are u-dependent and differ from ΔS_k(SRLS). A method that provides consistent, general, and accurate S_k, atomistic/mesoscopic in nature, has been developed. Its ability to provide new insights in protein research has been illustrated.