Local Ordering at the N-H Sites of the Rho GTPase Binding Domain of Plexin-B1: Impact of Dimerization

Local Structures in Proteins from Microsecond Molecular Dynamics Simulations: 2. The Role of Symmetry in GTPase Binding and Dimer Formation

The Rho GTPase binding domain of Plexin-B1 (RBD) prevails in solution as dimer. Under appropriate circumstances, it binds the small GTPase Rac1 to yield the complex RBD-Rac1. Here, we study RBD dimerization and complex formation from a symmetry-based perspective using data derived from 1 μs long MD simulations. The quantities investigated are the local potentials, u(MD), prevailing at the N-H sites of the protein. These potentials are statistical in character providing an empirical description of the local structure. To establish more methodical description, a method for approximating them by explicit functions, u(simulated), was developed in the preceding article in this journal issue. These functions are combinations of analytical Wigner functions, DL,K, belonging to the D2h point group. The D2h subgroups Ag and B2u are found to dominate u(simulated); the B1u subgroup contributes in some cases. The Ag (B2u) functions have axial or rhombic symmetry. For the first time, local potentials in proteins can be quantitatively characterized in terms of their strength (rhombicity) evaluated by axial Ag (rhombic Ag and B2u) contributions. Until now, the chain-segment [β3-L3-β4] and to some extent the α2-helix have been associated with GTPase binding. Here, we find that this process causes an increase (decrease) in the potential strength of β3 and β4 (the preceding L2 loop and the remote chain-segment [(α2-helix)-(α2/β5-turn)-(β5-strand)]), suggesting effects of counterbalancing and allostery. There is evidence for the L2 loop being associated with RBD-GTPase binding. Until now only the L4 loop has been associated with RBD dimerization. The latter process is found to cause an increase (decrease) in the potential strength and rhombicity of the L4 loop (the adjacent chain-segment [(α2-helix)-(α2/β5-turn)-(β5-strand)]), suggesting counterbalancing activity. On average, the RBD dimer features stronger local potentials than RBD-Rac1. The novel information inherent in these findings is mesoscopic in character. Prospects of interest include exploring relation to atomistic force-field parameters.

Local Structures in Proteins from Microsecond Molecular Dynamics Simulations: A Symmetry-Based Perspective

We report on a new method for the characterization of local structures in proteins based on extensive molecular dynamics (MD) simulations, here, 1 μs in length. The N-H bond of the Rho GTPase binding domain of plexin-B1 (RBD) serves as a probe and the potential, u(MD), which restricts its internal motion, as a qualifier of the local dynamic structure. u(MD) is derived from the MD trajectory as a function of the polar angles, (θ, φ), which specify the N-H orientation in the protein. u(MD) is statistical in character yielding empirical descriptions. To establish more insightful methodical descriptions, we develop a comprehensive method which approximates u(MD) by combinations of analytical Wigner functions that belong to the D2h point group. These combinations, called u(simulated), make it possible to gain a new perspective of local dynamic structures in proteins based on explicit potentials/free energy surfaces and associated probability densities, entropy, and ordering. A simpler method was developed previously using 100 ns MD simulations. In that case, the traditional "perpendicular N-H ordering" setting centered at Cα-Cα with (θ, φ) = (90, 90) and generally, featuring positive φ, prevailed. u(MD) derived from 1 μs MD simulations is considerably more complex requiring substantial model enhancement. The enhanced method applies to the well-structured sections of the RBD. It only applies partly to its loops where u(MD) extends into the negative-φ region where we detect nonperpendicular N-H ordering. This arrangement requires devising new reference structures and making substantial algorithmic changes, to be performed in future work. Here, we focus on developing the comprehensive method and using it to investigate perpendicular ordering settings. We find that secondary structures (loops) exhibit varying (virtually invariant) potentials with Ag, B2u, and B1u (Ag and B2u) D2h symmetry. Application to RBD dimerization and RBD binding to the GTPase Rac1 is described in the subsequent article. Applications to other probes, proteins, and biological functions, based on explicit local potentials, probability densities, entropy, and ordering, are possible.

Microsecond MD Simulations of the Plexin-B1 RBD: N-H Probability Density as Descriptor of Structural Dynamics, Dimerization-Related Conformational Entropy, and Transient Dimer Asymmetry

Amide-bond equilibrium probability density, P_eq = exp(-u) (u, local potential), and associated conformational entropy, S_k = -∫P_eq (ln P_eq) dΩ ─ln ∫dΩ, are derived for the Rho GTPase binding domain of Plexin-B1 (RBD) as monomer and dimer from 1 μs MD simulations. The objective is to elucidate the effect of dimerization on the dynamic structure of the RBD. Dispersed (peaked) P_eq functions indicate "flexibility" ("rigidity"; the respective concepts are used below in this context). The L1 and L3 loops are throughout highly flexible, the L2 loop and the secondary structure elements are generally rigid, and the L4 loop is flexible in the monomer and rigid in the dimer. Overall, many residues are more flexible in the dimer. These features, and their implications, are discussed. Unexpectedly, we find that monomer unit 1 of the dimer (in short, d1) is unusually flexible, whereas monomer unit 2 (in short, d2) is as rigid as the RBD monomer. This is revealed due to their engagement in slow-to-intermediate conformational exchange detected previously by ¹⁵N relaxation experiments. Such motions occur with rates on the order of 10³-10⁴ s^-1; hence, they cannot be completely sampled over the course of 1 μs simulation. However, the extent to which rigid d2 is affected is small enough to enable physically relevant analysis. The entropy difference between d2 and the monomer yields an entropic contribution of -7 ± 0.7 kJ/mol to the free energy of RBD dimerization. In previous work aimed at similar objectives we used 50-100 ns MD simulations. Those results and the present result differ considerably. In summary, bond-vector P_eq functions derived directly from long MD simulations are useful descriptors of protein structural dynamics and provide accurate conformational entropy. Within the scope of slow conformational exchange, they can be useful, even in the presence of incomplete sampling.

Microsecond MD Simulations of the Plexin-B1 RBD: 2. N-H Probability Densities and Conformational Entropy in Ligand-Free, Rac1-Bound, and Dimer RBD

Orientational probability densities, P_eq = exp(-u) (u, local potential), of bond-vectors in proteins provide information on structural flexibility. The related conformational entropy, S_k = -∫P_eq(ln P_eq)dΩ - ln ∫dΩ, provides the entropic contribution to the free energy of the physical/biological process studied. We have developed a new method for deriving P_eq and S_k from MD simulations, using the N-H bond as probe. Recently we used it to study the dimerization of the Rho GTPase binding domain of Plexin-B1 (RBD). Here we use it to study RBD binding to the small GTPase Rac1. In both cases 1 μs MD simulations have been employed. The RBD has the ubiquitin fold with four mostly long loops. L3 is associated with GTPase binding, L4 with RBD dimerization, L2 participates in interdomain interactions, and L1 has not been associated with function. We find that RBD-Rac1 binding renders L1, L3, and L4 more rigid and the turns β₂/α₁ and α₂/β₅ more flexible. By comparison, RBD dimerization renders L4 more rigid, and the α-helices, the β-strands, and L2 more flexible. The rigidity of L1 in RBDRAC is consistent with L1-L3 contacts seen in previous MD simulations. The analysis of the L3-loop reveals two states of distinct flexibility which we associate with involvement in slow conformational exchange processes differing in their rates. Overall, the N-H bonds make an unfavorable entropic contribution of (5.9 ± 0.9) kJ/mol to the free energy of RBD-Rac1 binding; they were found to make a favorably contribution of (-7.0 ± 0.7) kJ/mol to the free energy of RBD dimerization. In summary, the present study provides a new perspective on the impact of Rac1 binding and dimerization on the flexibility characteristics of the RBD. Further studies are stimulated by the results of this work.

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.