Coarse-grained biomolecular simulation with REACH: realistic extension algorithm via covariance Hessian

Abstract

Coarse-graining of protein interactions provides a means of simulating large biological systems. Here, a coarse-graining method, REACH, is introduced, in which the force constants of a residue-scale elastic network model are calculated from the variance-covariance matrix obtained from atomistic molecular dynamics (MD) simulation. In test calculations, the C(alpha)-atoms variance-covariance matrices are calculated from the ensembles of 1-ns atomistic MD trajectories in monomeric and dimeric myoglobin, and used to derive coarse-grained force constants for the local and nonbonded interactions. Construction of analytical model functions of the distance-dependence of the interresidue force constants allows rapid calculation of the REACH normal modes. The model force constants from monomeric and dimeric myoglobin are found to be similar in magnitude to each other. The MD intra- and intermolecular mean-square fluctuations and the vibrational density of states are well reproduced by the residue-scale REACH normal modes without requiring rescaling of the force constant parameters. The temperature-dependence of the myoglobin REACH force constants reveals that the dynamical transition in protein internal fluctuations arises principally from softening of the elasticity in the nonlocal interactions. The REACH method is found to be a reliable way of determining spatiotemporal protein motion without the need for expensive computations of long atomistic MD simulations.

FIGURE 1

(a) REACH force constants, k, of virtual (a1) 1-2, (a2) 1-3, (a3) 1-4, and (a4) nonbonded interactions in the myoglobin monomer as a function of pairwise distance, r. In the inset of a4 the averages of k within bins of 1 Å width are shown together with the corresponding fitted model function. (b) Mean-square fluctuation of each residue in myoglobin monomer derived from MD (dashed line), B-factor (dotted line), and REACH normal modes (solid line).

FIGURE 2

(a) Vibrational density of states from MD trajectory of myoglobin monomer along selected normal modes, (a1) mode 1; (a2) mode 100; and (a3) mode 400. The associated REACH normal-mode frequencies are also shown. In panel b, MD-derived frequencies are plotted as a function of REACH normal-mode frequencies. See text for details.

FIGURE 3

(a) REACH force constants, k, of virtual (a1) 1-2, (a2) 1-3, (a3) 1-4 interactions and (a4) nonbonded intramolecular and (a5) intermolecular interactions in myoglobin dimer are shown as a function of pairwise distance, r. In a6, the averages of k (intra, dot and inter, cross) within bins of 1 Å width are shown together with the associated fitted curves (intra, dotted curve and inter, dashed curve). (b1) Mean-square fluctuation of each residue in myoglobin dimer derived from MD (dashed curve) and REACH (solid curve). (b2) REACH mean-square fluctuation from the contributions of overall (thin solid curve), intramolecular motion (thick solid curve), and intermolecular motion (dotted curve).

FIGURE 4

Average of force constants, k, in myoglobin monomer as a function of temperature for virtual (a) 1-2, (b) 1-3, (c) 1-4, and (d) nonbonded interactions. In case of the nonbonded interactions the force constants at r < 10 Å were averaged over all the pairs. In panel e, the exponent in the nonbond force constant model, b (see text), is plotted. Linear fits are also shown, as dashed lines. (f) Mean-square fluctuation, x², from REACH normal modes calculated using temperature-dependent model force constants. Dotted line is linear fit for lower temperature data. MD-derived x² is also plotted, as dots. x² from REACH with the same force constants but with temperature-independent 1-4 and nonbonded force constants is shown as dashed curve. See text for details.