Substructure synthesis method for simulating large molecular complexes

Abstract

This paper reports a computational method for describing the conformational flexibility of very large biomolecular complexes using a reduced number of degrees of freedom. It is called the substructure synthesis method, and the basic concept is to treat the motions of a given structure as a collection of those of an assemblage of substructures. The choice of substructures is arbitrary and sometimes quite natural, such as domains, subunits, or even large segments of biomolecular complexes. To start, a group of low-frequency substructure modes is determined, for instance by normal mode analysis, to represent the motions of the substructure. Next, a desired number of substructures are joined together by a set of constraints to enforce geometric compatibility at the interface of adjacent substructures, and the modes for the assembled structure can then be synthesized from the substructure modes by applying the Rayleigh-Ritz principle. Such a procedure is computationally much more desirable than solving the full eigenvalue problem for the whole assembled structure. Furthermore, to show the applicability to biomolecular complexes, the method is used to study F-actin, a large filamentous molecular complex involved in many cellular functions. The results demonstrate that the method is capable of studying the motions of very large molecular complexes that are otherwise completely beyond the reach of any conventional methods.

Figure 1

Structures of actin. (a) Crystal structure of G-actin monomer that contains 375 amino acids (PDB code 1ATN) (21). (b) The 13-subunit repeat of F-actin filament established from fiber diffraction, i.e., the Holmes model (–24). The two helices are marked by different colors. To illustrate the overall shape, the atomic coordinates were blurred to 8-Å resolution to show the surface.

Figure 2

A one-dimensional example of SSM. (a) Fusing two 3-mass-point chains together. One of the two boundary points (3 and 1′) is sacrificed during synthesis. (b) Eigenvalues for the fused two 100-mass-point chains. Results from syntheses using all 100 substructure modes [199 modes after SSM, SSM(100/100)] and using 10 lowest-frequency substructure modes [19 modes after SSM, SSM(100/10)] are shown. The eigenvalues from SSM excellently match those from the exact solution, Exact(199). (c) Enlarged version for the first 19 modes in b. Both synthesis schemes have excellent results for the lowest-frequency modes before the 12th mode.

Figure 3

A three-dimensional elastic lattice. (a) The substructure lattice with each point mass illustrated as a sphere. The first bending (b) and twisting (c) modes for the assembled lattice are shown. The motional patterns of SSM-generated modes (SSM) are compared with those from an exact calculation (Exact) in an arbitrary displacement magnitude. For clarity, only one side of the lattice is drawn. (d) The convergence of the eigenvector of the bending mode in b. The spheres represent the positions of the centers of mass of the parallel layers from SSM, exact calculation, and a theoretical curve for a homogeneous elastic bar (31). The agreement is nearly perfect; i.e., each sphere is a superposition of three spheres.

Figure 4

Stereo pair for the interface of two substructures in the synthesis of an F-actin segment that contains two 13-subunit repeats. The 15 red spheres indicate the positions of the C_α-atoms that were chosen as the boundary points by the 5-Å distance search. The subunits from different substructures are shown in different colors. For clarity, only the subunits near the interface are shown, and the orientation of the filament is horizontal in the plane of the paper. The breaks in the graphic representation, especially in some loop regions, are due to the deviations of the experimentally refined structure (23, 24).

Figure 5

Lowest-frequency modes from SSM synthesis of an F-actin segment composed of two 13-subunit repeats. (a) Schematic illustration of typical low-frequency normal modes (bending, twisting, and stretching) for a homogeneous elastic rod. Two lowest-frequency modes are shown for each type of mode, with the arrow indicating the increase of frequency. Bending (b), twisting (c), and stretching (d) modes for F-actin are shown. In SSM, 300 low-frequency modes from each substructure were used. The two repeats are in different colors. Two opposite end-point structures are shown for each mode. (e) Comparison of the theoretical solution of a homogeneous elastic bar (Eq. 28) with the synthesized bending mode in b. The spherical dots represent the average position of the bent actin filament in the lower image of b by an arbitrary sampling density, and the central tube represents the theoretical solution. Their amplitudes were adjusted for a best match.

Figure 6

Convergence of the eigenvalues in SSM as a function of the number of substructure modes used under two boundary conditions. (a) Fifteen boundary points (C_α atoms) from the 5-Å distance search. (b) Thirty-six boundary points (C_α-atoms) from the 7-Å distance search. The eigenvalues are shown as deviations from their corresponding converged values, Δλ. For illustrative purposes, only the results for the first four vibrational modes are shown.