Simulation of F-actin filaments of several microns

Abstract

Here we report the results of applying substructure synthesis method to the simulation of F-actin filaments of several microns in length. The elastic deformational modes of long F-actin filaments were generated from the vibrational modes of the 13-subunit repeat of F-actin using a hierarchical synthesis scheme. The computationally synthesized deformational modes, in the very low-frequency regime, are in good agreement with theoretical solutions for long homogeneous elastic rods, which confirmed the usefulness of substructure synthesis method. Other low-frequency modes carry rich local deformational features that are unique to F-actins. All these modes thus provide a theoretical basis set for a description of spontaneously occurring thermal deformations, such as undulations, of the filaments. The results demonstrate that substructure synthesis method, as a method for computational modal analysis, is capable of scaling up the microscopic dynamic information, obtained from atomistic simulations, to a wide range of macroscopic length scale. Moreover, the combination of substructure synthesis method and hierarchical synthesis scheme provides an effective way in dealing with complex systems of periodic repeats that are abundant in cells.

FIGURE 1

Structures of actins. (a) Structure of G-actin monomer that contains 375 amino acids refined by a normal mode-based method (Tirion et al., 1995). The discontinuity of the structure in the loop region by MOLSCRIPT (Kraulis, 1991) comes from the small distortions in the original structure. (b) The Holmes model for a 13-subunit repeat of F-actin filament established from fiber diffraction (Holmes et al., 1990, 1993; Tirion et al., 1995) with two helical strands differently colored. To illustrate the overall shape, the atomic coordinates were blurred to 8-Å resolution.

FIGURE 2

Normal modes for a 13-subunit repeat of F-actin filament. The a–c′ represent the lowest-frequency modes for each type of mode: a and a′ are the bending modes (modes 7 and 10); b and b′ are the twisting modes (modes 9 and 12); and c and c′ are the stretching modes (modes 15 and 25). The two helical strands of F-actin are in different colors. Each mode is represented by two opposite end-point structures. The orientation of the filament in each mode is chosen to best illustrate that particular mode. (d) 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 solid arrow indicating the increase of frequency.

FIGURE 2

Normal modes for a 13-subunit repeat of F-actin filament. The a–c′ represent the lowest-frequency modes for each type of mode: a and a′ are the bending modes (modes 7 and 10); b and b′ are the twisting modes (modes 9 and 12); and c and c′ are the stretching modes (modes 15 and 25). The two helical strands of F-actin are in different colors. Each mode is represented by two opposite end-point structures. The orientation of the filament in each mode is chosen to best illustrate that particular mode. (d) 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 solid arrow indicating the increase of frequency.

FIGURE 3

(a) A schematic illustration of SSM when fusing two 3-mass-point chains together. One of the two boundary points (c and a′) is equalized (sacrificed) during synthesis. (b) Stereo pair for the interface of two substructures in the synthesis of a two 13-subunit F-actin segment showing the 15 red spheres (C_α-atoms) 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.

FIGURE 3

(a) A schematic illustration of SSM when fusing two 3-mass-point chains together. One of the two boundary points (c and a′) is equalized (sacrificed) during synthesis. (b) Stereo pair for the interface of two substructures in the synthesis of a two 13-subunit F-actin segment showing the 15 red spheres (C_α-atoms) 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.

FIGURE 4

Lowest-frequency modes from SSM synthesis of a two 13-subunit F-actin segment. (a) Bending, (b) twisting, and (c) stretching modes for F-actin. In SSM, 300 low-frequency modes from each substructure were used. Two substructures are in different colors. Each mode is represented by two opposite end-point structures. (d) Comparison of the theoretical solution of a homogeneous elastic rod (Eq. 28) with the synthesized bending mode in a. The dots represent the average positions of the bent actin filament in the lower image of a by an arbitrary sampling density, and the central tube represents the theoretical solution. Amplitudes were adjusted to achieve a best match.

FIGURE 5

The logarithms of eigenvalues of the synthesized modes from SSM-HSS as a function of the number of the 13-subunit repeat in the F-actin filaments, presented together with the theoretical results of homogeneous elastic rods of the same length. Two cases are shown, one with 300 substructure modes (abbreviated as Submodes; left panels), and the other with 800 submodes (right panels). The very lowest-frequency modes are shown for (a) bending, (b) twisting, and (c) stretching modes. The SSM and theoretical curves were normalized in each case by setting the first point equal to each other.

FIGURE 6

The motional patterns of several lowest-frequency bending modes for the 4.6 μm F-actin filament calculated by SSM-HSS. The indices of the modes are marked. The even indices are the ones that degenerate to the displayed one. The seventh mode is the lowest-frequency mode.