Nonlinear relaxation dynamics in elastic networks and design principles of molecular machines

Abstract

Analyzing nonlinear conformational relaxation dynamics in elastic networks corresponding to two classical motor proteins, we find that they respond by well defined internal mechanical motions to various initial deformations and that these motions are robust against external perturbations. We show that this behavior is not characteristic for random elastic networks. However, special network architectures with such properties can be designed by evolutionary optimization methods. Using them, an example of an artificial elastic network, operating as a cyclic machine powered by ligand binding, is constructed.complex systems ͉ protein machines ͉ molecular motors ͉ evolutionary optimization U nderstanding design principles of single-molecule machines is a major challenge. Experimental and theoretical studies of proteins, acting as motors (1-5), ion pumps (6-8), or channels (6, 9), and enzymes (10-14), show that their operation involves functional conformational motions (see ref. 15). Such motions are slow and cannot therefore be reproduced by full molecular dynamics simulations. Within the last decade, approximate descriptions based on elastic network models of proteins have been developed (16)(17)(18)(19)(20)(21). In this approach, structural elements of a protein are viewed as identical point particles, with two particles connected by an elastic string if the respective elements lie close enough in the native state of the considered protein. Thus, a network of elastic connections corresponding to a protein is constructed. So far, the attention has been focused on linear dynamics of elastic networks, characterized in terms of their normal vibrational modes. It has been found that ligand-induced conformational changes in many proteins agree with the patterns of atomic displacements in their slowest vibrational modes (refs. 22-25 and see also refs. 26 and 27), even though nonlinear elastic effects must become important for large deviations from the equilibrium (28, 29). The focus of this article is on nonlinear relaxation phenomena in elastic networks seen as complex dynamical systems.Generally, a machine is a mechanical device that performs ordered internal motions that are robust against external perturbations. In machines representing single molecules, energy is typically supplied in discrete portions, through individual reaction events. Therefore, their cycles consist of the processes of conformational relaxation that follow after energetic excitations. For a robust machine operation, special nonlinear relaxation dynamics is required. We expect that, starting from a broad range of initial deformations, such dynamical systems would return to the same final equilibrium state. Moreover, the relaxation would proceed along a well defined trajectory (or a low-dimensional manifold), rapidly approached starting from different initial states and robust against external perturbations. These attractive relaxation trajectories would define internal mechanical motions of the machine inside its operation cycle.This special confor...

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