From the FPU Chain to Biomolecular Dynamics

Abstract

The molecular dynamics simulations originally performed by Fermi, Pasta, and Ulam for an isolated one-dimensional chain with cubic anharmonicity had led afterwards to the discovery of stable coherent structures called “solitons”. Any study of the stability of solitons on such a one-dimensional lattice with respect to transverse motions of chain atoms or molecules requires introduction of a secondary structure realized for biological macromolecules in the form of a helix. In the simplest case of intermolecular interactions with spherical symmetry, the straightforward generalization of the Fermi-Pasta-Ulam chain to higher dimensions gives rise to the helical structure: zigzag in two dimensions and α-helix in three dimensions. The planar zigzag structure is provided by the first-and second-neighbor intermolecular bonds, whereas the helical structure in three dimensions requires for its stabilization, at least, three types of interactions. The coupled nonlinear field equations that describe longitudinal and transverse displacements of molecules in the helix backbone are studied. In particular, stable non-topological two- and three-component soliton solutions in two and three dimensions, respectively, are shown to exist. These solutions describe supersonic pulses of longitudinal compression propagating together with localized transverse thickening (“bulging”) and torsional stretching (twisting). Other, more specific, types of solitons are investigated in two dimensions for the zigzag backbone.