Chaotic Dynamics of Polyatomic Systems with an Emphasis on DNA Models

Hillebrand, Malcolm

Chaotic Dynamics of Polyatomic Systems with an Emphasis on DNA Models

Doctoral Thesis

2021

Abstract

In this work we investigate the chaotic behaviour of multiparticle systems, and in particular DNA and graphene models, by applying various numerical methods of nonlinear dynamics. Through the use of symplectic integration techniques—efficient routines for the numerical integration of Hamiltonian systems—we present an extensive analysis of the chaotic behaviour of the Peyrard-Bishop-Dauxois (PBD) model of DNA. The chaoticity of the system is quantified by computing the maximum Lyapunov exponent (mLE) across a spectrum of temperatures, and the effect of base pair disorder on the dynamics is investigated. In addition to the inherent heterogeneity due to the proportion of adenine-thymine (AT) and guanine-cytosine (GC) base pairs, the distribution of these base pairs in the sequence is analysed through the introduction of the alternation index. An exact probability distribution for arrangements of base pairs and their alternation index is derived through the use of Pólya counting theory. We find that the value of the mLE depends on both the base pair composition of the DNA strand and the arrangement of base pairs, with a changing behaviour depending on the temperature. Regions of strong chaoticity are probed using the deviation vector distribution, and links between strongly nonlinear behaviour and the formation of bubbles (thermally induced openings) in the DNA strand are studied. Investigations are performed for a wide variety of randomly generated sequences as well as biological promoters. Furthermore, the properties of these thermally induced bubbles are studied through large-scale molecular dynamics simulations. The distributions of bubble lifetimes and lengths in DNA are obtained and discussed in detail, fitted with simple analytical expressions, and a physically justified threshold distance for considering a base pair to be open is proposed and successfully implemented. In addition to DNA, we present an analysis of the dynamical stability of a planar model of graphene, studying the behaviour of the mLE in bulk graphene sheets as well as in finite width graphene nanoribbons (GNRs). The wellattested stability of the material manifests in a very small mLE, with chaos being a slow process in graphene. For both possible kinds of GNR, armchair and zigzag edges, the mLE decreases with increasing width, asymptotically reaching the bulk behaviour. This dependence of the mLE on both energy density and ribbon width is fitted accurately with empirical expressions.

Keywords

Mathematics and Applied Mathematics

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PhD / Doctoral

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