Study of nonlinear dynamic modes of native DNA via mathematical modeling methods

Abstract

One of the main problems in the nonlinear dynamics of DNA is the identification of physical mechanisms responsible for the mechanical features of its behaviour. The formation and interaction of open states in the DNA structure may be one such mechanism. In this paper we propose a mathematical model of native DNA, which allows us to describe its thermodynamic and kinetic properties taking into account the collective behaviour of the ensemble of open states. The concept of a microscopic open state in DNA and its associated parameter, the displacement vector of nitrogenous bases, are introduced. By averaging the base displacement vectors over the ensemble of states, the thermodynamic variable is determined. According to the statistical model of DNA in the self-consistent field approximation, the structural parameter of the system thermalization is derived. This parameter reflects the statistical self-similarity in the behaviour of the ensemble of open states. Regularities of "criticality" for different ranges of the structural parameter are established and phenomenological representations of the free energy in the framework of the Ginzburg-Landau approach are proposed. Numerical modelling of the dynamics of native DNA is carried out for different ranges of the structural parameter, which has allowed us to establish the types of self-similar solutions and the corresponding open-state collective modes. It is shown that the latter have the nature of finite-amplitude fluctuations in the form of breahters and autosoliton modes, as well as blow-up dissipative structures. The sensitivity of the model to initial conditions is investigated, and the influence of nonlinearity inherent in the model on the modelling results is established. The limitations and prospects of using the approach proposed in this work for mathematical modelling of native DNA are discussed.