Exact Solutions and hypothesis on Phase Transition in the Polyelectrolite Model of DNA

The two-dimensional generalization of the Polyelectolite model of DNA is proposed. It is reduced to the boundary problem for nonlinear quite integrable equation sinh-Gordon. In the linearisible version the exact solution is constructed and its asymp-totic is found. The soliton solutions of nonlinear equation are calculated that allowed tells about the possibility of the structural phase transition in the considered system (DNA+polyelectrolite) on the temperature. At the present time constructing and investigating mathematical models of DNA is an important problem of both theoretical biophysics and mathematical physics. Modern achievements in the field of nonlinear equations theory that occur, as a rule, in these models allow not only to set initial and boundary problems for such equations, but also to solve them in a number of cases. The given work is devoted to construction of exact solutions for a two-dimensional generalization of so-called polyelectrolyte model of DNA (PM DNA) [1]. In essence here we deal with solution of an inverse problem in the broad sense, i.e. it is to judge structure and properties of a DNA molecule from a field restoring within the framework of this model and also values of some functionals of it. It is necessary to note that PM represents the most concentrated description of role of Coulomb interactions which are of a great importance in formation of structure of a DNA molecule and in its functioning as well as in intrinsical structural transitions. One more important advantage of the model is possibility to operate with terms of statistical physics, which allows to implement a clear mathematical problem statement. The formulation of model and statement of the problem. In reality a DNA molecule is a strongly charged poly-ion, with two charges of electrons falling at every pair of nucleotides. The idea of PM consists in replacement of such system by an infinite and regular negatively charged cylinder of radius r 0 with a given surface charge density placed into solution of polyelectrolite. It is supposed that a processes of relaxation proceeds quickly enough, so only an equilibrium situation is considered. Besides here we are restricted ourselves by consideration of the case when a solution consists of unitary charged ions and electrons. Standard physical reasons result to a system of Poisson-Boltsmann equations written down at any point of volume outside the cylinder: div E = 4πρ, E = −grad U, ρ = e(n i − n e), (1) in …