Thermodynamics of interacting systems of DNA molecules

Abstract

We define a DNA molecule as a sequence of the numbers 1 and 2 and embed it on a path of a Cayley tree such that each vertex of the Cayley tree belongs to only one DNA and each DNA has its own countable set of neighboring DNA. The Hamiltonian of this set of DNA is a model with two spin values regarded as DNA base pairs. We describe translation-invariant Gibbs measures (TIGMs) of the model on the Cayley tree of order two and use them to study the thermodynamic properties of the model of DNA. We show that there is a critical temperature $T_{ \mathrm{c} }$ such that if the temperature $T\ge T_{ \mathrm{c} }$, then there is a unique TIGM, and if $T<T_{ \mathrm{c} }$, then there are three TIGMs. Each TIGM gives a phase of the set of DNA. In the cases of very high and very low temperatures, we find stationary distributions and typical configurations of the model.