Thermodynamics of interacting system of DNAs

TL;DR

This paper models DNA interactions on a Cayley tree using a spin system, analyzing phase transitions and thermodynamic properties with Gibbs measures, revealing a critical temperature where the number of phases changes.

Contribution

It introduces a novel spin-based model of DNA interactions on Cayley trees and characterizes phase transitions via translation invariant Gibbs measures.

Findings

Existence of a critical temperature T_c for phase transition.

Unique Gibbs measure for T ≥ T_c.

Three Gibbs measures for T < T_c.

Abstract

We define a DNA as a sequence of $1, 2$'s and embed it on a path of Cayley tree in such a way that each vertex of the Cayley tree belongs only to one of DNA and each DNA has its own countably many set of neighboring DNAs. The Hamiltonian of this set of DNAs is a model with two spin values considered as DNA base pairs. We describe translation invariant Gibbs measures (TIGM) of the model on the Cayley tree of order two and use them to study thermodynamic properties of the model of DNAs. We show that there is a critical temperature $T_{\rm c}$ such that (i) if temperature $T\geq T_{\rm c}$ then there exists unique TIGM; (ii) if $T<T_{\rm c}$ then there are three TIGMs. Each TIGM gives a phase of the set of DNAs. In case of very high and very low temperatures we give stationary distributions and typical configurations of the model.