Holliday Junctions for the Potts Model of DNA

Abstract

In this paper a DNA is considered as a configuration of 4-state Potts model and it is embed on a path of a Cayley tree. Then we give a Hamiltonian of the set of DNAs by an analogue of Potts model with 5-spin values \(0, \pm 1, \pm 2\) (considered as DNA base pairs and 0 means vacant) on a set of admissible configurations. To study thermodynamic properties of the model of DNAs we describe corresponding translation invariant Gibbs measures (TIGM) of the model on the Cayley tree of order two. We show that there are two critical temperatures \(T_{i, \mathrm c}\), \(i=1,2\) such that (i) If the temperature \(T> T_{1,\mathrm{c}}\) then there is at last one (TIGM), (ii) If \(T= T_{1,\mathrm c}\) then there are at least 4 TIGMs, (iii) If \(T_{2,\mathrm c}<T< T_{1,\mathrm c}\) then there are at least 7 TIGMs, (iv) If \(T=T_{2,\mathrm c}\) then there are at least 10 TIGMs, and (v) If \(T<T_{2,\mathrm c}\) then there are at least 13 TIGMs. Each such measure describes a phase of the set of DNAs. We use these results to study distributions of Holliday junctions of DNAs. In case of very low temperatures we give stationary distributions and typical configurations of the Holliday junctions.