MATHEMATICS

Gibbs measures of bubble coalescence in an interacting system of DNA molecules for the Ising–SOS model on a Cayley tree

U. A. Rozikov^abc, N. M. Khatamov^da, N. N. Malikov<sup>d

a</sup> Institute of Mathematics named after V. I. Romanovsky, Academy of Sciences of the Republic of Uzbekistan, ul. Universitetskaya, 9, Tashkent, 100174, Uzbekistan
^b National University of Uzbekistan named after Mirzo Ulugbek, ul. Universitetskaya, 4, Tashkent, 100174, Uzbekistan
^c Karshi State University, ul. Kuchabag, 17, Karshi, 180119, Uzbekistan
^d Namangan State University, ul. Uichi, 316, Namangan, 160119, Uzbekistan

Abstract: In this paper, two models of interacting DNA molecules are considered. The first is a (four-parameter) bubble coalescence model in interacting DNAs (shortly, BCI–DNA). The second is a (three-parameter) bubble coalescence model in a condensed DNA molecules (shortly, BCC–DNA). To study the thermodynamics of bubble fusion of these models, a method of statistical physics is developed. Namely, the Hamiltonian (defined by functions) of each model is determined and for specific functions of the Hamiltonian, their translation-invariant Gibbs measures (TIGM) are given.
In this work, such Hamiltonian functions are chosen that the model has the form of the Ising–SOS model. In this case, for the BCI–DNA model, such parameters are found that the corresponding Hamiltonian has up to three TIGMs (three phases of the system), which biologically means the existence of three states: “No bubble coalescence”, “Dominated soft zone”, “Bubble coalescence”.
For the BCC–DNA model, it is shown that for any (acceptable) parameters, this model also has up to three TIGMs, which biologically means the existence of three states: “No bubble coalescence”, “Dominated soft zone”, “Bubble coalescence”.

Keywords: DNA, bubble, configuration, Cayley tree, Gibbs measure, Ising–SOS model

UDC: 517.98

MSC: 82B05, 82B20, 60K35

Received: 24.10.2024
Accepted: 05.01.2025

DOI: 10.35634/vm250106

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