Geodesic
Thermodynamics of interacting systems of DNA molecules
Teoretičeskaâ i matematičeskaâ fizika, Tome 206 (2021) no. 2, pp. 199-209 Cet article a éte moissonné depuis la source Math-Net.Ru

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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, 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.
Keywords: DNA, temperature, Cayley tree, Gibbs measure. 
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     title = {Thermodynamics of interacting systems of {DNA} molecules},
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U. A. Rozikov. Thermodynamics of interacting systems of DNA molecules. Teoretičeskaâ i matematičeskaâ fizika, Tome 206 (2021) no. 2, pp. 199-209. http://geodesic.mathdoc.fr/item/TMF_2021_206_2_a4/

[1] B. Alberts, A. Dzhonson, D. Lyuis, M. Reff, K. Roberts, P. Uolter, Molekulyarnaya biologiya kletki, NITs “Regulyarnaya i khaotichnaya dinamika”, M.–Izhevsk, 2013

[2] D. Swigon, “The mathematics of DNA structure, mechanics, and dynamics”, Mathematics of DNA Structure, Function and Interactions, The IMA Volumes in Mathematics and its Applications, 150, Springer, New York, 2009, 293–320 | DOI | MR

[3] C. Thompson, Mathematical Statistical Mechanics, Princeton Univ. Press, Princeton, 1972 | MR

[4] U. A. Rozikov, “Tree-hierarchy of DNA and distribution of Holliday junctions”, J. Math. Biol., 75:6–7 (2017), 1715–1733 | DOI | MR

[5] U. A. Rozikov, “Holliday junctions for the Potts model of DNA”, Algebra, Complex Analysis, and Pluripotential Theory (Urgench, Uzbekistan, August 8–12, 2017), Springer Proceedings in Mathematics and Statistics, 264, eds. Z. Ibragimov, N. Levenberg, U. Rozikov, A. Sadullaev, Springer, Cham, 2018, 151–165 | DOI | MR | Zbl

[6] U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore, 2013 | MR

[7] U. A. Rozikov, F. T. Ishankulov, “Description of periodic $p$-harmonic functions on Cayley trees”, Nonlinear Differ. Equ. Appl., 17:2 (2010), 153–160 | DOI | MR

[8] Kh.-O. Georgi, Gibbsovskie mery i fazovye perekhody, Mir, M., 1992 | MR | Zbl

[9] L. V. Bogachev, U. A. Rozikov, “On the uniqueness of Gibbs measure in the Potts model on a Cayley tree with external field”, J. Stat. Mech. Theory Exp., 2019:7 (2019), 073205, 76 pp. | DOI | MR

[10] C. Külske, U. A. Rozikov, R. M. Khakimov, “Description of all translation-invariant splitting Gibbs measures for the Potts model on a Cayley tree”, J. Stat. Phys., 156:1 (2014), 189–200, arXiv: 1310.6220 | DOI | MR | Zbl

[11] C. Külske, U. A. Rozikov, “Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree”, Random Structures Algorithms, 50:4 (2017), 636–678, arXiv: 1403.5775 | DOI | MR