najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2021-12-5-563-568najo-509Research ArticlePHYSICSФИЗИКАHolliday junctions in the HC Blume–Capel model in “one case” on DNAСоединения Холлидея в модели HC Blume-Capel в «одном случае» на ДНКKhatamovN. M.KhatamovN. M.

4B, St. University, 100174, Tashkent

nxatamov@mail.ruInstitute of mathematicsUzbekistan202105082025125563568Copyright © Khatamov N.M., 20252025Khatamov N.M.Khatamov N.M.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/509

We consider a DNA as a configuration of HC Blume–Capel model and embed it on a path of Cayley tree. To study thermodynamic properties of the model of DNAs, we describe the corresponding translation-invariant Gibbs measures (TIGM) of the model on the Cayley tree. It is shown that, for k ≥ 2, for any temperature T > 0 there is a unique TIGM. Using these results, we study the distributions of the Holliday junctions DNA. For very high and very low temperatures, we give stationary distributions and typical configurations of the Holliday junctions.

Мы рассматриваем ДНК как конфигурацию модели HC Blume–Capel и встраиваем ее в путь дерева Кэли. Для изучения термодинамических свойств модели ДНК мы опишем соответствующие трансляционно-инвариантные меры Гиббса (TIGM) модели на дереве Кэли. Показано, что при k >= 2 для любой температуры T > 0 существует единственный ТИГМ. Используя эти результаты, мы изучаем распределения ДНК контактов Холлидея. Для очень высоких и очень низких температур мы приводим стационарные распределения и типичные конфигурации контактов Холлидея.

ДНКдерево Кэлимодель Блюма–Капелямера Гиббсапереход ХоллидеяDNACayley treeBlume–Capel modelGibbs measureHolliday junctionThe author thanks Professor U. A. Rozikov for his suggestions which were helpful to improve the readability of the paper.ReferencesAlberts B., Johnson A., et al. Molecular biology of the cell, 4th edn., Garland Science, New York, 2002.Alberts B., Johnson A., et al. Molecular biology of the cell, 4th edn., Garland Science, New York, 2002.Holliday R. A mechanism for gene conversion in fungi. Genet Res., 1964, 5, P. 282–304.Holliday R. A mechanism for gene conversion in fungi. Genet Res., 1964, 5, P. 282–304.Khatamov N.M. Holliday junctions in the Blume–Capel model of DNA. Theoretical and Mathematical Physics, 2021, 3 (206), P. 383–390.Khatamov N.M. Holliday junctions in the Blume–Capel model of DNA. Theoretical and Mathematical Physics, 2021, 3 (206), P. 383–390.Rozikov U.A. Tree-hierarchy of DNA and distribution of Holliday junctions. J. Math. Biol., 2017, 75, P. 1715–1733.Rozikov U.A. Tree-hierarchy of DNA and distribution of Holliday junctions. J. Math. Biol., 2017, 75, P. 1715–1733.Rozikov U.A. Holliday junctions for the Potts model of DNA. Algebra, Complex Analysis, and Pluripotential Theory, 2018, 264, P. 151–165.Rozikov U.A. Holliday junctions for the Potts model of DNA. Algebra, Complex Analysis, and Pluripotential Theory, 2018, 264, P. 151–165.Rozikov U.A. Thermodynamics of interacting systems of DNA molecules. Theoretical and Mathematical Physics, 2021, 2 (206), P. 174–183.Rozikov U.A. Thermodynamics of interacting systems of DNA molecules. Theoretical and Mathematical Physics, 2021, 2 (206), P. 174–183.Rozikov U.A. Thermodynamics of DNA-RNA renaturation. Inter. Jour. Geom. Methods Mod. Phys., 2021, 6 (18), 2150096.Rozikov U.A. Thermodynamics of DNA-RNA renaturation. Inter. Jour. Geom. Methods Mod. Phys., 2021, 6 (18), 2150096.Cirillo E.N., Olivieri E. Metastabilty and nucleation for the Blume–Capel model.Different mechanisms of transition. Journal of Statistical Physics, 1996, 83, P. 473–554.Cirillo E.N., Olivieri E. Metastabilty and nucleation for the Blume–Capel model.Different mechanisms of transition. Journal of Statistical Physics, 1996, 83, P. 473–554.Khatamov N.M., Khakimov R.M. Translation-invariant Gibbs measures for the Blume–Capel model on a Cayley tree. JMAG, 2019, 2 (15), P. 239–255.Khatamov N.M., Khakimov R.M. Translation-invariant Gibbs measures for the Blume–Capel model on a Cayley tree. JMAG, 2019, 2 (15), P. 239–255.Khatamov N.M. Translation-invariant extreme Gibbs measures for the Blume–Capel model with a wand on a Cayley tree. Ukrains’kyi Matematychnyi Zhurnal, 2020, 4 (72), P. 540–556.Khatamov N.M. Translation-invariant extreme Gibbs measures for the Blume–Capel model with a wand on a Cayley tree. Ukrains’kyi Matematychnyi Zhurnal, 2020, 4 (72), P. 540–556.Rozikov U.A., Ishankulov F.T. Description of periodic p-harmonic functions on Cayley trees. Nonlinear Diff. Equ. Appl., 2010, 2 (17), P. 153– 160.Rozikov U.A., Ishankulov F.T. Description of periodic p-harmonic functions on Cayley trees. Nonlinear Diff. Equ. Appl., 2010, 2 (17), P. 153– 160.Rozikov U.A. Gibbs measures on Cayley trees. World Scientific Publishing, Singapore, 2013.Rozikov U.A. Gibbs measures on Cayley trees. World Scientific Publishing, Singapore, 2013.Georgii H.O. Gibbs measures and phase transitions. De Gruyter stadies in Math, Berlin, 1988.Georgii H.O. Gibbs measures and phase transitions. De Gruyter stadies in Math, Berlin, 1988.

The authors declare that there are no conflicts of interest present.