Holliday Junctions for the Potts Model of DNA

  • Utkir RozikovEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 264)


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.


Cayley tree Potts model Gibbs measure Holliday junction 



The author thanks Prof. M.V.Velasco and the Department of Mathematical Analysis, University of Granada, Spain for financial support and kind hospitality during his visit to the university. He thanks referees for helpful comments.


  1. 1.
    Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., Walter, P.: Molecular Biology of the Cell, 4th edn. Garland Science, New York (2002)Google Scholar
  2. 2.
    Friedli, S., Velenik, Y.: Statistical Mechanics of Lattice Systems. A Concrete Mathematical Introduction. Cambridge University Press, Cambridge (2018)zbMATHGoogle Scholar
  3. 3.
    Gandolfo, D., Rahmatullaev, M.M., Rozikov, U.A.: Boundary conditions for translation-invariant Gibbs measures of the Potts model on Cayley trees. J. Stat. Phys. 167(5), 1164–1179 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Georgii, H.O.: Gibbs Measures and Phase Transitions. 2nd edn. de Gruyter Studies in Mathematics, 9. Walter de Gruyter, Berlin (2011)Google Scholar
  5. 5.
    Holliday, R.: A mechanism for gene conversion in fungi. Genet. Res. 5, 282–304 (1964)CrossRefGoogle Scholar
  6. 6.
    Külske, C., Rozikov, U.A., Khakimov, R.M.: Description of the translation-invariant splitting Gibbs measures for the Potts model on a Cayley tree. J. Stat. Phys. 156(1), 189–200 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Külske, C., Rozikov, U.A.: Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree. Random Struct. Algorithms 50(4), 636–678 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Rozikov, U.A.: Gibbs Measures on Cayley Trees. World Scientific Publishing, Singapore (2013)CrossRefGoogle Scholar
  9. 9.
    Rozikov, U.A.: Tree-hierarchy of DNA and distribution of Holliday junctions. J. Math. Biol. 75(6–7), 1715–1733 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Rozikov, U.A., Ishankulov, F.T.: Description of periodic \(p\)-harmonic functions on Cayley trees. Nonlinear Diff. Equ. Appl. 17(2), 153–160 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rozikov, U.A., Rahmatullaev, M.M.: On free energies of the Potts model on the Cayley tree. Theor. Math. Phys. 190(1), 98–108 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Swigon, D.: The Mathematics of DNA Structure, Mechanics, and Dynamics, IMA. Volumes in Mathematics and Its Applications, vol. 150, pp. 293–320 (2009)Google Scholar
  13. 13.
    Thompson, C.: Mathematical Statistical Mechanics. Princeton University Press, Princeton (1972)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute of MathematicsTashkentUzbekistan

Personalised recommendations