# Tree-hierarchy of DNA and distribution of Holliday junctions

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## Abstract

We define a DNA as a sequence of \(\pm 1\)’s and embed it on a path of Cayley tree. Using group representation of the Cayley tree, we give a hierarchy of a countable set of DNAs each of which ’lives’ on the same Cayley tree. This hierarchy has property that each vertex of the Cayley tree belongs only to one of DNA. Then we give a model (energy, Hamiltonian) of this set of DNAs by an analogue of Ising model with three spin values (considered as DNA base pairs) 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 is a critical temperature \(T_{\mathrm{c}}\) such that (i) if temperature \(T>T_{\mathrm{c}}\) then there exists unique TIGM; (ii) if \(T=T_{\mathrm{c}}\) then there are two TIGMs; (iii) if \(T<T_{\mathrm{c}}\) then there are three TIGMs. Each such measure describes a phase of the set of DNAs. We use these results to study distributions of Holliday junctions and branches of DNAs. In case of very high and very low temperatures we give stationary distributions and typical configurations of the Holliday junctions.

## Keywords

DNA Holliday junction Temperature Cayley tree Gibbs measure## Mathematics Subject Classification

92D20 82B20 60J10 05C05## Notes

### Acknowledgements

The author thanks Professor L. Bogachev and Department of Statistics, School of Mathematics, University of Leeds, UK; Professor Y. Velenik and the Section of Mathematics, University Geneva, Switzerland; Professor M. Ladra and the Department of Algebra, University of Santiago de Compostela, Spain for financial support and kind hospitality during his visits to these universities. He also thanks both referees and Katharina Huber for their suggestions which were helpful to improve readability of the paper.

## References

- Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P (2002) Molecular biology of the cell, 4th edn. Garland Science, New YorkGoogle Scholar
- Carlon E (2008) Thermodynamics of DNA microarrays. Stochastic models in biological sciences, vol 80. Banach Center Publications, Polish Academy of Sciences, Warsaw, pp 229–233Google Scholar
- Gandolfo D, Rakhmatullaev MM, Rozikov UA, Ruiz J (2013) On free energies of the Ising model on the Cayley tree. J Stat Phys 150(6):1201–1217MathSciNetCrossRefzbMATHGoogle Scholar
- Georgii HO (2011) Gibbs measures and phase transitions. de Gruyter Studies in Mathematics, vol 9, 2nd edn. Walter de Gruyter, BerlinCrossRefGoogle Scholar
- Hohng S, Zhou R, Nahas MK, Yu J, Schulten K, Lilley DMJ, Ha T (2007) Fluorescence-force spectroscopy maps two-dimensional reaction landscape of the holliday junction. Science 318:279–283CrossRefGoogle Scholar
- Holliday R (1964) A mechanism for gene conversion in fungi. Genet Res 5:282–304CrossRefGoogle Scholar
- Khrennikov AYu (1997) Non-archimedean analysis: quantum paradoxes, dynamical systems and biological models. Kluwer Academic Publishers, DordrehtCrossRefzbMATHGoogle Scholar
- Lehnert S (2008) Biomolecular action of ionizing radiation. Series in Medical Physics and Biomedical Engineering. Taylor & Francis, Boca Raton, FL, USAGoogle Scholar
- Mandel M, Marmur J (1968) Use of ultraviolet absorbance-temperature profile for determining the guanine plus cytosine content of DNA. Methods Enzymol 12(2):198–206Google Scholar
- Malo J, Mitchell JC, Vénien-Bryan C, Harris JR, Wille H, Sherratt DJ, Turberfield AJ (2005) Engineering a 2D protein-DNA crystal. Angew Chem Int Ed 44:3057–3061CrossRefGoogle Scholar
- Percus JK (2002) Mathematics of genome analysis. Cambridge studies in mathematical biology, vol 17. Cambridge University Press, CambridgeGoogle Scholar
- Rozikov UA, Ishankulov FT (2010) Description of periodic \(p\)-harmonic functions on Cayley trees. Nonlinear Diff Equ Appl 17(2):153–160MathSciNetCrossRefzbMATHGoogle Scholar
- Rozikov UA (2013) Gibbs measures on Cayley trees. World Scientific Publishing, SingaporeCrossRefzbMATHGoogle Scholar
- Suhov YuM, Rozikov UA (2004) A hard-core model on a Cayley tree: an example of a loss network. Queueing Syst 46(1/2):197–212MathSciNetCrossRefzbMATHGoogle Scholar
- Swigon D (2009) The mathematics of DNA structure. Mech Dyn IMA Vol Math Appl 150:293–320MathSciNetGoogle Scholar
- Tanaka F, Kameda A, Yamamoto M, Ohuchi A (2004) Nearest-neighbor thermodynamics of DNA sequences with single bulge loop. DNA computing, Lecture notes in computer science, vol 2943. Springer, Berlin, pp 170–179Google Scholar
- Thompson C (1972) Mathematical statistical mechanics. Princeton University Press, Princeton, NJGoogle Scholar
- Watson J, Hays FA, Ho PS (2004) Definitions and analysis of DNA Holliday junction geometry. Nucleic Acids Res 32:3017–3027CrossRefGoogle Scholar
- Xie P (2007) Model for RuvAB-mediated branch migration of Holliday junctions. J Theor Biol 249:566–573MathSciNetCrossRefGoogle Scholar