# 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.

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