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# Abstracts

## Dynamics and Phase Transitions for a Mutator Gene Model

Yakushkina T., Saakian D.B.2, Hu C.K.1

National Research University Higher School of Economics, Moscow, 101000, Myasnitskaya Str. 20, Russia, tyakushkina@hse.ru

1Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan

2A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Alikhanian Brothers St. 2,

3Yerevan 375036, Armenia

1 pp. (accepted)

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To analyze the mutator effect, we introduce the modifications of evolutionary models in Eigen and Crow-Kimura settings. This phenomenon is very important for understanding the evolution of cancer and RNA viruses \cite{lo10}. We formalize the evolutionary process on the basis of general Crow-Kimura model\cite{ck70}. In this study, we consider a genome, which conists of $(N+1)$ genes. Alleles are represented by $s_\tau = \pm 1, \tau = 0, \ldots, N$. The wild phenotype is defined by a genome with mutator gene with a state $s_0=1$, the mutant phenotype by $s_0=-1$. Consider the following system \begin{equation} \frac{dP_l(t)}{dt} = P_l(Nf(x)- N(\mu_1+\alpha_1)) + \mu_1(P_{l-1}(N-l+1) + P_{l+1}(l+1))+\alpha_2 Q_l N, \nonumber \end{equation} \begin{equation} \frac{dQ_l(t)}{dt} = Q_l(Ng(x) - N(\mu_2+\alpha_2)) + \mu_2(Q_{l-1}(N-l+1) + Q_{l+1}(l+1))+\alpha_1 P_l N,\nonumber \end{equation} where $P_l$ and $Q_l$ are the probabilities for the Hamming classes with $l$ mutations for the wild and mutant type respectively, $\alpha_i,\,i=1,2$ are mutations rates for the mutator gene, $\mu_i, i = 1,2$ are general mutation rates, $f(x)$ and $g(x)$ ($x=1-2l/N$) --- fitness functions. To investigate the model we use the Hamilton-Jacobi method proposed in\cite{sa07}.

In this work we provide analytical expressions for the maximum dynamics and for the mean fitness. We construct a phase portrait for the smooth fitness landscape. Fo linear, quadratic and random fitness functions we perform numerical analysis.

\begin{thebibliography}{100} \bibitem{lo10} \textit{ Fox E.\,J., Loeb L.\,A.} Lethal mutagenesis: targeting the mutator phenotype in cancer.\ --- Seminars in Cancer Biology, 2010, {\bf 20}, 353.

\bibitem{ck70} \textit{ Crow J.\,F. and Kimura M.} An Introduction to Population Genetics Theory.\ --- Harper Row, New York, 1970.

\bibitem{sa07} \textit{ Saakian D.\,B.} A new method for the solution of models of biological evolution: Derivation of exact steady-state distributions.\ --- J. Stat.\ Phys.\, 2007, {\bf 128}, 781. \end{thebibliography}

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