
Conference publicationsAbstractsXXIII conferenceDynamics and Phase Transitions for a Mutator Gene ModelNational Research University Higher School of Economics, Moscow, 101000, Myasnitskaya Str. 20, Russia, tyakushkina@hse.ru ^{1}Institute of Physics, Academia Sinica, Nankang, Taipei 11529, Taiwan ^{2}A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Alikhanian Brothers St. 2, ^{3}Yerevan 375036, Armenia
To analyze the mutator effect, we introduce the modifications of evolutionary models in Eigen and CrowKimura 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 CrowKimura 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_{l1}(Nl+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_{l1}(Nl+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 $g(x)$ ($x=12l/N$)  fitness functions. To investigate the model we use the HamiltonJacobi 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 steadystate distributions.\  J. Stat.\ Phys.\, 2007, {\bf 128}, 781. \end{thebibliography}
