Conference publications


XXIII conference

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,

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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