### Article

## The exact solution of the mutator model with directed mutations, linear fitness function, and finite genome length

In this study, we considered the model by Beckman and Loeb [Proc. Natl. Acad. Sci. U.S.A. 103 (2006) 14140] for the mutator phenomena. We construct an infinite population Crow-Kimura model with a mutator gene, directed mutations, a linear fitness function, and a finite genome length. We solved analytically the dynamics of the model using the generating function method. Such models provide realistic predictions for finite population sizes and have been widely discussed recently. The analytical formulas provided can be used to calculate the advantage of the mutator mechanism for the accumulation of mutations in cancer biology.

We propose a modification of the Crow-Kimura and Eigen models of biological molecular evolution to include a mutator gene that causes both an increase in the mutation rate and a change in the fitness landscape. This mutator effect relates to a wide range of biomedical problems. There are three possible phases: mutator phase, mixed phase and non-selective phase. We calculate the phase structure, the mean fitness and the fraction of the mutator allele in the population, which can be applied to describe cancer development and RNA viruses. We find that depending on the genome length, either the normal or the mutator allele dominates in the mixed phase. We analytically solve the model for a general fitness function. We conclude that the random fitness landscape is an appropriate choice for describing the observed mutator phenomenon in the case of a small fraction of mutators. It is shown that the increase in the mutation rates in the regular and the mutator parts of the genome should be set independently; only some combinations of these increases can push the complex biomedical system to the non-selective phase, potentially related to the eradication of tumors.

We investigate the dynamics of a molecular evolution model related to the mutator gene phenomenon. Here mutation in one gene drastically changes the properties of the whole genome. We investigated the Crow-Kimura version of the model, which can be mapped into a Hamilton-Jacobi equation. For the symmetric fitness landscape, we calculated the dynamics of the maximum of the total population distribution. We found two phases in the dynamics: a simple one when the maximum of distribution moves along a characteristics, and more involved one when the maximum jumps to another characteristic at some turnout point T.

Evolutionary games are used in various elds stretching from economics to biology. Most assume a constant payoff matrix, although some works also consider dynamic payoff matrices. In this article we propose a possibility of switching the system between two regimes with different sets of payoff matrices. Such a model can qualitatively describe the development of bacterial or cancer cells with a mutator gene present. A nite population evolutionary game is studied. The model describes the simplest version of annealed disorder in the payoff matrix and is exactly solvable. We analyze the dynamics of the model, and derive the equations for both the maximum and the variance of the distribution by applying the Hamilton-Jacobi formalism. Therefore, we present an exactly solvable version of an evolutionary game with annealed noise in the payoff matrix.

We construct an example of blow-up in a ”ow of min-plus linear operators arising as solution operators for a Hamilton…Jacobi equation @S/@t+|∇S|/ + U(x, t) = 0, where > 1 and the potential U(x, t) is uniformly bounded together with its gradient. The construction is based on the fact that, for a suitable potential de“ned on a time interval of length T, the absolute value of velocity for a Lagrangian minimizer can be as large as O(log T)2−2/. We also show that this growth estimate cannot be surpassed. Implications of this example for existence of global generalized solutions to randomly forced Hamilton…Jacobi or Burgers equations are discussed.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.