The rich phase structure of a mutator model

Abstract

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.

Evolutionary Games with Randomly Changing Payoff Matrices

“…1 to represent alleles with normal mutation rate and the lower chain to represent alleles with higher mutation rate. One can use Crow-Kimura model [43][44][45] on such chains to calculate the phase diagram of cancer 48) and dynamic behavior of a mutator gene model. 49) ( ¼ A; B; m ¼ AE1) in our HJE.…”

A solution of the Crow-Kimura evolution model on fluctuating fitness landscape

“…somehow the mutator phenomenon, where the mutation in a specific gene in the genome results in a drastic change of the mutation rates of other genes or their fitness landscape [32], so we again have an evolution with two fitness landscapes. In the case of the mutator, there is a wellformulated mathematical model with a system of 2(L + 1) equations (where L is the genome length) [32].…”

“…However, the accurate consideration of evolution on fluctuating fitness landscapes requires a complicated system of functional equations, it is a much harder mathematical tool than the one used in the mutator model [32]. The significant difference between the two phenomena lies in the fact that we have macroscopic transition rates for the mutator model, while our model of evolution on fluctuating fitness landscapes assumes stochastic transitions.…”