The mutator model with asymmetric transitions

We investigate the mutator model for the asymmetric transition rates between the wild-type and mutator type. When the mutator gene changes its type, both mutation rate of genome and fitness landscape are changed. We look at smooth symmetric fitness landscapes both for the wild type (normal allele of special gene) and mutator type (mutator allele). In some cases involving small degree of transition rates asymmetry (or not too large genome length) where we have smooth large genome length limit, the large system of ODE can be replaced by Hamilton–Jacobi equation. We derive here the analytical results for the mean fitness and population distribution in steady state including the finite size corrections. We have observed that some interesting oscillations arise involving longer genomes, and we cannot map the large system of ODE to the single partial differential equation HJE using a simple ansatz. We assume that the found counter-intuitive phenomenon should exist for evolution on fluctuating landscapes also, for asymmetric transition rates. Actually, the asymmetry of transition (mutation) rates is an important characteristics of evolutionary dynamics.