Muller's ratchet with compensatory mutations
arXiv:1108.4059 · doi:10.1214/11-AAP836
Abstract
We consider an infinite-dimensional system of stochastic differential equations describing the evolution of type frequencies in a large population. The type of an individual is the number of deleterious mutations it carries, where fitness of individuals carrying k mutations is decreased by αk for some α>0. Along the individual lines of descent, new mutations accumulate at rate λper generation, and each of these mutations has a probability γper generation to disappear. While the case γ=0 is known as (the Fleming-Viot version of) Muller's ratchet, the case γ>0 is associated with compensatory mutations in the biological literature. We show that the system has a unique weak solution. In the absence of random fluctuations in type frequencies (i.e., for the so-called infinite population limit) we obtain the solution in a closed form by analyzing a probabilistic particle system and show that for γ>0, the unique equilibrium state is the Poisson distribution with parameter λ/(γ+α).
Published in at http://dx.doi.org/10.1214/11-AAP836 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)