Muller's ratchet with compensatory mutations

Details Muller's ratchet with compensatory mutations Muller's ratchet with compensatory mutations

2011-08-19

arxiv.org/abs/1108.4059v2

Mathematics

Probability

28 pages, 3 figures

Scientific paper

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 $\alpha k$ for some $\alpha>0$. Along the individual lines of descent, new mutations accumulate at rate $\lambda$ per generation, and each of these mutations has a probability $\gamma$ per generation to disappear. While the case $\gamma =0 $ is known as (the Fleming--Viot version of) {\em Muller's ratchet}, the case $\gamma > 0$ is associated with {\em 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 (that is 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 $\gamma >0$ the unique equilibrium state is the Poisson distribution with parameter $\lambda/(\gamma + \alpha)$.

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