The Annals of Applied Probability

Muller’s ratchet with compensatory mutations

P. Pfaffelhuber, P. R. Staab, and A. Wakolbinger

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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) Muller’s ratchet, the case $\gamma>0$ 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 $\gamma>0$, the unique equilibrium state is the Poisson distribution with parameter $\lambda/(\gamma+\alpha)$.

Article information

Ann. Appl. Probab. Volume 22, Number 5 (2012), 2108-2132.

First available in Project Euclid: 12 October 2012

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Zentralblatt MATH identifier

Primary: 92D15: Problems related to evolution 92D15: Problems related to evolution 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60H35: Computational methods for stochastic equations [See also 65C30] 60H10: Stochastic ordinary differential equations [See also 34F05]

Muller’s ratchet selection back mutation Fleming–Viot process Girsanov transform


Pfaffelhuber, P.; Staab, P. R.; Wakolbinger, A. Muller’s ratchet with compensatory mutations. Ann. Appl. Probab. 22 (2012), no. 5, 2108--2132. doi:10.1214/11-AAP836.

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