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October 2012 Muller’s ratchet with compensatory mutations
P. Pfaffelhuber, P. R. Staab, A. Wakolbinger
Ann. Appl. Probab. 22(5): 2108-2132 (October 2012). DOI: 10.1214/11-AAP836


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 $\lambda/(\gamma+\alpha)$.


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P. Pfaffelhuber. P. R. Staab. A. Wakolbinger. "Muller’s ratchet with compensatory mutations." Ann. Appl. Probab. 22 (5) 2108 - 2132, October 2012.


Published: October 2012
First available in Project Euclid: 12 October 2012

zbMATH: 1251.92035
MathSciNet: MR3025691
Digital Object Identifier: 10.1214/11-AAP836

Primary: 60J70, 92D15, 92D15
Secondary: 60H10, 60H35, 60K35

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.22 • No. 5 • October 2012
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