A small noise approximation for Muller's Ratchet

Abstract

We consider an infinite system of SDEs with Fleming-Viot noise indexed by $k=0,1,2,\dots$, whose parameters $\hat{I}\pm ,\hat{I}»$, and $\hat{I}½$ are the (deleterious) selection coefficient, the (uni-directional) mutation rate, and a quantity which determines the size of the system's fluctuations. The SDE's unique weak solution $X(t)=(X_k(t){)}_{k=0,1,2,...}$ models what is known in population genetics as Muller's ratchet. Here, $X_k(t)$ stands for the frequency of individuals carrying $k$ deleterious mutations. Since the mutation process is uni-directional, $t\mapsto \inf \{k:X_k(t)>0\}$ is non-decreasing for almost every path of $X$, and we refer to an increase as a click of Muller's ratchet. A long standing question concerns the clicking rate of Muller's ratchet. Using Duhamel's principle for semigroups, we give a partial answer by approximating $E({\sum }_{k=1}^{\infty }k{X}_k(t))$ and $E(X_0(t))$ up to $O(1/\hat{I}{½}^2)$ for fixed $\hat{I}\pm$, $\hat{I}»$ and $t>0$. Our results suggest that $\ddot{I}ˆ:=\hat{I}½\hat{I}\pm e^{-\hat{I}»/\hat{I}\pm }$ is a crucial quantity also when the mutation/selection ratio $\hat{I}¸=\hat{I}»/\hat{I}\pm$ is moderately large: for large $\hat{I}½\hat{I}\pm$, clicking of the ratchet on the time scale $\frac{1}{\hat{I}}\pm \log \hat{I}¸$ becomes rare as soon as $\ddot{I}ˆ$ becomes large.

28 pages, 3 figures, 1 table