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 $α,λ$, and $ν$ 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 $E(\sum_{k=1}^\infty kX_k(t) )$ and $E\big(X_0(t)\big)$ up to $O(1/ν^2)$ for fixed $α$, $λ$ and $t>0$. Our results suggest that $ψ:=ναe^{-λ/α}$ is a crucial quantity also when the mutation/selection ratio $θ= λ/α$ is moderately large: for large $να$, clicking of the ratchet on the time scale $\frac 1α\log θ$ becomes rare as soon as $ψ$ becomes large.