A small noise approximation for Muller's Ratchet

We consider an infinite system of SDEs with Fleming-Viot noise indexed by k = 0, 1, 2, … whose parameters alpha, lambda, and nu 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)) 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 maps to inf{k: X_k(t) > 0} and 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 from k = 1 to infinity of kX_k(t)) and E(X_0(t)) up to O(1/nu^2) for fixed alpha, lambda and t > 0. Our results suggest that psi := nu alpha exp(-lambda/alpha) is a crucial quantity also when the mutation/selection ratio theta = lambda/alpha is moderately large: for large nu alpha, clicking of the ratchet on the time scale (1/alpha) log theta becomes rare as soon as psi becomes large.