Quasi-Equilibria and Click Times for a Variant of Muller's Ratchet

Abstract

Consider a population of N individuals, each of them carrying a type in N0. The population evolves according to a Moran dynamics with selection and mutation, where an individual of type k has the same selective advantage over all individuals with type k′>k, and type k mutates to type k+1 at a constant rate. This model is thus a variation of the classical Muller's ratchet: there the selective advantage is proportional to k′−k. For a regime of selection strength and mutation rates which is between the regimes of weak and strong selection/mutation, we obtain the asymptotic rate of the click times of the ratchet (i.e. the times at which the hitherto minimal ('best') type in the population is lost), and reveal the quasi-stationary type frequency profile between clicks. The large population limit of this profile is characterized as the normalized attractor of a "dual" hierarchical multitype logistic system, and also via the distribution of the final minimal displacement in a branching random walk with one-sided steps. An important role in the proofs is played by a graphical representation of the model, both forward and backward in time, and a central tool is the ancestral selection graph decorated by mutations.