Loss of least-loaded class in asexual populations due to drift and epistasis

We consider the dynamics of a non-recombining haploid population of finite size which accumulates deleterious mutations irreversibly. This ratchet like process occurs at a finite speed in the absence of epistasis, but it has been suggested that synergistic epistasis can halt the ratchet. Using a diffusion theory, we find explicit analytical expressions for the typical time between successive clicks of the ratchet for both non-epistatic and epistatic fitness functions. Our calculations show that the inter-click time is of a scaling form which in the absence of epistasis gives a speed that is determined by size of the least-loaded class and the selection coefficient. With synergistic interactions, the ratchet speed is found to approach zero rapidly for arbitrary epistasis. Our analytical results are in good agreement with the numerical simulations.

💡 Research Summary

The paper presents a comprehensive theoretical investigation of Muller’s ratchet in a finite, non‑recombining haploid asexual population that accumulates irreversible deleterious mutations. The authors model the fitness of an individual carrying k mutations as W(k) = (1‑s)^{k^{α}}, where s is the selection coefficient and α quantifies epistatic interactions: α = 1 corresponds to the classic non‑epistatic (multiplicative) case, while α > 1 represents synergistic epistasis (mutations interact to produce a greater fitness loss than expected from independence).

First, the deterministic dynamics of an infinitely large population are described by a quasispecies equation. In steady state, the frequency X_J(k) of the class with J + k mutations is derived. For the non‑epistatic case (α = 1) the well‑known Poisson distribution X_J(k) = e^{-U/s}(U/s)^k/k! is recovered, and the size of the least‑loaded class is n_0 = N e^{-U/s}. The ratchet click rate is then governed solely by n_0 and s; larger n_0 or stronger selection slows the ratchet.

When synergistic epistasis is introduced (α > 1), the fitness landscape becomes increasingly steep with J. The authors obtain exact expressions for α = 2 using modified Bessel functions, and for general α > 1 they derive asymptotic approximations showing X_J ≈ exp