Equilibrium Distribution of Mutators in the Single Fitness Peak Model

TL;DR

This paper presents an analytical model for the equilibrium distribution of mutator strains in unicellular populations, revealing a phase transition known as the repair catastrophe at a critical error probability.

Contribution

It introduces a tractable model based on the single fitness peak framework to analyze mutator equilibrium and identifies a phase transition in the mutation dynamics.

Findings

Identifies a phase transition at a critical repair error probability.

Provides a quantitative estimate for mutator fraction in E. coli.

Defines the conditions for the repair catastrophe in mutation rates.

Abstract

This paper develops an analytically tractable model for determining the equilibrium distribution of mismatch repair deficient strains in unicellular populations. The approach is based on the single fitness peak (SFP) model, which has been used in Eigen's quasispecies equations in order to understand various aspects of evolutionary dynamics. As with the quasispecies model, our model for mutator-nonmutator equilibrium undergoes a phase transition in the limit of infinite sequence length. This "repair catastrophe" occurs at a critical repair error probability of $ \epsilon_r = L_{via}/L $, where $ L_{via} $ denotes the length of the genome controlling viability, while $ L $ denotes the overall length of the genome. The repair catastrophe therefore occurs when the repair error probability exceeds the fraction of deleterious mutations. Our model also gives a quantitative estimate for the equilibrium fraction of mutators in {\it Escherichia coli}.