Mutation-selection models of sequence evolution in population genetics

Abstract

The equilibrium properties of a number of deterministic mutation- selection models of sequence evolution are investigated. Both two- and four-state sequences are considered, the mutation model is a single-step mutation model. Two types of fitness functions are studied, namely permutation-invariant fitness functions, where the fitness of a sequence depends only on the number of mutations, not on their location within the sequence, and Hopfield-type fitness functions, where the fitness of a sequence is determined by its similarity to a number of predefined patterns. Maximum principles to determine the population mean fitness in equilibrium are derived, where the maximiser gives also the ancestral mean genotype. These maximum principles are used to investigate the error threshold phenomenon, i.e., the phenomenon that for certain fitness functions the population changes at a critical mutation rate from a well localised to a delocalised distribution in sequence space. The error threshold phenomenon is investigated for a four-state model with permutation-invariant fitness functions and for a two-state model with Hopfield-type fitness functions. Both models yield ordered and disordered as well as partially ordered phases