The Expected Fitness Cost of a Mutation Fixation under the One-dimensional Fisher Model

Abstract

This paper employs Fisher’s model of adaptation to understand the expected fitness effect of fixing a mutation in a natural population. Fisher’s model in one dimension admits a closed form solution for this expected fitness effect. A combination of different parameters, including the distribution of mutation lengths, population sizes, and the initial state that the population is in, are examined to see how they affect the expected fitness effect of state transitions. The results show that the expected fitness change due to the fixation of a mutation is always positive, regardless of the distributional shapes of mutation lengths, effective population sizes, and the initial state that the population is in. The further away the initial state of a population is from the optimal state, the slower the population returns to the optimal state. Effective population size (except when very small) has little effect on the expected fitness change due to mutation fixation. The always positive expected fitness change suggests that small populations may not necessarily be doomed due to the runaway process of fixation of deleterious mutations.