Survival of the frequent at finite population size and mutation rate: bridging the gap between quasispecies and monomorphic regimes with a simple model

In recent years, there has been increased attention on the non-trivial role that genotype-phenotype maps play in the course of evolution, where natural selection acts on phenotypes, but variation arises at the level of mutations. Understanding such mappings is arguably the next missing piece in a fully predictive theory of evolution. Although there are theoretical descriptions of such mappings for the monomorphic (N << 1) and deterministic or very strong mutation (N [&gt;&gt;&gt;] 1) limit, given by developments of Iwasas free fitness and quasispecies theories, respectively, there is no general description for the intermediate regime where N ~ 1. In this paper, we address this by transforming Wrights well-known stationary distribution of genotypes under selection and mutation to give the probability distribution of phenotypes, assuming a general genotype-phenotype map. The resultant distribution shows that the degeneracies of each phenotype appear by weighting the mutation term; this gives rise to a bias towards phenotypes of larger degeneracy analogous to quasispecies theory, but at finite population size. On the other hand we show that as population size is decreased, again phenotypes of higher degeneracy are favoured, which is a finite mutation description of the effect of sequence entropy in the monomorphic limit. We also for the first time (to the authors knowledge) provide an explicit derivation of Wrights stationary distribution of the frequencies of multiple alleles.