Survival Probabilities of Mutant Alleles in Fine-Grained Environments

Abstract

A stochastic environment is said to be fine grained if the environmental heterogeneity is experienced by an individual within its own lifetime such that the average conditions are identical for all other individuals and for all generations. Because of the constancy of the average conditions, constant fitness models have been applied to fine-grained environments. If a large number of individuals independently sample the environmental heterogeneity, the realized sampling differences between individuals average out in the population as a whole. Two conditions under which this argument breaks down are investigated. The 1st condition is that for a newly arisen mutant allele the number of bearers of this mutation is very small initially even if the total population size is effectively infinite. Hence, small numbers prevent the "averaging out" of random fluctuations in fitness during the critical early generations that are known to strongly influence the ultimate fate of most mutations. Consequently, the impact of fine-grained heterogeneity on a newly arisen mutant is investigated for an infinite sized population. Such fine-grained heterogeneity usually decreases the chance of survival of an allele with respect to the comparable constant fitness case. The 2nd condition under which the averaging out of individual fitness differences do not occur is when the total population size itself is small. A diffusion approximation is used to investigate the probability of survival of an allele and mean time to fixation or loss starting from an arbitrary initial frequency in a finite-sized, haploid population. Fine-grained heterogeneity has its greatest impact when the initial allele frequency is small, and gradually converges to the constant fitness case as the number of copies of the allele present in the population increases. As in the infinite sized models, fine-grained heterogeneity usually decreases the chance of survival of an allele, but can actually increase the probability of survival of an allele that is deleterious of the average in a finite population.