The first arrival time and mean age of a deleterious mutant gene in a finite population

The mean and standard deviation of the first arrival time for a single mutant to reach a certain frequency and the mean age of a mutant persisting in a population have been studied using diffusion methods. These quantities are shown to be highly dependent on both the heterozygous effect and the population size. For partially recessive deleterious mutations, both the mean first arrival time and the mean age decrease with increasing selection coefficient against heterozygotes. For overdominant mutations, the mean age always increases very rapidly with increasing heterozygous advantage, while the mean first arrival time first increases rapidly with increasing heterozygous advantage to a maximum and then decreases rapidly with increasing heterozygous advantage. The standard deviation of the first arrival time is small while that of the age is large. The results of this study have been applied to study the case of the sickle cell anemia mutant in Africa. It is argued that the present prevalence may be explained without the necessity of quite so great a heterozygous advantage as .25 or higher as proposed by some workers. A reasonable range for the heterozygous advantage seems to be from .05 to .18.