Growth of deleterious mutant genes in a large population

Formulae for the arrival probability and first arrival time for a single mutant gene to reach a certain number have been obtained by using a continuous branching process. If the mean of the progeny number of heterozygotes is less than one the arrival probability increases with increasing variance of the progeny distribution whereas if the mean is greater than one the contrary is true. Since most human populations are growing at fairly high rates, the result indicates that the probability for a single mutation to grow to a large number is quite high. Numerical computations show that the mean first arrival time decreases with increasing variance of the progeny distribution of the mutant carriers. The results have been applied to investigate the case of the acheiropodia gene in Brazil. The hypothesis that all acheiropodia genes in Brazil were derived from a single mutation seems to be tenable.