A stochastic model concerning the maintenance of genetic variability in quantitative characters.

The mechanism by which genetic variability is maintained in natural populations for quantitative characters is not well understood. Many observations show that there is a considerable amount of genetic variability in a large population, and unless the character is closely correlated with fitness the optimum is usually near the mean, with fitness decreasing as the distance from the mean increases. Probably, Fisher' was the first to investigate a model in which the fitness was assumed to decrease in proportion to the squared deviation from the optimum. This model was also used by Haldane2 and Wright.3 Robertson4 has shown that if genes are maintained by overdominance, but act additively with respect to a quantitative character, then the optimum is at the mean and the decrease of fitness is proportional to the squared deviation from the optimum. In all these treatments the relation between the mutation rate and the amount of genetic variability maintained is either ambiguous or left out of consideration. The purpose of this paper is to propose a new model which enables one to make predictions about the relations between mutation rate, genotypic variance, and genetic load or amount of selective elimination involved in the maintenance of genetic variability. Assumptions and Mathematical Formulation.-The basic assumptions are: (1) At every locus involved with the quantitative character under discussion, mutation can produce an infinite sequence of alleles. Every mutation may produce a new allele different from the pre-existing ones. (2) The effect of a new allele on the quantitative character is only slightly different from the parent allele from which it was derived by a single mutational step. (3) The genes are additive with respect to their effect on the quantitative character. (4) The optimum phenotype is fixed, and fitness decreases in proportion to the squared deviation from the optimum. Consider a particular locus. We denote by x the average effect of an allele on the quantitative character, taking the optimum as the origin. We assume that by mutation an allele having an average effect x changes to another allele with an average effect x + t with probability density given by f(s). If we denote by p(x,t) the relative frequency of the alleles having an average effect x in a large population at time t (measured in generations), then the rate of change in p per generation is given by the sum of the following two components: (a) Change due to mutation: If JA is the mutation rate per gene per generation (assumed to be constant), the contribution to bp/bt by mutation is