Rate of evolution of a quantitative character

The rate of change of the mean of a quantitative character is evaluated exactly and also under the hypothesis of linear biparental regression. Generations are discrete and nonoverlapping; the monoecious population mates at random. The genotypic and environmental contributions to the character are additive and stochastically independent. The character is influenced by arbitrarily many multiallelic loci and has constant genotypic values; dominance, epistasis, and the linkage map are also arbitrary. The population is initially in linkage equilibrium, and there is no position effect. If the biparental regression is linear, then the regression coefficient is simply Vgam/V, and hence the single-generation change in the mean is deltaZ = (Vgam/V)S, where Vgam and V denote the gametic and total variances in the character and S designates the selection differential. The corresponding exact result is DeltaZ = (C/W) + (B/W2), where C, W and B represent the gametic covariance of the character and fitness, the mean fitness, and a correction term, respectively. If selection is weak, then DeltaZ approximately C/W. Furthermore, deltaZ = C/W if either there is no environmental contribution and the gametic effects are additive or the character is fitness itself. In the latter case, C is the gametic variance in fitness. Thus, even in linkage equilibrium, weakness of selection generally does not suffice to validate the linear-regression result. This conclusion holds even for additive loci.