Title:

The validity of the heritability concept in quantitative genetics

A substantial positive curvature was discovered in offspringparent regression for a bristle number in a laboratory population of Drosophila melanogaster. This initiated a theoretical investigation of the restrictions on the classical prediction equation in quantitative genetics. The main interest was in models of genetic and environmental variation which lead to asymmetrical responses in opposite directions in the first generation of selection, that is to nonlinearity in offspringparent regression. Nonlinearity was assessed by fitting a quadratic regression in an infinite random mating population. A genetic model with a small number of loci, each with an arbitrary number of alleles, was used. The effects of dominance, multiplicative interaction and unequality of loci were studied. When there are no environmental deviations, apart from the case of complete dominance, single parent and midparent regression were found to show similar curvature, so that in general the offspringparent regression between genotypic values has largest departures from linearity when there are rare, almost completely recessive alleles segregating at equal loci or in an analogous way directional recessivity and low averaged gene frequency over unequal loci. When the number of alleles is increased nonlinearity decreases. For the number of loci making equal contributions to the variation the amount of nonlinearity is roughly proportional to the number of loci. To study the effect of an additive independent environmental deviations various distributions for them were used. It was shown that when the offspringparent regression between genotypic values is linear and H1 is the ratio of genotypic to phenotypic variance, the regression of offspring on parental phenotype is linear only if the skewness of the environmental distribution is a proportion JH'/(tH') of that of the genotypic distribution. The more the skewnesses depart from this equality or smaller H2 is, the larger the departures from linearity are. The genotypic nonlinearity shows up only if H2 is very large. When the environmental deviations have a normal distribution, the largest departures from linearity are expected when there are rare and completely recessive alleles segregating at loci with large contributions. Models with dependence between genotypic and environmental distributions were also studied. Multiplicative interaction as such was shown to make only small contributions to nonlinearity, its effect being more substantial when there is a locus with a very large effect acting in a genetic background due to a very large number of loci with small effects. The use of Abplanalp's linear heritability estimates in checking the asymmetry of response were examined fitting a quadratic regression between sibs. Nonlinearity in halfsib regression was found to be the same as in the regression of offspring on single parent. Dominance and common environmental effects were shown to cause biases in fullsib estimates. The effects of linkage on sib on sib regression were discussed when there is a large number of multiplicative loci contributing to the variation.
