Genetics, Vol 121, 839-844, Copyright © 1989

INVESTIGATIONS

A Building Block Model for Quantitative Genetics

H. Tachida and C. C. Cockerham
Department of Statistics, North Carolina State University, Raleigh, North Carolina 27695-8203

We introduce a quantitative genetic model for multiple alleles which permits the parameterization of the degree, D, of dominance of favorable or unfavorable alleles. We assume gene effects to be random from some distribution and independent of the D's. We then fit the usual least-squares population genetic model of additive and dominance effects in an infinite equilibrium population to determine the five genetic components--additive variance {sigma}(a)(2), dominance variance {sigma}(d)(2), variance of homozygous dominance effects d(2), covariance of additive and homozygous dominance effects d(1), and the square of the inbreeding depression h--required to treat finite populations and large populations that have been through a bottleneck or in which there is inbreeding. The effects of dominance can be summarized as functions of the average, D, and the variance, {sigma}(D)(2). An important distinction arises between symmetrical and nonsymmetrical distributions of gene effects. With symmetrical distributions d(1) = -d(2)/2 which is always negative, and the contribution of dominance to {sigma}(a)(2) is equal to d(2)/2. With nonsymmetrical distributions there is an additional contribution H to {sigma}(a)(2) and -H/2 to d(1), the sign of H being determined by D and the skew of the distribution. Some numerical evaluations are presented for the normal and exponential distributions of gene effects, illustrating the effects of the number of alleles and of the variation in allelic frequencies. Random additive by additive (a*a) epistatic effects contribute to {sigma}(a)(2) and to the a*a variance, {sigma}{complex}, the relative contributions depending on the number of alleles and the variation in allelic frequencies. There are plausible situations where the contribution to {sigma}(a)(2) can be larger than that to {sigma}{complex}. When the number of alleles is large and there is little variation in allelic frequencies most of the variance is {sigma}{complex}. The effects of the genetic components on the additive variance within finite populations are discussed briefly.

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