Quantitative Genetic Components Within Restricted Populations

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Abstract
Breeders often work with populations with a restricted genetic base either from the maintenance of varieties as small finite populations or from the initiation of populations with a few individuals or inbred lines. It is important to know the consequences of these restrictions on quantitative variation in comparison to that for large noninbred populations. To accomplish this the quantitative genetic variation within restricted populations is formulated in terms of the quantitative genetic components in the noninbred ancestral population. Required to treat an additive, dominance and additive by additive model are six components: σ2a‐additive variance, σ2d‐dominance variance, d1‐covariance of additive and homozygous dominance effects, d2‐variance of homozygous dominance effects, h2‐ square of the inbreeding effect, and σ2aa‐additive by additive variance. Comparable components (starred, e.g. σ2a*) for restricted populations, as an average over replicates, are expressed as linear functions of the components in the ancestral population with coefficients of identity by descent measures. These results from operating directly on the effects and variances within populations for all the components confirm previous results from covariances of relatives for σ2a* and σ2d*. Examples of further self‐fertilization and of a few inbred founders are considered in some detail. In the latter case considerable simplification occurs in that all coefficients are functions of the coancestry coefficient. While it is informative to relate the quantitative genetic components in restricted populations to those in ancestral populations, there is no practical advantage over just defining the components for a specific population when attention is to be confined to that population.