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By expanding each term, factoring out 2pqa2 and then canceling, the equation simplifies to which after substitution of the expressions for VA and VD becomes

VA = 2pqa2

or, after substituting the definition of a from equation 10.22, to

Similarly, the dominance variance, or VD, is the variance in dominance deviation values in a randomly mating population. As before, the mean dominance deviation taken over all genotypes is zero so the variance in breeding value is the square of the mean dominance deviation for each genotype multiplied by the frequency of each genotype:

Expanding each term gives 4p2q2d2(p2 + 2pq + q2), which then gives the equation for the dominance deviation variance as

The separate expressions for VA and VD give us the means to estimate the total genotypic variance or

VG as

Note that VG is commonly referred to as the total genetic variance, even though it is the variance in genotypic values.

The components of the genetic variance and their relationship to the allele frequencies and the geno-typic values can be seen in Fig. 10.5. In all cases, VG is greatest at intermediate allele frequencies. When there is no dominance as in Fig. 10.5a, VG is made up exclusively of additive genetic variation. Without any dominance VA is greatest at intermediate allele frequencies where the total frequency of the two homozygotes equals the frequency of the heterozygotes under random mating. When there is dominance, as in panels (b) and (c), VG is made up of both additive and dominance genetic variation. Dominance variance is greatest when the frequency of all heterozygotes is the greatest, or p = q = 0.5 for a diallelic locus under random mating. Also notice that with dominance VG is low when the frequency of the recessive allele (A2) is low because the population has very few individuals with the -a genotypic value. This means that VG is made up mostly of the variance in +a and d genotypic values that are very similar or identical because of dominance. Since all of the illustrations

■ Additive genetic variance (VA)

Dominance genetic variance (VD)

■ Total genotypic variance (VG)

Figure 10.5 The additive (VA) and dominance (VD) components of the total genotypic variance (VG). In (a) there is no dominance (d = 0.0). Panel (b) has partial dominance (d = 5.25) and panel (c) has complete dominance (d = 10.5). In (b) and (c) the dominance component of the total genotypic variance is greatest when allele frequencies are equal because the frequency of heterozygotes is greatest. In all cases, the value of +a = 10.5.

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