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-2pa

### Number of At alleles in genotype

Figure 10.4 The relationship between average effect of an allele replacement, genotypic values, breeding values, and dominance deviations. The solid line represents the least-squares regression of genotypic value on number of alleles in a genotype in a population of 100 individuals with Hardy-Weinberg genotype frequencies. The slope of the line is equal to the effect of an allelic replacement or a. There is no dominance in (a) and partial dominance in (b). The breeding value is equal to the genotypic value adjusted by the effects of dominance on the progeny population mean as measured by the dominance deviation. All values are deviations from the population mean.

A1 allele) would change the phenotypic value of that group of individuals from -10.5 to zero. Likewise, changing all A1A2 genotypes to A1A1 genotypes would change the phenotypic value of that group of individuals from zero to 10.5. The average pheno-typic effect of increasing the number of A1 alleles is therefore 10.5 (or the slope of the regression line), so a = 10.5 as shown in Fig. 10.4.

Given the average effect of an allele replacement, we can then predict the expected phenotypic value of the progeny of any one genotype. Let's use the A2A2 genotype as an example. When A2A2 individuals mate, the mean value of the progeny population will decrease because progeny of A2A2 individuals receive the A2 allele that confers the lower genotypic value. The amount of the decrease in value is -2qa, which in Fig. 10.4a equals -2(0.5)(10.5) = -10.5. Figure 10.4b shows the impact of dominance. Even with dominance, when A2A2 individuals mate the mean value of the progeny population still decreases because their progeny receive the A2 allele. The decrease in phenotypic value is -2qa = -10.5. But dominance has caused the population mean to be 2.625 rather than zero. Another consequence of dominance is that when A2A2 individuals mate, some of the A2 alleles will pair with A1 alleles to make heterozygotes with the phenotype of d. Therefore, the average effects of two A2 alleles need to be adjusted by the dominance deviation. The dominance deviation for A2A2 is -2p2d, which in panel (a) equals zero and in panel (b) equals -2(0.5)2(5.25) =-2.625.

10.5 Components of total genotypic variance

Deriving the additive (VA) and dominance (VD)

components of genotypic variation. VA and VD are rel genotypic values.

### VA and VD are related to allele frequencies and

We are at last in a position to obtain expressions for the components of variance in phenotype due to additive genetic variation and dominance genetic variation. Recall from earlier in the chapter that additive genetic variation (VA) contributes to the resemblance between parents and offspring. Using the terminology of this section, we can say that VA is due to the variance in breeding values. Similarly, earlier in the chapter dominance genetic variation (VD) was described as being due to the effect, if any, of combining different alleles into a heterozygote genotype. We can now recognize VD as the variance in dominance deviations. These variances describe the spread or range of values in a population caused by breeding value or by dominance.

The previous subsections devoted considerable effort to developing expressions for average values. In particular, we obtained expressions for the average breeding value and average dominance deviation of each genotype. Obtaining these averages was important because an average is a critical part of a variance (see the Appendix for the definition of a variance). As was shown above, in a randomly mating population both the mean breeding value and the mean dominance deviation are zero. These are extremely useful results because they greatly simplify the expressions for the variance in breeding value and the variance in dominance deviation.

The additive variance, or VA, is the variance of breeding values in a randomly mating population. Since the mean breeding value taken over all genotypes is zero, the variance in breeding value is simply the square of the mean breeding value for each genotype multiplied by the frequency of each genotype:

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