M

The same logic can be applied to obtain the progeny mean values for the A1A2 and A2A2 genotypes under random mating.

Prepared with the expression for Mprogeny A A , we can now obtain an equation for the breeding value of the A1A1 genotype. Using the definition of the breeding value given in equation 10.26,

Breeding value aiai - 2(mprogeny a1a1 - m)

Substituting in the expression for mean progeny value in equation 10.30 and the expression for the population mean in equation 10.4 gives

Breeding value A1A1 = 2(ap + dq - (a(p - q) + 2pqd))

Rearrangement of this equation yields

(Readers carrying out the algebra can refer to Math box 10.1 for the steps that show d(1 - 2p) = d(q - p)). Notice that the q(a + d(q - p)) term is equal to the definition of the average effect of the A1 allele given in equation 10.13. Therefore, the breeding value of the A1A1 genotype is simply equal to two times the

Based on the definition of the breeding value as the difference between the progeny mean and the population mean, the breeding value of any genotype is simply the sum of the average effects of each of the alleles that make up the genotype. We can therefore use the definitions of the average effect in the last section to define breeding values for the three genotypes at a diallelic locus:

average effect of the A1 allele.

Because breeding values are made up of the average effects of the two alleles in a genotype, breeding values also depend on the population context in which they are measured. Thus, breeding values are not universal features of genotypes but rather depend on the population allele and genotypes frequencies.

The breeding value expressions tell us the mean value of the progeny that would be produced by each genotype under random mating. Look at the example breeding values for the three genotypes at the IGF1 locus in dogs (Table 10.5). Under the allele frequencies and zero dominance in case A, a population of progeny from an individual with an A1A1 genotype would have a mean of 10.5 kg relative to the midpoint. (In the absolute, the progeny would have a mean value of 30 kg, which is identical to the A1A1 genotypic value.) In case (a) with p = q = 0.5, an A1 allele would be paired with an A1 allele in half of the progeny and

Table 10.5 Examples of breeding values for the three IGF1 locus genotypes in dogs. Values are all in kilograms and relative to the midpoint value of 19.5 kg.

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