## Aa

Genotypic value

Change in genotypic value

Frequency of allele not replaced

Frequency-weighted change in value of allele replacement

Average change in value of allele replacement a - d

Figure 10.2 The derivation of the average change in value caused by replacing one A2 allele with an allele in those genotypes sampled at random from the population that contain at least one A2 allele. The mean change in value caused by an allelic replacement forms the basis of the average effect once it is multiplied by the frequency of the allele that is replaced.

to an A-A- genotype will change the value from d to +a, so the difference in value is a - d. Since the frequency of the A- allele that remains intact in the genotype is p, then the frequency of changing an A1A2 genotype to an A1A1 genotype is p. The frequency-weighted change in value when making this allelic substitution is therefore p(a - d). Similarly, when changing an A2A2 genotype to an A1A2 via replacement of one A2 allele, the value changes from -a to d, so the difference in value is d - (-a) or d + a. The frequency of the A2 allele that remains intact in the second genotype is now q, so the frequency-weighted change in value when making this allelic substitution is therefore q(d + a). The total average change in value when replacing an A2 allele with an A1 allele is the sum of these two separate changes in value or p(a - d) + q(d + a). (Note that the same result, except multiplied by -1, can be obtained by electing to replace an A1 allele with an A2 allele. The negative sign comes about because A2 is the allele that decreases value.)

Using algebraic manipulation similar to that in Math box 10.1, the expression for the average change in value due to an allelic replacement simplifies to a = a + d(q - p)

with a not bearing a subscript denoting the average change in value caused by an allele replacement.

Notice that a + d(q - p) also appears in equations 10.13 and 10.14. On a strictly mathematical basis, we could substitute a in these two equations to restate the expressions for the average effects of the

A1 and A2 alleles:

However, this reformulation makes intuitive sense in terms of our imagined ability to replace one allele with another in genotypes used to derive equation 10.22. When replacing an A2 allele with an A1 allele, the frequency of the A2 allele in the population is q and this is the frequency of allelic replacements under random mating (see Fig. 10.2). Likewise, p is the frequency in the population of the A1 allele being replaced with A2 alleles. The expression for a2 is negative because A2 is the allele that decreases value. Table 10.3 also gives computations of the average effect using equations 10.23 and 10.24, with identical results.

Problem box 10.2 Compute the allele average effect of the IGF1 A2 allele in dogs

Using Table 10.3 as a guide, compute the average effect in three ways: (1) as the difference between MA and the population mean M, (2) by the formula a2 = -p(a + d(q - p)) from equation 10.14, and (3) by a2 = -pa from equation 10.24. Be sure to compare your results with the average effects for the A1 allele given in Table 10.3 and explain why the average effects are the same or different with reference to dominance and the allele frequencies.

10.4 Breeding value and dominance deviation

• Deriving breeding values in a population.

• Breeding values under random and non-random mating.

• Deriving dominance deviations in a population.

• Dominance deviations under random and non-random mating.

A2A1*

The next step in understanding the Mendelian basis of quantitative trait variation is to move back to the level of the genotype. In Chapter 9, both additive (VA) and dominance (VD) genetic variation were identified as separate components of the total genetic variation in quantitative traits. In terms of population mean values, we can divide the genotypic mean value into its components due to additive effects of alleles and the dominance effects of genotypes

In words, this equation says that the mean genotypic value (G) is the sum of the mean breeding value (A) and the mean dominance deviation (D). As pointed out above, this assumes that there is no interaction genetic variance due to epistasis because we are working with a one-locus example. If there were epistasis, then there would also be a mean value for an interaction deviation due to the mean value of interactions among loci that make up genotypes.

This subsection will address how the mean pheno-type of a population of progeny depends on mating among the population of parents. It is important to first understand the motivation to predict what is called the breeding value of a genotype. When natural selection occurs, it is essentially the differential mating success of certain phenotypes. Humans achieve the same thing in domestic plants and animals through artificial selection by allowing only those individuals with preferred phenotypes to breed. To the extent that phenotype is a function of genotype, natural and artificial selection allow some genotypes to breed more often than others. A full understanding of how and why a given mean phenotype occurs in a population of progeny therefore requires an understanding of the consequences of mating in the parental generation. Here we will start with the genotypic value of an individual, and then track the frequencies and genotypic values of its progeny to predict the mean genotypic value of its progeny.

Breeding value of an individual Twice the mean value of the progeny that would be produced by a single genotype under random mating expressed as a difference from the population mean.

inherit one allele from each of two parents rather than a diploid genotype.) The impact of a single parent on the mean value of the progeny population could be measured in a manner akin to the average effect. Now, instead of a special slot machine, we could just take a single individual and have it mate with many individuals drawn at random from the parental population. Each progeny from these matings would inherit one allele from the focal parent and another allele from an individual drawn at random. The resulting population of progeny would have a mean value that could be measured. The breeding value of a genotype is two times the difference between the progeny mean value and the parental population mean value (M). Expressed as an equation:

where Mprogeny A A is the mean value of progeny produced when an individual of genotype AxAx is mated to many individuals drawn at random from the parental population. The difference between the means is multiplied by 2 because a parent's genotype possesses two alleles but its progeny inherit only one allele at a time.

The components that lead to an expression for Mprogeny for the A1A1 genotype are shown in Table 10.4. When the A1A1 genotype mates, it will encounter and mate with individuals of a given genotype in proportion to the frequency of that genotype in the population. An A1A1 individual is expected to mate with A1A1, A1A2, and A2A2 individuals with frequencies of p2, 2pq, and q2, respectively, since these are the Hardy-Weinberg expected genotype frequencies in the population. Each of the matings between an A1A1 genotype and another genotype will produce progeny with one or two genotypes. The phenotypic values of each of the progeny genotypes is also known. To obtain an expression for the mean phenotypic value of the progeny that result from the A1A1 genotype mating at random in the population (Mpr0geny a^), add up all of the progeny phenotypic values after weighting each one by its relative frequency among all of the progeny and by the frequency of mating pairs to obtain

Mprogeny a,a, = P2a + 2pq(1/ia + 1/2d) + q2d (10.2 7)

This expression can be simplified by first expanding the middle term

Parents pass on alleles and not genotypes to their progeny. (From the progeny point of view, individuals

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