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in Fig. 10.5 use a genotypic value of a = 10.5, each panel represents one case of the components of genotypic variance for IGF1 depending on the degree of dominance.

Across the three graphs in Fig. 10.5, the maximum VG progressively increases and the allele frequency where maximum VG occurs also shifts to lower values of p. This increase in maximum VG corresponds to an

Interact box 10.2 Components of total genotypic variance, VG

The additive and dominance components of the total genotypic variance can be interactively graphed using an Excel spreadsheet model on the textbook website. Values of a and d can be set in the model. Set a = 10.5 as in the IGF 1 example. Then set d to 0.0, 5.25, and 10.5 and view the graphs in each instance. The graphs should be identical to those in Fig. 10.5.

What would the graph of the components of the total genotypic variance look like with strong overdominance? Predict what the graph might look like and sketch it on paper. Then set d to a value that constitutes overdominance and view the graph. Was your prediction correct? What is the impact of strong overdominance on heritability? If a population with strong overdominance and allele frequencies near p = q = 0.5 experienced genetic drift or consanguineous mating, how would VA and the heritability change?

Math box 10.2 Deriving the total genotypic variance, VG

While the method of adding VA and VD to obtain the total genotypic variance seems reasonable, it does have an important assumption. The total genetic variance, VG, is the sum of VA + VD plus twice the covariance between VA and VD:

This covariance can be estimated by summing the genotype frequency-weighted product of the breeding value and the dominance deviation for each of the genotypes:

covad = p2(2qa)(-2q2d) + 2pq((q - p)a)(2pqd) + q2(-2pa)(-2p2d) (10.53)

After multiplying out each term to get covAD = - 4p2q3ad + 4p2q2(q - p)ad + 4p2q3ad (10.54)

and then factoring out 4p2q2ad to get covAD = -4p2q2ad(-q + q - p + p) (10.55)

it is then apparent that the covariance between the breeding values and the dominance deviations is zero.

The same overall result of the total genetic variance being a function of VA + VD + 2covAD can be shown by starting with the expression for the total genotypic variance based on the frequency-weighted variances for each genotypic value

VG = p2(a - M)2 + 2pq(d - M)2 + q2(-a - M)2 (10.56)

and then using definitional substitutions and algebra to obtain an equation with terms that represent the variance of the breeding values plus the variance of the dominance deviations and the covariance between the breeding values and dominance deviations.

increase in the maxima of both VA and VD. Complete dominance causes more total genotypic variation because the genotypic values in the population are only the extremes of +a and -a without a genotypic value that is intermediate. The variance in geno-typic values is at a maximum when there are equal frequencies of the +a and -a genotypic values. At p ~ 0.29 and q ~ 0.71 the combined frequencies of the A1A1 and A1A2 genotypes are equal to the frequency of the A2A2 genotypes, thus maximizing VG.

10.6 Genotypic resemblance between relatives

• Additive (VA) and dominance (VD) components of genotypic variation and resemblance of relatives.

• Resemblance of relatives depends on the probability that alleles or genotypes are shared.

• The covariance between mid-parent and offspring genotypic values.

Obtaining expressions for the components of the total genotypic variation is ultimately useful because the variance components provide the basis to predict the resemblance of genotypic values between populations of related individuals. This relationship can also be reversed, and the phenotypic resemblance between relatives can be used to estimate the components of the total genotypic variance for specific phenotypes. This section shows how the variance within or covariance between relatives are related to components of the genotypic variance. In practice, VA is usually estimated from the resemblance between relatives to then determine the heritability of a phenotype. See the Appendix for background on the covariance if necessary.

The expected covariance in genotypic values between relatives can be determined using the auto-zygosity of the relatives compared (Cotterman 19 74, 1983; Crow and Kimura 1970). The expected covari-ance between related individuals is if the parental population is not inbred (f = 0) and cov(x, y) = rVA + uVD

where x and y represent the phenotypic values in a population of individuals and r and u are fractions determined by probabilities of identity by descent for the individuals in groups x and y and for their parents. If the parents of individuals in group x are A and B and the parents of individuals in group y are C and D then

where f is the probability of autozygosity in a pedigree as explained in Chapter 2.

These coefficients have a clear biological interpretation based on the pedigrees of individuals x and y (Fig. 10.6). The r coefficient is twice the probability that individuals x and y inherit an allele that is identical by descent. The u coefficient is the probability that individuals x and y inherit the same genotype. The probability of inheriting the same genotype depends on the probability that both alleles in the genotypes of x and y are identical by descent. Two alleles could be identical by descent through the parents in the left

(a) General case

Progeny

(b) Half siblings

Parents A

: y r - 2fxy for bilinear relatives u - fACfBD + fADfBC

Progeny x y

The probability that individuals x and y both inherit an allele identical by descent from individual B is 1/4. Therefore, fxy = 1/4.

(c) Full siblings Parents Progeny y r = 2fx xy

The probability that individuals x and y both inherit an allele identical by descent from individual A or B is 1/4. Therefore, fxy = 1/4. r = 2fxy = 2(1/4) = 1/2

1 1 fBA

Figure 10.6 The expected covariance in genotypic values for relatives based on probabilities that individuals share alleles (r) and genotypes (u) that are identical by descent. (a) The pedigree for the general case. (b) Half siblings (half brothers and sisters) share parent B in common and are thus unilineal relatives. (c) Full siblings (full brothers and sisters) share both parents in common and are thus bilineal relatives. In general, bilineal relatives include dominance components in their expected covariances.

r - fxy for unilinear relatives

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