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positions of each pedigree (fAB) and at the same time through the parents in the right positions of each pedigree (fBD), giving fACfBD as the probability that the genotype is identical by descent. Alternatively, two alleles could be identical by descent through the parents in the left and right positions of each pedigree ( fAC) and through the two parents in the right and left positions of each pedigree (fBD), giving fAD fBC as the probability that the genotype is identical by descent. Since both outcomes can occur in the populations of x and y individuals, the total probability of genotypes being identical by descent is fAC fBD + fAD fBC. In all cases, if one parent of x and one parent of y are unrelated there is zero probability that the alleles they transmit to their progeny can be identical by descent.

Half and full siblings make instructive examples of how to determine the expected covariance between relatives. Figure 10.6 shows the pedigrees for these two cases. For both half and full siblings, the probability that individuals x and y inherit an allele identical by descent is Given that x and y have a parent in common, there is a probability of 1/2 that a given allele is transmitted from a common parent to individual x and an independent probability of 1/2 that a copy of the same allele is transmitted to individual y, giving f = (V2XV2) = 1/4. For half siblings, parents A, B, and C are unrelated so there is zero probability that they transmitted alleles identical by descent to their offspring. There is, however, a probability of 1/2 that both half sibs inherited the same allele from parent B, but this is only one allele and not a genotype. For full siblings, while A and B are unrelated, A and B are the parents of both x and y. It is therefore possible that x and y inherited the same allele from parent A and the same allele from parent B to produce genotypes that are identical by descent.

Half siblings resemble each other since 50% of individuals share one allele that is identical by descent. Full siblings have an even greater degree of resemblance since 50% of individuals share an allele identical by descent and at the same time 25% of individuals share a genotype identical by descent. The variance in genotypic value caused by alleles and genotypes corresponds to VA and VD. Therefore, the genotypic values of half siblings have a covariance V2 VA while the genotypic values of full siblings have a greater covariance of 1/2 VA + 1/4Vd. Table 10.8 gives additional examples of the expected covariance between various relatives based on the same logic used to obtain the covariances for half and full siblings.

Unilineal relatives such as half siblings can share only one allele in their genotypes that is identical by descent. This is the case since one of the parents of each individual are related and can provide an avenue for inheritance of an allele that is identical by descent. The other allele in the genotype cannot be identical by descent because they are inherited from two different parents who are unrelated. In contrast, bilineal relatives, such as full siblings, share both parents in common or have parents who are related. Bilinear relatives therefore have dominance components in their expected covariances because they have a chance of inheriting both alleles and genotypes that are identical by descent. Sharing alleles in common leads to phenotypic resemblance due to the additive phenotypic effects of alleles while sharing genotypes in common leads to phenotypic resemblance due to the phenotypic effects of genotypes (dominance and epistasis).

With knowledge of the genetic basis of covari-ance in genotypic values among relatives, we can look back to Fig. 9.8 to better understand why Abney et al. (2001) decided to estimate heritabilities in a

Hutterite population. The 806 individuals in that study descended from only 64 ancestors. That means that all individuals in the study had a non-zero probability of sharing two alleles that were identical by descent. The consequence is that the u coefficient in equation 10.57 was non-zero for all pairs of individuals in the study. This led to improved precision for estimates of dominance variance because comparisons between all pairs of individuals could be used to estimate the covariance between phenotype and the u coefficient. In contrast, the u coefficient is zero in randomly mating populations except for between pairs of full siblings, making estimates of dominance variance imprecise because the number of full siblings in one family is quite small.

Since the expected covariance between the mid-parent value and progeny values forms the basis of the parent-offspring regression, it is worth working through an additional method to obtain the covari-ance between mid-parent and offspring values. Deriving this covariance relies on a mathematical property of the covariance, or what some might call a math trick. The covariance can be expressed in a different form as

where O is the value of the offspring from each parental mating, P is the mid-parent value of a mating between two parental genotypes, and M is the population mean. In the case of parents and offspring, the average taken by multiplication by 1/n in equation 10.60 is instead accomplished by multiplying the product of P and O by the expected frequency of each parental mating. Table 10.9 gives the expected frequencies for each union of parental genotypes under random mating, along with the expected mid-parent and progeny values. Multiplying the appropriate three quantities from each row in Table 10.9 and then summing across rows cov(O,P) = p4a2 + 4p3q

With some algebraic manipulation, canceling of terms and substitution of the definition of the population mean, the covariance between the mid-parent and offspring values is

In terms of the mid-parent and offspring values, the covariance is then cov(O,P) = (OP) - M2

Since a = a + d(q - p) and VA = 2pqa2, after substitution the variance becomes cov(O,P) = ^ V

Table 10.9 Frequencies and mean values for parents and progeny used to derive the covariance between the

average value of parents (mid-parent value) and the average value of the progeny from each parental mating.

Parental mating

Parental mating

Mid-parent

Progeny genotype

Progeny

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