## Y

Alleles not identical by descent

Figure 2.15 The possible patterns of transmission from one parent to two progeny for a locus with two alleles. Half of the outcomes result in the two progeny inheriting an allele that is identical by descent. The a and a' refer to paths of inheritance in the pedigree in Fig. 2.14b.

progeny G. The probability of a given allele being transmitted along a path is independent of the probability along any other path, so the probability of autozygosity (symbolized as f to distinguish it from the pre-existing autozygosity of individual A) over the entire pedigree for any of the G progeny is:

P(b = d) P(a = b) P(a = a') P(a'= c) P(c = e) = (1/2)5(1 + fA) = 1/32(1 + fA) (2.19)

This probability would still be 1/2 no matter how many alleles were present in the population, since the probability arises from the fact that diploid genotypes have only two alleles.

To have a complete account of the probability that B and C inherit an identical allele from A, we also need to take into account the past history of A's genotype since it is possible that A was itself the product of mating among relatives. If A was the product of some level of biparental inbreeding, then the chance that it transmits alleles identical by descent to B and C is greater than if A was from a randomly mating population. Another way to think of it is, with A being the product of some level of inbreeding instead of random mating, the chances that the alleles transmitted to B and C are not identical (see Fig. 2.14b) will be less than 1/2 by the amount that A is inbred. If the degree to which A is inbred (or the probability that A is autozygous) is FA, then the total probability that B and C inherit the same allele is:

If Fa is 0 in equation 2.18, then the chance of transmitting the same allele to B and C reduces to the 1/2 expected in a randomly mating population.

For the other paths of inheritance in Fig. 2.14, the logic is similar to determine the probability that an allele is identical by descent. For example, what is the probability that the allele in gamete d is identical by descent to the allele in gamete b, or P(b = d)? When D mated it passed on one of two alleles, with a probability of 1/2 for each allele. One allele was inherited from each parent, so there is a 1/2 chance of transmitting a maternal or paternal allele. This makes P(b = d) = 1/2. (Just like with individual A, P(b = d) could also be increased to the extent that B was inbred, although random mating for all genotypes but A is assumed here for simplicity.) This same logic applies to all other paths in the pedigree that connect A and the since independent probabilities can be multiplied to find the total probability of an event. This is equivalent to the average relatedness among half-cousins. In general for pedigrees, f = (1/2)i(1 + FA) where A is the common ancestor and i is the number of paths or individuals over which alleles are transmitted. A trick is to write down the chain of individuals starting with the common ancestor and ending with the individuals of interest and count the individuals along paths of inheritance (not including the individuals of interest). That gives GDBACEG or five ancestors, yielding a result identical to equation 2.19.

Although it is useful to determine the inbreeding coefficient (autozygosity) for a specific pedigree, the more general point is to see mating among relatives as a process that increases autozygosity in a population. When individuals have common relatives, the chance that their genotype contains loci with alleles identical by descent is increased. Further, the inbreeding coefficient or autozygosity measured for a specific pedigree is identical in concept and interpretation to the departure from Hardy-Weinberg measured by the fixation index (assuming most of the deviation from Hardy-Weinberg expectations is caused by non-random mating and not other population genetic processes). Both express the probability that two alleles in a genotype are identical due to common ancestry.

The departure from Hardy-Weinberg expected genotype frequencies, the autozygosity or inbreeding coefficient, and the fixation index are all interrelated. Another way of stating the results that were developed in Fig. 2.12 is that f measures the degree to which Hardy-Weinberg genotype proportions are not met, due to inbreeding:

With consanguineous mating, the decline in heterozygosity is proportional to the increase in the inbreeding coefficient, shown by substituting He for 2pq in equation 2.20 and rearranging to give

where He is the Hardy-Weinberg expected heterozygosity based on population allele frequencies. Rearranging equation 2.21 in terms of the inbreeding coefficient gives:

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