Population Genetics. Matthew B. Hamilton

observed heterozygosity declines and the fixation index increases at different rates depending on the coancestry coefficient (f) of each mating type. Selfing was 100% self‐fertilization, while mixed mating was 50% of the population self‐fertilizing and 50% mating at random. Full sibling is brother–sister or parent–offspring mating. Backcross is one individual mated to its progeny, then to its grand progeny, then to its great‐grand progeny and so on, a mating scheme that is difficult to carry on for many generations. The fixation index at each generation for each mating scheme is based on the following recursion equations: selfing F_{t + 1} = ½(1 + F_t); mixed F_{t + 1} = ½(1 + F_t)(s) where s is the selfing rate; full sibling F_{t + 2} = ¼(1 + 2F_{t + 1} + F_t); backcross F_{t + 1} = ¼(1 + 2F_t).

Figure 2.14 Average relatedness and autozygosity as the probability that two alleles at one locus are identical by descent. Panel A shows a pedigree where individual A has progeny that are half‐siblings (B and C). B and C then produce progeny D and E, which in turn produce offspring G. Panel B shows only the paths of relatedness where alleles could be inherited from A, with curved arrows to indicate the probability that gametes carry alleles identical by descent. Upper case letters for individuals represent diploid genotypes and lower case letters indicate allele copies within the gametes produced by the genotypes. The probability that A transmits a copy of the same allele to B and C depends on the degree of inbreeding for individual A or F_A.

Autozygosity is measured by the coefficient of coancestry (sometimes called the coefficient of kinship) and symbolized as f, can be seen in a pedigree such as that shown in Figure 2.14. Figure 2.14a gives a hypothetical pedigree for four generations. The pedigree can be used to determine the probability that the fourth‐generation progeny, labeled G, have autozygous genotypes due to individual A being a common ancestor of both their maternal and paternal parents. To make the process simpler, Figure 2.14b strips away all of the external ancestors and shows only the paths where alleles could be inherited in the progeny from individual A.

      Allozygous genotype: A homozygous or heterozygous genotype composed of two alleles not inherited from a recent common ancestor.

      Autozygosity (f): The probability that two alleles in a homozygous genotype are identical by descent.

      Autozygous genotype: A homozygous genotype composed of two identical alleles that are inherited from a common ancestor.

      Coancestry coefficient (Θ): The probability that two randomly sampled gametes, one from each of two individuals, both carry a given allele that is identical by descent.

      Identity by descent (IBD): Sharing the same state because of transmission from a common ancestor.

      Relatedness: The expected proportion of alleles between two individuals that are identical by descent; twice the autozygosity.

To begin the process of determining the autozygosity for G, it is necessary to determine the probability that A transmitted the same allele to individuals B and C, or in notation P(a = a'). With two alleles designated 1 and 2, there are only four possible patterns of allelic transmission from A to B and C, as shown in Figure 2.15. In only half of these cases do B and C inherit an identical allele from A, so P(a = a') = 1/2. This probability would still be ½ no matter how many alleles were present in the population, since the probability arises from the fact that diploid genotypes have only two alleles.

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 panel B of Figure 2.14.

      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 Figure. 2.14b) will be less than ½ by the amount that A is inbred. If the degree to which A is inbred (or the probability that A is autozygous) is F_A, then the total probability that B and C inherit the same allele is:

(2.18)

If the parents of individual A are unrelated, then F_A is 0 in Eq. 2.18, and then the chance of transmitting the same allele to B and C reduces to the ½ expected in a randomly mating population.

      For the other paths of inheritance in Figure 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 ½ for each allele. One allele was inherited from each parent, so there is a ½ chance of transmitting a maternal or paternal allele. This makes P(b = d) = ½. (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 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 preexisting homozygote excess or deficit of the population individual A belongs to, or F_A) over the entire pedigree for any of the G progeny is:

(2.19)

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,