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Coancestry coefficient and autozygosity
ОглавлениеThe effects of consanguineous mating can also be thought of as increasing the probability that two alleles at one locus in an individual are inherited from the same ancestor. Such a genotype would be homozygous and considered autozygous since the alleles were inherited from a common ancestor. If the two alleles are not inherited from the same ancestor in the recent past, we would call the genotype allozygous (allo‐ means other). You are probably already familiar with autozygosity, although you may not recognize it as such. Two times the probability of autozygosity (since diploid individuals have two alleles) is commonly expressed as the degree of relatedness among relatives. For example, full siblings (full brothers and sisters) are one‐half related and first cousins are one‐eighth related. Using a pedigree and tracing the probabilities of inheritance of an allele, the autozygosity and the basis of average relatedness can be seen.
Figure 2.13 The impact of various systems of mating on heterozygosity (H) and the fixation index (F) over time. All populations have allele frequencies of p = q = 0.5 and initially are mating at random so heterozygosity equals 0.5 and remains at that level with random mating. As different patterns of mating among relatives occur in the four independent populations, 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 Ft + 1 = ½(1 + Ft); mixed Ft + 1 = ½(1 + Ft)(s) where s is the selfing rate; full sibling Ft + 2 = ¼(1 + 2Ft + 1 + Ft); backcross Ft + 1 = ¼(1 + 2Ft).
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 FA.
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 FA, then the total probability that B and C inherit the same allele is:
If the parents of individual A are unrelated, then FA 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 FA) over the entire pedigree for any of the G progeny is:
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 = (½)i(1 + FA) where A is the common ancestor and i is the number of paths or individuals over which alleles are transmitted. By writing down the chain of individuals and counting the individuals along paths of inheritance, we can determine the probability that a sample of two alleles, one from each individual, would exhibit both alleles identical by descent. That method gives GDBACEG or five ancestors for , yielding a result identical to Eq. 2.19.
We can use the method of tracing paths between ancestors to determine the coancestry coefficient, often symbolized by Θ, for any type of relationship. The pedigree in Figure 2.16 provides a set of examples of close relatives where we can determine the coancestry coefficient using paths of inheritance. In general, coefficients of coancestry for two individuals A and B can be determined using
Figure 2.16 A pedigree showing first (A and C are parent and offspring, C and D are full siblings), second (A and B are the grandparents of F and G, D, and E are half siblings), and third (F and G are cousins) degree relatives. The coancestry coefficient gives the probability that an allele sampled from each of two related individuals is identical by descent, defining the degree of relatedness.
where P is the number of ancestor–descendant paths connecting A and B, and F is the homozygote excess or deficit of the ancestor (Wright 1922; Thompson 1988). The inbreeding coefficient of an individual, or the probability of an individual inheriting two copies of the same allele, is a function of the coancestry of its parents. One example is the coancestry between one parent and an offspring, such as individuals A and C in Figure 2.16. There is one ancestor–descendant link between A and C, so the coancestry coefficient is
assuming that the parent A has F = 0. For the parent, the chance of sampling the allele that was transmitted to the offspring is ½. For the offspring too, the chance of sampling the allele inherited from that parent is ½. When combined, the chance of sampling the one allele that is identical by descent (IBD) between parents and offspring is (1/2)2 = 1/4. When considering both alleles in the offspring, one of them is IBD to one parent, so a parent and an offspring are ¼ + ¼ = ½ related.
The coancestry coefficient for full siblings, such as individuals C and D in Figure 2.16, is a case where individuals share two parents in common and therefore have two common ancestors to account for. Counting the paths C–A–D and C–B–D, we obtain
(2.22)
The chance that a given allele was transmitted from one parent to one offspring is ½, with a probability of (1/2)2 = ¼ of both full siblings inheriting the same allele from one parent. Because C and D share both parents and can inherit alleles identical by descent from both, we add the coancestries of each allele to give a relatedness of 1/4 + 1/4 = ½.
The coancestry coefficients for self‐fertilization and for full siblings explain the pattern of decreasing heterozygosity and increasing fixation indices over generations seen in Figure 2.13. For self‐fertilization, each generation has a coancestry coefficient of ½ when a self‐fertilized parent descended from unrelated individuals, making the fixation index equal to ½ after one generation of selfing. For a second generation of self‐fertilization, the coancestry coefficient remains ½ but now the parent has a higher probability of being homozygous because of the first generation of selfing. This makes the second‐generation fixation index F3 = ½(1 + ½) = 3/4.
In the case of full siblings, the recursion equation is Ft + 2 = ¼(1 + 2Ft + 1 + Ft). For full siblings, the coancestry coefficient is ¼ when their parents have no history of consanguineous mating among their ancestors, resulting in F2 = ¼. Taking two individuals from the second generation of full sibling mating as parents means that they each have a higher probability of being homozygous that is added to the constant ¼ probability of coancestry for full siblings. This gives a fixation index after three generations of full sibling mating of F3 = ¼(1 + 2[¼] + 0) = 3/8 (it is not until the fourth generation that the grandparents in generation t have a fixation index greater than zero).
It is useful to determine the coancestry coefficient for a specific set of relatives or pedigree as well as the change in the fixation index over generations for mating systems, especially in quantitative genetics. With the recent ability to obtain genome‐scale DNA sequences of numerous individuals, it is now possible to estimate coancestry directly from observed DNA polymorphism. Direct estimation of shared DNA using sequences highlights that traditional coancestry coefficients obtained from pedigrees are an average for a large sample of individuals and there can be substantial variation around these averages in the DNA‐level relatedness of an observed set of individuals (Ackerman et al. 2017). The general point is to understand that mating among relatives as a process that increases autozygosity in a population. When individuals have common relatives, the chance that they share alleles identical by descent is increased as is the chance that the genotype of one individual is homozygous.
The departure from Hardy–Weinberg expected genotype frequencies, the coancestry coefficient, the autozygosity, and the fixation index are interrelated. Another way of stating the results that were developed in Figure 2.12 is that F measures the degree to which Hardy–Weinberg genotype frequencies are not met due to departure from purely random union of alleles in diploid genotypes as expressed by f. To see this, imagine starting with a one locus with two alleles in a population at Hardy–Weinberg expected genotype frequencies with F equal zero. The expected frequencies of the three genotypes in progeny could be expressed as
