and one quarter of each homozygous genotype. Every generation, the frequency of the heterozygotes declines by one half while one quarter of the heterozygote frequency is added to the frequencies of each homozygote (diagonal arrows). Eventually, the population will lose all heterozygosity, although allele frequencies will remain constant. Therefore, assortative mating or self‐fertilization changes the pairing of alleles in genotypes but not the allele frequencies themselves.
Genotype frequencies change quite rapidly under complete assortative mating, but what about allele frequencies? Let's employ the same example population with p = q = 0.5 and Hardy–Weinberg genotype frequencies D = 0.25, H = 0.5, and R = 0.25 to answer the question. For both of the homozygous genotypes, the initial frequencies would be D = R = (0.5)2 = 0.25. In Figure 2.12, the contribution of each homozygote genotype frequency from mating among heterozygotes after five generations is H/2(1 − (1/2)5) = H/2(1–1/32) = H/2(31/32). With the initial frequency of H = 0.5, H/2(31/32) = 0.242. Therefore, the frequencies of both homozygous genotypes are 0.25 + 0.242 = 0.492 after five generations. It is also apparent that the total increase in homozygotes (31/32) is exactly the same as the total decrease in heterozygotes (31/32), so the allele frequencies in the population have remained constant. After five generations of assortative mating in this example, genotypes are much more likely to contain two identical alleles than they are to contain two unlike alleles. This conclusion is also reflected in the value of the fixation index for this example,
The fact that allele frequencies do not change over time can also be shown elegantly with some simple algebra. Using the notation in Figure 2.12 and defining the frequency of the A allele as p and the a allele as q with subscripts to indicate generation, allele frequencies can be determined by the genotype counting method as
(2.14)
as a function of genotype frequencies in the previous generation using substitution for D1 and H1:
(2.15)
which simplifies to:
(2.16)
and then recognizing that the right‐hand side is equal to the frequency of A in generation 0:
(2.17)
Thus, allele frequencies remain constant under complete assortative mating. As practice, you should carry out the algebra for the frequency of the a allele.
Under complete self‐fertilization heterozygosity declines very rapidly. There can also be partial self‐fertilization in a population (termed mixed mating), where some matings are self‐fertilization and others are between two individuals (called outcrossing). In addition, many organisms are not capable of self‐fertilization but instead engage in biparental inbreeding (mating between two different but related individuals) to some degree. In general, these forms of mating among relatives will reduce heterozygosity compared to random mating, although they will not drive heterozygosity toward zero as in the case of complete selfing. The rate of decline in heterozygosity can be determined for many possible types of mating systems, and a few examples are shown in Figure 2.13. Regardless of the specifics of the form of consanguineous mating that occurs, it remains true that mating among relatives causes alleles to be packaged more frequently as homozygotes (heterozygosity declines) and most forms of mating among relatives do not alter allele frequencies in a population. Negative assortative mating is an exception where allele frequencies can change depending on the initial allele frequencies (Workman 1964).
Coancestry coefficient and autozygosity
The 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).
