relatedness among parents, shown by substituting He for 2pq in Eq. 2.20 and rearranging to give
(2.24)
where He is the Hardy–Weinberg expected heterozygosity based on population allele frequencies. Rearranging Eq. 2.21 in terms of the autozygosity gives
(2.25)
This is really exactly the same quantity as the fixation index (Eq. 2.9)
(2.26)
With the understanding that we are considering only a single generation of mating and that the population initially had Hardy–Weinberg expected genotype frequencies, we see that the fixation index measures an excess of homozygosity exactly equal to the autozygosity. Sustained non‐random mating over multiple generations will continue to increase the homozygosity, and therefore, F will reflect the cumulative deficit of heterozygosity from all past mating among relatives. In contrast, f is defined on a per generation basis and reflects only the probability of identity by descent at mating in the most recent generation. This distinction can be observed in Figure 2.13 where F increases over time toward an equilibrium as a function of the coancestry coefficient due to each mating system.
We can also use the fixation index in Eq. 2.23 rather than the autozygosity to express deviations from Hardy–Weinberg expected genotype frequencies due to accumulated autozygosity caused by mating patterns such as freq(AA) = p2 + Fpq and freq(Aa) = 2pq − F2pq where F2pq is the amount of heterozygosity missing from (or added to) the Hardy–Weinberg expected heterozygote frequency with half of that total or Fpq being added to (or subtracted from) each of the homozygotes. It is important to note that this time F represents the correlation of allelic state within genotypes from all causes accumulated over time that alters genotype frequencies from their Hardy–Weinberg expected values. While the difference in notation between F and f seems minor, their biological interpretations differ substantially in this example.
Returning to Figure 2.13 reinforces the relationship of the coancestry coefficient, the fixation index, and the decline in heterozygosity in several specific cases of regular consanguineous mating. Remember that in all cases in Figure 2.13, the Hardy–Weinberg expected heterozygosity is 0.5 when mating is random and f = 0.
Phenotypic consequences of mating among relatives
The process of consanguineous mating is associated with changes in the mean phenotype within a population. These changes arise from two general causes: changes in genotype frequencies in a population per se and fitness effects associated with changes in genotype frequencies.
The mean phenotype of a population will be impacted by any changes in genotype frequency. To show this, it is necessary to introduce terminology to express the phenotype associated with a given genotype, a topic covered in much greater detail and explained more fully in Chapters 9 and 10. We will assign AA genotypes the phenotype +a, heterozygotes the phenotype d, and aa homozygotes the phenotype −a. Each genotype contributes to the overall phenotype based on how frequent it is in the population. The mean phenotype in a population is then the sum of each genotype‐frequency‐weighted phenotype (Table 2.10). When there is no dominance, the phenotype of the heterozygotes is exactly intermediate between the phenotypes of the two homozygotes and d = 0. In that case, it is easy to see that mating among relatives will not change the mean phenotype in the population since both homozygous genotypes increase by the same amount and their effects on the mean phenotype cancel out (mean = ap2 + aFpq + d2pq − dF2pq − aq2 − aFpq, where the heterozygote terms are crossed out since d = 0). When there is some degree of dominance (positive d indicates the phenotype of Aa is like that of AA while negative d indicates the phenotype of Aa is like that of aa), then the mean phenotype of the population will change with consanguineous mating since heterozygotes will become less frequent. If dominance is in the direction of the +a phenotype (d > 0), then mating among relatives will reduce the population mean because the heterozygote frequency will drop. Similarly, if dominance is in the direction of −a (d < 0), then mating among relatives will increase the population mean again because the heterozygote frequency decreases. It is also true in the case of dominance that a return to random mating will restore the frequencies of heterozygotes and return the population mean to its original value mating among relatives. These changes in the population mean phenotype are simply a consequence of changing the genotype frequencies when there is no change in the allele frequencies.
There is a wealth of evidence that the increase of homozygosity caused by mating among relatives has deleterious (harmful or damaging) consequences and is associated with a decline in the average phenotype in a population, a phenomenon referred to as inbreeding depression. Since the early twentieth century, studies in animals and plants that have been intentionally inbred provide ample evidence that decreased performance, growth, reproduction, viability (all measures of fitness), and abnormal phenotypes are associated with consanguineous mating. A related phenomenon is heterosis or hybrid vigor, characterized by beneficial consequences of increased heterozygosity such as increased viability and reproduction, or the reverse of inbreeding depression. One example is the heterosis exhibited in corn, which has led to the widespread use of F1 hybrid seed in industrial agriculture.
Table 2.10 The mean phenotype in a population that is experiencing consanguineous mating. The fixation index quantifying deviation from Hardy–Weinberg expected genotype frequencies is F, and d = 0 when there is no dominance.
| Genotype | Phenotype | Frequency | Contribution to population mean |
|---|---|---|---|
| AA | +a | p2 + Fpq | ap2 + aFpq |
| Aa | D | 2pq – F2pq | d2pq – dF2pq |
| aa | −a | q2 + Fpq | −aq2 – aFpq |
population mean: ap2 + d2pq – dF2pq ‐ aq2 = a(p‐q) + d2pq(1‐F)
