Population Genetics. Matthew B. Hamilton
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can be rearranged by noticing the first four terms all contain g11 which can be factored out to give
(2.36)
Recall that D = g11g22 − g12g21 and make the substitution to obtain
(2.37)
Next, notice that (g11 + g22 + g21 + g22) is the sum of all gamete frequencies and equals one. Making that substitution, we obtain
(2.38)
This final result shows that gamete frequencies in the second generation are a function of the gamete frequency we expect from multiplying the respective allele frequencies, increased or decreased by the product of the recombination rate and the amount of two locus disequilibrium. The expected frequency of the A1A1B1B1 genotype, for example, in the next generation is then (g11 ‐ cD)2, and it is not just a function of the product of the allele frequencies but also depends on the recombination rate and the amount of two locus disequilibrium. This is analogous to adjusting single locus H‐W expected genotype frequencies using F to account for one locus disequilibrium.
It is helpful to keep in mind that the term linkage disequilibrium is widely employed in the literature and has deep historic roots (e.g. Lewontin 1964), even though it is an imprecise label that confounds a pattern (two locus haplotypes or genotypes departing from the frequencies expected by the product of frequencies of alleles) and a process. Linkage disequilibrium is a misnomer since physical linkage only dictates the rate at which allelic combinations approach independent assortment or equilibrium. Processes other than linkage are responsible for the production of deviations from independent assortment of alleles at multiple loci in gametes. Using terms like gametic disequilibrium or two‐locus disequilibrium reminds us that the deviation from random association of alleles at two loci is a pattern seen in gametes or haplotypes. Although linkage can certainly contribute to this pattern, so can many other population genetic processes. It is likely that several processes operating simultaneously produce the two‐locus disequilibrium observed in any population, as illustrated by the pie chart in Figure 2.20.
Gametic disequilibrium is a central concept in formulating predictions for multiple locus genotype and haplotype frequencies in populations. Observations of the amount of gametic disequilibrium present in populations can then be used to identify the fundamental population genetic processes operating in populations. Thus, gametic disequilibrium forms the basis for a wide range of hypotheses to explain multiple locus genotype and haplotype frequencies, with gametic equilibrium or Mendel's second law serving as the null hypothesis. The numerous processes that maintain or increase gametic disequilibrium include those discussed in more detail the following sections.
Physical linkage
Linkage is the physical association of loci on a chromosome that causes alleles at the loci to be inherited in their original combinations. This association of alleles at loci on the same chromosome is broken down by crossing over and recombination. The probability that a recombination event occurs between two loci is a function of the distance along the chromosome between two loci. Loci that are very far apart (or on separate chromosomes) have recombination rates approaching 50% and are said to be unlinked. Loci located very near each other on the same chromosome might have recombination rates of 5 or 1% and would be described as tightly linked. Therefore, the degree of physical linkage of loci dictates the recombination rate and thereby the decay of gametic disequilibrium. Genome locations are often mapped in terms of their recombination frequencies with the measure centimorgan (abbreviated cM) or map unit (m.u.) where 1 cM is equivalent to a 1% recombination rate under a model that corrects for multiple crossovers called Haldane's map function (see Casares 2007).
Linkage‐like effects can be seen in some chromosomes and genomes where gametic disequilibrium is expected to persist over longer time scales due to exceptional inheritance or recombination patterns. Organisms such as birds and primates have chromosomal sex determination, with the well‐known X and Y sex chromosome system in humans. Loci located on X chromosomes experience recombination, whereas those on Y chromosomes experience no recombination. This is caused by the Y chromosome lacking a homologous chromosome to pair with at meiosis since YY genotypes do not exist. In addition, we would expect that the rate of decay of gametic disequilibrium for X chromosomes is about half that of autosomes with comparable recombination rates, since X recombination takes place only in females (XX) at meiosis, and not at all in males (XY). Organelle genomes found in mitochondria and chloroplasts are a case where gametic disequilibrium persists indefinitely since these genomes are uniparentally inherited and do not experience observable levels of recombination. There is variation in the rate of recombination among species, within and among populations and between sexes (Stapley et al. 2017), with “hotspots” that show elevated rates as well as areas of restricted recombination such that genomes may have marked heterogeneity in recombination rates. In humans, for example, patterns of haplotype polymorphism suggest that about 80% of all recombination events take place in a subset of only about 15% of the genome (Myers et al. 2006).
Natural selection
Natural selection is a process that can continuously counteract the randomizing effects of recombination. Imagine a case where two locus genotypes confer different rates of survival or different levels of reproduction. In such a case, natural selection will reduce the frequency of lower fitness genotypes, which will also reduce the number of gametes these genotypes contribute to forming the next generation. At the same time that natural selection is acting, recombination is also working to randomize the associations of alleles at the two loci. Figure 2.21 shows an example of epistatic natural selection acting to maintain gametic disequilibrium in opposition to recombination acting to establish gametic equilibrium. Natural selection at one locus can also impact the frequency of nearby loci that experience limited recombination. Both hitchhiking, where natural selection rapidly increasing the frequency of a beneficial genotype alters the frequency of linked loci, and background selection, where natural selection eliminates low fitness genotypes, can lead to linkage disequilibrium at loci adjacent to the loci experiencing selection. This is because natural selection has the potential to change the frequency of haplotypes more quickly than recombination can act to randomize the arrangement of alleles found together in the same haplotype. More on these natural selection topics can be found in later chapters.
Figure 2.21 The decay of gametic disequilibrium (D) over time when both strong natural selection and recombination are acting. Initially, there are only coupling (P11 = P22 = ½) and no repulsion gametes (P12 = P21 = 0). The relative fitness values of the AAbb and aaBB genotypes are one, while all other genotypes have a fitness of 0.5, a form of epistasis for relative fitness. Unlike in Figure 2.19, gametic disequilibrium does not decay to zero over time due to natural selection that is stronger than recombination.
The action of natural selection acting on differences in gamete fitness can produce steady‐states other than D = 0 even with free recombination. In such cases, the population reaches