Population Genetics. Matthew B. Hamilton

gamete frequencies cannot be negative). In this sense, D measures the deviation of gamete frequencies from what is expected under independent assortment. Since D can be either positive or negative, both coupling and repulsion gametes can be in excess or deficit relative to the expectations of independent assortment.

      Different estimators of gametic disequilibrium have different strengths and weaknesses (see Hedrick 1987; Flint‐Garcia et al. 2003). The discussion here will focus on the classical parameter and estimator D to develop the conceptual basis of measuring gametic disequilibrium and to understand the genetic processes that cause it.

      Gametic disequilibrium: An excess or deficit or absence of all possible combinations of alleles at a pair of loci in a sample of gametes or haplotypes.

      Linkage: Co‐inheritance of loci caused by physical location on the same chromosome.

      Recombination fraction: The proportion of “repulsion” or recombinant gametes produced by a double heterozygote genotype each generation.

      Now that we have developed an estimator of gametic disequilibrium, it can be used to understand how allelic association at two loci changes over time or its dynamic behavior. If a very large population without natural selection or mutation starts out with some level of gametic disequilibrium, what happens to D over time with recombination? Imagine a population with a given level of gametic disequilibrium at the present time (D_{t = n}). How much gametic disequilibrium was there a single generation before the present at generation n − 1? Recombination will produce c recombinant gametes each generation so that:

      (2.28)

      Since gametic disequilibrium decays by a factor of 1 − c each generation,

      (2.29)

      We can predict the amount of gametic disequilibrium over time by using the amount of disequilibrium initially present (D_t0) and multiplying it by (1 − c) raised to the power of the number of generations that have elapsed:

(2.30)

Figure 2.19 shows the decay of gametic disequilibrium over time using Eq. 2.30. Initially, there are only coupling gametes in the population and no repulsion gametes, giving a maximum amount of gametic disequilibrium. As c increases, the approach to gametic equilibrium (D = 0) is more rapid. Eq. 2.30 and Figure 2.20 both assume that there are no other processes acting to counter the mixing effect of recombination. Therefore, the steady‐state will always be equal frequencies of all gametes (D = 0), with the recombination rate determining how rapidly gametic equilibrium is attained.

      A hypothesis test that the observed level of gametic disequilibrium is significantly different than expected under random segregation can be carried out with:

      (2.31)

where N is the total sample size of gametes, is a gametic disequilibrium estimate, and p and q are the allele frequencies at two diallelic loci. The χ² value has 1 degree of freedom and can be compared with the critical value found in Table 2.5.

Figure 2.19 The decay of gametic disequilibrium (D) over time for four recombination rates. Initially, there are only coupling (P₁₁ = P₂₂ = ½) and no repulsion gametes (P₁₂ = P₂₁ = 0). Gametic disequilibrium decays as a function of time and the recombination rate (D_{t = n} = D_{t = 0}[1−c]ⁿ) assuming a single large population, random mating and no counteracting genetic processes. If all gametes were initially repulsion, gametic disequilibrium would initially equal −0.25 and decay to zero in an identical fashion.

Figure 2.20 A hypothetical partitioning of the contributions to the total population gametic disequilibrium (D) in a population caused by numerous population genetic processes. The finite sample of gametes or genotypes used to measure D can itself contribute to the disequilibrium observed, as can departure from Hardy–Weinberg expected genotype frequencies at single loci or within‐locus disequilibrium. The fractions of the total gametic disequilibrium attributable to each cause will vary depending on history of a population and the relative strengths of the multiple processes acting in a population.

One potential drawback of D in Eq. 2.27 is that its maximum value depends on the allele frequencies in the population. This can make interpreting an estimate of D or comparing estimates of D from different populations problematic. For example, it is possible that two populations have very strong association among alleles within gametes (e.g. no repulsion gametes), but the two populations differ in allele frequency so that the maximum value of D in each population is also different. If all alleles are not at equal frequencies in a population, then the frequencies of the two coupling or the two repulsion gametes are also not equal. When D < 0, D_max is the value of −p₁q₁ or − p₂q₂ that is closer to zero, whereas when D > 0, D_max is the value of p₁q₂ or p₂q₁ that is closer to zero.

      A way to avoid these problems is to express D as the percentage of its largest value:

      (2.32)

      This gives a measure of gametic disequilibrium that is normalized by the maximum or minimum value D can assume given population allele frequencies. Even though a given value of D may seem small in the absolute, it may be large relative to D_max given the population allele frequencies. A related and more commonly employed measure expresses disequilibrium between two loci as a correlation:

      (2.33)

      where ρ (pronounced “roe”) takes the familiar and more easily interpreted range of −1 to +1 (the disequilibrium correlation is sometimes given as ρ² with 0 ≤ ρ² ≤ 1) (Lewontin 1988). Analogous to the fixation index, the two locus disequilibrium correlation