Population Genetics. Matthew B. Hamilton

process of recombination. Sister chromatids touch at random points during meiosis and exchange short segments, a process known as crossing‐over (Figure 2.18).

Figure 2.18 A schematic diagram of the process of recombination between two loci, A and B. Two double‐stranded chromosomes (drawn in color and gray) exchange strands and form a Holliday structure. The cross over event can resolve into either of two recombinant chromosomes that generate new combinations of alleles at the two loci. The chance of a cross over event occurring generally increases as the distance between loci increases. Two loci are independent when the probability of recombination and non‐recombination are both equal to ½. Gene conversion, a double cross over event without exchange of flanking strands, is not shown.

      Linkage of loci has the potential to impact multilocus genotype frequencies and violate Mendel's law of independent segregation, which assumes the absence of linkage. To generalize expectations for genotype frequencies for two (or more) loci requires a model that accounts explicitly for linkage by including the rate of recombination between loci. The effects of linkage and recombination are important determinants of whether or not expected genotype frequencies under independent segregation of two loci (Mendel's second law) are met. Autosomal linkage is the general case that will be used to develop expectations for genotype frequencies under linkage.

The frequency of a two‐locus gamete haplotype will depend on two factors: (i) allele frequencies and (ii) the amount of recombination between the two loci. We can begin to construct a model based on the recombination rate by asking what gametes are generated by the genotype A₁A₂B₁B₂. Throughout this section, loci are indicated by the letters, alleles at the loci by the numerical subscripts, and allele frequencies by p₁ and p₂ for locus A and q₁ and q₂ for locus B. The problem is easier to conceptualize if we draw the two‐locus genotype as being on two lines akin to chromosomal strands

A1	B1
A 2	B 2

      This shows a genotype as two haplotypes and reveals phase or the sets of alleles packaged together on the same chromosomal strand (in contrast to writing the genotype as A₁A₂B₁B₂ where phase would be unknown). Given this physical arrangement of the two loci, what are the gametes produced during meiosis with and without recombination events?

A1B1 and A2B2	“Coupling” gametes: alleles on the same chromosome remain together (a term coined by Bateson and Punnett).
A1B2 and A2B1	“Repulsion” gametes: alleles on the same chromosome seem repulsed by each other and pair with alleles on the opposite strand (a term coined by Thomas Morgan Hunt).

      The recombination fraction, symbolized as c (or sometimes r), refers to the total frequency of gametes resulting from recombination events between two loci. Using c to express an arbitrary recombination fraction, let's build an expectation for the frequency of coupling and repulsion gametes. If c is the rate of recombination, then 1 − c is the rate of non‐recombination since the frequency of all gametes is 1, or 100%. Within each of these two categories of gametes (coupling and repulsion), two types of gametes are produced so the frequency of each gamete type is half that of the total frequency for the gamete category. We can also determine the expected frequencies of each gamete under random association of the alleles at the two loci based on Mendel's law of independent segregation.

Gamete	Frequency
	Expected	Observed
A1B1	p 1 q 1	g11 = (1 − c)/2	1 − c is the frequency of all coupling gametes.
A2B2	p 2 q 2	g22 = (1 − c)/2
A1B2	p 1 q 2	g12 = c/2	c is the frequency of all recombinant gametes.
A2B1	p 2 q 1	g21 = c/2

      The recombination fraction, c, can be thought of as the probability that a recombination event will occur between two loci. With independent assortment, the coupling and repulsion gametes are in equal frequencies and c equals ½ (like the chances of getting heads when flipping a coin). Values of c less than ½ indicate that recombination is less likely than non‐recombination, so coupling gametes are more frequent. Values of c greater than ½ are possible and would indicate that recombinant gametes are more frequent than non‐recombinant gametes (although such a pattern would likely be due to a process such as natural selection eliminating coupling gametes from the population rather than recombination exclusively).

      We can utilize observed gamete frequencies to develop a measure of the degree to which alleles are associated within gamete haplotypes. This quantity is called the gametic disequilibrium (or sometimes linkage disequilibrium) parameter and can be expressed by:

(2.27)

where g_xy stands for a gamete frequency. D is the difference between the product of the coupling gamete frequencies and the product of the repulsion gamete frequencies. This makes intuitive sense: with independent assortment, the frequencies of the coupling and repulsion gamete types are identical and cancel out to give D = 0, or gametic equilibrium. Another way to think of the gametic disequilibrium parameter is as a measure of the difference between observed and expected gamete frequencies: g₁₁ = p₁q₁ + D, g₂₂ = p₂q₂ + D, g₁₂ = p₁q₂ – D, and g₂₁ = p₂q₁