Population Genetics. Matthew B. Hamilton

of the correlation between the states of the two alleles found together in a two locus haplotypes. When ρ = 0 there is no correlation between the alleles at two loci that are found paired in gametes or on the same chromosome – the allelic states are independent as expected under Mendel's second law. If ρ > 0 there is a positive correlation such that if one of the alleles at one locus is an A, for example, then the allele at the second locus will have a correlated state and might often be a B allele. When ρ < 0 there is a negative correlation between the states of two alleles in a haplotype, such as if A is infrequently paired with B.

Thus far, we have approached gametic disequilibrium by focusing on the frequency of four gamete haplotypes. A helpful complement is to consider the gametes made by all possible two locus genotypes as shown in Table 2.12. This table is somewhat like the table of parental matings and their offspring genotype frequencies we made to prove Hardy–Weinberg for one locus, except Table 2.12 predicts the frequencies of gametes that will make up the next generation rather than genotype frequencies in the next generation. Most genotypes produce recombinant gametes that are identical to non‐recombinant gametes (e.g. the A₁B₁/A₁B₂ genotype produces A₁B₁ and A₁B₂ coupling gametes and A₁B₁ and A₁B₂ repulsion gametes). Only two genotypes – both types of double heterozygotes – will produce recombinant gametes that are different than parental haplotypes. These are the only two places where c enters into the expressions for expected gamete frequencies because recombination does not change the gametes produced by the other eight two locus genotypes.

Table 2.12 Expected frequencies of gametes for two diallelic loci in a randomly mating population with a recombination rate between the two loci of c. The first eight genotypes have non‐recombinant and recombinant gametes that are identical. The last two genotypes produce novel recombinant gametes, requiring inclusion of the recombination rate to predict gamete frequencies. Summing down each column of the table gives the total frequency of each gamete in the next generation.

Parental mating	Expected frequency of mating	Frequency of gametes in next generation
A1B1	A2B2	A1B2	A2B1
A1B1/A1B1	(p1q1)2	(p1q1)2
A2B2/ A2B2	(p2q2)2		(p2q2)2
A1B1/ A1B2	2(p1q1)(p1q2)	(p1q1)(p1q2)		(p1q1)(p1q2)
A1B1/ A2B1	2(p1q1)(p2q1)	(p1q1)(p2q1)			(p1q1)(p2q1)
A2B2/ A1B2	2(p2q2)(p1q2)		(p2q2)(p1q2)	(p2q2)(p1q2)
A2B2/ A2B1	2(p2q2)(p2q1)		(p2q2)(p2q1)		(p2q2)(p2q1)
A1B2/ A1B2	(p1q2)2			(p1q2)2
A2B1/ A2B1	(p2q1)2				(p2q1)2
A2B2/ A1B1	2(p2q2)(p1q1)	(1−c)(p2q2)(p1q1)	(1−c)(p2q2)(p1q1)	c(p2q2)(p1q1)	c(p2q2)(p1q1)
A1B2/ A2B1	2(p1q2)(p2q1)	c(p1q2)(p2q1)	c(p1q2)(p2q1)	(1−c)(p1q2)(p2q1)	(1−c)(p1q2)(p2q1)

We can relate two locus Hardy–Weinberg expected genotype frequencies to the recombination rate and two locus disequilibrium if we sum the columns to determine the expected gamete frequencies with the possibility of recombination. Focus on the column for the gamete A₁B₁. Summing the five terms in that column, we get

      (2.34)

      And expanding the two terms on the right gives

      (2.35)