Gametic disequilibrium

One of the most important properties of a two-locus system is that it is no longer sufficient to talk about allele frequencies alone, even in a population that satisfies all of the assumptions necessary for genotypes to be in Hardy-Weinberg proportions at each locus. To see why consider this. With two loci and two alleles there are four possible gametes:¹

Gamete	$A_1B_1$	$A_1B_2$	$A_2B_1$	$A_2B_2$
Frequency	$x_{11}$	$x_{12}$	$x_{21}$	$x_{22}$

If alleles are arranged randomly into gametes then,

$$\begin{eqnarray*} x_{11} &=& p_1p_2 \\ x_{12} &=& p_1q_2 \\ x_{21} &=& q_1p_2 \\ x_{22} &=& q_1q_2 \quad , \end{eqnarray*}$$

where $p_1 = \hbox{freq}(A_1)$ and $p_2 = \hbox{freq}(A_2)$. But alleles need not be arranged randomly into gametes. They may covary so that when a gamete contains $A_1$ it is more likely to contain $B_1$ than a randomly chosen gamete, or they may covary so that a gamete containing $A_1$ is less likely to contain $B_1$ than a randomly chosen gamete. This covariance could be the result of the two loci being in close physical association, but it doesn't have to be. Whenever the alleles covary within gametes

$$\begin{eqnarray*} x_{11} &=& p_1p_2 + D \\ x_{12} &=& p_1q_2 - D \\ x_{21} &=& q_1p_2 - D \\ x_{22} &=& q_1q_2 + D \quad , \end{eqnarray*}$$

where $D = x_{11}x_{22} - x_{12}x_{22}$ is known as the gametic disequilibrium.² When $D \ne 0$ the alleles within gametes covary, and $D$ measures statistical association between them. It does not (directly) measure the physical association. Similarly, $D = 0$ does not imply that the loci are unlinked, only that the alleles at the two loci are arranged into gametes independently of one another.