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Let’s consider two linked loci $A$ and $B$ that are both bi-allelic. In consequence, we have four different possible haplotypes $A_1B_1$, $A_1B_2$, $A_2B_1$, $A_2B_2$, which frequencies are $X_1$, $X_2$, $X_3$, $X_4$ respectively. Let the linkage disequilibrium $D$ be defined as:

$$D=X_1X_4 - X_2X_3$$

The change in haplotype frequencies through time is given by:

$$\delta X_1 = \frac{X_1(w_1 - \bar w) - Drw_{14}}{\bar w}$$

$$\delta X_2 = \frac{X_2(w_2 - \bar w) - Drw_{14}}{\bar w}$$

$$\delta X_3 = \frac{X_3(w_3 - \bar w) - Drw_{14}}{\bar w}$$

$$\delta X_4 = \frac{X_4(w_4 - \bar w) - Drw_{14}}{\bar w}$$

, where

  • $r$ is the recombination rate between the loci $A$ and $B$
  • $\bar w$ is the mean fitness (sum of the fitnesses of each haplotype weighted by their relative frequencies)
  • $w_{14}$ is the fitness of the (diploid) individual that has the 1st and the 4th haplotypes
  • $w_i = X_1w_{i1} + X_2w_{i2} + X_3w_{i3} + X_4w_{i4}$

The change in allele frequency of locus $A$ is $x$ and $1-x$. The change of allele frequency of locus $B$ is $y$ and $1-y$. Those changes are given by the addition of haplotypes frequencies:

$$\delta x = \delta X_1 + \delta X_2$$

and

$$\delta y = \delta X_1 + \delta X_3$$

Question

I don’t fully understand these formulae! Could you please help me making sense of them. Especially of the presence of the expression $-Drw_{14}$.


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