Linkage Disequilibrium - Role of Recombination

Role of Recombination

In the absence of evolutionary forces other than random mating and Mendelian segregation, the linkage disequilibrium measure converges to zero along the time axis at a rate depending on the magnitude of the recombination rate between the two loci.

Using the notation above, we can demonstrate this convergence to zero as follows. In the next generation, the frequency of the haplotype, becomes

This follows because a fraction of the haplotypes in the offspring have not recombined, and are thus copies of a random haplotype in their parents. A fraction of those are . A fraction have recombined these two loci. If the parents result from random mating, the probability of the copy at locus having allele is and the probability of the copy at locus having allele is, and as these copies are initially on different loci, these are independent events so that the probabilities can be multiplied.

This formula can be rewritten as

so that

where at the -th generation is designated as . Thus we have

If, then so that converges to zero.

If at some time we observe linkage disequilibrium, it will disappear in the future due to recombination. However, the smaller the distance between the two loci, the smaller will be the rate of convergence of to zero.