How fast recombination removes disequilibrium

Recombination breaks haplotypes apart

Linkage disequilibrium D is the excess of a gamete type over what independence predicts. It starts with some event, such as a new mutation or a mixing of two populations. After that, one process removes it: recombination. A parent that is heterozygous at both loci can make a recombinant gamete when a crossover falls between the loci. The recombination rate r is the chance per generation that it does.

Take the gametes of a double heterozygote. A fraction 1 − r are non-recombinant and keep the old haplotype combinations. A fraction r are recombinant, and their alleles are paired at random, which has zero association. Only the non-recombinant part carries D forward, so D(t + 1) = (1 − r) × D(t). Iterating gives D(t) = D₀ × (1 − r)^t.

A worked case with the defaults, r = 0.1 and D₀ = 0.15: after one generation D = 0.15 × 0.9 = 0.135. After two it is 0.1215. Each step removes the same fraction, 10%, of what is left. This is geometric decay, and the curve never reaches zero. The model has no drift, selection or mutation: the population is infinite and mates at random, so the only change in D comes from recombination.

Half-life

The time for D to fall to half its value is the half-life: solve (1 − r)^t = 1/2, so t = ln 2 / −ln(1 − r). For small r this is close to 0.69 / r. At r = 0.1 it is 6.58 generations, and D first drops to 0.075 or below at generation 7 (0.0717).

Half-life is the useful number because it does not depend on D₀. Whether the initial association is 0.15 or 0.05, it halves in the same time. It depends only on r, and r depends on the distance between the loci. A short distance means a small r and a long half-life. Markers 1 cM apart have r near 0.01, so their association lasts about 69 generations per halving.

Unlinked loci and the upper limit

The recombination rate cannot exceed 0.5. For loci far apart or on different chromosomes, an odd and an even number of crossovers are equally likely, and only an odd number makes a recombinant. At r = 0.5, D halves every generation: 0.075 after one, 0.0375 after two, 0.000146 after ten. Loci that are not physically linked lose their association in a few generations.

At the other end, r = 0 means no recombination. D stays at D₀ forever, since no gametes are remixed.

D has a ceiling. It cannot exceed 0.25 in absolute value, and only reaches it when both alleles at both loci have frequency 0.5 and only two gamete types exist. The module restricts D₀ to between −0.25 and 0.25. A negative D₀ decays towards zero from below at the same rate.

The module plots raw D, but the same factor applies to related measures. Recombination changes no allele frequency, so D max stays fixed, D′ = D / D max decays by (1 − r) per generation, and r² decays by (1 − r)^2 per generation.