Selective sweeps and hitchhiking

A beneficial mutation and its neighbours

The Natural Selection tutorial follows one locus in isolation. On a real chromosome the selected site has neighbours: thousands of neutral variants a few centimorgans away, each on the particular chromosome where it happened to arise. When a new beneficial allele A rises to fixation, every chromosome in the population comes to descend from the one on which A appeared, and the neutral alleles on that chromosome rise with it. This is genetic hitchhiking, and the whole episode is a selective sweep.

The Selective Sweep module simulates it forward in time. N = 200 diploids carry a chromosome of L = 8 cM with 600 standing neutral variants at mutation-drift balance, in linkage equilibrium. A appears at the centre; AA has fitness 1 + s and Aa has 1 + hs, with s = 0.1 and h = 0.5 at the defaults. Recombination follows a Haldane map. At fixation the module samples 100 chromosomes and reads out diversity, Tajima's D and haplotype homozygosity in windows along the map, next to a neutral control grown from the same starting population for the same number of generations.

Fixation is not guaranteed. A single new copy of A fixes with probability about 2hs = 0.1 (Kimura's formula gives 0.095 here); most copies are lost by drift while still rare. The module conditions on fixation: attempts that are lost restart from the same population and are shown as faint ghost paths, and the counter tells you how many failed.

Why the valley has a width

The rise is logistic, slow at both ends and fast in the middle, and with h = 0.5 it takes about (4/s)(ln 2Ns + γ) generations, 174 here. That is short compared with the roughly 4N = 800 generations over which neutral variation turns over, which is why the sweep leaves a mark: the neutral sites have no time to recover.

Recombination is what limits the damage. During the sweep a crossover between a neutral site and the selected site, in an Aa heterozygote, moves A onto a chromosome from the other background. The further a site is from the selected site, the more such crossovers happen before fixation, and the more of its original variation survives. For additive selection the expected diversity relative to neutral is π/θ ≈ 1 − (2Ns)^(−4r/s), where r is the recombination fraction to the selected site. The characteristic width is the recombination fraction r ≈ s/(4 ln 2Ns), which is 0.68 cM at the defaults; the module's own expectation puts the point where π/θ is back to one half at 0.34 cM from the site.

Stronger selection means a shorter sweep, fewer crossovers before fixation, and a wider valley. Weaker selection means a longer sweep that recombination has time to blur. The depth at the selected site itself depends on how often the population's chromosomes trace back to the original A carrier, which for a hard sweep is nearly always.

Footprints, hard and soft

Three statistics in the module's readouts are the ones sweep scans use on real genomes. π/θ is diversity relative to its neutral expectation; a valley centred on a site is the classic signal. Tajima's D compares the number of segregating sites with pairwise diversity; after a sweep the genealogy is star-like, variants are mostly young and rare, and D is negative. H12 is haplotype homozygosity with the two commonest haplotypes pooled; it peaks where one haplotype, or one haplotype and its close relatives, dominates the sample.

The neutral control matters more than it looks. Its windows hold a few dozen segregating sites each, so single windows scatter: the control's D at the centre is −0.21 with no selection at all, and its H12 differs from window to window. A scan calls a sweep where the sweep statistic is far outside what the control, or the genome-wide background, produces. One run is an anecdote; the comparison is the evidence.

A hard sweep starts from one new copy. A soft sweep starts from standing variation: A was already present, neutral or nearly so, on several chromosomes when the environment changed and it became beneficial. In the module the Soft option places A on round(f₀ × 2N) random chromosomes; with f₀ = 0.2 that is 80 of the 400, and the allele cannot be lost (P(fix) rounds to 1).

Several founder haplotypes now rise together, each carrying its own neutral neighbours. The sample at fixation descends from more than one of them, so the valley is shallower, D is less negative, and H12 is lower while H2/H1, the share of haplotype homozygosity not due to the single commonest haplotype, is higher. At f₀ = 0.2 the expected π/θ in the centre window rises from 0.42 to 0.85 and the expected number of founders among the 100 sampled chromosomes is 3.8. Soft sweeps are harder to detect for exactly this reason, and much of the debate about how adaptation usually happens is a debate about how often sweeps are soft.