Recurrent mutation in the ancestry of a rare variant

Abstract

Recurrent mutation produces multiple copies of the same allele which may be co-segregating in a population. Yet most analyses of allele-frequency or site-frequency spectra assume that all observed copies of an allele trace back to a single mutation. We develop a sampling theory for the number of latent mutations in the ancestry of a rare variant, specifically a variant observed in relatively small count in a large sample. Our results follow from the statistical independence of low-count mutations, which we show to hold for the standard neutral coalescent or diffusion model of population genetics as well as for more general coalescent trees. For populations of constant size, these counts are distributed like the number of alleles in the Ewens sampling formula. We develop a Poisson sampling model for populations of varying size, and illustrate it using new results for site-frequency spectra in an exponentially growing population. We apply our model to a large data set of human SNPs and use it to explain dramatic differences in site-frequency spectra across the range of mutation rates in the human genome.

Latent mutations in the ancestries of alleles under selection

“…If β/n ≪ 1, the ancestral process of tracing back to these mutations will be complete before the population has changed much in size and the results of Section 3.4 will hold. But this is clearly not the case for the gnomAD data in Wakeley et al (2023) and Seplyarskiy et al (2023). The distribution of n 1 in the non-Finnish European sample with n = 114K is well fit by β/n = 3.…”

“…Because r 2 → ∞, this approximation will hold long enough for the required fixed number of events among the A 1 lineages to occur, despite the rapid decrease of r 2 in the last line of (59). A proof of this is given in Wakeley et al (2023, Appendix). Theorem 4 addresses the corresponding issues for the model of Section 3.…”

“…The demographic function ρ could for example represent exponential population growth, in which case ρ ( t ) = ρ (0) e βt for some positive constant β . This model was used in Wakeley et al (2023) to illustrate the effects of rapid growth on neutral rare variation in humans. Here we allow that ρ ( t ) is piecewise continuous.…”

“…When α < 0, using (A.5) in (55) gives This differs from the neutral case (Wakeley et al, 2023) only by the possibility of virtual lineages. As in (57), these will be removed quickly if they are produced.…”

“…For single nucleotide sites, these population-scaled mutation rates have been estimated for many species, using average pairwise sequence differences and assuming constant population size, and are typically about 0.01 with a range of about 0.0001 to 0.1 (Leffler et al, 2012). Values for humans are smaller but they vary almost as widely among sites in the genome, with a mean of about 0.0008 and a range of about 0.0001 to 0.02 (Seplyarskiy et al, 2021(Seplyarskiy et al, , 2023Wakeley et al, 2023). In contrast, there is no reason to suppose that the selection parameter |α| is small (Eyre-Walker and Keightley, 2007;Chen et al, 2020;Agarwal et al, 2023).…”