Latent mutations in the ancestries of alleles under selection

We consider a single genetic locus with two alleles $A_1$ and $A_2$ in a large haploid population. The locus is subject to selection and two-way, or recurrent, mutation. Assuming the allele frequencies follow a Wright–Fisher diffusion and have reached stationarity, we describe the asymptotic behaviors of the conditional gene genealogy and the latent mutations of a sample with known allele counts, when the count $n_1$ of allele $A_1$ is fixed, and when either or both the sample size $n$ and the selection strength $\left|\alpha \right|$ tend to infinity. Our study extends previous work under neutrality to the case of non-neutral rare alleles, asserting that when selection is not too strong relative to the sample size, even if it is strongly positive or strongly negative in the usual sense ($\alpha \to -\infty$ or $\alpha \to +\infty$), the number of latent mutations of the $n_1$ copies of allele $A_1$ follows the same distribution as the number of alleles in the Ewens sampling formula. On the other hand, very strong positive selection relative to the sample size leads to neutral gene genealogies with a single ancient latent mutation. We also demonstrate robustness of our asymptotic results against changing population sizes, when one of $\left|\alpha \right|$ or $n$ is large.