The allele-frequency spectrum in a decoupled Moran model with mutation, drift, and directional selection, assuming small mutation rates

Abstract

We analyze a decoupled Moran model with haploid population size N, a biallelic locus under mutation and drift with scaled forward and backward mutation rates θ(1)=μ(1)N and θ(0)=μ(0)N, and directional selection with scaled strength γ=sN. With small scaled mutation rates θ(0) and θ(1), which is appropriate for single nucleotide polymorphism data in highly recombining regions, we derive a simple approximate equilibrium distribution for polymorphic alleles with a constant of proportionality. We also put forth an even simpler model, where all mutations originate from monomorphic states. Using this model we derive the sojourn times, conditional on the ancestral and fixed allele, and under equilibrium the distributions of fixed and polymorphic alleles and fixation rates. Furthermore, we also derive the distribution of small samples in the diffusion limit and provide convenient recurrence relations for calculating this distribution. This enables us to give formulas analogous to the Ewens-Watterson estimator of θ for biased mutation rates and selection. We apply this theory to a polymorphism dataset of fourfold degenerate sites in Drosophila melanogaster.

Fig. 1

Comparison of the exact versus the approximate probability of polymorphism in a sample of size $n=2$ (solid line). The dashed line shows equality.

Fig. 2

The exact probability (bars) and approximate probabilities (solid and stippled lines) for ${\theta }_0=0.2,{\theta }_1=0.2,\gamma =1.0$, (A) and (C) and ${\theta }_0=0.02,{\theta }_1=0.02,\gamma =1.0$, (B) and (D), respectively, for $N=25$, (A) and (B), and $N=100$, (C) and (D), respectively. For the stippled line, the constant was obtained by summing over formula (11); for the solid line the constant was approximated from only the monomorphic states (see the text for details).

Fig. 3

Likelihood of obtaining $y=(0,1,\dots ,5)$ individuals of the favored type in a sample without replacement of size $n=5$ from a population of size $N=500$ (thick line), $N=5$ (thin line) and the approximation in formula (39) (stippled line) with ${\theta }_0=0.5,{\theta }_1=0.3,\gamma =2$. Note that the sample from $N=5$ is slightly less affected by selection and thus more symmetric.

Fig. 4

Observed (bars) and inferred (lines) allele frequency spectra showing the GC-frequencies. A: unpolarized GC-frequencies at fourfold degenerate sites ($\gamma =2.05$). B: unpolarized GC-frequencies in introns ($\gamma =0.53$). C: polarized (AT to GC) spectrum of GC-frequencies at fourfold degenerate sites ($\gamma =2.58$). D: polarized (GC to AT) spectrum of GC-frequencies at fourfold degenerate sites ($\gamma =1.99$).