Polymorphisms and substitutions in D. melanogaster. Genomic categories are shown. (a) Four-fold synonymous sites in coding DNA (squares), and nonsynonymous changes in coding DNA (diamonds). (b) Intergenic regions. (c) Introns. (d) UTRs. Data points indicate frequency distributions Q̂(k) (dots) of conserved positions (k = 0), k-fold polymorphisms (k = 1, …, 11), and substitutions (k = 12) per unit sequence length, relative to a reference distribution Q₀(k) obtained as the best neutral fit for 4-fold synonymous sites. Maximum-likelihood model distributions Q(k) shown as lines to guide the eye:equilibrium model (κ = 0) (short-dashed lines) and demographic model (with a 100-fold reduced D. melanogaster population size between speciation and 0.65 of divergence time and time-independent selection, κ = 0) (long-dashed lines), producing a poor joint fit to all genomic categories (P < 10⁻¹⁷ and P < 10⁻¹⁰, respectively). Fluctuating-selection model, with parameters given in Table 1, produces a good joint fit to all genomic categories (solid lines).

Inference of adaptive evolution under fluctuating selection. (a) Statistical evidence for adaptation, as given by the likelihood score difference per locus, Δs, between the input model and the best equilibrium (κ = 0) model as a function of the input selection parameters σ and κ (input μ and t as in the Drosophila data sets below). Contours from bottom to top: 0 (thick line), 5 × 10⁻⁴, 5 × 10⁻³, 0.01, 0.015 (thin lines). (b) Analogous score difference Δs_MK/L based on the McDonald–Kreitman test. Contours as in a, no-inference region (ΔS_MK = 0) marked by stripes. (c) Fraction of adaptive changes α̃(σ, κ) as given by Eq. 17, which is correctly reconstructed by the full maximum-likelihood procedure for a sufficiently large number of loci. Contours from bottom to top: 0 (thick line), 0.1, 0.3, 0.7, 0.9 (thin lines). (d) Estimated fraction α̃_MK (9) using the McDonald–Kreitman test, which leads to a systematic underestimation. Contours as in c, no-inference region (α̃_MK = 0) marked by stripes.

Allele frequency evolution under constant and fluctuating selection. The distribution p(x, t) of a two-allele Fisher–Wright model involving mutations, selection, and genetic drift with parameters μ = 0.025, σ = 4, and a discretization value N = 400 (shown as dots) is obtained by numerical propagation using the exact continuous time Master equation. (a) Initial regime (t < 1) for constant selection: From an initially monomorphic allele-a population, the solution p(x, t)(shown here for t = 0.2, 0.4, 0.6, 0.8, 1.0, bottom to top) builds up the spectrum of “forward” polymorphisms towards the stationary shape p₊(x) (solid line) as given by Eq. 4. (b) Quasistationary regime (t > 1) for constant selection: The solution p(x, t) (shown here for μt = 0.05, 0.45, 0.80) follows the quasistationary form (thin lines) in excellent approximation and approaches the equilibrium distribution p_eq(x) (thick line) for large values of μt. (c) Quasistationary evolution of the coefficient function λ₊(t) for constant selection (dashed line) and fluctuating selection with κ = 0.5 (solid line), which follows Eq. 9 in each interval of constant selection for a given realization of switching events (shown as vertical lines).