Selective sweep at a quantitative trait locus in the presence of background genetic variation

Abstract

We model selection at a locus affecting a quantitative trait (QTL) in the presence of genetic variance due to other loci. The dynamics at the QTL are related to the initial genotypic value and to the background genetic variance of the trait, assuming that background genetic values are normally distributed, under three different forms of selection on the trait. Approximate dynamics are derived under the assumption of small mutation effect. For similar strengths of selection on the trait (i.e, gradient of directional selection beta) the way background variation affects the dynamics at the QTL critically depends on the shape of the fitness function. It generally causes the strength of selection on the QTL to decrease with time. The resulting neutral heterozygosity pattern resembles that of a selective sweep with a constant selection coefficient corresponding to the early conditions. The signature of selection may also be blurred by mutation and recombination in the later part of the sweep. We also study the race between the QTL and its genetic background toward a new optimum and find the conditions for a complete sweep. Overall, our results suggest that phenotypic traits exhibiting clear-cut molecular signatures of selection may represent a biased subset of all adaptive traits.

Figure 1.—

Dynamics at the QTL under linear directional selection. Exact recursions of Equations 11 and 13 (shaded line) are compared to approximate dynamics using Equations 24 and 25 (dashed line) and to approximate dynamics without background genetic variance (σ² = 0, dotted line). The assumption |a| ≪ |m| was either valid from the beginning (A) (m₀ = 10, a = 0.1, b = 1, ω = 0.05, σ = 1, ɛ = 0.001) or initially violated (B) (m₀ = 0.1, a = 1, b = 1, ω = 0.05, σ = 1, ɛ = 0.001). The measuring unit is arbitrary and is scaled to ω.

Figure 2.—

Dynamics at the QTL under Gaussian stabilizing selection. Exact recursions of Equations 11 and 13 (shaded line) are shown together with approximate expected dynamics using Equations 24 and 25 (dashed line) and expected dynamics without background genetic variance (dotted line). The situation where the assumption that |a| ≪ |m| is met from the beginning (m₀ = 2, a = −0.06, ω = 1, σ = 0.1, ɛ = 0.001) is shown in A, and that where it is initially violated (m₀ = 0.5, a = −0.1, ω = 1, σ = 0.05, ɛ = 0.001) is shown in B. Note that |a| cannot be very large relative to |m₀| in the presence of background genetic variation, not to overshoot the optimum.

Figure 3.—

Conditions for a complete “hard” sweep under Gaussian stabilizing selection. (A) Expected values of a (shaded area) that allow the focal mutation at the QTL to fix in the population as a function of the background standard deviation σ (parameters m₀ = 1, ω = 1, ɛ = 0.001). The dotted-dashed line is the maximum σ that allows fixation of the mutation, from Equation 26. Dynamics of (B) the mean value of the trait (arbitrary measuring unit, scaled to ω) and (C) the frequency p of the focal mutation A₁ near the limits of the range of parameters defined in Equation 26 are shown. Parameters are as in A, with σ = 0.12 and a = −0.15 (solid line) or a = −0.12 (dashed line). Expected threshold from Equation 26: a = −0.14.

Figure 4.—

Effective selection coefficient s_e for the hitchhiking effect, under Gaussian stabilizing selection. (A) Expected reduction of heterozygosity r_H plotted against the recombination rate with the locus under selection, in a selective sweep at a QTL with background genetic variation, under stabilizing selection. Expected pattern (shaded line) and fitted pattern under the assumption of a constant selection coefficient are shown. (B) Actual and estimated selection coefficients. Exact recursion (shaded line) and approximate value using Equation 22 (dashed line) for the growth rate ς of the mutation are shown. The dotted line denotes the effective selection coefficient estimated by fitting the approximated formula in Equation 10 to the expected heterozygosity pattern. The dotted-dashed shaded line is the mean selection coefficient estimated by the composite-likelihood method of Kim and Stephan (2002) on 100 coalescence simulations of selective sweep with a decreasing selection coefficient following Equation 24. Parameters are the same as in Figure 2A, except ɛ = 1/(4N_es_o) with N_e = 10,000 and s_o = −am_o/(σ² + ω²).