Effect of varying epistasis on the evolution of recombination

Abstract

Whether recombination decelerates or accelerates a population's response to selection depends, at least in part, on how fitness-determining loci interact. Realistically, all genomes likely contain fitness interactions both with positive and with negative epistasis. Therefore, it is crucial to determine the conditions under which the potential beneficial effects of recombination with negative epistasis prevail over the detrimental effects of recombination with positive epistasis. Here, we examine the simultaneous effects of diverse epistatic interactions with different strengths and signs in a simplified model system with independent pairs of interacting loci and selection acting only on the haploid phase. We find that the average form of epistasis does not predict the average amount of linkage disequilibrium generated or the impact on a recombination modifier when compared to results using the entire distribution of epistatic effects and associated single-mutant effects. Moreover, we show that epistatic interactions of a given strength can produce very different effects, having the greatest impact when selection is weak. In summary, we observe that the evolution of recombination at mutation-selection balance might be driven by a small number of interactions with weak selection rather than by the average epistasis of all interactions. We illustrate this effect with an analysis of published data of Saccharomyces cerevisiae. Thus to draw conclusions on the evolution of recombination from experimental data, it is necessary to consider the distribution of epistatic interactions together with the associated selection coefficients.

Figure 1.

Frequency f of allele 1 at a recombination modifier linked to a two-locus/two-allele system as a function of time. The graph displays the linear and the saturation phase; the inset displays the perturbation and the linear phase. Parameters are m = 0.001, r = 0, δr = 0.5 × 10⁻², and .

Figure 2.

Histogram for epistasis ɛ = w₁w₄ − w₂w₃ (A and C) and linkage disequilibrium D (B and D) for 10⁵ independent pairs of loci, where the fitnesses w_i are drawn according to the following distributions: Dashed lines refer to the mean values of D and ɛ, respectively. (A and B) First the double-mutant fitness w₄ is drawn uniformly between 0 and 0.5. Then the epistasis ɛ is drawn uniformly between −w₄ and w₄, resulting in a single-mutant fitness of . (C and D) First the single-mutant fitness w₂ = w₃ is drawn uniformly between 0.8 and 1. Then the epistasis ɛ is drawn uniformly between −() and (), resulting in a double-mutant fitness of .

Figure 2.

Histogram for epistasis ɛ = w₁w₄ − w₂w₃ (A and C) and linkage disequilibrium D (B and D) for 10⁵ independent pairs of loci, where the fitnesses w_i are drawn according to the following distributions: Dashed lines refer to the mean values of D and ɛ, respectively. (A and B) First the double-mutant fitness w₄ is drawn uniformly between 0 and 0.5. Then the epistasis ɛ is drawn uniformly between −w₄ and w₄, resulting in a single-mutant fitness of . (C and D) First the single-mutant fitness w₂ = w₃ is drawn uniformly between 0.8 and 1. Then the epistasis ɛ is drawn uniformly between −() and (), resulting in a double-mutant fitness of .

Figure 2.

Histogram for epistasis ɛ = w₁w₄ − w₂w₃ (A and C) and linkage disequilibrium D (B and D) for 10⁵ independent pairs of loci, where the fitnesses w_i are drawn according to the following distributions: Dashed lines refer to the mean values of D and ɛ, respectively. (A and B) First the double-mutant fitness w₄ is drawn uniformly between 0 and 0.5. Then the epistasis ɛ is drawn uniformly between −w₄ and w₄, resulting in a single-mutant fitness of . (C and D) First the single-mutant fitness w₂ = w₃ is drawn uniformly between 0.8 and 1. Then the epistasis ɛ is drawn uniformly between −() and (), resulting in a double-mutant fitness of .

Figure 2.

Histogram for epistasis ɛ = w₁w₄ − w₂w₃ (A and C) and linkage disequilibrium D (B and D) for 10⁵ independent pairs of loci, where the fitnesses w_i are drawn according to the following distributions: Dashed lines refer to the mean values of D and ɛ, respectively. (A and B) First the double-mutant fitness w₄ is drawn uniformly between 0 and 0.5. Then the epistasis ɛ is drawn uniformly between −w₄ and w₄, resulting in a single-mutant fitness of . (C and D) First the single-mutant fitness w₂ = w₃ is drawn uniformly between 0.8 and 1. Then the epistasis ɛ is drawn uniformly between −() and (), resulting in a double-mutant fitness of .

Figure 3.

Linkage disequilibrium (A) and strength of selection on the recombination modifier (B) in a two-locus system as a function of the single- and double-mutant fitnesses w₂ = w₃ and w₄. Note that in B the z-axis has been flipped to correspond to A. Different colors indicate different levels of epistasis. Other parameters are m = 10⁻³ (backward mutation rate = forward mutation rate), r₀ = 0, and δr = 0.5 × 10⁻³ (B only).

Figure 3.

Linkage disequilibrium (A) and strength of selection on the recombination modifier (B) in a two-locus system as a function of the single- and double-mutant fitnesses w₂ = w₃ and w₄. Note that in B the z-axis has been flipped to correspond to A. Different colors indicate different levels of epistasis. Other parameters are m = 10⁻³ (backward mutation rate = forward mutation rate), r₀ = 0, and δr = 0.5 × 10⁻³ (B only).

Figure 4.

The sensitivity of a modifier to selection coefficients is plotted as a function of the recombination rate between the selected loci r₀. The sensitivity is measured as Log₁₀(s₁/s₀), where s₁ is the selection on the modifier under weak selection (w₂ = w₃ = 0.99) and s₀ is the selection on the modifier under strong selection (w₂ = w₃ = 0.1), holding epistasis constant (ɛ = −0.01). Parameters are: m = 0.001; linkage between selected loci, r₀ variable, δr = 0.5 × 10⁻³; and linkage between the recombination modifier and the first selected locus fixed at 0.5 × 10⁻³.

Figure 5.

Strength of selection on the modifier in a two-locus system as a function of single-mutant fitness w₂. The selection on the recombination modifier is positive save for very weak selection on the single mutants (i.e., ). The double-mutant fitness is fixed at w₄ = 0.5. Other parameters are m = 10⁻³, r₀ = 0, and δr = 0.5 × 10⁻³.

Figure 6.

Time until linkage disequilibrium is within 10% of its expected value (T) as a function of w₂ = w₃ and w₄. Other parameters are m = 10⁻³ and r = 0.

Figure 7.

Strength of selection on the modifier linked to two independent pairs of loci. The first pair has fitnesses w₁ = 1, w₂ = w₃ = 0.99, w₄ = 0.9 and thus a negative epistasis ɛ₁ = −0.089. The second pair has fitnesses w₂ = w₃ = 0.9 and w₄ = 0.9² + ɛ₂, where the positive epistasis ɛ₂ is plotted on the x-axis. The dashed line refers to the case where positive and negative epistases are of the same strength (i.e., ɛ₂ = 0.089). Other parameters are m = 10⁻³, r₀ = 0, and δr = 0.5 × 10⁻³.

Figure 8.

(A) Fitness effects of single and double deleterious mutants from Szafraniec et al. (2003). (B) The selection induced on a modifier allele that increases recombination by 10%, as estimated by the leading eigenvalue minus one in a local stability analysis (Otto and Feldman 1997). Selection on a modifier is strongly influenced by a single pair of mutant alleles, with very weak effects in single mutants relative to double mutants (arrows).

Figure 8.

(A) Fitness effects of single and double deleterious mutants from Szafraniec et al. (2003). (B) The selection induced on a modifier allele that increases recombination by 10%, as estimated by the leading eigenvalue minus one in a local stability analysis (Otto and Feldman 1997). Selection on a modifier is strongly influenced by a single pair of mutant alleles, with very weak effects in single mutants relative to double mutants (arrows).