Rate of adaptation in sexuals and asexuals: a solvable model of the Fisher-Muller effect

Abstract

The adaptation of large asexual populations is hampered by the competition between independently arising beneficial mutations in different individuals, which is known as clonal interference. In classic work, Fisher and Muller proposed that recombination provides an evolutionary advantage in large populations by alleviating this competition. Based on recent progress in quantifying the speed of adaptation in asexual populations undergoing clonal interference, we present a detailed analysis of the Fisher-Muller mechanism for a model genome consisting of two loci with an infinite number of beneficial alleles each and multiplicative (nonepistatic) fitness effects. We solve the deterministic, infinite population dynamics exactly and show that, for a particular, natural mutation scheme, the speed of adaptation in sexuals is twice as large as in asexuals. This result is argued to hold for any nonzero value of the rate of recombination. Guided by the infinite population result and by previous work on asexual adaptation, we postulate an expression for the speed of adaptation in finite sexual populations that agrees with numerical simulations over a wide range of population sizes and recombination rates. The ratio of the sexual to asexual adaptation speed is a function of population size that increases in the clonal interference regime and approaches 2 for extremely large populations. The simulations also show that the imbalance between the numbers of accumulated mutations at the two loci is strongly suppressed even by a small amount of recombination. The generalization of the model to an arbitrary number L of loci is briefly discussed. If each offspring samples the alleles at each locus from the gene pool of the whole population rather than from two parents, the ratio of the sexual to asexual adaptation speed is approximately equal to L in large populations. A possible realization of this scenario is the reassortment of genetic material in RNA viruses with L genomic segments.

Keywords: advantage of recombination; clonal interference; speed of adaptation.

Figure 1

Log-mean fitness $\text{ln}\hspace{0.17em}{\overline{w}}_t$ of the infinite population model as a function of time for r = 0, r = 10⁻⁹, and r = 1 (from bottom to top) with U = 0.1 and s = 0.02. As argued in the text, the speed does not depend on r once r is nonzero.

Figure 2

Frequency distribution of the total number of mutations for the infinite population model at generations 895, 900, 905, …, 930 (left to right) with parameters r = 10⁻⁹, U = 0.1, and s = 0.02.

Figure 3

Speed of adaptation of finite asexual populations as a function of N on a double logarithmic scale for U = 10⁻⁶ and s = 0.01. The numerical solutions of Equations 12 and 13 are drawn for comparison with the simulation data. As anticipated, the ad hoc modification (Equation 13) provides a more accurate estimate.

Figure 4

Ratio of the sexual adaptation speed, v_s(r, U), to the asexual speed at half mutation rate, v_a(U/2), as a function of population size N. Recombination rates are r = 0 (open inverted triangle), 10⁻⁵ (open square), 10⁻⁴ (solid triangle), 10⁻³ (open triangle), 10⁻² (solid circle), 10⁻¹ (open circle), and 1 (solid square) from bottom to top, and U = 10⁻⁶ and s = 0.01 are used throughout. The scaling relation in Equation 14 predicts that v_s(r, U)/v_a(U/2) = 2. Note that two data sets for r = 0.1 (open circle) and r = 1 (solid square) are indiscernible.

Figure 5

Ratio of sexual to asexual speed of adaptation, v_s(r, U)/v_a(U), as a function of population size N on a semilogarithmic scale. Recombination rates are r = 10⁻⁵, 10⁻⁴, 10⁻³, 10⁻², 10⁻¹, and 1 from bottom to top, and U = 10⁻⁶ and s = 0.01 are used as before. As in Figure 4, the two data sets for r = 1 and r = 0.1 are hardly discernible.

Figure 6

Numerical verification of the Guess relation. The figure compares the logarithmic mean fitness divided by generation time, $\langle \text{ln}\hspace{0.17em}{\overline{w}}_t\rangle /t$, to the right-hand side of Equation 16. The data are obtained from simulations with r = 10⁻³, U = 10⁻⁶, and s = 0.01. For t → ∞, both curves intersect the ordinate at the same point, which equals the asymptotic speed of adaptation.

Figure 7

The mutation number imbalance (MNI) V vs. population size N for r = 0 (open inverted triangle), 10⁻⁵ (open square), 10⁻⁴ (solid triangle), 10⁻³ (open triangle), 10⁻² (solid circle), 10⁻¹ (open circle), and 1 (solid square) from top to bottom. Other parameter values are U = 10⁻⁶ and s = 0.01. The symbols for r ≥ 10⁻² essentially overlap. For comparison, the analytic prediction V_a = v_a/s for asexuals is drawn as a line.

Figure 8

Mean logarithmic fitness $\langle \text{ln}\hspace{0.17em}{\overline{w}}_t\rangle$ vs. time t in the presence (symbols) and absence (lines) of two-site mutations. The mutation schemes employed in the two cases are given in Equation A19 and Equation 9, respectively. The population size is N = 10²⁰ and the probability for a single mutation is U = 10⁻⁶. The two data sets are indistinguishable, which implies that multiple-site mutations do not play any role.

Figure 9

Ratio of the sexual adaptation speed in the three-locus model, v_s(r, U), to the asexual speed v_a at mutation rate U/3 as a function of population size N. Recombination rates are r = 0 (open inverted triangle), 10⁻⁵ (open square), 10⁻⁴ (solid triangle), 10⁻³ (open triangle), 10⁻² (solid circle), 10⁻¹ (open circle), and 1 (solid square) from bottom to top, and U = 10⁻⁶ and s = 0.01 are used throughout. The scaling relation in Equation 23 predicts that v_s(r, U)/v_a(U/3) = 3. Note that two data sets for r = 0.1 (open circle) and r = 1 (solid square) are indiscernible.

Figure 10

MNI for the three-locus model with recombination rates r = 0 (open inverted triangle), 10⁻⁵ (open square), 10⁻⁴ (solid triangle), 10⁻³ (open triangle), 10⁻² (solid circle), 10⁻¹ (open circle), and 1 (solid square) from top to bottom. Other parameter values are U = 1.5 × 10⁻⁶ and s = 0.01. The symbols for r ≥ 10⁻² essentially overlap.