How does epistasis influence the response to selection?

Abstract

Much of quantitative genetics is based on the 'infinitesimal model', under which selection has a negligible effect on the genetic variance. This is typically justified by assuming a very large number of loci with additive effects. However, it applies even when genes interact, provided that the number of loci is large enough that selection on each of them is weak relative to random drift. In the long term, directional selection will change allele frequencies, but even then, the effects of epistasis on the ultimate change in trait mean due to selection may be modest. Stabilising selection can maintain many traits close to their optima, even when the underlying alleles are weakly selected. However, the number of traits that can be optimised is apparently limited to ~4N_e by the 'drift load', and this is hard to reconcile with the apparent complexity of many organisms. Just as for the mutation load, this limit can be evaded by a particular form of negative epistasis. A more robust limit is set by the variance in reproductive success. This suggests that selection accumulates information most efficiently in the infinitesimal regime, when selection on individual alleles is weak, and comparable with random drift. A review of evidence on selection strength suggests that although most variance in fitness may be because of alleles with large N_es, substantial amounts of adaptation may be because of alleles in the infinitesimal regime, in which epistasis has modest effects.

Figure 1

The effect of selection on the mean and variance components in the presence of epistasis. Directional selection, β=0.2 (solid line) is contrasted with the neutral case (dashed line); shaded areas indicate ±1 s.d. The left panel shows the change in mean from its initial value and the right panel shows the additive variance, V_A, and the additive × additive variance, V_{A A} (lower pair of curves). Only the genic components of variance are shown; random linkage disequilibria make no appreciable difference on average. There are M=3000 loci, and N=100 haploid individuals. Alleles are given equal main effects but random sign . Sparse pairwise epistasis is represented by choosing a fraction 1/M of pairwise interactions, ω_{ι j}, from a normal distribution with s.d. . The trait is now defined as z=δ.γ+δ.ω.δ^T, where δ=±(1/2). Initial allele frequencies are drawn from a U-shaped β-distribution, mean p̂=0.2 and variance 0.2 p̂q̄. Individuals are produced by Wright–Fisher sampling from parents chosen with probability proportional to W=e^{β z}. For each example, three sets of allelic and epistatic effects are drawn and for each of those, three populations are evolved; this gives 9 replicates in all.

Figure 2

The mean and variance of offspring plotted against components of the parents' trait values. Top left: additive component of offspring, A_O, against the mean of the parents' additive component, A_P. The line represents A_O=A_P. Top right: the same, but for the additive × additive components. The line shows a linear regression. Bottom left: additive variance among offspring, V_A,O against the mean additive components of the parents, A_P. Bottom right: additive × additive variance of offspring against the mean additive × additive component of the parents. Lines in the bottom row show quadratic regressions. The example shows a nonadditive trait under selection β=0.2, with M=3000 loci and N=100 haploid individuals, as in Figure 1. At generation 20, 200 pairs of minimally related parents (F=0.165) were chosen, and 1000 offspring were generated for each pair. For each offspring, the components of trait value were calculated relative to the allele frequencies, p, in the base population. Defining genotype by X=0, 1, these components are A=ζ.(α+(ω+ω^T).(p−1/2)), AA=ζ.ω.ζ^T, where ζ=X−p.

Figure 3

The effective dimension of trait variation in short versus long term. Left: the fraction of variance explained by the largest 1, 2, …, eigenvectors for 10, 100, 1000 traits (black, blue, red, top to bottom), measured in the final generation. Right: the same, but for a population that contains all mutations that fixed over 50 000 generations (that is, an F2 between the ancestral and derived population). An additive infinite sites model was simulated, with free recombination, stabilising selection exp(−|z|²/(2V_s)), V_s=100, N=100 haploid individuals, and mutation rate U=0.1 per genome per generation. Mutations have magnitude |α| drawn from an exponential with mean 1 with random direction. In these simulations, the variance of each trait mean around the optimum is close to the predicted V_s/(2N)=0.5, causing a loss of fitness 1/(4N)=0.0025 per trait. A full colour version of this figure is available at the Heredity journal online.

Figure 4

Adaptation on a rugged landscape. Each panel plots the genetic variance against the trait mean for an additive trait under stabilising selection towards an optimum at zero; fitness is exp (−Sz²/2), with S=0.005. There are 100 loci each with two alleles and symmetric mutation μ=0.0005. Populations are evolved for 10⁴ generations from an initial β distribution with variance Fpq, with F=0.5, p=0.1, 0.5, 0.9 (black, blue, grey). Large dots show the final state for an infinitely large population (100 replicates), whereas small dots show results for a diploid population of N=3 × 10⁴ individuals. The upper panel is for equal effects (γ=1) and the lower panel for unequal effects, drawn from an exponential distribution with mean γ=1. A full colour version of this figure is available at the Heredity journal online.

Figure 5

Evolution of the additive variance under stabilising selection in the presence of epistasis. Top: variance components are plotted against the mutation rate, μ, for the epistatic model (solid lines show the mean, and grey areas the s.d.). The upper line shows the total genotypic variance, V_G, that is the sum of additive and nonadditive components (middle, lower lines). This is compared with the variance, V^*_A, under the corresponding additive model, starting from the same allele frequencies and the same additive effects; this is indistinguishable from V_G (upper line). The (small) s.d. among 10 replicates is indicated. Bottom: the variance of additive effects at the beginning and end; this is proportional to the mutational variance V^*_m. Fitness is 1−z²/(2V_s), V_s=5. The trait, z, is the sum of exponentially distributed main effects plus random pairwise interactions. There are M=1000 biallelic loci, but otherwise parameters are as in Figures 1 and 2. A single realisation of the genetic architecture is used with 10 replicates for each mutation rate starting from allele frequencies drawn from a β-distribution with mean=0.2 and variance 0.2. Simulations are deterministic and run for 50 000 generations. Linkage equilibrium is imposed, so that only allele frequencies are followed.

Figure 6

Comparison between quadratic and truncation selection on the deviation from a multitrait optimum. The left panel shows the distribution of distance from the optimum, r, for n=3, 10, 30, 100 traits (left to right), under quadratic stabilising selection; the upper curve shows the fitness, exp(−(S/2) r²). The right panel shows the same, but for truncation selection in which only individuals with r<2 reproduce. Simulations are of 100 haploid individuals, each with 100 unlinked loci; alleles have continuously distributed vectors of effects. The trait is the sum of effects of each locus. Mutation rate is 0.001 per locus, and adds a random Gaussian with s.d. σ=0.1 for each trait. Results are averaged over generations 4000 to 20 000. Under quadratic stabilising selection, the reduction in mean fitnesses is 0.014, 0.046, 0.127 and 0.318 for 3,…, 100 traits. In contrast, under truncation selection the loss of mean fitness (that is, the fraction of offspring with r>2) are 0.00247, 0.00642 and 0.0269 for 3, 10, 30 traits. With 100 traits under truncation selection, the population does not equilibrate: loci fix deleterious alleles, leading to a decline in fitness through Muller's Ratchet.

Figure 7

Each plot shows the increase in information, the fitness flux multiplied by 2N, and the cumulative variance in fitness, multiplied by 2N (red: ΔH, blue: 2N Φ, black: 2N V). Initially, the population is in the stationary state, with μ=0.0025 and a haploid population of N=50. The left plot shows an abrupt increase in selection to s=0.05, whereas the right plot shows a linear increase from s=0 to s=0.05 over 10 000 generations. In both examples, the increase in information (red) is smaller than the fitness flux (blue) that in turn is smaller than the cumulative variance in fitness (black). However, the increase in information is closer to the upper bound set by fitness flux when selection increases gradually. Numerical values are calculated using the Wright–Fisher transition matrix. A full colour version of this figure is available at the Heredity journal online.

Figure 8

The gain in information, ℐ, and the expected total heterozygosity, , plotted against initial allele frequency, p₀ (left and middle respectively). The right plot shows the ratio between the information gain and the expected total fitness variance. Selection strength is 4N_es=α=0.125, 0.25,…, 8 (black…red). In the limit α→0, the scaled ratio tends to 1/16, independent of p₀. A full colour version of this figure is available at the Heredity journal online.