Fig 1.
Positive epistasis enhances adaptation.
Wright-Fisher population of 500 genomes has been simulated for 20 generations, starting from uniformly wild-type (best-fit at E = 0) population. The adaptation rate A = dW/dt (bottom row) and substitution rate Vs0 = s0 df/dt (upper row) were averaged over 300 runs and are plotted as a function of the epistatic strength, E. The selection coefficient s0 and E are the same for all sites. Parameters in (a-h): |s0| = 0.2, total site number L = 300, mutation rate per genome μL is shown (colors). The binary connectivity matrix Tij is random with ~1 interaction per site. Each column corresponds to a different sign of s0 and E (shown). (a, d, e, h) In the two cases of reciprocal epistasis, the evolution rates demonstrate strong non-linear dependence on the epistatic strength.
Fig 2.
A pair of interacting sites in a long genome.
(left) Open and filled circles: wild type 0 and mutated allele 1. Red line: existing interaction. Black line: potential interactions between sites. Dashed line: negligible interaction. Grey box: the rest of genome. (B-E) Derivation of the universal footprint of epistasis explained in the text. W is total fitness, S(W-Wpair) is entropy of the rest of genome, and fii are the haplotype frequencies. Parameter E represents the relative strength of epistasis (Eq 1).
Fig 3.
Universal footprint of epistasis (UFE) predicts epistatic strength in a broad time range.
Value of E estimated from Eq 6 is plotted as a function of the actual value of E, where (a-d) correspond to different time points. Each dot represent a single Monte-Carlo run. Initial population is randomized with f = 0.5. Haplotype frequencies in Eq 6 are averaged over sites and pairs. Blue: known epistatic pairs. Red: the same number of randomly chosen pairs. Parameters: L = 300, s0 = 0.05, N = 500, μL = 0.5, ~1 interaction per site.
Fig 4.
Long genome of interacting pairs.
Top: Linked interacting pairs with different haplotypes and their fitness values Wij. (a—g) Flow chart of the derivation of the universal footprint of epistasis (see the text or S1 Appendix).
Fig 5.
Epistasis causes strong linkage disequilibrium: Analytics and simulation.
(a, b, c) Correlation coefficients D11, D10 and mutation frequency f are shown as a function of E. D11, D10, are calculated from Eq 6 using simulated values of fij and f averaged over sites and pairs. Color lines correspond to the average over 300 runs, and the shaded areas show the standard deviation among runs, for epistatic pairs (blue) and the same number of random pairs (red). Dotted black line is the analytic prediction. Parameters: N = 500, s0 = 0.05, L = 300, μL = 0.5, t = 800, f0 ≅ 0.1. Initial population is randomized with f = 0.5. Thus, simulation agrees well with analytic predictions.
Fig 6.
UFE is preserved for different topologies at moderate epistasis' strengths.
Here we show the dependences on E for (a, b) correlation coefficients, (c) mutation frequency f, and (d) UFE on E for the five topologies in Fig 6. UFE is the estimate of E (Eq 6) from haplotype frequencies f01, f11 derived analytically for each topology (S1 Appendix). Parameters: f0 = 1/100. Thus UFE is exact for the isolated pairs, and overestimates E at large E for other topologies. Asymptotic expressions are given in S2 Table.
Fig 7.
Examples of epistatic network.
Filled and open circles denote mutated and wild-type genomic sites, respectively. Epistatic interactions between sites are shown by black and red lines. Red lines show clusters of mutated sites. ki is the number of the cluster with i sites, bi is the bond number per cluster. Different topologies correspond to a) isolated pairs, b) isolated triple arches, where each site has two epistatic partners, c) double arches, where three sites are involved in two epistatic associations, d) long connected chain where each site forms two pairs, (a-d) show connection topology but not the actual site order. (e) Binary tree: possible site order and the equivalent tree structure.