(A) Mean log-fitness effect of mutations as a function of the distance to the optimum. From up to down: Q = 0.5, 1, 1.5, 2, 2.5, 3. (B) The distribution of epistasis among pairs of mutations for several values of Q and of the number of dimensions. The mean effect of mutation on logarithm of fitness was normalized to be equal to −0.2 in all conditions. (C) Impact of parameter Q on the average epistasis among pairs of mutations as a function of the mean effect of mutations on logarithm of fitness. Q takes values 0.5, 1, 1.5, 2, 2.5, and 3 with low values in light gray and higher values in darker gray (D) Impact of the distance (translated in log-fitness) to the optimum on the relationship between epistasis and effect of mutations for various values of Q. Black line, Q = 1.5, z = 0; dashed black line, Q = 1.5, z = −0.1; dotted black line, Q = 1.5, z = −0.2; gray line, Q = 2.5, z = 0; dashed gray line, Q = 2.5, z = −0.1; dotted gray line, Q = 2.5, z = −0.2.

(A) Fitness as a function of the distance to the optimum and parameter Q, and distance distributions at MSFDE. Top panel: log-fitness as a function of the distance to the optimum for Q_max = 2.5 (solid line) and Q_min = 0.8 (shaded line). Second, third, and fourth panels: density distribution of the distance to the optimum at MSFDE (“short-term equilibrium”) for a given Q: Q_max (solid line) and Q_min (shaded line). Second panel: strong selection (K_max < 1, N = 100, n = 25), an increase of Q is always advantageous. Third panel: intermediate selection (K_min < 1 < K_max, N = 100, n = 200). Fourth panel: intense drift (K_min > 1, N = 100, n = 700), a decrease if Q is always advantageous. (B) Distribution of Q at MSFDE as a function of phenotypic complexity. For a population size of 30, and phenotypic complexity of 10 (solid line), 15, 20, 25, 30, 40, 50, and 100 (dotted line), we see that for intermediate values of n, high and low values of Q have a high probability, which explains the dependency on initial conditions observed in simulations. (C) Impact of phenotypic complexity on the average value of Q for population sizes of 30 in the simulation model, with several initial values of Q. The average size of a mutation was 10. Each point is the average of 10 simulations.

(A) Impact of Q on the mean effect of mutation on log-fitness at MSFDE when selection favors low values of Q. Populations are assumed to be at the mean distance for their value of Q at MSFDE. For all sizes of mutations in phenotypic space, the average effect of mutation on fitness decreases with decreasing Q. n = 10, N = 100, and the standard deviations of the mutation size in phenotypic space are = 0.05 (solid line), = 0.1, = 0.2, = 0.3, and = 0.5 (dotted line). (B) Impact of Q on the mean effect of mutation on log-fitness at MSFDE when selection favors high values of Q. Depending on the size of mutations in phenotypic space, the average effect of mutation on fitness increases or decreases with increasing Q. n = 100, N = 30, and the standard deviations of mutation size in phenotypic space are = 0.05 (solid line), = 0.1, = 0.2, = 0.3, and = 0.5 (dotted line).

(A) Representation of Fisher's model in two dimensions. Here an organism is defined by a combination of its traits' x and y values; its fitness is a function of the distance of its phenotype (x₁, x₂) to the optimum genotype (0, 0), and therefore fitness isoclines are circular. Mutations move an organism from one point of the phenotypic space to another. (B) Impact of parameter Q on fitness decline. For distances to the optimum >1, low values of Q have a higher fitness than high values, while the opposite is true for distances <1. Q takes values 0.5, 1, 1.5, 2, 2.5, and 3 with low values in light gray and higher values in darker gray. (C) Schematic representation of the effect of a mutation for several values of Q: log-fitness as a function of the phenotype (x₁, x₂) in two dimensions for Q = 0, Q = 0.5, Q = 1, Q = 2, Q = 3, Q = 10. The thick black line represents the mean displacement caused by a mutation at the optimum, whereas the thin black line represents the mean displacement when away from the optimum [at the position denoted by the white (Q = 0, 0.5, 1, 2) or black dot (Q = 3, 10)]. The dotted white (Q = 0, 0.5, 1, 2) or gray (Q = 3, 10) line is the fitness isocline at this distance. Mutations tend to be more deleterious when Q increases. Note that for Q = 0, log-fitness is −1 away from the optimum and 0 at the optimum, as in models with a single-fit “master sequence.” The case Q = 10 (or Q → ∞) corresponds to a flat neutral space (for a distance smaller than unity) and lethal genotypes (for a distance greater than unity).

(A) Impact of population size on the average value of Q at MSFDE for several phenotypic complexities. Solid line, phenotypic complexity, n = 100; dashed line, n = 30; dotted line, n = 10. (B) Impact of phenotypic complexity on the average value of Q at MSFDE for several population sizes. Solid line, population size N = 100; dashed line, N = 30; dotted line, N = 10.

Impact of the mutation rate on the evolution of Q. The mutation rate was set either to 0.001 or to 0.1. The position of the switch from maximal to minimal value of Q was decreased when mutation rate increased. Here population size was 100, and several effects of mutation size gave the same result.

(A) Impact of Q on the mean epistasis among mutations at MSFDE when selection favors low values of Q (same conditions as in Figure 6A). (B) Impact of Q on the mean epistasis among mutations at MSFDE when selection favors high values of Q (same conditions as in Figure 6B).