Fitness variation in isogenic populations leads to a novel evolutionary mechanism for crossing fitness valleys

Individuals in a population often have different fitnesses even when they have identical genotypes, but the effect of this variation on the evolution of a population through complicated fitness landscapes is unknown. Here, we investigate how populations with non-genetic fitness variation cross fitness valleys, common barriers to adaptation in rugged fitness landscapes in which a population must pass through a deleterious intermediate to arrive at a final advantageous stage. We develop a stochastic computational model describing the dynamics of an asexually reproducing population crossing a fitness valley, in which individuals of the same evolutionary stage can have variable fitnesses. We find that fitness variation that persists over multiple generations increases the rate of valley crossing through a novel evolutionary mechanism different from previously characterized mechanisms such as stochastic tunneling. By reducing the strength of selection against deleterious intermediates, persistent fitness variation allows for faster adaptation through rugged fitness landscapes.

The shape of simulated steady-state population fitness distributions generated by stable fitness variation is insensitive to the shape of the input distribution of fitness effects, F. a-c. Simulated population-level relative fitness distributions with gamma (a), double-exponential (b), and centered Bernoulli (c) fitness effect distributions F. During each reproductive event, the daughters draw a new fitness alteration from the distribution F. Their new fitness is equal to the mother's fitness multiplied by this stochastic fitness alteration value. Different shapes of F may therefore affect the properties of the population fitness distribution at steady-state. However, as the fitness effect lifetime τ increases, we found the population fitness distributions converge to a single distribution, exemplified by the results given by the permanent fitness alteration model (τ = ¥). In particular, the modified Bernoulli fitness effect distribution (Methods) generates smoother steady-state fitness distributions when fitness effects are more stable, i.e. as τ goes to infinity. The distinctive shape and discontinuities of the fitness effect distribution have less of an impact on the overall fitness distribution as more fitness effects accumulate in the population, which occurs at higher values of τ. d. Steady-state population fitness distributions for τ = ¥ with different fitness effect distribution shapes F. Different distributions have the same variance V = 10 -4 . At longer fitness effect lifetimes τ, the shape of the overall fitness distribution is insensitive to the shape of F. All simulated distributions in this figure are estimated by smoothing data combined from 1000 separate simulations of N = 100 individuals each (100,000 individuals total) of 3600 generations. See Table 1 for other parameter values.  Figure 4, where a log-normal distribution was used instead. c, d. The final stage fixation probability (c) and the tunneling probability (d) of a population with a double exponential distribution of fitness effects (Methods) over different valley depths r1 are similar to those presented in Figure 4, where a log-normal distribution was used instead. Colors indicate values of τ; the same color scheme was used in all panels. See Table 1 for other parameter values. 10,000 simulations were run per condition, and vertical bars represent 95% confidence intervals.

Final stage fixation probability
Intermediate type fitness disadvantage r1 Tunneling probability Intermediate type fitness disadvantage r1 b.

Final stage fixation probability
Intermediate type fitness disadvantage r1 Tunneling probability Intermediate type fitness disadvantage r1  Table 1 for other parameter values. All panels reflect 10,000 simulations per condition. Vertical bars are 95% confidence intervals.

Final stage fixation probability
Initial mutation rate u1 a.

Tunneling probability
Initial mutation rate u1 b.

Final stage fixation probability
Second mutation rate u2 c.

Tunneling probability
Second mutation rate u2 d. As expected, the adaptation probability increases with the final type fitness advantage r2, for all values of r2 tested. Colors represent fitness effect lifetimes τ. b. The tunneling probability also increases with r2. A lower final-stage fitness r2 decreases the rate at which final stage individuals fix, making tunneling more difficult. Stable variation leads to lower tunneling probabilities at values of r2 that are sufficiently large to support stochastic tunneling in the zero-variance model. Colors represent the same fitness lifetime values τ as in (a). c. The effect of r2 changes for different values of r1 in the persistent nongenetic variation model, but almost no difference in the tunneling probability relative to the zero-variance Moran model was observed when τ = 1, i.e. fitness variation does not persist through reproduction. See Table 1 for other parameter values. 10,000 simulations per condition. Vertical bars are 95% confidence intervals. difference in tunneling probability relative to zero-variance  Table 1 for other parameter values. 10,000 simulations per point. Vertical bars are 95% confidence intervals. Here, the persistence length of nongenetic alterations is no longer deterministic, but is geometrically distributed with an expected lifetime τ. Under this model, a similar increase in the valley crossing rate is found, except that this effect is magnified at lower values of τ. b. The relative tunneling probability again decreases as the expected lifetime τ increases. The tunneling probability decreases more rapidly than in simulations with deterministic fitness effect lifetimes for increasing τ. See Table 1 for other parameter values. 10,000 simulations per condition. Vertical bars are 95% confidence intervals. Simulated tunneling probability (a) and adaptation probability (b) depending on both the relative intermediate fitness r1 and the estimated variance of the overall fitness distribution in the model with permanent nongenetic fitness variation (τ = ¥). The estimated variances on the x-axis describe the width of the overall simulated population fitness distribution, with higher estimated variances corresponding to more inter-individual variability in fitness. The estimated variances were calculated as the mean sample variance of the population fitness distribution from 1000 independent simulations, after 3600 generations. c, d. Simulated tunneling probability (c) and adaptation probability (d) depending on both the relative intermediate fitness r1 and the variance of the overall fitness distribution in the model where fitness changes do not persist through reproduction (τ = 1). In (c) and (d), the population fitness distribution has exactly the same variance V as the fitness effect distribution, so the fitness variance on the x-axis was not estimated by simulation. In all panels, simulated data c. d.