An illustration of our model. (a) The simplest case, K = 2. Here the wild-type has fitness 1, and there is a possible double-mutant with fitness 1 + s > 1. The single-mutant, however, has fitness 1 – δ₁ < 1. The mutation rate from the wild-type to the single-mutant is μ₀, while the mutation rate of the single-mutant to the double-mutant is μ₁. (b) A slightly more complex example where K = 3. The wild-type again has fitness 1, single-mutants have fitness 1 – δ₁, double-mutants have fitness 1 – δ₂, and the triple-mutant is advantageous with fitness 1 + s.

Schematic of the different parameter regimes for the case K = 2. In the sequential fixation regime, the population usually crosses the valley by first fixing the single-mutants. For N∣δ₁∣ < 1 (the neutral sequential fixation regime), the single-mutants are approximately neutral, and the expected time for the beneficial double-mutants to establish is just τ ≈ 1/μ₀. For larger Nδ₁ (the deleterious sequential fixation regime, between the dotted and solid lines), the process is slowed by selection against single-mutants, and τ ≈ 1/Nμ₀ρ₁, where ρ₁ is given by Eq. (29). For larger values of N or δ₁, the valley-crossing dynamics are dominated by tunneling. In the neutral stochastic tunneling regime, τ is given by the neutral limit in Eq. (28), while in the deleterious tunneling regime, τ is given by the strongly-deleterious limit of Eq. (28). In the neutral semi-deterministic tunneling regime, the initial increase of the single-mutant class is approximately deterministic, and τ is given by the first line of Eq. (35). In the deterministic regime, the dynamics are fully deterministic, and τ is given by Eq. (37). Note that the transitions between regimes are not sharp, but represent smooth crossovers. Note also that the parameter regimes describing weakly deleterious intermediates also apply to weakly beneficial intermediates (the region of the figure where δ₁ < 0).

Comparison of theoretical predictions for τ, the expected time for the beneficial mutants to establish, to simulation results. These plots are for the case K = 2, with double-mutants having selective advantage s = 0.1. The crosses are the simulation results, while the solid curves show the theoretical predictions. The different parameter regimes are indicated on the plots. In (a) and (b), population size N is varied while the other parameters are held constant. In (a), μ₀ = 10⁻⁵, μ₁ = 10⁻⁴, and δ₁ = 2×10⁻⁴, so that the single-mutants are effectively neutral. In (b), μ₀ = 10⁻⁵, μ₁ = 10⁻⁶, and δ₁ = 7 × 10⁻³, so that the single-mutants are strongly deleterious for intermediate population sizes. In (c), the selective disadvantage δ₁ of the single-mutants is varied, with N = 10⁵, μ₀ = 10⁻⁶, and μ₁ = 4 × 10⁻⁵. Note that the tunneling-based theoretical prediction Eq. (27) is accurate even for negative values of δ₁, corresponding to beneficial single-mutants. In (d), the rate of mutation from single- to double-mutants, μ₁, is varied, with N = 10⁵, μ₀ = 10⁻⁶, and δ₁ = 3 × 10⁻³.

Characteristic situations where there are several different pathways to a final advantageous genotype or genotypes. In each panel, the initial genotype is at the left, the final advantageous genotype or genotypes at the right, and all intermediate genotypes are assumed to be neutral or deleterious. In (a), the population initially has genotype abcd, and there are two entirely disjoint pathways by which evolution can proceed: either mutation A and then mutation B can occur, or mutation C and then mutation D. The overall rate at which a population acquires one of the two fitter genotypes equals the sum of the rates of acquiring each separately, regardless of the fitness of the intermediates. The probability the population acquires genotype AB before genotype CD is given by their relative rates. In (b), the population can acquire mutations A or B in either order, and then mutation C. Again, the overall rate equals the sum of the rates for the two possible pathways independently. In (c), the potential evolutionary trajectory branches after the population acquires mutation A. The population can then acquire mutations B and C, or mutations D and E, to reach an advantageous genotype. If genotype Abcde is strongly deleterious, the overall rate is the sum of the rates for each pathway. If Abcde is effectively neutral, the overall rate is lower than the sum of the rates for each pathway. In (d), the three mutations A, B, and C can be acquired in any order to reach advantageous genotype ABC. Because there are intermediate branching points, the overall rate is lower than the sum of the rates for each possible pathway if the single-mutants are effectively neutral.

Sketch of the fate of a mutant lineage which has selective disadvantage δ. Shown is the population size n of the lineage as a function of time t. With probability 1/T, the lineage reaches a size T in roughly T generations, and then drifts to extinction in another T generations. This gives rise to an overall weight W = T². The overall shape of a “typical” such trajectory is shown (in an actual case, the trajectory would be somewhat “jagged” due to stochastic fluctuations in n(t), but the overall shape would be roughly as shown). This is valid for T < 1/δ; the lineage will almost never reach a population size larger than 1/δ.

A typical example of the dynamics by which a beneficial triple-mutant (K = 3) is acquired, from an individual-based computer simulation. Shown in light gray is the population size of single-mutants, in darker gray is the population size of double-mutants, and the darkest gray represents triple mutants. Inset is a magnified view of the last few hundred generations in this simulated dynamics. The waiting time T₀ for the first successful single-mutant to arise is shown along with the time T₁ this single-mutant drifts before giving rise to the first successful double-mutant, the time T₂ this double-mutant drifts before giving rise to the first successful triple-mutant, and the time T₃ for this triple-mutant to establish. We see that the total time until the beneficial mutant establishes (T_e) is dominated by T₀. The total population size is N = 10⁵, the fitness parameters are δ₁ = δ₂ = 10⁻³, s = 0.1, and the mutation rates are μ₀ = 10⁻⁶, μ₁ = μ₂ = 10⁻⁴, which corresponds to a neutral stochastic tunneling regime.

Comparison of theoretical predictions for τ, the expected time for the beneficial mutants to establish, to simulation results. These plots are for the case K = 3, with triple-mutants having selective advantage s = 0.1. The crosses are the simulation results, while the solid curves show the theoretical predictions. The different parameter regimes are indicated on the plots. In (a), population size N is varied while the other parameters are held constant at μ₀ = 10⁻⁵ and μ₁ = μ₂ = δ₁ = δ₂ = 2 × 10⁻⁵, so that single- and double-mutants are effectively neutral. In the “stochastic neutral fixation-tunneling” regime, the most likely way for the population to cross the fitness valley is for single-mutants to drift to fixation, after which the triple-mutants establish through stochastic tunneling. In the “deterministic-stochastic-stochastic neutral tunneling” regime, the single-mutant class is treated deterministically and the higher mutant classes are stochastic. In the “deterministic-deterministic-stochastic neutral tunneling” regime, both single- and double-mutants are taken to be deterministic, with stochastic effects important only for the triple-mutants. The jumps in the theoretical prediction reflect points at which the different regimes do not fuse smoothly. In (b), the rate of mutation from double- to triple-mutants is varied, with N = 10⁵, μ₀ = 10⁻⁶, μ₁ = 4 × 10⁻⁴, δ₁ = 2 × 10⁻⁴, and δ₂ = 5 × 10⁻³. The “neutral-deleterious stochastic tunneling” regime refers to a region of parameter space in which single-mutants are effectively neutral while double-mutants are strongly deleterious; the other parameter regimes are named analogously. In (c), the fitness cost of the double-mutant is varied, with N = 5 × ×10⁵, μ₀ = 5 × 10⁻⁵, μ₁ = 3 × 10⁻⁵, μ₂ = 5 × 10^–, and δ₁ = 8 × 10⁻⁴. The “neutral deterministic-deleterious stochastic tunneling” regime refers to a region of parameter space in which single-mutants are effectively neutral and increase in number roughly deterministically while double-mutants are strongly deleterious.