Deterministic and stochastic regimes of asexual evolution on rugged fitness landscapes

Abstract

We study the adaptation dynamics of an initially maladapted asexual population with genotypes represented by binary sequences of length L. The population evolves in a maximally rugged fitness landscape with a large number of local optima. We find that whether the evolutionary trajectory is deterministic or stochastic depends on the effective mutational distance d(eff) up to which the population can spread in genotype space. For d(eff) = L, the deterministic quasi-species theory operates while for d(eff) < 1, the evolution is completely stochastic. Between these two limiting cases, the dynamics are described by a local quasi-species theory below a crossover time T(x) while above T(x) the population gets trapped at a local fitness peak and manages to find a better peak via either stochastic tunneling or double mutations. In the stochastic regime d(eff) < 1, we identify two subregimes associated with clonal interference and uphill adaptive walks, respectively. We argue that our findings are relevant to the interpretation of evolution experiments with microbial populations.

Figure 1.—

Quasi-species evolution of the populations . The numerical iteration of Equation 1 is shown for μ = 10⁻⁸, 10⁻⁶, and 10⁻⁴ (top to bottom) with L = 15, starting from all the populations at sequence σ⁽⁰⁾ in the fitness landscape explained in the text. The sequences with fraction ≥0.005 are shown.

Figure 2.—

Punctuated rise of the average fitness for fixed landscape and fixed initial condition in the quasi-species model with genome length L = 15. The solid line is the fitness W_max of the global maximum and the dashed line is e^−μLW_max with μ = 10⁻². The steps become more diffuse as μ increases, and the fitness level is reduced for the largest value of μ due to the broadening of the genotype distribution. Inset: average fitness plotted as a function of t/|ln μ| to show the scaling of jump times.

Figure 3.—

Evolutionary trajectories in a sequence space of length L = 6 with N = 2¹⁴, μ = 10⁻⁴ so that Nμ ≈ 1.64 and d_eff ≈ 1.05. The population fraction is denoted by X_rank(σ), where the σ_i's that do not change in the course of time are represented by a dash. Only the sequences with population fraction ≥0.05 are shown. In the initial phase, the three populations X₅₅, X₂₈, and X₅ occur in all of the above trajectories and have rather similar curves, supporting deterministic evolution. At later times, the population escapes the local peak with rank 5 via tunneling (top) and by a double mutation (center).

Figure 4.—

Evolutionary trajectories for μ = 10⁻³ (A) and μ = 10⁻² (B) with N = 2¹⁴ and L = 6. (A) The effective distance d_eff ≈ 1.4 and the population passes deterministically through the rank 28 sequence toward the global maximum. (B) Distance d_eff ≈ 2.1 and the population reaches the global maximum almost immediately.

Figure 4.—

Evolutionary trajectories for μ = 10⁻³ (A) and μ = 10⁻² (B) with N = 2¹⁴ and L = 6. (A) The effective distance d_eff ≈ 1.4 and the population passes deterministically through the rank 28 sequence toward the global maximum. (B) Distance d_eff ≈ 2.1 and the population reaches the global maximum almost immediately.

Figure 5.—

Stochastic trajectories for L = 15, N = 2¹⁰, μ = 10⁻⁴ with Nμ ≈ 0.10 and LNμ ≈ 1.54. The population passes through different routes in each case right from the beginning and at short times, several mutants at constant Hamming distance are produced simultaneously. Only the mutants that achieve a fraction ≥0.005 are shown in the plot. (Top) All the mutants shown belong to the same lineage; (center and bottom) while a fit mutant is on its way to fixation, a split in the lineage produced an even better mutant, thus bypassing the former one.

Figure 6.—

Population evolution when for L = 15, N = 2¹⁰ with μ = 10⁻⁵, 10⁻⁶, and 10⁻⁷ (top to bottom). The mutants with X_rank(σ) ≥ 0.005 are shown.