Modeling ecological diversification in rapidly evolving asexual populations.

(a) Individuals compete for ℛ substitutable resources in a chemostat-like environment. (b) Individuals acquire mutations that alter their resource uptake rates . Mutations that impact the normalized resource uptake strategy occur at rate Ua, while mutations that only increase the overall magnitude X = log ∑i ri occur at rate Ub. (c, d) Simulated dynamics for ℛ = 2 resources at small (c, N = 106) and large (d, N = 109) population sizes, where strategy mutations are sampled uniformly from the discrete strategies (or “ecotypes”) depicted at top. Muller plots show the relative abundances of each strain that reaches at least 5% frequency in the population. Strains are colored by their resource uptake strategies (top), with shading denoting different genotypes. Other parameters are Ub = 10−8, sb = 0.02, Ua = 10−7. (e) Total relative abundances of the ecotypes in panel d. (f) Shannon diversity (base e) of the individual strains in panel d (black) as well as the coarse-grained ecotypes in panel e (green). The dashed line depicts the maximum stable diversity possible in the rare mutation regime in panel c, when two coexisting strains are at equal frequencies. These data show that large populations cluster into a few coarse-grained ecotypes, even when their underlying strain diversity exceeds the competitive exclusion bound.

Clonal interference creates a priority effect that favors the resident ecotype.

(a) Schematic of a strategy mutation arising in a population with a single resident ecotype. The mutation arises in an individual from the resident fitness distribution (grey), which has a maximum width xc and moves to the right at rate v. Its descendants found a new ecotype (green) that competes with the resident ecotype as they both acquire additional mutations. (b) The probability that a strategy mutation “establishes” (i.e. reaches 10% frequency) as a function of its initial invasion fitness sinv. Green points show simulation results for N = 1010, Ub = 10−6, and sb = 0.01; for each value of sinv, the values of α1, α2, and β were chosen so that f * 0 = 1/ 2. Grey points show the establishment probabilities of constitutively beneficial mutations with the same values of sinv. (c) The median frequencies of the mutations in panel b T sw ≈ 1000 generations after first reaching 10% frequency. The dashed line denotes the expected frequency in the absence of clonal interference. (d) The long-term fixation probabilities of the mutations in panel c. Inset: the average length of time required for a mutation to transit from 10% frequency to either fixation (solid curves) or extinction (dashed).

Oscillatory invasion dynamics of new ecotypes.

(a) Replicate frequency trajectories of an invading ecotype with seco ≈ 0.03 and that arises on an initial genetic background with xinv (0) ≈ xc + sb. Simulated parameters are N = 1012, Ub = 10−6, and sb = 0.01, for which xc 4sb. Five random realizations are shown in solid lines, while the dashed line depicts an average over 50 independent replicates, which emphasizes the typical overshooting behavior between T * (Eq. 7) and Tsw ≈xc /v. Horizontal dashed lines show the original equilibrium frequency f * (Eq. 3) and the “fitness mortgaging” term from Eq. (9). (b) The oscillations become more pronounced in the asymptotic limit, where log N = 500, log Ub = −50, and xc ≈ 21sb. (c) The initial growth rates of successive fitness classes (Fig. S1, SI Section 3.3) for the ecotype trajectories in panel b, illustrating how the runaway growth is eventually stabilized ∼Tsw generations later.

Recurrent strategy mutations stabilize coexistence by replacing the lagging ecotype.

(a) Recurrent strategy mutations between established ecotypes can replace the resident lineage if the fitness differences grow sufficiently large. (b) Example simulation showing bistable ecotype frequencies on long timescales, with continuous lineage turnover on shorter timescales due to successful cross-invasions. Thick shaded curves represent overall ecotype frequencies, while thinner lines denote lineages founded by distinct strategy mutations. Ecotype parameters are sc = 0.1, ΔX = 0, and , with Uα = 10−11; the remaining parameters are the same as Fig. 3A. (c,d) Lineage turnover time (c) and overall frequency (d) of the trailing ecotype for different values of sc and sb. Symbols denote the median across simulations with the same N, Ub parameters as panel b. Dashed lines denote theoretical predictions (SI Section 3.6).

Clonal interference biases the metabolic structure of the population.

(a) Schematic of a mutational landscape with three possible resource strategies. (b) Ecotype frequency trajectories from three example populations with U1→ 2 /U1 → 3 = 3, 30, and 300, with U1 → 3 = 10−10 held fixed. The resource strategies satisfy α1 = 0.43, α2 = 0.61, and α3 = 0.69, with β = 0.5, while the remaining parameters are N = 1012, Ub = 10−6, and sb = 10−2. (c) The fraction of time (Φ) spent in the (α1, α2)-dominated state, for different values of α3 (color-scale) and U1→2 (x axis); all other parameters are the same as panel b. Inset: the relative mutation rate where the two lagging ecotypes are equally likely (Φ ≈1/2) for different values of U3. Clonal interference amplifies selection on small differences in resource consumption, which can overwhelm much larger differences in the relative mutation rates of different strategies.

Schematic of the definition of establishment fitness.

A lineage f0 (t) with a declining relative fitness x0 − vt produces beneficial mutations that found a new fitness class f1 (t). The establishment fitness x1 of the new fitness class can then be defined by the relative fitness it would have had at its back-extrapolated establishment time; see SI Sections 3.3 and 4 for more details.

Analog of Fig. 4C,D for strategy mutations with non-zero direct cost ΔX < 0

Populations in which strategy mutations incurred variable direct costs of relative strength ΔX/sb were simulated. Ecosystems with differing strengths of ecological feedback B2 are arrayed from left to right. All other parameters as in Fig. 4C,D. At sufficiently strong ecological feedback strength relative to the strength of within-ecotype evolutionary dynamics (by increasing sc left to right, or decreasing sb from purple to light blue), direct costs have negligible effect on the typical cross-invasion timescale.

Invasion of strategy mutations in populations with an exponential distribution of general fitnesses.

Probability of fixation of a strategy mutation with ΔX = 0, f * = 1/ 2, conditioned on establishing (reaching 10% frequency), in clonally interfering populations with an exponential distribution of general fitness effects (DFE) of scale σ. Each panel represents a different set of evolutionary parameters (N, Ub, σ), with the mean fixed effect also indicated. To compare to pure strategy mutations (green), conditional probabilities for fitness mutations of equivalent effect ΔX = sinv are also shown (grey). Insets shows the corresponding establishment probability for strategy and equivalent fitness mutations.