Figure 1.Negative frequency‐dependent interactions exist in a simple foraging game with multiple resources
A simple foraging game with a mixed equilibrium: a group of genetically identical foragers encounters two resources: A and (a less preferred resource) B. Each individual must specialize in consuming one or the other. We assume that individuals choose simultaneously and without knowledge of the intentions of other individuals. Because of resource limitations, each individual's fitness is a function of the actions of other individuals.
If all other members of the population adopt some pure strategy (e.g., “specialize in resource A”), an individual opting for the opposite pure strategy (e.g., “specialize in resource B”) gains a fitness advantage, and vice versa. Hence, a population of pure strategists will always be open to invasion by the opposite pure strategist. A stable equilibrium is reached when the population divides between the two resources such that both phenotypes have equal expected fitness. A genetic strategy that results in this phenotypic heterogeneity is evolutionarily stable. This simple scenario serves to illustrate why we might expect environments with multiple food sources to favor the evolution of mixed strategies.
Modeling this simple game in the context of a growing microbial population demonstrates negative frequency‐dependent selection between the two phenotypes (A and B) and the existence of the evolutionarily stable mix. Importantly, the stable mix is not identical to the optimal division of labor that would maximize population growth.
Indeed, with increasing disparity in growth rates between the phenotypes, the two solution concepts diverge. A higher growth penalty for phenotype B results in a higher phenotype B fraction in the growth‐optimal mix, but a lower fraction in the stable one. For details and parameters of the model, see Appendix Text S2 (Appendix Figs S16, S17, S18, S19, S20, S21 and S22).
Figure 2.Yeast GAL network ON and OFF phenotypes are mutually invasible in mixed glucose and galactose
Gene expression in the yeast GAL network is regulated in part by GAL4, GAL80, and GAL3 (full network not shown). The wild‐type W303 yeast GAL network adopts a mixed strategy in glucose and galactose, but a GAL‐OFF pure strategist can be engineered by inducing GAL80, whose protein product inhibits GAL expression. Likewise, a GAL‐ON pure strategist can be engineered by inducing expression of GAL3, which inhibits GAL80 in the presence of galactose.
YFP driven by a GAL1 promoter allows for determination of GAL activation states via flow cytometry. GAL activation histograms are shown for the engineered pure strategist strains and for the wild‐type W303 yeast GAL network. After incubation for 10 h in a mixed sugar environment (0.03% (w/v) glucose, 0.05% (w/v) galactose, 1 μg/ml doxycycline), GAL‐ON and GAL‐OFF pure strategists remain unimodally activated and inactivated, respectively, for the GAL network, while the wild‐type GAL network exhibits bimodal gene expression. To induce GAL3 and GAL80, cultures were initially incubated in 0.01% glucose and 1 μg/ml doxycycline for 24 h before being transferred to the mixed sugar environment.
GAL‐ON and GAL‐OFF phenotypes are mutually invasible in mixed sugar conditions. Population frequency of the GAL3‐induced (GAL‐ON) pure strategists (orange circles) cultured with GAL80‐induced (GAL‐OFF) pure strategists is plotted at the beginning and end of a 20‐h incubation in mixed sugars. Six biological replicates of each pure strategist were mixed at high (top panel), intermediate (middle panel), and low (bottom panel) initial GAL‐ON fractions. Each pure strategist invades the other when rare. Dashed lines indicate that frequencies are shown at beginning and end to convey the direction of overall frequency change. Actual temporal fitness changes are not linear through the course of the 20‐h competition; for more granular temporal fitness dynamics, see Appendix Fig S1.
Fitness, in number of generations, is shown for the two phenotypes as a function of the population frequency of GAL‐ON. Data is shown for sixty cultures and sixty initial fractions representing six biological replicate pairs. The crossing point indicates an evolutionarily stable coexistence of around 40% GAL‐ON for this mix of sugars.
The total population density for the mixed GAL‐ON/GAL‐OFF populations is plotted against the initial population frequency of GAL‐ON. The data are shown for 16 h after incubation, when populations dominated by GAL‐ON are already reaching saturating density (˜4 × 106/ml), but populations of intermediate or low numbers of GAL‐ON cells are still growing. Growth at 16 rather than 20 h is shown because by 20 h the high and intermediate GAL‐ON populations have saturated too much to show growth differences between them (see Appendix Fig S7 for final saturating densities). Panels (D) and (E) taken together indicate that the evolutionarily stable mix contains less initial GAL‐ON than is growth‐optimal from a population standpoint.
A, B The relative fitness of the GAL‐ON pure strategist and absolute fitness (in generations, as in Fig 2D) of both pure strategists are shown for 30 different populations at varying initial frequency of GAL‐ON. Data are shown for 0.05% galactose and two conditions: high glucose (0.03%, A) and low glucose (0.017%, B). The GAL‐ON pure strategist undergoes roughly the same number of divisions between the two conditions, while the GAL‐OFF pure strategists are more fit in higher glucose. Stable equilibrium frequencies (calculated by polynomial spline fitting of relative fitness data) are shown as black dotted lines. Lower glucose results in a higher frequency of GAL‐ON cells at the stable mix.
C. Equilibrium GAL‐ON pure strategist frequencies as a function of increasing glucose concentrations. Data are shown for high (0.05%, circles) and low (0.017%, triangles) galactose. All equilibria were calculated by polynomial spline fitting of relative fitness data (error bars are 95% confidence intervals; n = 3).
Figure 4.Wild‐type mixed strategist invades both pure strategists and is uninvasible by either
The relative fitness of the wild‐type mixed strategist over the GAL‐OFF pure strategist (A) and GAL‐ON pure strategist (B). is shown. Low frequencies of the mixed strategist invade strongly in populations dominated by either pure strategist. As expected of an evolutionarily stable mixed strategy, the relative fitness of the mixed strategist to the GAL‐OFF pure strategist approaches one in populations dominated by the mixed strategist. However, the mixed strategist does not display frequency dependence against the GAL‐ON pure strategist. Source data are available online for this figure.
Figure 5.Frequency dependence from a mixed resource environment drives the evolution of both phenotypic and genetic heterogeneity
A, B Eight replicates of each of the two specialist strains (GAL‐OFF, A, and GAL‐ON, B) were incubated in the presence of doxycycline and three separate sugar conditions: 0.1% glucose (blue), 0.1% galactose (orange), and a mixture of 0.03% glucose and 0.05% galactose (purple). Cultures were diluted 1,000× daily into fresh media after reaching saturation. To determine the composition of the evolved mixed populations, cultures were plated on agar and individual colonies were assayed for GAL activation in mixed sugars. (A) Starting from a glucose specialist strain and in the presence of galactose, a mutant pure strategist GAL‐ON strain arose spontaneously. In pure galactose, the strain eventually took over the population (orange), while in the mixed resource condition, it evolved toward a stable equilibrium with the GAL‐OFF strain (purple, right panel). (B) Starting from a galactose specialist strain in the presence of mixed sugars, the population similarly evolved to a stable mix of GAL‐ON and GAL‐OFF, but colony purification revealed that the population had evolved to a clonal population of mixed strategists rather than coexistence of pure strategists. See Appendix Figs S8 and S9 for YFP histograms of all mixed sugar replicates. Fraction GAL‐ON is not shown for the 1% glucose condition because in that condition, the GAL‐ON pure strategist adopts a very low‐level unimodal activation state straddling the ON/OFF fluorescence threshold (see Appendix Fig S10).