Fig 1.
Chemostat behavior represented in chemical space.
A. Schematic diagram of a chemostat occupied by a single microbial species. In the well-mixed medium (pale blue) of a chemostat, cells (orange ellipses) consume nutrients and grow. An influx of nutrients with fixed concentrations (blue and green arrows) is supplied at the same rate as dilution, keeping the medium volume constant. B. Visual representation of how a species creates its own chemostat environment. Background color indicates the growth rate of cells as a function of metabolite concentrations ca and cb, with the growth contour shown by the red curve. The flux-balance curve is shown in blue. Black curves with arrows show the time trajectories of chemostat simulation. C. Example of successful invasion of the indigenous species Blue by the invader species Red. A small amount of species Red is introduced to a steady-state chemostat of species Blue. Growth contours and steady-state environments of species Blue and species Red are shown as curves and dots in the corresponding colors (colored background indicates the “invasion zone” of Red, and represents the growth rate of Red in this zone). The supply condition is marked by a black circle. Black curves with arrows show the time trajectory of the invasion in chemical space. Inset: time course of species biomass in the chemostat during the invasion. D. Same as (C), except that because the supply condition (black circle) is different, the attempted invasion by species Red is unsuccessful.
Fig 2.
Metabolic models with substitutable nutrients can achieve a flat fitness landscape.
A. Example of a metabolic model with a trade-off in allocation of internal resources for import of two substitutable nutrients, with both nutrients contributing additively to growth. Species Red and species Blue allocate resources differently (indicated by parameter αa, see S1 Appendix). B. Growth contours and the steady-state environments created by Red or Blue alone, under the supply condition shown by the black circle. Black curve with arrows shows a trajectory in chemical space. Purple dot indicates the steady-state environment created by Red and Blue together. Lower inset: time course of species biomass. C. From upper panel to bottom panel: the fitness landscape created by Red alone (for the red dot in (B)), created by Blue alone (for the blue dot in (B)), and created by both species (for the purple dot in (B)). Diamonds mark the locations of Red and Blue strategies and their corresponding fitness in each fitness landscape. D. Growth contours and the species-specific steady-state environments for seven different species alone, under the supply condition shown by the black circle. Black curve with arrows shows a trajectory in chemical space. Lower inset: time course of species biomass in the chemostat. E. Population dynamics in a 10-dimensional chemical space. The chemostat is initially occupied by a species (Init) that has an arbitrarily assigned enzyme allocation strategy. Then, in the steady-state environment created by Init, the “opportunist” species (Opp 1) with the maximal growth rate in that environment is added to the chemostat. Subsequently, further opportunist species (Opp 2–9) are added sequentially to the steady states created by the existing consortia of species, until there is no further opportunist strategy with a growth rate higher than the dilution rate. F. The instantaneous growth rates of the 10 strategies from (E) under the steady-state environments created after adding each species to the existing consortium. Each white arrow indicates the addition of the new species with the fastest growth rate in that environment; black arrows indicate the change of the steady-state chemical environment caused by adding these new species.
Fig 3.
Multistability, chain of invasion, and non-invasible strategy.
A. Example of a metabolic model with a trade-off in allocation of resources for import of two essential nutrients, with the lower of the two import rates determining growth rate. Species Red and species Blue allocate resources differently (indicated by parameter αa, see S1 Appendix). B. Bistability of the system in (A) shown in chemical space. Black curves with arrows show the trajectories of simulations with different initial conditions. Inset: the fitness landscape created by species Red or Blue alone, with colors corresponding to the steady-state environments shown by colored dots in the main panel. C. The evolving fitness landscape. Fitness landscape created by species with different internal resource allocation strategies (marked by diamond shapes). Starting from species Blue, the species having the highest growth rate in the steady-state fitness landscape created by the “former” species is selected. This creates a chain of invasion from Blue to Light Green, Yellow, Deep Green, Deep Purple, all the way (intermediate processes omitted) to the species Black, which places itself on the peak of its own fitness landscape. The same procedure is also performed starting with species Red. D. Depiction of non-invasible strategies under different supply conditions. Black-white background indicates the maximal growth rate of the model in (A) under each environment, and the contour of maximal growth rates contains different strategies (represented by red-to-blue color). Growth contours of three species adopting one of the “maximizing strategies” are colored by their strategies. The supply conditions allowing these strategies to be “non-invasible” (supply lines) are marked by dashed black lines. E. Chain of invasion. Addition of the strategy with the fastest growth rate under the steady-state environment created by the existing consortium, as indicated in (C), is repeated 22 times. The sequentially added strategies are marked by colored circles, with the value of αa give for some representative strategies. An arrow from node j to node i indicates that strategy j can invade the environment created by strategy i.
Fig 4.
A. Metabolic model with a trade-off in allocation of internal resources for import of two nutrients plus their interconversion, with both nutrients necessary for growth. B. Three subclasses of maximizing metabolic strategies in chemical space are indicated by background color, and circles with arrows illustrate the metabolic strategies of each subclass. The maximal growth contours for four growth rates (0.1, 0.2, 0.3, 0.4) are marked by gray colors. C. Two maximizing strategies co-creating a non-invasible steady state. At dilution rate d = 0.2, the maximal growth contour and the corresponding maximizing strategies are shown as colored squares. At a discontinuous point of the growth contour, the supply lines of two distinct metabolic strategies (Red and Blue) span a gray region, where any supply condition (e.g. black circle) requires the two maximizing strategies to co-create the environment on the discontinuous point. Red and blue dots mark the environments created by species Red and species Blue alone, and the purple dot marks the environment co-created by Red and Blue. Black curve with arrows shows a trajectory in chemical space. Inset: competition dynamics of species Red and species Blue together with 10 other maximizing species with different strategies. D. The fitness landscapes for the three environments in (C) indicated by corresponding box colors. For class Green and Red, the strategy is represented by αa, for class Blue, the strategy is represented by αb.
Fig 5.
Species creating new nutrient dimensions and achieving evolutionarily stable coexistence.
A. Metabolic model with a single supplied nutrient S. Cells allocate enzymes to convert S into internal intermediate Iint and produce energy (denoted as “ATP”), export internal intermediate into the chemostat to become Iext, import external intermediate, or consume Iint to produce ATP. The growth rate is the sum of ATP production (see S1 Appendix). B. Three subclasses of maximizing metabolic strategies in chemical space are indicated by background color, and circles with arrows illustrate the metabolic strategies of each subclass. The maximal growth contours for three growth rates (0.2, 0.4, 0.6) are marked by black-to-white colors. C. At dilution rate d = 0.4, two maximizing strategies co-create a non-invasible environment. The maximal growth contour and the corresponding maximizing strategies are shown as colored squares. At a discontinuous point of the growth contour, the supply lines of two distinct metabolic strategies (Green and Blue) span a gray region, where any supply condition (e.g. black circle) requires two maximizing strategies to co-create the environment at the discontinuous point. The blue dot marks the environment created by species Blue alone, and the cyan dot marks the environment co-created by Blue and Green. Th black curve with arrows shows a trajectory in chemical space. Inset: time course of species biomass, with species Green added to the chemostat at time 100. D. The fitness landscapes for two environments in (C) indicated by corresponding colors of the boxes, reflecting the relationship between instantaneous growth rate and resource allocation strategy. For classes Blue and Red, the strategy is represented by αATP1; for class Green the strategy is represented by αimp. E. Same as (C), except that the dilution rate is d = 0.6. Inset: time course of species biomass, starting with Blue and Green, with species Red added to the chemostat at time 100. F. Same as (D), except corresponding to the two steady-state environments shown in (E).