Metabolic trade-offs and metabolic strategies

As discussed above, microorganisms need to allocate their limited internal resources into different cellular functions. In our models, we use a_j to denote the fraction of internal resources allocated to the j-th metabolic function, with representing a metabolic strategy. As a simple representation of the limited internal resources, an exact metabolic trade-off is assumed, such that ∑_jα_j = 1. All possible values of define a continuous spectrum of metabolic strategies, which we name the strategy space. One major goal of our work is to construct a general and intuitive framework for evaluating strategies for a broad range of different metabolic models and experimental conditions.

Geometrical representation of how strategies interact with the environment

One way to evaluate metabolic strategies is by comparing the competitiveness of “species” with fixed strategies in chemostat-type resource-competition models. In an idealized model of a chemostat (Fig 1A), p types of nutrients are supplied at rate d and concentrations , meanwhile cells and medium are diluted at the same rate d. The chosen values of d and constitute the “external condition” for a chemostat. Accordingly, the biomass density m_σ of species σ adopting strategy in the chemostat obeys: (1)

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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 c_a and c_b, 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.

https://doi.org/10.1371/journal.pcbi.1008156.g001

The concentration c_i of the i-th nutrient is a variable, influenced by its rate of consumption I_i per cell volume. In a chemostat occupied by a single species σ, the changing value of c_i satisfies (2) where r is a constant representing the biomass per cell volume (See S1 Appendix for details). A negative value of I_i corresponds to secretion of the metabolite from cells into the environment.

In this manuscript, we define as the “chemical environment”, and all possible values of constitute the “chemical space”. Eqs (1) and (2) represent a general chemostat model with a single species. The simplicity of the chemostat has inspired many theoretical studies of resource competition. Different model assumptions about how species grow (Eq (1)) and consume nutrients (Eq (2)) have produced a variety of intriguing behaviors and conclusions. However, the origins of these differences are not always simple to discern. To facilitate the evaluation of metabolic strategies, we next present a geometric representation that allows ready visualization of the feedback between species and the environment in a chemostat. Intuitively, the steady state created by a single species can be visualized by the intersection of two nullclines, derived from Eq (1) and Eq (2), respectively (details in S1 Appendix):

First, setting Eq (1) to zero leads to a p-dimensional version of the ZNGI, which we name the “growth contour”. For a given metabolic strategy , the growth-rate function maps different environments in the chemical space onto different growth rates (background color in Fig 1B). At steady state, the relation dm_σ/dt = 0 (Eq (1)) requires the growth rate to be exactly equal to the dilution rate (assuming nonzero cell density). Therefore, the contour in chemical space satisfying indicates all possible environments that could support a steady state of the strategy (red curve in Fig 1B, and orange, red, deep red curves in S1B Fig). This contour reflects how the chemical environment determines cell growth.

Secondly, the nullcline derived from Eq (2) reflects the impact of cellular metabolism on the chemical environment. At steady state, the nutrient influx should be equal to the summation of dilution and cellular consumption. When Eq (2) is set to zero, varying values of cell density m lead to different values of (Eq. (S5)) constituting a one-dimensional “flux-balance curve” in chemical space (blue curve in Fig 1B, and purple, cyan, and blue curves in S1A Fig).

At the intersection of the growth contour and the flux-balance curve, the steady-state chemical environment is created by the species σ (Fig 1B, red dot). Changes in conditions d and influence the shapes of the growth contour and the flux-balance curve separately, enabling clear interpretation of chemostat experiments under varied conditions (S2 Fig, details in S1 Appendix).

This graphical approach provides an intuitive understanding of how chemostat experiments can be controlled and interpreted (See S1 Appendix for details). More importantly, it enables an intuitive picture for the outcomes of competitions among multiple species.

Evaluating strategies by the rule of invasion, and the environment-dependent fitness landscape

We use the outcome of competition between species to evaluate metabolic strategies, assuming each species σ adopts a fixed strategy . We first focus on the outcome of invasion. Similar to several previous works that use invasion to assess the stability of a consortia [29, 34], here “invasion” is defined as the introduction of a very small number of an “invader” species to a steady-state chemostat already occupied by a set of “indigenous” species. The invasion growth rate is quantified by the instantaneous growth rate of the invader species at the moment of introduction. Unlike adaptive dynamics, the invasion growth rate is evaluated with respect to the chemical environment created by the indigenous species, rather than with respect to the population composition of the indigenous species [30–32].

In the chemical space, the outcome of an invasion can be summed up by a simple geometric rule, as demonstrated in Fig 1C and 1D. The growth contour of the invader (species Red) separates the chemical space into two regions: an “invasion zone” where the invader grows faster than dilution (green-colored region in Fig 1C and 1D), and “no-invasion zone” where the invader has a growth rate lower than dilution. If the steady-state environment constructed by the indigenous species (species Blue) is located within the invasion zone of the invader, the invader will initially grow faster than dilution. Therefore, the invader will expand its population and the invasion will be successful (Fig 1C). By contrast, if the steady-state chemical environment created by the indigenous species lies outside of the invasion zone, the invasion will be unsuccessful (Fig 1D, same species as in Fig 1C but with a different supply condition, and therefore a different steady state). (See S1 Appendix for details.)

In the steady-state environment created by the indigenous species, each strategy has an invasion growth rate. We define an environment-dependent “fitness landscape” as the relation between the invasion growth rate and the metabolic strategy of invaders (Eqs. (S8)-(S9), see S1 Appendix for details). Different indigenous species can create different chemical environments, and thus give rise to different shapes of the fitness landscape.

Mutual invasion, a flat fitness landscape, and unlimited coexistence

The rule of invasion allows for easy assessment of competition dynamics. The emergence of complex dynamics generally requires that competitiveness be non-transitive [33]. For example, if species Red can invade species Blue, that does not mean Blue cannot invade Red. Such mutual invasibility can be observed in substitutable-resource metabolic models, with a simple version illustrated in Fig 2A: two substitutable nutrients a and b, such as glucose and galactose, contribute linearly to biomass increase. Since a substantial investment of protein and energy is required for nutrient uptake, the model assumes an exact trade-off between the allocation of internal resources to import either nutrient. Specifically, a fraction of resources α_a is allocated to import a and a fraction (1−α_a) to import b. As shown in Fig 2B, while the steady-state environment created by Blue is located within the invasion zone of Red, the steady-state environment created by Red is also located within the invasion zone of Blue. According to the rule of invasion, each species can therefore invade the steady-state environment created by the other. In the face of such successful invasions, the only possible stable chemical environment for this system is at the intersection of two growth contours, where the two species can coexist.

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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 α_a, 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.

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The environment-dependent fitness landscape readily explains this coexistence: In this model, the fitness landscape is linear respect to α_a, with the slope positively correlated with the difference between the steady state value and (See S1 Appendix for details). In the steady-state environment created by Red (α_a = 0.6), a becomes scarce relative to b, and strategies with smaller α_a have higher fitness (Fig 2C, upper panel). In the steady-state environment created by Blue (α_a = 0.2), the slope of the fitness landscape changes its sign, and strategies with larger α_a have higher fitness (Fig 2C, middle panel). Therefore, each species creates an environment that is more suitable for its competitor, which leads to coexistence.

Typically in resource-competition models, the number of coexisting species cannot exceed the number of resources [35–38]. This conclusion can be understood intuitively from the geometric approach: The steady-state growth of a species can only be achieved on the growth contour of this species. The stable-coexistence of N species can thus only occur at the intersection of their N corresponding growth contours. Generally, an N-dimensional chemical space allows a unique intersection of no more than N surfaces, and the diversity of species is therefore bounded by the number of metabolites in the environment. This theoretical restriction on biodiversity, made formal as the “competitive exclusion principle”, contradicts the tremendous biodiversity manifested in the real world [25, 39, 40]. There have been a multitude of theoretical efforts to reconcile this contradiction [21, 22, 24, 25, 28, 41].

Under the very simplified assumption of exact trade-offs, a special property of this metabolic model is that all growth contours intersect at a common point (Fig 2D). In the environment co-created by Blue and Red (Fig 2B, purple dot), which is the common intersection point for all growth contours, the fitness landscape becomes flat (Fig 2C, bottom panel). Therefore, in this system, once any pair of species with a mutual-invasion relationship constructs the steady-state chemical environment together, all species become effectively neutral. Subsequent works showed that with spatial structure [42], even non-exact trade-offs can lead to high species abundance by partially “leveling the playing field” among different metabolic strategies, showing the potential of a nearly-flat fitness landscape to promote biodiversity.

In order to understand whether a microbial community will evolve towards or away from a flat fitness landscape, we extended the metabolic model to incorporate 10 substitutable nutrients. We started the chemostat with a species with an arbitrarily assigned strategy (Init), and let it come to steady state. Then, in the steady-state environment created by species Init, the “opportunist” species (Opp 1) with the maximal growth rate in that environment was added to the chemostat. Subsequently, in the steady states created by the existing consortia, we identified the fastest growing opportunist species (Opp 2–9) and added them sequentially to the chemostat, until there was no further opportunist strategy with a growth rate higher than the dilution rate (Fig 2E). During this process, the newly selected opportunist can always invade without replacing any of the existing species (Fig 2E). It is analytically provable that these opportunist species exhibit all-or-none resource allocation strategies (α_i = {0,1}) to maximize their growth rates (S3A Fig, S1 Appendix), and act as “keystone” species, similar to those defined in the work of Posfai et al., that expand the convex hull for coexistence [27]. As the opportunists specializing in different nutrients were selected and added to the chemostat one by one, more species start to acquire equal-to-dilution growth rates (Fig 2F), and the fitness landscape becomes more and more flattened (S3B Fig). Finally, this process of “evolution” of the community self-organizes towards multiple keystone species that completely flatten the fitness landscape and thus ensure unlimited coexistence.

This substitutable-resource metabolic model also suggests that non-stationary fitness landscape is the prerequisite for interesting ecological dynamics. Another example comes from a slightly modified metabolic model, where enzymes are also required to convert the imported raw materials into biomass, which can form a “rock-paper-scissors”-type invasion loop (See S1 Appendix and S4 Fig). This loop leads to oscillatory population dynamics with an ever-changing fitness landscape, similar to the oscillatory dynamics demonstrated by Huisman et al. [21].

Multistability and the chain of invasion

When species create environments that are more favorable for their competitors, mutual-invasion can occur. Can species create environments that are hostile to their competitors, and if so what will be the consequences?

Fig 3A shows a simple metabolic model with two essential nutrients a and b, such as nitrogen and phosphorus (see S1 Appendix for details). Similar to the model in Fig 2A, the model assumes a trade-off between the allocation of internal resources to import nutrients, so that a resource allocation strategy is fully characterized by the fraction of resources α_a allocated to import nutrient a. The growth rate is taken to be the minimum of the two input rates [43]. As shown in Fig 3B, two species, Red and Blue, each creates a chemical environment outside of the invasion zone of each other. According to the rule of invasion, neither can be invaded by the other. Therefore, the steady state of the community depends on initial conditions–whichever species occupies the chemostat first will dominate indefinitely. It is worth noting that despite the fact that coexistence is excluded in a single chemostat under this metabolic model, the ability of species to create a self-favoring environment allows the spontaneous emergence of spatial heterogeneity and coexistence in an extended system with multiple linked chemostats (S6 Fig). In ecology, the spontaneous emergence of spatial heterogeneity has been shown for species with the capacity to construct their own niches [37, 44], and the chain of chemostats provides a simple model for such spatial coexistence.

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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 α_a, 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 α_a 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.

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The nonmonotonic fitness landscape can produce a chain of invasion. From the perspective of the strategy-growth relationship (Fig 3B, inset), species Red (α_a = 0.65) creates a fitness landscape where small α_a is disfavored. Symmetrically, species Blue (α_a = 0.35) creates a fitness landscape where large α_a is disfavored. However, neither Red nor Blue sits on the top of the fitness landscape each one creates (Fig 3C). In the fitness landscape created by Blue, a slightly larger α_a (green diamond in Fig 3C) has the highest growth rate. Consequently, species adopting the Green strategy can invade Blue. Nevertheless, species Green is not on the top of its own fitness landscape as an even larger α_a (yellow diamond in Fig 3C) maximizes the growth rate in the environment created by Green. If we repeatedly perform the process of adding the fastest-growing species to the chemostat, as in the previous section, the newly added species always outcompetes and replaces the former species. A series of replacements by the fastest-growing species in the environment created by the former species creates a chain of invasion (Fig 3E), which finally leads to a balanced enzyme budget with α_a = 0.5. Yet, even for this final strategy that cannot be replaced by any other strategies (α_a = 0.5), it cannot invade strategies just three steps earlier (Fig 3E).

In this particular model after four steps of replacement, multistability appears. The species with α_a marked by Deep Purple, which is reached by the chain of invasion going from Blue to Green to Yellow to Deep Green cannot invade the original species Blue (Fig 3C). A similar relationship holds between Cyan and Red. Actually, any set of species in Fig 3E that are not directly linked by the arrow of invasion will exhibit multistability. This phenomenon highlights the difference between ecological stability and evolutionary stability: Ecologically, as both Blue and Deep Purple create a fitness landscape where the other species grows slower than dilution, they constitute a bistable system. However, evolutionarily, “mutants” with slightly larger α_a can invade Blue, eventually driving the system towards Deep Purple.

Non-invasible strategies

In this model, with symmetric parameters, the only evolutionarily stable strategy is α_a = 0.5 (black diamond in Fig 3C). This is the only strategy that locates itself on the top of the fitness landscape it creates, and therefore cannot be invaded by any other species. This simple model demonstrates a general definition of optimal (aka evolutionarily stable or non-invasible) strategies: those strategies that create a fitness landscape which places themselves on the top (Eq. (S10)).

A chemical environment defines a fitness landscape, and the steady-state chemical environment created by the species present is influenced by supply condition, dilution rate, and the details of cell metabolism. Therefore, different chemostat parameters and different metabolic models lead to different optimal strategies. In the following, we described a generally applicable protocol for obtaining the non-invasible strategies, using the metabolic model in Fig 3A as the example (Fig 3D, details in S1 Appendix):

First, under a chemical environment , the maximal growth rate (background color in Fig 3D) and the corresponding resource allocation strategy , defined as a “maximizing strategy”, can be obtained analytically or via numerical search through the strategy space (Eq. (S11)). and are independent of the chemostat parameters and d.

Second, the “maximal growth contour” for dilution rate d is defined as all chemical environments that support a maximal growth rate of d (Eq. (S12)). Different maximizing strategies exist at different points of the maximal growth contour, as shown by the colors of the curve in Fig 3D. In a multi-species ecosystem, possible steady states can only occur at the outermost surface of the multiple species’ growth contours, as highlighted in S3C Fig. By definition, the maximal growth contour is the outermost surface that envelops all growth contours, so that chemical environments on the maximal growth contour are outside of the invasion zone of any strategy. Therefore, if a species is able to create a steady-state environment on the maximal growth contour, it cannot be invaded.

Finally, different form different maximal flux-balance curves (Eq. (14)), which intersect with the maximal growth contour at one point . Species that adopt the maximizing strategy at create the environment , and are therefore immune to invasion. Under different , different species become non-invasible (e.g., orange, green, and blue growth contours in Fig 3D).

Conditions for evolutionarily stable coexistence

Given d and , the maximal growth contour and the maximal flux-balance curve are unique, therefore there is only one . Does the uniqueness of imply a single evolutionarily stable species? Or is coexistence still possible even in the face of evolution? In a recent work [28], this question was addressed by modeling a population of microbes competing for steadily supplied resources. Through in-silico evolution and network analysis, the authors found that multiple species with distinct metabolic strategies can coexist as evolutionarily-stable co-optimal consortia, which no other species can invade.

Using a simplified version of Taillefumier et al.’s model (Fig 4A), we employ the graphical approach to help identify the requirements for such evolutionarily-stable coexistence and the role of each species in supporting the consortium. In this model, at the cost of producing the necessary enzymes, cells are not only able to import external nutrients, but can also convert any one of the internal nutrients into any other. Meanwhile, nutrients passively diffuse in and out of the cell. The internal concentrations of nutrient a and nutrient b are both essential for cell growth (see S1 Appendix for detail). Therefore, metabolic trade-offs in this system have four elements: the fraction of internal resources allocated to import nutrient a (α_a) or nutrient b (α_b) and/or convert one nutrient into another (α_ab converts internal b into a, and α_ba converts internal a into b). Each species is defined by its internal resource allocation strategy .

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Fig 4. Non-invasible cartels.

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 α_a, for class Blue, the strategy is represented by α_b.

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Following the general protocol described in the previous section, we first identified the maximal growth rates and the corresponding strategy or strategies at each point in the chemical space, and generated maximal growth contours for different dilution rates (Fig 4B). The maximal growth contours are not smoothly continuous, nor are the corresponding strategies. In chemical space, three distinct sectors of maximizing strategies appear (Fig 4B, S7A Fig): When nutrient a is very low compared to b, the maximizing strategy is a “b-a converter” which imports b and converts it into a (blue sector, only α_b and α_ab are non-zero). Symmetrically, when a is comparatively high, the optimal strategy is a “a-b converter” (green sector, only α_a and α_ba are non-zero). Otherwise, the maximizing strategy is an “importer” which imports both nutrients without conversion (red sector, only α_a and α_b are non-zero). On the border between sectors, the maximal growth contour has a discontinuous slope.

Optimal coexistence occurs at these discontinuous points. If an environment point is located in a continuous region of the maximal growth contour, only one maximizing strategy exists for that environment (maximizing strategies along the maximal growth contour are indicated by colored squares in Fig 4C).Supply conditions that make the optimal strategy (i.e. allow to create the steady-state environment ) constitute the supply line for and .

The optimal coexistence of species Blue and species Red can be understood intuitively from the dynamic fitness landscape. Given a chemical environment, the relation between α_a and the growth rate of an importer (red curve) or an a-b converter (green curve), and that between α_b and the growth rate of a b-a converter (green curve) constitute the fitness landscape of species adopting different sectors of maximizing strategies (Fig 4D). In the environment created by species Blue (blue dot in Fig 4C), not only will some importers grow faster than Blue, species Blue (strategy marked by blue diamond) is not even on the fitness peak of its own class (Fig 4D, upper panel). Similarly, in the environment created by species Red, the strategy of Red is not at the top of the fitness landscape (Fig 4D, middle panel). By contrast, in the environment co-created by species Blue and Red (purple dot in Fig 4C), their strategies are at the top of the fitness landscapes of their own classes and at equal height. For all supply conditions in the gray region, species Blue and species Red jointly drive the nutrient concentrations to the discontinuous point of the optimal growth contour, and thereby achieve evolutionarily stable coexistence.

Species creating a new nutrient dimension, and evolutionary stability with or without cross-feeding

One possible solution to the competitive-exclusion paradox is the creation of new nutrient “dimensions” by species secreting metabolites that can be utilized by other species. For example, E. coli secretes acetate as a by-product of glucose metabolism. Accumulation of acetate impedes the growth of E. coli on glucose [45], but the acetate can be utilized as a carbon source, e.g. by mutant strains that emerge in long-term evolution experiments [46, 47]. Recently, several modeling works investigated community structures when microbes create metabolic niches by secreting metabolic byproducts, and found that cross-feeding is capable of supporting high ecological diversity [23, 48]. Nevertheless, as ecological stability does not guarantee evolutionary stability, can such coexistence survive ceaseless mutation and selection? Why doesn’t the producer in a cross-feeding pair adjust its strategy to retain all useful metabolites? And can evolutionarily stable coexistence occur due to secreted metabolites even without mutually beneficial cross-feeding?

To explore the possibilities of evolutionarily stable coexistence when species create new nutrients, we used a simplified model to represent multi-step energy generation with a dual-role intermediate metabolite (Fig 5A). A single chemical energy source S is supplied into the chemostat. The pathway for processing S consists of four relevant reactions driven by designated enzymes: External S can be imported and converted into intermediate I_int to generate ATP (with a corresponding fraction of the enzyme budget α_ATP1).The intermediate has a dual role in energy production: on the one hand, it positively contributes to ATP production via a downstream reaction (with a fraction of the enzyme budget α_ATP2); on the other hand, it negatively contributes to ATP production through product inhibition of the first energy-producing reaction. To deal with this negative effect of internal intermediate, cells may synthesize transporters (with a fraction of the enzyme budget α_exp) to export intermediate out into environment, where it becomes external intermediate I_ext. By this reaction, cells can increase the dimension of chemical space from one (S) into two (S and I_ext). Cells can also import I_ext into I_int (with a fraction of the enzyme budget α_imp), then use I_int as an energy source via the second reaction. (See S1 Appendix for details.)

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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 I_int and produce energy (denoted as “ATP”), export internal intermediate into the chemostat to become I_ext, import external intermediate, or consume I_int 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 α_ATP1; for class Green the strategy is represented by α_imp. 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).

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The metabolic strategy in this model has four components: . When we examine the maximizing strategies and maximal growth rates in the chemical space, three distinct classes of strategy emerge (Fig 5B). When S is abundant and I_ext is low, the maximizing strategies have only two non-zero components, α_ATP1 and α_exp (S7B Fig), meaning this class of species only imports S, for the first energy-generating reaction, then exports intermediate as waste. Therefore, we call strategies in this class “polluters” (blue section in Fig 5B, S7C Fig). When I_ext is high while S is low, the maximizing strategies have two different non-zero components, α_ATP2 and α_imp (S7B Fig), meaning this class of species relies solely on I_ext as its energy source. We call these strategies “cleaners” as they clean up I_ext from the environment, (green section in Fig 5B, S7C Fig). When there are comparable amounts of S and I_ext present, a third class of maximizing strategies appears: these cells neither export nor import intermediates, but rather allocate all their enzyme budget to α_ATP1 and α_ATP2 to carry out both energy-producing reactions. We call species in this class “generalists” (red section in Fig 5B, S7C Fig).

As shown in Fig 5B, on the borders between classes of strategies in chemical space, the maximal growth contours turn discontinuously. These points of discontinuity, as in the example in the previous section, are chemical environments corresponding to evolutionarily stable coexistence of species from distinct metabolic classes. The classes of optimally coexisting species change with dilution rate. When the dilution rate is low (d = 0.4, Fig 5C), at the discontinuous point of the maximal growth contour, the corresponding two maximizing strategies are one polluter (species Blue) and one cleaner (species Green). Their supply lines span a gray region where both species Blue and species Green are required to create a steady-state environment on the maximal growth contour. As by assumption we are only supplying the system with S, the supply condition always lies on the x-axis of concentration space. For the supply condition shown by the black open circle in Fig 5C, polluter Blue creates a chemical environment (blue dot) far from the maximal growth contour. When the cleaner Green is added to the system, not only does the biomass of Blue increase (inset), but also the steady-state chemical environment moves to the discontinuous point of the maximal growth contour (cyan dot), where both Blue and Green occupy the peaks of their fitness landscapes (Fig 5D). This result is consistent with the long-term evolution experiment of E. coli and also intuitive: polluter Blue and cleaner Green form a mutually beneficial relationship by, respectively, providing nutrients and cleaning up waste for each other, thereby reaching an optimal cooperative coexistence.

A quite different coexistence occurs at higher dilution rate (d = 0.6, Fig 5E). Growth contours at this dilution rate show two turning points, but neither are between the polluter and the cleaner class. One discontinuous point is between the cleaner class (green squares) and the generalist class (red squares), but the gray region spanned by the corresponding supply lines does not cover the x-axis and so does not represent an attainable coexistence when only S is supplied. The other discontinuous point is between the generalist class and the polluter class (blue squares). The gray region spanned by the supply lines of the corresponding two maximizing strategies of generalist class (species Red) and polluter class (species Blue) does cover the x-axis. Therefore, a supply condition with only S within the gray region (e.g., the black open circle) leads to the optimal coexistence of generalist Red and polluter Blue on the discontinuous point (purple dot), despite the fact that they do not directly benefit each other. Indeed, when the generalist Red is added to a system with polluter Blue and a cleaner Green, the cleaner Green goes extinct and the biomass of the polluter Blue decreases (inset). Nevertheless, the steady-state chemical environment is moved from a cyan dot lying inside the maximal growth contour to the purple dot lying on the maximal growth contour. In the environment of the cyan dot created by cleaner Green and polluter Blue, Blue is not on the top of the fitness landscape of the polluter class (Fig 5F, upper panel). By contrast, for the fitness landscape created by polluter Blue and generalist Red (Fig 5F, bottom panel), despite being lower in biomass, Blue occupies the top of the landscape. Therefore, the optimal coexistence of this polluter and this generalist does not arise from direct cooperation, but rather from collaborating to defeat other competitors.