As cellular growth is primarily due to protein biosynthesis, we consider biomass production to result from the incorporation of building blocks (amino acids or amino acid precursors) into biologically functional units (proteins). Specifically, we assume that biomass production requires $p$ types of building blocks and we denote by $b_i$, $1\le i\le p$, the concentration of block $i$ in cellular biomass. To maintain the stoichiometry of building blocks in biomass, microbes that grow at rate $g$ incorporate block $i$ at rate $g{b}_i$. As the incorporation of building blocks is limited by the internal availability of free building blocks, we model the growth rate as a function $g(c_1,\mathrm{\dots },c_p)$, where $c_i$ is the internal concentration of block $i$. To obtain a plausible functional form for $g(c_1,\mathrm{\dots },c_p)$, we consider the rate of incorporation of a building block to be proportional to its concentration. We also consider that building blocks are sequentially incorporated into biomass (e.g. via protein elongation). Then the time to produce a unit of biomass (e.g. a protein) is the sum of the incorporation times for each type of block $i$, which we take to be proportional to $b_i/c_i$, the ratio of the building-block concentration in cellular biomass to the internal free building-block concentration. The growth rate, which is proportional to the inverse of this time, therefore has the form

where $\gamma$ is a rate constant. For simplicity, we consider that all microbes use the same molecular machinery and building-block stoichiometry to produce biomass. Thus, we consider that the rate function $g(c_1,\mathrm{\dots },c_p)$ is universal, independent of the metabolic strategy used by a microbe to accumulate building blocks. As defined by Equation (1), $g(c_1,\mathrm{\dots },c_p)$ is an example of a growth-rate function satisfying the biologically relevant requirements that more internal resources yield faster growth, i.e. ${\displaystyle g(c_1,\dots ,c_p)}$ is an increasing function of its arguments, and that the relatively scarcest resources are the most growth-rate limiting, i.e. $g(c_1,\mathrm{\dots },c_p)$ has the property of diminishing returns. Importantly, our analysis and conclusions hold for all growth-rate functions that satisfy these natural requirements.

In order to accumulate block $i$ internally, a microbe can import block $i$ from the external medium or produce it internally via conversion of another building block $j$. Thus, to produce biomass, microbes can substitute resources for one another. We allow all possible imports and conversions. The quantitative ability of a microbe to import and convert building blocks constitutes its ‘metabolic strategy’, and corresponds to the cell’s expression of the enzymes that mediate building-block import and conversion. For simplicity, we assume that metabolic fluxes are linear in both enzyme and substrate concentrations. This assumption corresponds to enzymes operating far from saturation, which is justified during resource-limited growth. Specifically, denoting the external concentration of block $i$ by $c_i^{\mathrm{ext}}$, the enzyme-mediated fluxes associated with import and conversion of block $i$ have the form ${\alpha }_i{c}_i^{\mathrm{ext}}$ and ${\kappa }_{ji}{c}_i$, respectively, where ${\alpha }_i$ and ${\kappa }_{ji}$ are enzymatic activities, which are proportional to the number of enzymes dedicated to each metabolic process. In addition to these active fluxes, we include passive transport of building blocks across cell membranes via facilitated diffusion (Pi et al., 1991; Wehrmann et al., 1995). For simplicity, we model passive transport via a single leakage rate $\beta$, yielding a net influx $\beta (c_i^{\mathrm{ext}}-c_i)$ for building block $i$. As cells can only devote a certain fraction of their resources to the production of enzymes, we require the enzymatic activities of each microbe to satisfy a budget constraint,

where $E$ denotes the total enzyme budget. The metabolic strategy of a cell type $\sigma$ is specified by a set of enzyme activities $\{{\alpha }_{i,\sigma },{\kappa }_{ij,\sigma }\}$ that satisfy the budget constraint Equation (2).

Note that for given external building-block concentrations, a cell type is ‘optimal’, i.e. achieves the fastest growth rate, if no other cell type can achieve the same growth rate with a lower enzyme budget. If such a cell type existed, allocating the saved enzyme budget to importing more building blocks would yield a new cell type with higher internal building-block concentrations, and thereby faster growth.

Our model considers a very simplified coarse-grained description of metabolic pathways. In reality, the details of biochemistry play an important role in determining metabolic efficiency. While our modeling framework can be easily generalized to realistic metabolic networks, capturing the complexity of real metabolic pathways is not the purpose of the present analysis, which aims at general principles. Indeed, because our model is anchored in flux conservation, we expect our results concerning the emergence and benefit of division of labor in microbial communities to hold independent of specific pathway biochemistry.

We consider various cell types $\sigma$ growing in a homogeneous environment of volume $V$. We denote the dimensionless population count of cell type $\sigma$ by $n_{\sigma }$ and the total population count by $N={\sum }_{\sigma }{n}_{\sigma }$. In the volume $V$, we consider that the $p$ building blocks are steadily supplied at rates $s_i$ (concentration/time) and can be lost, e.g. via degradation and/or diffusion out of the volume, at a rate $\mu$. Each cell type processes building blocks according to its own metabolic strategy. Conservation of internal building block $i$ for cell type $\sigma$ prescribes the dynamics of the internal concentration $c_{i,\sigma }$ (see Appendix 1),

where the only nonlinearity is due to the growth function $g$. Populations of the various cell types exchange building blocks with the external resource pool via import and passive transport, and also via biomass release upon cell death (Simpson et al., 2007; Schütze et al., 2013). Conservation of extracellular building block $i$ prescribes the dynamics of the external concentration $c_i^{\mathrm{ext}}$ (see Appendix 2),

with cell-type-specific fluxes

where $\delta$ is the rate of cell death (assumed constant) and $f$ is the fraction of biomass released upon cell death. Per-cell fluxes ${\varphi }_{i,\sigma }$ contribute to changing the external concentration $c_i^{\mathrm{ext}}$ via a geometric factor $v/(V-Nv)$, the ratio of the average individual cellular volume $v$ to the total extracellular volume $V-Nv$. As intuition suggests, the smaller the number of cells of a particular type, the less that cell type impacts the shared external concentration via metabolic exchanges.

The inverse of the cellular death rate $\delta$, i.e. the lifetime of a cell, is much larger than the timescales associated with metabolic processes such as building-block-diffusion, conversion, and passive/active transport. This separation of timescales justifies a steady-state approximation for the fast-variables: ${\dot{c}}_{i,\sigma }=0$ and ${\dot{c}}_i^{\mathrm{ext}}=0$. With this approximation, Equations (3) and (4) become flux-balance equations for building blocks. Solving Equation (3) with ${\dot{c}}_{i,\sigma }=0$ yields the internal concentrations $c_{i,\sigma }(c_1^{\mathrm{ext}},\mathrm{\dots },c_p^{\mathrm{ext}})$ as functions of the external concentrations. In turn, solving Equation (4) with ${\dot{c}}_{i,\sigma }=0$ and using the functions $c_{i,\sigma }(c_1^{\mathrm{ext}},\mathrm{\dots },c_p^{\mathrm{ext}})$ yields the external concentrations $c_i^{\mathrm{ext}}(\{n_{\sigma }\})$, as well as the growth rates of cell types $g_{\sigma }(\{n_{\sigma }\})$, as functions of the populations of cell types. Hence, the population dynamics of the cell types is described by a system of ordinary differential equations

which are coupled via the external concentrations. Note that the population dynamics is driven and dissipative: building blocks are constantly both supplied to and lost from the external media, while cell death leads to loss of building blocks because only a fraction of biomass is recycled $(f<1)$. In particular, we expect the dissipative character of the dynamics to drive the microbial population toward a stationary state, with at most $p$ coexisting cell types due to the competitive exclusion principle (Hardin, 1960; Levin and Equilibria, 1970).

The population dynamics prescribed above can be simulated by standard numerical methods. If the number of distinct strategies initially introduced exceeds the number of resources, then over time some cell types will become ‘extinct’, i.e. $n_{\sigma }<1$. We exploit this property to simulate competition between distinct cell types: whenever a cell type $\sigma$ is driven to extinction, we replace it with another randomly sampled strategy ${\sigma }^{\prime }$ with $n_{{\sigma }^{\prime }}=1$. If the already present cell types are not optimal, newly introduced cell types ${\sigma }^{\prime }$ may increase in population at the expense of those present. In any case, we continue to introduce new cell types over the time course of the dynamics. The closer to optimality the already present cell types are, the smaller the probability that a new random strategy will successfully colonize. Eventually, at long times, the surviving population will consist entirely of optimal cell types and will no longer change. It is this final population that concerns us; we only simulate metabolic competition to gain insight into the final optimized population, which is independent of the specific dynamics of the simulation.

In the following, we characterize the enzyme distributions $\{{\alpha }_i,{\kappa }_{ij}\}$ of the cell types that are present in the final population established via competitive population dynamics for equal stoichiometry ($b_i=1$). This characterization requires us to define the ‘metabolic class’ of each cell type in terms of its set of utilized enzymes. In particular, two strategies $\sigma$ and ${\sigma }^{\prime }$ belong to the same metabolic class $ℳ$ if and only if ${\alpha }_i=0\iff {\alpha }_i^{\prime }=0$ and ${\kappa }_{ij}=0\iff {\kappa }_{ij}^{\prime }=0$, where ${\alpha }_i$, ${\alpha }_i^{\prime }$, ${\kappa }_{ij}$, and ${\kappa }_{ij}^{\prime }$ are enzyme activities. We will show that metabolic competition leads to the emergence of consortia of cell types belonging to specific metabolic classes. In our analysis, the term ‘consortium’ designates a community of distinct cell types that cannot be invaded, or equivalently, that cannot be outgrown by any other cell types at specific fixed supply rates. In particular, consortia are composed of co-optimal cell types. The term ‘cartels’ refer to special communities that, in addition of being consortia, can also pin down building-block concentrations at fixed values for a range of different supply rates. Such cartels comprise at least as many distinct cell types as there are resources.

We simulated the population dynamics Equation (6) subjected to continual invasion by metabolic variants and found that metabolic competition leads to stationary states. In our simulations, the volume of the colony $V$ is chosen so that the carrying capacity is $\approx V/v={10}^5$ cells and the dynamics is simulated for a duration of ${10}^5/\delta$, i.e. on the order of ${10}^5$ generations. Figure 2 shows three independent simulations for $p=3$ and supply rates $s_1=11,s_2=9,s_3=0$, starting with $24$ different initial metabolic strategies for each simulation. While $3$ types may coexist for extended periods according to the competitive exclusion principle, the $21$ other types have populations $n_{\sigma }$ that decay exponentially until extinction, i.e. until $n_{\sigma }\le 0.9999$. Upon extinction, a new type is introduced at $n_{\sigma }=1$.

Figure 2

Download asset Open asset

To begin each simulation, the $24$ randomly chosen cell types $\sigma$ are introduced at low population ($n_{\sigma }=1$) in the volume $V$ where the building blocks are abundantly supplied ($c_i^{\mathrm{ext}}\simeq s_i/\mu$). Cellular growth quickly depletes the external concentrations of building blocks, until the overall population of cell types nears carrying capacity. At this point, the external building-block concentrations plummet to low values for which the growth rate of each cell type approximates the fixed death rate, i.e. $g_{\sigma }\approx \delta$. This is when metabolic competition begins in earnest. From this point on, in each simulation, the external building-block concentrations tend to the same stationary values, with the two externally provided building blocks $1$ and $2$ stabilizing at numerically identical values and building block $3$ stabilizing at a lower concentration.

As external building-block concentrations approach their stationary values, virtually all newly introduced metabolic variants quickly become extinct because randomly chosen cell types are unable to compete with those already present. As a result, new cell types are continually introduced, for a total of more than ${10}^9$ different cell types during each simulation. Among these introduced metabolic variants, only a cell type that improves on an already present (nearly optimal) one can invade, and displace the existing cell type. In particular, for each displacement event, we find that the invading and invaded cell types employ almost identical distributions of enzymes and always belong to the same metabolic class. Moreover, during displacement events, the external building-block concentrations are only marginally perturbed, while the overall population of the invaded and invading strategy (e.g. the sum of crossing curves) is nearly constant. This behavior indicates convergence toward a stationary state with fixed building-block availability and with fixed populations of cell types. We confirmed the generality of this convergent behavior with additional simulations over a broad range of different supply rates.

Our numerical simulations indicate that competitive population dynamics subjected to continual invasion leads to the emergence of stationary states. To show that these stationary states emerge independently of how metabolic variants are introduced, we developed an iterative optimization algorithm that yields stationary states without relying on random sampling of cell types. Specifically, the algorithm iterates two optimization steps: First, given distinct cell types $\sigma$, the algorithm implements a Newton-Raphson scheme to compute the steady-state external building-block concentrations $c_i^{\mathrm{ext}}$ and the steady-state populations $n_{\sigma }$ within a relative precision ${10}^{-11}$. Second, given external building-block concentrations $c_i^{\mathrm{ext}}$, the algorithm adapts a gradient-based constrained-optimization algorithm (Wächter and Biegler, 2006) to compute the strategies of the cell types ${\sigma }^{\prime }$ which locally optimize cellular growth rate. Provided the growth rate of a locally optimal cell type ${\sigma }^{\prime }$ exceeds the previous steady-state growth rate by a relative difference $g_{{\sigma }^{\prime }}/\delta -1>ϵ={10}^{-9}$, we initiate a new iteration with a set of starting cell types made of the union of the surviving cell types of the previous iteration $\{\sigma |n_{\sigma }\ge 1\}$ and the newly optimized cell type $\{{\sigma }^{\prime }|g_{{\sigma }^{\prime }}>\delta \}$ with $n_{{\sigma }^{\prime }}=1$. The algorithm halts when no cell type ${\sigma }^{\prime }$ can grow at a rate that exceeds the steady-state growth rate $\delta$ by more than a relative difference $ϵ$, thereby yielding theoretically optimal cell types with high accuracy. By design of the algorithm, the cell types present in the stationary state achieve the optimal growth rate allowed by the external building-block availabilities.

Crucially, for fixed supply rates, this optimization algorithm yields steady-state cell types that are virtually identical to the final cell types obtained via simulations. Moreover, additional simulations show that when present among the initial types, optimal steady-state cell types resist invasion by the more than ${10}^9$ random metabolic variants introduced over the course of ${10}^5$ generations. To demonstrate that simulations effectively converge toward optimal stationary states, we define the ‘relative fitness’ of a cell type $\sigma$ as the ratio $g_{\sigma }(c_1^{\mathrm{ext}},\mathrm{\dots },c_p^{\mathrm{ext}})/{\max }_{{\sigma }^{\prime }}{g}_{{\sigma }^{\prime }}(c_1^{\mathrm{ext}},\mathrm{\dots },c_p^{\mathrm{ext}})$, where ${\sigma }^{\prime }$ denotes a theoretically optimal cell type. Figure 3 shows the evolution of cell-type relative fitnesses during simulations of competitive dynamics. When metabolic competition begins in earnest, all cell types have relative fitnesses smaller than one, indicating that optimal cell types would grow faster at identical external building-block concentrations. Ensuing displacement events by fitter cell types lead to an overall increase in the relative fitness of the surviving cell types. As a result, the relative fitness of all surviving cell types approaches one, demonstrating the convergence toward a stationary population of optimal cell types. Consequently, the final steady-state cell types resist invasion because no metabolic variant can outgrow them.

Figure 3

Download asset Open asset

At steady supply, our numerical simulations together with numerical optimization reveal that competitive population dynamics leads to the stable emergence of optimal cell types. The metabolic strategies of these optimal cell types exhibit network structures that are directly related to external building-block availability. In fact, the building-block supply determines whether many distinct cell types can be jointly optimal, i.e. whether a consortium emerges. For instance, as shown in Figure 4A, if the building blocks are supplied with equal stoichiometry ($s_1=s_2=s_3$), the one survivor is a pure-importer strategy that imports each building block. Because this pure-importer strategy is the single most efficient cell type when building blocks are equally abundant, no other cell type can coexist with it and so no consortium emerges. However, when building blocks are supplied with a different stoichiometry, (e.g. with $s_3=0$ in Figure 4B–D), the pure-importer strategy coexists with other cell types that produce the non-supplied building block by conversion, thereby forming consortia of optimal cell types. The stability and optimality of these consortia is a collective property: No pure-importer strategy can survive without the converting cell types, and without a pure-importer strategy, there is a wasteful external accumulation of converted blocks.

Figure 4

Download asset Open asset

Can the same metabolic consortia emerge for different building block supplies? To answer this question, observe in Figure 2 that for supply rates $s_1=11,s_2=9,s_3=0$, competition leads to a stationary state where the concentrations of both supplied building blocks are equal. Correspondingly, the optimal cell types have symmetric strategies with respect to the usage of block $1$ and block $2$. However, the cell type that converts the most abundantly supplied block is more numerous than its symmetric counterpart, allowing for a symmetric steady state despite asymmetrical supply. For symmetric supply $s_1=s_2=10,s_3=0$, Figure 4B shows that competition yields the same steady-state external concentrations, but with an equal population of each symmetric converter type. Thus, exactly the same consortium of cell types can emerge for different supply conditions, with different population counts but reaching the same external building-block concentrations. By definition, such consortia are cartels. Figure 4 shows the different consortia that can emerge for representative supply conditions, highlighting which ones are cartels. Both numerical simulations and optimizations confirm that microbial cartels emerge for a large range of supply conditions.

A metabolic class is defined as the set of strategies that utilize the same enzymes, i.e. for which a particular subset of enzymes satisfies ${\alpha }_i>0$ and ${\kappa }_{ji}>0$. In total, there are $p$ import enzymes and $p(p-1)$ interconversion enzymes, for a total of $p^2$ enzymes. Thus, in principle, there are at most $2^{p^2}$ metabolic classes according to whether or not each type of enzyme is present (${\alpha }_i>0$ or ${\kappa }_{ij}>0$). However, our simulations suggest that at steady state, the cell types that form consortia and achieve optimal growth belong to very specific metabolic classes: these ‘optimal’ classes utilize only a few, non-redundant metabolic processes (that is, many ${\alpha }_i$ and ${\kappa }_{ji}$ are zero).

Can we specify the network structures of optimal metabolic classes using rigorous optimization principles? Exploiting the linearity of metabolic fluxes, we adapt arguments from transport-network theory (Bohn et al., 2007) to achieve this goal for an arbitrary number of building blocks (see Figure 5 and Appendix 3). Our approach consists in gradually reducing the number of candidate metabolic classes by showing that some classes $ℳ$ cannot contain an optimal strategy. Specifically, we consider a representative strategy $\sigma$ in $ℳ$ with enzyme budget $E={\sum }_i{\alpha }_i+{\sum }_{ij}{\kappa }_{ij}$ at arbitrary external concentrations $c_i^{\mathrm{ext}}$. For the same external concentrations $c_i^{\mathrm{ext}}$, we show that one can always find a strategy ${\sigma }^{\prime }$ from another metabolic class ${ℳ}^{\prime }$ that achieves the same internal concentrations using a smaller enzyme budget $E^{\prime }={\sum }_i{\alpha }_i^{\prime }+{\sum }_{ij}{\kappa }_{ij}^{\prime }<E$. As the existence of a more ‘economical’ strategy ${\sigma }^{\prime }$ contradicts the optimality of metabolic class $ℳ$, we can restrict our consideration to metabolic classes other than $ℳ$.

Figure 5

Download asset Open asset

Using the above approach, we show that optimal metabolic networks process building blocks via non-overlapping trees of conversions (Figure 5A), each tree originating from an imported building block (Figure 5B), and each converted building block being obtained via the minimum number of conversions (Figure 5C). Intuitively, these properties ensure the minimization of waste (loss of building blocks via passive transport) during metabolic processing. Moreover, we show that optimal networks use a single building-block resource as precursor for conversions, i.e. there is at most one tree of conversions (Figure 5D). Thus, at steady state, requiring that a metabolic class is optimal, i.e. contains the fastest growing cell type, strongly constrains the graph of its metabolic network. These constrained graphs can be fully characterized and enumerated for $p$ building blocks: there are $p$ distinct graphs, each utilizing $p$ distinct enzymes, which defines a total of $1+p(2^{p-1}-1)$ metabolic classes after accounting for building-block permutations.

To find the composition of consortia, we must identify the enzyme distributions $\{{\alpha }_i$, ${\kappa }_{ji}\}$ within a metabolic class that yield the fastest growth for fixed external building-block concentrations. Knowing analytically the optimal enzyme distributions in each metabolic class allows us to characterize the structure of consortia at steady state via the maximum growth rate as a function of external building-block concentrations,

At competitive stationary state, the maximum growth rate must equal the death rate $\delta$ by Equation (6). Otherwise, either there is a cell type such that $g_{\sigma }-\delta >0$, yielding a diverging population $n_{\sigma }$, or $g_{\sigma }-\delta <0$ for all cell types, yielding a vanishing microbial population. Thus, solving $G(c_1^{\mathrm{ext}},\mathrm{\dots },c_p^{\mathrm{ext}})=\delta$ determines the set of steady-state external concentrations $c_1^{\star },\mathrm{\dots },c_p^{\star }$ for which an optimal strategy ${\sigma }^{\star }$ is present. By virtue of its optimality, the strategy ${\sigma }^{\star }$ achieves the fastest possible growth rate and is non-invadable at steady state. Consortia emerge for external building-block concentrations for which there is more than one optimal strategy, i.e. when there are distinct strategies ${\sigma }^{\star }$ for which the maximum-growth function is attained: ${\displaystyle g_{{\sigma }^{\star }}(c_1^{\star },\dots ,c_p^{\star })=G(c_1^{\star },\dots ,c_p^{\star })=\delta }$.

Obtaining analytical expressions for optimal enzyme distributions proves intractable for a nonlinear growth-rate function such as Equation (1). However, optimal distributions can be obtained analytically for the minimum model $g(c_1,\mathrm{\dots },c_p)=\gamma \min (c_1,\mathrm{\dots },c_p)$, which is closely related to Equation (1) (see Appendix 4). In Figure 6, we represent the corresponding set of external building-block concentrations compatible with steady states, together with the associated optimal metabolic classes. For $p$ building blocks (see Appendix 5), we find that there exist $p!$ microbial cartels, each with $p$ distinct cell types for well-ordered external concentrations, e.g. $c_1^{\mathrm{ext}}>c_2^{\mathrm{ext}}>\mathrm{\dots }>c_p^{\mathrm{ext}}$. In such cartels, cell type $1$ converts building block $1$ into the $p-1$ other building blocks, cell type $2$ converts building block $1$ into the $p-2$ least abundant building blocks and imports building block $2$, and so forth, and cell type $p$ has a pure-importer strategy. We also find that for degenerate ordering with $q-1$ equalities, e.g. $c_1^{\mathrm{ext}}=\mathrm{\dots }=c_q^{\mathrm{ext}}>c_{q-1}^{\mathrm{ext}}>\mathrm{\dots }>c_p^{\mathrm{ext}}$, there exist $(p-q)!C_p^q$ microbial cartels with $1+q(p-q)$ distinct cell types. In such cartels, cell type $q^{\prime }$, $1\le q^{\prime }\le q$ imports all blocks $1,\mathrm{\dots },q$ but only uses block $q^{\prime }$ as a precursor for blocks $j>q$, cell type $q^{\mathrm{\prime \prime }}$, $q<q^{\mathrm{\prime \prime }}\le 2q$ imports all blocks $1,\mathrm{\dots },q+1$ but only uses block $q^{\mathrm{\prime \prime }}-q+1$ as a precursor for blocks $j>q+1$, and so forth, and cell type $1+q(p-q)$ has a pure-importer strategy. Moreover, we find that cartels that share $p-1$ metabolic classes are joined by continuous paths in the space of external concentrations over which these $p-1$ shared metabolic classes remain jointly optimal. Such paths define a graph which characterizes the topological structure of cartels in relation to changes in external building-block concentrations (see Appendix 6). Importantly, our analysis shows that the above cartels emerge with the same graph structure for all growth-rate functions satisfying $g(c_1,\mathrm{\dots },c_p)\ge \gamma \min (c_1,\mathrm{\dots },c_p)$ for some $\gamma >0$ and having diminishing returns (quasi-concave property), which includes Equation (1).

Figure 6

Download asset Open asset

Microbial cartels only exist for specific external building-block concentrations (cf. the intersection points in Figure 6A and B). Can competitive population dynamics lead to these cartels for generic supply conditions? To answer this question, we compute the set of supply conditions compatible with the emergence of a cartel. We label a microbial cartel ${\mathrm{\Sigma }}^{\star }$ by its associated external concentrations $c_1^{\star },\mathrm{\dots },c_p^{\star }$, which satisfy a specific (possibly degenerate) order relation. At concentrations $c_1^{\star },\mathrm{\dots },c_p^{\star }$, cartel cell types $\sigma \in {\mathrm{\Sigma }}^{\star }$ jointly achieve the optimal growth rate and are therefore the only surviving cell types. The per-cell fluxes ${\varphi }_{i,\sigma }^{\star }$ experienced by these cell types take fixed values that can be obtained via Equation (5). Then, the resulting flux-balance equations for extracellular building blocks,

yield the supply rates as a function of the populations $n_{\sigma }>0$, $\sigma \in {\mathrm{\Sigma }}^{\star }$. In fact, Equation (8) defines the sector of supply rates compatible with the existence of the cartel ${\mathrm{\Sigma }}^{\star }$ as a $p$-dimensional cone. Crucially, although $c_1^{\star },\mathrm{\dots },c_p^{\star }$ specify isolated points in the space of external concentrations, the cartel sectors have finite measure in the space of supply rates, showing that cartels can arise for generic conditions.

In Figure 7, we illustrate the supply sectors associated with cartels. What is the relation between supply sectors and steady-state external concentrations? To answer this question, we consider in Figure 7A–E the case of $p=2$ building blocks and first focus on supply rates for which the optimal cell type is either a pure converter (white dots in Figure 7A–B) or a pure importer (white dots in Figure 7C–D). With only one cell type present and fixed $c_1^{\mathrm{ext}}$ and $c_2^{\mathrm{ext}}$, varying $n_{\sigma }$ in Equation (8) defines a half-line in the supply plane (cf. Figure 7B and Figure 7D). This half-line originates from the point $(\mu c_1^{\mathrm{ext}},\mu c_2^{\mathrm{ext}})$, which are the supply rates that first support a nonzero population $n_{\sigma }$ for steady-state concentrations $c_1^{\mathrm{ext}}$ and $c_2^{\mathrm{ext}}$. Thus, all supply half-lines originate from a surface that is identical to the steady-state external concentrations, simply rescaled by the external building block leakage rate $\mu$. Changing the supply rates transverse to such a half-line yields different steady-state concentrations $c_1^{\mathrm{ext}}$ and $c_2^{\mathrm{ext}}$. In particular, one can increase the supply of one block until another optimal type can invade, i.e. until one reaches the cartel-specific concentrations $c_1^{\star }$ and $c_2^{\star }$ (pink dots in Figure 7A and Figure 7C). At the corresponding point $(\mu c_1^{\star },\mu c_2^{\star })$ in the supply space, the pure-converter half-line and the pure-importer half-line have distinct direction vectors, i.e. co-optimal cell types $\sigma$ have distinct per-cell fluxes ${\varphi }_{\sigma }$: e.g. for $s_1>s_2$, a pure-converter $\sigma$ consumes block $1$ to produce and leak block $2$, i.e. ${\varphi }_{1,\sigma }>0$ and ${\varphi }_{2,\sigma }<0$. By contrast, a pure importer $\sigma$ consumes all building blocks according to the biomass stoichiometry, i.e. ${\varphi }_{1,\sigma }={\varphi }_{2,\sigma }$. For each point $(\mu c_1^{\star },\mu c_2^{\star })$ in the supply space, these distinct per-cell fluxes ${\varphi }_{\sigma }$ define conic regions where a pure importer can invade a pure-converter population and a pure converter can invade a pure-importer population, i.e. where a cartel is stable. These cones are therefore cartel supply sectors (pink and yellow sectors in Figure 7E).

Figure 7

Download asset Open asset

The above argument can be generalized for $p>2$ building blocks by considering the metabolic fluxes of optimal cell types (cf. Figure 7F for $p=3$). For all values of $p$, we find that supply sectors associated with cartels define non-overlapping cones (see Appendix 7). Moreover cones associated with two connected cartels, i.e. cartels that share at least $p-1$ metabolic classes, have parallel facets in the limit of large budget $E\gg \beta$. As a consequence, at a fixed overall rate of building-block supply $s=s_1+\mathrm{\dots }+s_p$, the fraction of supply conditions for which no cartel arises becomes negligible with increasing overall supply rate $s$. For instance, for large rate $s$, every building block has to be supplied at exactly the same rate for a single pure-importer strategy to dominate rather than a cartel. Therefore, for very generic conditions, a cartel will arise and drive the external building-block concentrations toward cartel-specific values, thereby precluding invasion by any other metabolic strategy. This ability to eliminate competition is reminiscent of the role of cartels in human economies, motivating the name ‘cartels’ for stable microbial consortia that include at least $p$ distinct metabolic strategies.

At supply conditions for which no cartel arises, an optimal cell type or a consortium of cell types dominates at steady state but these cell types cannot control external resource availability. Indeed, a consortium that is not a cartel cannot compensate for changes in supply conditions via population dynamics. In particular, simply multiplicatively increasing the building-block supply augments the steady-state biomass of a consortium but also modifies steady-state resource availabilities, and therefore the distributions of enzymes that optimally exploit these resources. In other words, for consortia that are not cartels, optimal metabolic strategies must be fine-tuned to specific supply conditions.

By contrast, within a cartel supply-sector, any increase of the building-block supply is entirely directed toward the cartel’s growth of biomass. Remarkably, it appears that microbial cartels automatically achieve maximum carrying capacity, i.e. they optimally exploit the resource supply. At steady state, the total number of cells $N$ is related to supply rates $s_i$ via the overall conservation of building blocks by

which implies that maximizing biomass yield at fixed supply rates $s_i$ amounts to minimizing the overall external building-block concentrations ${\sum }_i{c}_i^{\mathrm{ext}}$. In Figure 8, for $p=2$ building blocks, we use the preceding equivalence to show that in each cartel sector, no consortium can yield a larger steady-state biomass than the supply-specific cartel. For a homogeneous growth-rate function such as Equation (1), this result generalizes to arbitrary $p$ if we conjecture that (i) in a cartel supply sector, the cartel’s metabolic classes can invade any other consortium, and that (ii) the maximum-growth-rate function associated with a given metabolic class has the property of diminishing returns (see Appendix 7). Intuitively, conjecture (i) means that the emergence of a cartel does not depend on the history of appearance of distinct metabolic classes and conjecture (ii) means that beating diminishing returns requires a switch of metabolic classes. Together, these conjectures ensure that adding a new metabolic class when possible implies a decrease in the total abundance of building blocks, i.e. a better use of resources. Because a better use of resources is equivalent to a steady-state biomass increase (by virtue of building-block conservation), this establishes that competing microbes achieve the global collective optimum by forming cartels.

Figure 8

Download asset Open asset

Inspired by classic network-theory arguments, we show that at steady state the network of import and interconversion enzymes associated with optimal strategies is a directed forest of trees rooted in some or all of the external building blocks.

Consider a strategy $\sigma$ that has a cycle of order $2$, e.g. enzymes interconverting building block $i$ into building block $j$ and building block $j$ into building block $i$:

Altering the enzymatic activities according to ${\kappa }_{ij}\leftarrow {\kappa }_{ij}+\delta {\kappa }_{ij}={\kappa }_{ij}^{\prime }$ and ${\kappa }_{ji}\leftarrow {\kappa }_{ji}+\delta {\kappa }_{ji}={\kappa }_{ji}^{\prime }$ with

leads to the same net flux between $i$ and $j$. As a result, the net building-block fluxes are preserved for the same internal concentrations $c_i$ and $c_j$. The corresponding altered enzyme budget is $E^{\prime }=E+\delta E$ with

showing that reducing the enzymatic activity ${\kappa }_{ji}$ leads to a smaller enzyme budget if ${\kappa }_{ij}\ge 0$. Thus, one can always find a more economical strategy ${\sigma }^{\prime }$ that uses only one enzyme to perform an interconversion( the same reasoning applies to ${\kappa }_{ij}$). This shows that the network of interconversion reactions of optimal strategies has no cycle of order $2$, so we can restrict our consideration to metabolic classes satisfying ${\kappa }_{ij}{\kappa }_{ji}=0$ for all $i$ and $j$.

We can generalize the above argument to show that an optimal strategy cannot exhibit interconversion cycles of any order (see Appendix 3—figure 1). To see this, consider a interconversion network of a strategy $\sigma$ with no $2$-cycles and suppose that its undirected graph has a $3$-cycle, e.g. of the form

Then, altering the enzymatic activities according to

leaves the net internal fluxes unchanged, and the altered strategy ${\sigma }^{\prime }$ uses a budget $E^{\prime }=E+\delta E$ with

Thus, for generic conditions, one can always form a more economical strategy ${\sigma }^{\prime }$ by either reducing or increasing ${\kappa }_{ji}$, until one of the activities along the cycle becomes zero. This shows that the network of interconversion reactions of optimal strategies has no cycle of order $3$, so that we can restrict ourselves to metabolic classes satisfying ${\kappa }_{ij}{\kappa }_{ji}=0$ and ${\kappa }_{(ij)}{\kappa }_{(jk)}{\kappa }_{(ki)}=0$ for all $i$, $j$, and $k$, where ${\kappa }_{(ij)}=\max ({\kappa }_{ij},{\kappa }_{ji})$. The above argument directly generalizes to cycles of any order showing that, if a strategy is optimal, the undirected graph of its interconversion network has no cycles. Therefore, the undirected graph of an optimal interconversion network is a ‘forest of trees’. Observe that, in particular, this result implies that cycles are never optimal.

Appendix 3—figure 1

Download asset Open asset

In the following, exploiting the same procedure as above, we show that optimal networks satisfy an additional topological property: building blocks are accumulated internally from a unique external source (see Appendix 3—figure 2). In graph theory, the defining property of a forest is that, given any two nodes, there is at most one path joining them. For our network of import and interconversion enzymes, this property implies that, given two external building-block concentrations, e.g. $c_i^{\mathrm{ext}}$ and $c_j^{\mathrm{ext}}$, there is at most one chain of processes whose undirected path links $c_i^{\mathrm{ext}}$ and $c_j^{\mathrm{ext}}$. For simplicity, consider a strategy $\sigma$ for which there exists such a path of length $3$, e.g. corresponding to:

Then, altering the enzymatic activities according to

leaves the net internal fluxes unchanged and the altered strategy ${\sigma }^{\prime }$, uses a budget $E^{\prime }=E+\delta E$ with

Appendix 3—figure 2

Download asset Open asset

Thus, for generic conditions, one can always form a more economical strategy ${\sigma }^{\prime }$ by either reducing or increasing ${\kappa }_{ji}$ according to the sign of the term in between parenthesis in Equation (A35), until one of the activities along the path becomes zero. In all rigor, this last point requires that building-block import can be set to zero, which is not realistic in the presence of passive leakage. However, including passive imports does not change our result as long as the building-block internal concentrations are larger than the building-block external concentrations, which is the biologically relevant case. In any case, if a network of import and interconversion enzymes is optimal, there is at most one path connecting a building block to an external resource. Moreover, such a path has to exist and has to flow from the external building-block source for the internal building-block concentration to be non-zero. As a result, the network of import and interconversion enzymes forms a forest of directed trees rooted in the external building-block concentrations.

Using arguments inspired by network theory, we have shown that, in optimal networks of import and conversion enzymes, each building block is made from a single external source, via a single enzymatic chain. This constraint on the topology of optimal networks drastically reduces the number of metabolic classes to inspect for optimal strategies. In the following, we will show that it is possible to further restrict the set of candidate classes. Specifically, we will show that optimal strategies have a single non-trivial tree of depth one, i.e. the building blocks that are not directly imported are all made in one step from a single imported building block. In other words, optimal strategies convert at most one building block.

To prove the above claim, we first show that, for a complete set of import and interconversion enzymes, trees originating from an imported building block are at most of depth one (see Appendix 3—figure 3). Consider a strategy $\sigma$ that processes a precursor building block $i$ via a tree of interconversions of depth at least $2$. Necessarily, there is a building block $k$ made after two successive interconversions. Call $j$ the intermediary building block between $i$ and $k$. We thus have

where $c_i$, $c_j$, and $c_k$ denote steady-state internal concentrations. Now consider a strategy ${\sigma }^{\prime }$ that is the same as $\sigma$ except that $k$ is directly made from $i$, i.e. ${\kappa }_{kj}^{\prime }=0$ and ${\kappa }_{ki}^{\prime }>0$, and possibly ${\kappa }_{ji}^{\prime }\ne {\kappa }_{ji}$:

Choosing ${\kappa }_{ji}^{\prime }$ and ${\kappa }_{ki}^{\prime }$ such that

leaves the net fluxes into each internal building block unchanged and the altered strategy ${\sigma }^{\prime }$, which grows as fast as $\sigma$, uses a budget

Thus, one can always form a more economical strategy ${\sigma }^{\prime }$ by replacing a two-step synthesis process by a one-step synthesis process. Recursive application of the above argument shows that, if not imported, building blocks should be made from their imported precursor in as few steps as possible. In particular, we deduce that optimal networks have trees of depth one, i.e. any converted building block is made in one step from its precursor.

Appendix 3—figure 3

Download asset Open asset

To complete the proof of our claim, we now show that an optimal strategy uses at most a single imported building block as a precursor for the synthesis of non-imported blocks (see Appendix 3—figure 4). Consider a strategy $\sigma$ that uses two imported building blocks: building block $i$ as a precursor for $q_i$ blocks, and building block $j$ as a precursor for $q_j$ blocks. By symmetry, in optimal strategies, the $q_i$ blocks made from $i$ are processed identically at concentration ${\displaystyle c_i}$, as well as the $q_j$ blocks made from $j$ at concentration ${\displaystyle c_j}$. Thus, the internal flux-balance equations for the precursors read

while the internal flux-balance equations for the end products read

where ${\kappa }_{ki}$ and ${\kappa }_{lj}$ are the enzymatic activities associated with the conversion of $i$ and $j$ into an end product, respectively. Algebraic manipulations of the flux-balance equations allow one to express the enzymatic activities ${\alpha }_i$, ${\alpha }_j$, ${\kappa }_{ki}$, and ${\kappa }_{lj}$ in terms of the internal and external building-block concentrations and of the rate of biomass production $g$. For instance, we have: