Model principles and state equations.

The synthetic consortium considered here consists of two E. coli strains, one growing on glucose and producing the heterologous protein, and the other preferentially growing on acetate and thus removing this growth-inhibitory by-product from the environment [20]. While the overall structure of the model is similar to other population-based models of synthetic mutualistic consortia [28–33], it also differs on key points from previous work in order to account for regulatory mechanisms specific to E. coli. We here explain the model for the producer strain in detail and then appropriately modify the model for the cleaner in a later section.

The model for the producer is schematically represented in Fig 2A. In a well-defined minimal medium, glucose at a concentration G [g L⁻¹] is taken up at a rate [g gDW⁻¹ h⁻¹] (where gDW stands for “gram Dry Weight”). The substrate is converted partly into proteins and other macromolecules, which make up most of the biomass, and partly utilized for producing ATP and other energy carriers. In fact, around half of the carbon contents of the substrate is consumed by energy metabolism and leaves the cell in the form of CO₂ [41]. The conversion of glucose into biomass is thus characterized by the biomass yield coefficient Y_g [gDW g⁻¹]. Acetate overflow at a rate [g gDW⁻¹ h⁻¹] occurs when the glucose uptake rate reaches a threshold level [42, 43], thus increasing the acetate concentration A in the medium. In the absence of glucose, E. coli can utilize acetate as an alternative substrate, taken up at a rate [g gDW⁻¹ h⁻¹] and converted into biomass with a yield coefficient Y_a [gDW g⁻¹]. Accumulation of acetate in the medium inhibits bacterial metabolism and growth [15–17]. The producer strain differs from a wild-type E. coli strain in that it carries a plasmid for the inducible expression of a heterologous protein. As a consequence, the total biomass concentration B_tot [gDW L⁻¹] is the sum of the concentration of heterologous protein H [gDW L⁻¹] and the concentration of biomass actively involved in cellular growth and maintenance, the autocatalytic biomass B [gDW L⁻¹]. The fraction of biomass production assigned to the synthesis of heterologous protein is determined by the (dimensionless) product yield constant Y_h. The total biomass B_tot [gDW L⁻¹] decays at a rate k_deg B_tot, with degradation rate constant k_deg [h⁻¹]. In other words, non-growth-related maintenance of the biomass requires an expenditure of resources equal to k_deg B_tot [44].

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Fig 2. Model of the producer strain and its calibration.

(A) The model describes the uptake of nutrients, glucose and acetate, at rates and , respectively, as well as the pathways for acetate overflow (), production of autocatalytic biomass and heterologous protein in proportions determined by the yield coefficient Y_h, and degradation of biomass (k_deg). The bacterial population grows in a bioreactor operating in continuous mode, whereby the dilution rate D and the glucose concentration in the inflow G_in can be tuned (indicated in blue). Concentrations in the bioreactor are denoted by G for glucose, A for acetate, H for heterologous protein, and B for autocatalytic biomass. For clarity, regulatory interactions due to growth inhibition by acetate and carbon catabolite repression have been omitted from the figure. (B) and (C) Experimental data from [13] used to identify the parameter values of the system of Eqs 1–6. The data include measurements of the growth rate μ⁺, the glucose uptake rate , the (net) acetate uptake rate , and the acetate overflow rate during exponential growth of an E. coli wild-type strain in batch conditions in minimal medium with glucose as the sole carbon source (B left panel), acetate as the sole carbon source (B right panel) and in glucose with increasing concentrations of acetate added (C). The + symbol indicates that the measurements have been carried out during exponential growth in batch. The fit of the model to the data is shown as well (red curve). The R² values of the fit are 0.6, 0.77, and 0.77 (from left to right).

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The producer strain is assumed to grow in a bioreactor operating in continuous mode [38], that is, with a dilution rate D [h⁻¹] and a glucose concentration G_in [g L⁻¹] in the inflow. This gives rise to the following dynamical system of Ordinary Differential Equations (ODEs): (1) (2) (3) (4) with (5)

The term represents the biomass equivalent lost due to acetate overflow, that is, the biomass that would have been produced if the secreted acetate had instead been taken up by the cells. Note that the specific rate of biomass production per unit of biomass, including both heterologous protein and catalytic biomass, is given by . Since only catalytic biomass is actively participating in cellular growth, the total rate of biomass production is obtained by multiplying the specific rate with B.

Definition of growth rate.

The model of Eqs 1–5 allows the (specific) growth rate μ [h⁻¹] to be explicitly defined in terms of the reaction rates. By definition, the rate of change of the total biomass in the bioreactor equals the biomass produced due to the growth of the bacterial culture (μB_tot) minus the biomass flushed out due to dilution (DB_tot): (6)

Therefore, the specific growth rate μ represents the sum of the rate of change of total biomass per total biomass unit and the dilution rate (7)

From Eqs 3–5 it then follows that (8)

In the absence of heterologous protein production B_tot = B, and the growth rate is given by the balance between substrate assimilation and dissimilation and biomass degradation. Heterologous protein production, however, puts a burden on the growth of the population, by drawing away resources from catalytic biomass. Eq 8 brings out this burden in that a larger H leads to a smaller ratio B/B_tot lowering the specific growth rate. At steady state, as shown in the Methods section, we find from Eqs 3 and 4 that , with * indicating quantities at steady state. Then, Eq 8 reduces to (9)

This formulation emphasizes that, at steady state, the growth rate linearly depends on the product yield Y_h, as well as on the glucose and acetate uptake rates.

Definition of reaction rates.

The rate equations are functions of the nutrient concentrations and express the regulatory mechanisms at work. The rate laws are motivated by a combination of modeling assumptions and experimental evidence as explained below.

The uptake rates are defined in such a way that, when inserted into Eq 8, they lead to Monod-like dependencies of the growth rate on the substrate concentrations, modulated by terms accounting for regulatory effects [45, 46]: (10) (11)

k_g, k_a [g gDW⁻¹ h⁻¹] denote maximal uptake rate constants and K_g, K_a [g L⁻¹] half-maximal saturation constants. The inhibition term with the constant Θ_a [g L⁻¹] in Eq 10 represents the inhibitory effect of acetate on bacterial growth [15–17], whereas the inhibition term with the constant Θ_g [g gDW⁻¹ h⁻¹] in Eq 11 corresponds to carbon catabolite repression (CCR), that is, the repression of enzymes in the metabolism of acetate and many other secondary carbon sources while cells are growing on glucose [34, 47]. The strength of CCR is correlated with the phosphorylation state of the preferred glucose uptake system, the phosphotransferase system (PTS) [48], and therefore with the glucose uptake rate in our conditions. Note that, due to acetate inhibition, the glucose uptake rate may be low (and CCR partially relieved) even in the presence of high glucose concentrations in the medium. The exponents n and m shape the nonlinear effect of acetate inhibition and carbon catabolite repression, respectively.

When the glucose uptake rate exceeds a threshold l [g gDW⁻¹ h⁻¹], E. coli produces and secretes acetate as a fermentation by-product [42, 43]. This overflow metabolism has been explained as providing cells, in the presence of excess glucose, with a less efficient but also less costly way to produce ATP than through respiration [43]. Experimental data show that, above the threshold level, the acetate secretion rate is proportional to the excess glucose uptake rate: (12)

Model calibration.

The model has 14 parameters most of which have been measured or can be estimated from published data sets. The fact that these experiments have been carried out with different E. coli strains and in different conditions carries the risk of obtaining parameter values that are mutually inconsistent. In order to avoid this problem, we have calibrated the model as much as possible against a single, recently published data set for an E. coli wild-type strain grown in batch in minimal medium with different concentrations of glucose and acetate [13], that is, for conditions in which D = 0 h⁻¹ and Y_h = 0. Only the values for the biomass degradation constant k_deg, the threshold for acetate overflow l, and the half-maximal saturation constants K_g and K_a were taken from the literature. The results of the parameter estimation procedure, described in more detail in the Methods section, are plotted in Fig 2 and show an excellent fit of the model to the data, despite the apparent noise in the measurements, which prevents a perfect fit by any sufficiently smooth model. The parameter values are reported in Table 1 and were compared with other published values for consistency (Methods). In order to verify that the resulting model is well-posed, we also performed an a-posteriori identifiability analysis (S1 Fig), showing tight confidence intervals for the parameter estimates. In most simulations, the product yield coefficient was set to Y_h = 0.2.

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Table 1. Parameter values.

Values of the parameters of the model of Eqs 1–4 describing the producer strain, with rates as in Eqs 10–12, and the cleaner strain, with rates as in Eqs 14–16.

https://doi.org/10.1371/journal.pcbi.1007795.t001

Diauxic growth.

When growing in batch in minimal medium with glucose, E. coli bacteria attain a high growth rate and secrete acetate into the medium. Only after the glucose has been consumed, the cells start to take up acetate [12, 49]. We validated our model by ensuring that it reproduces this so-called diauxic growth behavior by simulating a batch experiment (D = 0 h⁻¹) in the absence of heterologous protein production (Y_h = 0). The results are shown in Fig 3A. The biomass concentration in the bioreactor B is seen to increase while the glucose concentration (G) decreases and acetate accumulates without being consumed (A). When the glucose is almost completely depleted (∼4 h), cells continue growth on acetate at a lower rate. As shown in the same plot, the simulations are in good agreement with the experimental data of Enjalbert et al. [13]. The capability to reproduce diauxic growth depends on the inclusion of a regulatory term for CCR [34, 47] in Eq 11, a feature absent from many previous models. The property of diauxic growth is preserved in the case of heterologous protein production (S2 Fig).

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Fig 3. Model validation.

(A) Time evolution of the concentration of glucose G (blue curve), acetate A (red curve), and biomass B (green curve) in a bioreactor operating in batch mode, with initial concentrations of 2.7 g L⁻¹ of glucose and 0.1 g L⁻¹ of biomass. The predicted time-courses are compared with the data from Enjalbert et al. [13] (dots). (B) Effect of lower growth yield Y_g, accounting for the use of an energy-dissipating LacY mutant, on the onset of acetate overflow, indicated by the overflow rate (black curves) reached at steady state in a bioreactor operating in continuous mode, with G_in = 20 g L⁻¹. A decrease in growth yield (Y_g = 0.27 gDW g⁻¹ for the mutant vs Y_g = 0.44 gDW g⁻¹ for the wild-type strain) causes acetate overflow to occur at a lower growth rate in the mutant and the rate of overflow to depend more strongly on the growth rate (steepness of the curve). The model predictions are compared with the data from Basan et al. [43] (dots). Since the onset of acetate overflow occurs at slightly different growth rates in the wild-type strain used for calibration and the wild-type strain used by Basan et al., we compare the relative changes in growth rates upon a decrease in growth yield. refers to the growth rate at which acetate overflow starts. (C) Predicted versus measured growth rate values for glucose uptake mutants (dots and diamonds, see legend) with different values for k_g that grow exponentially in batch in minimal medium with glucose (G(0) = 3.6 g L⁻¹). The growth rates of the mutants μ⁺ have been normalized with respect to the growth rate of the wild-type strain in the same conditions. Dashed lines indicate perfect predictions as a reference. The data are from Steinsiek et al. [52]. (D) Idem for acetate secretion rates of mutants () and wild type (). In all four plots there is a good or very good correspondence between the model predictions and the experimental data.

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Shift in acetate overflow in energy-dissipating mutant.

The proton-leaking LacY mutant described by Basan et al. [43] has an effect of energy dissipation and therefore a lower growth rate. One might naively expect that the decrease in growth rate would lower the amount of acetate secreted, but this is not the case as energy dissipation causes acetate overflow to start at a lower growth rate [43]. The model reproduces this observation in the case of steady-state growth in a bioreactor operating in continuous mode, without heterologous protein production (Y_h = 0), as shown in Fig 3B. In order to mimick the effect of the LacY mutation, the growth yield Y_g was decreased from its estimated value of 0.44 gDW g⁻¹ (Table 1) to 0.27 gDW g⁻¹. Intuitively, the onset of acetate overflow occurs at a lower growth rate, because for a lower growth yield more substrate is needed to produce a given amount of biomass. As a consequence, the glucose uptake rate must be higher to attain the growth rate set by the dilution rate. The model not only correctly predicts that acetate overflow starts at a lower growth rate, but also that the rate of acetate secretion increases more strongly with growth rate, as witnessed by the steeper curve for the lower Y_g value in Fig 3B. The reproduction of the shift in acetate overflow is a non-trivial achievement of the model and critically depends on the introduction of a regulatory term for acetate overflow in the model, Eq 12, a feature missing in many other models.

Like energy dissipation, the overexpression of a heterologous protein puts a burden on the metabolism of the host cell, leading to a lower growth rate and other problems for the metabolic engineering of the cell [36, 37]. The analogy between energy dissipation and protein overexpression can be further developed by deriving from Eq 9 the following relation between the yields Y_g, Y_h, the dilution rate D, and the threshold l for acetate overflow: (13)

In the derivation we used that μ* = D and overflow starts at , where and (due to carbon catabolite repression). Eq 13 shows that a glucose uptake rate equal to l occurs at lower D for both lower Y_g (energy dissipation) and higher Y_h (heterologous protein production). For an increase in product yield, the model indeed predicts the same shift in acetate overflow as for a decrease in growth yield (S2 Fig), consistent with reports in the literature [50].

Changes in growth rate and acetate secretion rate in glucose uptake mutants.

While the PTS is the main glucose uptake system of E. coli, a number of other systems are capable of transporting glucose into the cell, such as maltose and galactose transporters [51, 52]. Steinsiek et al. systematically tested during exponential growth in batch how the growth rate, the glucose uptake rate, and acetate secretion rate change in strains in which (combinations of) uptake systems have been deleted. The mutants are straightforward to simulate by means of the model bearing in mind that the deletion of an uptake system decreases the value of k_g, the maximum glucose uptake rate. More precisely, the observed relative decrease of the glucose uptake rate in a mutant strain leads to a decrease in the same proportion of k_g. The predicted growth rates and acetate secretion rates for the mutants correspond well to those measured experimentally by Steinsiek et al. (Fig 3C and 3D). As expected, for mutants strongly reducing glucose uptake the growth rates are low and no acetate overflow occurs. Mutations in glucose uptake systems form the basis for constructing a cleaner strain that does not or only weakly grow on glucose but instead takes up acetate at a high rate (see [20] and below).

Design of cleaner strain and model equations.

The cleaner strain is obtained from a wild-type E. coli strain by knocking-out the gene ptsG, encoding a major subunit of the glucose uptake system PTS [51], and transforming this strain with a plasmid enabling the inducible overexpression of the native gene acs, coding for the enzyme acetyl-CoA synthetase (Acs) [12]. It has been shown that deletion of ptsG reduces the glucose uptake rate 4-fold [52]. The possibility of the cell to continue to take up glucose at a reduced rate, through other, non-specific systems, is advantageous for our purpose, since it allows the cleaner to grow faster than on acetate alone and thus enlarges the coexistence range in continuous-mode cultivation (see below). Overexpressing acs has been shown to increase the growth rate of E. coli on acetate [53]. Unlike the producer, the cleaner strain does not express the heterologous protein of interest.

With the above design, the model of the cleaner strain is similar to that of the producer, except that Eq 4 can be eliminated and B_tot = B. The rate equations for the cleaner are the following, using the subscript _C to refer to concentrations and rates of the cleaner: (14) (15) (16)

The only modifications with respect to the producer strain are the replacement of k_g by k_ΔPTS [g gDW⁻¹ h⁻¹], k_ΔPTS < k_g, in Eq 14, and the addition of an extra term acounting for Acs overexpression in Eq 16, with maximal uptake rate k_Acs [g gDW⁻¹ h⁻¹], k_Acs > k_a, and half-maximal saturation constant K_Acs [g L⁻¹].

Model of consortium.

Next, we assemble the model for the protein producer and the acetate cleaner to obtain the model of the consortium (Fig 4). The two populations, growing in the same reactor environment, interact in a number of direct and indirect ways. Both the producer and the cleaner take up glucose from the environment, while in addition the cleaner is capable of assimilating acetate secreted by the producer. The removal of acetate cleans the environment from a toxic by-product inhibiting the growth of, especially, the producer. The mutualistic interaction structure of the consortium, in which both strains favor growth of the other, causes the consortium to escape the exclusion principle [54] and makes coexistence of the two strains possible.

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Fig 4. Model of the producer-cleaner consortium.

The model describes the protein producer and acetate cleaner strains, in green and hatched orange, respectively. The producer strain preferentially grows on glucose, whereas the cleaner has been modified in such a way as to prefer acetate over glucose. The genetic modifications (purple uptake arrows) include the deletion of the preferred glucose uptake system and the overexpression of a selected enzyme in acetate metabolism. The producer also expresses the heterologous protein of interest. The biomass concentrations of the producer and the cleaner are denoted by B_P and B_C, respectively. Reaction rates specific to the producer and the cleaner are also identified by the subscripts _P and _C, respectively (e.g., vs for the glucose uptake rate). The consortium grows in a bioreactor operating in continuous mode, whereby the dilution rate D and the glucose concentration in the inflow G_in can be tuned (indicated in thin blue). For clarity, regulatory interactions due to growth inhibition by acetate and carbon catabolite repression have been omitted from the figure.

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The dynamics of the system are thus characterized by the following system of ODEs, where the dependency of the rate expressions on the concentrations has been dropped for clarity: (17) (18) (19) (20) (21) completed by the following mass conservation equation: (22)

The growth rates of the producer and cleaner strains are defined by the following equations (23) which leads to expressions for μ_P, and μ_C analogously to Eq 8. S3 Fig shows an example simulation of the consortium, in conditions leading to coexistence of producer and cleaner strains.

Dependence of coexistence on D and g_in.

For ease of illustration, in this section, we consider the system with Y_h = 0, that is, in absence of heterologous protein synthesis. All other parameters of the consortium are fixed as in Table 1. The case Y_h > 0 will be considered below.

We sought the (real, nonnegative, stable) steady states of the system for a grid of values of the bioreactor inflow parameters in the region (D, G_in)∈[0, 0.7] h⁻¹×[0, 20] g L⁻¹. Assuming that both biomasses are initially present, we first used a numerical integration approach (see details in Methods). For every value of the pair (D, G_in), unique steady-state biomass concentrations and were found for any initial positive biomass concentrations. Uniqueness of the steady state was reconfirmed by an algebraic approach (see Methods). Absence of multistability (in particular, bistability) differs from the results in [31] and follows from the monotonic dependence of uptake rates, and therefore growth rates, on substrate concentrations (see Discussion).

The results from the numerical computation of and are summarized in Fig 5A (the same results from the algebraic approach are shown in S4 Fig). As expected, stable coexistence (nonzero and ) is only observed in a finite range of values of D. This range depends very mildly on the specific value of G_in above a small threshold of about 2 g L⁻¹. For a given value of G_in above this threshold, four different regimes are encountered through increasing values of D, corresponding to stable existence of the sole producer, coexistence, cleaner washout, and producer and cleaner washout. To discuss what characterizes these regimes, we rely on Fig 5C and 5E, showing steady-state values of several rates and biomass concentrations as a function of D.

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Fig 5. Steady-state analysis of the consortium in chemostat.

Left: Y_h = 0; Right: Y_h = 0.2. (A)-(B) Nature of the unique (stable) steady state as a function of D and G_in (hatched red: stable coexistence; green: stable existence of the producer only; black: washout of both strains). (C)-(D) For G_in = 20 g L⁻¹, for the producer (green dots with solid lines) and the cleaner (orange dots with dotted lines) strains, steady-state value of glucose uptake rate as a function of D; (E)-(F) Idem, for biomass and substrate concentrations. Consortium parameters are as in Table 1. Acetate uptake and overflow rates are shown in S6 Fig.

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For small values of D, is smaller than the overflow threshold l. Notably, in analogy with Eq 13, and bearing in mind that Y_h = 0, one finds that (24) that is, glucose uptake grows linearly with D (see Fig 5C). Absence of acetate overflow impairs cleaner growth due to its inefficient growth on glucose, so that only the producer remains in the bioreactor at this rate.

At a value D ≃ 0.34 h⁻¹ where Eq 24 reaches threshold l, acetate overflow occurs. In this regime, a comparatively small population of cleaners grows on acetate as soon as the overflow rate is sufficient to sustain growth at the corresponding rate D. The resulting cleaner and producer populations balance out in a way that guarantees the precise amount of environmental detoxification to enable growth of the producer at the same rate. The gentle decrease of the producer population size results from utilization of part of the resources to sustain the growth of the cleaner population.

For D exceeding 0.55 h⁻¹, acetate uptake becomes insufficient to support cleaner growth. Together with the faster utilization of glucose by the producer strain, this results in the washout of the cleaner strain. In absence of acetate scavenging, producer growth at this high rate is only possible at low acetate concentrations, that is, if the population excreting acetate is small. This is witnessed by the sudden drop of .

For D above the maximal producer growth rate on glucose, the producer population is also washed out. In view of the dependence of growth rate on glucose concentration via the Monod law (Eq 10), this threshold depends on G_in. Finally, for small values of G_in mildly depending on D and typically below 2 g L⁻¹, the producer population does not excrete enough acetate to support the cleaner growth at the corresponding rate. Of course, in absence of glucose (G_in = 0), the entire consortium is washed out.

Dependence of coexistence on Y_h.

How is coexistence affected by the synthesis of a heterologous protein? To address this question, we repeat the analysis of the previous section for the case where Y_h = 0.2, which is the value of product yield reported in Table 1. Steady-state results for and are illustrated in Fig 5B (a unique stable steady state is again found for every value of D and G_in, see S4B Fig). Compared with Fig 5A, a similar shape of the various coexistence domains is observed. Steady-state rates and concentration profiles qualitatively similar to the case Y_h = 0 are also obtained, as shown in Fig 5D and 5F. However, quantitative differences in the values of D and G_in supporting coexistence are observed.

Coexistence occurs for D > 0.25 h⁻¹. This value is smaller than the corresponding value for the case where Y_h = 0. This is explained by the fact that, for larger Y_h, the necessary acetate overflow to support cleaner existence starts at lower growth (dilution) rates (see the discussion following Eq 13). For sufficiently large values of G_in, coexistence persists up to about D = 0.48 h⁻¹, whereas the whole consortium is washed out at about D ≃ 0.52 h⁻¹. Both of these dilution rates are again smaller than their counterparts for Y_h = 0. This is because the metabolic burden associated with Y_h steers away part of the glucose uptake from producer growth, thus decreasing the maximal producer growth rate (Eq 9) as well as the size of the producer population. The latter implies less acetate excretion and thus a reduction in the maximal cleaner growth rate as well. It is important to note that the interval of dilution rates supporting coexistence, though shifted toward lower values, is comparable to that observed for Y_h = 0. Interestingly, in presence of heterologous protein synthesis, smaller values of G_in (about 1 g L⁻¹) are required to observe coexistence at some suitable dilution rate.

We can thus summarize our results with the intriguing observation that, whereas a larger product yield shrinks the survival domain of the producer, it does not shrink the survival domain of the cleaner. In other words, heterologous protein production does not impair coexistence despite causing a metabolic burden on the producer. However, overly large values of Y_H drastically reduce the ability of the producer to grow, to the detriment of the whole community. In fact, it can be seen that when the burden is so heavy that the cleaner outperforms the producer in growth on glucose, regimes where only the cleaner survives are possible (S5 Fig).

The consortium attains highest productivity in a coexistence regime.

For the same conditions as in Figs 5B and 6A reports a heatmap of the productivity of the consortium as a function of the bioreactor parameters D and G_in. Productivity is defined as the steady-state rate of outflow of protein H through the chemostat effluent (DH*). Productivity is indeed maximal in a coexistence regime. Highest values are obtained near the maximal dilution rates before the cleaner washout, and for high input glucose concentrations (Fig 6A). Productivity drops suddenly for dilution rates near the boundary of cleaner washout (D ≃ 0.5 h⁻¹). This is the result of a sudden decrease of the producer population (Fig 5F). Productivity is of course zero for larger dilution rates where the producer is also washed out. In summary, the highest productivity is obtained in a regime where coexistence of producer and cleaner is possible. In the next section, we investigate the question whether the consortium is indeed advantageous over a producer strain working in isolation.

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Fig 6. Performance of the heterologous protein production process in chemostat (Y_h = 0.2).

(A) Heatmap of the productivity of the consortium (DH*) as a function of glucose inflow G_in and dilution rate D. The boundaries of the domains of coexistence and of existence of the sole producer are reported from Fig 5B in dashed red and solid green lines, respectively. (B) For G_in = 20 g L⁻¹, productivity as a function of D for the consortium (filled circles), and for a producer growing in isolation (empty circles). The color code for coexistence (dashed red) or existence of the sole producer (green) is the same as in Fig 5B. Vertical dashed lines indicate different productivity domains (see Discussion in main text). (C) Same as B for the process yield ((DH*)/(DG_in)).

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The consortium attains higher productivity than the producer species alone.

Does the consortium outperform a producer population growing alone? Of course, productivity of the former differs from that of the latter only where coexistence is possible, which is also where the consortium attains highest productivity. The question then becomes whether in this region a producer growing in isolation would outperform the consortium.

In Fig 6B, for the same value of glucose inflow concentration of Fig 5D and 5F (G_in = 20 g L⁻¹), we compare productivity of the consortium and of the producer species in isolation as a function of the dilution rate D. For the intermediate dilution rates where coexistence is possible, two subdomains can be distinguished. For D between 0.26 h⁻¹ and 0.36 h⁻¹, productivity grows with D for both the consortium and the sole producer. However, since part of the resources are devoted to maintain the cleaner species, the producer species alone outperforms the consortium. For values of D between 0.36 h⁻¹ and 0.48 h⁻¹, because of the larger overflow at higher growth rates, the toxic effect of acetate becomes dominant. In absence of the cleaner, this results in a sudden drop of productivity due to a reduced size of the producer population (see S7 Fig). For the consortium instead, the gentle reduction in producer biomass is overcompensated by the increase of D. The net result is an increase in productivity (Fig 6B), up to the dilution rates where the producer biomass washes out. Crucially, the maximum value of the consortium productivity (about 0.7 g L⁻¹ h⁻¹ at D ≃ 0.44 h⁻¹) is about 14% larger than that of the producer alone (about 0.6 g L⁻¹ h⁻¹ at D ≃ 0.36 h⁻¹). In summary, the consortium outperforms the producer species alone in terms of productivity. As seen before, maximal productivity is obtained for high values of G_in. It is worth noting that the maximum is obtained at a dilution rate strictly smaller than the maximal rate supporting coexistence. For applications, this warrants persistence of the cleaner in spite of fluctuations of the system and modelling inaccuracies. Results along the same lines are obtained for all other values of G_in supporting coexistence.

Highest productivity corresponds to a smaller process yield.

How efficient is the conversion of substrate into product by the consortium? How does this relate with productivity? To address this point we computed the yield of the process, defined as (25) that is, product outflow rate versus substrate inflow rate at steady-state (subscript cont stands for continuous bioreactor, to distinguish from later analysis in fed-batch). Fig 6C shows the yield as a function of D for the same value of G_in as in Fig 6B. Note that, since (see Methods), it holds that . Therefore, the yield of the consortium is proportional to the producer biomass (compare Fig 6C with Fig 5F). The same holds for the producer alone. The highest yield is found at the highest dilution rate before overflow (D ≃ 0.26 h⁻¹), thus in particular, in absence of cleaner. Intuitively, this is because at that point, the largest producer biomass is obtained before resources are partly redirected into acetate overflow. For the consortium, this overflow sustains the cleaner growth. At rates above D ≃ 0.26 h⁻¹, for the producer alone, biomass loss is initially less pronounced than in the consortium, since no other species consumes glucose. However, at higher dilution rates, this results in strong acetate accumulation, whence a sharp loss of biomass and yield. Compared with Fig 6B one notices that, for the consortium, maximizing productivity is paid for in terms of a decreased yield. This trade-off takes place because maximal productivity is achieved in presence of the cleaner, that is, in presence of acetate overflow, which subtracts part of the available glucose from the protein production process. Utilisation of glucose is made even less efficient by the glucose uptake of the cleaner, which subtracts further resources from protein production. From a qualitative viewpoint, however, the same trade-off is encountered for a hypothetical cleaner strain without residual glucose uptake (see S8 Fig).

Numerical approach.

In this approach, stable steady states of the system are sought by numerical integration of Eq 27. Recall that, for an assigned initial condition x(0) = x₀, a (strictly) stable equilibrium of the dynamical system is given by the asymptotic value of the corresponding solution , that is, by , as long as this exists, it is unique and finite. In practice we obtain this by integrating Eq 27 from x₀ = 0 by the Matlab ODE solver ode15s over a sufficiently long time horizon (T = 10⁷ h), see S1 Software for more details. This procedure is computationally efficient and allows for fast exploration of the system over large sets of values of θ. However, for any given θ, if the system has several stable steady states, then the steady state eventually reached will depend on x₀. This has motivated the parallel use of a second, algebraic approach.

Algebraic approach.

For any given θ, the possible stable steady states of Eq 27 may as well be sought as the real, nonnegative solutions in x of the equation (28) such that the Jacobian matrix F(x|θ) = ∂f(x|θ)/∂x is strictly stable (that is, all its eigenvalues have negative real part). In the interest of clarity and simplicity, in what follows we drop θ from the notation, and instead write rates as functions of x. It can be verified by inspection of Eqs 10, 11, 14–16 that f(x) is a piecewise rational function. Indeed, the piecewise linear structure of Eqs 12 and 15 induces a finite partitioning of the state space X into the four subsets

Over each X_i, f(x) = N_i(x)/D_i(x), where N_i(x) and D_i(x) are two multinomial functions determined by the relevant form of Eqs 12 and 15 (for instance, the rate conditions that define X₂ imply that Eq 12 is equal to 0 and Eq 15 is equal to in that domain). For every i, the solutions of Eq 28 belonging to X_i correspond to the roots in X_i of N_i(x). Out of these solutions, for i = 1, …, 4, the steady states sought are the real nonnegative solutions such that F(x) is strictly stable.

Multinomial root finding is a well explored problem and specialized tools for enumerating all solutions exist [79]. In addition, both F(x) and the N_i(x) can be computed explicitly or symbolically at little cost. This suggests the following algorithm:

1. Compute N_i(x), with i = 1, …, 4, and F(x);
   For i = 1, …, 4:
2. Find the real nonnegative solutions of N_i(x) = 0;
3. Record solution x as a system stable steady state if and only if x ∈ X_i and F(x) is strictly stable.

In our implementation, which is optimized by suitable simplifications for the exploration of steady states with B_P = 0 or B_C = 0, symbolic calculations (step 1) are performed in Wolfram Mathematica, while multinomial root finding (step 2) is performed by the Matlab (numerical) routine psolve available at NAClab [80, 81]. In practice, step 3 simply amounts to checking that the evaluation of and at the candidate solution x agrees with the rate conditions defining X_i, thus avoiding the explicit calculation of the latter. In view of the complexity of the multinomials N_i(x), which comprise terms of order up to 33, this method is computationally intensive, yet it returns all possible steady states within the precision of the routine psolve.