Long-term homeostasis in microbial consortia via auxotrophic cross-feeding

Abstract

Synthetic microbial consortia are collections of multiple strains or species of engineered organisms living in a shared ecosystem. Because they can separate metabolic tasks among different strains, synthetic microbial consortia have applications in developing biomaterials, biomanufacturing, and biotherapeutics. However, synthetic consortia often require burdensome control mechanisms to ensure that consortia members remain at the correct proportions. Here, we present a simple method for controlling consortia proportions using cross-feeding in continuous auxotrophic co-culture. We use mutually auxotrophic E. coli with different essential gene deletions and regulate the growth rates of members of the consortium via cross-feeding of the missing nutrients in each strain. We demonstrate precise regulation of the proportions by exogenous addition of the missing nutrients. We also model the co-culture’s behavior using a system of ordinary differential equations that enable us to predict its response to changes in nutrient concentrations. Our work provides a powerful tool for consortia proportion control with minimal metabolic costs to the constituent strains.

Synthetic microbial consortia are collections of strains which can segregate metabolic tasks for efficient use in biomaterials, biomanufacturing, and biotherapeutics. Here, the authors present a method to maintain and tune the ratio of two co-cultured bacterial strains via growth medium manipulation.

Introduction

As the complexity of synthetic biological constructs continues to increase, researchers are moving away from single-strain systems¹. Multi-strain microbial consortia enable division of labor, which reduces the metabolic load on individual organisms and allows for the entire consortium to behave more efficiently^2–4. This benefit has led to advances in microbiome engineering^5–8, bioremediation^9–14, and bioproduction^2,3,15–18, making consortia engineering one of the fastest growing fields in synthetic biology. Most natural microbial communities are made of many species working together. These communities exhibit complex collective behaviors such as patterning^19–21, increased robustness^22–25, and co-evolution through horizontal gene transfer^26–28. While there are benefits to engineering natural communities, the complexity of interactions within these systems often makes this infeasible. Synthetic consortia offer the advantage of being simple while retaining the benefits of the division of labor found in natural communities. This makes synthetic consortia ideal both for increasing the efficiency of known biological solutions and for studying behaviors only seen in complex communities. The chief challenge, then, becomes creating those same emergent behaviors through rational design.

One of the largest barriers to synthetic consortia design is the problem of competitive exclusion. Without pressure to coexist, simple differences in growth rates lead to the loss of less fit consortium members²⁹. Even worse, many desirable behaviors (e.g., bioproduction^4,16,17) rely not just on coexistence but on specific population ratios. Maintaining such specific ratios requires precise control of each strain’s growth rate. Further complicating the matter, any mutation that increases fitness will break this control, making long-term ratio control difficult. An ideal ratio control strategy would be robust over time and suitable for a wide variety of applications.

Many existing population control strategies maintain member coexistence but lack in other design pillars like robustness and modularity. Broadly, these strategies define pairwise interactions to control the growth rate of all members^1,30. Toxin-antitoxin systems and antimicrobials have proven promising in a wide variety of contexts, but are sensitive to mutation and rely on the production of molecules that limit growth^31,32. Symbiotic relationships are simple, but are thought to be limited by the necessity of strict inter-dependence^33–35. Quorum sensing regulation has been popular for enabling tunability in recent years, but the limited number of existing quorum sensors in turn limits the number of applications that this method can be combined with^36–39. Mutualistic auxotrophy was one of the first population control methods used in synthetic microbial consortia³³. While it has been known for some time that this commensalism is capable of maintaining multiple members of a population in a variety of contexts^34,35,40, little has been done to study the stability or tunability of this control strategy due to the slow growth rates of auxotrophs.

Here, we present a simple and tunable mechanism for controlling consortia ratios using mutualistic E. coli auxotrophs. The ratio is stabilized by the excess production of the cross-fed metabolites and is tunable via the exogenous addition of each metabolite. We show that auxotrophs in the consortium converge to a stable population ratio in continuous co-culture as a result of balanced growth rates, and that this ratio is insensitive to inoculation conditions. We further show that we can tune this ratio by supplementing the deficient amino acids and increasing the growth rate of a constituent strain. We develop and validate a mathematical model that explains the mechanisms of co-culture dynamics and ratio control. When fit to data this model predicts co-culture ratios under previously untested conditions. Our results represent a powerful tool for controlling microbial consortia composition.

Results

System description

To test whether cross-feeding auxotrophs can regulate their relative population abundance in co-culture (Fig. 1a), we co-cultured two strains taken from the Keio collection⁴¹, ΔargC and ΔmetA. These strains have a kanamycin resistance marker in the place of the argC or metA genes, which are required for arginine or methionine production, respectively. They have been previously shown to grow mutualistically in minimal media via cross-feeding³³. To do so, the ΔargC strain produces excess methionine, which allows the ΔmetA strain to grow and, in turn, produce excess arginine, allowing the ΔargC strain to grow.

Fig. 1. A simple, tunable system for ratiometric control using mutualistic auxotrophs.

a Schematic of control system and corresponding model state variables. Two strains of E. coli containing chromosomal deletions in either the metA (C_m) or argC (C_a) genes are grown together in minimal media, where growth is only possible by the mutualistic cross-feeding of methionine (x_m) and arginine (x_a). b Schematic of continuous culture turbidostat setup. (i) OD readings of the cell culture are taken periodically. (ii) OD data are used to compute necessary dilution volumes to maintain a set OD at each time step. (iii) A mechanized syringe and pinch valve system pulls the calculated dilution volume of media and dispenses it into the culture tube. (iv) Every 6 h, dilution effluent is plated onto selective and nonselective media plates to estimate the population ratio. c, d Bulk culture growth rates of the ΔargC and ΔmetA strains as arginine or methionine is added to minimal M9 media, respectively. Each strain’s independent growth rate is tunable via the addition of its corresponding missing metabolite. Source data are provided as a Source Data file.

In any microbial consortium, the abundance of each constituent strain is dependent on its growth rate. Growth regulation is thus essential to ratio control, but difficult in practice due to its sensitivity to mutation. Most ratio control mechanisms rely on limiting each member strain’s growth, making any random increase in fitness fatal to the precise control of the entire system. We noted that the mutualistic relationship between the ΔargC and ΔmetA strains could overcome this sensitivity because of its robustness to potential mutations. Since each strain contained a whole-gene chromosomal deletion, it is unlikely that either strain would regain its metabolic function. However, ratio control also requires the ability to tune the growth rate of each strain to modulate their abundances. Beyond the cross-feeding capability, both strains exhibit robust growth in co-cultures, indicating that their growth rates in this environment are not substantially different from each other³³. Such similar growth rates indicate that, if their growth rates could be tuned, it should be possible to create a wide range of relative abundances. We hypothesized that the growth rate of each strain in this mutualistic system was determined by the availability of either arginine for the ΔargC strain or methionine for the ΔmetA strain.

To test if we could control the strain growth rates to tune their relative abundances using these nutrients, we grew each strain in minimal media monoculture at a range of either arginine or methionine concentrations varying from 10 nM to 10 mM (Fig. 1c, d). After overnight growth, each strain was inoculated into supplemented M9 media at equivalent cell densities as measured by OD₆₀₀. We monitored the cell density of each culture via plate reader and used the changes in OD over time to estimate the growth rate under each growth condition. In both cases, the strain’s growth rate was restored to doubling times of under one hour at the highest concentrations, demonstrating the dominant effect of the missing metabolite on each auxotroph’s growth rate. Additionally, each strain exhibited a wide range of growth rates, indicating the possibility for a wide range of co-cultured ratios.

System reaches steady state

With confirmation that the growth rates of these two strains could be tuned by the presence of either arginine or methionine, respectively, we next tested their ability to regulate their relative abundance in co-culture. To do so, we co-cultured the ΔargC and ΔmetA strains in a continuous culture turbidostat designed to maintain a constant OD₆₀₀ over the course of the experiment (Fig. 1b)⁴². By periodically recording the culture OD and comparing it to an established setpoint, the turbidostat could add reserved minimal M9 media to maintain the setpoint OD as necessary. Culture media in excess of 15 mL was allowed to flow into a waste container as effluent. We periodically collected and plated the individual turbidostat dilutions on plates containing either rich media or minimal media supplemented with methionine to estimate the relative abundance of the ΔmetA strain over the course of several days. We also kept a running log of both OD measurements and dilution volumes to estimate the co-culture growth rate.

To test whether the consortium converged to a stable ratio, we inoculated the strains at a range of initial ratios and monitored their relative abundances in continuous culture. The consortium came to around a ratio of 3:1 (ΔmetA:ΔargC) within 24 h (Fig. 2a–c). This ratio persisted over several days after stabilizing, and the consortium gave no indication of losing fitness (Supplementary Fig. 1a–c, Supplementary Fig. 2). We found that 99:1 and 1:99 OD ratio inoculations did not alter either the final ratio or the time to steady state (Fig. 2b, c). We also found that increasing the cell density by a factor of 3 had no effect on the steady state ratio (Supplementary Fig. 3a). This indicates that this ratio control mechanism is robust and does not require precise estimation of relative strain abundances prior to co-culturing.

Fig. 2. Mutualistic auxotrophy creates a stable population ratio in minimal media.

a ΔmetA population proportion in continuous culture of minimal M9 media. Cultures inoculated at  ~ 50:50 OD initial ratio. b, c ΔmetA population proportion in continuous culture of minimal M9 media. Cultures were inoculated at either 1:99 or 99:1 OD to test robustness of steady state stability. d–f Tuning the ΔmetA population proportion in continuous culture with the addition of arginine (arg) and methionine (met). We added both methionine and arginine to the turbidostat setup, allowing us to tune the strain ratio above and below our initial unsupplemented experiments. Proportions calculated as ratio of triplicate colony forming units counted on selective LB plates (total population) and M9 100 μM methionine plates (ΔmetA population) (N=3 technical replicates for each of 3 biological replicates). Icons represent ratio of mean colony counts, shaded areas represent the range of all possible proportions calculated from the combination of two sets of triplicate plate counts. See “Methods” for further details. Ratios of  > 1 are possible due to the possibility of growing more colonies on the M9 plate than on the LB plate. Source data are provided as a Source Data file.

System tunability

We next attempted to tune the steady state ratio of the system by supplementing the minimal media with arginine and methionine (Fig. 2d–f). Cells were grown under identical conditions to the prior experiments, except for added arginine or methionine at indicated amounts. These concentrations were much lower than those used in bulk culture experiments due to the continuous supply of metabolite, resulting in much higher total metabolite amount over the course of tuning experiments (Supplementary Fig. 4). While neither strain needed the other for its survival, as long as each experienced a sufficient growth benefit from the presence of the other, the co-culture could still reach a steady state that maintained both strains. This method allowed us to produce a wide range of stable population ratios. These included concentrations which allowed either strain to make up the majority (~90% of the total population) (Fig. 2d, f). Finer grained changes were also possible by adjusting the amount of added arginine and methionine. For instance, we were able to lower the ratio from the baseline  ~75% ΔmetA to  ~ 50% ΔmetA (Fig. 2e). The system was similarly tunable at higher cell densities, allowing ΔargC to make up  ~90% of the total population (Supplementary Fig. 3b). In both supplemented and non supplemented cases, we found that the system was quite precise, with variations in co-culture ratio that were indistinguishable from measurement noise. None of the supplemented metabolite concentrations resulted in complete strain loss, and all caused a net increase in combined co-culture growth rate (Supplementary Fig. 1d-f). We also found that the time between inoculation and steady state increased.This timescale depends on the difference between the two strain growth rates: one strain growing much faster than the other would cause the ratio to change more quickly than if one strain was growing only slightly faster than the other. Because supplementation increases the growth rates of both strains, the cells are closer to their maximum growth rate in the supplementation experiments. The two strains have similar maximum growth rates, so the supplemented nutrients will shift the system into a growth regime where there is less difference between the growth rates of the two strains, and thus the population ratio changes more slowly.

Model captures experimental steady state

The cell population ratio in this system depends on factors such as the growth rates of ΔmetA and ΔargC and rates at which they produce and consume methionine and arginine. To confirm our hypothesis about the mechanisms of ratio control and to predict the population ratio as a function of nutrient supplementation, we developed a mathematical model of the concentrations of ΔmetA (C_m), ΔargC (C_a), arginine (x_a) and methionine (x_m) in the turbidostat, and fit the resulting model to population ratio data shown in Fig. 2.

To construct our model, we assumed that ΔmetA cells release excess arginine into the environment at a constant rate β_a and consume methionine at a rate that is proportional to the cell growth rate, with proportionality constant η_m. We similarly assumed that ΔargC cells release methionine at rate β_m and consume arginine with proportionality constant η_a. As shown in Fig. 1c, d, the cell growth rates are increasing functions of metabolite. We therefore assumed that, after an initial adjustment period in the turbidostat, the dependence of cell growth rates can be modeled by Hill functions

$${\gamma }_a\left(x_a\right)=\frac{{\gamma }_{a,\max }{x}_a^{b_a}}{K_a^{b_a}+x_a^{b_a}}$$ 1

for ΔargC and

$${\gamma }_m\left(x_m\right)=\frac{{\gamma }_{m,\max }{x}_m^{b_m}}{K_m^{b_m}+x_m^{b_m}}$$ 2

for ΔmetA. Because the cells are grown in rich media prior to being added to the turbidostat, we hypothesize that the cells are in a different growth regime at the onset of the experiment. To account for this initial period, we assumed that the growth rate functions have the form

$$G_a\left(x_a,t\right)=g{e}^{-at}+{\gamma }_a\left(x_a\right)\left(1-e^{-at}\right)$$ 3

for ΔargC growth and

$$G_m\left(x_m,t\right)=g{e}^{-at}+{\gamma }_m\left(x_m\right)\left(1-e^{-at}\right)$$ 4

for ΔmetA growth in order to reflect the nutrient levels stored by the initial cell population decaying with first order kinetics. The resulting model is nonautonomous and reflects the assumption that both cell types grow at a constant rate g at the beginning of the experiment, due to leftover nutrients from their preparatory growth. After adjusting to the turbidostat environment over approximately 1/a hours, the growth rates are modeled using Hill functions, γ_a, and γ_m.

To model dilution in the turbidostat, we also assumed a dilution rate function δ(C_a, C_m, x_a, x_m) defined so that the total cell concentration is constant,

$$C_a\left(t\right)+C_m\left(t\right)=C.$$ 5

Because methionine and arginine were supplemented in the dilution liquid, we included two source terms that represent supplementation concentrations α of arginine and μ of methionine added to the culture at a rate equal to the system dilution rate. We applied the conservation relation that is consistent with this dilution rate and scaled the concentration C_m by C to obtain an equation for the evolution of the fraction of ΔmetA in the population,

$$c_m=\frac{C_m}{C_a+C_m}.$$ 6

In this scaling, the quantities Cc_m and C(1 − c_m) represent the concentrations of ΔmetA and ΔargC, respectively. The resulting three-dimensional system of equations for c_m and the two metabolite concentrations has the form

$$\frac{d{c}_m}{dt}=G_m\left(x_m,t\right)c_m-\delta \left(c_m,x_a,x_m\right)c_m$$ 7

$$\frac{d{x}_a}{dt}={\beta }_{a}C{c}_m-{\eta }_a{G}_a\left(x_a,t\right)C\left(1-c_m\right)-\delta \left(c_m,x_a,x_m\right)x_a+\delta \left(c_m,x_a,x_m\right)\alpha$$ 8

$$\frac{d{x}_m}{dt}={\beta }_{m}C\left(1-c_m\right)-{\eta }_m{G}_m\left(x_m,t\right)C{c}_m-\delta \left(c_m,x_a,x_m\right)x_m+\delta \left(c_m,x_a,x_m\right)\mu .$$ 9

Further details about the model are provided in the “Methods” section and in the “Supplementary Methods” section. We broadly describe the model in Supplementary Method 1, and a derivation of the nondimensionalized model can be found in Supplementary Method 2.

It is not immediately obvious, in the mathematical model or experimental system, that the cell populations will always reach one stable population ratio starting from any inoculation ratio. In Supplementary Method 3, we include a steady state analysis showing that the model has at most one steady state that represents coexistence of the two strains, and that this steady state is stable if it exists. These modeling results reflect the experimental finding that the system reaches the same stable population ratio from many different inoculation ratios.

To ensure that the model captures the system response to supplemented nutrients, we fixed the supplementation parameters α and μ to the values used in the experiments shown in Fig. 2, and obtained the remaining parameters by fitting the model to the 6 different population ratio datasets through a two-step Bayesian approach using the Metropolis-Hastings algorithm (Supplementary Method 4). The priors we used for Bayesian inference reflected our understanding of general parameter ranges based on the literature and our experimental results for cells in bulk culture. We assumed that the metabolite consumption rates must be smaller than the production rates, since metabolites must remain in the system even as dilutions occur in order for the cell population to survive. We further assumed that the production rate of methionine is lower than the production rate of arginine, because our bulk culture experiments show that the ΔmetA cells require lower metabolite concentrations to grow than ΔargC cells. Also based on these bulk culture growth rate experiments, we assumed that the half maximal concentration in the ΔargC growth rate expression (K_a) is much larger than the half maximal concentration in the ΔmetA growth rate expression (K_m). We use gamma prior distributions for the parameters mentioned above, with means that reflect these approximate relationships, and standard deviations that reflect the level of uncertainty we had in these relationships. For the remaining parameters in the growth rate functions, we use uniform prior distributions with lower and upper bounds that represent the biologically reasonable ranges for each parameter: We assumed that the maximal growth rate of one cell type is no more than twice that of the other, and that the Hill coefficients are between 0.5 and 10. For the nonautonomous portion of the growth rate functions, we assumed that the initial cell growth rate is no more than twice as high as the maximal steady state growth rates, and that the cells in the initial growth regime decay with a half life of between two and thirty hours after the onset of the experiment. These prior assumptions are listed in Supplementary Tables 1 and 2.

The posterior parameter distributions are shown in Supplementary Fig. 5 and Supplementary Fig. 6, and the mean of the posterior is shown in Supplementary Table 3. In Fig. 3, we show the six datasets along with the population proportion, c_m, obtained by solving the model equations numerically using parameter values sampled from the posterior distribution. The model correctly captures the steady state population ratios for all supplementation experiments. In Supplementary Fig. 7, we show the c_m trajectory obtained using posterior means, with the shaded region corresponding to one standard deviation of the inferred observational noise.

Fig. 3. Mathematical modeling of turbidostat data.

We used a Bayesian approach to fit our ODE model to the data shown in Fig. 2. a–f Model solutions fit experimental data well. Solutions shown were generated using the mean parameter values (black curves) or samples (gray curves) from the posterior parameter distribution (N=3 technical replicates for each of 3 biological replicates). Means of experimental proportions were calculated as in Fig. 2 by summing the plate counts from each plate type and taking the summed ratio. Source data are provided as a Source Data file.

To confirm that the assumption of an initial transient in growth rates is needed, we fit an autonomous version of the model to data. To do so, we assumed that there is no transient period, and that growth rates are determined solely by the Hill functions, γ_i(x_i) for i ∈ {a, m}. Parameters and solutions for this autonomous model can be found in Supplementary Fig. 8 and Supplementary Fig. 9. When fit to data, this model produced solutions with strong damped oscillations that were absent in experimental data. Thus, the data suggest that including a transient growth state is necessary.

Parameter estimates suggest that the production rate of arginine is approximately an order of magnitude greater than the production rate of methionine, which is supported by previous studies^43–46. In both bulk culture growth rate experiments and supplementation experiments, more arginine is needed to increase the growth rate of ΔargC cells than methionine is needed to increase the growth rate of ΔmetA. To increase the level of arginine in the model, we must either have a consistently high proportion of ΔmetA cells, or a high arginine production rate. As the proportion of ΔmetA cells is not always high, the model suggests that the arginine production rate must be larger than the methionine production rate. The growth rate Hill function parameters also indicate the relatively high concentration of arginine needed to increase ΔargC growth. The metabolite concentration required for half maximal growth of ΔargC cells (K_a) must be about 130 times the metabolite concentration required for half maximal growth of ΔmetA cells (K_m) for the model to fit the data. This is a marked difference from the cell growth rates in bulk culture, where the half maxima of the growth rate curves are within an order of magnitude of one another.

The other inferred growth rate parameters also shed light on the cell population behavior. The inferred maximum cell growth rates are similar (${\gamma }_{m,\max }=1.014{\gamma }_{a,\max }$), implying similar growth rates for the strains when nutrients are present in abundance. Moreover, the posterior distribution for the parameter ratio $\frac{{\gamma }_{m,\max }}{{\gamma }_{a,\max }}$ has a very small variance, confirming our hypothesis that the cell ratio c_m depends strongly on the balance of growth rates of the two cell types. Additionally, the inferred value of a, the parameter defining the timescale at which the initial growth regime shifts to the Hill function growth regime, was found to be 0.05/hour. This implies that the influence of the initial growth regime is cut in half after approximately 13.7 h, which would allow for approximately 14 cell generations influenced by this initial growth regime, assuming that cells are dividing at a maximum growth rate of about 1 division/hour.

Predictions and model validation

We next asked how well our model can predict experimentally observed strain ratios. To do so we first obtained steady state ratio prediction over a range of supplementation concentrations, μ and α. Figure 4a, b show that the model predicts the population ratio can be adjusted by varying arginine supplementation between 0 μM and approximately 1.5 μM, and methionine supplementation between 0 nM and approximately 15 nM.

Fig. 4. Model prediction and validation.

a, b Heatmaps of predicted steady state proportions of ΔmetA at different arginine and methionine concentrations. Ratios were obtained by finding steady state solutions of Eqs. (7–9). Gray dots indicate training data, with unsupplemented training data depicted outside the bounds of the heatmap. Black and white dots indicate predictions selected for experimental validation, with labels indicating the relevant panel in order from left to right. Lines enclose regions in parameter space where one strain is predicted to be extinct. c, d Predictions of the model compared to experimental data (N=3 technical replicates for each of 3 biological replicates). Numerical solutions were generated using the mean parameter values (black curves) or random samples (gray curves) from the posterior parameter distributions. Means of experimental proportions were calculated as in Fig. 2 by summing the plate counts from each plate type and taking the summed ratio. Both supplementation values moderately increased the overall co-culture growth rate (Supplementary Fig. 10). Source data are provided as a Source Data file.

We tested these predictions in two supplementation experiments. For the first experiment (Fig. 4c, Supplementary Figs. 10a and 11a), we chose 10 μM arginine and 12.5 nM methionine, where the model predicts a steady state ratio of approximately 0.9. Achieving this ratio would experimentally confirm that the system can maintain a high proportion of ΔmetA without eliminating ΔargC. For the second experiment (Fig. 4d, Supplementary Figs. 10b and 11b), we chose 1.5 μM arginine and 3 nM methionine, because the model predicts that ΔargC cells will take over the culture - a result we did not see in our initial experiments (Fig. 2). In both cases, experimentally observed ratios at steady state agreed well with model prediction.

Discussion

In this work, we demonstrated that mutualistic cross-feeding in E. coli is a powerful tool for controlling the population abundances in a two-strain co-culture. In this system, each strain’s growth rate is regulated by the abundance of its deficient amino acid; while supplementation of these amino acids allows both strains to grow, it is the cross-feeding of these amino acids in both supplemented and unsupplemented regimes that allows for stable ratios. In unsupplemented regimes, each strain’s only source of these amino acids is the other strain. In supplemented regimes, each strain benefits from the additional production by the other strain to a degree that their growth rates are still tightly regulated by the abundance of each strain. As a result, both cross-feeding and exogenous supplementation are crucial to the functionality of this system. While we used a pair of methionine and arginine auxotrophs, we believe that this tight control mechanism can be extended to other auxotrophic combinations that have been shown to grow mutualistically in minimal media^33,34. Indeed, previous results have demonstrated multi-population maintenance in larger synthetic consortia containing mutualistic auxotrophs^35,40, suggesting that such tight control may extend to consortia with more than two strains. This tool also has a relatively minimal metabolic cost; while unsupplemented regimes severely limit the growth of even a cross-feeding co-culture, this limit is mitigated by the addition of arginine and methionine (albeit at an increased economic cost). While it is unlikely that this control mechanism would still function under supplement regimes that restore growth rates to non-auxtrophic levels, further work should explore the limits of this growth enhancement.

Mutually auxotrophic ratio control depends on a lack of competition for other nutrient resources. In regimes with limited carbon sources, for example, the concentration of arginine and methionine would no longer have a dominant effect on the growth rate of each strain. While both strains could remain in regimes without any supplemented amino acids, tuning via supplementation would be much more difficult, as the relative range of non-competitive growth rates for each strain would be narrower.

Our mathematical model lends further support for the proposed mechanism of ratio control in this system. The model relies on simplifying assumptions about cell metabolism, mutualism, and dilution to a finite population size: We assumed that cells are initially in a constant growth rate regime determined by the conditions they experienced before transfer to the turbidostat. Upon transfer, cells shift to a regime where their growth rates depend on the nutrient they need to survive. The data support this assumption. However, a more detailed understanding of the metabolic mechanisms that determine growth in these two regimes requires further study. Additionally, our model relies on the assumption that the turbidostat continuously dilutes the culture to keep the cell concentration constant. In reality, dilutions occur every 5 minutes. These increments are likely small enough compared to the timescale of changes in ratio, so that our use of continuous dilution in our model is justified. Modeling populations that experience dilutions less frequently would require a different dilution term. Even with these simplifying assumptions, fitting our model to data gave us insight into how cells grow and exchange nutrients in the turbidostat. The model we use was developed with a level of complexity that balances mechanistic accuracy and predictive value with parameter identifiability and analytical tractability. Regardless of the metabolic complexity that influences the timescale of experimental timecourse data, the mathematical model successfully predicts the steady state population ratio and thus generates hypotheses about the system behavior in areas of parameter space that are difficult to test experimentally.

This work presents a tool for controlling and understanding population abundances in microbial consortia. While work remains to be done to demonstrate its compatibility with other engineering practices, such as classical metabolic engineering or other forms of consortia control, the ability to supplement these cultures with growth-enhancing metabolites without breaking ratio control reduces the toxicity risks seen when using other mechanisms. Overall, its simplicity and predictability make it a promising candidate for future advances in synthetic microbial consortia.

Methods

E. coli strains

All experiments were conducted in the ΔargC and ΔmetA strains taken from the Keio collection⁴¹ (parent strain E. coli K-12 BW1125), which both contain a kanamycin resistance gene in place of their deletion.

Growth media

In all experiments, both ΔargC and ΔmetA monocultures were first grown up overnight in LB medium containing kanamycin (kan; 50 mg/L) at 37C. Co-culture experiments were conducted in minimal M9 media, comprised of 1X M9 salts, 0.1 mM CaCl₂, 1 mM MgCl₂, 0.4% glucose, and 50 mg/L kanamycin⁴⁷. Arginine and methionine were added in relevant assays at indicated concentrations during media prep.

Growth rate assay

Overnight auxotroph cultures were grown for 1 h in rich media and serially diluted in M9 media to equivalent ODs, calculated at 0.005. These were then loaded into 96-well plates and grown in a Tecan Spark plate reader with incubation (37C) and shaking (216 rpm) with measurements taken every 10 minutes for 24 h. Growth rates were then estimated by fitting the background subtracted OD data to a logistic growth curve in MATLAB using the lsqcurvefit function. See the Source Data file for plate reader data.

Turbidostat

The turbidostat used in this work was adapted from a version created by the Klavins lab^42,48. Briefly, our turbidostat follows the design which can be followed at https://depts.washington.edu/soslab/turbidostat/pmwiki/, with the main exception being that we used a Raspberry Pi to control the behavior of the turbidostat, rather than a full computer, which further reduced the cost and enabled the building of multiple turbidostats in the same lab space.

Turbidostat assay

Overnight auxotroph cultures were grown for 1 h in rich media and then mixed at equi-OD ratios, unless otherwise indicated. In order to give auxotrophic cells time to begin cross-feeding while undergoing continuous dilutions, cells were not pre-grown in minimal media, thus leaving some leftover rich media metabolites for growth in the turbidostat. These were then loaded into the culture tube via syringe, which allowed for penetrating the rubber stopper without contaminating the culture tube. Cells were grown at 37C in pre-growth and in the turbidostat culture tube for all turbidostat experiments. Samples were taken by collecting effluent into a micro-centrifuge tube every 6 hours. At the same time, the culture tube was visually checked for evidence of biofilm formation. No biofilms were found during this work. Samples were serially diluted into M9 media in 1:10 dilutions 4 times, with at least 10 pipette mixing steps in between each dilution to minimize dilution variance. Samples were then added to both a rich media kanamycin plate and a minimal M9 media kanamycin plate supplemented with 100 μM methionine. This plating was done by adding 10 μL of diluted sample as a spot. The plate was then tilted to allow the spot to streak down the plate due to gravity, to avoid cell loss that would potentially result from a tactile spreading mechanism. Plates were grown at 37C either overnight (rich media) or for 48 h (minimal media). Colonies were then counted and recorded by hand. Each sample was plated in triplicate. We reported the population proportions using the ratio of the mean of each plate type. Because technical replicates between plate types had no inherent association, proportion errors were reported by first generating all 9 possible proportions by combining each set of triplicates for each sample. We then reported the range of these possible proportions, with the shaded regions in Fig. 2 bordered by the maximum and minimum proportion value calculated using this method. Plate counts and turbidostat recordings can be found in the Source Data file.

Mathematical model

The full model consists of a system of four ordinary differential equations that describe how the concentrations (in μM) of the following species change over time (in hours): ΔargC (C_a), ΔmetA (C_m), arginine (x_a) and methionine (x_m). The resulting ordinary differential equations have the form:

$$\frac{d{C}_a}{dt}=G_a\left(x_a,t\right)C_a-\delta \left(C_a,C_m,x_a,x_m\right)C_a$$ 10

$$\frac{d{C}_m}{dt}=G_m\left(x_m,t\right)C_m-\delta \left(C_a,C_m,x_a,x_m\right)C_m$$ 11

$$\frac{d{x}_a}{dt}={\beta }_a{C}_m-{\eta }_a{G}_a\left(x_a,t\right)C_a-\delta \left(C_a,C_m,x_a,x_m\right)x_a+\delta \left(C_a,C_m,x_a,x_m\right)\alpha$$ 12

$$\frac{d{x}_m}{dt}={\beta }_m{C}_a-{\eta }_m{G}_m\left(x_m,t\right)C_m-\delta \left(C_a,C_m,x_a,x_m\right)x_m+\delta \left(C_a,C_m,x_a,x_m\right)\mu .$$ 13

The dilution term, δ(C_a, C_m, x_a, x_m), is defined so that the total cell concentration remains constant, C_a + C_m = C, or $\frac{d{C}_a}{dt}+\frac{d{C}_m}{dt}=0$. Plugging in the definitions of $\frac{d{C}_a}{dt}$ and $\frac{d{C}_m}{dt}$ and setting their sum to 0 gives,

$$\delta \left(C_a,C_m,x_a,x_m\right)=\frac{G_a\left(x_a,t\right)C_a+G_m\left(x_m,t\right)C_m}{C_a+C_m}.$$ 14

As there are billions of cells in the turbidostat, we assume that stochastic fluctuations can be neglected. The turbidostat dilutes the culture every 5 minutes, but we assume that these intervals are sufficiently small compared to the timescale of changes of the different concentrations so that dilution can be modeled as occurring continuously in time. The conservation relation, C_a + C_m = C, also allows us to reduce this system to 3 equations, and scale the C_m equation by C to obtain (7)–(9).

Parameter fitting

We used the DEMetropolisZ sampler⁴⁹ in the PyMC v5.5.0 Python library⁵⁰ to fit our model to data. We assumed that the cells take about 15 h to shift to the Hill Function growth regime in the turbidostat (based on observations from the turbidostat dilution rates), so we fit the model parameters in two steps: First, we fit the autonomous model to data collected at time points greater than 15 h in datasets shown in Fig. 2a, d–f. Second, we fit the two remaining parameters of the nonautonomous model to data collected at time points between 0 and 15 h in the six datasets in Fig. 2, while setting the remaining parameters equal to the mean of the posterior in the first step. Code is accessible at (https://github.com/amandaalexander/AuxotrophicCrossFeeding).

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Author contributions

N.E.G. designed experiments and analyzed data. N.E.G. and X.P. performed experiments. A.M.A. and K.J. developed the model. C.P. assisted with experiments. A.J.H., R.N.A., and X.P. designed and built the turbidostat. M.R.B. oversaw the project. All authors contributed to discussion and development of the project and helped write the manuscript.

Peer review

Peer review information

Nature Communications thanks Pauli S. Losoi, and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Data availability

Supplementary Figs. 5, 6, and 8 contain samples from the posterior distribution of parameter values calculated using the Metropolis Hastings algorithm. While the nature of Markov chain Monte Carlo methods means that each run generates different samples from the posterior distribution, the plotted data are representative of the consistent results generated by this code. See Supplementary Note 1 for more details. Source data are provided with this paper.

Code availability

All computer code generate for this project is available through Github (https://github.com/amandaalexander/AuxotrophicCrossFeeding)⁵¹.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-025-63575-z.