Quantifying the effects of the division of labor in metabolic pathways

Abstract

1. Introduction

From the earliest observations of microbial organisms, it has been apparent that microbial consortia are ubiquitous in nature. In fact, naturally occurring ecosystems are almost exclusively organized as consortia. Recent metagenomic studies from the soil (Fierer and Jackson, 2006), to the ocean (Venter et al., 2004), to the human gut (Gill et al., 2006), have found that microbial communities are incredibly diverse, often consisting of thousands of interacting species. Subsets of these communities form consortia that act together to enhance their capabilities and survival (Eiteman et al., 2008).

Early theoretical ecology studies led to the development of the competitive exclusion principle (CEP), which states that the maximum number of species that can coexist in a system is equal to the total number of limiting (essential) resources (Hardin, 1960; MacArthur and Levins, 1964; Rescigno and Richardson, 1965). However, in nature we see many examples of multiple microbial species stably coexisting. Frequent explanations for coexistence in a natural population can be categorized into three main types: the development of multiple niches due to spatial heterogeneity or self-organized segregation, the system not being in equilibrium due to environmental fluctuations or external forcing, and the presence of inter- and intra-species interactions. Despite the limited direct applicability of CEP to natural systems, the clear mathematical formulation of CEP allows for significant insight by identifying which conditions of the CEP have been violated to lead to the observed coexistence of species. For example, by applying the CEP to the dynamics in a chemostat, we see that due to the consistent flow of nutrients and continuous mixing, the environment is kept constant and the development of different niches due to spatial heterogeneity is not possible; therefore, it must be some form of inter- or intra-species interactions such as crowding, chemical signaling, cooperativity, or mutual inhibition that lead to any observed coexistence. Studying these interactions in simple chemostat systems gives us a better understanding of the factors that maintain the diversity in naturally occurring microbial consortia.

Natural consortia are often found to form syntrophic systems, where the microbes depend on each other for survival, either by the production of required metabolic substrates or by the maintenance of chemically advantageous conditions (Schink, 2002). It is often observed that this syntrophic cooperation within microbial consortia increases their productivity and can allow the consortia to perform advanced functions that the microbial species are not capable of individually. These microbial interactions are known to be important in diverse areas including chronic medical infections (e.g. diabetic ulcers, Gardner et al., 2013; James et al., 2008), biofuel synthesis (e.g. biodiesel production, Peralta-Yahya et al., 2012; Zuroff and Curtis, 2012), environmental nutrient cycling (e.g. CO₂ sequestering, nitrification, Costa et al., 2006), bioprocessing (Shong et al., 2012), and wastewater treatment (Schink, 1997; Seitz et al., 1990a, 1990).

A frequently observed syntrophic system is a cross-feeding chain where microbes work together to perform the sequential degradation of complex compounds like lingocellulosic material (Schink, 2002). In these syntrophic cross-feeding systems a single substrate must be broken down in many steps, with one species catabolizing the available substrate and oxidizing it to produce a byproduct that the next species in the chain can consume. The intermediate byproducts in these systems are often found to be inhibitory. In this work we will consider the case where the intermediate byproduct inhibits growth. However, this more complicated system can be reduced to the case where there is no inhibition by taking the appropriate limit.

It has been observed experimentally that these syntrophic chain systems where the metabolic pathways are split among separate organisms (known as ‘microbial specialization’ or a ‘division of labor’) are more productive than a single organism with the equivalent metabolic capabilities (Bernstein et al., 2012; Wintermute and Silver, 2010). For example, if we compare a single organism that metabolizes A → B → C to a pair of organisms that metabolize A → B (organism 1) and B → C (organism 2), experimental observation has found the pair of organisms to be more productive than the single organism. Productivity is defined here as total biomass production per unit of input A.

An example of this division of labor that has been found to evolve repeatedly in different experiments occurs when E. coli is grown on a glucose substrate. The original population of E. coli can fully metabolize glucose (glucose → acetate → CO₂ (TCA cycle)), but when grown on glucose for many generations the population splits into two main subpopulations: microbes that preferentially consume glucose and produce acetate (glucose → acetate) and microbes that preferentially consume acetate (acetate → CO₂) (Rosenzweig et al., 1994; Rozen and Lenski, 2000; Treves et al., 1998).

In this paper we use standard chemostat modeling techniques to investigate whether the division of labor (splitting the pathways into two separate organisms) alone is sufficient to explain the observed increase in biomass, and if not, what other changes may be required. It is important to note that in contrast to the past work which has investigated the evolution of such cross-feeding systems, we will not consider the two systems in direct competition, instead comparing the maximum biomass (productivity) of the systems in isolation. This is motivated by industrial applications where the total productivity of the consortia, which is usually proportional to the biomass, is of primary interest.

This type of system has been studied mathematically, and it can be shown that for n species in a simple syntrophic cross-feeding chain, there is a stable coexistence steady state (Kreikenbohm and Bohl, 1986; Powell, 1985, 1986; Reilly, 1974). This simple system has been modified to include other forms of inhibition, external toxins, multiple substrates, and other forms of mutualism; in all cases, a stable, stationary, coexistence steady state is found (Aota and Nakajima, 2001; Burchard, 1994; Elkhader, 1991; Katsuyama et al., 2009; Sari et al., 2012). Previous research has focused on proving the existence and stability of coexistence steady-states. There has been no investigation of the productivity of these syntrophic chain systems. Some recent work (Doebeli, 2002; Estrela and Gudejl, 2010; Pfeiffer and Bonhoeffer, 2004) investigated the evolution of cross-feeding in microbial populations and found that there are a wide range of parameter values for which cross-feeding is seen to evolve. The aim of these evolution studies was to identify conditions for stable coexisting syntrophic chain systems to evolve and outcompete equivalent monocultures. They did not explicitly investigate the productivity of the systems that are found to evolve.

In our initial model, we assume that the metabolic dynamics of the pathways do not change when being split into separate microbes. In addition, we initially assume that the growth rate for the monoculture is a linear combination of the growth rates of the two pathways. This formulation allows us to obtain theoretical results and is effectively an upper bound on the growth rate of the monoculture microbe. Realistically, there are costs to utilizing both pathways at once for the monoculture, and there are changes that are known to occur to the metabolic pathway dynamics when the microbes specialize to a single substrate. For example, Pfeiffer and Bonhoeffer (2004) theorize that the evolution of cross-feeding could be due to the system minimizing the concentration of inhibitory intermediates and minimizing the concentration of enzymes it must produce, while maximizing the rate of ATP-production. Another explanation comes from Johnson et al. (2012) who find, by considering the biochemical conflicts which constrain the relationships between two metabolic processes that the division of metabolic pathways could be advantageous as it allows microbes to focus on producing a smaller number of enzymes and optimizing a smaller subset of pathways.

By starting with the assumption of no adaptation for the specialists and the maximal growth rate for the monoculture, we get a strong theoretical result. Then, by varying the growth kinetics (growth rates and substrate affinities) and yields between the monoculture and the binary culture systems, we are able to use our model to test possible explanations for the observed increase in productivity and to quantitatively investigate what changes are required for the division of labor to be advantageous (increased biomass). Considering the specific example of E. coli grown on a glucose substrate (Rosenzweig et al., 1994; Rozen and Lenski, 2000; Treves et al., 1998), we use standard Monod kinetics and measured experimental parameters to determine the conditions under which the model results match the experimentally observed increase in biomass.

The main result of this work is a comparison of the biomass production of a single microbe with full metabolic capacity to a syntrophic consortium of two specialized microbes each with a unique subset of the full metabolic chain, where the intermediate byproduct may be inhibitory. We prove that the monoculture system will always have higher biomass production (i.e., higher productivity) if there are no changes to the growth kinetics or yield of the pathways between the two systems. In a specific example, we show that increasing the growth rates or substrate affinities of the consortial pathways, a change that might be expected due to specialization, is not sufficient to generate the observed higher biomass production in the binary consortia. However, by varying the yields for the consortial pathways, higher biomass production can be achieved for the binary consortia relative to the monoculture. Yield changes are an expected result of microbial specialization because utilizing a smaller number of specialized pathways leads to reduced energy requirements. This result provides evidence for the argument that an increase in efficiency (yields) is a key factor underlying the observed productivity benefit of microbial consortia. Additionally, the finding that changing yields can lead to higher biomass production through specialization has important implications for industrial settings where higher biomass production is often highly desirable.

2. Model construction

To investigate the effects of separating metabolic pathways, we will analyze the simplest case of a syntrophic chain where there are only two microbial species. For clarity we will use a specific example in developing our notation, in this case the metabolism of glucose by E. coli. To construct the model we assume that the metabolism of a single provided substrate (glucose, G) can be split into two pathways. The first step (pathway) takes substrate G and produces an intermediate by-product A (acetate), creating new biomass at a growth rate μ₁(G, A). The intermediate by-product acetate A serves as a substrate for the second step (pathway), which has a growth rate μ₂(A). As well as being a substrate, acetate lowers the pH and is detrimental to both metabolism pathways in the cell, which slows the growth rates μ_i and thus slows the uptake of substrate. The lower pH caused by acetate production could also increase the energy required for cell homeostasis, which would affect the yield parameters. We do not consider this effect here, as if yield did decrease at higher A values, this would further disadvantage the binary consortia, and thus would not qualitatively affect our results but would greatly complicate the analysis.

We will consider the situation where the microbes are grown in continuous culture (chemostat). We compare the total biomass produced in two different systems: a binary culture system where the two pathways are in two separate microbes, and a monoculture system where the pathways are combined into a single microbe. These two systems are shown in Fig. 1. For the monoculture system, acetate that is produced through the first pathway is then available to be consumed through the second pathway. If the acetate consumption rate is lower than the acetate production rate, acetate will accumulate in the system, see Eqs. (4) and (7). In our model of the monoculture, we consider the case where acetate is consumed (metabolized) by the individual microbial cell that produced it, and the case where acetate is released from that microbe into the environment and then consumed by distinct microbes, to be equivalent. This is possible because this model does not include any cost involved in transporting acetate into or out of the cell. Including a cost for transporting acetate would disadvantage the binary consortia compared to the monoculture, and thus it would not qualitatively affect our results.

Fig. 1

Diagram showing the metabolic pathways in (a) the monoculture system with the microbes x and (b) the binary culture system with the primary (glucose) consumer x₁ and the scavenger (acetate consumer) x₂, where G is the primary substrate (glucose) and ...

3. Main results

In this section we present our main result, the proof of which can be found in the Appendix. The implications of this technical result are then demonstrated with an example from a realistic situation in the following sections.

4. Effect of modifications to the model on relative biomass

In chemostat experiments, a binary culture of the type analyzed here shows a biomass increase of around 20% over a wild-type monoculture system (Bernstein et al., 2012). As we have shown in Theorem 3, under our assumptions the monoculture system always has a higher biomass. Clearly this means that one or more of the assumptions that we make in Section 2 are incorrect. This is useful, as by examining the assumptions we made, we can investigate which of the effects that we did not include are most relevant.

One of the assumptions we made in formulating the initial model is that there is no change in the growth dynamics or yields of the metabolic pathways when the microbes specialize to a single substrate. However, as discussed already, there is evidence that the growth dynamics and pathway efficiencies will change, and we will investigate the effects of incorporating these changes in this section.

5. Specific example using Monod kinetics and typical values for E. coli

In this section, the results of this paper will be applied to the specific example of E. coli growing on glucose as the sole carbon source. Experimentally observed parameters for E. coli growth on glucose or acetate are summarized in Table 1. In the majority of the literature, E. coli were grown in aerobic conditions in which both pathways would be utilized. This will lead to overestimation of the parameters: μ_1,max, Y₁₁, Y₂₁, K_IP, and underestimation of the parameter r.

Table 1

Typical parameter values for E. coli grown on glucose or acetate as the carbon source. See Table D1 for further details of the strains used and the specific experimental conditions.

We can quantify the discrepancy between the actual yield of each pathway and the reported yield in wild-type aerobic conditions. If we assume that the growth rate from the first (anaerobic) pathway is given by μ₁ and the growth from the aerobic metabolism of acetate (acetyl CoA) is given by μ₂, then the discrepancy between the actual yield of each pathway and the reported yield in wild-type aerobic conditions will be

• The reported yield Y11WT (gBiomass/gGlucose consumed) will be overestimated by Y11WT=Y11(μ1+μ2)/μ1.
• The reported acetate production yield Y21WT (gBiomass/gAcetate produced) will be overestimated by Y21WT=Y21(μ1+μ2)/(μ1-γμ2).
• The observed ratio r^WT (gAcetate produced/gGlucose consumed) will be underestimated by r^WT = r(1 − γμ₂/μ₁).

From the equation for Y11WT it can be seen that if μ₂ = 0 (i.e. there is no metabolism of acetate, which occurs in anaerobic conditions), then Y11WT=Y11. Thus, in order to estimate the parameters of just the first pathway (max f(G) and Y₁₁), we use the yield and growth parameters from anaerobic experiments (Chen et al., 1997; Varma and Palsson, 1994; Xu et al., 1999). To estimate Y₂₁ we use the ratio r = 0.667 determined by the reaction stoichiometry and calculate Y₂₁ = Y₁₁/r. Substituting reasonable values for growth rates, we find that our choice of parameters leads to observed Y21WTrWT that match experimental data (Table 1). The observed growth and yield parameters for E. coli grown on acetate as the sole carbon source provide good estimates for max (m(A)) and Y₂₂. The chosen yields are given in Table 2.

Table 2

Selected yield values for monoculture and binary culture systems.

It is important to note that for the yield terms Y₁₁ and Y₂₂, which are the grams of biomass produced when one gram of substrate is consumed, only some of the energy from the substrate metabolism will go towards biomass synthesis. Depending on the efficiency of the pathway, the energy produced per gram of substrate will vary, and the energy required for cell homeostasis (i.e. not available for biomass production) also varies. Estimates for the ratio of energy produced that goes towards biomass production and the efficiencies of the pathways used vary depending on the E. coli strain and the environmental conditions (Carlson and Srienc, 2004; Han et al., 1992). We choose to use measured values for the total yield from the literature. These yields could vary with strain and condition, and we will consider this in Section 5.2.3. In all cases the yields are bounded above by the stoichiometry of the most efficient biomass production pathways assuming zero energy required for cell homeostasis, and our choices for yield parameters are below this bound and consistent with experimental findings.

Table 3

Selected parameter values for growth and inhibition functions given in Eqs. (20), (21), (22), dilution rates, and substrate inflow concentrations.

At low glucose concentrations it is known that glucose affinity increases (Ferenci, 1996; Helling et al., 1987), and the parameter K_G varies from O(0.01) to O(100) mg/L (Table D1) through the upregulation of higher affinity (and often slower or more energy intensive) pathways. In our model, K_G is a fixed parameter, and to demonstrate the results of this paper we choose an intermediate K_G value (see Table 3). This choice does not affect our result, as for equal substrate affinities the monoculture will outperform the binary culture for any choice of K_G (Theorem 3). We will investigate the effect of varying the glucose affinities between the monoculture and the binary culture in Section 5.2.2. However, as increases in glucose affinity are found to depend mostly on the growth conditions, we would expect any increase in glucose affinity (K_G) to affect both the monoculture and the binary culture similarly.

In E. coli there are two pathways for acetate assimilation: the phosphotransacetylase-acetate kinase (PTA-ACKA) pathway and the acetyl-CoA synthetase (acs) pathway. The PTA-ACKA pathway has lower substrate affinity, with K_m ≈ 0.4 – −0.6 g/L (Brown et al., 1977; Wolfe, 2005), but is more efficient requiring only one ATP for acetate assimilation. The PTA-ACKA pathway is found to be active until acetate gets to low concentrations (A « 0.15 g/L) (Kumari et al., 1995). In contrast, the acs pathway has higher substrate affinity, with K_m ≈ 0.01 g/L (Brown et al., 1977; Wolfe, 2005), but is more energy intensive, requiring two ATP for acetate assimilation. Kumari et al. (1995) find that both pathways are required for optimal growth on acetate across a range of concentrations, with the acs pathway being required for A < 0.6 g/L and the PTA-ACKA pathway being required for A > 1.5 g/L.

In our model, we use a single function m(A) and single yield term Y₂₂ to describe the growth kinetics and efficiency of growth on acetate; we do not include the two pathways separately. We choose growth parameters μ_2,max, K_A, and K_IS that match the experimentally observed growth in chemostat cultures which are utilizing both pathways. For example, the combined growth kinetics has a substrate affinity of K_m = 0.23 g/L which is midway between the K_m values of each pathway. We will investigate the effect of varying the acetate affinities between the monoculture and the binary culture in Section 5.2.2.

The yield parameter Y₂₂ is fixed, so we have chosen Y₂₂ = 0.4 which is at the lower end of the yields observed across a range of growth conditions (Kleman and Strohl, 1994). This is because we find that in our model acetate concentrations stay low and the yield will be determined predominantly by the lower efficiency, higher affinity acs pathway.

The dilution rate D and the glucose inflow concentration G_in are set by experimental conditions. We must select a dilution rate that satisfies non-wash-out conditions (from assumption [A6]: max(m(A)) > D, μ_1,max > D). This leads to a low dilution rate (D = 0.1) due to the low value of max(m(A)) > D seen experimentally. We also choose a high substrate inflow G_in » G₀, so that the system is in glucose-excess conditions (assumption [A5]).

Fig. 4

The functions f(G), I(A), and m(A) (20, 21, and (22), respectively), in the original (dimensional) variables, for the chosen parameter values.

5.2. Quantitative investigation of how to increase the binary culture’s relative productivity

In chemostat experiments, a binary culture of this type shows a biomass increase of around 20% over the wild-type monoculture system (Bernstein et al., 2012). In Fig. 5 we plot the curves that determine the equilibrium points of the systems. The biomass is determined by the values of G and A, given in Eqs. (15) and (16), thus in the (A,G)-plane, the curves G = c − A/(1 + γ), for a constant c = 1 − γx_c/(γ + 1), are the lines of constant biomass, x_c. In order for the binary culture biomass to exceed that of the monoculture, the binary culture system must be modified such that the point (A^b, G^b) crosses the dashed line in Fig. 5, or, alternatively, the monoculture system must be modified such that the point (A^m, G^m) crossed the dotted line in Fig. 5. (The only exception to this is if Y₁₁ or G_in are varied between the two systems, as these parameters appear in the non-dimensionalization of the biomass variables.)

Fig. 5

The curves G₁ and G₂, which determine the position of the monoculture equilibrium point(s) in the nondimensionalized system are shown in black solid lines, with the intersection labeled (A^m, G^m). The curves f(G)I(A) = 1 and m(A) = 1, which determine the ...

5.2.1. Varying growth rates independently

When microbes specialize to grow on a single substrate they are found to increase their growth rate. One explanation for this is that they are able to allocate more resources to synthesizing transporters that increase the substrate uptake rate. By making the substitutions in Eq. (18), we can use parameters, α₁ and α₂ to modify the growth rates independently and perform a more general investigation.

To increase the growth rate of the microbes in the binary culture compared to monoculture we set α₁ > 1, or α₂ > 1, or both. It is important to note that increasing α_i will increase the growth rate of the pathway i by the factor α_i across all substrate concentrations.

• If we only increase the maximal growth rate of the glucose metabolism, α₁, we cannot increase the total biomass of the binary culture to match that of the monoculture, even if α₁ → ∞.
• If we only increase the maximal growth rate of the acetate metabolism, α₂, the binary culture has the same biomass of the monoculture at α₂ = 1.69, i.e., the growth rate has to increase 69%.
• Finally, we allow both α₁ and α₂ to increase then we find multiple combinations of α₁, α₂ where the biomass has the same or larger mass than the monoculture. The combination of multipliers with minimal value of the sum α₁ + α₂ for which the total biomass of the binary culture matches that of monoculture is when we only increase the second pathway, so α₁ = 1 and α₂ = 1.69.

Increases of around 15–20% have been seen experimentally when microbes have been grown on a single substrate. The increases of 70% that are required here are not realistic. Moreover, Bernstein et al. (2012) observed a biomass increase of 20% in chemostat experiments. Even increasing α₂ → ∞ the binary culture biomass never reaches 20% more than the monoculture

5.2.2. Increasing the substrate affinity in the binary culture

In E. coli the experimentally observed glucose affinity varies by orders of magnitude in different experimental conditions (K_m = 0.0053–99 g/L Senn et al., 1994). This is because the bacteria have many different systems to transport and metabolize glucose as a substrate, that all have differing speeds and affinities. If we predict that the glucose specialist (x₁) in the binary system has an increasing substrate (glucose) affinity, we can investigate the effect this would have on biomass production by decreasing K_G in our model.

• Varying K_G we find that there is no decrease in K_G that will lead to the binary system having a higher biomass than the monoculture system. This is expected from our results in Section 5.2.1, since f(G) ≈ μ_1,maxG/K_G at low G, and there is no α₁ that leads to the binary system biomass exceeding that of the monoculture.

Additionally, the phenomenon whereby E. coli in low glucose conditions uses higher affinity transporters would be predicted to occur in both the monoculture and primary microbe in the binary culture, as this depends mostly on growth conditions. If glucose affinities increase but remain equal, the monoculture will still outperform the binary culture by Theorem 3.

If the acetate scavenger specialist (x₂) in the binary culture has an increased acetate substrate affinity, relative to the monoculture, we can investigate the effect this would have on biomass production by decreasing K_A and K_IS in our model.

• Varying K_A (and K_IS) we find that the binary culture has higher biomass when K_A and K_IS decrease to 54% of the monoculture values i.e. an increase in substrate affinity of 85%. This is to be expected from our results in Section 5.2.1, as increasing m(A) by 69% gave a higher biomass for the binary culture, and thus decreasing K_A (which increases the growth rate at low A values) should have a similar effect.

Note: we decrease K_IS concurrently with K_A in order to keep max (m(A)) constant, however, if we only decrease K_A and allow max(m(A)) to increase we see similar results (55% decrease cf. 54% decrease).

A twofold increase in the acetate affinity (halving the K_m value) is within realistic realms, and brings the K_m value of acetate metabolism to 0.12 g/L. However, the higher-affinity acs pathway, that would be required for an increased acetate affinity, is more energy intensive (lower yield); this means that although the increased substrate affinity would act to increase the biomass, the consequent decrease in yield would counteract this increase. In our model formulation, we cannot directly account for the use of different acetate assimilation pathways between the binary culture system and the monoculture in the comparison, as we do not include the two pathways separately. Furthermore, up-regulation of the higher affinity acs pathway has been observed in vivo in wild-type E. coli (Wolfe, 2005) growing on acetate as well, so we consider it difficult to justify a change that increases the binary culture acetate affinity by 85% more than that of the monoculture.

If we take our analysis one step further and attempt to match the observed biomass increase of 20% in the binary culture (Bernstein et al., 2012), we find that there is no increase in substrate affinity (K_m → 0) which can produce the required increase in biomass.

5.2.3. Varying form of the monoculture growth rate

When there is glucose available, E. coli will preferentially consume glucose due to catabolite repression until the acetate concentration gets too high, at which point the E. coli will switch to consuming acetate (Nicolaou et al., 2010; Wolfe, 2005). This effect is incorporated implicitly into our model, as for high G, low A, f(G)I(A) > m(A) and μ_WT ≈ μ₁. Whereas, for high A we find m(A) > f(G)I(A), and μ_WT → μ₂. However, there may be some additional trade-off in the ability of the cell to produce the enzymes required for each pathway, so the increased utilization of one pathway may decrease the growth rate of the other (Oh et al., 2002). We can incorporate this into the model by using the modified growth function for the monoculture system in Eq. (19).

In experiments it is observed that the utilization of the two pathways, and thus the value of the parameter β, depends on the concentration of acetate A (Wolfe, 2005). We will not assume a form for the dependence of β on A, however, by considering the existing experimental evidence we can determine whether the equilibrium acetate concentration and pathway utilization for a given β value is a realistic scenario.

The position in the (A,G)-plane of the equilibrium point for the modified monoculture system as β varies is shown in Fig. 6(a). For β values (0.127 < β < 0.371) the monoculture equilibrium point is to the left of the dotted line (binary culture biomass equivalence), and thus has a higher biomass. As β decreases below β = 0.127 the monoculture equilibrium G value increases, until the monoculture does metabolize enough glucose to survive and the equilibrium point disappears in a saddle-node bifurcation at β = 0.0805 (SN₁). As β increases from β = 0.371, the monoculture equilibrium point disappears in a saddle-node bifurcation at β = 0.708 (SN₂). The system becomes bi-stable at β = 0.653 (SN₃), shown in Fig. 6(b). The lower biomass equilibrium point persists and reaches the x₁-only equilibrium point of the binary culture (A₁, G₁) at β = 1.

Fig. 6

Panel (a) shows the position of the monoculture equilibrium point in the (A, G)-plane as β varies as a solid black line. The coexistence equilibrium point of the binary culture ( ASb,GSb) is shown as a filled black circle. The dotted line shows ...

At β = 1 the monoculture is the equivalent of the binary culture system with x₂ = 0, which means it has a lower biomass than the binary culture system coexistence equilibrium point (not shown in Fig. 6). As β decreases from 1, the advantage of being able to consume the acetate increases the monoculture’s biomass. We find that the binary culture only has higher biomass than the monoculture for β > 0.371. However, for β > 0.371 the monoculture reaches an equilibrium point with high acetate concentration, but E. coli are observed to switch to consuming more acetate when the acetate concentration increases (Wolfe, 2005), which means that these high β values (a higher utilization of the glucose consumption pathway f(G)I(A) over the acetate consumption pathway m(A)) are unrealistic.

As β→ 0, the monoculture growth rate is increasingly dependent on the second pathway, m(A), which consumes exclusively acetate A. Since glucose G is the only substrate provided to the chemostat, we expect that the monoculture will not survive. Indeed, for β<0.0805 the monoculture washes out and the biomass of monoculture (x=0) will be less than the biomass of the binary culture.

The optimal β value (to maximize growth rate Abrams, 1987; Egli et al., 1993; Lendenmann and Egli, 1922) will depend on the concentration of G and A. However, for monoculture equilibrium values as β varies the growth rate is set to D by the chemostat dynamics. If we instead predict that the optimal β value would lead to a higher biomass, we find peak monoculture biomass at β = 0.208 (x^m=2.71), and we reach the conclusion that the monoculture would be expected to have a higher biomass than the binary culture.

5.2.4. Varying yields

In this section we examine the effect of varying yields in our model of E. coli growing on glucose. If the microbes in the binary culture are able to reduce their energy requirements when they specialize to a single substrate, the yields Y₁₁ and Y₂₂ would increase. If we include the effect of the cost of transporting acetate into the cell, this would decrease Y₂₂. Here we will assume that the metabolic pathway stoichiometry is fixed, i.e., the ratio r = Y ₁₁/Y₂₁ (grams of acetate produced per gram of glucose consumed) is fixed. The effects of varying the yields are as follows:

• By increasing Y₁₁ (and Y₂₁ as r is fixed) in the binary culture system, that is, reducing the amount of glucose G the primary consumer microbe requires to make one g of biomass x₁, the total biomass of the binary culture can exceed the monoculture biomass. In this example we find that the two systems have equal biomass when ₁₁ = 1.03 Y₁₁ (and ₂₁ = 1.03 Y₂₁), which represents an increase in glucose pathway efficiency of 3%.
• By increasing Y₂₂ in the binary culture system, which corresponds to a reduction in the amount of acetate A the secondary scavenger microbe requires to make one gram of biomass x₂, the total biomass of the binary culture can exceed the monoculture biomass. In this example we find that the two systems have equal biomass with ₂₂ = 1.02 Y₂₂. This represents an increase in acetate pathway efficiency of 2%.

If the ratio r = Y₂₁/Y₁₁ is not fixed, we find that increasing Y₁₁ independently of Y₂₁, we can get the same total biomass with ₁₁ = 1.01 Y₁₁, and decreasing Y₂₁ independently of Y₁₁ we find that ₂₁ = 0.98 Y₂₁ gives the same total biomass in both systems. However, in order for Y₂₁ to decrease without Y₁₁ decreasing (or Y₁₁ to increase without Y₂₁ increasing), the glucose consuming microbe would have to utilize different metabolic pathways, with different stoichiometry, that produce more A per G consumed. In experiments, it is observed that wild-type E. coli can modify their metabolic strategies depending on the environmental conditions, specifically, in excess acetate the monoculture is seen to increase the flux through the acetate-metabolism pathway m(A) (Wolfe, 2005). This would be equivalent to varying β in our modified monoculture system (Eq. (19)); the stoichiometry of the pathways would not change. In this application, we argue that it is unrealistic to modify Y₂₁ independently of Y₁₁.

Finally, we can generate the 20% increase in biomass observed in Bernstein et al. (2012) if we increase by the efficiency of the glucose-metabolism pathway (Y₁₁ and Y₂₁) by 57%, or increase the efficiency of acetate-metabolism pathway, Y₂₂, by 34%.

6. Discussion

In this paper we describe a minimal model of a syntrophic chain system in which the metabolism of a substrate is divided into two distinct organisms. We compare this to a monoculture, which is able to completely metabolize the substrate. Under very general assumptions we show that the monoculture will always produce larger biomass than the equivalent binary syntrophic chain system. This finding does not agree with experimental findings from microbial consortial systems, which show higher biomass for consortial systems (Bernstein et al., 2012). Two key assumptions in the creation of our minimal model are that there are no metabolic advantages to dividing the metabolic steps into separate organisms, and that there is no cost to the monoculture for using multiple substrates concurrently. Since the model under these assumptions does not reproduce the increased biomass in the consortium that is observed experimentally, this leads us to conclude that one (or both) of these assumptions is incorrect and allows us to investigate the effect of making these assumptions more realistic. By modifying the minimal model we are able to test hypotheses and investigate the effects of adaptations that can develop in a syntrophic chain system, clarifying the mechanisms that contribute to the experimentally observed advantage. We find that the division of metabolic pathways must have an effect on both the growth and yield of the system.

If we assume that in the binary culture the pathways are able to metabolize the substrate faster (increasing the growth rate), we find that it is sometimes possible to have a higher biomass in the binary culture system. However, considering a specific example from E. coli using typical parameters, the growth rate increases required are much larger than those observed experimentally when an organism adapts to a single food source. Considering instead the possibility of increases in substrate affinity, we find that, for the specific example of E. coli, a twofold decrease in the acetate K_m value for the binary culture relative to the monoculture would lead to higher biomass. However, in similar low substrate conditions acetate affinity increases would be seen in both the monoculture and binary culture, and even if the acetate specialist in the binary culture could increase its substrate affinity, this would not necessarily increase the productivity, as the higher affinity acetate assimilation pathway has a lower yield.

We also test a hypothesis that the efficiencies of the metabolic pathways in the binary culture are increased, as the specialists do not have to produce the enzymes related to the other pathways, reducing the amount of energy required for enzyme synthesis and maintenance, and thereby increasing the proportion of energy that can go towards biomass production (Oh et al., 2002). By varying the yield, it is possible to increase the binary culture’s biomass to match that of the monoculture with an increase of only 2–3% in the parameters Y₁₁ or Y₂₂. This magnitude of yield change is within the range of what is observed experimentally, and we consider this to be the most likely cause of the observed productivity increase in the binary culture.

Comparing our results to past work on the question of whether organisms maximize growth rates or efficiency, we find that two main theories are the Maximum Power Principle (MPP) (DeLong, 2008; Lotka, 1922; Sciubba, 2011) and the Resource Ratio Theory (RRT) (de Mazancourt and Schwartz, 2010; Miller et al., 2005; Tilman, 1982, 1988). The MPP states that systems strive to increase ‘power’ within the system constraints. This has been used to successfully predict the outcome of two species competition experiments (DeLong, 2008). The RRT states that the system/species that can use as much as possible of the resources available (have the lowest amount of ‘wasted’ substrate) will win in a competition experiment (or will evolve) (de Mazancourt and Schwartz, 2010). In summary, MPP argues for maximizing substrate uptake rate, whereas RRT argues for optimizing the growth at low substrate concentrations. In a chemostat with fixed yield, the highest substrate uptake will correspond to the highest growth rate, and these two will be consistent. Thus, in evolution (or competition) in a chemostat, maximizing growth rate or maximizing substrate uptake rate may be advantageous strategies in order to survive.

In this work we are motivated by the observation of increased biomass in a division of labor system (Bernstein et al., 2012), which has important implications for industrial settings. We find that yield maximization should be the advantageous strategy for increasing the biomass production of a chemostat system. In all our work we consider the binary and monoculture systems in isolation, and investigate and compare the optimal productivity (biomass production) of these systems. We are not considering the evolution of these division of labor systems, as there is much work already investigating this and it is not the focus of our work. We are also not considering competition of these two systems, thus the arguments from RRT and MPP are not directly applicable to this situation. Some work has found that evolutionary advantage depends on the stress conditions, with some papers even finding that the ‘optimal’ robust state has lower biomass (Pfeiffer and Bonhoeffer, 2004). This is not unexpected as up-regulating pathways with higher substrate affinities is often more costly (decreased yield), but this enables survival at lower substrate concentrations and thus resistance to invasion.

For E. coli communities, the cross-feeding syntrophic chain system studied in Section 5 has been seen to evolve repeatedly (Rosenzweig et al., 1994; Rozen and Lenski, 2000; Treves et al., 1998). This suggests that there must be some increase in growth rate or substrate affinity following specialization in order for the system to out-compete the original monoculture. When synthetically engineered, this cross-feeding system has been found to have higher productivity (Bernstein et al., 2012). Our model shows that increases in growth rate or substrate affinity are not sufficient to explain this observed increase in productivity, and that an increase in the efficiency (yield) of the syntrophic system must have occurred.

An interesting contradiction that remains is that in the unmodified model ASb>Am for all parameters and growth functions, but in experiments the concentration of the inhibitory intermediate A is observed to be lower in a syntrophic chain system than in a monoculture. If we assume that an increase in efficiency when organisms specialize to a single substrate is sufficient to explain the observed increase in biomass, we find that there is no decrease in ASb in the binary culture when we increase the yield Y₂₂ and only a small decrease when we increase the yield Y₁₁. However, increasing the growth rates in the binary system does decrease ASb. This suggests that increases in both yield (increased efficiency) and growth kinetics play a role in metabolic specialization. By applying our model to experimental data, we will be able to investigate these factors in future work.

Appendix A. Binary culture equilibrium points existence and stability

In this appendix we prove Theorem 1.

A.2. Stability of equilibrium points

We introduce the following variables:

w1(t)=G(t)+x1(t),

(31)

w2(t)=G(t)+γx2(t)+A(t).

(32)

Note that the system obeys a conservation law of material consumed to biomass produced. The rate equations for these new variables are

w.1=1-w1

(33)

w.2=1-w2.

(34)

which means that the set Ωb:={(x1,x2,G,A)∈ℝ+4:x1=1-G,γx2=1-G-A} is an exponentially attracting two-dimensional invariant subspace of the full ⁴ phase space. The asymptotic behavior of the system is determined by the dynamics on Ω^b, given by

G.=1-G-f(G)I(A)x1(G),

(35)

A.=-A+f(G)I(A)x1(G)-γm(A)x2(G,A),

(36)

with x₁(G) = 1 −G, x₂(A, G) = (1 −G −A)/γ. In this reduced system, we consider the region where x₁ ≥0, x₂ ≥0, given by G ≤1 − A in the positive quadrant. The Jacobian of this reduced system is

J(G,A)=[-1+fI-∂f∂GIx1-f∂I∂Ax1-fI+∂f∂GIx1+m-1+f∂I∂Ax1-γ∂m∂Ax2+m],

(37)

where x₁ = 1−G, x₂ = (1−G−A)/γ. The trivial equilibrium point (x₁, x₂, G, A) = (0, 0, 1, 0), I(0) = 1 and m(0) = 0 (from Assumptions [A3] and [A4]) and the Jacobian becomes

J(G,A)=[-1+f(1)0-f(1)-1],

(38)

which has eigenvalues λ = , −1, f (1) −1. From Assumption [A2] in Section 2.3 we know that 0<f (1)< μ_1,max. When f (1)>1 the trivial equilibrium will have three negative eigenvalues (two from the components normal to Ω^b and one from the 2D reduced subsystem) and one positive eigenvalue, thus it is an unstable saddle. The condition f (1)>1, which corresponds to f (G_in) >D in the original dimensions, is ensured by assumption [A6]: the trivial equilibrium point will be unstable as long as the substrate inflow is in excess.

The boundary equilibrium point (fI=1, x₂ = 0) has the Jacobian

J(G1,A1)=[-∂f∂GIx1-f∂I∂Ax1-1+∂f∂GIx1+m-1+f∂I∂Ax1+m].

(39)

From this we find

Tr(J)=-[(∂f∂GI-f∂I∂A)x1+(1-m)]

and

Det(J)=(1-m)[∂f∂GI-f∂I∂A]x1(G).

The boundary equilibrium point undergoes transcritical bifurcations when m(A) = 1, i.e., when A1=ASb and A1=AUb. For A1<ASb, m(A¹)<1, Det (J)>0, Tr (J)<0, and (Tr² −4 · Det)>0 and the equilibrium point is a stable node. For ASb<A1<AUb, m(A¹)>1, Det (J)<0 and the equilibrium point is an unstable saddle. For AUb<A1, m(A¹)<1, Det (J)>0, Tr (J)<0, and (Tr² −4 · Det)>0 and the equilibrium point is a stable node again. The vector (A, G) = (1, −1) is a stable eigenvector which lies on the x₂ = 0 invariant line: G = 1−A.

For the co-existence equilibria (fI=1, m=1) the Jacobian becomes

J(G,A)=[-∂f∂GIx1-f∂I∂Ax1-∂f∂GIx1f∂I∂Ax1-γ∂m∂Ax2],

(40)

with

Tr(J)=-[(∂f∂GI-f∂I∂A)x1(G)+γ∂m∂Ax2(G,A)]

and

Det(J)=∂f∂GIx1(G)γ∂m∂Ax2(G,A).

For ( ASb,GSb) in the region x₂ >0, Tr (J)<0 and Det (J)>0, and (Tr² −4 · Det)>0 so the equilibrium point is a stable node. For ( AUb,GUb) in the region x₂ >0, Det (J)<0 and the equilibrium point is an unstable saddle. These equilibrium points go through transcritical bifurcations at x₂ = 0 when they pass through the boundary equilibrium point. Details are shown in Fig. A2.

Fig. A1

Qualitative sketches of the growth and inhibition curves and positions of equilibria for the case where there are two co-existence equilibrium points. Panel (a) shows the function m(A) showing A_max and the relative positions of ASb and AUb. Panel (b) shows the function I(A) with the relative positions of ASb and AUb. Panel (c) shows function f(G) and the relative positions of GSb and GUb. The position of G₀ defined by f(G) = 1 is shown as well. Panel (d) shows the curves f(G)I(A) = 1 (solid black curve) and m(A) = 1 (dashed black lines). The co-existence equilibrium points projected to (A, G) plane are at the two intersections of these curves. The dotted black line G = 1−A (together with the solid black curve f (G)I(A) = 1) determines the position of the boundary (x₂ = 0) equilibrium point projected into the (A, G) plane. The position of G₀ defined by f(G) = 1 is shown as well.

Fig. A2

Qualitative sketches of the six possible cases based on assumptions [A1]–[A4] in Section 2.3. Equilibrium points are marked with a circle for stable equilibria and a triangle for unstable equilibria. In the shaded gray region (G>1−A) any solutions are physically unrealistic as x₂ <0. The trivial (x₁ = 0, x₂ = 0) equilibrium point at (0,1) is shown. For all cases, there is a boundary (x₂ = 0) equilibrium point (A¹, G¹) which, when projected into the (A, G) plane, lies at the intersection of the dotted black line G = 1−A together with the solid black curve f (G)I(A) = 1. The curves f (G)I(A) = 1 (solid black curve) and m(A) = 1 (dashed black lines) are shown, the position of co-existence equilibrium points projected to (A, G) plane is at the intersections of these curves. The vertical asymptote of the curve f (G)I(A) = 1 is labeled Acritb. The position of G₀ defined by f (G) = 1 is shown as well.

Appendix B. Monoculture equilibrium points

In this appendix we prove Theorem 2.