Metabolic enzyme cost explains variable trade-offs between microbial growth rate and yield

Abstract

Microbes may maximize the number of daughter cells per time or per amount of nutrients consumed. These two strategies correspond, respectively, to the use of enzyme-efficient or substrate-efficient metabolic pathways. In reality, fast growth is often associated with wasteful, yield-inefficient metabolism, and a general thermodynamic trade-off between growth rate and biomass yield has been proposed to explain this. We studied growth rate/yield trade-offs by using a novel modeling framework, Enzyme-Flux Cost Minimization (EFCM) and by assuming that the growth rate depends directly on the enzyme investment per rate of biomass production. In a comprehensive mathematical model of core metabolism in E. coli, we screened all elementary flux modes leading to cell synthesis, characterized them by the growth rates and yields they provide, and studied the shape of the resulting rate/yield Pareto front. By varying the model parameters, we found that the rate/yield trade-off is not universal, but depends on metabolic kinetics and environmental conditions. A prominent trade-off emerges under oxygen-limited growth, where yield-inefficient pathways support a 2-to-3 times higher growth rate than yield-efficient pathways. EFCM can be widely used to predict optimal metabolic states and growth rates under varying nutrient levels, perturbations of enzyme parameters, and single or multiple gene knockouts.

Fig 1. Rate/yield trade-offs and calculation of growth-optimal fluxes.

(a) Rate/yield spectrum of Elementary Flux Modes (EFMs) (schematic drawing). In the scatter plot, EFMs are represented by points indicating biomass yield and maximal achievable growth rate in a given simulation scenario. Pareto-optimal EFMs are marked by red squares. The set of Pareto-optimal flux modes (black lines) contains also non-elementary flux modes. An EFM may be Pareto-optimal when compared to other EFMs, but not when compared to all possible flux modes (e.g. the EFM below the Pareto front marked by a the pink square). Growth rate and yield are positively correlated in the entire point cloud, but the points along the Pareto front show a negative correlation, indicating a trade-off. (b) Enzyme cost of metabolic fluxes. The space of stationary flux distributions is spanned by three EFMs (hypothetical example). The flux modes, scaled to unit biomass production, form a triangle. To compute the enzyme cost of a flux mode, we determine the optimal enzyme and metabolite levels. To do so, we minimize the enzymatic cost on the metabolite polytope (inset graphics) by solving a convex optimality problem called Enzyme Cost Minimization (ECM). (c) Calculation of optimal flux modes. The enzymatic cost is a concave function on the flux polytope, and its optimal points must be polytope vertices. In models without flux bounds, these vertices are EFMs and optimal flux modes can be found by screening all EFMs and choosing the one with the minimal cost.

Fig 2. Metabolic strategies in E. coli metabolism.

(a) Network model of core carbon metabolism in E. coli. Each Elementary Flux Mode (EFM) represents a steady metabolic flux mode in the network, scaled to a unit biomass flux. Reaction fluxes defined by the EFM max-gr are shown by colors. In our reference conditions—i.e. high extracellular glucose and oxygen concentrations—this EFM allows for the highest growth rate among all EFMs. Some of the cofactors in the model are not shown. (b) Statistics of biomass-producing EFMs. (c) Spectrum of growth rates and yields achieved by the EFMs. The labeled focal EFMs are described in Table 1, and their flux maps are given in Figures 25-30 in S1 Text. Pareto-optimal EFMs are marked by squares; the Pareto front is shown by a black line. The plot reveals a positive correlation between growth rate and yield, despite the inevitably negative correlation among Pareto-optimal EFMs. See Figure 24 in S1 Text for a detailed view of the Pareto front and how it was sampled.

Fig 3. Uptake and secretion fluxes across EFMs.

(a) Oxygen uptake (scaled by glucose uptake). Flux values are shown by colors in the rate/yield spectrum (same points as in Fig 2b). The EFMs with the highest growth rates consume intermediate levels of oxygen. The other diagrams show (b) acetate secretion, (c) lactate secretion and (d) succinate secretion, each scaled by glucose uptake. Acetate secretion and O₂ uptake versus biomass yield are shown in Figure 9 in S1 Text.

Fig 4. Growth rates and rate/yield trade-offs depending on glucose and oxygen levels.

(a) Predicted growth rates and biomass yields of aerobic EFMs, at reference oxygen level (0.21 mM) and at a lower level (2.1 μM). Pareto-optimal EFMs are marked by dark triangles. Since changing oxygen levels affect the growth rate, but not the yield, points move vertically between the two conditions. Statistical distributions of growth rates across EFMs are shown in Figure 10 in S1 Text. (b) Oxygen-dependent growth rates for the five focal EFMs and the measured flux distribution. The oxygen level directly affects the catalytic rate of oxidative phosphorylation (reactions oxphos and sdh): lower oxygen levels require higher enzyme levels for compensation, to keep the fluxes unchanged. The non-respiring EFM ana-lac shows an oxygen-independent growth rate. In all other focal EFMs, the growth rate increases with the oxygen level and saturates around 10 mM. max-gr, which uses a higher amount of oxygen, has a steeper slope and loses its lead when oxygen levels drop below 1 mM. The corresponding changes in enzyme allocation are shown in Figure 18 in S1 Text. (c) Growth rate as a function of glucose and oxygen levels (“Monod surface”). For a closed approximation formula, see Section 4.6 in S1 Text. (d)-(f) The same plot, with oxygen uptake, acetate secretion, and lactate secretion shown by colors. Distinct areas represent different optimal EFMs (compare Figure 13 in S1 Text). The optimal EFMs for strictly anaerobic conditions are depicted in Figure 15 in S1 Text (b).

Fig 5. Predicted protein investments.

(a) Predicted protein demands for the EFM max-gr at reference conditions. (b) Predicted protein demand for the EFM max-gr at varying glucose levels and reference oxygen level. The y-axis shows relative protein demands (normalized to a sum of 1). The dashed line indicates the reference glucose level (100 mM) corresponding to the pie chart in panel (a).

Fig 6. Growth rates achieved with two variants of glycolysis.

(a) Glucose- and oxygen-dependent growth rates predicted for wild-type E. coli. Same data as in Fig 4(c), but shown as a heatmap. E. coli can employ two variants of glycolysis: the Embden-Meyerhof-Parnas (EMP) pathway, which is common also to eukaryotes, and the Entner-Doudoroff (ED) pathway, which provides a lower ATP yield at a much lower enzyme demand [21]. (b) A simulated ED knockout strain that must use the EMP pathway. The heatmap shows the relative growth advantage of the wild-type strain (i.e. of reintroducing the ED pathway to the cell). The ED pathway provides its highest advantage at low oxygen and medium to low glucose levels. (c) Growth advantage provided by the EMP pathway. The advantage is highest at glucose concentrations below 10 μM. (d) Comparison between the two knockout strains. Blue areas indicate conditions where ED is more favorable, and red areas indicate conditions where EMP would be favored. The dark blue region at low oxygen and medium glucose levels may correspond to the environment of bacteria such as Z. mobilis, which uses the ED pathway exclusively [50]. The same data are shown as Monod surface plots in Figure 21 in S1 Text.