Systematic evaluation of objective functions for predicting intracellular fluxes in Escherichia coli

Abstract

To which extent can optimality principles describe the operation of metabolic networks? By explicitly considering experimental errors and in silico alternate optima in flux balance analysis, we systematically evaluate the capacity of 11 objective functions combined with eight adjustable constraints to predict (13)C-determined in vivo fluxes in Escherichia coli under six environmental conditions. While no single objective describes the flux states under all conditions, we identified two sets of objectives for biologically meaningful predictions without the need for further, potentially artificial constraints. Unlimited growth on glucose in oxygen or nitrate respiring batch cultures is best described by nonlinear maximization of the ATP yield per flux unit. Under nutrient scarcity in continuous cultures, in contrast, linear maximization of the overall ATP or biomass yields achieved the highest predictive accuracy. Since these particular objectives predict the system behavior without preconditioning of the network structure, the identified optimality principles reflect, to some extent, the evolutionary selection of metabolic network regulation that realizes the various flux states.

Figure 1

Central carbon metabolism of E. coli. The 10 reactions that describe the actual systemic degree of freedom are indicated in red arrows. These 10 reactions are expressed as 10 split ratios, where each of the 10 reactions that consume a cellular metabolite is divided by the sum of all producing reactions. The corresponding metabolites are indicated in red, whereas the 10 split ratios are shown in blue rectangles. Abbreviations: ACA, acetyl-coenzyme A; ACE, acetate; ACL, acetaldehyd; ACP, acetyl-P; AKG, alpha-ketoglutarate; CIT, citrate; DHP, dihydroxyacetone-P; ETH, ethanol; E4P, erythrose-4-P; FBP, fructose-1,6-bi-P; FOR, formate; FUM, fumarate; F6P, fructose-6-P; GAP, glyceraldehyde-3-P; GLX, glyoxylate; G6P, glucose-6-P; ICT, isocitrate; KDG, 2-keto-3-deoxy-6-phosphogluconate; LAC, lactate; MAL, malate; OAA, oxaloacetate; PEP, phosphoenolpyruvate; PYR, pyruvate, 6PG, 6-phosphogluconate; P5P, pentose-5-P; QUH, ubiquinone; QUH2, ubiquinol; S7P, seduheptulose-7-P; SUC, succinate; 3-PG, 3-phosphoglycerate; xt, external.

Figure 2

Absolute range of in silico variation in individual split ratios due to alternate optima for the maximization of biomass yield (A) and maximization of ATP yield (B) objectives without additional constraints.

Figure 3

Predictive fidelities for aerobic and anaerobic batch cultures in minimal medium with glucose (arbitrary units). The results were obtained by minimization and maximization of the standardized Euclidean distance of the 10 split ratios for the reference flux solutions (Table II). The four oxygen constraints were not implemented for anaerobic batch cultures. Predictive fidelities above 0.1 are not shown. Crosses and red dots signify that the range of the predictive fidelity is less than 1%. Red dots and bars highlight predictive fidelities without additional constraints. Bars signify the predictive fidelity range for that particular combination of objective function, constraint and environmental condition. For the sole case of nitrate respiration, the upper oxygen uptake rates were translated into corresponding upper bounds for nitrate uptake. Objective functions and constraints are defined in Tables III and IV.

Figure 4

Scatter plots of in vivo and in silico split ratios in selected batch cultures. Letters A–F refer to the cases indicated in Figure 3. Horizontal and vertical error bars indicate experimental and computational variances, respectively. Blue circles indicate that the predicted value is more than 0.25 units different from the experimental reference value, hence were poorly predicted. Split ratios that contain zero fluxes cannot be calculated due to singularities in the error propagation, hence a default accuracy of 5% was assumed. Split ratios with a value above 0.05 are indicated.

Figure 5

Predictive fidelities for glucose- and ammonium-limited chemostat cultures (arbitrary units). The results were obtained by minimization and maximization of the standardized Euclidean distance of the 10 split ratios for the reference flux solutions (Table II). Predictive fidelities above 0.1 are not shown. Crosses and red dots signify that the range of the predictive fidelity is less than 1%. Red dots and bars highlight predictive fidelities without additional constraints. Bars signify the predictive fidelity range for that particular combination of objective function, constraint and environmental condition. Objective functions and constraints are defined in Tables III and IV.

Figure 6

Scatter plots of in vivo and in silico split ratios in selected chemostat cultures. Letters A–F refer to the cases indicated in Figure 5. Horizontal and vertical error bars indicate experimental and computational variances, respectively. Blue circles indicate that the predicted value is more than 0.25 units different from the experimental reference value, hence were poorly predicted. Split ratios that contain zero fluxes cannot be calculated due to singularities in the error propagation, hence a default accuracy of 5% was assumed. Split ratios with a value above 0.05 are indicated.

Figure 7

Sensitivity analysis of (A) the predictive fidelity (arbitrary units) and (B) acetate secretion on the oxygen uptake constraint in aerobic batch cultures given no assumption on the P-to-O ratio constraint (i.e., in practice a P-to-O ratio of 2 is chosen). The upper bound for the maximal oxygen uptake rate (5–19 mmol/g h) was varied, whereas the minimal experimentally reported maximal oxygen uptake rate (Varma et al, 1993; Varma and Palsson, 1994; Xu et al, 1999) is indicated in gray. Blue rectangles are for the maximization of ATP yield, red dots and bars for the maximization of biomass yield and green triangles for the maximization of ATP yield per flux unit. In (A), bars indicate the range of the predictive fidelities, whereas a dot, rectangle or triangle indicates a unique solution. Predictive fidelities above 0.05 are not shown. In (B), the line indicates the experimentally determined value of acetate secretion of 58 mmol/g h (Perrenoud and Sauer, 2005).

Figure 8

Hierarchical cluster trees based on the Euclidean distance among specific agreements ρ for the five objective functions considered in Figures 3 and 5, including all constraint combinations under the six conditions. Difficult to predict groups of split ratios are highlighted by black lines. Groups of nodes were assigned where the linkage among the nodes was less than 0.7, when the linkage was normalized to values between 0 and 1.