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

Robert Schuetz^1,2,a, Lars Kuepfer^1,aa & Uwe Sauer^1,2

^aThese authors contributed equally to this work

^aPresent address: Bayer Technology Services, Leverkusen 51368, Germany

Article highlights

Synopsis

Based on a long history of biochemical and lately genomic research, metabolic networks, in particular microbial ones, are among the best characterized cellular networks. Most components (genes, proteins and metabolites) and their interactions are known. This topological knowledge of the reaction stoichiometry allows to construct metabolic models up to the level of genome scale (Price et al, 2004). Experimentally, sophisticated ¹³C-tracer-based methodologies were developed that enable tracking of the intracellular flux traffic through the reaction network (Sauer, 2006). With the accumulation of such experimental flux data, the question arises why a particular distribution of flux within the network is realized and not one of many alternatives?

Here, we address the question whether the intracellular flux state can be predicted from optimality principles, with the underlying rational that evolution might have optimized metabolic operation toward particular objectives or combinations of multiple objectives. For this purpose, we performed a systematic and rigorous comparison between computational flux predictions and available experimental flux data (Emmerling et al, 2002; Perrenoud and Sauer, 2005; Nanchen et al, 2006) under six different environmental conditions for the model bacterium E. coli. For computational flux predictions, we used a constraint-based modeling approach that requires a stoichiometric model of metabolism (Stelling, 2004). More specifically, we employed flux balance analysis (FBA) where objective functions are defined that represent optimality principles of network operation (Price et al, 2004). This approach has been applied successfully to predict gene deletion lethality (Edwards and Palsson, 2000a, b; Forster et al, 2003; Kuepfer et al, 2005), network capacities and feasible network states (Edwards 2001, Ibarra 2002), but in only few cases to predict the intracellular flux state (Beard et al, 2002; Holzhütter, 2004).

While different objective functions were proposed for different biological systems (Holzhütter, 2004; Price et al, 2004; Knorr et al, 2006), by far the most common assumption is that microbial cells maximize their growth. To address this issue more generally, we evaluated the accuracy of FBA-based flux predictions for 11 linear and nonlinear objective functions that were combined with eight adjustable constraints. For this purpose, we constructed a highly interconnected stoichiometric network model with 98 reactions and 60 metabolites of E. coli central carbon metabolism. Based on mathematical analyses, the overall model could be reduced to a set of 10 reactions that summarize the actual systemic degree of freedom.

As a quantitative measure of how accurate the experimental data are predicted, we defined predictive fidelity as a single value to quantify the overall deviation between in silico and in vivo fluxes. By comparing all in silico predictions to ¹³C-based in vivo fluxes, we show that prediction of intracellular steady-state fluxes from network stoichiometry alone is, within limits, possible. An unexpected key result is that no further assumptions on network operation in the form of additional and potentially artificial constraints are necessary, provided the appropriate objective function is chosen for a given condition.

While no single objective was able to describe the flux states under all six conditions, we identified two sets of objectives for biologically meaningful predictions without the need for further constraints. For unlimited growth on glucose in aerobic or nitrate-respiring batch cultures, we find that the most accurate and robust results are obtained with the nonlinear maximization of ATP yield per flux unit (Figure 1). Under nutrient scarcity in glucose- or ammonium-limited continuous cultures, in contrast, linear maximization of the overall ATP or biomass yields achieved the highest predictive accuracy.

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.

Full figure and legend (192K)Figures & Tables index

Since these identified optimality principles describe the system behavior without preconditioning of the network through further constraints, they reflect, to some extent, the evolutionary selection of metabolic network regulation that realizes the various flux states. For conditions of nutrient scarcity, the maximization of energy or biomass yield objective is consistent with the generally observed physiology (Russell and Cook, 1995). The meaning of the maximization of ATP yield per flux unit objective for unlimited growth, however, is less obvious. Generally, it selects for small networks with yet high, albeit suboptimal ATP formation, which has three biological consequences. Firstly, resources are economically allocated since expenditures for enzyme synthesis are, on average, greater for longer pathways. Secondly, suboptimal ATP yields dissipate more energy and thus enable higher catabolic rates. Thirdly, at a constant catabolic rate, a small network results in shorter residence times of substrate molecules until they generate ATP. The relative contribution of these consequences to the evolution of network regulation is unclear, but simultaneous optimization for ATP yield and catabolic rate under this optimality principle identifies a trade-off between the contradicting objectives of maximum overall ATP yield and maximum rate of ATP formation (Pfeiffer et al, 2001).

MORE ARTICLES LIKE THIS

These links to content published by NPG are automatically generated.